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David N. Weil October 22, 2013 Lecture Notes in Macroeconomics Section 1: Consumption Several ways to approach this subject. 1. Note that “saving” and “consumption” are really the same question: that is, you get a certain amount of income, and you can save it or consume it. So can’t think about one without thinking about the other. 2. This topic is really part of both the long run and the short run analysis. In the long run, the saving rate determines the level of output (or the growth rate or output). But in the short run, the determination of consumption is also important for studying the business cycle. 3. Consumption theory is one of the most elegant branches of economic theory. Much of the approach taken here to consumption is taken elsewhere in economics to e.g. fertility, schooling, health, etc. Thus these tools (and the problems with them) are far more general than it might appear. 4. In all of this section of the course, we will be treating labor income as exogenous (Note: “exogenous” does not mean “constant” or “certain.”) We will also mostly treat interest rates as being exogenous, but also look at some cases of endogenous interest rates. You may recall the approach taken to consumption in many undergraduate macro textbooks is to think about a “consumption function” that relates consumption to disposable income: C = C(Y-T) [where note that we are using c as both the name of the function and the name of

the thing it is determining.] often this is written in a linear form: C = c0 + c1(Y-T) where the little c’s are coefficients. c1 is, of course, the marginal propensity to consume. [picture] This is often called the Keynesian consumption function. Keynes wrote that c0>0 and 0<c1<1 due to a “psychological law” -- essentially that when you do not have a lot of income, you focus on

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immediate needs; but when you have satisfied these, you look more to the future and save. Ways to test: look cross sectionally; look at short time series. (flesh his out) Both of these looked good for the Keynesian consumption function. Two problems with the Keynesian view: 1. Empirical: What does this model predict will happen to the rate of saving as a country gets richer?

where C/(Y-T) is often called the average propensity to consume. So the Keynesian consumption function says that as a country gets richer the saving rate should rise. This just doesn't work. The saving rate is pretty constant over long periods of time.12 2. Theory: Think about the act of saving: you are moving consumption from one period to another. Thus saving should be viewed explicitly as an intertemporal problem. So for example, the MPC should depend on why your income has gone up. Put another way, the consumption function should have a lot more than just today's income in it -- for example, it should have tomorrow's income in it. So we want a model of saving behavior that is more based on fundamentals. To build a such a model, we start with the question: Why do people consume? Answer: Because it makes them happy. We represent the idea that consuming makes people happy with a utility function.

1 Historical note: This wasn't actually known for sure when Keynes wrote: Simon Kuznets, who invented national income accounting – i.e. how to measure GDP and stuff -- discovered the approximate constancy of the US saving rate over a period of 100 or so years. His discovery set off a flurry of work on consumption in the 1950s that culminated in Friedman and Modigliani’s contributions. Interestingly, in most other developed countries (summarized in Angus Maddison’s work) the saving rate has risen over time -- although probably not in the way that Keynes' model predicted. 2The Kuznets finding can be put another way. If we go to the data (say annual data on income and consumption for a country over time) and run the regression C = c0 + c1 Y, we will get the result that in short samples the estimated value of c1 will be smaller than it will be in large samples. When we talk about the Permanent Income Hypothesis we will see why this is true.

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Y -T - C C cs = = 1- = 1- - cY -T Y -T Y -T

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By utility function, we just mean some function that converts a level of consumption into a level of utility. U = U(C) [picture] Why should the utility function be curved? Try to motivate intuitively: think about the marginal utility of additional consumption. Seems like this goes down. Most of the interesting things that we can say about utility come from thinking about two issues: how we add up utility over many different periods of time, and how we deal with the expected utility when there is uncertainty. Adding up Utility Over Time How do we add up utility across time? Well essentially, we can just take the sum of individual utilities. Say that we are considering just two periods. Let U( ) be the “instantaneous” utility function. Then total utility, V, is just V = U(C1) + U(C2) (In a little while we will introduce the notion of discounting, by which utility in the future may mean less to us than utility today. But for now, we will ignore this idea.) What does our understanding about the utility function say about the optimal relation between consumption at different periods of time. Say, for example, that we have $300 to consume over two periods (and we temporarily ignore things like the interest rate): How shall we divide it up? The answer is that we would want to smooth it -- that is, consume the same amount in each period. The way to see this is to look at the marginal utility of consumption. Suppose that we consumed different amounts in different periods. Then the marginal utility of consumption would be lower in the period where we consumed more. So we could consume one unit less in that period, and one unit more in the period where the marginal utility was higher, and our total utility would be higher. Utility under Uncertainty Now let’s consider a case where there is only one time period, but in which there is uncertainty about what consumption will be in that period. Suppose, for example, that I know that there is a 50% chance that my consumption will be $100 and a 50% chance that my consumption will be $200. How do we calculate my expected utility? There are two ways that you might consider doing it: could take the expected value of my utilities, or the utility of my expected consumption.

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V = .5*U(100) + .5*U(200) or V = U (.5*100 + .5*200) The first of these methods of calculating utility from a probabilistic situation is called Von Neumann - Mortgenstern (VNM) utility. This is the approach that we always use. The second method is called wrong. How do we know the VNM utility is the right way to think about utility when there are different possible states of the world? Here is a simple demonstration: Suppose that you can have either $150 with certainty, or a lottery where you have a chance of getting either $100 or $200, each with a probability of .5. Which would you prefer? Almost everyone would say they prefer the certain allocation. This is a simple example of risk aversion. But notice that if we chose the second technique for adding up utility across states of the world, we would say that you should be indifferent. The fact that uncertainty lowers your utility is called risk aversion. Notice that risk aversion is a direct implication of the utility function being curved. (The mathematical rule that shows this is called Jensen’s inequality: if U is concave, then U(E(C)) > E(U(C)), where E is the expectation operator.) If the utility function were a straight line then the utility of $150 with certainty would be the same as the utility of a lottery with equal chances of getting $100 and $200. A person who indeed gets equal utility from these two situations is called risk neutral.3 What are the consequences of risk aversion? Clearly this is the motivation for things like insurance, etc. Similarly, this is why in financial theory we say that people trade off risk and return: to accept more risk, an investor has to be promised a higher expected return. The Relation between Risk Aversion and Consumption Smoothing Now we get to the really big idea: risk aversion and consumption smoothing are really two sides of the same coin: they are both results of the curvature of the utility function. If the utility function were linear (and so the marginal utility of consumption constant) then people would not care about smoothing consumption, and their expected utility would not be lowered by risk. This will be important for many reasons: among them is that even when we are talking about a world with no uncertainty, we will often use the idea of risk aversion to measure the curvature of

3One can come up with many instances of risk neutrality or even risk-loving (i.e. more uncertainty raises utility) behavior, such as participating in lotteries, flipping a coin with your friend for who will buy coffee, etc. However, it is unlikely that these exceptions tell us much about the vast majority of consumption decisions.

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the utility function. The CRRA Utility Function We will often use a particular form of the utility function, called the Constant Relative Risk Aversion utility function.

where σ > 0. Note that if σ > 1, then the CRRA formulation implies that utility is always negative, although it becomes less negative as consumption rises. This does not matter (as long as whether you are alive or not is not subject to choice) although it often gets students confused. Note that in the special case where σ=1, the CRRA utility function collapses to U(C) = ln(C).4 σ is called the coefficient of relative risk aversion and it measures, roughly, the curvature of the utility function. If σ is big, then a person is said to be risk averse. If σ is zero, the person is said to be risk-neutral.5

4Proof: First re-write the utility function by adding a constant: u(c) = (c1-σ-1)/(1-σ). Think of this as a function of σ: g(σ) = (c1-σ-1)/(1-σ) re-write as

ln-( (c) )c - 1eg( ) = 1-

since g(1) = 0/0, we apply L’Hopital’s rule

lnlim ln

ln- (c)-c (c)eg( ) = = (c)

1 -1

.

5 The Coefficient of Relative Risk Aversion is defined as

U (C)C

-U (C)

It is easy to verify that this is constant in C for the CRRA utility function.

1-t

tCU( ) = C1-

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To see how σ measures the curvature of the utility function, we can calculate the elasticity of marginal utility with respect to consumption, that is

So the bigger is σ, the more rapidly the marginal utility of consumption declines as consumption rises [picture]. And the larger is this change in marginal utility, the greater is the motivation for consumption smoothing, insurance, etc. As an exercise, we can show this by calculating the amount that a person is willing to pay to avoid uncertainty. For example, calculate the value x such that the utility of $150-x with certainty is equal to the utility of a 50% chance of $100 and a 50% chance at $200. How does x change with σ? We solve:

We can use a calculator to find the value of x for different values of σ. By thinking about what value of x seems reasonable, we can then decide what is a reasonable value for sigma (see table). For example, if σ = 6, then x=35.8, so a person would be indifferent between a 50% chance of consuming $100 and a 50% chance of consuming $200, on the one hand, and certain consumption of $114.20, on the other. σ x 1 8.6 (log utility) 2 16.7 3 23.5 4 28.8 5 32.9 6 35.8

dUUdc = c = -

U Uc

1- 1- 1-(150 - x = .5* + .5*) 100 200

11- 1-

1-x = 150 -(.5* + .5* )100 200

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Note that even though log utility is probably not reasonable a priori (based on the above) or empirically, we use it a lot because it is so convenient. Empirical estimates of σ probably average around 2-3, but there is no agreement. Some anomalies in finance (such as the “equity premium puzzle”) can only be explained with what seem like unreasonably high values of σ. We can also do an exercise to show how the willingness to pay to avoid risk depends on the size of the risk. Suppose that we take the above example and now multiply the size of consumption in the two states of the world by some amount z. We can again solve for the amount that a person is willing to pay to avoid that risk:

( [ ( ]( ) ) ( ) )1 1

1- 1- 1- 1-1- 1-x = z150 -(.5 + .5 z z 150 -(.5 + .5z100 200 ) 100 200 )

Where the term in square brackets is just what you were willing to pay to avoid the original uncertainty. In other words, the amount that you are willing to pay to avoid uncertainty relative to certainty depends only on the risk relative to the certain outcome. If you are willing to pay $10 to avoid uncertainty of $50 relative to a base of $150, then you are willing to pay $10,000 to avoid uncertainty of $50,000 relative to a base of $150,000. That is why this formulation of the utility function is called Constant Relative Risk Aversion (CARA). The other utility function that we use a lot is Constant Absolute Risk Aversion (CARA).

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We can do the same exercise as above, considering a lottery where the outcomes are either, $100 or $200, each with probability 50% or else $150-x with certainty. Setting the utilities equal to solve for x we get

1 0.5 0.5

0.5

0.5

This equation gives an implicit value for x. Now, suppose that we change the value of consumption prior to the introduction of risk (i.e. the $150), without changing the size of the risk. In other words, instead of consumption being 150+50 or 150-50, we make it so that consumption is 1500+50 or 1500-50. Looking at the derivation above, it should be clear that the value of x will not change. That is why this is called constant

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absolute risk aversion. So, suppose that we start in a case where a person is willing to pay 10 to avoid a risk of 50, starting with a base of 150. That is, she thinks that 140 with certainty give the same expected utility as 100 and 200, each with probability 50%. Under CRRA, we know that she would also view 1400 with certainty as having the same utility as 1000 or 2000, each with probability 50%. Under CARA, we know that she would view 1490 as having the same utility as 1550 and 1450, each with probability 50%. But how much would she be willing to pay to avoid the risk of 1000 and 2000? The answer is that it must be more than 100. The way to see this is that for any utility function with negative second derivative, the utility loss from uncertainty rises with the size of the uncertainty. So if she is willing to pay 10 to get rid of uncertainty of 50 (plus or minus), she must be willing to pay more than 100 to get rid of uncertainty of 500 Fisher Model So now we look more formally at an intertemporal model of saving. The simplest model is the two-period model of Irving Fisher. People live for two periods. They come into the world with no assets. And when they die, they leave nothing behind. In each period they have some wage income that they earn: W1 and W2. Similarly, in each period, they consume some amount C1 and C2. The amount that they save in period 1 is S1 = W1 - C1. S1 can be negative, in which case they are borrowing in the first period and repaying their loans in the second period. For the time being assume that they do not earn any interest on their savings or pay any interest on their borrowing. So the amount that they consume in the second period is C2 = S1 + W2 that is, in the second period you consume your wage plus your savings. We can combine these two equations to get the consumer's intertemporal budget constraint: C1 + C2 = W1 + W2 We can draw a picture with C1 on the horizontal axis and C2 on the vertical axis. The

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budget constraint is a line with a slope of negative one. Note that the budget constraint runs through the point W1, W2 -- you always have the option of just consuming your income in each period. The Y and X intercepts of the budget constraints are both W1 + W2. Consumer can consume any point along this line (or any beneath it, but that would be waste). What is saving in this picture? Show which points involve saving or borrowing. So where does the person choose to consume? Well, clearly along with a budget constraint we are going to need some indifference curves. Say that his total utility (V) is just the sum of consumption in each period: V = U(C1) + U(C2) Where U() is just a standard utility function. To trace out an indifference curve, consider a point where C1 is low and C2 is high. At such a point, the marginal utility of first period consumption is high, and that of second period consumption is low. So it would take only a small gain in C1 to make up for a big loss of C2 in order to keep the person having the same utility. So the indifference curve is steep. Similarly, when C1 is large and C2 is small, the indifference curve is flat.6 So it has the usual bowed-in shape. So optimal consumption is where the budget constraint is tangent to an indifference curve. We can also solve the problem more formally, setting up the lagrangian:

and taking the first order conditions: dL/dC1 = 0 = U'(C1) - λ ===> λ = U'(C1) dL/dC2 = 0 = U'(C2) - λ ===> λ = U'(C2) so C1 = C2 6 More formally, one can use the implicit function theorem. Let F(C1,C2) = U(C1) + U(C2). Then for F(C1,C2) = k (where k is some constant):

1

2

2 1C

1 2C

d U ( )FC C= - = -d U ( )C CF

1 2 1 2 1 2L = U( ) + U( ) + ( + - - )C C W W C C

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Combining this with the budget constraint gives: C1=C2= (W1 + W2 )/2 , which is not so shocking, when you think about it. We can use this simple model to think about consumption in the face of different circumstances. What happens if income rises? This will shift out budget constraint. Consumption in both periods will rise. What happens to saving? Answer: it depends on which periods income went up. -- Suppose that your current income falls but that your future income rises by exactly the same amount. How should consumption change? How about saving? Already, we can see some problems with Keynes' way of looking at consumption. Consumption depends not just on today's income but on future (or past) income. Interest rates Now we make the model slightly more complicated by considering interest rates: let r be the real interest rate earned on money saved in period 1 -- or the interest rate paid by people who borrow in period one. Now the definition of saving is still: S1 = W1 - C1 but consumption in the second period is now: C2 = (1+r) S1 + W2 or combining these: W1 + W2/(1+r) = C1 + C2/(1+r) Can draw diagram as before Y intercept is (1+r)W1 + W2,

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X intercept is W1 + W2/(1+r). The budget constraint still goes through the point (W1, W2), which we call your “endowment point” -- that is, if you consume your income in each period, that is a feasible consumption plan no matter what the interest rate is. What happens to the budget constraint when the interest rate changes?? Answer: it rotates around the endowment point. What does this do to saving in the first period (ie to consumption in the first period?) Answer is: it depends. First, let’s look at what happens in the case where the person was initially saving. Remember from micro that there are two effects: the income and the substitution effect. Income effect is that we can get onto a higher indifference curve. This tends to raise C for both periods. Substitution effect: consumption in the second period has gotten cheaper. This tends to lower first period consumption and raise second period consumption. Upshot is that in this case, can't tell what happens to first period consumption, or first period saving. What if person had had negative saving in the first period, and then interest rate goes up? Now income and substitution effects work in the same direction, so that first period consumption will fall, and saving will rise. Discounting We might want to introduce some discounting of utility experienced in the future. For example, suppose that Θ is some discount factor that we use for discounting future utilities. V = U(C1) + U(C2)/(1+Θ) Now we can once again solve for the optimal path of consumption with both interest and discounting. We set up the Lagrangian:

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and get the first order conditions dL/dC1 = U'(C1) - λ dL/dC2 = U'(C2)/(1+Θ) - λ/(1+r) which can be solved for: U'(C1)/U'(C2) = (1+r)/(1+Θ) this is one equation in the two unknowns of C1 and C2. It can be combined with the budget constraint to give two equations in two unknowns, and so can be solved for the two values of C. To do this, however, one needs to know the exact form of the utility function. This is done in one of the homework exercises. Liquidity Constraints What happens if there are constraints on borrowing? What does the budget constraint look like now? For person who would have wanted to save anyway, no big deal. But for person who would have wanted to borrow, they will be at corner. We say that such a person is "liquidity constrained." Example of a college student. What will such a person's consumption be? Just their current income. So they will look a lot more like the Keynesian model, except that the MPC will be one. Differential interest rates It may also be the case that the interest rate for borrowing is different than the interest rate for saving -- presumably the rate for borrowing will be higher. What will the budget constraint look like in this case? It will be kinked at the endowment point. In this case, there are three possible optima: either tangent to one of the arms, or at the kink point. Interesting result is that if the optimum is at the kink point, then small changes in one or both interest rates will not affect the optimal level of consumption.

2 2 21 1 1

U( )C W CL = U( ) + + + - - C W C1 + 1 + r (1 + r)

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Extension to More than Two Periods Now we can easily extend the model to an arbitrary number of periods: Consider a person planning consumption over periods 0...T-1. (labeling the periods this way is just slightly more convenient) She faces a path of wages {W0, ....WT-1} she gets utility according to an instantaneous utility function U(C), which is discounted at rate Θ. That is

She faces interest rate r on any assets (negative or positive) that she has. In particular, call At the assets that she has at the beginning of a period. This is equal to At = (1+r)*(At-1 + Wt-1 - Ct-1 ) She starts life with zero assets: A0 = 0. We also impose the rule that she must have zero assets at the end of her life -- that is AT = 0 (where AT = (1+r)(AT-1 + WT-1 - CT-1) . Put another way, in the last period of life she spends her earnings plus any accumulated assets [or less any accumulated debts). Dying in debt is not allowed. How will we derive her inter-temporal budget constraint? Start by writing down the expression for assets in each period A1 = (1+r)(W0 - C0) [since A0 = 0 ] A2 = (1+r)(A1 + W1 - C1) = (1+r)(W1-C1) + (1+r)2(W0 - C0) etc... AT = (1+r)(WT-1 - CT-1) + (1+r)2(WT-2 - CT-2) + ... (1+r)T (W0 - C0) We divide all the terms in this last expression by (1+r)T, and note that it is equal to zero, to get

Notice that we have gotten rid of all of the A's. This expression can be re-arranged to say that the

T -1

tt

t=0

U( )CV =(1+ )

T -1

t tt

t=0

-W C0 =(1+r )

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present discounted value of consumption is equal to the present discounted value of wages.

This is the intertemporal budget constraint... which looks a lot like the two period version derived above. An Aside: The Budget Constraint in Continuous Time We can also derive a similar intertemporal budget constraint in continuous time. The evolution of assets is governed by the differential equation: d A(t)

= A = rA(t) + w(t) - c(t)dt

This can be solved, along with the initial condition A(0)=0, to give t

r(t-s)

0

A(t) = (w(s)- c(s)) dse

This just says that assets at time t are the present values of the past differences between wages and consumption. Assets at the end of life are zero, that is, A(T)=0. So setting t=T in the above equation, T T

r(T -s) r(T -s)

0 0

w(s) ds c(s) dse e =

Multiplying by e-rT T T

-rs -rs

0 0

w(s) ds c(s) dse e =

[End of Aside] Now with our budget constraint and our utility function, we can do a big Lagrangian....

to solve this we would just find the T first order conditions which, combined with the budget

T -1 T -1

t tt t

t=0 t=0

W C=(1+r (1+r) )

T -1 T -1

t t tt t

t=0 t=0

U( ) -C W CL = + (1+ (1+r) )

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constraint, would allow us to solve for the T+1 unknowns: λ and the T values of consumption. In many cases this is a big mess to solve, but we can get far by just looking at the FOCs for consumption in two adjacent periods, t and t+1:

these two can be combined to give

this is a key condition that relates consumption in adjacent periods. Notice that even if we don't know the full solution to the consumer's problem (that is, what the level of consumption in each period should be), we know that this condition should hold. There is a huge amount of intuition built into this expression, so it is worth thinking about for a while. Let's start on the intuition by showing how we could have gotten a similar result without calculus: Suppose that I have a discounted utility function, and that the interest rate is zero. I have some set amount of total consumption that I want to do. How will I divide it between the periods? To see the answer: consider a path of consumption (C0, C1,...) Suppose that I want to know whether this path of consumption is optimal. Well, suppose that I consider consuming slightly less (call it one unit, for convenience) in period zero, and then consuming the same amount more in period one. How much would I lose? answer: U'(C0) How much would I gain? answer: U'(C1)/(1+Θ)

tt t

t

dL U ( ) 1C= - = 0d (1+ (1+ r) )C

t + 1t + 1 t + 1

t + 1

dL U ( ) 1C= - = 0d (1+ (1+ r) )C

t

t +1

U ( ) 1+rC =U ( ) 1+C

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Note that the (1+Θ) comes from the fact that utility that I get in the second period is not worth as much to me as utility in the first period. Now if one of these was bigger than the other, then clearly the path of consumption that I was considering was not the optimal one. So everywhere along the optimal consumption path, it will be the case that U'(Ct) = U'(Ct+1)/(1+Θ) So what does this say about the optimal path of consumption in the presence of discounting? I says that the marginal utility of consumption must be rising. So therefore consumption must be falling along the optimal path. Now imagine that we have an interest rate to contend with (and forget about discounting for a second): Whatever we don't spend will grow in value at an interest rate r. Again consider some allegedly optimal path of consumption. Suppose that I were to move one unit of consumption from period 0 to period 1 would lose: U'(C0) would gain U'(C1) (1+r) Suppose that these two were not equal -- then clearly you were not on the optimal path. So the condition for being on the optimal path is U'(C0) = U'(C1) (1+r) So what has to be happening to consumption in this case? The marginal utility must be falling -- so consumption must be rising. So now say that I want to characterize the optimal path of consumption in the case where I have both an interest rate r and a rate of time discount Θ. Clearly the first order conditions relating every two adjacent periods' consumption will be:

So what does this tell us? Suppose that Θ is greater than r? Then the marginal utility of consumption in period t+1 is higher than the marginal utility of consumption in period t, and so consumption must be falling. What if r>Θ? What if they are equal? So interest and discounting work against each other.

t

t+1

U ( ) 1+rC = U ( ) 1+C

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If we did know the exact form of the utility function, we could go further. For example, if we know that the utility function is of the CRRA form

then U'(C) = C-σ and so the first order condition can be re-written

Before we discuss the interpretation of this first order condition, we can derive a similar one in continuous time. To re-write the first order condition with CRRA utility in continuous time: First note that for small values of x, the approximation ln(1+x) x (or alternatively, 1+x ex ) is fairly accurate. So for (1+r) we write er, and same for Θ. [being completely accurate, the r that we use in continuous time, the “instantaneously compounded” interest rate, is not exactly equal to the r used in discrete time.] so we can rewrite the first order condition as

re-write this allowing the unit of time used to be a parameter:

1-t

tCU( ) = C1-

1

t +1

t

1+rC = 1+C

1/r1/t +1 r-

t

c e = = ( )ec e

1/t + t (r- ) t

t

c = ( )ec

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where if Δt=1 then we have the previous equation. define c as the time derivative of consumption: c = dc/dt.

