The Life-Cycle Permanent-Income Model:

Extensions of the Ss Model

Lucas Zalduendo MMSS Senior Thesis Adviser Robert J. Gordon

______________________________

Many people were instrumental in the development of this thesis. To my friends and family I owe thanks for

the love and support they have shown me at every step of this process. To Bruno Strulovici and Martin

Eichenbaum I owe thanks for helpful comments that guided me along my journey. To Mark Witte and Bob

Gordon I owe thanks for all of the above plus special thanks for their role in my intellectual development as

budding macroeconomist. And to Abigail Jacobs, thank you for 4 wonderful years and counting.

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Spending on durable goods has long been identified as a key feature of business cycle dynamics.

Comprising a significant percentage of GDP and being highly correlated with business cycle fluctuations,

its not difficult to see why a large body of literature has risen up to the task of better understanding

durable goods consumption. Yet despite large strides elucidating some aspects, many problems remain.

Chief among these problems is the difficulty in replicating the empirical dynamics of durable goods

spending in theoretical macroeconomic models.

Many frameworks have been used to address this modeling problem, but two of the most prominent

bodies of literature build off the so called stock adjustment models and the Ss models. After highlighting

some of the pros and cons of these two frameworks, we opt for the greater flexibility granted by the Ss

model framework. From this flexible foundation, we present and examine two extensions. The first

extension will allow us to model the consumption of multiple goods of varying durability and presents a

strong endogenous force propagating the co-movement of nondurables and durables. This endogenous

force could go a long way in fixing the so called co-movement problem, prevalent in the modeling

literature. The second extension will allow us to model the consumption of durable goods under risk.

Some simple empirics will reveal that this extension correctly predicts, contra to the random walk

theory of consumption, that factors such as current unemployment and uncertainty play a significant

role in future durable goods consumption.

Section 1 introduces a brief overview of consumption optimization theory starting with the LCH/PIH.

Section 2 focuses on the stock adjustment literature attempting to incorporate a significant durable

goods component. Section 3 looks at the Ss literature addressing the same topic by re-deriving a simple,

more flexible model developed by Blinder and Bar-Ilan. Section 4 examines some problems with the

standard Ss models and Stock Adjustment models and will highlight the greater flexibility that the Ss

model allows in addressing modeling problems. Section 5 expands the model to account for multiple

durable goods. Section 6 expands the model to account for risk and tests the implications empirically.

Section 7 concludes.

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1 Introduction

The permanent income-life cycle hypothesis (PIH) developed by Franco Modigliani and Richard

Brumberg (Modigliani & Brumberg, 1954) and Milton Friedman (Friedman, 1957) postulate how a

rational economic agent should behave when making consumption decisions. The primary assumption

underpinning the models is that consumers take into account their life-time income and decide how

much to consume each period based on this life-time income and a desire to keep their utility flows

“smooth.” The PIH has been frequently put to empirical test, with evidence both for and against put

forward. Despite the wide appeal of its logic, it still has a few apparent shortcomings when tested

empirically.

One critical limitation of this theory is that it applies only to consumption in a utility sense

whereas the macroeconomic data available is typically of expenditures. The distinction is important

because it implies that consumers can have “smooth” utility paths over time, but “unsmooth”

expenditure paths. For example, take an economy that experiences a large negative wealth shock. As an

extreme example, assume no one buys a new car during the month following the large wealth shock.

This reduction in expenditure is contra to the PIH, but the consumption path is not. This is because

people still “consume” or “use” their old vehicles. In this way, expenditures can be expected to be more

volatile than true “consumption” or “utility.”

When working with non-durable goods, this distinction is of no great concern, since goods

purchased in one period are typically consumed in that same period. For this reason, expenditures on

non-durables tend to be a very good proxy for consumption of non-durables. Thus, empirical evaluation

of the PIH with regards to non-durable goods consumption is fairly straight forward and often favorable

to the PIH (Hall, 1978).

Unfortunately, the evaluation of these theories with regards to durable goods consumption is less

straight forward and, typically, unfavorable to the basic PIH (Mankiw, 1982). This problem arises in large

part from the fact that even if a person doesn’t buy a durable good in a given period, they still use (i.e.-

consume) goods bought in previous periods. Thus, expenditure on durable goods tends to be a poor

proxy for consumption of durable goods. In empirical evaluation, this can be problematic because the

macroeconomic data we observe is expenditure on goods, not consumption of goods.

These modeling problems are magnified by their importance in understanding business cycles due

to the strong, procyclical nature of durable goods consumption (Black & Cusbert, 2010). The prominent

shortcoming of many models of durable goods consumption, coupled with its prominence in business

cycles, has resulted in a rich literature attempting to supplement PIH models with improved dynamics

for durable goods expenditures. The next section reviews one of the initial models that attempted to

reconcile the PIH with a durable goods component: the stock adjustment model.

2 Literature Review: Stock Adjustment Model

One standard model of consumption is the stock adjustment model. These models arose long

initially in theories of private firm inventories. The most famous and comprehensive review and

development of these methods is the famous book Planning Production, Inventories, and Work Force, by

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Holt, Modigliani, Muth, and Simon (1960). The parrallels between firm investments and consumer

purchases of durable goods were quickly realized and the stock adjustment models were soon used to

model consumption.

When dealing with durable goods, these models assume that consumers own a stock of durable

good that depreciates, each period, by some constant factor, δ. Each period, the consumer replaces the

depreciated stock by buying new stock until reaching the desired level of durable good stock. The

transition equation for the stock of goods looks something like:

Where K is the realized stock, K* is the desired stock; the subscripts, t and t+1, denote the time

period. Another assumption that is often made is that consumers only close the gap between their

current stock and desired stock by a constant fraction, γ. This γ makes the model dynamics better mimic

real world data. The simplicity of these models, both in intuitive appeal and implementation, made

them fairly popular; however, despite the wide appeal, there remain a number of questionable

properties.

First, it is often assumed that the speed of adjustment, γ, is constant. The motivation for

including the γ term, is to create “habit formation.” In practice, this means that after a large shift in

permanent income, the rate of change in consumption is large and then declines as we return to the

steady state. Intuitively, this is what allows the nice “hump-shaped response of consumption;” however,

though it is a justifiable term in order to create desirable consumption dynamics that mimic real world

data, it is admittedly “arbitrary” (Christiano, Eichenbaum, & Evans, 2005). Ideally, we would have a

stronger justification for the dynamics of the speed of adjustment.

