Long-Run Labor Supply and the Elasticity of Intertemporal Substitution for Consumption Susanto Basu Miles S. Kimball University of Michigan July 2000 Abstract: The fact that permanent increases in the real wage have very little e ect on labor supply implies a parameter restriction in the consumption Euler equation augmented by predictable movements in the quantity of labor. This parameter restriction is not rejected by aggregate U.S. data. The implied estimate of the elasticity of intertemporal substitution is around .35, and is signi cantly di erent from zero. This estimate is robust to di erent instrument sets and normaliza- tions. After accounting for the e ects of predictable movements in labor implied by the restriction, there is no remaining evidence in aggregate U.S. data of excess sensitivity of consumption to current income. We would like to thank Tom Bishop for able research assistance. Susanto Basu's work on this paper was supported by the National Science Foundation. Miles Kimball's work on this paper was supported by the National Institute of Aging. KEYWORDS: Consumption, Euler equation, Labor Supply, King-Plosser-Rebelo utility func- tion. I. Introduction Traditional theology wrestles with the inconsistency of three propositions that are mutually inconsistent: (A) God is all powerful (B) God is all good (C) evil is real. A logically consistent theology must qualify at least one of these propositions. Consumption theory poses a signi cantly less important conundrum with the same logical structure. Currently, we suspect there are many economists thinking about consumption who would like to maintain simultaneously that (A) con- sumption and labor are additively separable in an additively time-separable utility function, (B) the elasticity of intertemporal substitution of consumption is relatively low|well below 1, and (C) income and substitution e ects on labor supply cancel, so that a permanent increase in the real wage will have little e ect on the labor supply of a household that relies entirely on labor income. But these three propositions are logically inconsistent. To see the logical inconsistency, assume additive separability of consumption and labor from each other and additive separability across time. For much of consumption theory this has long been the default assumption. Usually additive separability between consumption and labor is an implicit assumption made by omitting labor from the analysis as anything other than a source of income. Given (1) the assumption of additive separability between consumption and labor and across time, (2) empirical estimates of the elasticity of intertemporal substitution have repeatedly found quite low values. To be speci c, as recommended by Hall (1988), consider the IV estimation of the equation ln(Ct ) = s(rt ; ) + t + t;1 (1) using as instruments appropriately lagged variables that should be uncorrelated with the time- averaged rational expectations error t + t;1 . C is consumption, r is the real interest rate, is the utility discount rate. The parameter s is the elasticity of intertemporal substitution for consumption. Hall (1988) gets point estimates of s equal to .1 or .2 that are not signi cantly di erent from zero statistically. With the same maintained assumptions of additive separability, but identifying s by asking respondents to choose among hypothetical consumption paths rather than by observed responses to actual interest rate uctuations, Barsky, Kimball, Juster and Shapiro (1995) nd a value for the elasticity of intertemporal substitution s in the same range as Hall (1988). Let us take these results at face value and choose s = :2, which on the high end of what can be justi ed by either Hall (1988) or Barsky, Kimball, Juster and Shapiro (1995). With s = :2, and the maintained assumption of additive separability between consumption and labor, the implied period 1 utility function would be of the form u(C N ) = ; 4C 4 ; v(N ), where v(N ) is a convex function of 1 N . The implied real consumption wage is PC = ; uN (C N ) = C 5 v 0 (N ). Per capita consumption W uC (C N ) c has roughly doubled in the 35 years since 1960 (a growth rate of approximately 2% per year). The average number of work hours per person N has stayed fairly constant|or if anything has slightly increased over that period (which in interaction with the convexity of v(N ) would slightly increase v0 (N )). Thus, this functional form implies, counterfactually, that the real consumption wage should have increased by a factor of 25 = 32 over that time period!Even if s were as high as :333, this kind of exercise would imply an eight-fold increase in the real wage. An alternative way to state the same problem is that with consumption and the real wage both roughly doubling over this time period, the swift decline in the marginal utility of consumption implied by s = :2 should have led to a marked reduction in work hours as households satis ed the most pressing consumption needs and then turned to additional leisure when it became di cult to nd additional attractive consumption. Such an outcome, with the income e ect of the higher wage exceeding the substitution e ect was quite conceivable, but it didn't happen. Indeed, Keynes, in \The Economic Prospects for our Grandchildren," predicted a large increase in leisure during the remainder of the century. No a priori principle prevents the income e ect from exceeding the substitution e ect as Keynes guessed it would, but the lack of a strong trend in labor hours in the face of an enormous joint trend in wages and consumption indicates something close to cancellation between income and substitution e ects on labor supply. The macroeconomic literature on home production1 questions the standard interpretation of income and substitution e ects cancelling. The other alternative for explaining long-run labor supply facts is that the rate of technological progress in home production is the same as the rate of technological progress in market production. We will discuss the issue of home production more below. But the most important point is that the hypothesis of technological progress in home production at just the right rate can only explain the trend facts. In addition to long-run growth facts, a great deal of both cross-sectional and panel evidence analyzed by labor economists that indicates that the elasticity of labor supply with respect to a permanent increase in the real wage is very small. To summarize, the typical approach has been to maintain additive separability between con- sumption and labor. That maintained assumption leads to an estimate of the elasticity of intertem- 1 See for example Benhabib, Rogerson and Wright (1991), Greenwood and Hercowitz (1991), Greenwood, Roger- son and Wright (1995), Campbell and Ludvigson (1997), McGratten, Rogerson and Wright (1997), Canova and Ubide (1998) and Baxter and Jermann (1999). 2 poral substitution in consumption which would make the income e ect of a permanent wage much stronger than the substitution e ect of a permanent wage increase. This implication is at variance with at least three types of evidence about long-run labor supply: (1) the lack of a strong trend in weekly hours in the face of a dramatic trend in the real wage, (2) the fact that households say at the 75th percentile of wages work on average about as much as those with wages in the 25th percentile, and (3) the fact that permanent wage shocks to an individual on average do not appear to have much a ect on work hours. In the face of the impressive evidence for approximate equality of the income and substitution e ects on labor supply of a permanent increase in the real wage, our approach is to make this equal- ity of income and substitution e ects on labor supply a maintained assumption when estimating the elasticity of intertemporal substitution in consumption. With this maintained assumption, we nd a di erent value for the elasticity of intertemporal substitution under the maintained assump- tion of additive separability between consumption and labor, but the elasticity of intertemporal substitution is still signi cantly di erent from 1. Thus, we reject additive separability between consumption and labor. While an income e ect of permanent wage increases that is much larger than the substitution e ect is at serious violence with evidence on long-run labor supply, we see no serious problem with abandoning additive separability between consumption and labor. Indeed, as we will discuss in the conclusion, additive nonseparability between consumption and labor helps to make sense of a wide variety of economic phenomena beyond those that motivate us to consider this nonseparability in the rst place. Historically, we suspect that one of the greatest recommendations of the assumption of additive separability between consumption and labor has been simplicity. In this paper, we hope to show among other things that the price in added complexity is quite reasonable. In order to make the issues introduced by additive nonseparability with income and substitution e ects on labor supply cancelling as clear as possible, we illustrate our approach in the context of what is otherwise simple log-linearized consumption Euler equation estimation with additive time separability in the style of Hall (1988). At this point, more complicated consumption empirics would obscure the intuition for the issues we most want to clarify. As for theory, the key idea here of imposing the parameter restrictions implied by cancellation of income and substitution e ects on labor supply can be applied to much more general models2 (and we hope it will be), but the case of additive time separability 2 More general models of interest include models with consumer durables, models with habit formation in con- sumption and habit formation or durability of leisure. 