Thus the growth rate of consumption is given by

The numerator and denominator of the last expression are both zero when Δt is zero, so we apply L'Hopitals rule, taking derivatives of top and bottom with respect to Δt:

Evaluating at Δt=0, we end up with

Interpretation of the FOC In both discrete and continuous time, the FOC says the same thing: the rate at which consumption should fall or grow depends two things: first, the difference between r and theta; and second on the curvature of the utility function.

lim t + t t

t 0

- c cc = t

0 0 0

lim lim lim

t + tt + t t1/(r- ) t

t

t t tt

c-c c -1c ( -1)ect = = = c t tc

1-1(r- ) t (r- ) t1

( (r - ))e e

1

c 1

= ( r - )c

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-------------------------------------- Completing the solution Often all we need to look at is the first order condition. But if we want to complete the solution to the lifetime optimization problem, we can. The FOC tells us how consumption in adjacent periods compares. So given one value of consumption (say, consumption in the first period), we can figure out consumption in all periods -- that is, the entire path of consumption. [note, by the way, what will happen to the FOC if r changes over time. This condition would then have to be re-written with r(t) in it, but would be otherwise the same.] From here, it is simply a matter of finding the value of consumption in the first period that satisfies the budget constraint. Completing the solution is easiest in the case of continuous time where we let the time horizon (i.e. T) be infinite. Note that there are some technical problems that can crop up in considering infinite time as opposed to just letting T be very large. For example, we can’t impose the “no dying in debt” condition (A(T) = 0 ), and instead have to impose a different condition (often called the “no Ponzi game condition” that I will not discuss here. For our purposes, it is sufficient to state that the infinite PDV of consumption has to equal the infinite PDV of wages. Consider a simple case where w(t) = 1 for all t. Utility is CRRA, and r are given. The first order condition for consumption growth can be integrated to give (1/ )( )( ) (0) r tc t c e The budget constraint is thus

(1/ )( )

0 0

(0)rt rt r te dt e c e dt

Notice that for this budget constraint to make sense, we have to have that the right hand side is finite. If r>theta, then consumption is growing, but it must be growing slowly enough so that its PDV is finite. Thus we assume:

1( )r r

Integrating the budget constraint….

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1 (0)

(1/ )( )

c

r r r

1

(0) 1r

cr

From this we see

If θ>r, then initial consumption is above 1, in which case consumption asymptotes to zero The bigger is σ, the closer is initial consumption to 1.

Endogenous Interest Rates and an Open Economy Application We can use the two period model of consumption to draw a helpful picture. Suppose that we graph the interest rate on the vertical axis, and the level of (first period) saving on the horizontal, with zero somewhere in the middle of the horizontal axis. What is the relation? Obviously, the position of the curve will depend on the values of w1 and w2. (as well as the parameters of the utility function). The bigger is w1 and the smaller is w2, the higher will be saving at any given interest rate. But what about the shape of the curve overall? We know that if saving is negative, then an increase in the interest rate will raise the amount of saving -- we know this because in this case the income and substitution effects are aligned. For zero saving, we also know that the curve is upward sloping. But for positive saving, we don’t know. The curve may well bend backward. Question: what determines the degree to which the curve can bend backward? Answer: the degree of risk aversion! Why? The degree of risk aversion tells us how the person trades off smoothing of consumption for taking advantage of the interest rate to get more consumption in a later period. If a person is very risk averse, then he wants very smooth consumption. In this case, the curve will end up bending backward Now suppose that we have a two-period world, and we are thinking about a country, rather than an individual.

21

Quick review of open economy national income accounting: From this we derive the standard national income accounting equation Y = C + I + G + NX one problem: is Y GDP or GNP? The answer is that we can make it either one; as long as we define imports and exports appropriately. In fact, for (almost) all of this course, the distinction will not matter. When we think about capital flows, we will be thinking not about portfolio investment or foreign direct investment (FDI) but rather about debt (denoted B). In this case, there will be no foreign ownership of factors of production, and so GDP and GNP will be the same. Y = C + I + G + NX Y - C - G = national saving = I + NX (Y - T - C)+ (T-G) = national saving private saving + gov't saving = national saving = I + NX Define Bt as net foreign assets at time t. The Current Account is the change in net foreign assets. It is equal to NX plus interest on the assets we hold abroad, minus interest on the debt that we owe foreigners. In discrete time: CA = Bt+1 - Bt = rBt + NXt In continuous time: CA B rB NX So for our thinking about capital flows between countries, there are going to be a variety of assumption that we can make about the different pieces. Nature of openness (for this course, the only type of openness we will think about is capital flows.): closed economy: NX is zero; r is endogenous. small open economy: r is exogenous and fixed at r*, the world level, which is exogenous; NX is endogenous.

22

large open economy: economy is large enough to affect the world interest rate, so r=r*, but r* is endogenous. Also, if this is a two-country world, then NX = -NX* Well: a person saving in the first period and consuming more than his income in the second period is exactly equivalent to running a CA surplus in the first period and a CA deficit in the second period. (Even though the world only lasts for two periods, we can think of the requirement that people do not die in debt as meaning that B3 = 0.) There is another way that we can think about this same issue, which is more international. Suppose that there is no trade between countries. Then, since there is no government and no investment, W=C in both periods. Note that this is not just a case of liquidity constraints in the standard sense. Rather, since everyone is identical, there will be no borrowing or lending. But (key observation): there can still be an interest rate! We think of the interest rate as being the level that clears the market for loans -- which will clear at the level where there neither borrowing or lending. This is called the “Autarky interest rate” To figure out the Autarky interest rate, we can just go back to the first order condition, but now we know that consumption has be equal to Y, and so we can just substitute it: U'(Y1)/U'(Y2) = (1+r)/(1+Θ) Now, here is the big result: ===> If the autarky interest rate is lower than the world interest rate, then the open economy will run a current account surplus in the first period. And if the autarky interest rate is higher than the world interest rate, then the economy will run a current account deficit in the first period. Intuitively, this is pretty obvious. We can also show it graphically [The autarky interest rate is what arises in the closed economy version of our model. It is the place where the curve derived above crosses zero. So we can also see here the result about the interest rate!] ---- Large Open Economy model Now we can do a large open economy model, making r (= r*) endogenous. We just draw

23

two versions of the saving vs interest rate diagram that we derived above, and look for the interest rate where saving in one country is the negative of saving in the other. etc. Intuition building problem: Let’s look at the large open economy model with an infinite number of periods, instead of just two. Let’s think about two equally sized open economies. Equally sized in the sense that they have the same endowment income. Y1,t = Y2,t =Y for all t we forget about G and I θ1 < θ2 Two countries start with B=0. What will the equilibrium look like? The key to figuring this out is to realize that the interest rate cannot remain constant! (At least if we assume that consumption can’t be negative). ===> In the long run, we know that the interest rate will be equal to the θ1. We can trace out the path of interest rates and net assets pretty easily (at least graphically!). [Exercise: think about the solution if preferences are CARA (with c ≥0 )instead of CRRA] The PIH and the LCH the model just presented in very standard. The PIH and LCH are two ways of making the same point. Permanent Income Hypothesis Developed by Milton Friedman Rather than focusing on the whole life cycle, the PIH thinks about shorter period changes in income. The PIH starts by separating income into two parts Y = YP + YT (note, could have used Y-T here...)

24

permanent income is the part of income that you expect to persist into the future, sort of like your average future income. Transitory income is the other part of income - the part that is different from the average (note that it can be positive or negative in a given year). Take a person with a job. Their permanent income is their salary. If in some year they get a bonus, or if in some year they have a smaller salary for some reason, that is positive or negative transitory income. Think about the following two changes in my income. One month I get a letter saying that I have won $1000 in the lottery. Is that a change in my permanent or transitory income? What about if I get a letter from the dean saying that my salary is higher by $1000 a month? How will my consumption change in each of these scenarios? This was Friedman's insight. Your consumption should just depend on your permanent income. To the extent that transitory income is different from permanent income, you will just use your saving to make up the difference. Let's take an example: suppose you looked at two people, both of whom earned the same amount -- say $100,000 in a given year. One is a businessman, for whom this is the regular salary. The other is a farmer, who has very unstable income, and for whom this was a good year. Which should have higher saving? So in the PIH, the consumption function is roughly, C = *Yp Now we can go back and see how the PIH explains the facts about the consumption function that Keynes failed on. First think about the long run: over the long run, when income increases, this is clearly a change in permanent income. So consumption and saving will just be constant fractions of income in the long run. What about in the short run (or looking across households)? What would you see if all of the variation in income that you looked at was transitory? Then there would be no relation between C and Y -- the short run consumption fn would be flat. What if some of the variation were transitory and some permanent? Then you would see what is present in the data. (see homework problem). We can also give this result an econometric interpretation. Suppose that a researcher has collected income and consumption data from a large population. Consumption in the population is

25

determined by the permanent income hypothesis: C = Yp, where Yp is permanent income and

0<<1. Permanent income in the population is given by Yi =pY + ρi, where ρ is distributed

normally, with variance σ2p The researcher does not observe permanent income, however. She only

observes current income, which is related to permanent income by Yci = Yp

i + εi. εi is transitory income. It is distributed normally, with variance σ2

t. There is zero covariance between permanent and transitory income. The researcher estimates the “consumption function”

ˆi iC Y

The relationship between estimated beta and the true value is given by

2

2 2ˆ p

p t

(this is the formula for attenuation bias or measurement error bias. You can think of observed income as permanent income measured with error). This explains the better fit of Keynes in the short run than in the long run, and also the fact that the Keynesian consumption model fits well in cross-section. In these cases, measured income is equal to permanent income plus a lot of noise, so there is a lot of attenuation bias so beta hat is well below beta, and correspondingly alpha is biased upward. One current issue in macro is what Friedman meant -- or what is the truth -- about how far into the future one should look in thinking about "permanent" income. Is it for the rest of your life? For your life and your children's lives? Or is it for some shorter period, like the next 5 years? This will turn out to be important in some of the questions we look at below. Life Cycle Hypothesis Due to Franco Modigliani7 One direction to go with the analysis of consumption presented above is to look more realistically at what determines saving of people in the economy.

7 Once, when asked exactly what the difference was between the LCH and the PIH, Modigliani replied that when the model fit the data well it was the LCH, and when it didn’t it was the PIH.

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Lifetime budget constraint is:

Now think about your income over the course of life (where we start life at the beginning of adulthood). The biggest thing that you will notice is that there is a big change at retirement -- your income goes to zero. [picture] Now think about your preferences. We know that in you are going to want to have smooth consumption -- for example in the case where the interst rate is equal to the discount rate, you will want constant consumption. [picture] What is the relation between the income and the consumption lines? Well, if the interest rate is zero, then the areas under them have to be the same (that is, the sum of lifetime income has to be the same as the sum of lifetime consumption). If the interest rate is not zero, it is a little more complicated -- what matters is the present discounted value of income is equal to the PDV of consumption. What does this model say about a person's assets over the course of life? [picture] The LCH is also concerned with the total wealth of all of the people in the economy. Why is this so important? Because, for a closed economy, the capital stock of the economy is made up of the wealth of the people in the economy. And, as you know from studying growth, the capital stock is really important. Indeed, the OLG model is the life cycle model, with a very simple life cycle. We can see the aggregate amount saved in the economy by just adding up each age groups saving or dissaving, multiplied by the number of people who are that age. What does this say should be happening to the saving rate of the US as the population ages? We can also see the effect of social security on saving or total assets in the economy. Social Security lowers income during the working part of life, but raises it during the retirement part of life. So it lowers the saving rate (and level of wealth) at any given age.

T T

t tt t

t=0 t=0

C W=(1+r (1+r) )

27

[note -- we will talk about the empirical implications of this model and how well they stand up later.] Income growth and Savings in the Life Cycle model [Could be added: do this with flat wage profile (or cross section) and do explicit examples] How does the growth rate of income affect the saving rate in the life cycle model? Specifically, if we compare two countries that have the same θ and r, and the same age structure, but different growth rates of wage income, which will have higher saving rate. Answer: it depends on the form of income growth. Two cases to look at. 1) Suppose that the shape of the life cycle wage profile is the same in the two countries (it could be flat, or hump shaped, or whatever). Then in the high growth country, the growth rate of wages between successive generations must be larger. This means that if we look at a cross section of the population by age, the growth rate of aggregate wages will be reflected in it, i.e. the youngest people will have relatively higher wages in the high growth country. 2) Suppose that the cross sectional profile of wages in the two countries is the same. Then any individual’s lifetime wage profile will reflect this aggregate growth; in this case, people in the high wage growth country will have rapidly growing lifetime wage profiles. (Of course there could be a mixed case in between 1 and 2 as well) Cases 1 and 2 yield very different results. Case 1: here, the lifetime profile of the saving rate is unaffected by growth. The aggregate saving rate is just a weighted average of this, where the weights depend on the number of people and their income. Since young do saving and are richer when growth is higher, higher growth will raise the aggregate saving rate! Case 2: Now, higher growth affects the saving rate. Specifically, it lowers the saving rate of the young. It also means that working age people (who are saving) earn more than did old people (who are dis-saving) -- the effect which we saw in case 1 tends to raise the saving rate. For reasonable parameters, the lowering effect dominates, so higher income growth lowers the saving rate. Which case is right? Probably 2 is closer to the truth. For example, wage profiles do not depend on aggregate growth rate of income.

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Ricardian Equivalence We will now talk about some of the implications of the optimal consumption/saving models that we have discussed. Later we will look at more direct empirical evidence. The most controversial implication is the so-called Ricardian Equivalence proposition (which was mentioned, and dismissed, by David Ricardo, and was given its modern rebirth by Robert Barro). Consider the effect of changes in the timing of taxes. To do so, let’s look at the simplest model with taxes, one with just two periods. Let T1 and T2 be taxes in the first and second periods. Lifetime budget constraint is now:

Now consider a change in tax collections that leaves the present value of tax collections unchanged:

For example, if Z is positive (the usual case that we will think about), this would mean that we were cutting taxes today, and raising them in the future. What does this do to the budget constraint?

You can see that the Z's will just cancel out, and the budget constraint is left unaffected. What about savings, though? Since the budget constraint has not changed, first period constumption will not change. But saving of the people in this economy is equal to S = Y1 - T1 - C1 So if we reduce taxes by Z, we should raise saving by the same amount. So does the capital stock go up by Z? No: because the government is going to have to borrow to finance its tax cut. In fact, it is going to have to borrow exactly Z (or, if it was running a deficit already, it will have to borrow Z more dollars).

2 221 11

( - )C Y T+ = - +C Y T1+ r 1+ r

1 2 = - Z = (1+ r)ZT T

2 221 11

( - [ +(1+ r)Z])C Y T+ = - [ - Z] +C Y T1+ r 1+ r

29

The government will issue bonds, paying interest r, and people will hold them instead of capital -- so the amount of capital will not change. (just like giving people a piece of paper with "bond" written on one side and "future taxes" written on the other.). Notice that although people who hold the bonds think of them as wealth, as far as the economy is concerned they are not "net wealth," since they represent the governments liabilities, which will in turn be payed by the people. This is essentially all there is to the Ricardian Equivalence idea. -- idea has generated a huge amount of discussion among economists. -- natural application is the explosion of the US government debt in the 1980's and again in the 2000's. One way to look at it is: Y = C + I + G + NX Y - C - G = national saving = I + NX (Y - T - C)+ (T-G) = national saving private saving + gov't saving = national saving = I + NX Ricardian equivalence says that if we cut T, it will lower gov't saving, but raise private saving by an equal amount. -- Can also look at Ricardian equiv in the life cycle model.... -- Similarly, in PIH, tax cuts and increases are just transitory shocks; they do not affect permanent income, and so do not affect consumption. -- Note that Ricardian equivalence is about the timing of taxes -- it does not say that if the government spending increases this should have no effect on consumption. That is, Ric Equiv says that you care about the present value of the taxes you pay. Government spending, either today or tomorrow, will affect this present value, and so affect consumption. For example, if the govt fights a war today, your consumption will fall, because you will have to pay for the war. But whether the war is tax financed or bond financed will not matter to your consumption today. [but note that the response of consumption will depend on how long you expect the extra spending to last]. Potential problems with Ricardian Equivalence: -- different interest rates. If the government can borrow for less than the rate at which people can, then gov't debt may expand budget constraint (at least for borrowers).

30

-- If people are liquidity constrained in first period consumption, then government borrowing will raise their consumption (show in fisher diagram). -- If people are myopic whole thing doesn't wash. This is probably true, but hard to model. --If people are life-cyclers, and will not be alive when the tax increase comes along, then their budget constraints will be expanded and they will consume more. Later generations will get extra taxes and consume less. This objection has generated the most debate, and often discussions of Ricardian Equivalence lapse into discussions about intergenerational relations. Before going along this path, we should note that even if this objection were true, most of the present value of any tax cut today will be paid back by people who are alive today; in which case even if there were no relations between generations, Ricardian Equivalence would be mostly true. The intergenerational argument in defense of Ricardian Equivalence goes: Since we see people leaving bequests to their children when they die, we know that they must care about their children's utility. Now suppose that we take money away from their children and give it to them. Clearly, if they were at an optimum level of transfer before, they will just go back to it by undoing the tax cut (by raising the bequest that they give). Much ink has been spilled attacking this proposition. For example: -- Can specify the motive for bequests in a number of ways: if parents get utility from the giving of the bequest, rather than from their children's consumption (or utility), then a shift out in the parent's budget constraint will lead them to consume more of both bequests and consumption today. Slight variation (Bernheim, Shlieffer, and Summers) is that bequests are payment for services (letters, phone calls) from kids. Same result in response to a tax cut. -- Alternatively, can argue that bequests are not for the most part intentional, but rather accidental. Consider the life cycle model with uncertain date of death. This model will be covered later. When you see it (with all its discussion of bequests, annuities, etc.), remember why it is relevant to the debate about Ricardian equivalence. -- Interaction of precautionary savings and Ricardian Equivalence (Barsky, Mankiw, and Zeldes, AER.) -- Don't do in lecture -- just do in HW. (precautionary saving will be discussed below). One more thing to think about with Ricardian Equivalence: What if people were completely myopic, and never expected to pay back their tax cut. Note that if they were following our usual consumption smoothing models, they would still raise their consumption only very slightly in response to a tax cut (since they would spread their windfall out over the whole of their lives). So Ricardian Equivalence is still almost true in such a case: for example, if people had 20 years left to live, and the real interest rate were 5%, and they kept consumption constant, then a tax cut of $100 that they never expected to pay back would increase consumption by approximately $8. This is pretty close to the zero dollar increase predicted by Ricardian Equivalence. By contrast, if one

31

believed in a Keynesian consumption function (where empirically estimated MPC's are in the rough neighborhood of .75), then there would be a $75 increase in consumption. [but, of course, if RE were true and the tax cut were perceived to be permanent (due to a cut in government spending), then C would rise by the full amount of the tax cut]. Deep thought: Suppose that I look at data on the path of consumption followed by some person (or household). What are the characteristics that I can expect to see in it, assuming that the household is behaving according to lifetime optimization model described above. I want to argue that one of the most important is that the level of consumption will never “jump,” by which I mean that it will never change dramatically from period to period. When consumption does change, it will be because of the difference between theta and r. So if we do observe consumption jumping up or down, what are we to conclude from it? I will list some possibilities, but it will take us a while to cover them. But you should see in the list that they are all violations of the simple model presented above. 1)Liquidity constraints -- we had been assuming that these didn't exist 2) New information -- we had been assuming a world with certainty. 3) “non-convex budget sets,” specifically things like means tests -- we had been assuming these

away since we made income exogenous. Liquidity constraints under certainty: Let's return to the issue of liquidity constraints that came up when we looked at the two period model. Suppose that you have data on the income and consumption of a large number of individuals, over a long period of time. Each individual is assumed to have known in advance (that is, from the beginning of the sample period), what her income would be for the rest of her life. Individuals in this data set chose their consumption to maximize a usual utility function, with u'(c)>0, u''(c)<0. They were able to save money at a real interest rate which was exactly equal to their time discount rate. However, they were not able to borrow money at all. Individuals did not have smooth income. That is, for each individual there was a good deal

32

of year-to-year variation in income. Furthermore, some individuals had income that rose over the course of the time period examined, while others had income that fell over the time period. Remember, however, that each individual knew in advance what the time path of her income would be. What would you expect the data on consumption by individuals to look like? In particular, discuss the following two points: First, what general statements can you make about what the time paths of consumption of individuals can look like. What sorts of paths can you rule out? What sorts of paths would you expect to see? Second, what will be the relationship between changes in income and changes in consumption experienced by individuals? Are large decreases in consumption going to occur in the same years as large decreases in income? Will large consumption increases come in the same years as large income increases? 1) consumption can never jump down. When theta = r, consumption can never fall at all. 2) when consumption jumps up, it must be the case that assets are zero. If theta=r, then it must be the case that assets are zero when consumption rises at all. Uncertainty So far, we have looked at consumption only in a certainty framework. We now look at the effects of uncertainty. Let's start by reviewing what we mean by expectation. Let x be a random variable, with probability density function f(x). Then the expectation of x is

The relation between the actual realization of a random variable x and its expectation can be written as x = E(x) + ε where ε is a random variable with mean zero. First cut at uncertainty: lifespan uncertainty

-

E(x)= x f(x) dx

33

In our presentation of the life cycle model, we assumed that the date of death was known. In reality, of course, there is a good deal of uncertainty. To incorporate this into the LC model, we apply the insight that, if you are not alive, you get no utility from consumption. Let Pt be the probability of being alive in period t. Then an individual maximizes

0

( )

(1 )

Tt t

tt

PU c

note that we still allow for Θ to measure pure time discount. Consider the problem of a person who may die over period 0..T. Assume that there is no advanced warning. The formal problem is

s.t. At = (1+r)(At-1 + Wt-1 - Ct-1) At ≥ 0 for all t A0 given Note that here, W, C, and A are the paths of wages, consumption, and assets that the person will have if they are alive. That is, since the only uncertainty in this model is when you will die, and that uncertainty is not resolved until it happens, you might as well plan out your whole conditional paths of consumption and assets from the beginning (put another way: no new information arrives until it is too late to do anything about it). Note that the constraint on assets differs from before: as before we say that you cannot die in debt. With certain lifespan, this implies that you have to have zero assets at period T. But now, it implies that you have to have non-negative assets at all periods! Maximization problems of this form are generally unpleasant (it will be discussed a little below, when we look at the “Buffer Stock” model of saving). To get around it, assume that we are looking at an elderly person with no labor income, who has only some initial stock of wealth. Such a person would never let wealth become negative, because then she would have zero consumption for the rest of her life.