Second, and more problematic, is the counter-intuitive nature of the model with regards to the

dynamics for individual consumers. Holt, Modigliani, Muth and Simon (1960) show that the SA model

can be justified rigorously by quadratic costs of adjustment; however, as Bliner and Bar-Ilan (1988) point

out, this implies that consumers partially adjust their stocks over several periods. This is problematic

because there is no apparent reason, either theoretical or empirical, why people should adjust holdings

in several small steps rather than all at once. Some attempts to rationalize the assumption of quadratic

adjustment costs can fall flat upon closer inspection. For example, Bernanke (1984) points out that “it

takes time to shop for and acquire a new car”. But this, as blinder and Bar-Ilan point out, implies the

existence of transactions costs, not adjustment costs.

Third, a related problem is, at least on a micro level, the stock adjustment model of durable

goods consumption seems infeasible because, more often than not, consumers cannot buy fractions of

goods. For, example when a TV’s image brightness begins to depreciate or the neighbors the Joneses

get a new 60 inch TV, a consumer cannot go and buy 1/10th of a new TV to restore the brightness or

stretch out the now relatively lowly 52 inch TV.1 Yet, in the stock adjustment model, this is the only kind

1 One could point out that some depreciated goods could be repaired. If a TV controller breaks, one could easily go

out and buy a new controller. This is a fair argument, but one can retort that many repairs are impractically costly (restoring brightness to a TV) or cost more than the good depreciated (e.g.- replacing one plate from a set of

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o f purchase; agents wishing to raise their stock of durable goods can go out and buy whatever fraction

of good they wish. This kind of illogical purchasing mechanism has long been identified in the literature.

For example, Bather (1966) observes that lumpy behavior arises naturally when durable goods

purchases entails a fixed transaction cost. However, the inflexibility of the stock adjustment model led

some authors to create their own frameworks (see for example, Wykoff 1973).

A final problem, and the most relevant to this paper, arises from a difficulty in reconciling the

PIH with the stock adjustment model. In 1978, Robert Hall postulated that if the PIH were true, then

consumption should follow a random walk. Hall declared this hypothesis was supported by a regression

showing that consumption seemed to follow an ARIMA(1,0,0) process. However, this claim would soon

be questioned by N. Gregory Mankiw who would point out that Hall had ignored durable goods in his

theory. In his response to the seminal paper by Robert Hall on the random walk model of consumption,

Mankiw postulates that if the PIH/Life Cycle hypothesis holds and consumption can be modeled via a

stock adjustment model, then expenditures on durable goods should follow an ARIMA(1,0,1) (Mankiw,

1982; Hall, 1978). Unfortunately, when put to empirical test, Mankiw (as well as many researchers since)

observe contradictory results. Where the theory predicts a statistically significant moving average

coefficient, the data seems to reveal a non significant moving average term. Controlling for different

factors, such as income or wealth yields some improvement, but even when the moving average term

appears significant, the empirical data implies implausibly high rates of depreciation, or worse, a

negative rate of depreciation (Blinder & Bar-Ilan 1987). This inconsistency, between theory and data,

has been referred to by some as Mankiw’s paradox and has stimulated a rich literature attempting to

elucidate a solution.

One of the largest branches of literature attempting to resolve the seeming disparity between

the PIH and the incorporation of durable goods replaces the stock adjustment framework in favor of the

more flexible Ss framework.

3 Deriving a Basic Ss Model

Ss models were adopted for modeling durable goods consumption as a response to the weak

micro-foundations present in the Stock Adjustment models. One prototypical example, and the one we

will build upon in this paper, is the Ss model developed by Blinder and Bar-Ilan (1987). Their model is

built upon 2 key assumptions:

a) The market for durables is characterized by important, lumpy transaction costs.2

b) Durable goods have the property of non-combinability (Lancaster, 1979).

Lumpy transactions costs can be attributed to a variety of possible origins, including search costs

(Bernanke, 1984), asymmetric information between buyers and sellers (Akerlof, 1970), or simply the

dinnerware will cost more than the original purchase because of discounting of bulk purchase). It would be suffice to agree that, more often than not, durable goods purchases are large and infrequent. 2 Here, as in the literature, lumpy transactions costs refer to the expenditure on durable goods being marked, on

an individual level, by large, one time shocks (i.e. purchases).

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large and explicit upfront costs associated with durable goods (e.g. down-payments on a new house or

car) (Bar-Ilan & Blinder, 1988). The non-combinability assumption is to prevent marginal purchases that

are acceptable in the typical stock adjustment model. For example, when a refrigerator breaks, we

cannot go buy a fraction of new refrigerator to replace it. Instead, we must either repair the refrigerator

or replace it with a new one.

To get a better understanding of Ss models of durable goods consumption, we now re-derive a

model similar to the one developed by Blinder and Bar-Ilan. The basic mechanism consists of an

individual consumer buying a good of stock value Sn. The subscript n denotes that this is the nth good

purchased. This good depreciates until a lower bound value sn given by:

Here, tn denotes the time at which the nth good was purchased and tn+1denote the time at which the nth

good is sold and replaced by the n+1th good. 3 Additionally, δ is exponential rate of depreciation for the

good. With this transition function of durable good stock, we can proceed to calculate the present value

of the utility gained during the life time of the nth good given by:

Where Un is the discounted lifetime utility of the nth good, u(*) denotes the instantaneous utility of

holding the durable good stock given by (1), and γ denotes the exponential discount rate. Summation

over all lifetime purchases of durables, rearranging, and use of the specific functional form for utility of

aSk will yield4,

(3)

With the life-time present value utility function in place, we need to establish a budget constraint.

Assuming a price p and interest rate r, the discounted cost of the nth good is,

For simplicity, p and r are assumed to be constant. The income gained from selling the nth good at time

tn+1 is assumed to be a fraction q of the remaining stock.

3 Note that this utility function exists in continuous time.

4 This is the same functional form adopted from Bar-Ilan and Blinder (1987).

(1)

(2)

(4)

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The difference between (4) and (5) becomes the net cost of the nth durable good. Summation over all

durable goods yields the following budget constraint.