3 is the obvious baseline case. Because we hope to have readers come away with a new perspective on the consumption Euler equation, we will present our theory (Section II) and evidence (Section III) before discussing the extensive previous literature on the consumption Euler equation (Section IV) and on home production in macroeconomics (Section V). II. Theory As alluded to above, in order to focus on the main issue of interactions between consumption and labor, we maintain the assumption of a representative consumer who has an additively time- separable von Neumann-Morgenstern utility function with a constant utility discount rate: 1 X ;j Vt = E e u(Ct+j Nt+j ): t j =0 For convenience in the estimation, the time interval will be one quarter. Time aggregation up from continuous time will be handled in the usual way in the estimation by lagging the in- struments an extra quarter and allowing for an MA(1) structure to the error term of the Euler equation. However, for clarity of exposition, this theory section will use discrete time and ignore time aggregation. Operationally, \imposing cancellation of the income and substitution e ects of a permanent wage change on labor supply" means choosing from the set of utility functions that yield a real wage proportional to consumption times some function of the quantity of labor: W (C N ) = ; uN (C N ) = Cv0 (N ) (2) PC u C Integrating this partial di erential equation indicates that felicity (the period utility function) must be of the form u(C N ) = (ln(C ) ; v(N )) for some monotonically increasing function U. At the risk of belaboring the obvious, note that with this form of the felicity function, 0 0 (ln( ; uN = ; ;v (N0)(ln(C )C )v;N()) )) = Cv0(N ): u 1 ; ( v N C C 4 Thus, the monotonically increasing function does not a ect the atemporal rst-order con- dition. only a ects intertemporal substitution between now and the future, not atemporal substitution between consumption and labor. The reasonable additional assumption of a constant elasticity of substitution in consumption when the quantity of labor is held constant narrows the utility function down to the King-Plosser- Rebelo form, which we write conveniently as C 1; ;1)v(N ) : u(C N ) = e( (3) 1; We also write s = 1= where s now represents the labor-held-constant elasticity of intertertemporal substitution in consumption, as will be apparent below. The marginal utility of consumption is ; uC (Ct Nt ) = Ct e( ;1)v(N ) For the consumption Euler equation, we will need the natural logarithm of t : ln(uC (Ct Nt )) = ; ln(Ct ) + ( ; 1)v(N ) (4) While it is helpful to assume constancy of the elasticity of substitution s as consumption C trends upward, the lack of a strong trend in N allows us to deal with the function v(N ) non- parametrically by using a Taylor expansion around the average value of N , which we label N .3 Optimal choice of consumption implies the Euler equation uC (Ct;1 Nt;1 ) = Et;1 e(rt ; ) uC (Ct Nt ) Programmatically, we want to focus on just the rst-order terms of the Taylor expansion. Moreover, assuming homoscedasticity of the stochastic process for ln(C ) and N , even the second- order departures from certainty equivalence only contribute a constant to the right-hand side of 3 Given good data on the uctuations in W , we could be slightly more exact (\slightly" in the sense of second- order), but we consider the short-run uctuations in the observed real wage to be unreliable as indicators of the short-run uctuations in the marginal disutility of labor. Our approach has the advantage of relying only on the long-run average value of (after-tax) C . WN P C 5 this equation. Accordingly, we can act as if the natural logarithm can be interchanged with Et;1 , yielding ln uC (Ct;1 Nt;1 )] = Et;1 frt ; + ln uC (Ct Nt )]g + higher-order terms Substituting in the King-Plosser-Rebelo form of the utility function yields ; ln(Ct; ) + ( ; 1)v(Nt; ) = Et; 1 1 1 rt ; ; ln(Ct ) + ( ; 1)v(Nt ) + higher order terms Dividing through by , rearranging, writing 1 = s, and making rational expectations error term t explicit, ln(Ct ) = s(rt ; ) + (1 ; s) v(Nt ) + t + higher order terms (5) (Note that the rational expectations error term involves surprises in all three terms: consumption growth, the ex post real interest rate and a function of labor.) The nal step is to use a rst-order Taylor expansion of v(N ) in ln(N ) around the trend level of labor N . Since v(N ) = v(eln(N ), by the chain rule, v (N ) v (N ) + N v0 (N ) ln(N ) ; ln(N )] Assuming that in the long run, the household can optimize labor supply, the intratemporal rst-order condition W=PC = Cv0 (N ) applied to the long run implies that WN N v 0 (N )= = (6) PC C where it is important that w be the after-tax wage seen by the household and the right-hand side, which we denote as , is calibrated as a long-run average value of PCN . Thus, can be W C treated as a constant in the estimation. is a number known from long-run labor supply facts. For example, below, our preferred value is = :8. The rest of the estimation takes place conditional on a particular value of . Substituting the constant known from long-run labor supply facts into the log-linearized Euler equation and using small c and n to denote the natural logarithms of consumption C and labor N yields 6 c = s(rt ; ) + (1 ; s) n + t + higher order terms: One more rearrangement shows that this is a very simple IV estimation: c; n = constant + s rt ; n] + t (7) In this equation, we have nally omitted the higher-order terms, except those that can be absorbed into the constant term. It is clear now that our estimation does more than simply add the growth in labor to the consumption Euler equation. There is a non-trivial linear restriction between the coe cient on the real interest rate and the coe cient on the coe cient on the growth in the quantity of