It is better to have a permanent income than to be fascinating. Oscar Wilde ( )

Transcription

1 6. CONSUMPTION AND TAX SMOOTHING It is bette to have a pemanent income than to be fascinating. Osca Wilde ( ) The hadest thing in the wold to undestand is income tax. Albet Einstein ( ) This section focuses on theoies descibing aggegate consumption spending and how it is elated to income and inteest ates. Consumption is an impotant component of aggegate demand, and hence cental to ou undestanding of business cycles. As we ll see, one of the leading theoies of how households consume also can be thought of as a theoy of how govenments set taxes. (a) Budget Constaints and Pesent-Value Accounting We ll begin with some popeties of dynamic budget constaints and the notion of pesent-value budget balance, since it is elevant to lifetime budgeting. Conside the sequence budget constaint: a t+1 =(1+)a t + y t c t =(1+)a t + s t (1) whee thee is no uncetainty, is constant, and a is wealth, y is income, c is consumption, and s is saving. Note that y t excludes inteest income, which is given by a t. Sometimes we wite t fo the inteest ate applying fom peiod t to t + 1, although one might want to alte that notation to t+1, depending on the natue of the specific financial investment. To avoid confusion, let me mention that thee ae othe possible ways to wite down the timing in the budget constaint. A second one is a t+1 =(1+)(a t + y t c t ) and a thid one is: a t+1 =(1+)a t + y t+1 c t+1. Technically, which one you use depends on the time at which the vaiables ae measued, but nothing in ou economic findings depends on which one we use, as long as we use it consistently. Solving this diffeence equation gives the coesponding pesent-value budget constaint. Solving backwads gives: a t = (1 + ) i s t 1 i (2) i=0 227

2 povided that lim i (1 + ) i a t i = 0, which we shall assume. wealth is sufficient fo this limit to be zeo. Stating life with zeo An altenative is to solve the diffeence equation the othe way ound. This gives, a t = (1 + ) j 1 s t+j (3) j=0 povided that lim j (1 + ) j a t+j = 0, which we shall assume. This tansvesality condition ules out unlimited lending o boowing (bubbles) and hence dying in debt. Equation (2) simply says that cuent wealth aises fom past saving. Equation (3) says that cuent wealth can be used to finance futue dissaving. It is feasible (satisfies the budget constaint) to consume moe than income (i.e. c t+j >y t+j ) in some futue peiod if a t > 0. In that case s t+j < 0so s t+j > 0 which is consistent with a t > 0 in equation (3). So equation (3) is the pesent-value constaint, conditional on cuent assets. Combining the two esults gives: (1 + ) i 1 s t+i =0, (4) i= which is the lifetime pesent-value budget constaint. The same accounting applies to a govenment. Suppose that b t is govenment debt o bonds outstanding, and s t is the pimay suplus, which equals t t g t, whee t is evenue and g is spending. The standad budget constaint o financing identity is: b t = b t 1 + g t 1 t t 1 so that spending and inteest payments in excess of evenue must be financed by issuing new bonds. This becomes b t+1 =(1+)b t 1 s t 1 since a suplus educes liabilities. Note the diffeence fom equation (1). Fo this agent, the govenment, we ae measuing liabilities, athe than assets i.e. b t = a t fo the govenment (although b t is an asset, such as a bond, fo the pivate secto). The analogues to equations (2)-(3) now descibe how past deficits lead to positive debt and how inheited debt constains futue fiscal policy. The tansvesality condition ules out debt bubbles, in which the govenment meets the inteest payments on its matuing debt by issuing moe debt. Howeve, with an infinite hoizon the debt does not need to be paid off it can gow at ate less than. See if you can show that the tansvesality condition ules out a pemanent, pimay deficit and that the govenment can un a pemanent deficit inclusive of inteest payments e.g. s = 0, with outstanding debt. What would an appaent failue to satisfy the tansvesality constaint mean? One possibility is that the economy is dynamically inefficient. A second possibility is that one 228

3 of the seies has been mismeasued, since accounting fo govenment assets and liabilities is not easy. A thid possibility is that a ejection shows that the cuent patten of fiscal policy cannot be sustained. Finally, exactly the same accounting applies to a county, and now links the stock of foeign debt to the tade balance. (b) Eule Equation Evidence In studying consumption so fa all we have done is wite down the lifetime budget constaint. To lean which consumption plan (of the many that satisfy the constaint) is chosen, we shall next look at an optimization poblem. We shall suppose that agents ae not constained in labou o cedit makets (i.e. thee ae no liquidity constaints). Suppose that an agent chooses {c t } to maximize the following functional: T E 0 t=0 β t u(c t ) It is called a functional because it is a function (specifically, a discounted sum) of utility functions at each time t. u is the peiod o instantaneous utility function; this functional is additively sepaable in the u s. β is the discount facto, whee β =1/(1 + θ), and θ is the discount ate. A positive θ eflects some impatience o time pefeence. The function u sums up the degee of substitution between consumptions (of some composite good) in diffeent peiods and also chaacteizes (though its concavity) the degee of isk avesion within a peiod (moe on this below). In seveal pats of the couse we ve focused on labou supply and consumption given pices and now we do the same sot of thing i.e. we focus on the household s plans athe than seek a complete geneal equilibium. The esults depend on the objectives and constaints, and by woking them out we lean about those objectives and constaints indiectly. We imagine the consume as maximizing this functional subject to the budget constaint: a t+1 =(1+)a t + y t c t, whee y t denotes labou income. To avoid confusion, let us use the tem life-cycle/pemanent-income hypothesis (lcpih) to efe to the idea that obseved aggegate consumption can be descibed as the outcome of this kind of optimization. I should wan you that this may not be standad. Some wites use the tem to efe to a paticula linea example (fom quadatic utility) which we shall see in a moment. Just as in the two-peiod model (which is a special case) one of the fist-ode necessay conditions fo a maximum in this optimization poblem is the Eule equation: u (c t )=E t β(1 + t+1 )u (c t+1 ) 229