(6)

where W represents present value of lifetime income. Using (3) and (6) we have enough to infer some

interesting characteristics of Ss models. In particular, the fact that the lifetime utility (3) is homogeneous

of degree k and the lifetime budget constraint (6) is homogenous of degree 1 will allow us to extend this

model to the case of multiple goods and to incorporate the element of uncertainty.

4 Problems Present in the Ss Model and Stock Adjustment Model

Before congratulating ourselves on the success of this model in its micro-founded tractability and

convenient properties (namely homogeneity), we would be wise to point out the shortcomings of this

model. We will see that the Ss framework provides a flexible set of tools to modify the standard Ss

model and address these shortcomings, even if sometimes the solutions to the modified model are not

analytically tractable due to non-linearity. On the other hand, Stock Adjustment model will often leave

us with no recourse other than to ignore these shortcomings all together. The two biggest problems

present with both models are the explicit assumption of a constant depreciation rate and the implicit

assumption of a continuous spectrum of goods upon which consumers are excessively willing to shift

consumption.

Constant Rate of Depreciation

The constant depreciation rate, often represented by δ, has two fairly undesirable implications to

both Stock Adjustment models and Ss models. In particular, the constant depreciation rate forces,

quietly and unsuspectingly, an unreasonable convex path for durable goods stocks to follow. A common

argument for these convex paths is that a good, for example a car, decreases in value as soon as you

drive it off the lot. This is intuitively appreciable, and the example of a driving a new car of the lot

depreciating dramatically in value is commonly accepted without much hesitation. However, while the

resale value may decline dramatically, there is zero reason for the value to the consumer of the car to

depreciate in this fashion.

The first unreasonable implication is that the biggest decline in utility comes in the first month of

owning a good. To observe how unreasonable this is, continue the example of automobiles. At best,

one could argue that as soon as the “new car smell” disappears, people are suddenly struck with a

(5)

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severe reduction in satisfaction.5 Of course, this happens in the first month. Do we expect a similar

reduction in happiness during the second month? A constant δ implies that we should. Perhaps we

expect a concave depreciation function instead of a convex one. After all, many goods are still quite

useful up until the day they break down completely and have to be replaced. The second implication of

a constant convex depreciation rate is that more expensive goods last proportionally longer than their

inexpensive counterparts. It is perhaps quite reasonable to accept the fact that more expensive goods

are more durable, but to what degree should we hold this assumption. For example, do we expect a

$2,000 laptop to last much longer than a $1,000 laptop of the same brand, style, and series? Almost

surely not.6 In a more extreme example, more expensive goods can have a “faster” depreciation rate.

For example, a refrigerator with an ice dispenser is actually more likely to break earlier than one without

since the ice breaker is a relatively fragile component.

How do the Ss and Stock Adjustment models fare in accommodating more flexible depreciation

rates? In the Ss model we can replace the initial equation (1) with a new path of depreciation:

Where,

t is the time that has passed after the initial purchase

D is the total expected duration of a good. When t>D the good provides 0 utility.7

S0 is the stock of a durable good at the initial purchase

St is the stock of a durable good at time t after the initial purchase

θ is an arbitrary parameter the modeler can set. We assume this is greater than 0.

Note that many desired properties are satisfied by this depreciation function. The function is in

continuous time. The function concavity can be adjusted at will by changing the θ parameter to greater

or less than 1. (Note that if θ>1, the function is convex as is typically assumed. If θ<1, the function is

concave.)8 The duration a good lasts can be adjusted simply by adjusting D. Finally, the function is

independent of the value of S0, our initial stock of good. Solving an Ss model with this transition function

5 The author was surprised to find that you can purchase an aerosol can of “new car smell” for just over 3 dollars.

This has other implications for the implicit value of δ. 6 The astute reader may retort that these are not the same product, and thus their durability is not meant to be

compared. But with regards to the consumer’s utility function, I would argue that they are effectively the same product as they present largely substitutable goods. Suppose a consumer has a need for 1 laptop, then the $2,499 and $1,199 MacBook Pros are substitutes for the same need, but we expect the differences in processing power and memory to be independent of the durability of the laptops. 7 In the worked examples we will always assume that goods last long enough that a consumer can purchase a new

good before the old one yields 0 value. This assumption is made for simplicity but should not affect the principle results in any appreciable way. 8 Another type of model that appeared in the 1960s was the “one hoss shay” model. This model would postulate

that durables would last a specified period of time and then depreciated all at once. Though we do not enter a discussion of this model at this time, it may be usefull to point out that the depreciation function just presented converges towards the “one hoss shay” depreciation as θ approaches 0.

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is not analytically tractable, but some intuition on dynamics can be gleaned. On the other hand, the

Stock Adjustment model leaves us completely in the dark since it necessitates a constant rate of

depreciation from one period to the next. In particular, you cannot “follow” the depreciation path for a

single durable good.

The second problem present in both SA and Ss models is the implicit assumption of a continuous

spectrum of goods upon which consumers are excessively willing to shift consumption. This is often

overlooked assumption has dramatic implications for the plausibility of a number of models. First, let’s

clarify more formally what we are addressing and why the assumption exists. In both, SA and Ss models,

consumers can decide to consume any level of good along a continuum of goods. Typically, the goods

can be any real number greater than 0. So if a consumer wants to, she can literally optimize her

consumption down to a penny’s worth of marginal value. However, it seems obvious that when a

consumer observes desirable goods, there is a rather large discrete jump between one durable goods

and the next “better” version of the good. No retailer contains a continuum of durable goods; even if

such a continuum of goods existed, no retailer is large enough to hold a continuum of durable goods in

stock. But the assumption exists because of the mathematical convenience and seemingly innocuous

nature of the assumption. After all, consumers do have a fairly large number of choices.

