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Labor Supply in the Past, Present, and Future: A Balanced-Growth Perspective

Timo Boppart

Institute for International Economic Studies and Center for Economic and Policy Research

Per Krusell

Institute for International Economic Studies, University of Gothenburg, National Bureau of Economic Research, and Center for Economic and Policy Research

The absence of a trend in hours worked in the postwar United States is an exception: across countries and historically, hours fall steadily by a little below 0.5% per year. Are steadily falling hours consistent with a stable utility function over consumption and leisure under balanced growth of the macroeconomic aggregates? Yes. We fully character- ize the class of such functions and thus generalize the well-known “balanced-growth preferences” that demand constant (as opposed to falling) long-run hours. Key to falling hours is an income effect (of steady productivity growth on hours) that slightly outweighs the substi- tution effect.

I. Introduction

We propose a choice- and technology-based theory for the long-run be- havior of the main macroeconomic aggregates. Such a theory—standard

We thank seminar participants at Arizona State University, University of Chicago, Ei- naudi Institute for Economics and Finance, London School of Economics, New York Uni- versity Stern School of Business, National Bureau of Economic Research (Economic Fluc- tuations and Growth program), University of Oslo, Paris School of Economics, Sveriges Riksbank, and University College London for many valuable comments and suggestions. We particularly thank Nicola Fuchs-Schuendeln for valuable comments and the editor and anonymous referees for very helpful feedback. Kasper Kragh-Sørensen provided valuable

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balanced-growth theory, specifying preferences and production possibil- ities along with a market mechanism to be consistent with the data—al- ready exists, but we argue that it needs to be changed. A change is re- quired because of data on hours worked that we document here: over a longer perspective—going back 100 years or more—and across many countries, hours worked are falling at a remarkably steady rate: roughly half a percentage point per year. Figure 1 illustrates this fact for a set of countries and for hours on the intensive margin (the extensive margin is rather stationary; we discuss this and other data sources at length in this paper). This finding contrasts with the postwar United States, where hours per capita are well described as stationary, but this period is an ex- ception to earlier US history and to postwar data from other countries. The persistent fall in hours worked is not consistent with the prefer-

ences and technology used in the standard macroeconomic framework. Our proposed alteration of this theory is very simple and, on a general

FIG. 1.—Hours worked per worker. The figure shows data for the following countries: Belgium, Denmark, France, Germany, Ireland, Italy, the Netherlands, Spain, Sweden, Swit- zerland, the United Kingdom, Australia, Canada, and the United States. The scale is log- arithmic, which suggests that hours fall at roughly 0.57% per year. Source: Huberman and Minns (2007). Maddison (2001) shows a similar systematic decline in hours per capita. A color version of this figure is available online.

research assistance. Boppart thanks Vetenskaprådet (grant 2016-02194) and Krusell thanks the Knut and Alice Wallenberg Foundation for financial support. Data are provided as sup- plementary material online.

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level, obvious: we point to steadily increased productivity over very long periods and propose preferences over consumption and leisure such that, during such an experience, income effects on hours exceed substitution effects. We impose additional structure by summarizing the long-run data as (roughly, at least) having been characterized by balanced growth. On a balanced-growth path, our main economic aggregates—hours worked, output, consumption, investment, and the stock of capital—all grow at constant rates. Characterizing the data as fluctuations around such a path is only an approximation, but it is roughly in line with the last 150 years of data for many developed countries. Hence, we ask, Is there a stable utility function such that consumers choose a balanced-growth path, with con- stant growth for consumption and constant (negative) growth for hours, if labor productivity grows at a steady rate? We assume time additivity and constant discounting, as in the standard-preference setup. Given that the historical movements in hours per worker swamp those in participation, we also consider only the intensive margin of labor supply. We find that there are preferences that have the desired properties. Our main result fully characterizes the class of such preferences. The modern macroeconomic literature is based on a framework with

balanced growth and constant hours worked, to a large extent motivated with reference to postwar US data; see, for example, Cooley and Prescott (1995). Our main point here is not to take fundamental issue with this practice, and from a high-frequency perspective our proposed utility spec- ification is quantitatively similar to the preferences normally used. As for the discussion of hours from a historical perspective, there is significant recognition in the macroeconomic literature that hours worked have in- deed fallen over time. For example, several broadly used textbooks point to significant decreases over the longer horizon, often with concrete ex- amples of how hard our grandparents worked; see, for example, Barro (1984) or Mankiw (2010). In a well-known piece, John Maynard Keynes also speculated that hours worked would fall dramatically in the future— from the perspective he had back then (see Keynes 1930). Keynes thus imagined a 15-hour workweek for his grandchildren, supported by steadily rising productivity. As it turned out, Keynes was wildly off quantitatively, but he was right qualitatively (on this issue).1 Finally, in his recent hand- book chapter on growth facts, Jones (2016) also points to the tension be- tween the typical description of hours as stationary and the historical data. The picture that arises from looking at a broader set of countries

strengthens the case for falling rather than constant hours, and going further back in time reinforces this conclusion. With our eyeballing, at least, a reasonable approximation is actually even more stringent: hours worked are falling at a rate that appears roughly constant over longer pe- riods (though of course with swings over business cycles, etc.). This rate

1 For a discussion of Keynes’s essay from today’s perspective, see Pecchi and Piga (2008).

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is slow—somewhere between 0.3% and 0.5% per year—so shorter-run data will not suffice for detecting this trend, to the extent that we are right; to halve the number of hours worked at this rate requires around 175 years. Moreover, Bick, Fuchs-Schuendeln, and Lagakos (2018) re- cently updated the hours-income correlations across countries. Their find- ings are broadly in line with the long-run time-series data: people work much less in richer countries. Thus, in the United States over the last more than 150 years, as hours

have fallen, output has grown at a remarkably steady rate, mainly inter- rupted only by the Great Depression and World War II. Over this long period, all the other macroeconomic balanced-growth facts also hold up very well; we review these data briefly in section II.C. Thus, as output is growing at a steady rate, hours are falling slowly at a steady rate. Our interpretation of these facts is that preferences for consumption and hours belong to the class we define. This preference class is very similar to that used ubiquitously in the macroeconomic literature: that defined in King, Plosser, and Rebelo (1988). King, Plosser, and Rebelo showed that the preferences they put forth, often referred to as KPR or, perhaps more de- scriptively, balanced-growth preferences, were the only ones consistent with an exact balanced-growth path for all the macroeconomic variables with the restriction to constant hours worked. The class of preferences that we consider here is thus strictly larger in that it also allows hours worked to change over time at a constant rate along a balanced path. The period utility function in KPR is a power function of cv(h), where

c represents consumption, h represents hours worked, and v is an arbi- trary (decreasing) function. We show in our main theorem 1 that the broader class has a similar form: period utility is a power function of cvðhcn=ð12nÞÞ, where n < 1 is the key new preference parameter. In partic- ular, n can be interpreted as the fraction of a 1 percentage point produc- tivity gain that the representative household chooses to convert into more leisure as opposed to more consumption along a balanced-growth path. In terms of gross growth rates, if productivity grows at rate g, then hours grow at rate g–n, whereas consumption grows at g12n. For n > 0, the factor cn=ð12nÞ captures the stronger income effect: as consumption grows, there is an added “penalty” to working (since v is decreasing). Our preference class nests KPR, which corresponds to n 5 0. Quantitatively, given that growth in productivity per hour has proceeded at a rate around 2% per year, the rate of decline in hours implies that n must be calibrated to be around 0.2. Section II looks at hours worked over different time horizons and in

different countries. This section also examines the intensive and exten- sive margins and argues that over a longer horizon, almost the entire fall in total hours is accounted for by the former. In addition, section II briefly reviews the long-run facts for aggregates, with a focus on the United States, so as to motivate our balanced-growth perspective. In section III, we first

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contrast the standard-preference class used to match constant long-run hours with a simple function that is actually consistent with falling hours— namely, that proposed in MaCurdy (1981), which features a constant Frisch elasticity of labor supply. The main theorem of the paper, theorem 1, is found in section IV, where we lay out the precise balanced-growth restric- tions. It states what kind of utility function is needed for households to choose balanced-growth consumption and labor sequences. The proof of the theorem is in appendix A. The proof relies heavily on two lemmata— one characterizing the implications of balanced-growth choices for the consumption-hours indifference curves and one for consumption curva- ture—and we discuss those results in some detail in the main text. Sec- tion V discusses a number of specific functional forms that are useful in applications and comments on some of their properties. Section VI briefly discusses two remaining empirical challenges: the US postwar data and the cross-sectional distribution of hours within an economy. Sec- tion VII concludes.