Max 0

( )

(1 )

Tt t

tt

PU c

34

We set up the lagrangian:

The first order condition relating consumption in adjacent periods is

we can re-write the second part of the right hand side as

where small ρt is the probability of dying in a given period conditional on having lived to that age [ρt=(Pt - Pt+1)/Pt] The FOC is

Note that the path of consumption that we are solving for here is the path that the person will follow if she is alive. So the probability of dying functions just like a discount rate in this case. ------------------------------------ A note of realism: Obviously, the probability of dying rises with age. Empirically, the probability of dying in old age turns out to conform very closely to a log-linear specification: ln(rho) = β_0 + β_1 age This regularity is known as “Gomperetz's rule” What will consumption paths of people look like given that this is true?

T T -1

t t t0t t

t=0 t=0

U( )C CPL = + -A(1+ (1+r) )

tt +1

t +1t

U ( ) 1+C P = U ( ) 1+ rC P

t t +1 t t +1 t t +1t

t +1 t +1 t +1

+( - ) -P P P P P P = = 1+ 1+P P P

t +1 t t

t

(1+ )(1+ ) (1+ + )U ( )C = U ( ) (1+ r) (1+r)C

35

Suppose that initially, theta+rho<r. Them consumption should be rising. But over time, rho will rise, and consumption will begin to fall. So there should be these hump-shaped paths of consumption. ------------------------------------ Note, by the way, that even though I said that new information can be one of the reasons for a jump in consumption, this simple model with uncertain lifespan does not get you any jumps in consumption -- since when the new information arrives there is nothing that you can do about it! But you could get a jump in consumption if information arrived about the date of death or about future probabilities of death. Simple examples:

1) An individual can live either two or three periods, each with probability 50%. At the end of period 1, she finds out whether or not she will be alive in period 3. Let r=theta=0. There is no labor income, and she has initial wealth endowment W0 . She has log utility . Solve for her first period consumption (actually, just set up the problem). The key to solving this problem is to realize there is no single value for “period 2

consumption.” She will have different consumption in the state of the world where she finds out she will live two periods vs. 3. We could solve this problem two different ways. One would be to just define two different values called c2,live and c2, die , then there are two budget constraints:

,

,

And then maximize expected utility subject to them. Expected utility is

0.5 , , You could do this with a Lagrangian (incorporating both budget constraints), which you would maximize w.r.t. the four choice variables c1 , c2, die , c2, live , and c3. The other, more intuitive way, in this simple case, would be as follows. First, observe that c3 will be equal to c2, live so we have c2, die = (W0 – c1) and c3 = c2, live = (W0 – c1)/2 Now we need to know the relationship between marginal utility in the first period and expected marginal utility in the second. Answer: they had better be equal, for optimality. So we can directly write:

36

1.5

1 112

2) Suppose an individual is born at time zero with wealth W0. She gets no labor income. Time is continuous. Let r=theta=0. Initially, her probability of death is zero. She knows starting at time zero that at some later time, t, she will learn her probability of death (which will be constant thereafter). Suppose that there are two possible probabilities she will learn:

0 , where each will happen with probability 50%. We can solve this problem backward: first, figure out consumption after the information is revealed at time t as a function of Wt in each of the two possible states of the world. Then figure out Wt as a function of initial consumption c0 (which in this simple case is the same as ct -- that is, consumption just before the information is revealed.). Then impose the FOC, which is that the marginal utility of consumption just before the information is revealed has to be equal to the expected marginal utility of consumption just after.

This gives us a very important rule that we can generalize: consumption can be expected to jump, but it has to be the case that the marginal utility of consumption just before the expected jump is equal to the expected marginal utility of consumption just after. Annuities The person faced with the above problem will almost certainly die holding assets. Only if she lives as long as was remotely possible ex-ante will she die with zero wealth. Assuming that she does not value leaving a bequest, what could make her better off? Answer: an annuity. Consider a cohort of people with a probability of dying , and some market interest rate, r. Suppose that a company makes a deal with each person, saying: "Give me your money, and I will pay you some rate of interest z, but if you die before next year I will get to keep you money." What would z have to be such that the insurance company earned zero profits? Pays: Earns (1+z)(1- ) = (1+r) z = (1+r)/(1-ρ) - 1 (1+r+ρ) -1 = r+ρ An annuity is an example of such a contract. You give the company money, and they pay you a yearly payment until you die. (Actual annuities are not like the ones described here, in that you pay a lump sum up front, and then they pay you a constant level of income each year. )

37

What is the consumption path of an old person with access to an annuity? The FOC is just

so in the case where θ=r, the person would have flat consumption even though her probability of death was rising. Thus the payments from a real life annuity are consistent with the consumption path that a individual with r=θ would choose. ) Cost of Lifespan Uncertainty (Ryan Edwards) [should this and Bommier be moved to after becker etc.?] Why this interesting: We know that dying is bad (since you miss out on utility). Here we look at another bad, which is uncertainty about when you die. One way this is bad is that you may have money left over at the end of life. However, annuities can take care of that problem. But it turns out that there is still a cost. Consider a person born at time zero with some known survival curve P(t). For simplicity, we will give him some initial wealth A(0) and no labor income. Also for simplicity, we set the interest rate r and the time discount rate θ to be equal and greater than zero. There is a perfect market for annuities. This means that the man will have flat consumption. The actual level of that

consumption will depend on the survival curve (it would be ( ) (0)r A if mortality were constant at rate rho, for example). But we will not worry about this. Call c* the optimal level of consumption. His instantaneous utility is thus U(c*). We will assume that this is positive (Recall that for a CRRA utility function, instantaneous utility can be negative unless you add some constant in front of it. We will get back to this issue in a little bit). If he lives to age T, then his lifetime utility is

0

( *)( *) (1 )

Tt Tu c

e u c dt e

But from the perspective of the beginning of life, T is a random variable. So his expected lifetime utility is the integral of the above thing over all the possible realizations of T. call f(T) the probability that he will die at age T and call V the expected utility:

0

( *)(1 ) ( )Tu c

V e f T dT

t+1

t

U ( ) (1+ + )C = U ( ) (1+ r + )C

38

0

( *)(1 ( ) )Tu c

e f T dT

So from this expression we can see directly that uncertainly in the lifespan makes you worse off [draw a picture of e^(-theta T); it is convex to the origin, so uncertainty raises the expected value by Jensen’s inequality].

Now, suppose that we know something about the uncertainty of life span. It turns out that the age of death (viewed from birth) has a hump shaped distribution that is not really normal, but is not too far off. So let’s take it as normal. The standard deviation of age of death is around 15 (the mean, which is to say life expectancy at birth, is around 75). So let T be distributed N(M,S2 )

We will use the fact that if x is normally distributed with variance 2x , then

2( ) / 2( ) xE xxE e e

(This is a fact that is often very useful when paired with the CARA utility function in all sorts of problems involving uncertainty). This in turn implies

2 2( ) /2( ) xE xxE e e

So then we can write expected lifetime utility as

2 2 / 2( *)(1 )M Su c

e

Now we can ask, what variation in M would be equivalent to eliminating uncertainty in lifespan (i.e. setting S2 to zero.8 The answer is

2

2

S

If we choose theta=.03, and for S=15, this gives 3.38 years. So this is how much you would be willing to reduce mean life to eliminate uncertainty.

8 I am doing a slight cheat here in holding c* constant. However, I could come up with a way of justifying that if I really needed to.

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Notice that all the “benefit” here comes from discounting, which gives us the convexity. An Alternative View of Lifetime Utility (Bommier, “Mortality, Time Preference, and Life Cycle Models, working paper, 2006) The model of lifetime utility with uncertainty that we have been using is the standard one, first developed by Yaari (1965). The model is (in continuous time)

0( ) ( ) ( ( ))tE V P t e u c t dt

[Note, I am using exponential time discounting, but Bommier uses a more general time discounting where instead of the exponential term there is just some term ( )t which represents the weight on utility from a period, which we assume is non-declining] This formulation comes, in turn, form applying the usual Von-Neumann Morgenstern model of expected utility under uncertainty to a model of utility form a certain lifetime: Utility from a certain lifespan T is :

0( ) ( ( ))

TtV T e u c t dt

To get expected utility, we just integrate this over all the possible life spans (Where F(T) is the PDF of lifespans)

0( ) ( ) ( )E V V T F T dT

Bommier proposes the following alternative model for lifetime utility in the case of certainty:

0( ) ( ( ))

TW T u c t dt

Where () is a function that we will (in the usual case) assume has positive first and negative second derivatives.

40

He argues that this function looks different than what we are used to, but that it meets the same axioms that we want. For example, it says that the marginal rate of substitution between consumption at any two points in time is independent of consumption at other points in time and of the length of life, that more years of consumption make us happier, that more consumption makes us happier, etc. The big thing that this formulation does not have is a pure time discount rate. This is an old argument. Back in 1928, Ramsey famously argued against a pure time discount rate in the absence of mortality uncertainty (he said that it arose from “weakness of the imagination.”) Similarly Pigou (1920) says that pure time discount is “wholly irrational.” It is not clear whether Ramsey and Pigou meant these as statements about what was an appropriate model of human behavior (that is, a positive view) or as normative statements. But anyway…. The justification for the function is along the lines that a person gets “filled up” with instantaneous utility, so that further increments do not do as much as initial increments. It is sort of the lifetime equivalent of the explanation for the curvature of the instantaneous utility function. For example, when you choose what books to read, you read the great ones first, and the less great, and so on. Or similarly, you might spend $100 on a restaurant meal every night, but some of those will give you more pleasure because they are a new experience. An interesting difference between the Bommier formulation and the standard one is regarding “temporal risk aversion.” Consider some consumption levels c1<C1 and c2<C2

Now consider two sets of lotteries. Lottery # 1 gives (c1, c2) and (C1, C2) each with equal probability. Lottery #2 gives (c1, C2) and (C1, c2), each with equal probability. Loosely speaking, Lottery # 1 gives you a chance of winning in both periods or losing in both periods, whereas in lottery #2 you always win in one and lose in one. According to the Yaari formulation, you are indifferent between lottery #1 and lottery #2. According to the Bommier formulation (assuming is concave) you prefer lottery #2. This seems (I guess) more reasonable [at least until we introduce stuff like habit formation later on.] Another advantage of the Bommier formulation is that it separates risk aversion from the intertemporal elasticity of substitution. In the standard model, the curvature of the instantaneous utility function determines both of these. In the Bommier formulation, the function affects risk aversion but has nothing whatsoever to do with intertemporal elasticity of substitution. Now let’s think about lifetime uncertainty. In the absence of lifetime uncertainty, the Bommier formulation implies that if there is a positive interest rate, consumption will be rising. Indeed, without lifetime uncertainty, we don’t ever need to know anything about the function – we just maximize the thing inside it (which is

41

standard utility without time discounting) and we are done. [Note that in the case of certainty, there is no reason to have the good meals or read the good books early in life, which is why you would never get a declining path of consumption. You might get this if you allowed for utility from memory, but then you would also have to allow for utility from anticipation… but this is all not the point.] The Bommier formulation becomes more useful when we allow for lifetime uncertainty. Since is a function, we can’t just pass the expectation sign through the integral. So expected lifetime utility is

0 0

( ) ( ( )) ( )T

E W u c t dt F T dT

[We adopt the same setup as for the Yaari model, in which death is uncertain and unpredictable, so one simply picks a feasible consumption path at time zero and sticks with it until death] To see how this affects consumption, think about the example of reading books. If you know that you will live exactly 80 years and can read one book per year, then with the Bommier formulation, it doesn’t matter in which order you read the books. But if lifespan is uncertain, then you will start with the best and read down the list from there. Similarly, with time discounting would have done the same thing. So the point is that once we allow for mortality uncertainty, the Bommier formulation gives us “discounting-like” behavior. Unlike the Yaari model, however, the effect of mortality on consumption paths is not just like incorporating a higher time discount rate. In fact, there is no simple closed form solution for consumption paths (I think). The whole thing has to be solved numerically. This makes it less attractive…. But Bommier says that now that we all have computers, this should not be a big obstacle. The Value of Being Alive vs. Dead and its implications for Convergence of Full Income (very simplified discussion of Murphy and Topel, NBER 11405, and Becker et al, “The Quantity and Quality of Life, AER March 2005). The starting point for estimates of the utility of being alive vs. dead is people’s willingness to trade off risks of death for money. This can be seen in e.g. the wage premium required to get an individual to take a risky job, or the willingness of people to pay for safety features of a product. We generally look at the effects of small changes in the probability of death, but to make such

42

measures useful, we blow them up to the “value of a statistical life.” For example, if a person is indifferent between paying $1000 and taking a 1 in 10,000 risk of death, then the value he is putting on a statistical life is $10,000,000. Estimates of the value of a statistical life in the US are around $6,000,000. We consider a very simple setup. An individual has constant mortality probability . He never retires, and has constant wage w. The interest rate r and time discount rates are equal and greater than zero. Finally, there is an annuity market, so that the interest rate that the individual can earn on his savings is r + . These conditions deliver the result that the individual will want flat consumption, and since he is born with zero assets, he will just consume his wage at every instant. The individual has CRRA utility with coefficient of relative risk aversion sigma. In addition, the individual has utility just from being alive. His instantaneous utility function is

1

1

cu

Now, consider a case in which the individual has the opportunity to trade a very small risk to his life for more money. For example, he can spend $500 more to take a safe flight vs. a risky one. We consider a trade between life and risk that leaves the individual indifferent. [Note that the constant mortality assumption buys us that willingness to trade life risk for money is not a function of age. In real life, old folks should be less willing to pay to take a safe flight.] Let be the probability of dying, and let x be the amount of extra consumption that he gets. Since x is small, it does not affect the marginal utility of consumption (it doesn’t matter if we imagine him consuming it all at once or spreading it out over the rest of his life.) The cost in terms of expected life utility lost is

1

1( )

0

1

1t

w

we dt

The gain is the marginal utility of consumption, which is w , multiplied by the extra consumption, x. Putting these together,

1

1w

xw

The term /x is called “the value of a statistical life.” [Note, in this derivation, there is a missing (1-ε) term that does not matter as long as ε is small. Really, you only get the extra marginal utility if

43

you are alive] We can rearrange this to solve for alpha

1

( )1

x ww

From here, we can just plug in numbers. I use the following (mostly from Becker et al.) W = c = $26,000 this is GDP per capita in the US

/x = $2,000,000 this is on the low end of estimates for the US. (This is roughly what is implied by Becker et al.’s formulation). = .8 This is their reading of the literature. It seems too low to me, but no one has a good estimate r = = .03

.02 (this gives a 50 year life expectancy) Putting these together gives a value of = -8.81. Becker et al., using a fancier approach, get a value of -16.2. [Note that the value of depends on . Since their 1 , utility from consumption is positive, and so can be negative. For 1 , utility from consumption is negative, so has to be positive.] In what follows below, I will use their value. Given a value of alpha, we can ask at what level of consumption an individual is indifferent between being alive or dead. That is, setting utility to zero

1

01

c

1/(1 )( (1 )) $357c

[Note: the level of consumption that gives indifference between being alive and dead is incredibly dependent on . To demonstrate this, I did the following. Using the above setup, including =.8, I chose the value of life so that I replicated Becker et al.’s value of (this involved setting the value of life to around $1.5 million). This then replicated their value of the indifference level of

44

consumption. Holding the other values of the parameters constant, I then changed to 3. The implied value of the indifference level of consumption is roughly $9,900! In fact, if is 10 (admittedly an unreasonable value), then the break even level is around $18,000, implying that being a grad student is no better than being dead! The intuition for this large effect of is that when is large, the marginal utility of consumption falls rapidly as the level of consumption rises. If sigma is big, then it means that reductions in consumption raise the marginal utility of consumption a lot, so sufficient reductions in consumption very rapidly get you to the point where utility from being alive is zero. (That is, if you are willing to take any life risk at all, the U’ must be not too small relative to the utility of being alive. So then if U’ is not too small and lowering c raises U’ a lot, then at some point lowering c will also make it not worth being alive.] Implications for Economic Growth We usually look at economic growth by looking at growth rates of GDP per capita. But if people get utility from being alive as well as consumption, we should consider their “full income.” (Note: we don’t look at growth by looking at growth of utility. Why not? Because utility is not observable.) Consider an indirect utility function of an individual with annual income y(t) and survival function (probability of being alive in year t) of S(t)

0

( , ) max ( ) ( ( ))tV Y S e S t u c t dt

s.t.

0 0

( ) ( ) ( ) ( )rt rte S t y t dt e S t c t dt

[Fix up notation. Look back at paper.] Note that we are assuming a perfect annuity market, so that expected lifetime income is equal to expected lifetime consumption. Define Y as the PDV of expected lifetime income. So the indirect utility is a fn of Y and S. Consider a country at two points in time, with lifetime incomes Y and Y’ and similar survival functions S and S’.

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We are interested in the extra income that we would have to give the person so that he would have the same utility he had in the second period, but with the mortality rates observed in the first. Call this extra income W(S,S’) V(Y’ + W(S,S’),S) = V(Y’, S’) The growth rate of full income is the change in actual income plus this imputed change in income (I think that this is called “compensating variation,” which is discussed more below in the context of Jones and Klenow) G = [Y’ + W(S,S’)]/Y – 1 (a few adjustments, not discussed here, have to be made to turn this from PDVs into annual income growth). 1960

life expect

1960 GDP p.c.

2000 Life Expect

2000 GDP p.c.

Value of life expect gains (in terms of ann income

Growth Rate GDP p.c.

Growth Rate full income

Poorest 50% of countries in 1960

41 896 64 3092 1456 3.1% 4.1%

Richest 50%

65 7195 74 18162 2076 2.3% 2.6%

So poorest countries get big income growth boost Jones and Klenow “Beyond GDP” Here is a whole writeup of a paper on this (and many other) topic. As of 2013, this is still a working paper. You can find it on Charles Jones’s web site: Utility:

v(l) is utility from leisure; log utility from consumption is used for convenience – this can be done

46

away with but then don’t get such clean expressions; ubar is the utility from being alive. Mortality stuff S(a) is the probability of living to age a (from birth). e is life expectancy at birth. They calculate this thing p which is “the probability that Rawls is alive and gets to consume in this year”

They normalize the utility of not being alive to zero, so they write:

So they will basically multiply utility of “alive people” by life expectancy (they drop the divided by 100 part for convenience). Critique: what is going on here? First, notice that another way to describe what they are

calculating is expected utility from birth, with zero time discount factor (where we are ignoring life cycle type stuff and just giving everyone the same per-period consumption; and we are also assuming that people will have their whole lives at this constant level of consumption. So it is sort of utility for a synthetic person.) What they are calculating is not the happiness of people that you will meet on the street, because it includes the people of a cohort who are not actually alive. This raises all sort of philosophical issues. For example, holding consumption per person constant, this formulation says that utility is the same in a country where everyone dies as 40 as in a country where half the people die at birth and half at 80. Seems like the right model could have some sort of “investment” in utility by society in people or by people in themselves.

Inequality: Assume that consumption is distributed lognormally, with mean consumption c and standard deviation of log consumption sigma.

ln ln2

Variance of log GDP ranges from about 0.4 to 1.2 in the data (and is lower in rich countries on average). So going from a high to a low variance of GDP raises (E(ln (c ) ) by about .4 – that is the same as raising average income by exp(.4), which is about raising it by 50%. Income-metric measure of utility differences

A) Equivalent Variation

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By what factor do we have to adjust Rawl’s consumption in the US to make him indifferent between being born in the US and being born in country i?

Plugging these numbers into the utility function and re-arranging, we can get:

Rewrite this as

Where in practice the ln(y) will move to the other side. Compensating Variation By what factor would be have to change income of Rawls in a poor country to make him indifferent between being born there and being born in the US?

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Difference between CV and EV is only in the stuff in parenthesis in the first term.

Equivalent variation weighs differences in life expectancy by a country’s own flow utility,

while compensating variation weighs differences in life expectancy by US flow utility.

Data and calibration Leisure Assume 16 hours/day time endowment (sleep doesn’t count)

So looking at just adults (issue?) There is some utility from leisure thing that I don’t understand, but it doesn’t matter much. Ubar This is key parameter, utility of being alive. They derive it from willingness to pay to avoid risks to life (see Econ 207 notes). hey look at the literature and based on it set the value of statistical life for a 40 year old in the US at $4 million. This give ubar of 5.54. They find that ubar is big enough so that even in the poorest countries, Rawls is ex-ante

happier being alive then dead. J&K say that this is a good thing in terms of their theory making sense. However, given inequality, there will be some fraction of the population in poor countries that is ex-post less happy being alive than if they had never been born. This seems like a problem.

My view: measuring Ubar from rich countries and importing it to poor countries bad for two reasons. 1) it is just always bad to extrapolate to a very different part of the data based on functional form. 2) I think that “comparison utility” matters here. When a person in the US takes some risk to life in return for consumption, he/she is thinking about doing the consumption themselves, given what they are already used to consuming. A US person

49

might well think that with consumption below $x, it would not be worth living, because he/she is used to consuming much more than this. But a person in a poor country is used to consuming less, so they are likely to be happier with $x. What we really need is either A) value of a statistical life derived from poor places or B) decisions made by pre-birth Rawlsian souls about how many dollars of extra consumption they are willing to trade for a small risk of not being born at all.

Results

50

51

Prinz and Weil (as yet unwritten) paper on Value of a Statistical Life Present the initial puzzle here and the solution later on. The model of JK or BSL implies that for a certain level of consumption, it is better to be

dead than alive. For levels of consumption that are above the critical level, VSL is low for two reasons: first,

VSL is always low for poor people because they can’t pay a lot; but second, VSL is low because the utility of being alive is not very high.

We can calculate an interesting object: VSL / consumption. [derive this] For our standard (JK or BSL) formulation, this rises with income When we look at the data, we see that VSL/consumption does not vary much at all with the

level of consumption! Implications of Curved Utility for Health Care Expenditures (based on Hall and Jones, QJE 2006) Here is a core exam question from 2007 about this issue: A man is born at time zero. Time is continuous. He receives exogenous, constant income at a rate of y per period that he is alive. The man cannot borrow or save. He can use his income for two things: he can purchase a consumption good, which gives him utility, or he can spend it on health, which raises his probability of staying alive (note that spending on health is not considered part of consumption in this setup. In the real world, health spending is considered part of consumption, although it really shouldn’t be.) Spending on the consumption good is denoted c; spending on health is denoted h. Thus the budget constraint is:

y = c(t) + h(t). His pure rate of time discount is zero. However, he only gets utility when he is alive. The man’s probability of dying at any point in time, (t), is a function of his health spending at that point in time. Specifically,

(t) = 1 / h(t) His instantaneous utility is ( ) ln( ( ))u t c t

52

Note: you do not have to do any fancy dynamic optimization, Hamiltonians, etc. to solve this problem.