labor. This restriction comes from facts about long-run labor supply. Before going on to estimation, it is worth pausing to ask if we can give a more intuitive expla- nation for this linear restriction on the consumption Euler equation with labor. In the equation, it is evident that the degree of nonseparability required|as indicated by the size of the coe cient on n|becomes greater as the elasticity of intertemporal substitution drops further below one. One way to understand this is as follows. As illustrated in the introduction, low values of the elasticity of intertemporal substitution in consumption mean that the marginal utility of consumption falls rapidly with growth in consumption. Without any interaction between consumption and labor in the utility function, this swift decline in the marginal utility of consumption would lead households to want more leisure unless the real wage increased markedly. What happens in the case of the King-Plosser-Rebelo utility function is that consumption and labor are complements, so that the increased level of consumption expenditures makes labor more pleasant. To tell a story, with the extra expenditures, things at home can be taken care of pretty well despite all of the hours spent at work this makes households willing to continue working the same work week even as they become richer. Note that complementarity between consumption and labor goes both ways. To use intro- spection to check the plausibility of the complementarity that arises with the King-Plosser-Rebelo utility function when s < 1, one can consider equivalently (1) whether an increase in work hours would lead to an increase in the marginal utility of expenditures or (2) whether an increase in expen- ditures would reduce the marginal disutility of work. Both are re ections of the same cross-partial derivative inequality uCN > 0 . 7 III. Evidence Data We use quarterly, seasonally-adjusted, aggregate U. S. data from 1949:1 to 1998:3. Our measure of consumption comprises non-durable consumption plus services, per capita. Our measure of per- capita hours is total hours worked by all persons (civilian and military), from unpublished BLS sources, divided by the non-institutional population over 16 plus members of the military. When we augment the regressions with disposable income, we use aggregate disposable personal income as de ned by NIPA, per capita. The real interest rate is computed as the after-tax nominal rate on three-month U. S. Treasury bills minus in ation in the price index of non-durable consumption and services. We took our measure of the average marginal tax rate from Stephenson (1998) since Stephenson's calculations extend only to 1994, we assumed that the average marginal rate for all subsequent years equals the 1994 value.4 According to the theory, equals labor income divided by nominal consumption expenditure. Taking nominal wages and salaries from the National Income Accounts and dividing by nominal spending on non-durable consumption and services gives an average ratio of 0.90. But we should use prices as perceived by the consumer, so we should de ne using the after-tax wage. Multiplying the numerator by our average marginal tax series reduces the mean to 0.77. We thus use = 0:8 as our preferred value, but check our results for several other values. Results Due to time aggregation of the data, the error term in our estimating equation has a MA(1) structure. Thus, instead of equation (7), we actually estimate: c; n= + s(r ; n) + t + t;1 (8) (Consumption and labor are in logs, as indicated by the small letters c and n. The constant potentially includes higher order terms such as precautionary saving e ects from a homoscedastic time series process for c.) Given the potential ambiguities about the precise value of , we estimate the equation for four values of ranging from 0.6 to 1.2. We use twice-lagged values of c, n and as instruments. The results are in Table 1. We consistently estimate values of the IES signi cantly greater than zero, unlike Hall (1988) and most subsequent work in this area. Depending on the value of the estimate of the elasticity of intertertemporal substitution s ranges from 0.3 to 0.5. The value s=.5 corresponds to the utility function 4 Since average marginal rates are very stable, this procedure should not create signi cant problems. 8 v(N ) ;e C , while s = :333 corresponds to the utility function 2v (N ) ; e2C 2 : We also report the rst-stage F -statistics in Table 1. According to Staiger and Stock (1997), these indicate that we are unlikely to have weak-instrument problems. The Hansen J -statistic indi- cates that we cannot reject the overidentifying restrictions, except, perhaps, for = 1:2, indicating that this value of may be signi cantly too high. We now check the robustness of our results. First, we show that our results are robust to alternative instruments sets. (We use some of these instruments later in the paper, when we investigate the robustness of our result to adding disposable income to the consumption Euler equation.) In addition to the variables we use as instruments in Table 1, we use the twice-lagged change in disposable income, y(;2), and the twice-lagged ratio of consumption to disposable income