II. Hours Worked over Time and across Countries

We now briefly go over the hours data across time and space. In an online appendix, we discuss a significantly larger set of data series.

A. Hours over Time

We first discuss US data and then look at other countries. From a long-run perspective, hours worked in the United States have been falling. Figure 2 illustrates this with both an intensive-margin measure and a measure for total hours worked. In figure 2A, we show establishment data on hours worked per worker, indicating a remarkably stable falling path; all graphs plot the logarithm of hours so that the stability can be interpreted in per- centage terms. Of course, like in all macroeconomic time series, there are large deviations during the Great Depression and World War II, but aside from these periods—and including the recent data—hours per employee keep falling. As for total hours (per population aged 14 and older), hence including the participation margin, we again see the decline at a very sim- ilar rate, except for the recent period 1980–2000, which we discuss in sec- tion VI in some detail. Figure 1 shows data on the intensive margin from Huberman and Minns

(2007) for a set of developed countries. Have these steady downward trends petered out? Turning to a slightly different sample of developed coun- tries for the postwar period, consider figure 3: total hours worked per per- son of working age (15–64), as in Rogerson (2006). We see that a horizon- tal line is not a good approximation here either; rather, a country fixed

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effect regression suggests that hours fall at roughly 0.45% per year. There is significant heterogeneity: the United States (solid line) marks an excep- tion in that its total hours worked have now flattened out, and Canada showsa similarpattern.Intheonlineappendix,wedetailhowthebreakdown

FIG. 2.—US hours in the long run. A, Regressing the log of hours on a constant and year gives a slope coefficient of 20.00315 in the full sample (and 20.00208 for the years 1970– 2015). This graph shows an updated series of the data in Greenwood and Vandenbroucke (2008). B, Regressing the logarithm of hours worked on time gives a slope coefficient of 20.00285. Source: Ramey and Francis (2009). A color version of this figure is available online.

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on total hours for the United States can be summarized by a steady down- ward trend in the intensive margin but—since 1950—a relatively steeply increasing participation rate that roughly cancels the downward trend in the intensive margin.2 Moreover, the increased participation rate is ac- counted for more than 100% by women (the male participation rate has fallen slightly). The online appendix also shows that over the long run, the participation rate has remained remarkably stable for any given coun- try, though these rates differ significantly across countries. Figure 3, finally, is not a biased sample of countries. The online appendix uses the equiv- alent graph for all countries with available data and reveals almost the same average rate of decline. Can the falling trend in hours worked be explained by demographics

or by the rise in schooling? Restricting attention to the United States, in another graph in the online appendix, we hold hours worked of different age groups constant at their 2005 values and check whether the observed changes in the age structure can account for the falling hours. The effect implied by the demographic change is nonmonotonic and overall very

FIG. 3.—Selected countries’ average annual hours per capita aged 15–64, 1950–2015. Source: Groningen Growth and Development Centre Total Economy Database for total hours worked and OECD for the data on the population aged 15–64. The figure is compa- rable to the ones in Rogerson (2006). Regressing the logarithm of hours worked on time gives a slope coefficient of 20.00393. A color version of this figure is available online.

2 In the online appendix we also display data from a time-use survey showing decreasing total hours worked even for the postwar United States.

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small.3 Furthermore, the online appendix also shows data from Ramey and Francis (2009) on time attending school and studying at home, mak- ing clear that average weekly hours of schooling increased by less than 2 hours in total over the period 1900–2005 and cannot therefore account for the drop in hours worked.4

To sum up: over >100 years, hours have been falling in all developed countries. In the postwar data, hours are still falling in most countries. In countries where hours are rather stable, such as Canada or the United States, the stability is accounted for by a dramatic increase in the partic- ipation rate. Hours per worker show a clear downward trend in all coun- tries, and participation rates do not show a clear trend over time in devel- oped countries; thus, the theory we develop here focuses on the intensive margin.5 We conclude that, purely in terms of trend extrapolation, if the participation rate stabilizes in the United States, hours will continue to fall. In fact, since the Great Recession, the participation rate fell, as did hours worked per working-age population.

B. Hours Worked in the Cross Section of Countries

For the cross section of countries, our theory predicts that labor pro- ductivity (or GDP per capita) should be negatively correlated with hours worked. Winston (1966) establishes such a fact in a sample of 18 coun- tries and estimates the elasticity of hours worked with respect to the wage rate to be 20.11. Bick, Fuchs-Schuendeln, and Lagakos (2018) document a negative correlation for a sample that includes developing countries and estimate an elasticity of log hours on GDP per hour of 20.15.6 We il- lustrate the same pattern for 1955 and 2010 in the online appendix: in each case, there is a negative correlation for our pooled sample.

C. Balanced-Growth Facts

Last, we review the basic “stylized facts of growth” for the United States. These data have been key in guiding the technology and preference spec- ifications in macroeconomic theory and remain instrumental in the the- ory we present here.

3 The baby boomers entering prime working age can partially explain the observed in- crease in hours since the 1980s.

4 Time-use studies also indicate that leisure has indeed increased (i.e., the time spent on “home production” has not risen).

5 Clearly, one would expect a theory based on the income effect exceeding the substitu- tion effect along a balanced-growth path to also affect participation. In another paper, we examine a labor-supply theory like that considered here but with an extensive margin as well; see Boppart, Krusell, and Olsson (2017).

6 With labor productivity growth of 2.5% per year, this slope coefficient suggests that hours worked decrease at about 0.375% per year.

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Figure 4A and 4B show how output and consumption grew at a very steady rate. Figure 4C and 4D show that the consumption-output ratio and the capital-output ratio remained remarkably stable. (In the online appendix, we display the capital-output ratio over a longer time horizon and an additional balanced-growth fact often imposed in the literature: constant factor income shares.) Our main takeaway message from figure 4 is that—in the style of Kaldor (1961)—we would like our macroeconomic framework to be consistent with these facts. Given our theory and our balanced-growth focus, one would also want

to establish balanced-growth facts for a cross section of countries. A sys- tematic such study is beyond the scope of this paper. However, in the online appendix, we take a preliminary look at the relation between labor productivity growth and hours worked—since our theory stipulates that hours fall because productivity rises—for a set of 21 countries over 1955– 2010. We find that productivity growth is clearly negatively correlated with the growth rate in hours worked among the countries in the sample, so long as one excepts Korea (which experienced very high productivity growth but close to zero labor growth).

FIG. 4.—Balanced growth. Source: Bureau of Economic Analysis and Maddison Project. A color version of this figure is available online.

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III. Standard Utility and a Contrasting Example

We now discuss whether the data illustrated above can be rationalized based on a stable utility function over consumption and leisure. We begin in section III.A by reminding the reader of the utility functions ordinarily used in macroeconomic analyses when restricting attention to balanced- growth paths with constant hours. We thus present King, Plosser, and Rebelo’s (1988) formulation. In section III.B, we give an example outside this class that admits hours falling at a constant rate along a balanced- growth path. The general analysis follows in section IV.

A. King, Plosser, and Rebelo (1988)

Consider time-additive preferences with a period utility function uðc, hÞ, where c represents consumption and h represents hours worked. King, Plosser, and Rebelo (1988) show that balanced growth with constant hours worked is obtained if and only if

uðc, hÞ 5 c � vðhÞð Þ12j 2 1

1 2 j if j ≠ 1,

logðcÞ 1 log vðhÞ if j 5 1:

8>< >: (1)

Here v can be any function that satisfies the usual regularity conditions. The KPR class has dominated the applied macroeconomic literature; in this literature, it is considered paramount to use a framework that is con- sistent with a balanced-growth path.7

Within the KPR class, two special cases stand out. One is the Cobb- Douglas case: uðc, hÞ 5 ðcð1 2 hÞkÞ12j=ð1 2 jÞ for j ≠ 1 and (replacing the j 5 1 case) uðc, hÞ 5 logðcÞ 1 k logð1 2 hÞ. This case, which is ob- tained by setting vðhÞ 5 ð1 2 hÞk in (1), restricts the elasticity of substitu- tion between consumption and leisure to be one and is part of the Gor- man class; that is, the marginal propensities to consume and work are independent of wealth. The second often-used case of KPR preferences is

uðc, hÞ 5 log cð Þ 2 w h 111=v

1 1 1=v , (2)

which follows by setting j 5 1 and vðhÞ 5 exp½2wh111=v=ð1 1 1=vÞ�. The parameter v > 0 is then the constant Frisch elasticity—that is, the per- centage change in hours when the wage is changed by 1%, keeping the marginal utility of consumption constant.

