Note Part 1

8/3/2019 Note Part 1

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Part 1: Individual Values and Choices

We want a formal and reproducible description of how individuals value choiceobjects, in a way that it is identifiable from actual choices.

1 Preferences

1.1 General Formulation and Assumptions

X: set of all the possible/conceivable alternatives or objects of choice, whichmay or may not be available or affordable.

: preference relation defined over X, with being its asymmetric componentand being its symmetric component. Similar to , >, =, but not quite (why?).

Given any x, y X,

x y is read as x is at least as good as y.

x y is read as x is better than y.

x y is read as x is as good as y, or the decision maker is indifferent betweenx and y.

Two properties are assumed throughout.

1. Completeness: For all x, y X, either x y or y x holds. It is possible that both x y and y x hold, and then we write x y.

2. Transitivity: For all x,y,z X, if x y and y z then x z.

1.2 Preference over Consumptions

Assume that there are just two goods. Sounds stupid, but this is enough forunderstanding the point.

Mostly we take the set of all the possible alternatives to be X = R2+, the non-negative quadrant.

x = (x1, x2) is read as...

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Notice two underlying assumptions

Divisibility

Homogeneity

Two applications

Consumption over time

Uncertain consumption

Indifference Curves: Graphical description of preference

Examples

Perfect substitution

Perfect complements

Smooth preferences between the above two

Two more properties are assumed on preference over consumptions.

1. Monotonicity: For all (x1, x2) R2

+ and c > 0,

(x1 + c, x2) x and (x1, x2 + c) (x1, x2).

2. Convexity: For all (x1, x2), (y1, y2) R2+ with (x1, x2) = (y1, y2),

(x1, x2) (y1, y2) implies(

x1 + y12

,x2 + y2

2

) (x1, x2).

Basic properties of indifference curves

You can draw them = Completeness They do not cross. = Transitivity Downward-sloping = Monotonicity Not thick = Monotonicity Better set is convex = Convexity

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1.3 Marginal Rate of Substitution

Measure of subjective relative value

To illustrate, consider the case of perfect substitution. How valuable is good 1 for you, compared to good 2?

Restate

To get one more unit of good 1, how much of good 2 are you willing to giveup?

Marginal rate of substitution of good 2 for good 1= amount of good 2you are willing to give up, to obtain one extra unit of good 1.

The roles of good 1, good 2, respectively, are fixed throughout the course.

Why marginal?

Slope changes point by point.

Look at local slope of the indifference curve

At (x1, x2), to get additional x1 units of good 1, you are willing to sacrificex2 units of good 2, so that

(x1, x2) (x1 + x1, x2 + x2)

Per one more unit of good 1, you are willing to sacrifice good 2 by

x2x1

Taking the limit,

MRS(x1, x2) = dx2dx1

1.4 Implication of convexity on MRS

Diminishing MRS: A good is relatively less important as it is more abundant.d

dx1

(dx2

dx1

)< 0

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2 Representation of Preference, a.k.a. Utility Func-

tion

2.1 Definition

Definition 1 A function u : X R represents preference defined over X ifx y u(x) u(y).

for all x, y X. Implication

x

y

u(x) > u(y).

x y u(x) = u(y).

2.2 One preference may have different representations

Consider

u(x1, x2) = x1x2 v(x1, x2) = x1x2 + 6

v(x1, x2) = 2x1x2

v(x1, x2) = 2x1x2 + 6

v(x1, x2) = ln(x1x2) = ln x1 + ln x2

In general, consider any f monotone (increasing) function.

Proposition 2.1 If u represents , then v = f(u) also.

Ordinal Utility: It is no more than a representation of a ranking, and theassigned numbers have no quantitative meanings.

Q: Then why do we consider such a thing if it has no quantitative meaning?A: It makes analysis much more operational than directly looking at preference

does. Youll see.

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2.3 Examples

Perfect substitutes

Perfect complements

Cobb-Douglas

2.4 Marginal Utility

How can we measure the impacts of increasing amounts of goods?

Keeping in your mind that it has no economic content per se, considerMU = increase of utility when you add one unit of good

1 good caseu

x=

u(x + x) u(x)x

Taking the limit, you getdu(x)

dx

or simply u(x).

In the 2 goods case, how?