A. Characterize the optimal paths of spending on health and consumption. You should not do any math at all to answer this part of the problem. Simply state in words what these paths look like and how you reached this conclusion.

B. Write out expected lifetime utility as a function of c(0). Solve this integral to get the PDV

of lifetime utility as a function of c(0).

C. Use the expression you just derived to find an implicit expression for the optimal value of c(0). You will not be able to solve this explicitly.

D. Suppose that income is

y = 2e, where e is the base of natural logarithms (2.71828….). Solve for the exact values of h and c at time zero. [Hint: Try writing the answer to part C in terms of the ratio h/c or c/h.] E. [You can solve this part without solving parts C or D.] Consider several individuals who have different levels of income. Each individual’s income is constant. How does the share of income devoted to health vary with the level of income? (You could answer this by applying the implicit function theorem to your answer from part C.) Based on this answer, what should we think about the fact that rich countries spend more of their income on health than do poor countries? Sketch of answer:

1 1

0 0ln( ) ln( ) ln( )( )y c y cU c e c e c y c

Derivative w.r.t. c implies (y-c)/c = ln(c) = h/c Other things to do with this: What is implied risk behavior as income gets really low?

Willing to risk life Willing to risk wealth if there is an option not to be alive.

Relate this to Stone Geary – very different implications.

SG says that risk aversion toward wealth rises as wealth falls.

53

Precautionary saving We now ask a different question about uncertainty: how does uncertainty affect behavior? It is easy to show that uncertainty makes you worse off. In the single period case, we know that, if c is some random variable: E(u(c)) < u(E(c)) . So, if you could pay to reduce uncertainty, you would do so (for example, buying insurance). But now we want to ask a different question: does uncertainty raise saving? Let's examine the question in a simple two period model. For simplicity, we will assume that there is no discounting and no interest rate. In period 1 you get income Y and consume C1. In period 2, you participate in some lottery: with probability .5 you get some amount L given to you. With probability .5 you have the same amount taken away from you. So your consumption in period 2 is C2 = Y - C1 + L with probability .5 and Y - C1 - L with probability .5 Your only choice variable is C1. We will want to answer the question: how does saving (or first period consumption) change when the size of the lottery changes (that is, as uncertainty is increased). Expected utility is just: E(V) = U(C1) + E(U(C2)) (Notice that we don't need to put an expected value symbol in from of C1, since it is known). = U(C1) + .5*U(Y - C1 + L) + + .5*U(Y - C1 - L) We maximize this by taking the derivative with respect to C1 and setting it equal to zero: 0 = U'(C1) - .5*U'(Y - C1 + L) - .5*U'(Y - C1 - L) The optimal value of C1 is the value that solves this equation. That is, the equation implicitly tells us C1 as a function of L. What we care about is how C1 changes as L changes. To find this derivative, we use the implicit function theorem (See Chaing). If we write this as 0 = F(C1,L), the

54

implicit function theorem tells us that :

Notice that since U''(C)<0, the term in the denominator that is in square brackets is positive, and so the whole denominator is negative. So the key term is: U''(Y - C1 + L) - U''(Y - C1 - L) When will this term be positive? When U'''(C) > 0. In this case, the whole expression is negative, and so

in which case there is precautionary saving. Assuming U'''>0 is what we have been doing in drawing the marginal utility curve as [picture] The intuition is that spreading out consumption in the second period raises the expected value of the marginal utility of consumption in the second period. This means that in taking part of consumption away from the first period and moving it to the second period (ie by saving more), you can, in expectation, get higher marginal utility. This result about U'''>0 being necessary for precautionary saving is not unique to the two period model -- it holds generally. Many of our favorite utility functions, such as log, CRRA, and CARA have positive third derivatives, and thus imply precautionary savings. A utility function that doesn't imply precautionary savings is quadratic utility: U(C) = ß0 + ß1C - ß2 C

2 Here the first derivative is positive (for low enough C), the second negative, and the third is zero. (We know that quadratic utility cannot be globally correct, since it implies that marginal

1

L1 1 1

1 1 1C

d -.5[U (Y - + L)-U (Y - - L)]C C CF= - = -dL U ( )- .5[-U (Y - + L)-U (Y - - L)]C C CF

1dC < 0

dL

55

utility becomes negative at some point. But it can still serve as a useful approximation as long as we restrict our attention to a limited range of values of C) We often use quadratic utility for mathematical simplicity, and also sometimes precisely because it gets rid of precautionary savings. If there is no precautionary savings (and also no precautionary dis-savings), then the economy or consumer is said to display “certainty equivalence.” That is, they act as if future income at every period t were certain to be equal to E(Yt). ====> Another application of this idea is precautionary childbearing [expand this; point out that positive third derivative is a natural assumption.] ====> The general way to deal with these sort of uncertainty problems is via dynamic programming, which you will see later in your math course. 3rd approach to uncertainty: Stochastic Income (leading up to Hall’s Euler Equation approach) Example of a stochastic process and permanent income. Suppose that we are the consumer, and we have some form of expectations about future income: what is the optimal level of consumption that we should choose. [This example is taken from Abel's chapter in Handbook of Monetary Economics.] We consider optimal consumption for a person with income that follows a stochastic process:

1 1( )t t tw w w w where ε is iid mean zero. Describe what is meant by this stochastic process. If α is zero, then income is iid with mean w bar. If α is one, then income is a random walk. More generally, the size of α tells us how persistent the income process is -- that is, how long a shock lasts. Consider a person with 1) quadratic utility 2) r = Θ 3) infinite horizon

56

1) will give us "certainty equivalence." That is, consumption will be the same as if there was no uncertainty (even though utility will be lower). 2) will give us flat desired consumption 3) This is for convenience -- things would look almost the same with a "long" horizon. So for certainty equivalence, we simply have to set consumption today at the level that would be sustainable in expectation. That is, if all future values of ε were zero. so the budget constraint is

notice that the subscript on consumption is t, while on wages it was s. Of the two terms on the right hand side, the first is just assets (ie wealth), and the second is the pdv of future wages -- which we could call "human wealth." We use the rule that

so the left hand side can be replaced with ct((1+r)/r) so ct = (r/(1+r))*( A + HW) (or, in continuous time, c = r*(A + HW) -- this should be fairly intuitive looking. It just says that if you want to have constant consumption, you should just consume the interest on your wealth, where wealth is actual plus human).

t st ts-t s-t

s=t s=t

c w = + A E(1+r (1+r) )

s

s=0

1 x1- x

=

57

Notice that if w were constant, then the pdv of future wages would be w /r -- but because

of out timing assumption, HW would include today's wages, so HW would be w ((1+r)/r), and so consumption would be

c = (r/(1+r))At + w Now we can apply our same rule to the right had side of the equation. First re-write wt as

w + (wt - w )

The sum of expected is just ((1+r)/r)* w . As to the other term, from the stochastic process, we know that

Et (wt+1 - w ) = α(wt - w )

Et (wt+2 - w ) = α2(wt - w ) etc. So we get

re-arrange and get:

This says that the amount by which current income affects consumption depends on the

persistence of income. The term /(1 )r r is the MPC, which will be very low unless income is very persistent. The Euler Equation Approach to Consumption In the above example, consumption ends up being a function of only current income and current assets, with the MPC out of current income depending on the degree of persistence in the

s-tt

tt s-ts=t

1+r 1+r ( - w )w = + w + c Ar r (1+r )

tt t

r r = + w + ( - w )c wA

1+ r 1+ r -

58

time series (ie on α). The above approach used a particularly simple stochastic process for income (ie an AR(1)). However, income could follow a much more complex stochastic process ( for example Autoregressive Moving Average process, called ARMA -- you can learn about these in a time series course.) In such a case, to predict future income we need not only know today’s income, but also income from several past periods. Since consumption depends on the PDV of income, this would imply that consumption was also a function of not only today’s income, but also income from several past periods. Going even further, there might be things other than past income that were observable today and which also helped predict future income. For example, -- suppose that we are looking at data on an individual. Say that she is a professor. Then her future income growth depends on the number of publications. So her optimal consumption depends on her number of publications. -- Suppose that we have data on people’s professions (plumber vs surgeon). Different professions have different wage profiles (flat for plumbers, rising for surgeons). Since the individual knows this, he should take it into account in figuring out the PDV of future wages, and thus this should affect consumption today as well. -- if we are looking at the aggregate economy, there may be pieces of data available at time t that tell us about expectations of future income, for example the value of the stock market or the consumer confidence index. These should affect consumption today as well. The upshot is that consumption today should be a function of wages today and of lots of things that predict how wages will change in the future. So one could imagine estimating a very complicated “consumption function” that incorporated all of this kind of stuff on the right hand side. And indeed, people do this. The problem is that it is very hard to interpret what you get in terms of any theory. Now, suppose that instead of looking at the level of consumption, you look at the change in consumption, that is, c(t+1) - c(t). What do we expect to be the predictive power of all of this stuff (current income, past income, other stuff) that went into the consumption function? After all, this stuff went into the consumption function in the first place because it helped to predict how income would change over time. The answer is that under the PIH, this stuff should have no predictive power at all! So now suppose that you tried to predict consumption in period t+1 using consumption in period t as well as other information available at time t. The test that Hall proposed is that nothing else available at time t should matter. This turns the usual sort of econometrics that we do on its

59

head: usually, we do empirical work hoping that what we look at will matter (as judged by a t statistic, for example). Here, the theory succeeds if other things do not come in significantly. More formal description of the Euler Equation approach: Consider the consumption plan formulated at some date t. That is, consider a person at time period t who is deciding on consumption today. We know that, along this consumption path, the relation between Ct and the Ct+1 is

where now consumption in period t+1 has an expectation sign in front of it because it is not actual consumption in that period, but just what is expected to be consumed.9 Note that all of the things on the right hand side are known already at time t. Recall that the relation between the actual realization of a random variable x and its expectation can be written as x = E(x) + ε where ε is a random variable with mean zero. Or in this case,

That is, the actual marginal utility of consumption at time t+1 will be equal to the expected marginal utility plus a mean-zero error. If we assume that r= theta, then we can combine these last two equations to get

9 This first-order condition relating consumption in adjacent periods is also known as the Euler equation, and so this approach to studying consumption is often called the “Euler Equation Approach.”

t t+1 t

1+[U ( )] = U ( )C CE

1+ r

tt+1 t+1 t+1U ( )= (U ( ))+C CE

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This says that the marginal utility of consumption follows a random walk. Of course we can’t observe the marginal utility of consumption -- we can only observe consumption itself. Now, assume quadratic utility: U(C) = ß0 + ß1C - ß2 C

2 then U'(c) = ß1 -2ß2 C So the above equation becomes

or

where the epsilon is slightly differently scaled, but is the same mean zero error term. So if utility is quadratic, consumption should follow a random walk. Now, consider what should happen when we regress the change in consumption on all of the “stuff” observed in period t that went into the consumption function for the level of consumption (ie current and past income, consumer confidence, the stock market, papers published, etc.). All these things predicted changes in income, but they should not predict changes in consumption. The reason is that changes in income that they predicted were already taken into account when optimal consumption in period t was calculated. This is the test of the PIH proposed by Robert Hall. Specifically, suppose that you found some variable that plausibly predicted changes in income, and which also came in significantly when you regressed the change in consumption on it (and was in the information set of people at time t when they made their consumption decisions). This would violate the PIH. Hall's test of the permanent income hypothesis is so clever because it can be performed by an observer (i.e. the econometrician looking at data) who knows very little about how the consumption decision is being made. For example, we don't have to know anything about what the person's expectations of future income are.

t+1 t t+1U ( )= U ( )+C C

t+1 t t+11 2 1 2- 2 = - 2 + c c

t+1 t t+1= +C C

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(One of the clever aspects of the Hall approach is that it does not rely on being able to actually estimate the consumption function. The essence of the “Lucas Critique” is that the consumption function should not be stable when the economic environment changes, and so estimating consumption functions went out of style. Since the Lucas Critique has not yet been presented in this course, this comment will not be very meaningful the first time you read it. But later in the course it should make sense.) More generally, if r is not equal to θ, or if utility is not quadratic, the implication of the PIH is that consumption should just be predictable by last period's consumption and some constant growth term. If there are any other ways in which consumption is predictable, then the LCH/PIH does not hold. Empirical evidence on Euler equation results: it turns out that the question gets complicated. Can see Hall for a summary. An Aside: Tax Smoothing The discussion earlier of Ricardian Equivalence took the movements of taxes from one period to another as being exogenous. But a related literature has asked what is the optimal pattern of taxes. That is, given that the government has a certain amount of spending that it wants to do, how should it arrange taxes to finance it? To answer this question we have to think a little harder about the question of how the taxes are collected. We often assume that taxes were lump sum -- for example, $1 per person. Why is this the easiest assumption to make? In this case, taxes do not have any effect on behavior other than through the budget constraint. (In fact, even if taxes fall on labor income, if the supply of labor is inelastic, then there is still no distortion, so taxes might was well be lump sum.) In the case of lump sum taxes, we have seen that the timing of taxes may not matter at all. But in the real world, very few taxes are lump sum. Instead, taxes are levied on income or consumption -- that is, you have to pay the tax when you engage in economic activity. We have learned in Micro that such taxes are distortionary. As an example, consider a tax on wages. We have some supply for labor and some demand for labor, an equilibrium price and quantity, and an associated consumer and producer surplus (note that in this case the "consumer" is the firm, while the "producer" is the worker.). Now the government imposes a tax on wages -- it doesn't matter if this tax is imposed on the worker or the firm, the result is the same. The tax imposes a wedge between what the producer gets and what the consumer pays. In equilibrium, quantity goes down. We can show the areas of consumer surplus, producer surplus, and government revenue.

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Note that a little triangle got lost -- this is the loss of efficiency due to the tax. This is the "dead weight loss." Now suppose we wanted to graph a function that related the loss of efficiency per unit of revenue collected. When the tax rate is zero, revenue is zero. But the marginal revenue from imposition of a tax is high, while the marginal efficiency loss is zero. As the tax rate rises, each rise in the tax rate costs more in efficiency, and earns less revenue. So the cost in efficiency per tax dollar collected is an increasing and convex function (that is, its second derivative is positive). Now you should see the relation between the government's problem in choosing tax collections over time and the individual's problem of choosing consumption over time. Individual has a concave utility function, and wants to maximize utility. Government has a convex "distortion" function, and it wants to minimize total distortions. In both cases, the optimal thing is to smooth. So in the presence of distortionary taxes, if Ricardian Equivalence holds, gov't wants constant taxes. And all of the Permanent Income type thinking that we are used to doing for consumption should also hold for taxes: for example, suppose that the government has to fight a war. Should it pay for it by taxing, or by borrowing? Clearly, this is a temporary change in it's spending. We know that an individual's consumption should not react much to a temporary change in income; similarly, taxes should not react much to a temporary change in spending. (Note: We have been assuming that the interest rate is invariant to all of this moving around of taxes. Maybe that is right for a small open economy fighting a war. But for a closed economy, there will be a reaction of the interest rate to the fact that the government wants to borrow a huge amount during a war. So all of the conclusions of this section have to be revised). Notice that given what we said about tax smoothing by an optimizing government, the random walk property should also apply to taxes: nothing other than today's taxes should predict tommorrows taxes. So if r=θ, there should never be any predicatble changes in taxes. (See Barro article on this topic). Precautionary Savings and Stochastic Income Now we look at a similar problem of solving for optimal consumption when income is stochastic.10 But this time we are interested in precautionary savings, so we no longer set up the problem with certainty equivalence. Recall the difference between risk aversion and precautionary savings. Risk aversion says that you are made worse off by uncertainty. This is just due to u''<0. Precautionary saving says that

10Blanchard and Fischer, p. 288-291 and notes from Cecchetti.

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you will change your behavior (i.e. save more) in response to uncertainty about future income. This requires that u'''>0. For this problem we will use a utility function where u'''>0. However, we are not going to be able to use our favorite CRRA utility function, because in this case it is not analytically tractable. Instead we use the Constant Absolute Risk Aversion utility function:

[(move this earlier?) What is the difference between CRRA and CARA? Consider the question of how much someone is willing to pay to get rid of a certain risk. For example, how much would you pay to get rid of a risk that consumption will go up or down by $100, each with probability 50%. Say that the base level of consumption is $500 (so that consumption might be either $400 or $600), and you find that a person is willing to pay $30 to eliminate the risk (that is, they are indifferent between $470 with certainty and the lottery of $400 or $600). Now suppose that we raise their base consumption to $1,000 but keep the size of uncertainty constant: their consumption will be either $900 or $1,100. How much will they pay to insure this risk. Under CARA, the answer would still be $30. Under CRRA, it would be less than $30, since the risk relative to consumption has declined. Under CRRA, it would be the case that they would pay $60 to insure a risk of $200 on a base of $1,000..... extend this example? [Put in Kimball measure of prudence.] Notice that a property of CARA is that when consumption is zero, the marginal utility of consumption is finite. By contrast, for CRRA, when consumption is zero the marginal utility of consumption is infinite. Indeed, the CARA utility function is defined for negative consumption, while CRRA is not. This makes CARA suitable (mathematically) for problems like the one we will do now, where consumption may sometime end up being negative in order to satisfy a lifetime budget constraint. Whether this makes sense economically is a different question. Consumption can't really be negative, but on the other hand, the property of the CRRA utility function that marginal utility is infinite at zero consumption may overstate the importance of that state of the world... even a low probability of infinite marginal utility will dominate optimization. In fact, if consumption gets near zero in advanced countries, other mechanisms kick in... we will address these below.] We will assume that θ=r=0 for convenience. We also assume a finite horizon. We assume that income follows a random walk: Yt = Yt-1 + εt

- c-1

U(c) = e

- c - c 2 - cu = u = - u = e e e

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Where ε is distributed N(0,σ2) Notice that income can become negative. So the optimization problem at time t is:

where assets and income at time t are known, subject to the constraint that AT=0. I will state the solution, and then prove that it is correct. The solution is:

We will first check that it satisfies the first order condition. Given zero interest and discount rates, the FOC is just: First we will solve for ct+1 in terms of things that we know at time t. The evolution of assets if given by

Max s

T -1- c

ts=t

-1 eE

2

t tt

1 (T - t -1) = + - c A Y

T - t 4

tt t+1U ( ) = [U ( )]c cE

t+1 t t t = + - cA A Y

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Now, we write ct+1 in terms of quantities at time t+1, being very careful to write in the proper time index!

We can substitute in for assets and income at time t+1:

So and

2t t t t

1 (T - t -1)= + - + -A Y A Y

T - t 4

2t

T - t -1 (T - t -1)= + A

T - t 4

2

t+1 t+1t+1

1 (T - t - 2) = + - c A Y

T - t - 1 4

2 2t tt+1 t+1

1 (T - t - 2) = + + + -c A Y

T - t 4 4

2t tt+1 t+1

1 (T - t - 3) = + + -c A Y

T - t 4

2

t+1 t t+1 = + + c c2

2

t t+1 t( ) = + c cE2

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Notice that this is similar to the Hall random walk result. From the utility function: But we care about the expectation of this. Luckily, there is a rule that we can use in this case: Let x be some random variable distributed normally with mean E(x) and variance σ2

x. Then Notice that this rule corresponds to Jensen's inequality: the exponential function is convex, so the expectation of exponential of a random variable is greater than the exponential of the expectation of the random variable (say that five time fast). For the current problem, ct+1 is a random variable when viewed from the perspective of time t. We already solved for its expectation, and its variance is just σ2.

we know that So

t+1- c

t+1U ( ) = c e

2xx E(x) + /2E( ) = e e

t+1

tt+1 t+1Var(- )c- (- ) + c cE

2t tt+1(U ( )) = ( ) = c e eE E

22

t tt+1 t+1 t(- ) = - ( ) = - -c c cE E2

2 2 2

t+1 t+1Var(- ) = Var( ) = c c

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So the first order condition checks! Checking the budget constraint is much easier. We only have to check that if we follow the suggested policy, end-of-life assets will be zero. As usual, T-1 is the last period of life, and so we just check that assets at the beginning of period T would be zero: we substitute into the formula for assets in period t+1 above, letting period t be period T-1:

So what do we learn from this? Most important result is that expected consumption rises over time. Further, the speed with which it rises depends on the degree of uncertainty: σ2. This confirms the result of our two-period model of precautionary saving. We can derive a testable prediction from the model of precautionary saving: Think if you followed a cohort of people over their lives, where each person had uncertain income, but on average there was no uncertainty. What would you expect to see for average consumption of the people in the cohort? -- it would be rising over time. Now suppose that we grouped people into occupations. Within each occupation, there is an expected wage profile (say: flat for mechanics, rising for brain surgeons, etc.) Should the slope of the wage profile affect the slope of people's consumption profiles? No, of course. But suppose that occupations differed in their degrees of uncertainty? We could look at people in risky vs non-risky occupations. The model predicts that people in less risky occupations should have lower average rates of consumption growth. This is a generally applicable point: there are many situations where on average there is not much uncertainty, but where at the individual level there is a good deal of uncertainty. Some examples: health, date of death, success in career, etc. Macro-economists used to ignore this sort of uncertainty, precisely because at the societal level the law of large numbers them go away. But if they affect behavior, then the uncertainty is relevant.

t- c

t t+1 t(U ( )) = = U ( )c e cE

2T T-1 T-1 T-1 T-1 T-1 T-1T-1

1 (T-(T-1)-1) = + - = + - + + = 0CA A Y A Y A Y

T-(T-1) 4

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Liquidity constraints under uncertainty: if θ>r, this leads to buffer stock saving (do more formally?). Here is the simplest buffer stock model:

E(w) is constant, but there is variance (iid): t tw w with iid shock. time horizon is infinite (or long) θ > r liquidity constraint == if there was no uncertainty and no liquidity constraint, would get a declining path of consumption == if there was uncertainty but no liquidity constraint, would also get a declining path, adjusted for shocks (or could have a rising path due to precautionary savings, I guess). == if liquidity constraint but no uncertainty, would get consumption = wages as constrained optimum. == if both, it is complicated. We have to solve by dynamic programming. (flesh this out!) Define cash on hand as w + A -- that is, all that you have available to spend this period. Since income is iid, this is all you can base your decision on. So we are looking for c as a function of cash on hand.

===> handout picture from Deaton book. (Deaton’s parameters: 10% , 5%r ) To be added (maybe): non-linear budget sets. In real life, the assumption that income is completely exogenous to consumption choices does not hold. One of the most important feedbacks from consumption to income is in the form of means tested government programs, which provide income to you only if you are poor enough to qualify. Examples are Medicaid, Food Stamps, Welfare, financial aid for college, etc.11 11 There is also an important feedback at the aggregate level: the more that people in a country save, the higher will be the capital stock, and thus the higher will be wages. We will deal with this feedback extensively when we study growth.