7 Here and in what follows, we focus on hours worked rather than leisure. Leisure could be computed as 1 2 h, where 1 is the time endowment, here normalized to one, and the preferences could of course then instead be expressed as a function of c and leisure.

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B. An Example with Falling Hours

The KPR class does not admit hours falling at a constant rate. So suppose we look at a case outside their class: the rather familiar function

uðc, hÞ 5 c 12j 2 1

1 2 j 2 w

h111=v

1 1 1=v , (3)

which was proposed in MaCurdy (1981). Notice that this case is a gener- alization—allowing consumption curvature different from the log case— of the most commonly used constant Frisch formulation in KPR: (2). A consumer facing a wage rate wt at time t would thus have an intratemporal first-order condition reading

wtc 2j t 5 wh

1=v t :

Is this equation consistent with balanced growth, in particular with hours falling at a constant rate? Suppose that wages grow at rate g > 1, con- sumption grows at rate gc, and hours grow at gh, all in gross terms. For the first-order condition to hold at all points in time, we then need

gg 2j c 5 g

1=v h :

On the kind of balanced-growth path considered in typical macroeco- nomic models, we would have g 5 gc, which is consistent with this equa- tion and implies that hours grow at the gross rate gh 5 g

ð12jÞv. However, unless j 5 1, this suggestion would not be consistent with the budget or the aggregate resource constraint: wtht would be growing at a rate dif- ferent from ct. Rather, for labor income and consumption to grow at the same rate, we need ggh 5 gc. Using this in the previous equation, we in- stead obtain g12j 5 g

1=v1j h , so that gh 5 g

vð12jÞ=ð11vjÞ. This example shows that hours fall over time if j > 1. Consumption will thus grow at rate gc 5 gð11vÞ=ð11vjÞ. Finally, the Euler equation is standard, since uðc, hÞ is additive, so under a constant interest rate, it can be met for a constant-growth con- sumption path. Thus, this utility function can rationalize falling hours worked under balanced growth.

IV. Theory

A. Balanced Growth: Technology and Preferences

We now set up our formal analysis. We first state the balanced-growth re- strictions from the perspective of the aggregate resource constraint. The workhorse macroeconomic framework has a final-good resource con- straint given by

Kt11 5 F Kt, AtLthtð Þ 1 1 2 dð ÞKt 2 Ltct, (4)

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where capital letters refer to aggregates, lowercase letters refer to per cap- ita values, and F ðKt, AtLthtÞ is a neoclassical production function. Here L represents population, h represents hours worked per capita, and d is the depreciation rate. Growth is of the labor-augmenting kind, be- cause of the Uzawa theorem.8 We thus assume constant exogenous tech- nology and population growth; that is, At 5 A0g

t, and Lt 5 L0h t.

We assume that preferences are additively separable over time with a constant discount factor b. In line with the KPR setting, the instantaneous utility, uðct, htÞ, is assumed to be stationary. Then, households (whether infinitely or finitely lived) maximize ::: 1 uðct, htÞ1buðct11, ht11Þ 1 ::: sub- ject to a sequence of budget constraints at11 5 ð1 1 rtÞat 1 htwt 2 ct, where a denotes per capita wealth.9

On a balanced-growth path, K and c grow at constant rates. Feasibility of such a path thus requires ðAt11=AtÞðht11=htÞðLt11=LtÞ5ðLt11=LtÞðct11=ctÞ 5 Kt11=Kt (see [4]). Hence, since At11=At 5 g and Lt11=Lt 5 h, a balanced- growth path implies a constant ht11=ht. When labor productivity (alternatively, the real wage per hour) changes

at a constant gross rate g > 0, consumption needs to grow at the same rate as labor income for growth to be balanced. The derivations above led to gc 5 ggh, where gc is the gross growth rate of consumption and gh that of hours worked. We thus seek preferences such that gc and gh are deter- mined uniquely as a function of the growth rate in (real) wages. Thus, we parameterize preferences with a constant n so that gc 5 g

12n and gh 5 g

2n.10 A value of n greater (smaller) than zero then corresponds to the income effect of increasing productivity on hours being stronger (weaker) than the substitution effect along a balanced-growth path. The special, standard case n 5 0 is of interest, but we focus mainly on n ≠ 0. Thus, on a balanced-growth path, for all t, ct 5 c0g

ð12nÞt and ht 5 h0g 2nt,

for some values c0 and h0. One can think of c0 as a free variable here, deter- mined by the economy’s, or the consumer’s, overall wealth, with h0 pinned down by a labor-leisure choice given c0. We are interested in an interior solution of the consumption and labor

supply decision that is consistent with a balanced-growth path.11 Such a solution requires some regularity conditions that we comment on further below. Two first-order conditions are relevant in the optimization. The labor-leisure choice requires

8 Grossman et al. (2017) discuss an interesting alternative case. 9 The budget here assumes no changes in the household size over time. We omit the

time constraint ht 1 lt 5 1, where l denotes leisure per capita and both h and l are nonneg- ative, because we focus on interior solutions.

10 With n ≥ 1, the theory would predict decreasing (or constant) consumption as the wage rate increases; we rule this case out.

11 We analyze the extensive margin in Boppart, Krusell, and Olsson (2017).

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2 u2 ct, htð Þ u1 ct, htð Þ

5 wt,

where wt, the return from working one unit of time, grows at rate g: wt 5 w0g

t.12 On a balanced-growth path, we thus need this condition to hold for all t. In our theorem below, we also require that preferences admit a balanced-growth path for all w0 > 0. That is, we are looking for prefer- ences that imply first-order conditions that admit a balanced path for consumption and hours at growth rates g12n and g2n, respectively, regard- less of the (initial) level of the wage rate relative to consumption. The intertemporal Euler equation, where r is the interest rate, reads

u1 ct, htð Þ u1 ct11, ht11ð Þ

5 b 1 1 rt11ð Þ;

b > 0 is the discount factor. When the economy grows along a balanced path, we would like this condition to hold for all t, and we need the right- hand side to be equal to an appropriate constant, which moreover may depend on the rate of growth of consumption and hours. We denote this constant R and discuss its dependence on c, h, and g below. In the sub- sequent analysis, we switch from sequence to functional notation. Thus, we omit t subscripts and instead specify the balanced-growth conditions as a requirement that the paths of all the variables start growing from ar- bitrary positive values (that meet the typical nonlinear restrictions im- plied by the first-order conditions); they can be scaled arbitrarily.

B. Balanced Growth Using Functional Language

Our balanced-growth requirements on utility can be stated as follows. Assumption 1. The utility function u has the following properties:

for any w > 0, c > 0, and g > 0, there exists an h > 0 and an R > 0 such that, for any l > 0,

2 u2 cl

12n, hl2nð Þ u1 cl

12n, hl2n � � 5 wl (5)

and

u1 cl 12n, hl2nð Þ

u1 cl 12n

g 12n, hl2ng2n

� � 5 R, (6) where n < 1.

12 In a decentralized equilibrium, this return denotes the individual wage rate including potential taxes and transfers. Similarly, the return on saving that we discuss below should be taken to be net of taxes and transfers.

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That is, we must be able to scale variables arbitrarily, but of course con- sistently with the balanced rates, and still satisfy the two first-order condi- tions. The scaling is accomplished using l (for wages/productivity), l12n

(for consumption), and l2n (forhours). Our main theorem will thus char- acterize the class of utility functions u consistent with these conditions. Our theorem will not provide conditions on convexity of the associated maximization problem (of the consumer or a social planner); obviously, however, conditions must be added such that the first-order conditions are indeed sufficient. We briefly discuss this issue in section IV.C.5.

C. The Main Theorem

Our main result gives necessary restrictions on utility for producing bal- anced growth. Theorem 1. If uðc, hÞ is twice continuously differentiable and satis-

fies assumption 1, then (save for additive and multiplicative constants) it must be of the form

u c, hð Þ 5 c � v hc n= 12nð Þð Þð Þ12j 2 1 1 2 j

for j ≠ 1, or

u c, hð Þ 5 logðcÞ 1 log v hcn= 12nð Þ � ��

where n is an arbitrary, twice continuously differentiable function. The proof relies crucially on two lemmata, one characterizing the mar-

ginal rate of substitution (MRS) function between c and h and one char- acterizing the curvature with respect to consumption: the relative risk aversion in consumption (RRAc) function. The proof then uses these lemmata to derive the final characterization. The proofs of the lemmata and details of how to use them to complete the proof of the theorem are contained in appendix A. Here we state and make some comments on the lemmata as well as on the overall method of proof.