Slice, and look at good 1 only:

MU1 = increase of utility when you add one unit of good 1, keeping the

amount of good 2 the same

u

x1=

u(x1 + x1, x2)

u(x1, x2)

x1

Taking the limit, you getu(x1, x2)

x1

which is the slope along x1-axis

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Similarly for 2:u(x1, x2)

x2

The law of diminishing MU has no economic content It depends on which representation to use

u(x1, x2) = x21x22 andv(x1, x2) = ln x1 + ln x2

Which way of measurement is independent of which representation to use? =MRS

2.5 From MU to MRS

Local change of u by (x1, x2) = (x1 + dx1, x2 + dx2):

du =u(x1, x2)

x1dx1 +

u(x1, x2)

x2dx2

For keeping du = 0,u(x1, x2)

x1

dx1 +u(x1, x2)

x2

dx2 = 0

Then,MRS(x1, x2) = dx2

dx1=

u(x1,x2)x1

u(x1,x2)x2

Notice: Beware of the numerators and the denominators.

MRS is independent of which representation to use, whereas MU is dependent.Why? Take one representation u(x1, x2), then

M RS(x1, x2) =

u(x1,x2)x1

u(x1,x2)x2

Now take another representation f(u(x1, x2)). Then,

MRS(x1, x2) =

f(u(x1,x2))x1

f(u(x1,x2))x2

=f(u(x1, x2)) u(x1,x2)x1f(u(x1, x2)) u(x1,x2)x2

=

u(x1,x2)x1

u(x1,x2)x2

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Try with the above example.

1: Consider the preference represented by u(x1, x2) = xa1x

b2.

(a) What is the marginal utility of good 1 at (x1, x2)? MU of good 1 is given as the partial derivative w.r.t. x1. Thus,

u(x1, x2)

x1= axa11 x

b2

(b) What is the marginal utility of good 2 at (x1, x2)?

MU of good 2 is given as the partial derivative w.r.t. x2. Thus,

u(x1, x2)

x2= bxa1x

b12

(c) What is the marginal rate of substitution of good 2 for good 1 (I mean dx2/dx1)at (x1, x2)?

Recall that


=

u(x1,x2)x1

u(x1,x2)x2

By plugging the results of (a) and (b) in, we obtain

M RS(x1, x2) =axa11 x

b2

bxa1xb12

=ax2bx1

2: Let me try with another representation v(x1, x2) = ln(xa1xb2) = a ln x1 + b ln x2.

What does it change and what does not?

(a) What is the marginal utility of good 1 at (x1, x2)?


v(x1, x2)

x1=

a

x1

(b) What is the marginal utility of good 2 at (x1, x2)?


v(x1, x2)

x2=

b

x2

(c) What is the marginal rate of substitution of good 2 for good 1 (I mean dx2/dx1)at (x1, x2)?

Recall that


=

u(x1,x2)x1

u(x1,x2)x2

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By plugging the results of (a) and (b) in, we obtain

MRS(x1, x2) =ax1b

x2

=ax2bx1

,

which is the same as what we obtained previously.

3 Optimal Choice

3.1 General Definition

Definition 2 Let X be the universal set and be a preference defined over X.

Given an opportunity set A X, its element x A is said to be optimal in A for if x y for all y A.

Equivalently saying, x A is optimal in A for if there is no y A such thaty x.

Optimal choice is described as if the consumer is solving the problem:

maxx

u(x)

subject to x A

Interpretation

3.2 Optimal Consumption Choice

p and m given

Budget set:

B(p,m) = {x R2+| p1x1 + p2x2 m}

x B(p,m) is optimal in B(p,m) if x y for all y B(p,m).

In other words, x B(p,m) is optimal in B(p,m) if there is no y B(p,m)such that y x.

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3.4 Some important exceptions/extreme cases

Perfect Substitutes Multiple optima and corner solution

Perfect Complements Interior optimum where the tangency condition fails

4 Revealed preference

How can you say that choices are based on preference maximization?

Start from a series of observed choices, ask whether it can be explained bypreference maximization.

: Choice function that describes the series of input-output data.When a set of alternatives B is given, it selects (B) B. Ties are excluded for simplicity of exposition.

Definition 3 is rationalizable if there is a complete/transitive ordering over X

such that

(B) y

for all B and all y

B.

Definition 4 satisfies contraction if for every B, C with C B and (B) C,

(C) = (B).

For simplicity, here assume X is finite.

Theorem 4.1 is rationalizable if and only if it satisfies contraction.

Choices not rationalized by a single preference:

1. Framing, reference points and loss aversion

2. Self-control problem

3. Choices driven by anticipated regret, e.g., minmax regret choice

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5 Demand

5.1 How does your choice change when the price/income

change?