c(;2) ; y(;2). The results are in Table 2. We nd that the estimated s is not sensitive to the instrument set used. All the results say that s is about one-third three of the four point estimates are within 0.02 of one another, and the standard error are all roughly 0.10. The rst-stage t is reasonably good in all cases, and in no case can we reject the overidentifying restrictions at conventional signi cance levels. Second, we investigate the restrictions imposed by the King-Plosser-Rebelo functional form that we have assumed so far. In particular, it is possible that the signi cant IES that we have estimated so far is due to the correlation between n and c, and does not re ect the e ect of the real interest rate on consumption growth. We check this hypothesis by estimating several less- constrained versions of our equation, for our preferred value of = 0:8. We report the results in Table 3. We fail to reject the null hypothesis that the coe cient on r makes no additional contribution beyond the combination r ; n at the 5% level. The p-values for the test of the null hypothesis hover around 10% for our four instrument sets (.090, .105, .143 and .081). To the extent that the point estimates do not obey the restriction, they disobey in the direction of unpredictability of c. We then run our regressions \in reverse"|that is, we regress r ; n on c ; n, which normalizes the coe cient of r ; n to 1, instead of normalizing the coe cient of c ; n to 1 as in the previous regressions. Assuming additive separability, which in equation (8) is mechanically 9 equivalent to setting equal to zero, Campbell and Mankiw (1989) nd that both the forward regression and the reverse regression often yield coe cients close to zero. They interpret this nding as a speci cation test indicating that the standard rational representative-agent model of consumption is incorrect. By contrast, in Table 4B we nd that our reverse regressions yield estimates of 1=s that are quite consistent with the estimates of s from the forward regressions in Table 4A. For example, for = 0:8, we estimate s to be 0.36 from the forward regression and 0.47 from the reverse regression. Since 0.47 is within the 95 percent con dence interval of our forward estimate, we do not think that the Campbell-Mankiw speci cation test rejects our model. On the other hand, the reverse estimate of the standard = 0 consumption Euler equation, 0.75, is not in the con dence interval of its forward estimate, con rming Campell and Mankiw's original nding. However, we prefer estimates of s based on the forward regression, because the reverse regression has poor rst stage t (for all instrument sets we examine in Table 2, not just for the baseline result we report). The rst-stage F -statistic for the reverse regression ranges from 2 to 3, which is well within the danger zone identi ed by Staiger and Stock (1997). The reason is fairly intuitive-in the reverse regression, the right-hand-side variable is approximately the change in consumption minus the change in labor hours. Both of these series are procyclical and both are less volatile than output, so it is not surprising that their di erence is di cult to predict. We now proceed to compare our model of consumption based on non-separable leisure to Campbell and Mankiw's (1989) hypothesis of rule-of-thumb consumption. Campbell and Mankiw augment the standard ( = 0) consumption Euler equation with disposable income, and nd a positive, signi cant coe cient on disposable income. They interpret this coe cient as the frac- tion of consumption that is done by rule-of-thumb consumers who consume a fraction of their current disposable income (or, perhaps, the fraction of consumption due to liquidity-constrained consumers). They nd estimates as large as 0.5, suggesting that up to half of all consumption is done by rule-of-thumb consumers. In Table 5 we augment our basic estimating equation (8) with disposable income, for = 0:8. We use the various instrument sets we explored in Table 2, since some of the additional instruments may be better at predicting income growth than our basic set of variables. The results generally support our model, and do not support the hypothesis of signi cant rule-of-thumb consumption. In all cases the disposable income variable is insigni cant. In three of the four cases the coe cient is negative, which has no meaningful interpretation in the rule-of-thumb context. In three of the four cases, our estimate of s is signi cant and quite close to the values we estimate for our basic speci cation. The one exception is for our standard instrument set, which is not surprising since that instrument set is not chosen for its ability to predict future 10 income. The largest instrument set, which adds the twice-lagged consumption-income ratio to our basic instruments, yields an estimate of s that is exactly the same as our original estimate in Table 1 and an insigni cant, negative coe cient on disposable income. IV. Relationship to the Consumption Euler Equation Literature Campbell and Ludvigson (2000), in summarizing previous empirical literature on the consump- tion Euler equation, write that \: : : aggregate data o er no evidence of any important nonsepara- bility between