1. Consumption-Hours Indifference Curves

Lemma 1 characterizes the shape of the within-period indifference curves. Lemma 1. If uðc, hÞ satisfies (5) for all l > 0 and for an arbitrary c > 0

and w > 0, then its MRS function, defined by u2ðc, hÞ=u1ðc, hÞ, must be of the form

u2 c, hð Þ u1 c, hð Þ

5 c1= 12nð Þq hcn= 12nð Þ � �

, (7)

for some function q.

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Notice here that in the long run, hcn=ð12nÞ will be constant, so that the argument of q will not change over time. The proof technique for lemma 1 is very similar to that for Euler’s theorem. The indifference curves are illustrated using three graphs in figure 5.

Dashed lines are expansion curves (as wages change); black depicts n > 0, and gray depicts the standard n 5 0 case. Figure 5A shows the consumption- leisure trade-off, whereas figure 5B and 5C show the trade-off in consumption-hours space, with the latter using logarithms. The expansion path for n > 0 is clearly downward sloping in c 2 h

space, defining the key characteristic that the income effect of a wage change dominates the substitution effect in the absence of unearned in- come.13 When this path is linear in logs, we moreover have preferences allowing hours to fall at a constant rate as consumption grows at a con- stant rate.14

2. Curvature

Next, let us characterize curvature of u with respect to c with lemma 2. Lemma 2. Under assumption 1, the RRAc, 2cu11ðc, hÞ=u1ðc, hÞ, must

satisfy

2 cu11 c, hð Þ u1 c, hð Þ

5 p hcn= 12nð Þ � �

for some function p.

FIG. 5.—Consumption-leisure trade-off. The figure panels abstract from unearned in- come. A color version of this figure is available online.

13 Thus, without unearned income, this characteristic amounts to a negative Marshallian wage elasticity of hours. With unearned income, our preferences still feature an income effect that exceeds the substitution effect when wages are scaled (by l) so long as there is a simultaneous scaling of unearned income (l12n). This simultaneous scaling precisely mimics the behavior on a balanced-growth path.

14 The slope in fig. 5C is 2ð1 2 nÞ=n. For n < 0, the substitution effect would be stronger, and the expansion path would be upward sloping.

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The proof of the lemma is straightforward; it involves differentiation of the Euler equation with respect to l, the use of lemma 1, and some ma- nipulations. As above, let us point out that in the long run (i.e., along a balanced-growth path), hcn=ð12nÞ is constant. Thus, the RRAc will be constant. However, in general, its long-run level is endogenous, and over shorter time horizons, it will not be constant. The term “relative risk aversion” (RRA) here is used for convenience; it

is appropriate only to the extent that we consider a gamble where hours are not allowed to adjust.15

3. Proof Structure and Some Comments

The structure of the overall proof, based on the lemmata, proceeds in two steps that are similar in nature. First, we use lemma 2 to integrate over c to obtain a candidate for u1; this can be accomplished straightforwardly since the left-hand side of the lemma can be expressed as the derivative of log u1 with respect to log c. The integration with respect to one vari- able delivers an unknown function (a constant) of the other variable. This function can then be restricted by comparison with the characterization in lemma 1 (a cross-check). Second, after the first integration and cross- checking, with its implied restrictions, we integrate again with respect to c from the obtained u1 to deliver a candidate for u. Then, as in the previous step, another function of h appears, and it too needs to be cross-checked with lemma 1 and thus further restricted. This then completes the proof. Although we were motivated by data displaying increasing productivity

growth and falling hours, the proof does not assume that g > 1 or n ≥ 0. Potentially, the model could thus generate an increasing h at a constant rate as productivity increases steadily, and we see an example of this below. Interestingly, and to our surprise, we could not find a full proof of the

KPR result in the literature.16 In particular, in the proofs we have looked at, the fact that the RRAc is constant along a balanced path is taken to mean that this constant is exogenous (i.e., given by a preference param- eter j and independent of h). A nontrivial aspect of our proof is to show this presumption to be correct.

4. Sufficiency

Theorem 2 gives the converse of theorem 1: with the utility function in the specified class, the first-order conditions for optimization are consis- tent with balanced growth.

15 The appropriate risk aversion concept in typical applied dynamic models with valued leisure is based on the value function; see Swanson (2012).

16 We would be very grateful if someone could point us to a proof somewhere, because we may well have missed it.

labor supply in the past, present, and future 133

Theorem 2. Assume that n < 1. If uðc, hÞ is given by

u c, hð Þ 5 c � v hc n= 12nð Þð Þð Þ12j 2 1 1 2 j

for j ≠ 1, or

u c, hð Þ 5 logðcÞ 1 log v hcn= 12nð Þ � ��

where v is an arbitrary, twice continuously differentiable function, then it satisfies assumption 1. Since this proof is much less cumbersome than that for the main the-

orem, and since it involves the manipulations necessary in applied work based on the preference class we identify here, we include it in the main text. Proof. For easier notation, let x ; hcn=ð12nÞ. The two first derivatives

then read

u1 c, hð Þ 5 1

c 1 1

1 2 n

v0 xð Þ v xð Þ x

� � c � v xð Þð Þ12j,

u2 c, hð Þ 5 1

v0 xð Þ v xð Þ x c � v xð Þð Þ

12j :

Dividing the latter by the former, we obtain

u2 c, hð Þ u1 c, hð Þ

5 c

v0 xð Þx=v xð Þ 1 1 nv0 xð Þx= 1 2 nð Þv xð Þð Þ :

By multiplying c by l12n and h by l2n, we find that this expression increases by a factor l. We have thus reproduced the first part of assumption 1— that is, the intratemporal first-order condition on a balanced-growth path. By evaluating u1ðc, hÞ=u1ðcg12n, hg2nÞ, we obtain gjð12nÞ—that is, an expression that is independent of c and h, and hence c and h can be scaled arbitrarily. By letting R 5 gjð12nÞ, we therefore see that the second condi- tion of assumption 1 is also verified. QED

5. Utility Maximization under Explicit Constraints

Our two theorems together state necessary and sufficient conditions for our utility function to be consistent with balanced growth as represented by an interior solution given by the first-order conditions in assumption 1. Thetheoremsarethus designedstrictly to characterize preferences.Whether an exact balanced-growth path exists, as a competitive equilibrium or the solution to a planning problem, with preferences in the defined class, is a different (though of course closely related) question. The answer de- pends on features of the constraints facing the consumer or planner.

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For example, if the marginal product of capital is not high enough for the given productivity-growth rate when the capital input is at zero, the balanced-growth version of the Euler equation could not be satisfied. Moreover, to ensure sufficiency of a maximum based on the first-order conditions requires not only features of the constraints but also additional assumptions on preferences ensuring existence, interiority, and sufficiency. A general treatment of these issues is beyond the scope of the present study, and as far as we can tell, these conditions do not crucially relate to whether n is nonzero. To be sure, however, we have verified that these con- ditions are met in the specific examples we discuss below.17 To be clear, however, these issues were kept aside in our analysis precisely in order to provide as general, and hence as powerful, a theorem 1 as possible, one that can be applied to different contexts.

V. Utility Functions for Applied Use

This section discusses the applied use of our proposed preference class. This involves discussing the value of n, special parametric forms, and connections to utility functions used in the literature, along with some properties.

A. Calibration

We first briefly discuss how to calibrate our preferences to long-run data and then discuss an often-used case: the MaCurdy preferences.

1. What Is n?

We add a new element, n, to standard preferences as represented by KPR’s balanced-growth function (setting n 5 0 in theorem 1, we obtain the KPR class in [1]). The new element regulates the size of the additional-income effect—that is, that above and beyond the income effect present in the KPR class. With n > 0, hours fall when productivity rises as the income ef- fect dominates the substitution effect on a balanced-growth path. Similarly, if n < 0, hours rise when productivity rises. Along a balanced-growth path, productivity grows at (gross) rate g, hours fall at rate g2n, and consump- tion as well as output grow at rate g12n. Thus, n can be interpreted as the fraction of a 1 percentage point productivity gain that the representative household chooses to convert into more leisure, as opposed to more con- sumption, along a balanced-growth path. We can use the time-series data for an estimate of its value. In the data, as we have seen in section II, hours fall as productivity rises, so the empirically relevant case is n > 0. Thus,

17 Details can be obtained from the authors upon request.

labor supply in the past, present, and future 135

with productivity and hours growing annually at, say, 2% and 20.4%, re- spectively (these numbers are rough averages), we would obtain an esti- mate of n from 0:996 5 1:022n—that is, a n around 0.2. With n 5 0:2 and g 5 1:02, the 0.4% fall in hours per year makes per capita consumption grow by 1.6% in the long run. Clearly, depending on the time period, the set of countries, and the precise data definitions, the resulting n would differ, and our different calculations suggest that a value of 0.2 or perhaps a little below is reasonable. The key point is that when restricting the anal- ysis to long-run facts, the class of preferences we propose is not larger; it is different, thus necessitating a n of around 0.2 rather than n 5 0.