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[flesh out simplest example : see former homework problem below ] 33. Consider a world in which people live for two periods. In the first period of life, person i has income of xi. In the second period, all people have income of zero. Individuals can save money at a real interest rate of zero. They cannot borrow. They are born and die with zero assets. Individuals have log utility, and their time discount rate is zero. There is a government program which has the following setup: if the sum of your income and your savings (from the previous period, if any) in a given period is less than a cutoff level, c, then the government will give you enough money so that you can consume c. If they sum of your income and savings in a period is higher than c, then the government will not give you anything. Describe the relationship between the first-period saving rate (that is, saving divided by first period income) and first period income. If you can't get exact solutions, try drawing a picture and discussing how you think it might be done. Alternative forms of the utility function We have been assuming time-separable preferences -- so utility at a given time depends on just consumption at that time. Is this reasonable? Quite possibly not: two people with the same consumption might have different happiness depending on what they were used to. There is nothing radical about non-time-separability (unlike some other alternatives to the usual utility function) -- when we use time separable preferences it is really only for convenience. One form of time non-separability is durability [think about food or vacations]. If utility from consumption is durable, then the more I consume in period t, the lower my marginal utility of consumption is in period t+1. [Note: if c in one period makes me happy in another period without affecting the marginal utility of consumption, then we can just add this happiness back into the initial period’s utility and we are (almost) back to time separability.] We will look at another form of non-separability called habit formation, in which consuming more in a period raises the marginal utility of consumption in the future.12

12 It is perfectly possible for durability and habit-formation to coexist at different time horizons. For example, if I consume a fine French meal at noon, it lowers the marginal utility of French food at dinner time, because I am still full. But it also raises the marginal utility of French food in future days because I will not longer be full and now have acquired a taste for good food.

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Consider a utility function like

where z is the quantity of habit. Notice that habit makes you worse off. If gamma is zero, then habit is irrelevant. If γ =1, then only the ratio matters. So γ indexes the importance of habits. The closer it is to 1, the more people care about consumption relative to habit. The closer it is to zero, the more people care about the absolute level of consumption. For example if γ=.5, then a person with consumption of 4 and habit stock of 4 will have the same utility as a person with consumption of 2 and a habit stock of 1. Habit in turn evolves according to the level of consumption: zt+1 = ρct + (1-ρ) zt The bigger is ρ, the faster habits adjust. We can rewrite this as an infinite sum:

10

(1 )it t i

iz c

So this says that the smaller is ρ, the more consumption in the distant past matters for today’s habit stock. (As an aside: we have been taking “habit” or “what you are used to” to depend on your own past consumption. Another possibility is that it depends on the consumption that you observe around you. This is the idea of “keeping up with the Jonses.”) This can also be incorporated into a utility function. The classic case was Dusenberry, who formulated the “relative income hypothesis” in an attempt to explain the same facts that Friedman successfully explained by the PIH.) Anyway, these things are a mess to solve, but they may provide better fit to reality that the function that we use. Interesting thoughts: do people understand their own future habits? See Keynes on Economic Prospects! Discuss COW. Example of tenure, loss of limbs, etc. Also example of lunch.

1-c

zU(c,z) = 1-

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Discuss Robert Frank stuff New Material to be added: More formal discussion of optimization in habit formation model. Can do something based on Deaton formulation (see deaton book or Paxson eer): quadratic sub utility (muellbaur form) gives simple first order condition even in stochastic model. Or can go over COW Also discuss Stevenson and Wolfers re: Easterlin Paradox

Discuss here: the Prinz-Weil solution to the VSL puzzle.

Inter-Temporal Consistency Go back to time-separable preferences. You will see if/when we you study monetary policy that one of the key problems that faces a policy maker is time inconsistency (example: negotiating with terrorists; nuclear war; monetary policy). Question: In a world of certainty (which we have been doing so far), would you ever want to re-optimize. That is, if you can make all of your decisions at time one, are you then happy to stick with them? The classic paper that deals with this problem is Stroz (1956), which I confess I haven't read. Stroz shows that the only form of discount factors which does not lead to inconsistency of the lifetime utility function is our usual exponential discounting (i.e. dividing utility in period t by (1+ θ )t in discrete time, or multiplying by e^{- θ t} in continuous. Any other form of discounting leads to the paradox that the relative importance of two periods= consumption depends on what point in time the problem is being viewed from [picture] Some people (such as Angus Deaton in his book) think that Stroz’s result shows that any form of discounting other than exponential is irrational, and thus should not be considered. David Laibson takes on this challenge: argues that preferences are not, in fact, exponential, and further that there is loads of evidence of inconsistency. examples of inconsistency: revisions of plans, diets, Christmas clubs and other forms of hand-tying, etc.

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Laibson claims that the natural discount function is hyperbolic: so discount the immediate future a lot, but not much between immediate and far future. [note: “hyperbolic” is not the mathematically correct term for what Laibson posits, but the term has stuck.] The actual utility function that he uses in his paper is

where ß<1 hyperbolic discounting means that my (today) self will want to precommit: forced saving, purchase of durables, etc. Laibson then models a game between different “selves” at different periods. Notice that the selves will want to pre-commit later selves. For example, I might want to save my money in a way that it can only be taken out with one period's advance request... (Thaler’s proposal: save more tomorrow) Laibson also argues that the recent decline in the saving rate comes from the ability to get around these sort of precommitment devices. [add stuff from Thaler?] [for somewhere: talk about Epstein-Zinn]. Some Other Applications of the Basic Consumption Model Mbiti / Weil: Discount Rates and M-Pesa (see Article in May 2013 AER; also slides from that presentation) “Nudging Farmers to Use Fertilizer: Theory and Experimental Evidence”

Duflo, Kremer, and Robinson. (AER 2011)

Fertilizer is a good thing.

1

1

T -t

t + tt t=1

= u( ) U c cE u( )

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Using one half tsp per plant raises yield by $54 per acre at a cost of $40 per acre – a 36% return (for less than half a year of time).

Why don’t farmers use? When asked, the overwhelming response is that they don’t have the money to buy it. [note “traps” are not a possibility here, since fertilizer quantity is perfectly divisible.]

Farmers want to be able to use fertilizer. In one study, of the farmers who participated in a demonstration project on fertilizer use, 97% said they intended to use fertilizer next season (when asked at the end of the program, i.e. at harvest), but only 37% did so.

Their model is a big mess, so here is my simplified version:

Period 0: before harvest – will be relevant later.

Period 1: farmer harvest previous crop. Receives income x>1.

Period 2: farmer plants maize and may choose to fertilize it if he has sufficient money to buy fertilizer

Period 3: farmer harvests maize.

Farmer has present biased utility where beta is the discount applied to all future periods. For simplicity, there is no other time discounting (i.e. theta (in my notation) is zero – except that they call it delta). So in period 1

U = U(c_1) + β [U(c_2) + U(c_3) ]

In period 2: U = U(c_2) + β (U(c_3))

Application of fertilizer is discrete – either do or don’t do it. This is not due to a technological constraint, since fertilizer can be purchased in small quantities and applied to part of the crop. But anyway, they say we see that farmers generally do none or all/most.

Price of fertilizer is 1.

No borrowing.

Define y as the incremental yield from application of fertilizer. So will harvest (1+y) if apply fertilizer.

Between periods 1 and 2, there is a non-fertilizer investment that the farmer can do which yields return r. This could be investing in a business or something.

We assume that (1+ y) > (1+r) . That will be important later.

Utility is just equal to consumption (I am a bit confused about this, and about how to think about utility in period 2, but I think that we can just think that this is the utility calculus that applies to money from this maize or something. In any case, this linear utility will give us that people are at corner solutions of either shifting all consumption or none).

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There are two values of beta: 1> β_H> βL

They assume that βL is low enough that an impatient guy will not want to invest in the off season investment:

Beta_L (1+r)<1

And patient farmers do :

Beta_H (1+r) >1

Further assumptions:

Beta_h *(1+ y) >1

This says that a patient farmer will prefer to buy one unit of fertilizer to get himself extra yield y. Notice that he would ideally like to buy the fertilizer in period 2, after he has earned a return of r form investing in period 1. However, the condition says that even if he had to buy in period 1, he would prefer to do this.

[By the way, another interesting case would be

Beta_h * (1+y) * (1+r) > 1 > beta_h *(1+ y) > beta_h (1+r)

This would imply that a naïve but always patient guy would want to buy fertilizer in period 1 so that he could use it in period 2 and consume more in period 3. However, once period 2 came around, he would no longer want to follow that plan – and that therefore in period 1 a sophisticated guy would want to bind his hands, or else he would just consume in period 1]

Next, we have

(1+r) βL (1+y ) < 1

This says that an impatient guy would not want to save in period 1 in order to buy fertilizer in period 2.

Now, they say that there are three types of guys: always impatient, always patient, and stochastically impatient. The stochastic guys can move from one state to another. The stochastic guys who start off impatient are not very interesting, since they just consume everything in period 1. The stochastic guys who are patient in period 1 have to form an estimate of the probability of being impatient (βL) in period 2.

Under the assumption above, a patient guy who knew that he would be impatient in period 2 would buy fertilizer in period 1, in order to bind his own hands (this is because (1+y)>(1+r) )

However, if the guy thinks that there is a low probability of his being impatient, then he will invest in the liquid asset, intending to buy fertilizer in period 2.

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So… you can have a case where patient people underestimate the probability of becoming impatient. Then they will invest their money in the liquid asset in period 1, but then when period 2 rolls around, they will not buy fertilizer.

DKR then think about giving subsidies that reduce the price of fertilizer in periods 1 or 2.

They find that there are some period 1 subsidies that can induce a beta_h farmer to purchase fertilizer then, rather than waiting. That will depend on the subsidy relative to (1+r) as well as the subjective probability of the farmer switching. [calculate expression]

Indeed, even if the farmer thought with 100% probability he would be patient in period 2, a subsidy that sets the price to (1/(1+r)) would be sufficient to get him to buy in period 1.

If the subjective probability is wrong, there can be a case where the subsidy will lead to the right outcome, etc.

By contrast, a period 2 subsidy would have to be much bigger to get farmers to buy fertilizer. In particular, we know that a subsidy that sets the price to (1/(1+y)) would not be sufficient.

What they did:

All this in western Kenya. Do pre-survey, find that very few farmers buy fertilizer at any time other than right before it is to be used.

Then they have some interventions. Basic one is to come to farmers at harvest time, when they are flush with cash, and offer to sell them a voucher (for cash now) for fertilizer to be delivered at the appropriate time. The free delivery is a subsidy, but it is only small because they say it really isn’t very hard to get fertilizer.

Second, they come to farmers earlier (before harvest – i.e. period zero) and say: “we are going to come around and offer to sell you a voucher for fertilizer and free delivery. When do you want us to come: right at the harvest, when you are flush with cash, or later on, when it is time to apply the fertilizer?”

Findings:

1) Fertilizer higher in group of farmers offered the program than in a comparison group (45 vs. 34% -- this difference gets bigger in a regression that controls for other observables)

2) Once treatment is eliminated, fertilizer use of the treated group goes right back to being the same as the untreated group! So it is not lack of knowledge or simply being “too poor to buy fertilizer” that leads to low uptake.

3) Of the farmers offered a choice of when to have the voucher person come to them 44% wanted them to come at harvest time, and of these 46% bought fertilizer 52% wanted them to come later, and of those 39% bought fertilizer.

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(I think that the important thing is not the difference between fertilizer uptake here, but rather that a very large number of people wanted the early visit, which was a means of tying their hands. Indeed, maybe these were exactly the people with self control problems, so absent early delivery, their uptake would have been lower than the ones who chose late delivery.) From their more complicated model:

Findings

71 percent stochastically present biased (and somewhere in the model is their mis-perception)

16 percent always patient

13 percent always impatient

Conclusion: they say that small subsidies, with the appropriate timing, can be a “nudge” that get farmers to do the right thing. In the context of African agriculture, this is a midpoint between the view that farmers need large fertilizer subsidies (the old philosophy, recently re-introduced in Malawi) and the view that the market will take care of everything, and that people optimize (identified in the paper as the Chicago view). Conclusion is in line with the Thaler/Sunstein “Nudge” view.   

Extending the Basic Consumption Model to Encompass Labor Supply (This is not a consumption topic at all, but it uses the same basic model, so I am putting it here) If we look at business cycles in the US, we see big co-movements in GDP and total hours (or average weekly hours). Real Business Cycle theory attempts to explain these co-movements via labor supply choices of optimizing households So why should hours vary over the business cycle? To see why, let's look at our basic intertemporal optimization problem, this time allowing for labor supply to be a choice variable. Let's solve the individual's problem, taking interest rates and wages as exogenous. define N as the amount of leisure the a person consumes per period. We get utility from consumption and leisure, so the instantaneous utility function is Ut = U(Ct, Nt). Each period, the person has an endowment of time (say the total number of hours they are awake), which she can divide between leisure and work. Call the total endowment 1, and so time spent working is (1-Nt). There is some exogenous wage, wt, so total labor income in period t is just wt(1-Nt).

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There is some interest rate, r, which we will just hold as constant for now. There is also some discount rate, Θ. So the individual's problem is

s.t.

so we set up the old Lagrangian:

to solve, we would just differentiate this with respect to the T different values of C, the T different values of N, and λ. This would give us as many FOC's as unknowns, so we could solve. We will just

c 0 0 c 0 00

dL1) = ( , ) - = 0 ===> ( , )=U C N U C N

dC

-1 -1c 1 1 c 1 1

1

dL 1+2) = ( , )(1+ - (1+r = 0 ===> ( , )=) )U C N U C N

d 1+rC

n 0 0 0 n 0 0 00

dL3) = ( , ) - = 0 ===> ( , )=U C N w U C N w

d N

1 1

1 1 1 1 111

14 (1 ) (1 )

1n n

dL) = ( , ) - r = 0 ===> ( ,N )=U C N w U C w

d rN

look at the FOC's for C and N in periods 0 and 1 (obviously it could be any two periods). combining (1) and (2) gives us the usual FOC for consumption:

1

MaxT

-tt t

t=0

U( , )(1+ )C N

1 1T T

-t-tt t t

t=0 t=0

(1+ r (1- ) (1+ r) )C N w=

1 1T T

-t -t -tt t t t t

t=0 t=0

L = U( , )(1+ + (1- ) (1+r - (1+r) ) )C N N w C

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combining (3) and (4) gives us a FOC for the relation between leisure today and tomorrow:

This says that if w1 is higher than w0, then you want higher marginal utility of leisure in period 1 -- and the way you get higher marginal utility of leisure is to have less of it. So you work more in response to higher wages. We can apply the same intuition that we developed about intertemporal consumption smoothing to the question of intertemporal leisure smoothing. That is, the key question is the intertemporal elasticity of substitution -- the willingness of people to move consumption or leisure from one period to another. As with consumption, the key is the curvature of the utility function -- if there is rapidly declining marginal utility of leisure, then people will not readily substitute leisure from one period to another in response to higher wages. On the other hand, if there is fairly constant marginal utility of leisure, then in response to wage changes, people will be willing to shift leisure around -- and then you will get a big response of labor input to changes in the real wage. We can also consider the effects of changing the interest rate on labor supply: raising the interest rate that holds between period 0 and 1 makes us want to work more in period 0 relative to period 1. Point of all of this -- when you have a productivity shock, the wage goes up. This will cause people to intertemporally substitute. Lucas example: how much extra do you have to pay people to move their vacations? This first order condition is clearly the key to thinking about how labor input should vary over the business cycle: when there is a good shock, output goes up because people work more, and because people are more productive. [aside here or elsewhere: we can get an even bigger effect if we allow for time-non-separable utility from leisure: say that the utility of leisure today depends on how much leisure you consumed yesterday (or last quarter, or last year, or whatever). An example of such a utility function is the one used in Kydland and Prescott's model:

c 1 1

c 0 0

( , ) 1+U C N =( , ) 1+rU C N

n 1 1 1

n 0 0 0

( , ) 1+U C N w=( , ) 1+rU C N w

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current effective leisure is a weighted average of current and past leisure. With such a specification of utility of leisure, if there is extra-high productivity today, and regular productivity tomorrow, you may work more than usual today, but then less than usual tomorrow. So this will exacerbate the fluctuation-causing power of productivity shocks. ] Finally, we can also look at the “static first order condition,” derived from looking at FOCs (1) and (3):

This just says that the marginal utility of consumption in a period has to be equal to the marginal utility of leisure in that period times the wage -- or else people would just work more or less. We can think of all of these relations -- between consumption in adjacent periods, between leisure in adjacent periods, and between consumption and leisure in the same period -- arranged as follows: c0 --- c1 --- c2 --- etc. | | | | | | n0 --- n1 --- n2 --- etc. It is clear that some of the restrictions are redundant -- that is, you do not need to know them to pin down all of the values of c and n. For example, if you satisfy all of the dynamic c and n FOCs, and just one of the static FOCs, then you can be sure that you are satisfying all of the other static FOC's. Or if you are satisfying the dynamic c FOCs, and all of the static FOCs, then you don't need to check the dynamic FOCs for n. Business cycle effects The static FOC is w Uc = Un. We can think about doing some things with this. First, think about a permanent change in w – that is, a country going from having low w to having high w. What does that do to labor supply? Answer is that it is unclear: there are income and substitution effects that go in opposite directions. Indeed, if utility from consumption is ln ( ), then

1

ln ln 1 -it-it

i=0

u(c,l)= ( )+ - )c l (

n 0 0 0 c 0 0( , )= ( , )U C N w U C N

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these balance out exactly, so there is no change in labor supply. Second, think about a temporary (one period) change in w. If you are optimizing consumption over the life cycle, this will have almost no effect on c, and thus on Uc. Therefore, for the static FOC to hold, you have to have an increase in Un – that is, more labor supply. That is the essence of the RBC story for why labor supply goes up in booms. Third, think about adding a temporary increase in government spending (like a war) to the model. You can think of this as an exogenous reduction in consumption in a period (lump sum tax). This would raise the marginal utility of consumption. In order to make the static FOC hold, marginal utility of leisure has to go down (people work more). Further, to make the dynamic FOC hold, the interest rate between current period and the future has to go up. This is the RBC version of fiscal (spending) policy affecting output. Non-convexities in labor supply One problem with the fit of the simple RBC labor supply model presented here is that if you looked at the hours more carefully, you would see that it does not match reality in the following way: in the model, variation in hours comes from everyone reducing labor input by a little (everyone has to do the same thing, since they are all representative Robinson Crusoes). In the real economy, by contrast, variation in total hours is mostly accomplished by some people moving between full time employment and unemployment, while employed people work a fairly constant number of hours. If H is total hours, h is hours per worker, and N is the number of workers, then it is true that: Var(ln(H)) = Var(ln(N)) + Var(ln(h)) + 2 Cov(ln(N),ln(h)) Gary Hansen shows that 55% of the variance in log H is due to the variance of log N, 20% is due to the variance of log h, and the remaining 25% comes from the covariance term. This problem is solved by Gary Hansen as follows.... Start with the production function mapping hours of work into "hours of labor services." We have been assuming implicitly that this was just linear -- a straight line from the origin. So we could just graph the marginal benefit of working as a horizontal straight line. This will just be the marginal utility of consumption times the wage. Note that for an individual considering how much to work on a given day (or month, or short period of time), the extra consumption due to working more will be spread over the entire lifetime -- and so the marginal utility of extra income will not decline with working more in a given day. Thus the marginal benefit of working will just be a horizontal line. Against this we would put the marginal utility of leisure increasing with the number of hours worked, and find the number of hours the person would choose. We would then consider different wages (which would shift up the marginal benefit of working), and get the individual's labor supply curve -- which would be the same for the individual as for the economy in a RBC world.

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But we could think that there was a different mapping of hours- supplied to productive work: say there is some fixed, unproductive cost (commuting, dressing up, having coffee and gossiping). So the graph showing the marginal benefit of working is a step function, where the wage affects the height of the step part. Now there are several possibilities: if the marginal utility of leisure is always higher than the marginal benefit of working, they you never work. If they cross, it would be natural to conclude that the person would want to work the number of hours given by the point where they cross -- but this could be wrong! Because in addition to checking interior maxima, we also have to check end-point values. In this case, we have to check whether the individual might not be better off working zero hours. To see this, graph the contribution to total lifetime utility of a day's labor and leisure as a function of hours worked. (This is just the sum of the two pieces: utility from leisure and consumption). It slopes down initially, then up to a local max (at the crossing of the marginal benefit curves derived above), then down again. Changing the wage will not shift the left-most point on the curve, but it will shift the rest of the curve up and down. Depending on the wage, either zero labor supply or the interior local max will be chosen -- and for one level of the wage, individuals will be indifferent between the two. Call this wage w*. Now draw the labor supply curve for the economy (made up of identical individuals). Labor supply is zero below w*, then jumps up, after which it is upward-sloping. Assume that labor demand is generated the usual way by firms hiring labor such that the marginal product is equal to the wage -- it slopes down. What do we make of a situation where demand crosses supply at w*? At this wage, workers are indifferent between working and not working -- and so we can imagine a variety of devicies that make sure that just the right fraction of them decide to work such that supply equals demand. Now let labor demand shift around due to productivity shocks -- the wage will remain constant at w*, while the quantity of labor shifts around. Further, quantity shifts will occur due to movement of people in and out of working, rather than adjustments of hours! This is the basis of the Hansen model. These fixed costs of going to work are an example of a non-convexity, in this case in production. We could also introduce non convexities in preferences: working an infinitesimal number of hours will give you much less utility than working zero hours. For example, if you work even a little, you can't go to Florida, have to wear a tie, etc. Non-convexities in preferences would produce the same results as the story presented here. [Aside: Looking beyond business cycles, some sort of non-convexity may be the explanation for the way we do retirement: all at once. Most models (with declining productivity or increasing disutility of working), would say that you have a gradual retirement since you set the wage equal to the marginal utility of leisure. As you get older, wage falls gradually (or the marginal utility of leisure rises

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gradually), and so, without a non-convexity, we should observe people gradually increasing leisure.] [behavioral economics aside] Camerer et al., “The Labor Supply of New York City Cab Drivers” QJE May 1997. Data from taxi driver trip sheets. Record all the trips taken, how long the cab driver worked, how much earned. Taxi drivers are a good subject for the study for two reasons:

1) they have control over their own hours: they can decide when to go home. (For regular workers, there are coordination issues).

2) their wage varies from day to day due to weather and other demand factors. Wage is calculated by dividing all fares collected in an hour by number of taxis working that hour. Two key findings:

hourly wage is variable across days. The average hourly wage ranged from $13 to $19. wages positively correlated within days. If the hourly wage is higher early in the day, it is likely

to be high later in the day as well. This is because demand for taxi rides is higher on rainy days, etc.