4 No matte what assets ae available fo tansfeing wealth between time t and time t +1, in equilibium thei ates of etun must satisfy this equation. O, fo the household is exogenous and choices of endogenous c must satisfy this equation. So depending on ou emphasis we can view these equations (with ates of etun on vaious assets) eithe as a theoy of consumption and saving decisions o a theoy of asset etuns, as we saw in section 2. In fact they ae both, but it is convenient to take one pespective at a time. Let us conside seveal examples of this Eule equation. Example 1 u(c t )=ac t bc 2 t with a>0, u > 0 and u < 0; and suppose that θ =, a constant. Then the Eule equation is: c t+1 = c t + ɛ t+1 E t ɛ t+1 =0 This is a vey special example with a constant inteest ate and quadatic pefeences which says that, given c t, no othe infomation available at time t should help pedict the value of c t+1. The idea is that agents base consumption on lifetime o pemanent income. They do this now, and hence c t summaizes all infomation available now on futue income pospects. If c t+1 diffes fom c t it must be due to new infomation not available at time t. One of the majo developments in dynamic, stochastic macoeconomics (as mentioned in section 5) has been the insight that complete solutions to models need not be found in ode to test them. The fist-ode condition given hee can be used to test the optimization model without a complete solution fo c t in tems of vaiables such as expected futue income and inteest ates (let alone exogenous shocks). Since it elates one endogenous vaiable to anothe, instumental vaiables methods ae used to estimate it. Hall (1978) noticed that the Eule equation implies that discounted maginal utility is a matingale: futue changes in maginal utility ae not pedictable fom anything in the agent s cuent infomation set. He tested this by methods made familia by applied eseach in maco based on ational expectations and in finance on efficient makets. If we knew what maginal utilities wee, call them mu t = u (c t ), and if the ate of inteest wee constant, we could estimate egessions of the fom mu t+1 = γ 1 mu t + γ 2 z t + ɛ t+1 whee z t is anything in the agent s date-t infomation set and ɛ t+1 is a foecast eo with conditional mean zeo: E t ɛ t+1 = 0. The fist paamete, γ 1, equals β(1+) and is expected to be close to one, especially fo shot time intevals. The theoy implies γ 2 = 0, which can be used as the basis of a statistical test. Hall made this opeational by assuming that maginal utility is linea in consumption, (as would be the case with quadatic utility) so that a simila equation can be estimated with c t eplacing mu t. Hall consideed thee candidates fo z. The fist, consumption lagged moe than once, had little pedictive powe fo changes in consumption. The second, stock pices, povided 230

5 modeately stong evidence against the andom walk model. Highe stock pices tend to be associated with lage changes in consumption. This tuns out to be a obust esult, and may suggest eithe that the fist-ode condition is an impefect desciption of aggegate data o that the assumption of constant inteest ates is inadequate. The thid choice of z was lagged income. Hall found that lagged changes in income lead to an impovement in pedictions of changes in consumption. Let us conside this ole fo cuent income fom the pespective of an altenative model suggested by Hall. Suppose thee ae two goups of consumes in the economy. The fist (goup a) consume accoding to the lcpih (quadatic vesion), and eceive a faction 1 µ of aggegate income. Fo them, E t c a t+1 = c a t is a good appoximation. The second goup simply consumes all thei income, c b t = µy t. If income is a fist-ode autoegession, i.e. y t+1 = ρy t + v t+1, then we can deive the behavio of aggegate consumption, c t = c a t + c b t, as follows: E t c t+1 =E t c a t+1 +E t c b t+1 = c a t + µe t y t+1 =(c t c b t)+µρy t = c t + µ(ρ 1)y t. The intuition is that changes in the second goup s consumption ae pedictable if changes in thei income ae. This leads to pedictability in aggegate consumption as long as µ>0 and ρµ 1. Fo Canada the evidence suggests that ˆµ =.20 o that 20% of consumption is by consumes in the second categoy. I have simplified this stoy in a way which leads to a negative coefficient on y t, which is not found empiically but that would change with a moe ealistic time seies model fo y t, involving some gowth o tend. This povides an intepetation of income in the egession: some faction of the population is liquidity constained. The idea is to conduct empiical tests of Eule equations using vaiables z t which should be elated to liquidity constaints, and to see whethe they can pedict the innovation o eo in the Eule equation, which should be unpedictable. Unfotunately, most of these tests do not specify an altenative hypothesis. The Eule equation (as we have seen) could fail fo seveal easons. The pecise natue of the constaint (it could be that the boowing ate exceeds the lending ate, o that thee is a quantity constaint on boowing) affects the implications. Moeove µ should pobably not be viewed as an estimate of the faction of the population who ae constained (since thee is not a complete model with two agents and tade hee an economy with µ constained agents and 1 µ unconstained is diffeent fom µ of a constained economy and 1 µ of an unconstained one), but simply as a measue of the failue of this vesion of the lcpih. Example 2 u(c t )=lnc t. 231

6 The Eule equation is 1 [ β(1 + t+1 ) ] = E t. c t c t+1 So fa we have examined implications of the optimising model fo the popeties of consumption. We mentioned that time vaiation in inteest ates would complicate tests of the lcpih. But it also is possible to test the Eule equation and the coesponding theoy by examining the elation between consumption and inteest ates, as section 2 descibed fo two-peiod models. One of the most inteesting aspects of macoeconomics concens the inteaction of financial makets and eal decisions, and the elation between consumption and asset etuns is pehaps the most fundamental example of this inteaction. As we have noted befoe, some cae with the timing notation is equied. The measue is the inteest ate fom time t to time t+1. Fo cetain assets (e.g. equities) that etun is uncetain at time t and the uncetainty is not esolved until time t + 1; hence t+1 is the appopiate notation. Fo othe assets (e.g. T-bills) the ate of etun is known at time t so that witing t is appopiate. Conside the log pefeences of example 2 at least some empiical wok suggests that this may not be fa wong. Fo simplicity I shall ignoe uncetainty fo a moment: c 1 t = β(1 + t )c 1 t+1 c t+1 c t = β(1 + t+1 ) ln( c t+1 c t )=lnβ + t+1 I have used the appoximation ln(1 + x) x fo small x. This equation says that the gowth ate ( ln c t ) of consumption should be equal to a constant (a small negative numbe if β (0, 1)) plus the inteest ate. In peiods in which the inteest ate is high consumption should be gowing apidly. The idea is simply that if thee is a lage etun to saving (defeed consumption) then consumption will be postponed. This specific popety depends on the logaithmic pefeences, which mean that thee is a geat deal of intetempoal substitution in consumption. One can imagine altenative (e.g. Leontief) pefeences in which c t+1 can in no way substitute fo c t in that case the gowth ate of consumption will be insensitive to maket oppotunities o pice changes as chaacteized by t+1. In this example, the one-fo-one esponse aises fom the log function. Moe geneally, the degee of esponse can tell us something about pefeences. This equation can be examined eadily in time seies data fo diffeent peiods and fequencies. Often thee is little evidence of intetempoal substitution in consumption. Example 3 u(c t )=ct 1 α /(1 α). This peiod utility function often is used in studies of consumption and of asset pices. The coefficient of elative isk avesion is u (c)c/u (c) = α. The paamete α is positive, fo 232