2. Calibrating the MaCurdy Preferences

The value of n is key for the long-run behavior of hours, but of course the form of the function v can matter greatly for many other aspects of how much people choose to work: for the level of hours (and therefore out- put and other macroeconomic aggregates), for transitional dynamics, and for how hours respond to shocks of various kinds. We now look at the special case in our introductory example: the MaCurdy formula- tion in equation (3), uðc, hÞ 5 ðc12j 2 1Þ=ð1 2 jÞ 2 wh111=v=ð1 1 1=vÞ. This function is often used in applied contexts in the business-cycle lit- erature; see, for example, Galí and Monacelli (2005). Its attractiveness is that two important elasticities are controlled by two separate parame- ters: j and v. In particular, the intertemporal elasticity of substitution (IES) is constant and equal to 1=j, and the Frisch elasticity of labor sup- ply is equal to v. As is well known, with j ≠ 1, preferences of the MaCurdy form are not part of the KPR class. For this reason, as already discussed in section III.A, a large part of the macroeconomic literature restricts itself to a unitary IES and sets j 5 1, so that the first term of the utility func- tion simply becomes log c.18

Under the MaCurdy formulation, a 1% increase in productivity will make hours fall by ðj 2 1Þ=ðj 1 1=vÞ percent. This parametric expres- sion thus corresponds to the n in our general formulation. Given a cali- brated value for n, this means that the two curvature parameters j and v must be related. Figure 6 illustrates this restriction applied on a balanced path, one setting n 5 0 (KPR, where j 5 1 is unavoidable) and one with n calibrated to 0.2. Thus, for the latter value, any point on the downward- sloping curve is admissible.

18 For example, Shimer (2010, 2) uses this function in his book and writes, “This formu- lation imposes that preferences are additively separable over time and across states of the world. It also imposes that preferences are consistent with balanced growth—doubling a household’s initial assets and its income in every state of the world doubles its consump- tion but does not affect its labor supply. . . . I maintain both of these assumptions through- out this book.”

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Thus, a lower willingness to accept a growing consumption path (a higher j) must, for a given n > 0, go hand in hand with a lower willingness to accept a falling-hours path (a lower v). The calibration of the model can thus admit high values of j if labor is hard to substitute over time. One way to calibrate j here would thus be the indirect way: choose the Frisch elasticity according to available studies and back it out given n 5 0:2. Fig- ure 6 shows the implied values of j for the average micro and macro esti- mates of the Frisch elasticity from Chetty et al. (2011). Interestingly, using an Aiyagari-style model, Heathcote, Storesletten, and Violante (2014) es- timate a MaCurdy function and find j 5 1:7 and v 5 0:5, which implies a n just below 0.2.19

B. Parametric Forms

A broad parametric preference class within theorem 1 is as follows:

u c, hð Þ 5 c 12j 1 2 a hcn= 12nð Þð Þb

� �d 2 1

1 2 j , (8)

FIG. 6.—Combinations of elasticities. Combinations of the RRAc, j, and the Frisch elas- ticity, v, in the functional form (3) that deliver (i) constant hours (n 5 0) and (ii) hours falling at rate g20:2. With g 5 1:02 and n 5 0:2, hours worked decline at around 0.4% per year. With n 5 0:2, the figure highlights the j implied by the average micro and macro estimates for the Frisch elasticity from Chetty et al. (2011) of 0.82 and 2.84, respectively. A color version of this figure is available online.

19 It is easy to show that the present model is consistent with an Aiyagari-style setting.

labor supply in the past, present, and future 137

where a, b, and d are constants to be discussed later. This function is obtained by choosing a particular functional form for v: vðxÞ 5 ½12 axb�d=ð12jÞ, with x ; hcn=ð12nÞ. This parametrization can straightforwardly be generalized further, for example, by changing the functional form of the squared brackets of x (say, by adding an additional x term to some different power). However, as we show below, many interesting cases can be viewed as special cases of (8). We thus view it as a natural starting point that is useful in applied work. The special cases appear as follows (it is straightforward to do the algebra to see how each functional form emerges):

1. MaCurdy (1981): ðc12j 2 1Þ=ð1 2 jÞ 2 wh111=v=ð1 1 1=vÞ. Ob- tained by setting a 5 wð1 2 jÞ=ð1 1 1=vÞ, b 5 1 1 1=v, d 5 1, and finally, n 5 ðj 2 1Þ=ðj 1 1=vÞ.

2. Generalized log-log: logðcÞ 1 k logð1 2 fhcn=ð12nÞÞ. Obtained by set- ting a 5 f, b 5 1, and d 5 kð1 2 jÞ and then letting j → 1.

3. A case of Greenwood, Hercowitz, and Huffman (1988): the quasi- linear

u c, hð Þ 5

c 2 wh121=n= 1 2 1=nð Þ½ �12j 2 1 1 2 j

if j ≠ 1,

log c 2 w h121=n

1 2 1=n

� � if j 5 1,

8>>>< >>>:

(9)

obtained by setting a 5 wð1 2 jÞ=ð1 2 1=nÞ, b 5 1 2 1=n, and d 5 1 2 j.

The MaCurdy function has already been discussed in some detail. The generalized log-log formulation is used in section V.C.3 on transitional dynamics below. As we show there, this case inherits the analytical conve- nience of the standard log-log case when combined with a Cobb-Douglas production technology and 100% depreciation and in fact can be viewed as our generalization of this setting.20 The quasi-linear Greenwood- Hercowitz-Huffman preferences (referred to as GHH) feature a constant Frisch elasticity equal to 2n; clearly, here a n < 0 is needed. These prefer- ences preclude any income effect on hours worked. With a substitution effect alone, GHH preferences imply increasing hours as the wage rate increases. GHH preferences are nonhomothetic, but they are part of the Gorman class, and they imply an RRAc, 2uccðc, hÞc=ucðc, hÞ, which de- pends on hcv=ð12nÞ.21 Quasi-linear preferences are widely used in the

20 In the online appendix, we derive the regularity conditions for this case. 21 Note also that the preferences (9) are well defined only with 1 > wx2ð12nÞ=n=ð1 2 1=nÞ,

and consequently additional restrictions are required.

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applied theory and labor literatures, where the household problem is of- ten assumed to be static and j can be set to zero without loss of generality. Finally, setting n 5 0 in (8), we also obtain a parametric KPR class that en- compasses the well-known forms used in the applied business-cycle liter- ature, including the second constant Frisch elasticity case in Trabandt and Uhlig (2011); we discuss this case in section V.C.2.22

C. Features

Within the proposed parametric class (8) there are many possible func- tions that are all consistent with balanced growth under falling hours but that differ in important ways. Some of these properties are now dis- cussed. We begin with curvature, then turn to the Frisch elasticity of labor supply, and finally discuss qualitatively different transitional dynamics in two special cases of the model where we also make specific assumptions on technology for the sake of tractability.