Key test is to see how hours vary with the wage. Finding: drivers work less on high wage days! If drivers worked a fixed number of hours per day (rather than working less when wage is high), holding average hours constant, they would raise their wage income by 5%. If they worked more on high-wage days (with an elasticity of 1), again holding their total hours constant, then they would raise their average wage by 15.6%. What should the elasticity be? We can figure this out by holding the marginal utility of income constant (since we are looking at a single day) Max Uy*w*(1-N) + N^(1- σ ) / (1- σ ) get d ln(N) / d ln(w) = -1/ σ But note that N was leisure B the elasticity of work effort w.r.t the wage will be different (but not much, if they are roughly equal)

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So why do they have the wrong sign? authors suggest “daily income targeting” as a way of dealing with self control issues don’t have to save money from good days to bad

-- gives an easy rule for when to quit. Else would be temptation to say that any given day was a bad one. Similar to writing a set number pages per day! -- more experienced drivers did better in optimizing [end of behavioral aside] Empirical evidence on LCH/PIH In this section I will present a grab-bag of pieces of evidence relating the LCH/PIH. Since the theory is a little amorphous and the data is imperfect, there is never going to be any one test that is completely convincing. Three points to looking at all of these tests: First, get an overall impression of how good the whole approach is. A second goal is to give some examples of how one applies data to the testing of a theory. Third, to see how theory can be twisted around to make it fit the data. 1.1 Tests of the LCH looking at the wealth of old people. If one takes the life cycle model seriously, and doesn't think that inter-generational relations are important, then one would expect to see people's assets declining in old age. Since there is uncertainty, we do not expect to see assets hit zero on the day before death, but we still expect to see them going down. Many people have tried to test the LCH by seeing if old people have negative saving rates. The answer tends to be that they either keep constant assets or they dissave slightly. To justify this with just uncertainty, one would have to posit extreme risk aversion. 1.2 Variation on this test is to ask whether a bequest motive can explain this failing of the pure LCH. Test is to look seperately at the behavior of people with and without children, under the assumption that people with childen have stronger bequest motives. Result is that there is no difference between the behavior of the two: this is bad news for the intergenerational version of the LCH. 2. Tests of the life cycle model as an explanation for the aggregate capital stock. One point of the LCH is to explain the size of the aggregate K stock. Can test this by simulation. Start with realistic parameters for discount rate, interest rate, wage path over course of life cycle, population growth, etc. Then can figure out what each person's wealth should be. Sum this up across people, and compare it to aggregate income [also derived by summing up individual income across age groups]. This gives you a wealth/income or K/Y ratio. Now compare this to what we observe in the economy

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[roughly 3]. Result: ratio delivered by model is too small. [cite: White?] In other words, the LCH does not explain most of the capital stock. 3. Relation between lifetime income and consumption profiles. Note that we don't expect consumption profile to be flat, since tastes may change over lifetime. For example, when you have kids, you may want to spend more. Similarly, when you are old, you may have less utility from consumption (ie can't go on vacations). The key prediction of the LCH/PIH is that your consumption profile should be invariant to the shape of your income profile. That is, two people with the same discounted lifetime value of income but different patterns in which it is received should have the same consumption profile. For example, an athlete who makes all his money early in life, vs a brain surgeon who doesn't start earning until late in life, but then makes a lot. This is one of the things that Carroll and Summers look at in their paper. They find that, breaking people down by occupation, the averages income and consumption profiles are, in fact, very similar. They interpret this as meaning that the LCH/PIH is just wrong. One counter argument to their results: suppose that utility is a function of some combination of consumption and leisure. What is the price of leisure? It is just the wage. So when wages are high, then leisure is expensive, and so people consume less of it. This may raise the marginal utility of consumption, and so people will consume more (since, ignoring interest and discounting, they are setting the marginal utility of consumption equal in every period.). Key the sign of d2u / dc dL (homework problem). This model is particularly appropriate if much of what we measure as “L” is really home production. High wage people, and people in the high wage part of their careers, substitute out of L and into more c. For example, we often see a big drop in consumption at the (predictable) time of retirement, when there is predictable fall in wage income. Is this a violation of the LCH? Maybe not, since people are using more time in order to shop around and get more units of goods for their consumption dollars. 4. Evidence on the size of assets: one can just look directly at the size of assets that people hold. Ask: are they holding as much wealth as the LCH says that they should be? Find that, in fact, people hold little wealth. For example median financial wealth (that is, not counting house or pension or NPV of future Social Security) of families with head aged 45-54 was $4,131 in 1983. So it doesn't look like life cycle saving. But there are some mitigating points here. First, most people have their retirement saving done for them in the form of SS and pensions. So maybe they are even saving too much in these forms, and don't want to save more -- maybe they would even save less, if they could. Also, people hold

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houses, which are very valuable. But in any case, assets seem so close to zero that it is hard to believe that they represent some optimum choice [note: it is possible that people set their pensions so that non-pension assets are zero...]. 5. Campbell and Mankiw -- time series tests of the response of consumption to income. As we have seen above, there is no problem with consumption responding to changes in income. It may simply be that the change in income contains information about future changes in income. But suppose that we could look only at predictable (in advance) changes in income. PIH/LCH says that consumption should not change in response to these. This is what Cambell and Mankiw look at Yt = total disposable income. Suppose divided into two streams: to PIH people (group 2) and to people who consume current disposable income (group 1). λ goes to group one. C1,t = Y1,t = λ Yt Y2,t = (1-λ) Yt in differences:

While for the PIH groups C2,t = Y2,t

p = (1-λ)Ytp

1 Where the epsilon is an innovation to permanent income

where mu is the trend growth rate (due to OLG and growth) and epsilon is the innovation in _permanent_ income. (and is orthogonal to past...)

Of course this can't be estimated directly, since change in Y is correlated with error in permanent

1,t t1,t = = C Y Y

2,t t= +(1- )C

tt t= + +(1- )C Y

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income. (If there were never any shocks to permanent income, then ε would be zero all the time, and we could estimate the equation directly). The solution is to find instruments for Delta Y -- lagged things that predict changes in income. The solution is to use twice-lagged income -- see my notes for 284 for further discussion (or see the article itself or Deaton's book). They estimate λ to be around .5 -- so half of all consumption is done by rule-of-thumb (or liquidity constrained) consumers. (This does not mean that half of people are this way -- since it is probably the poorer people who are more likely to be liquidity constrained, more than half of actual people will be liquidity constrained if half of consumption is done by such people.). Shea (not the paper discussed in the Romer book) Same framework, but examine the possibility that liquidity constraints are the problem. Divide changes in income into pos and neg. It should be for Pos where consumption follows income -- so it should have a larger lambda. Unfortunately, there are few drops in income, and fewer predictable (at least at the agg level) More realistically: divide into above and below trend. Table: perverse finding: consumption responds more to predictable changes in income when income is going down than when it is going up. Kubler Ross? (see quote). Wilcox changes in SS benefits announced at least 6 (and usually 8) weeks in advance. Under PIH, there should be no change in consumption in the months in which change actually occurs. Coefficients say that there is a 1.4% change in consumption for a 10% increase in SS benefits. Note this does not say that consumption of the SS recipients goes up by 14% of their income change -- this is total consumption. 1986 SS payments 200 billion, PCE 2.8 trillion. So by point estimate, more than dollar for dollar

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spending. Wilcox also finds that the increase is biggest in durable goods (and of these, biggest in autos) -- this seems like people are buying a car with their extra cash flow! Many extensions of this work possible: look at consumption of old-people sensitive goods (big, slow cars, eg) or by location... New Material on Consumption More evidence for and against the PIH: Souleles (1999) Consumption rises when income tax refunds are received, even though this is predictable. Adams et al. AER Mar 2009, car purchases by low income people. Even though interest rate is 20% or more, most pay mininimum down payment. Also peak in purchases in Feb, at time of tax refunds. (nice pictures); sort into how big their EITC was; show that groups that get bigger refunds have bigger purchase spikes. Parker ( 1999) Consumption rises when take home pay rises as a result of cessation of Social Security payments. This consumption is concentrated in durables and goods that can more easily be postponed. (also discuss first of the month papers) (this would be a good place to discuss M-Pesa material if not discussed already) -- Shapiro and Slemrod (AER 2003, and an earlier paper): looked at survey evidence from two episodes. In 1992, there was a reduction in withholding but no change in eventual tax liabilities. In 2001 there was a change in law that cut taxes semi permanently (for 10 years), and a rebate of excess withholding. In both cases, they asked households how they planned to deal with the extra money: consume, save, reduce debt (which is the same as saving). Very odd results. In 1992, 40% of households said they would mostly consume the extra take-home pay; in 2001, 22% of households said that they would mostly raise spending. But this is the opposite of what theory says: there should be little or now consumption out of the first change, and (ignoring long-term Ricardian considerations) the second really was a semi-permanent increase in disposable income.

Parker, Souleles, Johnson, and McClelland (2011, NBER WP 16684) Study fiscal

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stimulus payments of 2008. Payments were large: 300-600 per for individuals, $600-1,200 for couples, with an addition $300 per child. Find that consumers spent between 50 and 90 percent of payments, with the largest part of spending going to durables. How can they measure? Answer: there was a random component to the timing of the payments that households received, based on the last two digits of the social security number. The authors used the Consumer Expenditure Survey, where they could look at consumption spending and knew SSN (and date) so they could look at whether households had received payment or not.

monthly income is relatively constant throughout the year, but consumption expenditure rises significantly in December (in US, 21 percent.) This is actually good news for the PIH, even though consumption is non-smooth. The reason consumption rises is obviously because of preferences for holiday spending. The fact that consumption responds to this, rather than matching income, is thus good news for PIH.

Browning and Collado (2001): in Spain, the majority of workers receive a double paycheck in June and December -- but there are some workers who do not receive such payments. (This depends on the worker’s job -- it is _not_ like a bonus that is uncertain.) Finding is that the seasonal pattern of expenditure is the same in two groups. This is a big win for PIH.

[Possible reconciliation: in the Spanish case, the cost of not smoothing would be large. In the US cases, the non-smoothness is small, so no big cost from just having consumption move with income. One-time Payments: B 1950, unanticipated payments to subset of US veterans holding National Service Life Insurance policies. MPC = .3 - .5. Similarly, reparations payments from Germany to certain Israelis 1957-58, MPC = .2 (these were very large payments, equal to about one year’s income). Carroll points out that when these cases were originally analyzed, they were seen as victories for the Friedman PIH since the MPC was much less than one. But in modern view, they are failures, since the MPC is much higher than .05. Carroll’s point is that the modern PIH is not the same as the original Friedman PIH. Browning and Crossley, JEP, Summer 2001. Also Carroll JEP Summer 2001. Yet more to be added on this: Stephens AER 2003 “3rd of tha Month…”, Toward reconciling all (some?) of this stuff:

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1. Liquidity constraints prevent borrowing 2. Many people would like to borrow (either because income is growing rapidly and/or because they have a high discount rate). 3. People don't hold zero assets because of a precautionary motive: income is stochastic and they may get a bad draw. 4. Optimal strategy (ala Deaton and Carroll) is to hold a buffer stock of assets -- say a few months worth of income. [could do a little exercise to show this... or go over Deaton’s version.] 5. How explain the aggregate capital stock? ==> rich people are not subject to the above model. They hold most of the wealth. They are not subject to any of the models that we know. 6. Maybe some room for LCH/PIH people in the middle 7. Maybe more folks would optimize if they had to

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Problems

1. [core exam, 2006] A researcher is trying to determine the parameters of a woman’s utility function. He knows that she has CRRA utility, but he does not know the value of her coefficient of relative risk aversion, , or her time discount rate, . The woman is infinitely lived. Time is continuous.

The researcher has presented the woman with different possible paths of consumption, asking which was preferred to which. The woman answered that she was indifferent among the following three paths of consumption: c(0) = 1, g=.01 c(0) = 2, g = 0 c(0) = 4, g = -.0025 where c(0) is initial consumption and g is the annual growth rate of consumption. Based on this information, solve for the woman’s values of and . Note: solving this in general would require a computer. However, I will make things easier by telling you that is equal to either 2 or 3.

2. Consider a two period Fisher model in which utility in each period is given by the function

U(C) = C1-σ/1-σ Solve for the derivative of saving with respect to the interest rate: dS/dr (this is just the negative of the derivative of first period consumption w.r.t. the interest rate). What condition on the values of w1 and c1 guarantees that dS/dr is positive? Assuming that this condition does not hold true, how (informally) does the value of σ affect whether dS/dr is positive or negative.

3. [midterm 2005] An individual is born at time zero with assets of 100. She will live forever. Time is continuous. She will receive no labor income. There is no uncertainty. She has CRRA utility with coefficient of relative risk aversion ( ) equal to 2. The interest rate is 5%. Her initial consumption is 3.

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Solve for the value of .

4. [midterm exam, 2005] A woman is born at time zero and will live forever. Time is continuous. She has initial assets of zero. She can borrow or lend at interest rate r=0.04. Her time discount rate is θ = .02. Here wage at time zero, w(0) = 1. Here wage grows at a rate of 2% per year. She has CRRA utility, with a coefficient of relative risk aversion of 2. Solve for her consumption at time zero.

5. [Midterm exam, 2010] Mr. X is born at time zero with initial assets of 200. He

receives no labor income. He has a pure time discount rate of 4%. Mr. Y is born at time zero with initial assets of 50. He also receives no labor income. His pure time discount rate is zero. Time is continuous. The interest rate is 2%. There is no mortality. Both men have CRRA utility, with coefficient of relative risk aversion equal to two.

A) After how many years will the assets of the two men be equal? You should be able to

answer with an actual number (using a small approximation), but if that doesn’t work, you can just write down the correct expression.

B) After how many years will the consumption of the two men be equal? In this case, you should supply the correct expression but you will not be able to (easily) approximate the number of years.

6. [midterm exam, 2008] A woman has Stone-Geary utility of the logarithmic form: ( ) ln( ) 0u c c c c She is born at time zero and will live forever. Assume that 0r . She has labor income that is constant and equal to w per period. Assume that w c . A) Write out the discrete-time first order condition relating consumption in two adjacent time periods t and t+1. B) Solve for consumption at time zero. This can be done by either solving the whole problem in discrete time, or switching to continuous time. If you switch to continuous time, you will have to “guess” or derive the FOC for consumption in continuous time. However, I find it easier to do this than to solve in discrete time. [Hint that was not on the midterm exam: things are much easier if midway through the problem you define a new variable like c c c .]

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7. A man lives for 40 periods. He has constant absolute risk aversion utility: He earns a wage of 100 in each period. The interest rate is r and the time discount rate is . He is born with zero assets and will die with zero assets, but he is able to borrow during his life (i.e. he is not liquidity constrained). There is no uncertainty. A) Derive (or state) the first-order condition relating consumption in adjacent periods of his life. B)Assume that α=1, =.05, and r=0. Derive his optimal first period consumption. Note: you will

have to use the approximation that ln(1+x) x, which holds true for values of x near zero.

8. [midterm 2005] An individual is born at time zero. Time is continuous. The individual has instantaneous utility function

ln( )u c if alive

= 0 if dead

(Note: this is the same as the utility function being u = ln (c) + where =0). The individual has initial assets A(0) of 20. She has no labor income. There is no uncertainty. The interest and time discount rates are both zero. The individual has a maximum life to 20 years. She can choose to stop living earlier, however. Solve for her optimal path of consumption. You should calculate the exact level of c(0).

9. A researcher has collected income and consumption data from a large population. Consumption in the population is determined by the permanent income hypothesis: C = α Yp,

t- c

t

-1U( ) = c e

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where Yp is permanent income and 0<α<1. Permanent income in the population is given by Ypi =

pY + ρi, where ρ is distributed normally, with variance σ2p The researcher does not observe

permanent income, however. She only observes current income, which is related to permanent income by Yc

i = Ypi + εi. εi is transitory income. It is distributed normally, with variance σ2

t. There is zero covariance between permanent and transitory income.

The researcher estimates the “consumption function” Ci = ß0 + ß1 Y

ci

What value will she get for ß1, the marginal propensity to consume? How will ß1 compare to α? Under what circumstances will it be a good estimate, and under what circumstances a bad estimate?

10. [midterm exam, 2005] There are two kinds of people in a country: blue and red. Each person has annual labor income given by the following equation:

, , ,i c t c i i tw w

Where cw is the mean permanent income of people of color c (c=red or blue), i is a random

variable that determines the permanent income of person i relative to the mean for his color group, and ,i t is transitory income for individual i in year t. i and ,i t are distributed iid normally with

means of zero and standard deviations and . Assume that red bluew w .

A researcher has collected data on income, color, and consumption for a large number of individuals in a single year. The researcher looks at how saving rates of red and blue people compare, holding income constant (that is, the researcher looks at the saving rates of red and blue people who have the same income). What will the researcher find? Why? Discuss how the relative sizes of and will affect the magnitude of the researcher’s findings.

Problems with Liquidity Constraints

11. Consider the following version of the Fisher model. Individuals live for two periods. They can borrow at a real interest rate of 100% (that is, if they borrow one dollar in period 1, they must repay two dollars in period 2). They can lend at a real interest rate of zero. Their preferences are given by:

U = ln(C1) + ln(C2)

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Find optimal first period consumption for the following three individuals: Ms X: W1 = 32 W2 = 32 Ms Y: W1 = 0 W2 = 64 Ms Z: W1 = 24 W2 = 40

12. [final exam, 2006] A woman is finishing graduate school and deciding on her career. She has zero assets and is infinitely lived. The two careers she can choose are labeled A and B. Wages in career A begin at w0,A and grow at rate 2% per year. Wages in career B begin at w0,B and grow at rate 4% per year.

She can save at interest rate zero, but she cannot borrow. She discounts the future at rate 4%

per year. She has CRRA utility with coefficient of relative risk aversion . Solve for the ratio of initial wages in the two careers such that she is indifferent about which to choose.

13. (midterm, 2001) In our previous analysis of differential interest rates we assumed (realistically) that the interest rate on borrowing was higher than the interest rate for saving. Now we make the opposite assumption, which is not all that realistic.

Consider a person who lives for two periods. The interest rate for saving is 100%, so that one dollar saved in period one turns into two dollars in period 2. The interest rate on borrowing is zero. A woman has labor income w1 = 2 and w2 = 3. She has log utility and a time discount rate of zero. Draw the set of feasible consumption possibilities. Show how to solve optimal consumption in each period. It turns out that in figuring out optimal consumption, the very last step requires the use of a calculator.

14. Mr. A and Mr. B have the same preferences (that is, the same instantaneous utility function and the same discount rate). Both are born at time zero and die (with certainty) at time T. Both face the same interest rate. Each is born with zero assets and dies with zero assets. Both are also liquidity constrained: they are never allowed to have negative assets. They have different lifetime wage profiles. There is no uncertainty: both individuals know in advance their entire lifetime wage profiles.

2

95

It is observed that Mr. A's consumption grows at a constant (positive) rate over the course of his life, while Mr. B's consumption declines at a constant rate over the course of his life. The rate at which Mr. B's consumption declines is smaller than the rate at which Mr. A's consumption grows. However, the present discounted values of lifetime consumption for the two men are the same. Which man has higher lifetime utility? Explain.

15. [core exam, 2004] A woman is born at time zero and will live forever. Time is continuous. She is born with zero assets. Her instantaneous utility function is of the CARA form:

- c1U(c) = - e

where α =1. She discounts the future at rate θ =0.10. She can save at interest rate zero. She cannot borrow. Her labor income is exogenous. From time zero to time t=10, her labor income is two per period. From time t=10 to time t=20, her labor income is one per period. After time t=20, her labor income is two per period. Solve for her path of consumption. What is her consumption immediately before and after t=10? What is her consumption immediately before and after t=20? If there are any jumps or inflections in her time path of consumption, calculate the exact point in time at which they take place. Note: the First Order Condition for consumption with CARA utility is

1c = (r - )

. This is analogous to the condition that we look at in the more common case of CRRA utility, which is

c 1 = (r - )

c

96

16. (Final exam, 2001) This question was going to be on the Core exam, but I realized that it was way too hard. So I have broken it down into steps and given you instructions as you go along.

A woman has labor income that is constant at one. Time is continuous. She is liquidity constrained, so that she can never have negative assets. She is born with zero assets. From time zero to time s, she faces an interest rate of zero. Starting at time s, she will be able lend at interest rate r>0. Her time discount rate is θ, where r> θ >0. Her instantaneous utility function is logarithmic. Sketch her optimal time path of consumption. Under what conditions will optimal consumption at time zero be equal to one? A. As a first step, solve for the path of optimal consumption starting at time s. Solve for consumption at time s, c(s), as a function of her assets at time s, A(s). You should be able to find a closed-form expression for c(s). Make a graph with A(s) on the horizontal axis and c(s) on the vertical axis. B. Now consider what happens before time s. It turns out that there are several possible paths for consumption in the period before time s, depending on the values of s, r, and θ. Sketch out these possible paths. C. Based on your answer to part B, draw a graph with A(s) on the horizontal axis and c(s) on the vertical axis, showing how assets at time s are related to consumption at time s. Show (with words or arrows) how the different points on this curve are related to the different consumption paths in part B. D. Putting together your answers to parts A and C, along with the usual condition that consumption cannot jump, will give the optimal values of consumption and assets at time s. E. Holding r and θ constant, how will changing s affect the value of consumption at time zero? Specifically, under what conditions will consumption at time zero be equal to one? Problems with Endogenous Interest Rates

17. Consider a world with two countries, which have equal sized populations. Income is exogenous and is identical in the two countries. There is no means of storing output, so income in a given period has to be consumed in that period. There is no population growth.

Both countries have CRRA utility of the form

97

1-cu(c) = 1-

. But the two countries have different values of σ. Specifically, σ1 > σ2. In both countries, the time discount rate is zero. Income per capita at time zero is equal to one. Following this it grows at constant rate g. A. Suppose that there were no trade between the countries. What would the interest rate be in each country? B. Suppose that starting at time zero, the two countries are able to trade with each other. Sketch out what the path of consumption per capita will look like in each country. Also sketch out the path of the world interest rate. Toward what level will the world interest rate asymptote? What will the asymptotic growth rates of consumption in the two countries be, and what will be the asymptotic ratio of consumption in the two countries? How will consumption at time zero in the two countries compare? Note: doing this problem with fancy math is neither required nor recommended.

18. Consider an economy in which there are two kinds of people: red and green. There are equal numbers of each kind of person. The economy lasts for a two periods. In the first period, red and green people each receive income of one apple per person. In the second period, red people receive income of one apple per person, and green people receive income of zero.

Both red and green people have log utility, and the time discount rate is zero. There is no way that apples can be stored between periods. What is the market-clearing interest rate that will be changed on loans between period 1 and period 2? Also, what will the per-capita borrowing and lending of red and green people be?

19. [Core Exam, 2005]. Consider a world with two countries, which have equal sized populations. Time is continuous. Labor income per capita in each country is constant, exogenous, and equal to 1. There is no means of storing output. There is no population growth. People in both countries have the following instantaneous utility functions

98

-cu(c) = - if c 0e

= - if c < 0

Individuals in country i discount the future at rate θi (i =1,2), where 0< θ1 < θ2 Starting at time zero, individuals in each country are allowed to borrow or lend at the market-clearing world interest rate. Note: you do not have to write out any equations at all in order to get full credit for this problem, but you should carefully describe the principles that determine the answer you give. Because this problem is very difficult, I have broken it down into smaller steps.