Your choice is a function of price/income.

Demand Function:

x1 = x1(p1, p2, m)

x2 = x2(p1, p2, m)

Function of three variables into two variables

CD preference, separable preference

5.2 Response to income changes

5.2.1 Normal and Inferior Goods

Normal Good: Demand increases when income increases.

Good 1 is a normal good ifx1m

> 0.

Inferior Good: Demand decreases when income increases

Good 1 is an inferior good ifx1m

< 0.

CD example: Both are normal.

5.3 Response to price changes

5.3.1 Own-price change

Change of demand for a good when its own price changes

Ordinary Good: Demand decreases when its own price increases.

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Good 1 is an ordinary good ifx1p1

< 0.

Giffen Good: Demand increases when its own price increases. Good 1 is a normal good if

x1p1

> 0.

CD example: Both are ordinary.

5.3.2 Cross-price change

Change of demand for a good when the price of another good changes

Good 1 is a gross substitute of good 2 ifx1(p1, p2, m)

p2> 0.

Good 1 is a gross complement of good 2 ifx1(p1, p2, m)

p2< 0.

CD preference, separable preference

5.4 Substitution and income effects

What did I mean by putting gross in the previous definition of substitutionand complementarity?

Effect of price change is decomposed into two, the change of relative price, and

the change of real income.

First is called substitution effect, second is called income effect.

In terms of pure effect, substitution effect of own-price change is always non-positive, and the substitution effect of cross-price change is always non-negative.

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6 Choice over time

6.1 Lifetime consumption and saving

For simplicity, assume two dates, today and tomorrow (or young/old).

(x1, x2) You consume x1 today and x2 tomorrow.

Endowment: (1, 2) Today you earn 1 units of consumption good, and 2tomorrow.

Assume no inflation for a moment. Price level is 1 at each period.

Interest rate r You save 1 x1. If its negative, you are borrowing.

Consumption tomorrow is x2 = 2 + (1 + r)(1 x1)

Two alternative expressions of the budget equation:

(1 + r)x1 + x2 = (1 + r)1 + 2

andx1 +

x21 + r

= 1 +2

1 + r

(1 + r)1 + 2 = Future value of lifetime income amounts of consumption tomorrow if you save all.

1 +21+r

= Present value of lifetime income

amounts of consumption today if you borrow up to the limit.

6.2 Inflation

Inflation rate

Price level today = 1Price level tomorrow = 1 +

Nominal interest rate r

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Today you save 1 x1.

Budget constraint is

(1 + )x2 = (1 + )2 + (1 + r)(1 x1)

or equivalently,

x2 = 2 +1 + r

1 + (1 x1)

Real interest rate is defined by

1 + =1 + r

1 +

Approximately, r

BC is written as(1 + )x1 + x2 = (1 + )1 + 2

in the future value form, and

x1 +x2

1 + = 1 +

21 +

in the present value form.

Since inflation can be adjusted without loss, we focus on the no-inflation caselater on.

6.3 More on present value

An IBM stock gives you 40 units of the share today and 60 units tomorrow, interms of consumption good. When the interest rate is 20%, how much of value

does it have, in terms of todays consumption?

Present value of (40, 60) is

40 +60

1 + 0.2= 90

No Arbitrage Condition = (Under certainty,) Asset price should be equalto its present value.

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The case of more than 2-periods: When the interest rate is r, present value ofthe sequence (1, 2, 3, , T) is

1 + 11 + r

2 +(

11 + r

)23 + +

(1

1 + r

)T1T =

Tt=1

(1

1 + r

)t1t

1. Here is a bond which pays you $10 every year and matures in 20 years. Upon

maturity, it pays you $100. What is the present value of this bond when the

interest rate is 5%?

2. When the interest rate is 5%, what is the present value of a bond which gives

you $30 of coupon every period?

3. UT-Econ. Co. earns $5 million every year. Assuming that the interest rate is

10%, what is the present value of this company?

6.4 Preferences

6.4.1 Impatience and consumption smoothing

Which do you prefer, (10, 5) or (5, 10)?

Impatience: Future consumptions are discounted.

Which do you prefer, (10, 0) or (5, 5)?

Consumption smoothing: Fluctuation is disliked.

6.4.2 Discounted utility

Discounted utility: u is called discounted utility when it has the form

u(x1, x2) = v(x1) + v(x2)

: discount factor, 0 < 1 v(): period utility index

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6.4.3 Properties of period utility index

Period utility index explains attitude toward consumption fluctuation/smoothing.