1. Consumption Curvature

In a time-additive setting without an hours choice, balanced-growth pref- erences imply that the coefficient of RRA, 2u00ðcÞc=u0ðcÞ, is constant and inversely related to the IES of consumption. However, in a context where leisure is valued, it is more difficult to define these concepts. A natural measure of risk aversion would define a lottery over consump- tion and hours or over wealth; Swanson (2012) discusses this question in detail. Thus, what we defined as our RRAc function above, 2uccðc, hÞc= ucðc, hÞ, is not the most natural measure of risk aversion: it is defined as lot- teries over consumption only, keeping h fixed. Turning to its characteri- zation, lemma 2 shows that it is endogenous—it is a function of hcn=ð12nÞ— but it is constant along the balanced-growth path. Unlike when n > 0, in the standard KPR setting, the RRAc is exogenous: Proposition 1. Given the preferences specified in theorem 1, with

n 5 0, the RRAc is independent of c and h: it equals j. With n ≠ 0, how- ever, it can (but will not necessarily) depend on hcn=ð12nÞ. Proof. For the KPR class, this is verified straightforwardly. For the case

n ≠ 0, two cases are dealt with in the text below: one where the RRAc is decreasing in hcn=ð12nÞ and one where it is constant and exogenous (and equal to j). QED The MaCurdy utility function represents a case with n ≠ 0 for which the

RRAc is still exogenous and equals j. However, under GHH, for example, it is straightforwardly verified that the RRAc is endogenous. For some applications, perhaps especially in asset pricing, it may be valuable to

22 This is obtained as a special case of setting d 5 j.

labor supply in the past, present, and future 139

consider preferences where the RRAc is decreasing in the consumption- hours aggregate hcn=ð12nÞ: here, booms involve lower consumption curva- ture. One can imagine different functional forms here, but one possibil- ity is uðc, hÞ 5 c12j=ð1 2 jÞ 2 whec22j=e.23 This is an extension of the MaCurdy case and allows an RRAc that decreases in x 5 hc

n=ð12nÞ 5 hc1=e

when w > 0, j > 2, and e > j 2 1. The IES of consumption measures how the growth rate of consump-

tion is affected by a change in the real interest rate. As for risk aversion, several definitions are possible when there is also an hours choice. In the MaCurdy case, however, where utility is additive in consumption and hours, we find that d logðct11=ctÞ=d logð1 1 rÞ, where r is the net real in- terest rate between t and t 1 1, must equal 1=j—that is, one divided by the RRAc.

2. Frisch Elasticity

The Frisch elasticity of labor supply is defined as the percentage change in hours when the wage rate is changed by 1%, keeping the marginal utility of wealth constant. It captures the desire to intertemporally substi- tute hours worked when there is a frictionless market for borrowing and lending: if a period is short, most of the extra labor earnings will be spent at other times, so the current marginal utility of wealth will (almost) not be affected by the hours choice. The functional form in theorem 1 implies that the Frisch elasticity is

constant along the balanced-growth path. To see this, note that we can write u2ðc, hÞ 5 z1ðxÞc12j=h, u11ðc, hÞ 5 c2j21z2ðxÞ, u22ðc, hÞ 5 z3ðxÞc12j=h2, and u12ðc, hÞ 5 z4ðxÞc2j=h, where z1, z2, z3, and z4 are some functions of x 5 hcn=ð12nÞ. Inserting these expressions into the formula for the Frisch elasticity, u2ðc, hÞu11ðc, hÞ=fh½u22ðc, hÞu11ðc, hÞ 2 u12ðc, hÞ2�g, one immedi- ately sees that the Frisch elasticity depends only on x, which is constant along a balanced-growth path. Its balanced-growth value will, in general, be endogenous and depend on model parameters; in a model with het- erogeneous agents, it would also differ across rich and poor, and so on. For the KPR formulation, Trabandt and Uhlig (2011) provide a theo-

rem specifying for which subclass of KPR the Frisch elasticity is constant globally—that is, independent of ðc, hÞ: it is constant under uðc, hÞ 5 logðcÞ 2 wh111=v=ð1 1 1=vÞ and under uðc, hÞ 5 c12j=ð1 2 jÞ½12 kð1 2 jÞh111=v=ð1 1 1=vÞ�j (for j ≠ 1 and k > 0), but it is constant for no other function.24 Our formulation, for n ≠ 0, appears harder to characterize fully in this regard, but there are at least two cases in which the Frisch elasticity is constant. One is our GHH formulation of section V.B,

23 This is obtained by setting d 5 1 in (8) and defining the other parameters appropriately. 24 Recall that we nest the former as a special case of the latter using (8).

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for which the Frisch elasticity is equal to 2n.25 Another case is obtained with the MaCurdy function.

3. Transitional Dynamics: Two Simple Examples

What are the implications for hours worked based on a neoclassical growth framework when productivity or wages do not grow at a constant rate or when the interest rate is not constant? The postwar period is a case in point where many economies experienced adjustments following the war disruptions. The restriction to KPR preferences actually still leaves open, even qualitatively, how hours would move under a transitional pe- riod. We now briefly discuss economy-wide dynamics from the perspective of our utility class. We then also need to make assumptions on technology. Under some additional conditions, as discussed in section IV.C.5, the

preferences in theorem 1 guarantee the existence of a balanced-growth path for any neoclassical production function. Along this path, the growth rates of all the variables are a function only of the rate of techni- cal change g and population growth h, as well as our new preference pa- rameter n. For transition dynamics, it is useful to detrend any growing (or shrinking) variables with their respective growth rates. Detrended cap- ital—the only state variable in our one-sector model—is k̂t ; Kt=ðhtgð12nÞtÞ, whereas we define detrended hours worked and consumption as ĥt ; ht=g

2nt and ĉt ; ct=gð12nÞt, respectively. Thus, on the balanced path, k̂ is constant, and this value, as usual, is determined jointly by the Euler equa- tion, which delivers k̂=ĥ as a function of preference and technology pa- rameters, and the first-order condition for hours, which involves k̂ and ĥ, where for each equation the resource constraint has been used to elim- inate the consumption variable. For given model parameters, the transi- tional dynamics can then be solved for numerically using standard meth- ods—for example, using linearization around the steady-state value k̂. It is beyond the scope of the present analysis to fully explore how the prefer- ences we propose alter transition dynamics in well-known models. Rather, we will illustrate different possibilities with simple examples where the dy- namics can be solved for in closed form. The main case focuses on pref- erences with j 5 1; the second looks at the MaCurdy class. As we shall see, the transitional dynamics for hours are qualitatively different in these two cases. Both cases involve 100% depreciation of capital, which does not appear restrictive to us since the focus here is on growth and the time

25 Interestingly, although the Frisch elasticity for any given function uðc, hÞ generally does not feature invariance with respect to monotone transformations—the elasticity for u is different from that for f(u), where f is monotone—it is actually invariant in the GHH case.

labor supply in the past, present, and future 141

period can be chosen to be long enough that such an assumption is not so unrealistic. Cobb-Douglas production with j 51.—Assume that j 5 1, the produc-

tion function is Cobb-Douglas—that is, K at ðgththtÞ12a—and there is 100% depreciation. We then obtain the following planner first-order conditions (using detrended variables):

hg 12nk̂t11 5 k̂

a t ĥ

12a t 2 ĉt,

2 1 2 að Þ k̂t ĥt

� �a 5

ĉt

ĥt

v0 xtð Þxt=v xtð Þ 1 1 nv0 xtð Þxt= 1 2 nð Þv xtð Þð Þ

� � ,

ĉt11 ĉt

1 1 nv0 xtð Þxt= 1 2 nð Þv xtð Þð Þ 1 1 nv0 xt11ð Þxt11= 1 2 nð Þv xt11ð Þð Þ

5 ba

g 12n

ĥt11

k̂t11

� �12a ,

with xt 5htc n=ð12nÞ t . It is straightforward to guess, and verify, that a constant

saving rate of abh—that is, ĉt 5 ð1 2 abhÞk̂at ĥ12at —a constant xt 5�x, and an hours supply of ĥt 5 ½k̂2nat �x12n=ð1 2 abhÞn�1=ð12naÞ satisfies these condi- tions, with

2 1 2 a

1 2 abh v �xð Þ 1 n

1 2 n v0 �xð Þ�x

� 5v0 �xð Þ�x (10)

determining the solution for �x.26 The speed of convergence 2∂ log ðk̂t11=k̂tÞ=∂ log k̂t is given by ð1 2 aÞ=ð1 2 anÞ and is consequently strictly increasing in n. Obviously, not any function v can be used here, and in the online appendix we use the generalized log-log formulation under which (10) can be solved explicitly for �x.27 Since the Frisch elasticity and theconsumption curvature are functions of xt alone, the closed-form so- lution also implies that they are constant even along the transition. Under the above solution with n > 0, if the initial capital stock is below

its balanced-growth level, hours worked will be above their balanced- growth level. This case thus suggests a rationalization for the high and steeply decreasing hours worked in France or Germany after World War II given the low initial physical capital stock. Of course, the precise shape of the transitional dynamics depends on the functional form of preferences and technology, so the example here is meant only as an il- lustration, and the dynamics do not generalize; for example, below we construct a case where x adjusts over the transition but where detrended hours are constant. MaCurdy preferences with constant elasticity of substitution (CES) produc-

tion.—Under MaCurdy preferences and j ≠ 1, a Cobb-Douglas production

26 The closed-form solution is also obtained in the presence of TFP variations, predict- able or not. For n ≠ 0, hours then respond to TFP but x and the saving rate are constant.