A. Suppose that at some point in time, c>0 in both countries. What must the interest rate be? B. Suppose that the interest rate is the value that you derived in part A. What does the path of

consumption (at that point in time) look like in each country? C. Explain why based on the above, the interest rate cannot permanently have the value you

derived in part A. D. Suppose that at a point in time, consumption is zero in country 1 and positive in country 2. What

must the interest rate be? What do the paths of consumption look like in each country? E. Is it possible that the interest rate will have the value you derived in part D from time zero until

infinity? Explain. F. Suppose that at a point in time, consumption is zero in country 2 and positive in country 1. What

must the interest rate be? What do the paths of consumption look like in each country? G. Is it possible that the interest rate will have the value you derived in part F from time zero until

time infinity? Explain. H. Putting all of the above together, sketch out the possible time paths of consumption in each

country as well as the world interest rate. If there are different possible cases, you should briefly describe these.

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20. [core exam, 2008] [note: this question has three parts.] An economy is populated by a continuum of individuals. The economy last for two periods. Each individual is endowed with one unit of the consumption good in the first period and one unit of the consumption good in the second period. The good cannot be stored. Individuals can loan the consumption good to each other at market clearing interest rate r.

Individuals have instantaneous utility functions of the CARA form

1

( ) cU c e

A) Individuals all have the same value of . However, they differ in terms of their time

discount rates. Specifically, , the time discount rate, is distributed uniformly on the interval [0,0.1]. Solve for the equilibrium value of r.

Note: for all of the parts of this question, you should use the approximation that ln(1+x)x, which is generally OK for x near zero.

B) Now suppose that varies among individuals as well. Specifically let and be jointly

uniformly distributed on the interval [1, 2] , [0,0.1] . Once again, solve for the equilibrium interest rate.

C) Now suppose that and are distributed over the same range as in part B, but they are not jointly uniform. Specifically, the marginal distributions of and are uniform, but two variables are perfectly correlated, so that the individual who has the lowest value of also has the lowest value of , and so on. Explain how (i.e. higher or lower) and why the interest rate in this case will differ from part B. You do not have to solve for the interest rate. Problems with Uncertainty

21. [midterm exam 2008] A woman has assets A0 at the beginning of time zero. She has no labor income. Time is discrete. She has a constant mortality hazard . In other words, if she is alive in period t, the probability that she will be alive in period t+1 is (1 ) . The interest rate is r>0. Her pure time discount rate is zero. Her utility function is of the CRRA form:

100

1

( )1

cU c

Calculate the ratio of consumption to assets in period zero. (Note: you can solve this problem by doing some infinite sums, but there is a much easier and more elegant way if you apply economic reasoning.)

22. [midterm exam, 2003] A) A woman has assets A(0) at time zero. Time is continuous. She has no labor income. The

interest rate and time discount rate are both zero. Her instantaneous utility function is u(c) = ln(c). She has a constant probability of death ρ>0. Solve for her initial level of consumption, c(0).

B) A woman has assets A(0) at time zero. Time is continuous. She has no labor income. The

interest rate and time discount rate are both zero. Her instantaneous utility function is u(c) = ln(c). From time zero to time 1, she has zero probability of death. Starting at time 1, she has a constant probability of death ρ>0. Solve for her initial level of consumption c(0).

C) A woman has assets A(0) at time zero. Time is continuous. She has no labor income. The

interest rate and time discount rate are both zero. Her instantaneous utility function is u(c) = ln(c). From time zero to time 1, she has zero probability of death. At time 1, she will have a constant probability of death. However, she will not find out that probability until time one. Specifically, what she knows at time zero is that the probability of death starting at time 1 will be ρ1 with probability 50% and ρ2 with probability 50%. Solve for her initial level of consumption, c(0) [Note: to solve this problem you have to deal with a nasty quadratic equation. You shouldn=t try to solve this. Just derive it, then say what you would do with the solution if you could solve it.]

23. [core exam, 2011] A woman is born at time zero, with initial assets of A0 >0. Time

is continuous. From time zero until time R (retirement), she has zero probability of death. From

time R onward, her probability of death is constant at per unit time.

From time zero to time R, she earns labor income at a rate of w per unit time. From time R onward, she has no labor income. The interest rate is zero. Her pure time discount rate is zero. She has log utility. Solve for her initial consumption in terms of A0, R, w, and .

101

24. Consider the problem of an old person trying to decide how much to consume as she approaches the end of her life. The interest rate is r, there is no time discounting, and she has no labor income. She has some initial amount of wealth, A0. She faces a constant probability, p, of dying each year. Thus her probability of being alive in year t is (1-p)t. Of course she only gets utility if she is alive. Her utility function is

A. What is the relationship between her coefficient of relative risk aversion (σ) and the expected size of the bequest that she will leave? Be sure to distinguish between the case where p>r and the case where p<r. Explain. B. Suppose that σ = 1 , so that the woman has log utility. Solve for her assets if she is still alive at the beginning of period t, At , as a function of p, r, t, and A0 .

25. [final exam, 2008] A large cohort of individuals is born at time zero (you can think of it as a continuum). Time is continuous. Individuals have a constant probability of death 0. Individuals do not know in advance when they are going to die. Their pure rate of time discount is

0. Individuals are infinitely risk averse. That is, they behave as if they have CRRA utility in the limit as the coefficient of relative risk aversion approaches infinity. At birth, each individual is endowed with wealth W0. They receive no labor income. Individuals can save at an exogenous interest rate . There are no annuities. When an individual dies, his remaining wealth is distributed evenly among all surviving individuals. Solve for the path of consumption.

26. [midterm exam, 2006] This question has five parts, of varying difficulty. You do not have to get the early parts of this question right in order to correctly answer the later ones.

1-

t t

t=0

cE(U)= (1- p )1-

102

A) A woman does all of her consumption in one period. Her utility function is

1

1

cU

, where 2

Call w her certainty level of income. If she pays x dollars, she gets to earn w with certainty, so that

her consumption is equal to w-x. If she does not pay x, then she will earn either 2w or 2

w, each

with probability 50%. Solve for the level of x such that she is indifferent between paying for certainty or not. Express your answer in terms of the ratio of x to w.

B) Suppose that a woman has the following utility function

1

1 if alive1

0 if dead

cU

where 2. Solve for the level of consumption at which she is indifferent between being alive or dead. C) Consider a woman who has a utility function as in part (B) and faces the uncertainty described in part (A). Suppose also that after she observes her income, she has the option of no longer being alive, and thus getting zero utility. For what range of values of w will the answer that you derived in part (A) still be correct? Explain very briefly.

D. Consider the setup from part (C). Solve for the level of w at which the woman’s value of x, that is, her willingness to pay for certainty, is exactly zero. Note: you should ignore values of w that are less than or equal to zero. Note #2: I am not asking you to solve in general for x as a function of w, even though this is one possible way to solve the problem.

E. Consider yet another utility function, the Stone-Geary utility function which is

103

1( )

1

cU

,

where 2 and 1 . Consider the form of uncertainty described in part (A), but assume that there is no possibility of voluntarily dying. Draw a graph showing willingness to pay for insurance as a fraction of income (x/w) on the vertical axis and w on the horizontal axis. Consider only values of w>2. Just sketch the rough form of this curve, showing what it asymptotes to as w gets large and what it looks like in the neighborhood of w=2. You do not have to derive the actual formula for willingness to pay. Indeed, you can actually answer this question without doing any math at all.

27. [final exam, 2005] A man is born at time zero. Time is continuous. He has a probability of dying, , that is a function of his age. Specifically,

( ) .01t t To be clear, this means that the man has a 1% per year chance of dying when he is one year old; a 2% per year chance of dying when he is two years old (if he didn’t die already), and so on. The man’s instantaneous utility is logarithmic. His pure time discount rate, , is negative. In other words, ignoring the possibility of death, he cares more about the future than about the present. Specifically, = -.05. The man has wage that is constant and equal to one per period. He can save at an interest rate of zero. He cannot have negative assets. His initial assets are zero. Sketch out the man’s paths of consumption and assets. You do not have to solve for the exact path or the exact value of c(0). However, you should label as much of the path as you can. Specifically, you should point out any inflections, jumps, maxima or minima, etc. You should also give the dates of any of these points if you can, and if you can’t give the exact date, then you should say whatever you can (for example, “this jump takes place after date x and before date y.”) 27.5 [Core exam, 2012] This question is about the value of annuities. A man is born at time zero with wealth W. He receives no labor income. The interest rate is zero. Time is continuous. He has constant probability of death >0. His time discount rate is zero. He may not go into debt. The man has two options. First, he can consume out of his wealth and ignore the annuity market. If he chooses this option, then any remaining wealth that he has when he dies will be thrown away. His second option is to put all of his wealth into an annuity account. If he chooses this option, then a fraction 0 1 of his wealth will be taken away, and he will receive the actuarially fair interest rate on the money in the annuity account.

104

Your goal is to solve for the value of such that he is indifferent between these two options. You should be able to get close to an actual number for this, that is, “about xx percent.” Note: you do not need to know the value of to answer this question. A) Assume that he has log utility. You will have to use the following fact:

1

B) Assume that he has CRRA utility with coefficient of relative risk aversion equal to two.

28. An individual is born at time zero with initial wealth 0w . Time is continuous. He

has no labor income. The interest rate is zero and his time discount rate is also zero. He has instantaneous utility

( ) ln( ( ))u t c t where is the utility of being alive. The individual has constant hazard of death . For some strange reason, the individual can choose before he is born. However, he cannot change the value of once he is living. A) Assume that there are no annuities. Solve for the optimal value of . Note: to solve this problem, you will have to know the following helpful fact (which I wouldn’t expect you to know off the top of your head):

-xt2

0

1 t dt = e

x

B) Now assume that there are actuarially fair annuities. The annuity rate of interest that an individual receives will be based on his true value of (that is, the value that he chose). Solve for the optimal value of . [Note: I have given you all the information about the annuity market that I think you will need. However, if you feel that you need to make additional assumptions about the annuity market, you should explicitly state what these are.]

29. (final exam, 2008) There is a group of individuals who all have wealth A0 at time zero. The individuals have no labor income. Time is continuous. Every individual i has a constant mortality hazard i that is known to him, but not to outside observers. Everyone knows

the distribution of .

105

Individuals all have log utility and pure time discount rate 0 . There are two investment

options. First, individuals may put their money in a standard asset that pays interest r= . Any wealth that is left over when the individual dies is thrown into the sea.

Alternatively, individuals may participate in an annuity market at time zero, but not at any other time. That is, they can turn over their full endowment to the annuity company at time zero in return for a constant stream of payments equal to 0

ar A , where ar is the annuity interest rate. To make

things easier, we will assume that a person with an annuity cannot save any of his annuity payments. He must consume them as they arrive.

A) Consider the problem of the individual. What is the condition on i and ar so that an

individual will choose the annuity vs. not? Note: to solve this problem, you will have to know the following helpful fact (which I wouldn’t expect you to know off the top of your head):

-xt2

0

1 t dt = e

x

B) Now consider equilibrium in the annuity market. Annuity firms invest the initial payments

they received at rate r. Solve for the expected profit (measured at time zero) on an annuity that pays rate ar that is sold to an individual with mortality probability i .

C) Annuities are supplied by zero-profit firms that are risk neutral. The CDF for individual

mortality probabilities is ( )f . Define * as the cutoff level of individual mortality, above which the individual will find it optimal to purchase an annuity. Write down the condition relating * and ar that produces zero expected profits in the annuity market.

30. [Midterm exam, 2008) A woman has instantaneous utility of the form ( ) ln( )u c c

Time is continuous. The interest rate and her pure time discount rate are both zero. The woman is indifferent between two career options. She can be a coal miner earning a wage of w1 and experiencing a constant probability of dying of 1 , or she can be a factory worker earning a wage of

w2 and experiencing a constant probability of dying of 2 , where w1>w2 and 1 2 0 .

There is an annuity market in which she can save money at interest rate that reflects the death

106

probability associated with her job. A) [20 points] Solve for . B) [15 points] Assume 1 =.02, 2 =.01, w1= e, w2 = 1.

Calculate the value of a statistical life if the woman decides to be a coal miner and if she decides to be a factor worker. Recall that the value of a statistical life is the ratio of the payment she would receive to the instantaneous (one time) risk of death she would accept if she was indifferent to making such a trade. You should assume that the risk of death in question is very small.

31. [Final Exam 2010] An individual is born at time zero with zero assets. Time is continuous She has log utility and a pure time discount rate of negative five percent per year. In other words, except for the effect of mortality she cares more about the future than the present. The interest rate is zero per period. She has mortality hazard given by the equation

1000

a

where a is her age and is the annual probability of death. In other words, when she is 10 years old, her mortality hazard is 1% per year; when she is 20 years old, the mortality hazard is 2% per year, and so on. She has receives a constant income flow of one unit per period. She cannot go into debt. I don’t want you to solve for her complete path of consumption, because that turns out to be really hard. Instead, I want you to characterize the paths of consumption and assets as well as you can. In particular, you should describe the equations that you would solve or the method that you would use to find c0. What are the conditions that hold at different critical points? At what ages do consumption and assets peak, etc.

32. [Midterm exam, 2010] A researcher was assigned by the government to calculate the value of statistical life by offering individuals sums of money in return for taking very small risks to life. However, the researcher did not understand economic theory (or his instructions), so instead he offered individuals sums of money in return for taking large risks to their lives. Your job in this question is to calculate the properly defined value of a statistical life.

The individual in question had a constant mortality probability ρ=.02, pure time discount rate θ=.03, and faced an interest rate r=.03. He is assumed to have instantaneous utility of the form U(c) = ln(c) + α The individual has no labor income. He has $1,000,000 in assets. There is a perfect annuity market. He has no utility from bequests. Asked what amount of money would be required to get him to take a 50% chance of instant death,

107

the individual replied that he would require $999,000,000. The researcher thus concluded that the value of a statistical life was almost two billion dollars.

A) [16 points] Calculate the individual’s value of α.

B) [16 points] Calculate the (properly measured) value of a statistical life for this individual.

Note: when I gave this problem on the midterm, I included a table of logarithms.

33. An individual lives for two periods with certainty and for a third period with probability 50%. He finds out at the end of period one (after he has done his consumption) whether he will die at the end of period two or at the end of period three. He has earnings of one unit in the first period, and no labor income thereafter. His instantaneous utility function is

U(Ct) = Ct

1-σ/(1-σ) He can borrow or lend at an interest rate of zero, and has a time discount rate of zero. He cannot die in debt. A. Derive the equation that you would solve to get his optimal first period consumption. You don't need to solve it. B. A government program is introduced that takes away τ from every worker in the first period of life, and pays 2τ to all people who survive until the third period of life. What is the value of τ that maximizes expected utility? (You don't have to do algebra to get this answer -- you can just reason it out and explain your reasoning).

34. [Midterm exam, 2010] A. A closed economy is populated by a large number of identical individuals. People live for two periods. They have log utility with a time discount rate of zero. In the first period, everyone has labor income of two units. In the second period, everyone will either have labor income of one unit, or everyone will have labor income of three units, each with probability 50%. People do not know until the beginning of period two what that labor income level will be. Individuals can lend or borrow from each other at interest rate r. There is no way of moving output from one period to another other than lending or borrowing from someone else. Solve for the equilibrium value of r. B) Consider the same setup as in part A, but now instead of all individuals having the same labor income in period two, suppose that half of the people have labor income of one and half have labor income of three – however, people still do not learn until the beginning of the second period what their labor income will be. Solve for the equilibrium value of r. (Note: we assume that there is no risk sharing, insurance, or state-contingent contracts. That is, there is no arrangement by which people who are lucky enough to have good wage outcomes transfer income to people with bad

108

outcomes.) C) Now, consider a similar setup with three periods. In periods one and two, everyone will have labor income of two units per period. In period three, half the people will have labor income of one and half will have labor income of three. However, people find out at the beginning of period two what their period three income will be. As above, there are no state contingent contracts and no way to store output between periods other than by lending it to someone else. Solve for the equilibrium r that will hold between periods one and two, and also for the equilibrium r that will hold between periods two and three.

35. [midterm exam, 2002] A man has a maximum lifespan of three periods. In the first two periods, he is alive with probability one. At the beginning of the second period (before he has made his consumption decision), he finds out what his health status is. There are two possible values of health status, “healthy” and “unhealthy,” which each occur with a probability of 0.5. If he is healthy, then he will be alive in the third period with probability one. If he is unhealthy, he will be alive in the third period with probability 0.5. The information about whether he will be alive in the third period is not revealed until after second period consumption has taken place.

The interest rate and time discount rates are zero. He has labor income W in the first period and zero in all subsequent periods. His instantaneous utility function is logarithmic. Solve for his optimal first period consumption.

36. A person lives for two periods. In the first period she has income of 8 dollars. In the second period, she has income of either 0 or 8 dollars, each with probability .5. The interest rate is zero.

Her instantaneous utility function is: U(c) = c - .05*c2 Her discount rate is zero. There is a test that can tell the person what her second period income will be before she makes her first period consumption decision. If she does not take the test, then she will not know her second period income until after the first period is over. The test costs 2 dollars. Calculate whether she should take the test or not.

37. Consider a two-period model in which people work during both periods. The number of hours is fixed at 1 in each period. People start off with no assets, and die with no assets. They

109

can borrow or lend at an interest rate of zero, and there is no time discounting. Their instantaneous utility function is U(C)=ln(C).

In the first period, everyone's wage is $10. In the second period, half of the people will make $15, and half will make $5. But in the first period, people do not know to which group they will belong.

A) The government is considering cutting taxes by $1 per person in the first period, and increasing taxes in the second period to pay back the debt. Taxes in the second period can either be lump sum ($1 per person) or proportional (10% of each person's wage).

How would each of these programs affect people's utility and the national saving rate? B) A test is invented that will tell people at the beginning of period 1 what their income will be in period 2. How would the availability of this test affect average utility of people in the population? How would it affect total saving in the first period?

38. [midterm 2005] An individual lives two periods. His first period income, w1 is certain. His second period income, w2 is distributed N( w , 2 ). The interest rate and time discount rate are both zero. He has CARA utility:

( ) cu c e

A. [10 points] Derive the first order condition linking consumption in the first and second periods of life. You will have to use the fact that if x is normally distributed with variance

2x , then

2( ) / 2( )

xxE xE e e

Hint: write lifetime utility in terms of first period consumption.

B. [5 points] Solve for first period consumption

C. [5 points] Consider two individuals, one named Homer and one named Mr. Burns. Homer has w1 = 20,000. Mr. Burns has w1 = 20,000,000. Second period incomes (and uncertainty) are the same for Homer and Mr. Burns. Suppose that initially, second period income is 20,000 with no uncertainty. Solve for first period consumption for each man. Now suppose that in the second period income is normally distributed with a mean of 20,000 and a variance of 10,000. Calculate the effect of this rise in uncertainty on each man’s first period consumption.

110

D. [5 points] How would your answer to C have differed if the men had CRRA utility? You should answer qualitatively. Do not try to calculate the exact answer.

39. [midterm exam, 2008] A person lives for two periods. In the first period she has income of A. In the second period, she has income that is a normally distributed random variable

with mean of zero and variance of 2 . She does not learn her second period income until after

she has done her first period consumption. The interest rate is zero. Her time discount rate is zero. Her utility function is of the CARA form

1( ) cu c e

A) [15 points] solve for her first period consumption You will have to use the fact that if x is normally distributed with variance 2

x , then

2( ) / 2( )

xxE xE e e

B) [5 points] Economic conditions change in such a fashion that A falls and 2 goes to zero.

Given these new conditions, the individual finds it optimal to choose exactly the same level of first period consumption that she chose in part A. What is the new value of A?

[10 points] Is her expected utility higher under the conditions of part A or part B? Show how you derived this answer.

40. [final exam, 2006] Consumption in period t+1 is a normally distributed random variable with mean c and variance 2

c . Consumption at time t is c . Individuals have CARA

utility of the form ( ) cU c e They have a time discount rate of zero. A. [10 points] What is the risk free interest rate that holds between periods t and t+1?

111

Note: you will have to use the rule that if x is a normally distributed random variable then 2x-x -E (x) + / 2E( ) = e e

B. [10 points] What is the effect of increasing 2

c on the interest rate? Explain in economic terms

why this is true.

41. An individual faces the following problem. He lives for three periods. In the both the first and the second period, his income is 1 per period. In the third period, his income is either 1 or 2, each with a probability of .5. Further, he will find out at the end of the first period (that is, after he has done his first period consumption, but before he has done his second period consumption) what his income in period 3 will be. The interest rate and the person's time discount rate are both zero. His instantaneous utility function is U(Ct)=ln(Ct). How would you solve for his consumption in the first period. (Don't worry about solving any nasty quadratic equations that arise: just show the equation that you would solve.)

42. [midterm exam, 2008] A man is born at time zero and will live forever. Time is discrete. r= =1 (note this means that the interest rate between periods is 100%.)

In the first period of life (period zero), his labor income is 1. In the next period (period 1) there is a coin flip: with probability 50%, his labor income will be 1 again. With probability 50% he will lose his job and his labor income will be it will be zero. If he loses his job, he never has labor income again. If his income in period one is 1, then again in period two there will be a coin flip, and he has a 50% chance of losing his job (and never finding a new one) and a 50% chance of keeping his job. This goes on forever. The man has quadratic utility. You should assume that the utility function is well behaved (i.e. u’>0, u’’<0) for the entire range relevant to this problem. (It turns out that technically this assumption cannot be correct, but you should ignore this technicality). A) [10 points] Solve for consumption in period zero. B) [10 points] What are the two possible levels of consumption in period 1? C) [10 points] How many different possible levels of consumption are there in period n? D) [10 points. Hard!] List all the possible values of consumption in period n.

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43. [midterm exam 2004] A man will live for three periods. He has time discount rate of

zero. His initial assets are A0. He has no labor income. He has CRRA utility with a coefficient of relative risk aversion of 2.

The interest rate between periods 0 and 1 is zero. In period 1 (before he makes his consumption decision for that period) he will find out what the interest rate between periods 1 and 2 will be. Specifically, it will be zero with probability 50% and 3 with probability 50%. Solve for his first period consumption. [Note: this final expression for c0 is sort of ugly. You should at least derive the equation that implicitly gives c0 as a function of A0.] 43.5 [Core exam, 2012] Consider an individual who can spend his money on consumption or on

health. He is born at time zero and endowed with wealth W. He receives no labor income. Time is continuous. He will die at date T, which is endogenous. In particular, he can affect the length of his life by spending an amount H at time zero (think of this as investment in health, but it is all done at the beginning of life). The relationship between health spending and length of life is given by

T = H

The interest rate and time discount rate are both zero. He cannot go into debt. There is no uncertainty. His instantaneous utility function is

U = ln (c ) +

A) Solve implicitly for the optimal level of spending on health, H. You can’t get a closed form solution, but you should be able to get an equation that you can interpret.

Hint: You should not be doing complicated math to get this answer.

B) It has been frequently observed that as countries have gotten richer, the fraction of income devoted to health has risen. Looking at the equation that you derived in part (A), does the model presented here provide a good explanation for that phenomenon? You should be able to answer this question in three or four sentences, without doing any math. Indeed, doing math in this case is very difficult.