The most dramatic implication of this in Ss models is that, in equilibrium, people hold goods for

the same amount of time, regardless of permanent income. So if a person has been planning to hold a

new car for 4 years and then experiences a negative wealth shock she must “downgrade” to a less

expensive model to hold for 4 years. Holding the more expensive car for 5 year intervals to save money

is not an option in the model, though intuitively we might expect it to be the case in reality. Making the

set of available options discrete is doable in an Ss model. The simplest case where there is one good

shows that all consumption decisions become a function of simply how long to hold a good before

replacing it. Extensions to n discrete options are also feasible, but presents analytic problems, namely, it

becomes more difficult to keep convexity of the decision set. On the other hand, the Stock Adjustment

model again leaves us in the dark with no easy way to optimize the amount of time a good is held. This is

in large part due to the discrete nature of time steps in the Stock Adjustment model and partly due to

the inability to track individual durable goods.9

These two shortcomings of the basic Ss model illustrate one of the most desirable properties of

the Ss model: adaptability. Many authors have taken advantage of this adaptability to incorporate their

own elements into the model, from Janice Eberly’s incorporation of uncertainty dynamics (Eberly, 1992)

to the model I will present in section 5 incorporating multiple durable goods of varying durability.

(Note: Appendix 1 has a section discussing another more general problem of the Ss framework and the

need to distinguish between cyclical and non-cyclical dynamics.)

9 Recall that new durable goods purchases are combined with leftover durable good stock. The new stock then

depreciates at a rate independent of whether the goods in the stock are old or new.

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5 Extension of the Model to Account for Multiple Durable Goods

In this section, I show that the Ss model developed above can be extended to an arbitrary number

of goods and that this extension yields some very important results. In particular, I will show differing

dynamics in the short and long run. In the short run, an optimizing agent will respond to a wealth shock

by greater changes in consumption of more durable goods relative to less durable goods. In the long

run, an optimizing agent will change consumption of goods by equal amounts. This distinction will

present a possible solution to the co movement problem present in many macroeconomic models. I

begin with a proof of the two durable good case and then use induction to prove the extension to N

durable goods.

A. Theory

The Case of 2 Durables

Consider a consumer that maximizes the consumption of 2, additively separable durable goods. The

utility function for each good assumes the form of a standard Bar-Ilan utility function for a durable good.

Let SD be the stock level of a durable good when purchased. Let SL be the stock of a less durable good

when purchased (The rate of depreciation for SL is higher than for SD). Thus, the utility function for the

consumer is

(7)

where U(*) is the standard Bar-Ilan utility function for a durable good. If we observe the utility function

in equilibrium we know that the standard Euler condition should hold, i.e.

(8)

where WD is the permanent wealth dedicated to consuming the SD good and WL is the permanent

wealth dedicated to consuming the SL good. Now apply the chain rule to get:

(9)

Now assume that a wealth shock forces us to a new equilibrium. Let α be the factor by which SD

changes and β be the factor by which SL changes. Note that both α and β have to be greater than 0. Our

new Euler condition is the following:

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(10)

Recall that U(*) is Homogeneous of degree k in S. Also, note that S is homogeneous of degree 1 in W.

As a consequence of Euler’s Homogeneous function theorem, if a function, f, is homogeneous of degree

k, the derivative of the function, f’, is homogeneous of degree k-1.

(11)

Note that the only way to maintain equality going from (9) to (11) is for α=β. Since α and β represent

the same shift in consumption of each good and since the budget constraint is homogeneous of degree

one in the amount of good bought, this implies that amount of wealth dedicated to each good declined

by the factor α=β.

The Case of N Durables

The case for N Durables can be proven via induction. Take A, B, and C, as the subscripts for each of 3

goods. We can re-write (9) as,

(12)

Assuming some wealth shock we get,

(13)

We can prove, using our proof for 2 goods, that any pair wise comparison of and will show they

are equivalent. Its easy to see how to extend the model to an arbitrary number of goods.

B. Implications and Data

(Note: Appendix 1 has a section discussing another more general problem of the Ss framework and the

need to distinguish between cyclical and non-cyclical dynamics. Interpreting the implications of this

extension will be easier after understanding Appendix 1)

Intuitively, the above proofs predict an X% wealth shock to permanent income will result in an

X% shift in the equilibrium (i.e.- long term) expenditures of each durable goods, regardless of their

differing durability. However, as in the standard one good model, there exists a transition phase that

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causes short term dynamics to differ from these long term equilibrium dynamics. In particular, note that

an X% reduction in wealth dedicated to durable good results in a greater delay the more durable the

good is. To see this, note that an X% reduction in expenditure translates to an X% reduction in the

replacement trigger level. The time it takes a durable to depreciate from the old replacement trigger to

the new replacement trigger is the “extra time” between purchases (Bar-Ilan, 1987). How long it takes

for the good to depreciate this extra X% is a function, exclusively, of the depreciation rate (i.e.-

durability) of the good.10 The more durable the good, the longer it will take to shift to the new lower

bound. As a result, the short term will experience a larger decline in consumption of more durable goods

relative to less durable goods.

One interesting and illuminating case to look at is where one good is durable and one good has

such a high rate of depreciation that it is effectively a non-durable good. The durable good behaves, as

described above, with dramatic, short-term dynamics differing from steady long-term dynamics. The

non-durable good behaves the same way except that the short-term dynamics are essentially eliminated

because it takes virtually no time to reach the desired replacement trigger level. As a result, the

consumption of the non-durable good is almost exclusively determined by the desired S. Bar-Ilan and

Blinder show that the desired stock follows a random walk. As a result, this model predicts that the

durable component will have significant moving average components while the non-durable good

follows something more akin to a random walk. This is favorable to the findings of many authors such as

Robert Hall and Greg Mankiw.

The fairly strict reliance on the functional form of utility and the constant rate of depreciation

lead us to question the results. Its unlikely the above results are a perfect representation of consumer

optimizations; however, its worth asking if the core dynamics are supported by the empirical evidence.

The dynamics described above do seem to be supported by the data, at least on a superficial

level. The BEA’s NIPA tables separate durable goods expenditures into four main categories: “Motor

vehicles and parts,” “furnishings and household equipment,” “recreational goods and vehicles,” and

“other durables.” Looking at the log changes for each of these categories can give us an idea of how

volatile these components are in percentage terms. Table 1 shows summary statistics for the log

changes in the durable goods categories listed above as well as for “clothing and footwear”,11

“nondurables”, and “food and beverages”. Roughly speaking, these deviations fall in line with our

expectations as more durable goods experiencing greater volatility than less durable goods; however,

solid conclusions are difficult to derive because of the inconvenient nature of the data.