27 Recall that we analyze the generalized log-log case above.

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function does not admit a closed-form solution for the neoclassical growth model; in particular, the saving rate will not be constant. However, one can show that the result in Koulovatianos and Mirman (2007) can be extended to the MaCurdy case (and thus with endogenous hours): under full depre- ciation, a closed-form solution with a constant saving rate along the transi- tion path is obtained when jε 5 1, where ε is the degree of substitutability between capital and hours in a CES production function. Thus, a constant saving rate can be optimal for very high consumption curvature, and hence a low intertemporal consumption substitutability, so long as the production function features a correspondingly low elasticity of substitution between inputs. How are hours chosen in this case? It is easy to show that detrended

hours, ĥ, will be constant during transition. That is, unlike in the case we looked at first, where the hours-consumption composite hcn=ð12nÞ is constant and hours vary with the capital stock, here detrended hours re- main constant whereas hcn=ð12nÞ moves monotonically. The saving rate re- mains constant along the transition and is equal to 1 2 hðabÞ1=j. Given CES production, factor shares also change along the transition.28

VI. Empirical Challenges

We have argued that the preference class proposed here can rationalize some broad patterns in the aggregate data across countries and over time. Our own view is that this rationalization is not just coherent but also a plausible explanation. In this section, we briefly discuss some empirical challenges that do remain. One is postwar US data, which we have men- tioned are an exception internationally; what can account for this? Our second question concerns the theory’s implication for the cross section of consumers: does it fit the facts here as well?

A. Hours Worked in the Postwar United States

The purpose of the short present section is not to provide an ambitious and full account of postwar US data but rather to suggest some possibil- ities. Let us first refer back to figure 3 and in particular look at the United States as contrasted with Germany and France. Clearly, hours fell at a fast rate in these European economies—indeed, at a much faster rate than in the broader cross section of countries that we looked at in the data sec- tion of the paper: the cross-country average is a rate of decline in hours

28 The closed-form solution with a constant saving rate extends to a case where there are additional movements in Harrod-neutral technology. An above-trend such technology value would then cause ĥ to rise above its balanced-growth value.

labor supply in the past, present, and future 143

that lies in between that of the United States and those in Germany and France. First, independently of the sign of n, higher taxes and a larger transfer

system would lower the hours worked, and so if these policy variables di- verged between two regions, one would expect hours worked to diverge as well. This argument was put forth in Prescott (2004) and Rogerson(2006, 2008); these authors argue that the gap in hours worked that opened up over the postwar period reflects relative changes in taxes and transfers: there have been upward movements of tax rates in Europe relative to those in the United States.29

Second, there has been demographic change, with the baby boom stand- ing out as a major factor. In the online appendix, we show the implied movements of aggregate US hours due to demographics based on a me- chanical view of labor supply by age (using current hours per adult by age to project backward). Clearly, demographics account for a fall in labor supply after the 1950s and then a turnaround only in the second half of the 1970s and thus can be an important factor to take into account. Third, another potential explanation is that median wages have not

grown much in the United States over the period. That is, if the vast ma- jority of the population does not experience wage growth, roughly con- stant total hours are consistent with our theory. Thus, the well-documented increase in wage inequality from the late 1970s onward—which is a period where wages actually rose—is a possible third factor. A fourth observation is that women’s labor-force participation increased more in the United States than elsewhere.30 Many factors can potentially explain this fact—for exam- ple, changes in workplace norms, the relative wage of women, or household production technologies.

B. Cross Section of Individuals

The empirical discussion so far has exclusively considered aggregates. Two (related) questions thus seem relevant: (1) Will aggregates behave in accordance with representative-agent theory if the microeconomic re- ality features significant heterogeneity? (2) Are the available data on individuals consistent with our theory where, on the balanced-growth path, income effects exceed substitution effects?

29 Interestingly, Ohanian, Raffo, and Rogerson (2008) look at the developments in a number of Organization for Economic Cooperation and Development (OECD) countries, along with tax-rate data, and also point to falling hours. They interpret this phenomenon as transitional and model it with Stone-Geary preferences as in the approach taken by Bick, Fuchs-Schuendeln, and Lagakos (2018).

30 In a follow-up paper (Boppart, Krusell, and Olsson 2017), we look at all the factors mentioned here and tentatively conclude that they all play a role but that the quantitatively most important one is women’s increased labor-force participation.

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1. Addressing the Microeconomic Data

Beginning with the second question, one might first think that our the- ory, which by design features income effects that exceed substitution ef- fects on the balanced-growth path, also implies a negative wage-hours cor- relation across individuals at a point in time (absent heterogeneity in other dimensions). Matters are not so straightforward, however. In the presence of other income—and almost all individuals have some other in- come—wage increases typically act nonmonotonically on hours: a higher wage raises hours worked for low wage levels (when wage income is rela- tively unimportant for consumption, hence limiting the income effect) but lowers hours for high enough wages.31 Thus, one should not expect a negative wage-hours correlation even when controlling for assets. In- stead, the proper experiment designed to test our theory would scale wages up while at the same time scaling up nonwage income appropriately, be- cause our preference relation dictates that the income effect dominates the substitution effect as the economy is scaled (as it is on a balanced path).32 A second complication in looking at cross-sectional data is that any comparison must involve permanent wage differences (we discuss the role of temporary changes below). The third complication, of course, is that individuals may be heterogeneous in dimensions other than wages or assets. Thus, when approaching microeconomic data, all these issues need to be taken into account. It is much beyond the scope of the present paper to do so, but we nevertheless briefly comment on some key prop- erties of these data and point to some indications that make us hopeful that our theory is consistent with the cross-sectional facts as well. To start with, a commonly stated suspicion is that there is a positive cor-

relation between wages and hours in the cross section of individuals (if nothing else, one would reach this conclusion by introspection); a robust such correlation would potentially challenge the theory, notwithstanding the remarks above. Is this suspicion borne out, however, when looking more carefully at the data? The Panel Study of Income Dynamics (PSID) is useful for this task and reveals a coefficient of 20.012 (insignificant) when simply running log wages against log hours (using men and the 2002 sample); the R2 is 0.000. There are really two challenges based on these observations. One is to identify factors that counteract the negative wage-hours prediction that must apply for at least some households, given our model. Second, and more importantly in a broader sense, as it is well known that it is very difficult to explain much of the overall cross-sectional

31 Using the MaCurdy function, this statement follows straightforwardly from the first- order condition and the budget in a static decision problem.

32 If wages are scaled by l, nonwage income must be scaled by l12n.

labor supply in the past, present, and future 145

variation in hours even in regressions of hours on wages controlling for a set of observables (e.g., age, sex, education, etc.), the challenge in under- standing people’s work behavior goes much deeper. In particular, it seems to point to significant unobservable heterogeneity. We now briefly com- ment on some explanatory factors—both observable and unobservable— and their implications. A number of observables do have statistically significant effects on

hours across individuals: to mention some, there is a life-cycle pattern in hours (first rising, then flat, then declining), people with children work less, women work less than men, and highly educated individuals work slightly more than less educated individuals. There are various pos- sible explanations behind these findings. One is that the “effort cost” of working—think of the parameter w in MaCurdy’s preference specifica- tion given by (3)—depends on these observables, such as age and par- enthood. More subtly, permanent heterogeneity in personal traits in this regard can be a powerful source: some individuals simply have a low ef- fort cost of working, making them accumulate human capital in terms of both formal training and job skills, thus simultaneously delivering high hours and high wages. Relatedly, one can perhaps also interpret hetero- geneity in how much people like working or accumulating skills through the same parameter.33 Another possible mechanism is that suggested by Aaronson and French (2004) and Erosa, Fuster, and Kambourov (2016): there is a wage (penalty) premium for working part-time (overtime), per- haps due to a nonconvexity in production. A complicating factor in un- derstanding individual data is that the discernible correlations have changed over time—for example, as documented by Costa (2000), in addition to the fact that women have increased their hours massively relative to men both on the intensive and the extensive margin. Moving to factors that can be examined more closely from our theoret-

ical perspective and that also work in the direction of a positive corre- lation between hours and wages, consider intertemporal substitution. When wages move over time (perhaps even randomly), it is individually rational to smooth consumption and work harder when wages are high, provided that the individual has access to the appropriate assets. In par- ticular, for an individual with a steadily rising wage profile, our theory would predict higher hours worked when wages are higher, unless the in- dividual is prevented from moving wealth over time (say, due to binding borrowing constraints) or has a zero Frisch elasticity. To illustrate this point in some quantitative detail, in particular paying attention to the role of assets, let us consider a worker who experiences a (particularly