market consumption and labor hours : : :. For example, Campbell and Mankiw nd that although there is substantial predictable variation in hours, it is not signi cantly related to predictable consumption growth as it should be if utility over leisure and consumption were addi- tively nonseparable. This evidence suggests that consumption and nonmarket hours can be well characterized by an additively separable utility function over consumption and nonmarket time, or, more generally, over consumption and some function of nonmarket time, as would be the case in models with home production." We disagree. First, even when we do not impose the parameter restriction implied by King- Plosser-Rebelo preferences, we nd a signi cant relationship between consumption growth and both the real interest rate and predictable movements in labor. This shows up in Table 3 as a coe cient on n signi cantly di erent from -1, since the left-hand-side variable is c ; n. To stack the deck against ourselves even more, we performed unrestricted \horse-race" IV regressions of c on both n and y with a wide variety of instrument sets and both with and without the real interest rate r in the regression. To summarize a large number of results. Without the real interest rate r in the regression, n and y do equally well by a t-statistic metric, but neither n or y ever have a coe cient signi cantly di erent from zero for any of our four instrument sets. (The minimum p-value is greater than .11.) With r in the regression, n does better than y.5 Both are insigni cant for the rst two instrument sets, but in the last two instrument sets, which include the lagged consumption/income ratio, n has a two-tailed p-value of :056 and :050, while y has a p-value of :362 and :688.6 The relative performance of n and y for the last instrument set7 can also be seen by comparing the last two scatter plots. Based on 5 Strangely enough, there is some tendency for y to do better when using instrument sets that emphasize lagged n and for y to do better when using instrument sets that emphasized lagged y and the lagged consumption to income ratio. 6 The instrument set c(;2), r(;2), y(;2), c(;2) ; y(;2) makes n signi cant with or without r (p-values of .011 with r and .031 without r) while y is insigni cant (p-values of .345 with r and .096 without r). 7 We consider the last instrument set most appropriate for this exercise since it includes both n(;2) and c(;2) ; y (;2). 11 these results, we maintain that an analyst who was indi erent between n and y a priori would have no reason to prefer y to n based on these horse-race regressions. But second, we do not think one should be indi erent a priori between augmenting the con- sumption Euler equation with n or with y. The burden of the rst two sections of this paper is to show why theory makes it almost mandatory theoretical grounds to have n appear in the consumption Euler equation. By contrast, the closest reasonable empirical substitute|adding y to the consumption Eu- ler equation|involves taking the grave step of abandoning the Permanent Income Hypothesis. Campbell and Mankiw (1989), believing that one could not reasonable avoid including y in the consumption Euler equation, write \The failures of the representative consumer model documented here are in some ways unfortunate. This model held out the promise of an integrated framework for analyzing household behavior in nancial markets and in goods markets." We believe that appropriate inclusion of labor in the formula for the marginal utility of consumption holds great promise for improving the empirical performance of the Permanent Income Hypothesis in a wide variety of economic contexts. In the consumption Euler equation context, theory not only mandates the inclusion of n, it mandates the size of the coe cient on n given the coe cient on the real interest rate r. We are not adding any free parameters. Therefore, we believe that the appropriate test for \excess sensitivity" of consumption to current income in aggregate data is the one reported in Table 5. V. Relationship to the Macroeconomic Literature on Home Production Although there is no doubt that home production is an important area of research for mi- croeconomics, we will argue in this section that macroeconomic analysis can ordinarily ignore the details of home production without serious loss. To see this, consider a household with an underlying utility function given by U (X ), where X is a vector of goods produced in home production. Some examples of possible elements of X are being well rested, being well fed, being entertained, etc. The household production function for the vector X is X = F (Q L Z) where Q is a matrix with rows showing all the di erent ways of using up the vector of marketed consumption goods, L is a row vector of di erent possible ways to spend time away from work, and Z is a vector of the technology for home production. Conditional on the total quantity vector 12 C of marketed consumption good used and time N spent at work and the technology for home production Z , the household solves max U (F (Q QL L Z )) s.t. X Qij = Ci j and X Lj + N =T j where T is the total time endowment. Since the total time endowment unit time is xed, the