For example, two categories that may fall somewhat out of line from our expectations are

“Motor Vehicles and Parts” and “Clothing and Footwear” which are, respectively, more volatile and less

volatile than our model might suggest. Since automobiles are the most expensive category in our list, we

might expect the excess volatility in automobile expenditures to be a function of cost. Indeed, Eberly

(1990) estimates that about half of automobile consumers face binding liquidity constraints. This could

add increased responsiveness to wealth shocks and thus increase volatility in automobile expenditures.

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This is, in turn, a function of the constant rate of depreciation. Otherwise, the level of the durable would need to be taken into account. This leads us to question the validity of the model, still, it is nice to know the dynamics are roughly in line with intuition and expectation. 11

The BEA categorizes “Clothing and Footwear” under the non-durable.

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The other unusual category is “clothing and footwear.” Those of us who have certain articles of clothing

for several years may think of clothes as very durable goods. However, we should remember that

around a quarter of clothing expenditures comes from clothing for children under 16 who outgrow old

clothes on a regular basis. On top of that, around 50% of clothing expenditures comes from women’s

expenditures (leaving the last 25% for men). It’s likely that cultural norms of style and fashion result in a

higher turnover in clothing amongst women since last year’s clothing may be out of fashion, or wearing

the same dress to multiple events can be seen as a fashion faux pas.

Additional problems interpreting Table 1 arise from the mixed bag that is inherent in each

category. For example, “recreational goods and vehicles” has items ranging from tennis rackets to jet-

skis; it is impossible to determine the effective “durability” of this category without more information.

Even then, the exercise would not be easy. However, at least in the categories where durability is

comparable, in particular the durable category versus the nondurable category, the dynamics seem to

be appropriate.

On the other hand, support for the long run dynamics implied by the model seems to be less

favorable. In particular, the core conclusion that the distribution of income among goods remains equal

is off the mark. In particular, Table 1 demonstrates that, relative to other goods, basic necessities such

as food and clothing has risen at a slower rate than “luxuries” such as recreational goods and vehicles.

This means that, over time, the percentage of total expenditures spent on basic necessities has

decreased. This is in part due to the increasing total wealth in the US. Additionally, consumer surveys

have shown that there exist some general similarities on a micro level in how people spend their money

but that people of differing wealth levels do tend to partition their income differently. The most likely

explanation for this discrepancy is the model’s identical utility function for all goods. This assumption

was central to the appeal to the properties of homogeneous functions but also relatively unrealistic.

Despite these shortcomings, this simple extension to multiple goods provides a powerful

mechanism by which an economy can simultaneously reduce consumption of durable goods and

nondurable goods. In particular, the relatively greater volatility of durable goods to nondurable goods is

consistent with the empirical data and may remain despite relatively greater changes in prices for

durable goods.

6 Extension of the Model to Account for Uncertainty (well, risk

actually)

One important factor that seems to be missing from the standard Ss model is a way to incorporate

uncertainty. In principal, the Permanent income hypothesis implies that all relevant information for

predicting future consumption is already incorporated into the last period’s consumption. This is known

to many as the Random Walk12 hypothesis postulated by Robert Hall (1978). Models that incorporate

durable goods, such as those derived by Mankiw(1982) and Bar-Ilan(1987) also predict statistical

significance of lags of income. Empirical tests presented by Bar-Ilan and Blinder(1988) seem to concur

12

Technically the process is a martingale not a random walk, but the literature usually refers to it as a random walk so we do the same here.

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with this assessment. However, these models are both deterministic. This section examines

incorporating risk to extend the basic Ss model to explain why uncertainty should, at least theoretically,

play a significant role in predicting consumption of durable goods. Empirical tests will then show that

these components are, in fact, statistically significant. A story similar to this one is used as an

explanation behind the empirical correlation between unemployment and consumption in much of the

literature on precautionary savings (Malley & Moutos, 1996). This section differs in that we use the Ss

framework to capture these dynamics specifically for durable goods.

A. Theory

To simplify the discussion, we will discuss the hypothetical dynamics of risk during a recession.

We assume, for simplicity, that at any given moment, an agent believes he has a given probability of

becoming unemployed and suffering a large, negative wealth shock. In particular, the agent thinks of the

event of becoming unemployed as a Poisson process. We could also assume that the agent’s underlying

Bayesian probability in this poison process is changing, perhaps in response to the changes in

unemployment. If an agent sees many people around him being fired, he may think he is more likely to

be next. If an agent sees fewer people around him being fired, he may think that his own risk of

unemployment is diminished.

This Poisson process thus leads our agent to the unfortunate realization that the optimal

consumption decision depends on the state of the world. In one state of the world, he remains gainfully

employed and so should not alter his consumption path; in the other state, he becomes unemployed

and should target a new consumption path. Let SE be the target stock our agent would prefer to buy if

he is employed and let SU be the target stock our agent would prefer to buy if he becomes unemployed

such that SU<SE.

Before adding the uncertainty term, let SE=SU*α where α>1.

The following represents the utility function the agent if he buys a good according to his original

schedule:

+

(14)

Where is the extra reduction in permanent wealth due to misallocation by buying high. The following

represents the utility function of the agent if he decides to wait and only replace the good once it

reaches the low threshold.

(15)

15

+

Where is the extra increase in permanent wealth starting after the first purchase due to the

insufficient consumption in the first period.

These equations essentially say that in the case were the agent knows he retains his job, he prefers to

buy according to his original planned income. In the case, were the agent know he will lose his job, he

prefers to wait before buying a new durable good. As the probability of losing one’s job increases, the

expected utility of waiting increases relative to the expected utility of proceeding as normal.

So we have some support, at least on a theoretical level, that uncertainty of this flavor should be

important in determining future consumption. This theoretical finding may not be intuitively hard to

swallow but it is at odds with common theories of consumption, most notably Mankiw and Bar-Ilan and

Blinder’s deterministic models.

B. Old Regressions with Spurious Results

As a baseline, we reproduce the standard simple auto-regression run by Bar-Ilan and Blinder (1992).