7 concavity. This class of instantaneous utility functions ae efeed to as constant elative isk avesion o CRRA utility functions. In fact, example 3 genealizes example 2; if α =1 then u(c) = ln(c). The Eule equation is: c α t = E t [β(1 + t+1 )c α t+1 ]. If I again ignoe uncetainty, lnc t = ln β α + t+1 α so that the elasticity of intetempoal substitution ( ln quantity divided by ln pice) is the invese of the coefficient of elative isk avesion. Given these pefeences, if consumption esponds vey little to changes in the inteest ate that implies high isk avesion o low intetempoal substitution. Wijanto (1991) examined egessions such as lnc t = γ 0 + γ 1 lny t 1 + γ 2 t, in quately Canadian data fom 1953 to 1986; is the 3-month, ex post eal Teasuy-Bill ate, c t is eal pe capita consumption on non-duable goods and sevices, and y t is eal pe capita disposable income. He finds that ˆγ 1 is oughly 0.2 (which ejects the andom walk model) and that ˆγ 2 is oughly.08 so that α is That implies a lot of isk avesion o a little intetempoal substitution in consumption. What do the indiffeence cuves look like, fo consumption in adjacent time peiods? This esult means that fiscal and monetay policies which influence (afte-tax) may not have vey much effect on saving. Notice also that the theoy links the vaiance of consumption gowth to the vaiance of inteest ates. Infomally, va( ln c t ) ( 1 α )2 va( t ). We know that inteest ates ae moe vaiable than consumption gowth ates, which implies that α also must be lage. Sometimes consumption is said to be too smooth elative to asset pices. That simply means that the value of α, fo example, which can match the consumption vaiability with that of inteest ates is implausibly lage. Example 4 o CARA utility. The Eule equation is: u(c t )= exp( αc t )/α, exp( αc t )=E t [β(1 + t+1 )exp( αc t+1 )]. With this fom of utility function we can solve fo the closed-fom consumption function if we assume that income shocks ae nomally distibuted. 233

8 Studying an Eule equation is easy, but if tests eject it then we don t necessaily know how to efomulate the model. To lean moe, we can study the consumption function. In some special cases we can solve fo this function analytically. (c) Consumption Functions With quadatic utility and a constant inteest ate the Eule equation is linea, so it s easy to combine it with the budget constaint. That constaint is: o, ewiting, a t + a t = E t (1 + ) j 1 s t+j j=0 (1 + ) j 1 E t y t+j = j=0 (1 + ) j 1 E t c t+j. When E t c t+1 = c t it s also tue that E t c t+j = c t, using the law of iteated expectations. Thus, c t = [a t + (1 + ) j 1 E t y t+j ], which is the consumption function. j=0 Unde the lcpih lifetime income mattes and not its composition. The individual consumes the annuity value of expected lifetime labou income and cuent wealth. To see what I mean by annuity value, suppose that thee is no labou income but only financial wealth a t, as if the agent has etied but will live foeve, like a tust fund. How much should be consumed? Hee c t = a t while the budget constaint is Combining these gives: j=0 a t+1 = a t (1 + ) c t a t+1 = a t (1 + ) a t = a t This pudent (and infinitely-lived!) consume spends the net inteest income but maintains wealth constant. In this special example, the maginal popensity to consume out of wealth is the constant inteest ate,. To make the theoy testable, we need to model the expectations of futue labou income, using the foecasting tools we developed in section 3. Suppose, fo example, that y t = ρy t 1 + v t E t 1 v t = 0 and ρ < 1 Recall that in this case: E t y t+j = ρ j y t. 234

9 The consumption function now is: y t c t = [a t ρy t (1 + ) ] = a t + 1+ ρ y t. Execise: Wite this poblem as a dynamic pogamme, and solve fo the consumption function using the guess-and-veify method. (This is exactly like using Method A instead of Method B in section 3). We still face an obstacle in testing this consumption function: assets a t ae vey difficult to measue accuately. But we can avoid this obstacle with a tick. Lag the consumption function and multiply it by 1 + : (1 + )c t 1 = (1 + )a t 1 +(1+) 1+ ρ y t 1. The tick is that the oiginal consumption function included a t while this vesion includes (1 + )a t 1. But the budget constaint is: a t (1 + )a t 1 = y t 1 c t 1. Thus if we subtact the vesion that is lagged and multiplied by (1 + ) fom the oiginal, then only the flows of income and consumption will emain. The esult of this subtaction is: c t = c t ρ (y t ρy t 1 ). When you ecall that y t ρy t 1 = v t, ou esult looks suspiciously like ou oiginal Eule equation, in which the change in consumption was unpedictable. But we now have a deepe undestanding of evisions in the consumption plan: ɛ t = 1+ ρ v t = (1 + ) j 1 (E t y t+j E t 1 y t+j ). j=0 It may take you a few minutes to confim this last esult. The idea is that a shock to cuent income, v t, leads to evisions in foecasts of futue income because the income seies is pesistent. So to see how much to change consumption, you fist have to ecalculate the pesent value of income. Then the change in the annuity value is the change in consumption. In the consumption function we ve found, ρ = 1 makes all income changes pemanent and ρ = 0 makes them completely tempoay. Thus actual consumption changes depend 235