max- imum value is only a function of C and N . Thus, for this household engaged in home production, we can de ne u(C N Z ) by u(C N Z ) = max U (F (Q L Z )) QL s.t. X Qi =C i and X Li + N = T: i We maintain that the reduced form utility function u(C N Z ) contains all of the information needed for macroeconomics. The only importance for considering home production is the theoretical one of establishing the a priori plausibility for various forms of the utility function u(C N Z ). For example, thinking about home production may make one more eager to allow for more than one consumption good and may legitimate the exogenous e ects of Z , which in the reduced form utility function look like preference shocks. But considerations about the household production function are on a par with any other reasoning about the form of the reduced form utility function u(C N Z ). For example, the main costs and bene ts of going from one type of marketed consumption good to two (say adding consumer durables to the nondurables and services we have been concentrating 13 on) are there in roughly the same degree whether or not one worries about home production, and can be considered by looking directly at the behavior implied by di erent forms of the reduced form utility function u(C N Z ). And from the macroeconomic point of view, the key costs and bene ts of adding preference shocks to a reduced form model of household behavior remain little changed by considerations of household production. If, for the sake of parsimony, the reader grants us the simpli cation of having only a single marketed consumption good, the argument we give in the rst two sections for the King-Plosser- Rebelo utility function remain. Since the evidence from labor economics about the e ects of a permanent increase in the real wage on N indicates that households at a given point in time, with a given value of Z , show cancellation of income and substitution e ects, which we think of operationally as the equation W = Cv0 (N Z ), the utility function must have the form C 1; ( ;1)v (N Z ) u(C N Z ) = (9) 1; e : Following through the same exercise as in section II, we arrive at the same estimating equation, except that linear terms involving the household technologies Z are added to the error term:8 X c= + sr + (1 ; s) n) + (1 ; s) j zj + t + t;1 (10) j . The di culties preference shocks can cause for Euler equation estimation are well known. If one believes in important household preference shocks, the best way to interpret our results is to think of our estimation as assuming that preference shocks (or equivalently, home production technology shocks) cannot be predicted by our instruments. To round out this section on home production, we should mention that the paper closest in spirit to ours is Baxter and Jermann (1999): \Household Production and the Excess Sensitivity of Consumption." Indeed, our model is the special case of theirs when there is no home production. Given the large share of pure leisure in their utility function, which is modeled in the King-Plosser- Rebelo form, we suspect that many of their results are not so much due to anything special about home production but rather to the e ects we discuss in this paper. We think readers of both our 8 The reason a rst-order approximation should be OK for the e ects of Z is that when s 6= 1 and v 6= 0, any N Z strong trend in Z should cause a trend in optimal N over time that is not observed. In the dimensions where vN Z = 0, so that v is additively separable in labor and the production technology, a trend in Z will not cause a trend in labor N , so we cannot rule out a trend in this dimension of Z , but the additive separability means that variation in in that dimension would be hard to distinguish from heteroscedasticity in that dimension j of Z . 14 paper and theirs will nd the two papers complementary, even though we would interpret their results di erently than they do. VI. Conclusion Departing from the assumption of additive separability, which is typically made more for convenience than from conviction, we estimate the elasticity of intertemporal substitution while imposing the King-Plosser-Rebelo functional form needed for balanced trend growth of consumption and the real consumption wage. This mode of estimation is not the same as simply including the quantity of labor in a consumption Euler equation in an unrestricted way. Additive separability is easily rejected, but our restriction to King-Plosser-Rebelo preferences is not rejected. We nd that such an estimate, which e ectively combines information about the responsiveness of consumption to uctuations in the real interest rate and the quantity of labor and information about the low- frequency behavior of labor and the real wage, yields an estimate of the elasticity of intertemporal substitution of about .6. Moreover, we nd that the interactions between consumption and labor induced by the lack of additive separability allow one to solve a problem one nds in a Hall (1988)- type regression: estimates of the elasticity of intertemporal substitution that di er dramatically depending on which variable in the instrumental variable regression has its coe cient normalized to 1. This problem can be viewed as a particularly serious failure of the overidentifying restrictions of the IV estimation. In view of our results, omitted variable bias from leaving the quantity of labor out of the equation can account for this failure. Omitted variable bias can also account for Campbell and Mankiw's (1989,1991) nding that predictable movements in disposable income are related to predictable movements in consumption. We nd no evidence for such a relationship once labor is properly included in the regression. 