As it is more curvy, the consumer has more preference for consumption smooth-ing.

Period utility index is a component of utility function, and NOT a utilityfunction.

Utility function is ordinal (no quantitative meaning), but period utility indexis cardinal (having quantitative meanings).

Still, it has nothing to say about how happy you are.

Recall that one preference can have many utility functions for it. It allows anyincreasing transformation.

i.e., if u represents , f(u) does also.

But period utility index does not allow arbitrary. You cannot in general do anoperation like

f(v(x1) + v(x2)) = f(v(x1)) + f(v(x2))

For example, compare v(x) = x and v(x) = v(x) = x.Assuming = 1, compare (100, 0) and (49, 49).

When v is more concave, you want more consumption smoothing.

One preference can have essentially only one period utility index. In otherwords, even if one preference has several period utility indices, they must be

congruent.

If you change the curvature of period utility index, it changes preference forconsumption smoothing. So, transformations are allowed only in the way that

you dont change the curvature.

Remember, still the overall representation is ordinal. That is, for any trans-formation f,

f(u(x1, x2)) = f(v(x1) + v(x2))

represents the same preference as u(x1, x2) does.

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Theorem 6.1 Suppose v is a period utility index which represents the preference

in the discounted utility form. Then av + b does also, where a > 0.

Moreover, if v and v are two period utility indices representing the same preference

in the discounted utility form, there must be a, b with a > 0 such that v = av + b.

We say, period utility index is unique upto positive linear transformations.

The case of more than 2-periods:

u(x1, x2, x3, , xT) = v(x1) + v(x2) + 2v(x3) + + T1v(xT)

=Tt=1

t1v(xt)

Assume the utility index is v(x) = x and = 0.9

Two options. One is that you consume 10 units every period. The other one isthat you consume 25 units in odd periods and 4 in even periods. Which option

do you prefer?

6.5 Choice

Utility maximization:

maxx1,x2

v(x1) + v(x2)

subject to x1 +x2

1 + r= 1 +

21 + r

Tangency condition:

M RS(x1, x2) =v(x1)

v (x2)= 1 + r

Interpretation

Solve for x1(r, 1, 2) and x2(r, 1, 2). Saving = 1 x1(r, 1, 2)

What determines saving?

Try with v(x) = ln x.

Try with v(x) = x.

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7 Choice under risk

7.1 Risk and Uncertainty

The distinction attributed to Frank Knight:

1. Risk: Outcomes will occur with known or estimable probability.

2. Uncertainty: Outcomes will occur with a probability that cannot even

be estimated.

We focus on risk.

7.2 Risk attitudes

What do you care about in choosing between bets?

Expected value of the return?

1. Which do you prefer, (100; 0.5, 0; 0.5) or (50; 1)?

2. St. Petersburg paradox

3. Why do people buy a lottery? Expected return is extremely low.

Are they rational?

Three attitudes

1. Risk averse: If I can get the amount equal to the expected return for

sure, thats better for me.

2. Risk neutral: Only the expected return matters for me.

3. Risk loving: Under the same expected return, I would rather bet.

In general, expected return is not the only factor.

Need to incorporate risk attitudes. = Expected Utility

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7.3 Expected utility: Basic idea

Assume the largest possible prize is 100 and the smallest is 0.

Assign v(100) = 1 and v(0) = 0, without loss of generality (in the senseexplained later).

Consider a lottery (100; 0.5, 0; 0.5).

How much of prize if you can get for sure is equally preferable to this bet? Certainty Equivalent

Say, CE is 25, which is not necessarily the expected return 50.

To be consistent with (100; 0.5, 0; 0.5) (25, 1), assign

v(25) = 0.5v(100) + 0.5v(0) = 0.5

Consider another lottery (100; 0.6, 0; 0.4). What is the CE now?

Say, CE is 36, again which is not necessarily the expected return 60.

To be consistent with (100; 0.6, 0; 0.4) (36, 1), assign

v(36) = 0.6v(100) + 0.4v(0) = 0.6

Repeat this procedure. Eventually you get a function

v : [0, 100] [0, 1]

Consider a lottery (25; 0.5, 81;0.5). What is the CE?

You solve for x such that v(x) = 0.5v(25) + 0.5v(81) For example, suppose v(x) = x/10. Then, whats the CE for (25; 0.5, 81;0.5)?

= solve for x such that x/10 = 0.525/10 + 0.581/10.You get x = 49, whereas the expected return is 53.

How much is the CE for the St. Petersburg lottery?