10 in a specific way on contempoaneous and past labou income. This shouldn t supise you. With these pefeences the agent ties to smooth consumption and to set it based on expected lifetime income. The eason why c t diffes fom c t 1 is that new infomation on lifetime income aives between t 1 and t. That infomation is contained in y t, which allows the agent to update foecasts fo futue incomes as well. Of couse, that updating is not pedictable at time t 1. The model can be tested by unning a set of linea egessions: c t = c t 1 +( 1+ ρ )(y t ρy t 1 )+e t y t = ρy t 1 + v t whee e t eflects vaious possible misspecifications. Notice that the theoy esticts this simple egession system quite seveely. Thee ae thee egessos in the fist equation (the consumption function) and one in the second (the income foecasting equation) but only two paametes to be estimated, namely ρ and. The system is oveidentified and can thus be tested by seeing whethe two values fo the paametes can explain all fou educed-fom coefficients. What is the evidence? An influential study by Flavin (1991) found that the coefficient on y t 1 was too lage to be consistent with the theoy, so consumption is moe closely tied to past labou income than the theoy pedicts. She efeed to this finding as excess sensitivity of consumption. A complementay finding by Campbell and Deaton (1989) was that consumption also displays excess smoothness. Recall ou esult that c t =( 1+ ρ )v t Then taking the vaiances of each side: ( ) 2va(vt va( c t )= ) 1+ ρ In most aggegate data (with estimates of and ρ) the actual value of va( c t ) is smalle than the ight-hand side of this equation. In that sense, consumption is too smooth to be consistent with the lcpih. To take an example, suppose that ρ = 1. In that case labou income is said to have a unit oot o to be integated. The idea is that the foecasting equation fo income is just a diffeence equation with an eo tem added, and the oot of that equation is one. If that is so then, solving backwads: y t = v t + ρy t 1 = v t + v t 1 + v t so that shocks (the v s) to labou income ae pemanent. In pactice estimates of ρ ae nea one, so that the coefficient in the egession of c t on v t should be lage (also one). 236

11 That makes sense: if innovations to one s labou income ae pemanent then one should adjust one s consumption by a lage amount. Howeve, the actual egession suggests that consumption is not sufficiently esponsive to these innovations in labou income. In summay, the empiical evidence suggests that the quadatic vesion of the lcpih is inconsistent with the evidence because consumption changes ae ovely sensitive to expected income (i.e. can be pedicted by y t 1 ) wheeas othe evidence suggests that consumption changes ae not sensitive enough to unanticipated income. One possible explanation is that thee ae liquidity constaints; anothe is that moe complex (nonlinea) vesions of the model, pehaps with vaiation in inteest ates, ae needed. In the case of quadatic pefeences we have been able to find both the Eule equation and the consumption function. But emembe that example 1, with quadatic utility and a constant = θ, is not vey geneal. In paticula: In the closed-fom solution fo c t given above the cuent level of consumption depends only on expected futue income and not on uncetainty about that income. This is the popety of cetainty equivalence. With othe functional foms fo u (in which u 0) the optimal c t also depends on the vaiability of the income steam. Typically, geate uncetainty (with the same expected value) about futue income leads to educed consumption cuently and moe pecautionay saving. We have assumed that thee ae no taste shocks, that is, that the utility function itself is not subject to andom movements. Hence, we egad data as geneated by a budget line shocked against a constant indiffeence cuve. Taste shocks may explain Chistmas shopping and othe seasonal effects but they ae assumed not to explain business cycles. We have assumed that the utility functional is additively sepaable ove time; thus utility at time t depends only on consumption at time t. Pefeences inconsistent with this assumption would include those in which thee is habit pesistence, fo example. This assumption of sepaability becomes inceasingly tenuous as the obsevation inteval (indexed by t) becomes small. You happiness this aftenoon may not depend on what you ate last yea but may depend on this moning s beakfast. We have also assumed that thee is one souce of consumption, say a single good. But c t should epesent the flow of consumption sevices, wheeas in empiical tests we must geneally identify it with expenditues on consumption goods. If these goods have some duability then this match may be misleading. So it seems sensible to test the model with data which exclude spending on duables. Measuement eo in consumption o income also will affect tests. We also have ignoed leisue i.e. assumed that the peiod utility function is additively sepaable in goods. Suppose that u = u(c t,l t ) whee l t is leisue at time t. If consuming the consumption good(s) and consuming leisue ae complementay activities then u(c t,l t )/ c t, the maginal utility in the fist-ode conditions above, will depend on leisue as well as consumption of the good. In that case, the model will hold pedictions/estictions fo the joint behaviou of leisue and consumption ove time. Thee can be a pemanent income theoy of leisue, fo example. 237

12 We have omitted vaiation in inteest ates. (d) Tax Smoothing In section 2 of this couse we studied the effects of a change in tax timing when thee ae lump sum taxes. One esult was that unde Ricadian equivalence thee would be no eason fo a govenment to choose any paticula path fo the budget deficit. Next, we ll conside this choice when the only taxes available ae distoting income o commodity taxes. The esult will be a set of pedictions called the tax smoothing model. We ll continue to take the sequence of govenment spending as given. Suppose that the deadweight loss of consume suplus fom taxation in a given peiod is a convex function, L, of total tax evenue t. This assumption means that tax take of $100 in one yea and $100(1 + ) in the next is pefeable to one (equivalent in pesent value tems) of $200 in the fist yea. In these cicumstances, smooth taxes will be best, just as unifom commodity taxes acos goods ae sometimes best in public finance theoy. Agents have access to pefect capital makets fo tansfeing income though time; the idea is athe that the elasticity of labou supply doesn t vay ove time, o costs of collection ae convex (o agents can legally avoid taxes in a given yea by tansfeing income acoss yeas). Suppose, fo simplicity though this is not necessay, that total output, y, though not welfae, is independent of the popotional tax ate t and constant ove time. And all vaiables ae eal, fo simplicity. Suppose that the steam of govenment expenditues {g t : t =0, 1, 2,...} is given. The govenment solves: subject to: min E t t=0 β t L(t t ) b t+1 + t t =(1+ t )b t + g t ; t t = τ t y t whee g is measued exclusive of inteest payments. The Eule equation is: L t = E t β(1 + t+1 )L t+1 which is exactly like the pemanent-income model of consumption, but with diffeent labels on the vaiables. Fo example, conside the case in which L is quadatic and β = (1 + ) 1. In that case, E t τ t+1 = τ t o, τ t+1 = τ t + ɛ t+1 and E(τ t ɛ t+1 ) = 0 so that thee is tax smoothing. This positive theoy pedicts that deficits will be used to smooth tax ates ove time. 238