C) Suppose that the optimal value of H/W is 1/2. What is the value of ?

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44. [final exam, 2008] Consider the Bommier utility function. The pure time discount rate is zero. An individual who lives for T periods (from zero to T-1) has lifetime utility:

1

0( )

T

tV u c

.

Assume that ( ) ln( )t tu c c

and

( ) xx e Suppose that individuals live for two periods with certainty. They are endowed in period zero with assets A0. They earn no labor income. They earn interest on their savings are rate r.

An economist is trying to understand consumption. She believes (correctly) that people have a pure time discount rate of zero. She believes (incorrectly) that the people have CRRA utility of the standard form (that is, not the Bommier form), with coefficient of relative risk aversion . She tries two experiments in order to estimate .

A) [10 points] Suppose that the economist varies r among individuals and looks at how the

path on consumption responds. What will she conclude is the value of the coefficient of relative risk aversion?

B) [10 points] Suppose that the economist attempts to estimate by looking at choices with

respect to lotteries. Specifically, before individuals receive their time zero endowment, A0, she is able to offer them choices of lotteries. For example, she offers them proposals like “you can have A0=150 with certainty, or you can have A0=200 or A0=100+x, each with probability 50%.” She then varies x until people are indifferent between the two lotteries. What value of will she estimate? [Hint: write lifetime utility in terms of initial assets.]

45. [final exam, 2008] Consider the Bommier utility function. The pure time discount

rate is zero. An individual who lives for T periods (from zero to T-1) has lifetime utility:

1

0( )

T

tV u c

.

Assume that ( ) ln( )t tu c c

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and

( ) xx e

Consider the problem of a person who will live either one or two periods. She is endowed at time zero with assets A0=1. The interest rate is zero and there are no annuities. She cannot die in debt. At the end of period zero, she dies with probability one half. If she does not die after period zero, then she dies at the end of period 1 with certainty. Solve for her first period consumption.

46. [core exam, 2009] A group of individuals is born at time zero. Time is continuous. The death rate is constant at ρ>0.

At birth, each individual is endowed with wealth W(0), which is equal for all individuals.

There is no labor income. Individuals save in capital that pays return r. They cannot borrow or die in debt. Individuals have pure time discount rate θ, and log utility. Assume r=ρ= θ >0. Consider two different scenarios.

1) Individuals have no information about their dates of death, but there is an actuarially fair annuity market. Individuals who deposit money in an annuity receive interest equal to r+ρ if they live.

2) Individuals are informed at birth of the date of their death.13 There is no annuity market.

A) Describe what the paths of consumption look like in the two different scenarios. Discuss both average consumption and individual consumption paths. You do not have to derive precise expressions, but you can if you like.

B) In which scenario is expected utility (not conditioning on an individual’s date of death)

higher? Explain why. [Note: you do not need to grind through a whole lot of math to answer this question. Economic reasoning should be sufficient. On the other hand, Boris said that he found the math easier than intuition.]

C) Suppose that we consider a Bommier style utility function of the form

13 Note that there is no inconsistency between the death rate being ρ and people being informed of their own dates of death. In our normal model we assumed that there was no advanced information about who would die. But you could just as well imagine that everyone was told in advance when they would die and that those known dates of death were such that a fraction ρ died every period.

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0ln( ( ))

TtV e c t dt

Where ' 0 , '' 0 and V is the lifetime utility of someone who lives T years. How might the results of part B be altered in this case? Explain economically why this would be so.

Problems with Nonlinear Budget Sets

47. [midterm, 2006] Consider the following two-period model. A man has labor income of W in the first period and zero in the second period. He has log utility. His time discount rate is zero. He can save at an interest rate determined by how much he saves. Specifically, for saving less than $10, he earns an interest rate of zero. If he saves more than $10, all of his savings earns an interest rate of 100%. (Note that the high interest rate does not only apply to his savings above $10. So for example, if he saves $11 in period 1, he will be able to consume $22 in period 2.)

A. Solve for the value of W for which the man is indifferent between saving enough to earn the high interest rate and not doing so. Note: you should get as close to the exact solution as you can, even if this involves some algebra. In particular, if you end up with an expression that yields multiple solutions, you should show how all but one can be ruled out. B. Solve for the value of W at which the consumption of a person facing the interest rates described above would be the same as the consumption of a person who could earn 100% interest no matter how much he saved. C. Based on your answers to (A) and (B), draw a graph with the first period saving rate on the vertical axis and W on the horizontal axis.

48. A woman lives for two periods. In the first period she has income of 3. In the second period, she will have income of zero. The interest rate and the time discount rate are both zero. She cannot borrow. Her utility function is U(ct) = ln(ct).

There is a government welfare program that provides a consumption floor of cmin in the second period. In other words: if she does not have enough money left over to afford to consume cmin , then the government will give her enough money so that she can afford to consume it. A)Obviously, for high enough values of cmin , the consumer will decide to consume all of her wages

in the first period and go on welfare in the second period. For low enough values of cmin she

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will choose consumption as if the welfare program did not exist. Calculate the critical value of cmin for which she is indifferent between these two strategies.

B)Now suppose that there is only a 50% chance that she will be alive in the second period.

Calculate the critical value of cmin at which she will be indifferent between the two strategies discussed in part A.

49. People live for two periods. In the first period they have income of 1, in the second period they have income of zero. The interest rate and the time discount rate are both zero. The utility function is U(ct) = ln(ct).

There is a 50% probability that there will be a government welfare program in the second period of people’s lives. If there is such a program, it will provide a consumption floor of cmin. In other words: if they do not have enough money left over to afford to consume cmin , then the government will give them enough money so that she can afford to consume it. Solve for the critical level of cmin such that, if cmin is below this level, people’s choice of first period consumption will not be affected by the existence of the program. The algebra on this question may get a little tedious. If you can’t solve it, then show clearly what equation you would solve to get the answer.

50. An individual has wealth A0 at time zero. She will live infinitely, and will not receive any income. The interest rate is zero. She is unable to borrow. Her instantaneous utility function is U(c) = ln(c). She has a discount rate of Θ>0. Time is continuous.

There exists a government welfare program that works in the following manner. If a person has any wealth at all, she will receive nothing. If her wealth is equal to zero, then she will receive cmin. Figure out her optimal consumption strategy. Solve for her initial consumption, c0. Hint #1: Try thinking about the case where Θ=0. Hint #2: Do not try thinking about this problem in discrete time. Doing so will make you insane.

51. [Core exam, 2001] A person is born at time zero and will live forever. He is born with assets of 10. His labor income is one per period. His time discount rate is 5%. His instantaneous utility function is of the CARA form:

U(c) = -e-c

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There are two interest rates in the economy. For borrowing, the interest rate is 5%. For saving, the interest rate is zero. Solve for his optimal path of consumption. What is the consumption at time zero? Draw a picture of the time path of consumption, showing any inflection points, etc. Note: Solving this problem in continuous time requires optimization techniques that we did not cover in my class. It can be more easily solved in discrete time. However, to do so, you have to use the approximations that ln(1+x) x and ln(1/(1+x)) -x, both of which hold for x near zero.

52. [Core Exam, 2005] An individual lives for one period with certainty and may live into a second period with probability ρ, where 0 <ρ <1. He knows the value of ρ. He does not find out whether he lives in the second period until after he has done his first period consumption. He has labor income w=2 in the first period of life, and no labor income in the second period of life.

His first period utility is ln(c1) His second period utility is ln(c2) if c2 ≥ c1 ln(c2) - 1 if c2 < c1 The time discount rate and interest rate are both zero. Make a graph showing his optimal first period consumption as a function of ρ. If there are any notable jumps or kinks in this function, you should write out the implicit equation for the value of ρ at which they take place. You do not have to solve explicitly for the value(s) of ρ.

53. [Core exam, 2011] A man does all of his consumption in one period. He has initial wealth of $100. Before he gets to do his consumption, there is some random bad event that occurs with probability 20%. If that event occurs, he loses $80. If the event doesn’t occur, he doesn’t have to pay anything. Whatever is left, he gets to consume. He has CRRA utility with coefficient of relative risk aversion 2 .

Before it is observed whether the event occurs, the man can buy insurance.

A) Suppose that the insurance plan works as follows. For each $1 that the man spends on insurance, he receives a payment of $2 in the state of the world where the bad event occurs. Notice that this is not actuarially fair insurance. Solve for his optimal insurance purchase.

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B) Suppose instead that there is a different insurance policy offered. The man must pay a fixed cost of $z in order to have the right to buy insurance. Once he spends $z, he is entitled to buy actuarially fair insurance in whatever quantity he wants. Solve for the value of z for which he is indifferent between buying insurance and not buying insurance. (Note: I want an actual number of dollars for z, not an expression involving coefficients. However, the math does not work out as neatly as I would have liked in this case, so it is OK to end up with an expression involving just numbers that you would have to plug into a calculator to solve. Even in this case, you should write down approximately how big z is.)

Problems with nonstandard utility functions

54. [midterm exam 2002] A woman lives for two periods. Her wage income is W in the first period and zero in the second period. The interest rate and time discount rates are zero.

She has a habit-formation utility function of the following form: ut = ln( ct - γ zt) γ < 1, where zt is the habit stock period t. We assume that habit stock in a period is equal to last period’s consumption zt+1 = ct . We further assume that habit stock in the first period of life is zero. Solve for optimal consumption in the first and second periods of life.

55. [midterm exam, 2008] A country is populated by a large number of identical individuals. Individuals live for two periods. All individuals live for the same two periods (this is not an overlapping generations model). Their utility functions are

21 1/ 2

ln( ) lnc

V cz

Where z is the habit stock in the second period. The interest rate is zero. Individuals have wage w in period 1 and zero in period 2.

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A) [10 points] Consider the case where the individual’s habit stock in the second period is his own consumption in the first period. Solve for consumption in each period.

B) [10 points] Consider the case where the individual’s habit stock in the second period is the

average level of consumption in the first period. Solve for consumption in each period.

56. [final exam, 2003] A person will be alive for three periods, labeled 1, 2, and 3. The interest rate is zero. He has initial assets A1. His utility function in period t (for t = 1, 2) is of the “hyperbolic” form,

ln ln3

t t ss=t+1

= ( ) + ( ) U c c

where β<1. Solve for his path of consumption in periods 1, 2, and 3 under the following two scenarios: A) [10 points] At time 1, he can decide his consumption in the current period as well as all future periods -- that is, he can pre-commit himself to a lifetime path of consumption. B) [10 points] At time 1, he can decide his consumption in the current period, but cannot pre-commit his future consumption path.

57. [midterm exam 2004] A woman is born at time zero and will live for exactly 9 periods, labeled t=0, 1, ...8. She has initial assets A0. The interest rate and time discount rate are both zero. She has hyperbolic preferences of the form

ln lnT -1

t t ss=t+1

= ( ) + ( )V c c

where β= 2. A) Suppose that at time 0 she chooses her entire consumption path in periods 0-8. Further, there will be no subsequent revisions in her plans -- that is, her hands will be tied in future periods. Solve for consumption in each period. B) Suppose that in each period the person makes a consumption plan as if she were able to bind the hands of her future selves, and then does her consumption in that period according to the plan. So, for example, in period 0 she will do the consumption that found in part A. But further suppose

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that in each future period, she is able to re-make her plans (and she again assumes, incorrectly, that her future selves will stick to that plan.). Solve for her consumption in each period 0-8. You don=t have to literally solve for each period, but rather show the general rule that will generate these values.

58. [midterm exam, 2008] An individual is born at time 0 and will live for exactly T periods (in other words, the last period of life will be T-1). He has initial assets A0, and receives no wage income. He is not allowed to hold negative assets. The interest rate is zero.

The individual has time t preferences given by the equation

1

1

1ln( ) ln( )

2

T

t t ss t

V c c

At time t, an individual can only choose his consumption for that period. He cannot “bind the hands” of his future selves. A) [8 points] Solve for the consumption of an individual who has assets AT-2 at the beginning

of period T-2 (the second to last period of life).

B) [7 points] Do the same for an individual who has assets AT-3 at the beginning of period T-3, and for an individual who has assets AT-4 at the beginning of period T-4.

C) [8 points] Based on the above answers, what is the consumption of an individual who has

assets of A0 at the beginning of period 0?

D) [7 points] What is the consumption in period T-1 of the individual who had assets of A0 at time zero? (The answer is actually a very simple expression).

59. [midterm exam, 2003] A woman is born at time zero and will live until time T. Time is continuous. There is no uncertainty. She is born with zero assets and will die with zero assets. Her wage per unit of time is constant and equal to w. The interest rate is zero.

The woman has an unusual instantaneous utility function. It is u(c(t)) = α c(t) if c(t) ≤c = αc + β (c(t) -c ) if c(t) > c

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where α > β. Her lifetime utility is given by

T- t

0

V = u(c(t)) dte

where θ > 0. In the four parts of the question below, I ask you to solve for the optimal path of consumption under different assumptions about the values of the parameters. In each case, you don=t have to give me exact values, although you may if you want to. Mostly, I want you to show me a picture of what the path looks like and briefly explain its key features. A) Suppose that w = c and α e -θT > β . Solve for the optimal path of consumption. Also, explain why the assumption that α e -θT > β is important. B) Suppose that w > c and that α e -θT > β . Solve for the optimal path of consumption. C) Suppose that w < c and that α e -θT > β . Solve for the optimal path of consumption. D) Suppose that w = c and that α e -θT < β. Solve for the optimal path of consumption. Problems not elsewhere classified

60. [midterm exam 2004] Time is continuous. A man is born at time zero and will live forever. He has initial assets A(0) = 100. He has no labor income. Assume that r=0 and θ=0.5.

There are two things on which the man can spend his money. He can buy consumption goods (denoted c) or he can give his money to charity (denoted g). His instantaneous utility is given by

lnU(c,g) = (c) + g Note that utility from gifts to charity has non-decreasing marginal utility, because it is assumed that

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one individual’s contributions have an infinitesimal effect on the world’s suffering. Solve for his optimal paths of consumption and charitable contributions. Problems with Variable Labor Supply (these come from notes on RBCs) (way too many of these relative to the importance of this material)

61. Consider a one-period model in which individuals make a labor- leisure decision. Individuals are endowed with one unit of time.

Utility is given by:

where n is leisure and c is consumption. People take the wage rate, w, as exogenous. They live for only one period, and consume all of their earnings. Show how the effect of the wage on labor supply depends on the value of σ. For what values of σ will an increase in the wage have a positive effect on labor supply, and for what values will it have a negative effect?

62. (final exam, 2002) Consider a world in which the only form of output is fruit that falls from trees. The time period is days. The fruit falls from the trees only on odd numbered days. Specifically, on each odd numbered day, one unit of fruit per capita falls. The fruit can be eaten on the day it falls or stored to be eaten at any day in the future, with zero depreciation (there is no integer constraint, that is, people can eat or store fractions of a fruit.)

The population is composed of identical, infinitely lived people with log utility and time discount rates θ >0. At time period 1 (the first period in which this world exists), there is no existing stockpile of fruit. Solve for the interest rate that holds between odd and even periods and the interest rate that will hold between even and odd periods.

ln1-cU = (n) + > 0

1-

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63. (midterm exam, 2008) Consider an infinitely lived individual born at time zero with no assets. Time is continuous. The wage rate is constant and equal to w per unit time. He is endowed with one unit of time per period.

The instantaneous utility function is

ln( ) ln( )U c n Where n is leisure. The individual can borrow or lend at rate 0r . He discounts the future at rate

.r A) [15 points] Solve for the lifetime paths of consumption and leisure. B) [10 points] Sketch the lifetime path of assets. In particular, you should be able to say what the level of assets does initially (goes up, goes down, stays constant) and explain why. You should also be able to say what it asymptotes to and why.

64. (midterm exam, 2008) Consider a woman with the utility function described above. Now assume that 0r . Further, assume that wages grow over time. Specifically, the wage is given by

( ) (0) gtw t w e ,

where r>g>0. A) Describe her paths of consumption, leisure, and assets. B) (hard): to what value does the ratio of assets to wage asymptote?

65. (Final Exam, 2010) A woman has a job picking mushrooms out in the forest. Her within-period utility function is V = ln(c) + ln(1-n) Where c is consumption and (1-n) is leisure. Her production of mushrooms depends on a stochastic term A:

t t tY A n

tA can take two values: 1 and 2, each with probability 50%. Call the states of the world corresponding

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to these outcomes s for sunny and r for rainy (mushrooms grow better in the rain).

A) [5 points] Suppose that the woman cannot borrow or save. Solve for her consumption and leisure in each state of the world.

B) [5 points] Suppose that the woman can borrow or save at an interest rate equal to her time discount rate. A new value of A is drawn every day, and the woman’s life is long enough that we can assume that the law of large numbers holds. Solve for her consumption and leisure in each state of the world. [Note: Depending on how you solve this, it might end up with a nasty quadratic equation, which you can leave un-solved. However, you may want to try solving a different way which yields a simple solution.]

C) [5 points] Suppose that the woman cannot borrow or save. However, prior to each period, she can sign an insurance contract. The contract specifies that if the state is A=2, then she will pay the insurance company x dollars, and if the state is A=1 the insurance company will pay her x dollars. She can choose x. Solve for her consumption and leisure in each state of the world. Also solve for the value of x that she chooses.

66. (Core exam, 2003) A woman is born at time zero and will live forever. She is born with zero assets. She can borrow or lend at interest rate r>0. She is endowed with one unit of time per period, which she can use for working or leisure. Time is continuous. Her lifetime utility function is

ln ln- t

0

U = [ (n) + (c)] dte

where θ is the time discount rate, n is leisure, and c is consumption. Her wage at time zero, w(0), is equal to one. Following time zero, her wage grows at rate g. The relation between r, θ, and g is as follows:

r= g =

2

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Solve for her optimal lifetime paths of consumption and leisure. Specifically, solve for c(0) and n(0), and describe mathematically the paths of consumption and leisure after time zero. Note: You can solve this problem by using a lot of math, but you don’t need to. You should be able to do it by simply putting together a bunch of things you already know.

67. [Final exam, 2008] An individual is born at time 0 and will live to time T with certainty. She can save at interest rate zero. Her time discount rate is zero. Time is continuous. She is endowed with one unit of time per period, which she divides between work and leisure.

At the beginning of her life, the individual will work. However, at any point in life she many “retire,” which means having full time leisure. Once the individual retires, she may not return to work. Her wage per unit of labor input is constant at w. Her utility function is U = ln(c) + ln(n) – γ if working = ln(c) + ln(n) if retired , where γ is the cost (in terms of utility) of not being retired.

A) [10 points] What is the condition on the values of w, γ, and T such that the individual will choose not to retire before dying? [Note: you can solve this part of the problem directly using a perturbation style of argument. Or you can solve it using some of the results from part B.]

B) [10 points] In the case where it is optimal to retire before dying, write down the first order condition for the optimal retirement age (don’t try to solve it). Here are some hints for solving part B of the problem:

Think about what the path of consumption will look like. Also think about what the path of leisure in non-retirement periods will look like.

With these two facts, it should be pretty easy to write down the lifetime budget constraint and function for total lifetime utility. You can substitute the budget constraint into the lifetime utility function.

Using the static FOC, you can simplify the lifetime utility function by writing it solely in terms of the retirement age, R.

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You can differentiate this w.r.t. retirement age in order to find the optimum; then you can check what condition is required for optimal retirement age to be less than T.

68. (final exam, 2001) A person will live for one period. Her utility function is U = ln(n) + ln(c) where n is leisure and c is consumption. She is endowed with one unit of time that she can spend working or enjoying leisure. Let w be the wage per unit of time. There is uncertainty about what the wage will be. Consider two scenarios: in scenario A, the person has to specify what her consumption will be before she gets to see the value of w. In scenario B, the person has to specify what her leisure will be before she gets to see the value of w. In which of these scenarios will her utility be higher? Explain why. You will notice that I haven’t given you the usual set of information in this problem. Specifically, I haven’t told you about the distribution of w. This is intentional. You can answer the problem with only the information given here.

69. [Core Exam, 2005] Consider the following setup. There is one period during which a man may work, consume, and enjoy leisure. He is endowed with one unit of time, which can be spent either working or having leisure. His utility function is

U = c1/2 + n, where c is consumption and n is leisure. His wage is uncertain. With probability .5 it will be w=1; with probability .5 it will be w=0.

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Prior to the period in which consumption takes place, and prior to observing what his wage will be, the man has the opportunity to buy insurance against a low wage by selling a share of his labor income. Call the share that he sells α. The firm that buys the labor income agrees to pay a fixed amount X to the man in the state of the world where w=0, but nothing in the state of the world in which w=1. In return, the man will pay the firm some share α of his labor income in the good state of the world. There is a perfectly competitive market of buyers who will purchase this labor income. Buyers are risk neutral and earn zero expected profits. The share of his income that the man sells is fully observable by buyers, and the man must commit to sell this share and no more before any transactions take place. The man cannot pre-commit to work a fixed fraction of his time, however. What is the optimal fraction of his labor income that the man should sell? Note: the algebra in this question is more messy than usual. In particular, depending on how you solve it, you may end up with a quadratic equation with two roots, only one of which is an optimum. You may leave the solution in the form of this quadratic equation. You do not have to determine which of the roots is an optimum. Alternatively, you can simply write down an implicit equation for α.

70. [final exam, 2006] An economy is composed of a large number of identical individuals. Individuals live for one period in which they work and enjoy leisure. Their utility is given by the following function:

ln( ) lnc

U nc

Where n is leisure, c is the average level of consumption, and 1 1 .

Individuals are endowed with two units of time (that will just make the numbers come out easier). The wage is equal to one.

A. [5 points] Solve for the equilibrium levels of consumption and leisure. B. [5 points] Solve for the levels of consumption and leisure chosen by a social planner who

wants to maximize average utility. C. [5 points] Explain in economic terms any difference between your answers to parts A and

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B. Your explanation should touch on the interpretation of the utility function.

71. [Core Exam, 2005] Consider a world in which people live two periods. Each individual has one child and one parent. All individuals in a generation are identical.

In the first period of life, people are children living in their parent’s house. During this period, individuals make no decisions. Their parents pay for their consumption. In the second period of life, individuals are endowed with one unit of time, which they split between work and leisure. The wage is equal to 1 per unit time. Their utility function of someone who is in the second period of life in period t is U = ln(n2,t) + ln(c1, t-1 + c2, t) where n2,t is his leisure, c2,t is his consumption during period 2 of his life, and c1, t-1 is the consumption that he did when he was young. Parents face the constraint that they must provide their child with the same level of consumption that they have, that is c1, t = c2, t. Analyze the dynamics of second period consumption. In particular, find the steady state level of consumption, and show how consumption varies over time if it starts at some level other than the steady state. Explain briefly (roughly 3-4 paragraphs) what the particular characteristic(s) of the utility

function lead to these consumption dynamics and why they do so. Also discuss whether this characteristic of the utility function is reasonable, and what an alternative would be. How would such an alternative formulation of the utility function affect the steady state and dynamics of the model?