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7.4 Expected utility: Formulation

Lottery, denoted by p = (x1;p1, , xn(p);pn(p)): You get x1 with prob. p1, ,

and xn(p) with prob. pn(p), where n(p) denotes the number of outcomes whichp assigns positive probabilities.

: Preference over lotteries (risk preference)

p q

e.g., (70; 0.6, 20;0.4) (40;0.55, 30;0.2, 10;0.25).

Expected utility: Utility function u represents in the form

u(p) =

n(p)k=1

pkv(xk)

where v is called von-Neumann Morgenstern utility index

Four axioms which characterize the expected utility theory

1. Completeness: For all p, q, either p q or q p holds.

2. Transitivity: For all p,q,r, p q and q r imply p r.

3. Continuity: preference does not jump

4. Independence: For all p,q,r and 0 < < 1,

p q p + (1 )r q + (1 )r.

Intuition?

Theorem 7.1 satisfies completeness, transitivity, continuity and independence if

and only if there exists a function v such that

p q n(p)k=1

pkv(xk) n(p)k=1

qkv(yk)

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7.5 Properties of the vNM index

It explains the decision makers risk attitude

Risk neutral when v is linear, v(x) = ax + b, since

nk=1

pkv(xk) =n

k=1

pk(axk + b) = an

k=1

pkxk + b = v

n

k=1

pkxk

depends only on the expected returnn

k=1pkxk.

Risk averse when v is concave, i.e.,

v(x1) + (1 )v(x2) < v(x1 + (1 )x2).

Risk averse when v is convex, i.e.,

v(x1) + (1 )v(x2) > v(x1 + (1 )x2).

What about the following?

v(x) = ln x

v(x) = x2

v(x) = ex

v(x) = ex

vNM index is a component of utility function, and NOT a utility function.

Utility function is ordinal, but vNM index is cardinal.

Recall that one preference can have many utility functions for it. It allows anyincreasing transformation.

i.e., if u represents , f(u) does also.

But vNM index is not allow arbitrary transformation. You cannot in generaldo an operation like

f

n

k=1

pkv(xk)

=

nk=1

pkf(v(xk))

For example, consider v(x) = x and v(x) = v(x) = x.Compare (100; 0.5, 0; 0.5) and (50; 1).

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The two correspond to different risk preferences. First is risk-neutral, butsecond is risk-averse.

One risk preference can have essentially only one vNM index. In other words,even if one risk preference has several vNM indices, they must be congruent.

If you change the curvature of vNM index, it changes risk preference. So,transformations are allowed only in the way that you dont change the curvature.

Remember, still the overall representation is ordinal. That is, for any trans-formation f,

f(u(p)) = fn

k=1

pkv(xk)represents the same preference as u(p) =

nk=1pkv(xk) does.

Theorem 7.2 Suppose v is a vNM index which represents risk preference in the

expected utility form. Then av + b does also, where a > 0.

Moreover, if v and v are two vNM indices representing the same risk preference in

the expected utility form, there must be a, b with a > 0 such that v = av + b.

We say, vNM index is unique upto positive linear transformations.

7.6 Application

Hereafter, fix the number of possible outcomes equal to two, fix the probabilityvalues as well (equal to (1, 2)), and just vary outcomes.

Thus the preference over lotteries reduces to a preference over two-dimensionalvectors, which is represented in the form

u(x1, x2) = 1v(x1) + 2v(x2)

7.6.1 Buying insurance

Your initial income is 400.

Youll hit a car-crash with probability 0.01.

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If you hit a crash, you lose 400.

There is an insurance available. It costs 2 per 1 unit, but it pays you back 80

if you hit a crash.

When your risk attitude is described by the vNM index v(x) = ln x, how manyunits of insurance will you buy? What if you are risk neutral?

General setting: Probability of hazard , initial income m, loss by hazard l,insurance cost c, return r.

The problem:

maxx

v(m l cx + rx) + (1 )v(m cx)

First-order conditionv(m l cx + rx)

v(m cx)r c

c=

1

Perfect Insurance ifr

c=

1

Then,

x =l

r

and you end up with m clr

, regardless of whether you hit a crash or not.

7.6.2 Portfolio choice 1

Two assets, safe asset and risky asset.

Return of safe asset is 1.05, but that of risky asset is 1.2 with probability 0.7

and 0.8 with probability 0.3.

You have an income of 100 to be saved.

When your risk attitude is described by the vNM index v(x) = ln x, how willyou allocate your income? What if you are risk neutral?