13 This Eule equation can be tested statistically. Moeove, it can be combined with the govenment budget constaint just as was the case in finding the consumption function. In this case, we shall find a pemanent-spending theoy of tax ates. One way to think about the implications of this theoy is to intoduce unexpected tempoay o pemanent shocks to g and tace out the behaviou of τ. Let us begin with a tempoay shock. Duing a wa, when g is tempoaily lage, the govenment should un a deficit and then pay it off with a seies of small supluses late. The numeical value of the tax ate will be detemined by the govenment budget constaint, given the steam of expected futue govenment puchases. Ty an example to see how this woks. You will see that the esponse to anticipated shocks may povide a sticte test of the theoy. Next, if thee is a pemanent incease in g the tax ate should jump up to eflect this new infomation. In these thought expeiments the pesent value of govenment spending is used in the budget constaint to give the level of the constant (expected) tax ate. A govenment following this stategy would incease τ so as to aise t one-fo-one in esponse to a pemanent incease in g. But the esponse to a tempoay incease in g would be smalle, as the necessay evenue is collected smoothly ove time and the initial deficits ae paid off. If thee is an expected tempoay incease in g then tax ates should ise by a small amount as soon as the futue incease is expected, again to smooth τ. Thus this positive theoy of taxes becomes a theoy of deficits, if we think of govenment spending as being given. This model fits the empiical facts that {t t } tends to be smoothe than {g t } and that was ae often deficit-financed. But, one woies that the sequence {g t } should come fom the same optimization poblem. We geneally do not model fims as deciding on investment decisions and then consideing how to finance them, and one suspects that many spending decisions ae influenced by cuent tax evenue and tax ates and also by deficits. Moeove, shocks which cause cycles in this model ae unelated to the convexity of L by assumption that is, a shock which causes income/output to fall does not affect the elative elasticities of demand that undelie calculations of optimal tax ates. Also notice that optimal tax ates ae unifom acoss peiods so that deficits ae countecyclical but not fo conventional Keynesian easons. The tax smoothing model teats the authoities as solving a poblem of commodity taxation only. When thee is debt and capital in the economy the govenment s incentives may be moe complex than we have assumed so fa. Fo example, when the govenment has outstanding debt it may levy a lump-sum tax simply by defaulting on its debt. Likewise, if thee is a positive capital stock a lump-sum tax may be levied on it. Of couse, agents will be awae of the govenment s incentive to default on its debt o to tax capital, and that may discouage them fom holding debt o investing in capital. In these cicumstances the govenment is said to face a poblem of time consistency. This simply means that if the govenment calculates time paths of taxes optimally by taking the behaviou of the pivate secto as given then it may have incentives to change its announced policy e.g. to pomise zeo taxes on capital to encouage investment and then subsequently to enege and tax capital since it is supplied inelastically. 239

14 Howeve, the implication of this incentive fo time inconsistency is not that we should obseve eneging. In a ational expectations equilibium agents will undeinvest in capital if the govenment cannot bind itself not to apply capital levies in the futue. The benevolent govenment has an incentive to tax capital now since this will not affect output o allocations in the economy and simultaneously pesuade those investing in physical capital that it will not esot to this souce of evenue in the futue. Thus the optimal tax poblem involves the seach fo means of commitment on the pat of the govenment. If the govenment can find a method of binding itself in futue actions then cuent agents may be made bette off. Sometimes, the govenment will have means to bind itself. Fo example, suppose that people hold less money than would be optimal because they believe the govenment will expand the money supply late, thus eaning evenue fom inflation since it is a lage debto in nominal tems. If the govenment issues only indexed debt, it will signal its emoval of its own incentive to esot to the inflation tax and will thus encouage moe money-holding, which may be efficient (of couse, the matuity composition of govenment debt may depend also on capital maket impefections which it may ty to exploit in ode to finance its debt most cheaply). Establishing a cental bank also may seve this pupose. A simila poblem in fiscal policy aises with egad to pivatisation. Suppose that the govenment has two aims in pivatising a public entepise (such as Ai Canada) namely, aising evenue and pomoting competition. If it wishes to aise a lot of evenue it will advetise and auction the entepise in a egulatoy envionment that makes it a monopoly and theefoe attactive to biddes seeking pofit. If it seeks to encouage competition it will then enege and ensue deegulation. The govenment s inability to pohibit futue deegulation may inhibit its ability to aise evenue cuently. Rational expectations equilibia (o sequential equilibia in games between the pivate and public sectos) do not involve pedictable U-tuns. Typically they involve a wide vaiety of equilibia, some of which can be sustained without commitment devices. Moe on these games of stategy as models of policy (and on the stategic ole of policy coodination and signalling) is found in you fiendly neighbouhood micoeconomics couse. Fo futhe eading Chapte 7 of David Rome s Advanced Macoeconomics (1996) suveys some topics in aggegate consumption. I wamly ecommend section 6.2 of Blanchad and Fische s Lectues on Macoeconomics (1989) and Angus Deaton s book Undestanding Consumption (1992) to PhD students. Blanchad and Fische discuss pecautionay saving, neglected in these notes. The oiginal wok on the andom walk model was by Robet Hall in Stochastic implications of the life cycle pemanent income hypothesis, Jounal of Political Economy (1978) , which M.A. students should ead. The coesponding consumption 240