They use disposable income data from the BEA to examine what information lags of income can tell us

about durables goods consumption. They do this by simply regressing number of new cars bought on

one lag of number of cars bought and multiple lags of disposable income to get something like:

(55.65) (1.64) (-0.32) (0.31) (-2.58)

(16)

Though initially promising, a closer look reveals that this relationship is not consistent over time. This is

most evident when we divide the data into 2 parts. Looking at the first half of the data, from 1967-1990

we get,

(22.54) (1.91) (-0.77) (-0.31) (-0.55)

(17)

Alternatively, looking at the second half of the data, from 1990 to 2010 we get,

(25.50) (0.51) (0.96) (1.23) (-4.48)

(18)

A Chow test strongly rejects the null hypothesis that coefficients are the same for both time periods.

Another indicator of instability, as opposed to a one time structural change, comes from figure 4. After

running the above regression on a rolling, 10 year window of data, the coefficients of each lag of income

are stored and plotted in figure 4. It is fairly evident, even to the naked eye, that the coefficients are

16

fairly unstable. Though some of the instability seems correlated to recessions, a lot of the instability

seems random or difficult to attribute to a particularly notable economic event. On more than one

occasion, we can even see the coefficients change signs. We can choose a handful the sharp changes in

coefficients as natural places to “divide” the data so that we can run a series of chow tests. All the Chow

tests strongly reject the null hypothesis that the coefficients are the same for each time period.

A potential cause for this instability is that the model is poorly specified. For example, Table 2

illustrates more clearly the existence of parameter redundancy (Shumway & Stoffer, 2010). Table 2

shows that no matter how many lags of income are added to the regression, the sum of the coefficients

for each lag summate to about (-.06). At the same time, the statistical significance attributed to each lag

is diluted as more lags are added. This, coupled with the strongly autoregressive nature of income

implies a miss-specification of the model. In particular, little predictive information is added by having

more than 1 lag of income (or at most 2). Further, this coefficient of (-.06) implies a negative correlation

between disposable income and consumption of automobiles. This is completely contrary to any theory

of consumption, Ss or otherwise. This spurious result arises from the steady increase of real disposable

income over time coupled with the steady decrease in the number of automobiles purchased over time,

as discussed in Appendix 1. Since income levels are clearly the wrong way to incorporate income into

our regressions, we will choose to incorporate the first difference of income in future regressions. This

falls more closely in line with accelerator models of consumption than with the permanent-income

hypothesis, but its incorporation will be interesting since it will be significant until elements of

uncertainty are added to the model. Regardless, we must keep these complications in mind when

incorporating new elements into our regressions.

Another spurious regression claiming to validate a theoretical model comes from Ricardo

Caballero’s (1990) paper on the expenditure of durable goods. Tam Bang Vu (2006) presents some

hesitations with Caballero’s results. Caballero (1990) hypothesizes that the underlying reason for the

lack of a significantly negative Moving Average coefficient in Mankiw’s ARIMA(1,1) process lies in the

consumers' slow adjustment of their durables expenditures. Caballero then runs an ARMA(1,5) and

ARMA(1,8) processes to test this theory. He finds that the sum of moving average effects is statistically

significant although all individual moving average effects are insignificant. However, as Vu points out,

Caballero does not explain why the slow adjustment results in statistical insignificances of the individual

MA coefficient estimates but the statistical significance of their sum. Moreover, the MA(5) and MA(8)

components in his models are not equivalent to the missing MA(1) in Mankiw's ARMA(1,1) model.

Vu is right to question these results, particularly given the lack of theoretical support and heavy

reliance on intuitive appeal. Unlike the Blinder & Bar-Ilan regression from above, the Caballero

regression makes no claims as to the stability of the coefficients, noting in fact that only the summation

is of importance. According to the intuition Caballero presents, each additional MA term should bring

value of the sum of the coefficients of the MA terms closer to -1. He runs an ARIMA(1,8) and shows that

the sum of the MA terms is -.51 and is statistically significant. He argues that despite still being far from -

1, this sum of coefficients provides evidence of slow adjustment in durables. Unfortunately, Caballero’s

regression has not stood the test of time.

Table 3 runs multiple ARIMA process on the log of durable goods with a varying number of MA

terms. The rightmost column shows the summation of these coefficients. The first thing to notice is that

17

the addition of 20 years of data has made these coefficients positive. The second thing to notice is that

the addition of more coefficients causes the sum of coefficients to increase, rather than decrease.

Neither of these facts agrees with the results Caballero presents.

The above two examples of spurious regressions are indicative of two large problems in the

literature on modeling durable goods. First, the empirical evidence is often at odds with several

theories, even when it seems like it may be supportive. Second, the literature often misses these

discrepancies due to either a lack of sufficient data, or empirical “tests” that are not rigorous enough.

With these problems fresh in our memory, we will test our own theory’s empirical theories.

C. Regressions Incorporating Unemployment and Uncertainty

The first question we ask is in the vein of the literature: Do multiple lags of unemployment affect

new automobile purchases? The standard Permanent Income Hypothesis model says no, but the model

we derived incorporating uncertainty says that both the change in unemployment from month to month

as well as, perhaps, the change in change is important. As a result, the model would predict that at least

the first difference in unemployment be statistically significant. Below is this regression with z-values in

parenthesis.

(142) (1.50) (2.08)

(-4.01) (-0.29) (-25.40)

(19)

As the model predicts, the first difference of unemployment, denoted by ΔU, is significant at the 1% and

negatively correlated. The second difference of unemployment, denoted by ΔΔU, is oddly not significant,

though the level of unemployment is. Additionally, the change in income is significant at the 5% level.

Changes in income seem to be insignificant once the unemployment terms are added. Perhaps the

biggest surprise in this regression is the highly significant and highly negative Moving Average

coefficient. Though not quite as close to -1 as Mankiw and others desire, this Moving Average coefficient

is much larger than previous studies have shown.