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7.6.3 Portfolio choice 2

Two states of the world.

Two assets, risky asset 1, risky asset 2. No safe asset.

Return table:

State 1 State 2

Risky asset 1 1.2 0.9

Risky asset 2 0.8 1.4

You can make a safe asset by combining the two assets. Hedging

7.6.4 Trading Arrow securities

Two states of the world.

Two securities, security 1 pays 1 unit of consumption if state 1 occurs andnothing if state 2 occurs, vice versa for security 2. called Arrow securities

Return table:

State 1 State 2Security 1 1 0

Security 2 0 1

Risky asset 1 in the previous example can be viewed as a combination of 1.2units of Arrow security 1 and 0.9 units of Arrow security 2.

When (p1, p2) is the price pair of such securities and (1, 2) is the pair of initialsecurity holdings, the choice problem is described by

max 1v(x1) + 2v(x2)

subject to p1x1 + p2x2 = p11 + p22

This reduces to the abstract optimal consumption problem.

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7.7 Anomalies

7.7.1 Violation of the independence axiom

Example 7.1 (A version of Allais paradox):

(A)

A1: a sure win of 30,

A2: a 80% chance to win 45 (and zero in 20% of the cases).

(B):

B1: a 25% chance to win 30

B2: a 20% chance to win 45.

Example 7.2 Normative violation Society should randomize for the sake of fair-ness.

7.7.2 Framing effects

Example 7.3 Consider the following two choice problems

(A)

A1: a sure gain of 20,

A2: a 70% chance to gain 30 and 30% chance of no gain.

(B) You are initially given 30.

B1: a sure loss of 10,

B2: a 70% chance of no loss and 30% chance to lose 30.

8 Monetary Evaluation of Consumptions

In many of economic analysis, we focus on the market for a single commodityand evaluate consumption of it in the monetary term. This is called partial

equilibrium analysis.

But how can this be well-grounded?

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8.1 Classical (naive) view

Utility = Money

Marginal Utility = Willingness to pay for an incremental consumption

Consumers problemmax U(x)px

with the assumption U(0) = 0.

The maximization conditionU(x) = p

which determines the (inverse) demand curve.

The maximized consumers surplus is

U(x) px

Problems

1. How can you measure (marginal) utility in monetary terms?

We learned that utility representations have no quantitative meaning.

2. In the first place, there is no money in our model yet.

3. We just learned that demand depends on income as well. Where is income?

8.2 Modern explanation

WTP should be described as marginal rate of substitution, not as marginalutility.

To illustrate, consider two goods.Good 1 = the good in consideration

Good 2 = Numeraire good, or reference good (can be positive or negative)

Interpretation: composite of all the other commodities, that is, income to be

spent on them.

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Underlying assumption: There are actually very many goods, and good 1 isa tiny tiny part of the entire set of goods, and the composite of all the others is

taken to be enormously abundant compared to this tiny piece, hence only the

relative gain/loss of it does matter.

Quasi-linear preference, represented in the form

u(x1, x2) = v(x1) + x2

No income effect on good 1: Indifference curves are parallel along the

x2-axis. WTP is independent of x2. Why? Recall the underlying assumption.

Willingness to pay = Willingness to give up the numeraire good for an extraunit of the good in consideration

This is exactly the marginal rate of substitution of the numeraire good for

the good in consideration.

MRS has the formMRS(x1, x2) = v

(x1)

Consumers surplus is measured in terms of the numeraire good.

Normalize p2 = 1. Only p1 varies.

Suppose you originally have (x1, x2). For extra x1 units of the good in con-sideration, the WTP should satisfy

v(x1 + x1) + x2 W T Px1 = v(x1) + x2

That is,

W T P =v(x1 + x1) v(x1)

x1Taking the limit,

W T P(x1) = v(x1)

For the x1-th unit of consumption, the marginal surplus measured by thenumeraire good is

W T P(x1) p1 = v(x1) p1

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Consumers surplus of having x1 units under the price p1 is

x1

0

(v(x1) p1) dx1 = v(x1) v(0) p1x1

8.3 Demand and inverse demand

Recall the normalization p2 = 1. Only p1 can vary.

The consumers problem ismaxx1,x2

v(x1) + x2

subject to p1x1 + x2 = m

The optimality condition isv(x1) = p1

Demand for the good in consideration depends only on p1.

By solving the above for x1, we get the demand function for the good, x1(p1).

In the Marshallian convention, we plot the inverse demand function

p(x1) = v(x1)

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