15 function was studied in moe technical wok by Majoie Flavin, The adjustment of consumption to changing expectations about futue income, Jounal of Political Economy (1981) Excess smoothness was diagnosed by John Campbell and Angus Deaton in Why is consumption so smooth? Review of Economic Studies (1989) On liquidity constaints, see Fumio Hayashi s Tests fo liquidity constaints, in Tuman Bewley, ed. Advances in Econometics (1987) fifth wold congess, volume 2. Canadian evidence on the pemanent income hypothesis is povided by Tony Wijanto in Aggegate consumption behaviou and liquidity constaints, Canadian Jounal of Economics (1995) and in Testing the pemanent income hypothesis: the evidence fom Canadian data, Canadian Jounal of Economics (1991) The most inteesting cuent eseach on consumption uses panel data, and leaves the epesentative-agent model behind. Fo an example see Angus Deaton and Chistina Paxson s Intetempoal choice and inequality, Jounal of Political Economy (1994) Fo good intoductions to tax smoothing see Sagent s essay Intepeting the Reagan deficits, Fedeal Reseve Bank of Minneapolis Quately Review (Fall 1986) and Rao Aiyagai s How should taxes be set? Fedeal Reseve Bank of Minneapolis Quately Review (Winte 1989) V.V. Chai descibes the commitment poblem in Time consistency and optimal policy design, Fedeal Reseve Bank of Minneapolis Quately Review (Fall 1988). Fo the undelying game theoy, begin with chapte 15 of Vaian s Micoeconomic Analysis (1992). 241

16 Execises 1. Show that adding white noise pefeence shocks to the quadatic utility model with β(1 + ) = 1 to give u(c t )= (α ɛ t c t ) 2 intoduces a seially coelated (in fact, fist-ode moving aveage) eo in the change in consumption. 2. Find the Eule equation in the two-peiod model with boow > lend. 3. Suppose that pefeences ae quadatic and inteest ates ae constant, in the life cycle/pemanent income model. Thus the Eule equation is given by c t = c t 1 + ɛ t ; E t 1 ɛ t =0. Suppose that ɛ t aises fom evisions in expectations, as follows: ɛ t = (1 + ) j E t y t+j (1 + ) j E t 1 y t+j. j=1 Finally, suppose that income evolves accoding to y t = ρy t 1 + v t. Suppose that =0.01 and that ρ =0.8 in the economy. Suppose that an economist oveestimates the pesistence in the income pocess, and thinks that ρ = 0.9. Can that mistake lead to findings of appaent (i) excess sensitivity (i.e. too lage a esponse to anticipated o lagged income) (ii) excess smoothness (i.e. too small a esponse to unanticipated income)? 4. Suppose that labou income evolves as follows: y t = η + ρy t 1 + u t, whee u t iid(0,σ 2 ). Suppose that consumption follows fom the linea-quadatic vesion of the lcpih: c t = c t 1 + ɛ t, ɛ t =( 1+ ) (1 + ) j (E t y t+j E t 1 y t+j ). j=0 Finally, suppose that ρ = 1. This is sometimes called a unit oot. The idea is that fo counties like Canada income may gow on aveage (i.e. in this case y t = η + u t ) athe than being stationay aound some mean. (a) Find ɛ t, the innovation in consumption, as a function of u t, the innovation in income. 242

17 (b) Find the vaiance of c t and hence the atio va c t /va y t. Does this model accod oughly with empiical evidence? Answe (a) E t y t+j = y t + jη = η + y t 1 + u t + jη, and E t 1 y t+j = y t 1 + η(j + 1). So thei diffeence is simply u t.thus ɛ t =( 1+ ) (1 + ) j u t = u t j=0 (b) Thus c t = u t so its vaiance is equal to the vaiance of income gowth. This doesn t match up with the facts, whee one tends to find that consumption is smoothe than income i.e. a atio less than one. This is simply the excess smoothness puzzle. 5. This question examines tests of the linea-quadatic vesion of the lcpih model of aggegate consumption. Suppose that labou income evolves as follows: y 0 =0 y t =0.9y t 1 + ɛ t ; t =1, 2, 3,...,50; y t =0.5y t 1 + ɛ t ; t =51, 52, 53,...,100. with ɛ t iin(0, 0.05). Notice that income becomes less pesistent in the second half of the peiod. We have witten the pocess in levels, but the same thing could be done with ates of change to allow fo gowth. Suppose that =0.05, a constant. Suppose that c 0 =0 and c t = c t 1 +( 1+ ρ )(y t ρy t 1 ). ( ) (a) On a compute, geneate one eplication of the labou income seies. (b) Using the ρ that applies fo each time peiod, calculate the seies fo consumption. (c) In this geneated data, do a Hall-type test of the andom walk model of consumption, by egessing c t on a constant and y t 1. (d) Now assume that an econometician does not ealize that the labou income pocess has changed. Said econometician obseves the income seies you have geneated in pat (a) and the consumption seies you have geneated (with the appopiate shift in ρ) in pat (b). This peson egesses y t on y t 1 fo the 100 obsevations to find ˆρ. This peson knows that ρ =0.05 and then geneates {ĉ t } using equation (*), with this and ˆρ. Plot the eos {c t ĉ t }. 6. This question studies the popeties of aggegate consumption fom the pespective of the life-cycle/pemanent-income hypothesis. Suppose that a epesentative agent maximizes EU =E (1 + ) t (b c t ) 2, t=0 243