We may ask if this unemployment data captures all the consumption changes resulting from

uncertainty or risk. One way to test this is by using the Michigan Consumer Sentiment Survey which asks

respondents, among other questions, is it “a good time to buy large durable goods” and why. Among the

reasons people list for not buying large durable goods is “uncertain future.” The percentage of

respondents citing “uncertain future” as a reason for not buying durable goods is thus stored each

month in a convenient monthly time series. Thus, it should be fairly easy to test to see whether the

percentage of people reporting uncertainty as a factor in not buying large durable goods is important in

18

predicting consumption of large goods even after. Removing the unemployment terms and replacing

them with the percentage of people reporting uncertainty about the future yields:

(72.30) (1.76) (-1.46)

(-25.34)

(20)

Curiously, uncertainty is statistically insignificant at the 10% level and its coefficient is decidedly very

small. The regression results imply that a 1% rise in respondents listing uncertainty as a cause for

postponing purchases of big ticket items would lead to only a 0.2% decrease in the number of car units

sold. Adding back in the unemployment data yields little new information.

(140) (1.46) (-.27)

(1.85) (-3.63) (-0.26) (-25.15)

(21)

Though the theoretical model presented at the beginning of this section predicted that the first

difference in unemployment would be significant in predicting unemployment, it did not predict that the

“uncertainty variable” discussed above would be statistically insignificant. This leads us to question

whether the motivation in the model was well placed.

One problem may be that the Michigan Consumer Sentiment Survey may not have most

accurate data. People’s declarations of uncertainty may be different from their own decision process.

Additionally, uncertainty is not a binary state as in the survey. The theory points out that one can be

very uncertain about the future and greatly postpone the purchase of goods, or one can be moderately

uncertain about the future and not postpone as much.13 Another problem may be the lack of

persistence in uncertainty relative to the first difference of unemployment. Simple autocorrelations and

partial autocorrelations show that the first differences of unemployment are more positively correlated

at longer lags than the first differences of uncertainty, which are in fact, slightly negatively correlated.

These characteristics of the uncertainty data may lead us to believe that changes in unemployment are

better indicators of the true uncertainty felt in the economy. On the other hand, it is distinctly possible

that changes in the unemployment are not signaling uncertainty, but the shift of people from

uncertainty to the certainty of a negative wealth shock. In this way, the unemployment data may be a

proxy for people’s inability to buy new goods rather than their desire to postpone consumption due to

uncertainty.

13

These concepts correspond to changing values of λ, or changing values in the underlying probability of the Poisson process.

19

Though these questions plague the theory, we may also ask how the regressions look for the

general class of durable goods. After all, it is likely that automobiles are more closely tied to variables we

are not observing in the model such as the availability of cheap credit or other financial constraints.

To do this, we need only to replace the number of units bought with durable goods expenditures. The

durable goods consumption looks like:

(838) (2.53) (-1.92)

(2.15) (-3.86) (-.02) (-16.52)

(22)

Perhaps because of the differing dynamics of various durable goods (as demonstrated in the Ss model

expansion to multiple goods), the same ARIMA model that worked so well for predicting new car

purchases falls short in predicting aggregate expenditure on durable goods. In particular, the coefficients

for the difference terms for the unemployment terms shrink dramatically and only the first difference of

unemployment and first difference of income are significant of at the 5% level. The uncertainty term

becomes statistically insignificant. The only desirable feature is that the moving average is still significant

and still more so than in much of the literature.

Fortunately, applying the model to nondurable goods consumption fails even worse:

(922) (-1.08) (000)

(-1.19) (-1.21) (.73) (-.55)

(23)

In this case, only the autoregressive lag of nondurable consumption in the previous period is significant.

Even the moving average term becomes irrelevant.

7 Conclusion

This paper examines two extensions of the Ss framework. The first extension provides a simple,

analytically tractable framework for understanding the co-movement of durable and nondurable goods

consumption. This framework could be particularly useful in attempts to resolve the co-movement

problem prevalent in much of the macroeconomic modeling literature. The second extension provides a

simple framework for understanding the role of uncertainty in delaying durable goods expenditures.

20

Though the data supporting these results is generally supportive, it fails to convince us that these

extensions are accurate representations of the empirical dynamics. Instead, the important contribution

of these extensions is in their mathematical convenience owed to simplicity and flexibility.

Future research could attempt to incorporate these dynamics into larger macro-models; only then

could we evaluate their true utility. Additionally, there exists a growing literature that attempts to

consider the financial constraints of consumers in durable goods markets. Things like access to credit

and interest rates have significant effects on a consumer’s set of available options, and the flexible Ss

framework seems like an ideal foundation upon which to add these frictions.

Tables & Figures

Table 1. Summary Statistics for Quarterly Log Changes in Various Consumption Categories, 1947Q1-2011Q2

Variable Mean Standard Deviation Furnishings and Durable Household Goods

1.3% 3.3%

Motor Vehicles and Parts 1.6% 7.0% Durables 1.6% 3.6% Recreational Goods and Vehicles 1.9% 2.8% Other Durable Goods 1.6% 2.4% Clothing and Footwear 1.2% 1.6% Nondurables 1.4% 1.2% Food and Beverages 1.2% 1.0%

Table 2. Number of New Cars Regressed on Multiple Lags of Income , 1967-2010 Coefficient on

Consumption Coefficient on Lags of Income Sum of

Income Coefficients

Lag 1 Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 N = .861 -.059 -.059 N = .857 .597 -.660 -.063 N = .855 .691 -.236 -.518 -.063 N = .852 .826 -.185 .161 -.863 -.061 N = .847 .903 -.134 .177 -.403 -.607 -.064 N = .848 .898 -.143 .173 -.408 -.673 .089 -.064

21

Table 3. Log of Durable Goods Regressed on Multiple Moving Average Coefficients, 1947-2010

Coefficient on Consumption

# of Moving Average Coefficients Sum of Moving Average Coefficients

Lag 1 N = .999 4 0.395 N = .999 6 0.540 N = .999 8 0.541 N = .999 10 0.790 N = .999 12 0.909

0

2000

4000

6000

8000

10000

12000

0

5000

10000

15000

20000

25000

30000

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Figure 1. New Car Consumption, 1967 to 2010

Thousands of CarsBought (Right Axis)

Average Cost of Cars Bought (Left Axis)

22

Figure 2. Dynamics of New Vehicle Consumptions During Recessions

23

0

50,000,000

100,000,000

150,000,000

200,000,000

250,000,000

300,000,000

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

0.0

0.5

1.0

1.5

2.0

2.5

Figure 3. Number of Vehicles Registered, 1967-2010

Number of Vehicles perHousehold (Right Axis)

Total RegisteredVehicles (Left Axis)

24

-4

-3

-2

-1

0

1

2

3

1975 1980 1985 1990 1995 2000 2005

Figure 4. Rolling ARIMA Coefficients (10 Year Window), 1975 m1-2010 m12

Lag 1 Income

Lag 2 Income

Lag 3 Income

Lag 4 Income

25

Appendix: The Importance of Non-Cyclical Components in Durable

Goods Consumption

Before testing new extensions of the Ss model, it is important to pause and discuss an often

overlook, yet important characteristic of durable goods consumption data and its implications for

attempts to model it. In this section, we will distinguish between cyclical and non-cyclical components of

consumption. This distinction will allow us to better understand and test the theoretical implications of

our Ss models against empirical data.