15 Bibliography Barsky, Robert, Miles Kimball, Thomas Juster and Matthew Shapiro, \Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in the Health and Retirement Survey," NBER Working Paper # 5213 (August 1995). Baxter, Marianne and Urban J. Jermann, \Household Production and the Excess Sensitivity of Consumption to Current Income," American Economic Review 89 (September 1999), 902-920. 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Hall, Robert E., \Intertemporal Substitution in Consumption," Journal of Political Economy 96 16 (April 1988): 339{357. King, Robert G., Charles I. Plosser, and Sergio T. Rebelo, \Production, Growth and Business Cycles: I. The Basic Neoclassical Model," Journal of Monetary Economics 21 (March 1988): 195{232. Mankiw, N. Gregory, \The Permanent Income Hypothesis and the Real Interest Rate," Economics Letters 7 (1981), 307-311. Mankiw, N. Gregory, Julio J. Rotemberg and Lawrence J. Summers, \Intertemporal Substitution in Macroeconomics," Quarterly Journal of Economics 100 (February 1985), 225{251. McGratten, Ellen, Richard Rogerson and Randy Wright, \An Equilibrium Model of the Business Cycle with Household Production and Fiscal Policy," International Economic Review 38 (May 1997), 267-290. 17 Table 1: Estimates of s ∆c − τ ⋅ ∆n = µ + s(r − τ ⋅ ∆n) + εt + θ ε t−1 Estimated s p-value of restrictions First-stage F 0.30 0.6 0.40 13.6 (0.11) 0.36 0.8 0.24 11.6 (0.11) 0.42 1.0 0.12 10.4 (0.10) 0.49 1.2 0.06 9.7 (0.10) Notes: Instruments are ∆c(−2 ), ∆n(−2 ), and r (−2). p-value is for the test of over-identifying restrictions. Table 2: Different Instrument Sets ∆c − τ ⋅ ∆n = µ + s(r − τ ⋅ ∆n) + εt + θ ε t−1 Instrument Set Estimated s p-value of restrictions First-stage F ∆c(−2 ), ∆n(−2 ), 0.36 0.24 11.6 r (−2 ) (0.11) ∆c(−2 ), ∆y(−2 ), 0.34 0.39 10.9 r (−2 ) (0.10) ∆c(−2 ), r (−2 ), 0.30 0.31 10.2 c(−2 ) − y(−2 ) (0.12) ∆c(−2 ), r (−2 ), 0.35 0.34 8.8 ∆n (−2 ), (0.10) c(−2 ) − y(−2 ) Notes: p-value is for the test of over-identifying restrictions. Table 3: Robustness Checks Included Variables r − τ ∆n 0.52 (0.16) -0.35 0.30 r (0.20) (0.12) -0.61 τ ∆n (0.18) Notes: Dependent variable is ∆c − τ ⋅ ∆n . is set to 0.8. Instruments are ∆c(−2 ), ∆n(−2 ), and r (−2). All regressions include a constant. Table 4A: Forward Regressions ∆ c − τ ⋅ ∆n = µ + s(r − τ ⋅ ∆n) + εt + θ ε t−1 Estimated s 0.30 0 (0.11) 0.36 0.8 (0.11) Table 4B: Reverse Regressions r − τ ⋅ ∆n = µ + (1 s)( ∆c − τ ⋅ ∆ n) + εt + θε t−1 Estimated 1/s Implied s 1.32 0 0.75 (0.44) 2.13 0.8 0.47 (0.54) Notes: Instruments are ∆c(−2 ), ∆n(−2 ), and r (−2). All regressions include a constant. Table 5: Adding Disposable Income ∆c − τ ⋅ ∆n = µ + s(r − τ ⋅ ∆n) + β ∆y + εt + θ εt−1 Instrument Set Estimated s Estimated ∆c(−2 ), ∆n(−2 ), 0.53 -0.67 r (−2 ) (0.31) (0.62) ∆c(−2 ), ∆y(−2 ), 0.41 -0.47 r (−2 ) (0.18) (0.46) ∆c(−2 ), r (−2 ), 0.28 0.02 c(−2 ) − y(−2 ) (0.14) (0.27) ∆c(−2 ), r (−2 ), 0.36 -0.13 ∆n (−2 ), (0.14) (0.27) c(−2 ) − y(−2 ) Note: is set to 0.8. 5 5 5 4 8 5 5 4 4 7 69 6 855 3 75 6 6 9 88 5 6667 65 9 7 5 58 776 9 96 8 6 8 54 855597586 9 9 9 7 6 7 95955 76 8 8 8 786 56 6 8 8 Dc - 0.8*Dn, predicted 7 5 7 5 5 79 69989 8 9 7 67 9 99 49 9 66 6 2 5 78 7 5 5 5 8 959758 9 8 9 975 7 6 66 7 66 6 9 76 59 5 9 6 8 89 7 88 89 8 8 5 7 7 767 8 6 5 79 6 98 8 76 8 6 8 7 8 8 77 77 76 77 8 5 8 6 8 8 8 7 5 5 1 6 5 5 5 7 0 -1 -2 -3 -4 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 r - 0.8*Dn, predicted 5 4 6 5 3 5 5 6 5 66 86 6 66 9 8 5 4 5 5 7 55 6 556 69 99 6 5 86 58 5 6 8 8 8 7 8 76 6 59 69 6 7 6 9 57 688 6 786 69 8 8 8 5 96 88 9 4 5 78 5 7 58 9 8 6 7 6 99 6 6 88 7 74 7 57 98 6 79599 9 5785 97 66965 8 6 8 7 755 9 986 9 7 6 8 87 569 8 8 2 8 97 9 9 8 8 77 9 5 5 7 8 8 9 5 7 8 78 8 7 7 778 95 9 9 5 7 77 75 4 7 5 6 5 Dc, predicted 7 7 9 9 5 1 7 7 7 8 5 5 0 -1 -2 -3 -4 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 r, predicted 5 4 5 6 3 5 5 5 5 7 5 5 77 5 6 66 5 6 5 79 5 7 68 69 85 7 6 6 66 5 7 7 55 6 8 7 4 5 898 9 76 8 6 5 77566 696 8 Dc - 0.35*r, predicted 8 7 68 7 7 57 9 88 5 5 66 7 56967 7658 9 9 2 5 7 8 7597 6 87 5 7 96 7 5 7 6 4 588 66679699 99 9 8 8 8 77 6 5 56 9 5 978 8 5 77 9 8 9 8896 9 8 8 88 4 999 6 99 7 7 45 9 7 5 86 8 5 7 5 67 7 87 9 8 8 99 765 8 9 8 8 88 5 8 1 5 5 8 0 -1 -2 -3 -4 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Dn, predicted 5 4 5 6 3 55 5 5 77 7 7 66 6 6 7 7 8 5 6 5 65 5 5 7 57 9 895 6 5 7 7 8 9 876 8 4 586 6 9866 5 7 8 75 6 56 696 6 65 6 Dc - 0.35*r, predicted 9 8 8 7 7 888585 6 66 6 6 7 6 2 7 5 77 7 75 8 9 9 5 87 9 95986 995 56 5 7 8 6 8 99 6 7 9 57996 8 96 8 8778 9785 59 59 6 6 4 8 8 9899 9 5 7 9 6 5 4 5 7 7 58 9 8 7 8 8 9 88 8 6 8 6 4 77 7 9 98 8 85 6 59 1 5 5 8 0 -1 -2 -3 -4 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Dy, predicted