18 subject to the budget constaint a t =(1+)(a t 1 + y t 1 (1 τ t 1 ) c t 1 ), whee b is a constant, a is assets, y is labou income, τ is an income tax ate, and c is consumption. (a) Find the Eule equation. (b) Find the consumption function. (c) Suppose that τ is constant but that y t = λy t 1 + η t, with E t 1 η t = 0. Specialize you answe to pat (b) by eplacing expectations with foecasts fom this time seies model. Assume that λ (0, 1). (d) Assets ae difficult to measue. So to deduce a test of this model, next wite c t in tems of c t 1, y t, and y t 1. (e) Find the vaiance of the change in c t. Show how that vaiance depends on the pesistence in shocks to labou income. (f) Suppose that the fiscal authoities hope to incease consumption spending by educing the income tax ate. Which will have a lage effect, a tempoay tax eduction o a pemanent one? (g) Descibe any diffeences between the effects of a tax cut that is announced in advance and one that comes as a supise. Answe (a) (b) (c) (d) (e) E t c t+1 = c t c t =( 1+ )[a t +E t (1 + ) i y t+i (1 τ t+i )] i=0 c t =( 1+ )[a t + (1 τ)y t(1 + ) ] (1 + λ) (1 τ) c t = c t 1 +( 1+ λ )(y t λy t 1 ) (1 τ) c t =( 1+ λ )η t 244

19 So the vaiance of is the vaiance of η t times the squae of the tem in backets. Inceases in λ incease this vaiance. (f) By the same logic as in pat (e) a pemanent cut will have a lage effect. (g) Whethe tempoay o pemanent cut, PV of consumption change is PV of tax cut, so no effect on that. But if announced in advance then the effect stats now and is smoothed out in that sense. 7. Empiically, one finds that savings ates diffe significantly acoss counties. This question exploes how isk in etuns to savings and also in income might affect the decision to save. To make this as simple as possible, conside a single save who seeks to maximize expected utility ove two peiods: u(c 1 )+E 1 u(c 2 ). Suppose that the utility function is of logaithmic fom: u(c i ) = ln(c i ); i =1, 2. In peiod 1 s 1 = y 1 c 1, while in peiod 2 c 2 = y 2 + s 1 (1 + ). The consume takes income and inteest ates as given. Suppose that y 1 = 1, that y 2 can takes on one of two values: ɛ and ɛ, each with pobability.5. Also suppose that takes on one of two values:.10 + η and.10 η, each with pobability.5. Thus isky income is modelled by using a lage value of ɛ and isky etuns on saving ae modelled with a lage value of η. The andom vaiables η and ɛ ae independent. (a) Show the effect on saving of inceasing income isk (ignoe etun isk). (b) Show the effect on saving of inceasing etun isk (ignoe income isk). (c) Do you findings have any policy implications? (d) Do the esults in pats (a) and (b) depend on the assumption of log utility? Answe (a) Fist check on income isk. Use the budget equations in the Eule equation: 1 1 s =1.10(.5 ɛ +1.10s +.5 ɛ +1.10s ) See how inceasing ɛ affects s. Foα = 1 (log utility) lage ɛ aises s (this is pecautionay saving). (b) Next check on inteest isk. By the same method 1 1 s =[.5( η) s( η) +.5(1.10 η) s(1.10 η) ] Obviously thee is no effect. (c) To pomote savings, add to income isk. educes income isk will educe savings. Fiscal policy (such as tax timing) which 245

20 (d) Unde CRRA esult (a) still holds, but esult (b) depends on whethe α>o < Suppose that consumption of nonduables and sevices can be descibed by the lcpih, with quadatic utility. Also ignoe vaiation in inteest ates and assume that the aveage inteest ate equals the discount ate. Thus the Eule equation is: E t c t+1 = c t. Suppose that the budget constaint of a typical household can be witten as: a t =(1+)(a t 1 + y t 1 c t 1 ), whee y is labou income and a is assets. Suppose that both c and y ae measued in logs so that c, fo example, is a gowth ate. (a) Find the consumption function, in tems of cuent and expected futue labou income. (b) Now suppose that labou income tends to evolve as follows: y t = λy t 1 + ɛ t, whee ɛ t iid(0,σ 2 ). Descibe a set of linea egessions that could be used to test the theoy. (c) Suppose that λ = 1. Find the atio of the vaiance of consumption gowth to the vaiance of income gowth. Does this pediction match the empiical evidence fo most counties? (d) Does empiical evidence suggest that one can ignoe vaiation in inteest ates in descibing vaiation in consumption gowth ove time? Answe (a) The consumption function is: c t =( 1+ )[a t +E t (1 + ) i y t+i ] i=0 which gives (b) c t = c t 1 +( 1+ )[ (1 + ) i (E t y t+i E t 1 y t+i )] i=0 c t = c t 1 +( 1+ λ )(y t λy t 1 ) y t = λy t 1 Then discuss identification. 246

21 (c) If λ = 1 then va y = σ 2. c t =( 1+ λ )(y t y t 1 )=ɛ t, so the vaiance atio is 1. In fact, we find atios less than 1 fo most counties (consumption is smoothe than income; see section 1). So the lcpih faces an excess smoothness puzzle. (d) Hall evidence (and similaly fo Canada) suggests that thee is vey low intetempoal elasticity of substitution (Leontief-type indiffeence cuves) so that ignoing may not matte. But thee is some limited suppot fo consumption-based asset picing models. 9. An election is expected and the authoities hope to stimulate consumption spending by cutting taxes. They want to know whethe to cut taxes in both peiods (of a two-peiod model) o to announce that only the fist-peiod tax ate will be lowe. One pundit agues that the effect will be geatest if people know that the tax cut is tempoay, because then they will concentate thei spending. Anothe claims that a pemanent cut will have a much lage effect. You ae to advise them on which plan will be most expansionay. Conside a two-peiod model of household consumption spending: subject to max U =(c 1 c 2 1/2) + β(c 2 c 2 2/2) c 1 + c 2 /(1 + ) =y(1 τ 1 )+y(1 τ 2 )/(1 + ), so that income is the same in each peiod. Assume that β(1 + ) = 1. In what follows, we shall use this budgetting poblem to find the effects of the two diffeent tax cut poposals, taking income and inteest ates as given (they could be affected in geneal equilibium since this is not an endowment economy, but we shall ignoe such effects). (a) Find the Eule equation linking c 1 and c 2. (b) Use the Eule equation and the budget constaint to solve fo c 1 in tems of incomes and the inteest ate. (c) Will c 1 incease moe if t 1 is educed o if both t 1 and t 2 ae educed? Answe (a) The Eule equation is: c 1 = c 2. (b) Using the consumption function: c 1 = [ 1+] [ y(1 τ1 )+y(1 τ 2 )/(1 + ) ] 2+ (c) dc 1 /dτ 1 = [ 1+] ( y) < 0 2+ dc 1 /dτ 2 = [ 1+] ( y)[1/(1 + )] <