Figure 1 exemplifies the types of incomplete arguments used to justify Ss models over Stock

Adjustment models. At first glance, these graphs seem supportive of the Ss models since it seems

apparent the number of units sold is more volatile than the average price paid, just as the Ss model

would predict.14 This fact is actually pointed out by several authors as favorable to the Ss model. For

example, Bar-Ilan and Blinder in particular point to the statistical significance of number of units in

predicting future consumption compared to the statistical insignificance of average expenditure per car

(Bar-Ilan & Blinder, 1988). However, this figure presents a very distant view of the dynamics at work.

Figure 2 can be thought of as a case study illustrating the less apparent nature of the response

of vehicle consumption during recessions. The green line is expenditure. The red line is average

expenditure per car bought. The blue line is the number of cars bought. Each graph shows a recession

plus the 6 months before and the 6 months after the recession. All numbers are indexed to 100 at 6

months before the recession. The most interesting, and perhaps surprising, element in these graphs is

the number of new cars purchased drops down to around 80% of the pre-recession levels and sits there

for several months at a time before rising again. This dynamic actually presents a problem for

interpreting this phenomenon with Ss models.

The problem is that standard Ss models can only explain these persistent drops in the number of

units sold in one of two ways. First, a sustained drop could be the result of a series of bad news that

“surprises” optimizing agents each month. In this way, some fraction of people would increasingly delay

their purchases each month. This seems unlikely though, as these declines would need to “surprise”

agents each period even well after a recession has ended when one would imagine some notion of

stability has been achieved. A second explanation for a sustained drop in the number of units bought is

the existence of a very large and negative wealth shock such that no one buys cars for a very long time.

This is more intuitive to interpret with the Ss model, but is also problematic since the number of cars

bought drops, even in the deepest of recessions, only to 60% of pre-recessionary levels instead of the

0% predicted by the model.

14

Back of the envelope calculations seem to indicate that about 80% of the change in expenditures on new cars is a reflection of changes in the number of new cars bought whereas 20% seems to be a reflection of changes in the average expenditure per car.

26

To help reconcile this phenomenon with the Ss framework we introduce the idea that

consumption of durables can be deconstructed into a cyclical component and a natural, “built-in”

component. This “built-in” component can be thought of as the purchases that are made to replace cars

that are no longer usable and must be replaced. The idea of a “built-in” deterioration, separate from

cyclical factors, is not new and was discussed by Richard Parks in the late 70s (Parks, 1977); (Parks,

1979). This view attempts to capture potential changes in the durability of assets over time (e.g.-

different car model years would have different durability). Alan Greenspan and Darrel Cohen wrote a

paper that attempted to tease out more carefully the “built-in” component15 from the cyclical

component and make better predictions of future vehicle scrappage rates16, sales, and stocks

(Greenspan & Cohen, 1996). Variants of these methods could be applied to our data, but the purposes

of this paper, it will suffice to believe that some fraction of durable goods consumption is a result of

goods requiring replacement and thus outside the realm of agent optimization problems and a cyclical

component which can be approximately modeled by agent optimizing Ss models.

A common first reaction to calculating this “non-cyclical” rate of consumption of automobiles is

to look at scraping rates. The scraping rate is the rate at which old cars are taken out of circulation and

torn apart for spare metals. Unfortunately, several authors, most notably (Greenspan & Cohen, 1996)

and (Parks, 1977), have pointed out the very pro-cyclical nature of scrapping rates. This is indicative that

scrapping rates are not just vehicles breaking down because of age. Indeed, the decision to scrap a car

is, more often than not, made by used car dealers whose inventories of used cars grow too large. In this

way, when times are good, older cars are more likely to be scrapped as more people buy new cars, but

when times are bad, older cars are more likely to continue being used as people postpone purchasing

replacements.

In our data set, the “built-in” component can be seen most clearly in viewing the decline of the

number of new cars bought in figure 1 against the rise of the total stock of vehicles in figure 3. The

declining rate of new vehicle consumption from 1970-2010, coupled with the fact that the stock of cars

in the United States has grown at a fairly consistent nominal rate from 1970-2008, indicates that fewer

new vehicles are needed to maintain the same growth of total vehicle stock. This is indicative of

increased durability of vehicles held. Other factors that may contribute to the decline in number of new

cars bought could be a saturation of the market for vehicles. This can be seen in the non-linear shape of

the number of cars per household plot in figure 3. With these factors in mind, we can see the motivation

behind the idea of slow changing underlying trend component.

To tease out the non-cyclical component, we assume that there exists some natural rate of

consumption, unaffected by cyclical factors. This component is, intuitively, composed of both the “built-

in” depreciation forcing people to buy new cars, but also a certain component which is people who are

in financially stable positions and do not respond as dramatically to cyclical events.17 Noting the

downward trend in new cars bought, we attempt to account for this drift by applying a Hodrick-Prescott

filter. We use a very strict λ to account for the fact that we do not expect the durability of vehicles to

decline dramatically from year to year. Using this filter we can detrend the data on new cars. For

15

Greenspand and Cohen call this the “engineering” component. 16

Scrappage rate is the rate at which cars are taken out of circulation and torn apart for spare parts or metals. 17

This is not to say that the second point is very important. It is only worth pointing out because it reminds us that the natural rate of consumption is not only a product of forced consumption.

27

robustness, several different trends could be deducted including various Hodrick-Prescott filters and

even OLS lines of best fit; however, they all yield similar results.

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