22 So a pemanent cut will be moe expansionay; fom the pemanent income hypothesis. But in a two-peiod model if g does not fall then t 2 must ise when t 1 falls; so one could also take that into account. 10. Show that fo CRRA utility the coefficient of elative isk avesion equals the invese of the intetempoal elasticity of substitution. Answe Fist, η = ln(c t) ln(p t ), because elasticities ae positive. The Eule equation (without uncetainty) is: whee p t 1 /p t =1+ t 1.Thus so that, taking diffeences, η = 1/α. t 1 = βp t 1c α t, p t c α α ln( c t c t 1 ) = ln(β) ln( p t p t 1 ) 11. Studying consumption functions usually equies some assumptions about labou income. Suppose that households can budget with a constant inteest ate such that 1 + = 1/β, whee β is thei discount facto. Also suppose that they have quadatic utility so that E t c t+1 = c t. They face a budget constaint given by a t =(1+)(a t 1 + y t 1 c t 1 ), whee c is consumption, y is labou income, and a is wealth. (a) Solve fo cuent consumption in tems of lagged consumption and cuent and expected futue labou income. (b) Now suppose that y t = µ t + ɛ t, whee ɛ t is distibuted iid(0,σ 2 ). Thus labou income has a time tend. Replace expectations in you answe to (a) with foecasts fom this desciption of labou income, and so solve fo c t in tems of obsevable vaiables. (c) Does empiical evidence suppot this model? 248

23 Answe (a) (b) c t = c t (1 + ) j [E t y t+j E t 1 y t+j ] j=0 c t = c t ɛ t (c) Thee is evidence against the Eule equation by itself. Then thee is excess sensitivity and excess smoothness 12. Pehaps ejections of the pemanent-income hypothesis fo aggegate consumption ae due to ou misepesenting expectations. Suppose that the inteest ate is constant and that β(1 + ) = 1. Thee ae two time peiods, denoted 1 and 2. The epesentative agent maximizes: EU = (a c 1 ) 2 βe 1 (a c 2 ) 2 subject to c 1 + E 1c 2 1+ = y 1 + E 1y (a) Deive the Eule equation linking consumption expenditues in the two time peiods. (b) Solve fo consumption functions fo both c 1 and c 2. (c) Now suppose that E 1 y 2 =0.6y 1. Show that an investigato who oveestimates the pesistence in income and assumes E 1 y 2 =0.8y 1 will find appaent excess sensitivity of c 2 to y 1. Answe (a) (b) Then E 1 c 2 = c 1 c 1 = (y 1 + E 1y 2 1+ ) c 2 = c 1 + ɛ whee ɛ is the income supise. (c) We know that c 2 = c 1 + ɛ = c 1 +(y 2.6y 1 ). 249

24 Thus c 2 = c 1 +(y 2.8y 1 )+.2y 1 which will look like excess sensitivity to the investigato. 13. Imagine that a household deives utility fom a stock of duable goods, denoted d t, accoding to: E t i=0 (1 + θ) i (λd t+i γd 2 t+i), whee λ and γ ae constants. The stock of duables evolves accoding to: d t =(1 δ)d t 1 + c t, whee δ is the depeciation ate, and c t is expenditue on duables. The household s assets evolve accoding to a t+1 =(1+ t )(a t + y t c t ), whee y t is labou income. (a) State the fist-ode conditions fo an optimum. (b) Suppose that = θ and that δ = 0. What ae the time seies popeties of d t and c t? (c) Can duability pehaps account fo evidence of excess smoothness in aggegate consumption expenditues? Answe (a) plus the budget constaints. ( 1+) E t (λ 2γdt+1 )=λ 2γd t 1+θ (b) Now d t follows a andom walk, and c t is white noise. (c) I don t think so. Fist, most tests use consumption of nonduables and sevices. Second, although cetainly c t will have lowe vaiance than in the usual model, suppose that y t = ρy t 1 + ν t, as in the notes. Then ecognize that c t is now the innovation in maginal utility. So c t = ( ) (yt ρy t 1 ), 1+ ρ (what is called ɛ t in the notes). Suppose that ρ = 1. Then va d t = vac t = va y t.if c t is white noise then the vaiance of c t c t 1 will be lage than the vaiance of c t = ν t. So this cannot explain findings of excess smoothness. 250

25 14. Detailed tests of the pemanent income hypothesis involve statistical evidence on both consumption, c t, and labou income, y t. Suppose that the Eule equation is: and that the budget constaint is: c t =E t c t+1, a t =(1+)(a t 1 + y t 1 c t 1 ). Finally, suppose that labou income evolves accoding to: y t = µ + ɛ t, ɛ t iid(0,σ 2 ). (a) Solve fo the consumption function which elates consumption to cuent income and lagged consumption and income (i.e. and does not equie obsevations on assets). (b) Descibe the coss-equation estictions which could be used to identify paametes and test the model. (c) Descibe how this famewok could be used to test fo excess sensitivity and excess smoothness. Answe (a) The consumption function is: c t = y t µ. (b) The estictions ae simple then. Thee is a common µ in both equations. The oveidentification allows a test. The two equations should have the same innovation vaiance, too. (c) To test fo excess sensitivity one would include y t 1 sepaately in the consumption equation. To test fo excess smoothness, one would compae the vaiances of c t and y t, which should be equal hee because all income changes ae pemanent. 15. Suppose that a typical household s assets, a t, evolve as follows: a t =(1+)(a t 1 + y t 1 c t 1 ), whee y t is labou income and c t is consumption. The household ties to set consumption so that: c t = E t c t+1, i.e. the andom walk model applies. (a) Find the consumption function in tems of cuent assets and cuent and expected futue labou income. 251