33 A combination of these channels at least appears to be useful in understanding our own hours choices.

146 journal of political economy

steep) fourfold real wage increase over her lifetime. For example, let the worker begin with an hourly wage of $25 (at age 20) and climb to an hourly wage of $100 (at age 55). Suppose first that the worker does not have access to any means of transferring wealth over time. Further- more, let us assume that the individual has no unearned income and so is literally “hand to mouth.” With n 5 0:2, our theory would then pre- dict that the fourfold increase in the wage rate would imply that hours fall by a gross rate of 420.2—that is, by about 24%—over her lifetime. Now con- sider the alternative scenario whereby the individual knows her future wage path at the beginning of time and has no constraints on borrowing and lending. Furthermore, assume that the market interest rate, which is important for the calculation to follow, will be constant and consistent with a balanced-growth rate of consumption of 2% per year. This implies consumption growth by a factor 2 between ages 20 and 55 for this worker. Let us also assume that preferences take the MaCurdy form.34 Then a fourfold increase in the wage rate of the individual implies a growth rate in hours worked between ages 20 and 55, given by

h55 h20

5 22j4ð Þv:

With j 5 1:55 and v 5 0:82 (in line with the microeconomic estimates in Chetty et al. 2011 and n 5 0:2), we obtain h55=h 20 5 1:29. In other words, it is optimal for the individual to increase her hours worked over the 35 years by 29% (say, from 35 to 45 hours). Thus, we see that the ability to smooth consumption matters greatly, giving us a range of predictions between 224% and 129%. The lower bound, based on no ability to smooth, would moreover rise in the presence of nonwage income, as the income effect of a wage change is then less potent. One could simi- larly also quantify the effects of imperfect insurance in case the wage fluc- tuations are random and/or the expectations about future wages are not fully rational, giving further bounds. Based on these calculations, we find it plausible that the intertemporal substitution channel does play a signif- icant role, but a formal test against data is also beyond the scope of the present work. Let us take stock. First and foremost, our understanding of the cross

section of hours worked is very incomplete: unobservable factors seem to explain a very large part of the variation in the data, leaving a corre- lation close to zero between hours and wages. This, however, does not mean that our theory is not operative: we argued above that there are

34 For the hand-to-mouth calculation, the specific utility function employed does not matter, as long as it is in our class.

labor supply in the past, present, and future 147

plausible factors that generate a positive hours-wage correlation while still being fully consistent with our theory and its balanced-growth predic- tions. In fact, structural estimation of the MaCurdy formulation of pref- erences is carried out in Heathcote, Storesletten, and Violante (2014), who use PSID data and control for a number of observables; they indeed find a power on consumption (j) that exceeds one (which, as pointed out above in sec. III.B, implies falling hours in a balanced-growth con- text). Remarkably, their estimated MaCurdy parameters j and v imply a value of n that is also quantitatively in line with the aggregate time- series data. Another recent study—Bick, Fuchs-Schuendeln, and Lagakos (2018)—merges microeconomic data from a number of different coun- tries and finds a negative correlation between individual hours and wages, once country fixed effects are included; they take their result as evidence that income effects dominate substitution effects on the level of individual utility. Thus, some existing evidence from estimates using cross-sectional data speaks in favor of our hypothesis, but to be sure, there are also inher- ent difficulties in providing convincing tests of the theory. As pointed out above, the ideal experiment would change a wage permanently, along with the consumer’s wealth being appropriately scaled, and then look at the hours response. Several parts of this experiment are obviously very diffi- cult to control. In addition, in the short run, one is unlikely to see the the- ory play out perfectly due to various frictions making it difficult for house- holds to adjust their hours and due to institutional rigidities, norms, and so on. Further investigations of hours differences across individuals should, in our view, simply be high on the agenda for future research.

2. Extending the Model to Allow for Heterogeneity

The other main question raised above involves the possible failure of ag- gregation in models of the present kind. Indeed, preferences in the class we propose here do not aggregate even under complete markets (except in very special cases). However, it is easy to show that at least some cases are perfectly consistent with a constant percentage decline in hours under balanced growth. One such case is where allindividuals have the same utility function—in the class defined here—but where they differ in wealth and wages and where the latter are random and there is no insurance except through saving, as in Huggett (1996).35 If, in addition, different house- holds had different utility functions—say, varying in their n’s—we suspect that one would still obtain a limit stationary distribution with aggregate characteristics similar to those coming out of a representative-agent model. Such a model would likely not behave like a true representative-agent

35 We demonstrated this in an earlier version of this paper.

148 journal of political economy

model in its short-run dynamics but would be consistent with the down- ward trend in hours under balanced growth.

VII. Conclusions

We have presented an extension to the standard-preference framework used to account for the balanced-growth facts. The new preference class admits that hours worked fall at a constant rate when labor productivity grows at a constant rate, as we have also documented the data to show across time and space. The new preference class intuitively involves an income effect of productivity on hours worked that, along a growth path where unearned income also grows at its balanced rate, exceeds the sub- stitution effect to an extent captured by the parameter n > 0. A 1 per- centage point productivity gain admits an equal consumption gain, but under our setting individuals choose to benefit from this gain by also low- ering hours worked: hours fall by n percent and consumption rises by only 1 2 n percent. We believe that our new preference class has potentially interesting

implications in a range of contexts. As for growth theory and growth em- pirics, note that on our balanced path, the main macroeconomic aggre- gates (output, investment, consumption) grow at the gross rate g12n > 1 (ignoring population growth), that is, at a rate lower than productivity and in a way that is determined by the preference parameter n. Notice also that from a development perspective, falling hours worked is not a sign of economic malfunctioning but rather the opposite: it is the natural outcome given preferences and productivity growth, and it instead illus- trates clearly how output growth is an incomplete measure of improved welfare over time in the development context (see Jones and Klenow 2016): leisure grows as well, and from the perspective of our labor-supply theory, joint output and leisure growth is what we should expect. Interest- ingly, our theory says that growth theory probably should not abstract from labor supply (which is typically set to 1 in models); rather, it seems an important variable to model, as it determines the growth of long-run output in conjunction with the process of technical change. Does our preference class have something to say about business-cycle

analysis? We cannot identify any immediate substantive implications, but it is clear that our model can be amended with shocks and transformed into a stationary one (including a detrending of hours) that can be ana- lyzed just like in the standard literature. The preference class consistent with hours falling at a constant but low rate is a bit different from the stan- dard one. If one uses the MaCurdy utility function, one can admit an ar- bitrarily low elasticity of intertemporal substitution of consumption (1=j), namely, if the Frisch elasticity (v) is also very low: the connection between these two elasticities is determined quantitatively by the value of n.

labor supply in the past, present, and future 149

Other areas where the new preference class may be interesting to en- tertain include asset pricing and public finance. For asset pricing—as we showed in this paper—it is possible to have attitudes toward risk behave qualitatively differently, and possibly more in line with data, than using standard balanced-growth preferences; we demonstrated this with an ex- ample in our utility class featuring countercyclical consumption curvature. These same features would potentially also help explain portfolio-choice patterns across wealth groups.36 For public finance, the sustainability of programs such as social security and debt service in the future depends greatly on how hours worked (and of course productivity) will develop. We build the explanation for the secular drop in hours into prefer-

ences. What about institutional factors? We take the view that over a long horizon, they must be endogenous and thus must be responding to pref- erences. Moreover, the facts—an approximately constant rate of decline in hours worked—are too stark not to suggest a “deep,” and time- and space-independent, explanation. In the short to intermediate run, of course, institutional factors, as well as a variety of frictions, may prevent hours from following what is implied from our theory of labor supply. What, then, are alternative theories that could explain why hours fall?

Could an alternative theory explain the past without contradicting the constant-hours presumption of the standard macroeconomic model? Other formulations are possible, such as the Stone-Geary formulation pro- posed in Bick, Fuchs-Schuendeln, and Lagakos (2018), following Atkeson and Ogaki’s estimates (1996). Whether the transition dynamics in such a model are slow enough to generate the long-term, constant percent de- cline in hours observed in the data is an open question. Another approach is to drop the stationarity of preferences and to instead build trends di- rectly into them, as in Mertens and Ravn (2011), or indirectly, based on an explicit model of human capital accumulation and home production, as in Hercowitz and Sampson (1991). It is an open and interesting ques- tion as to under what conditions home-production models can be con- sistent with falling hours and balanced growth.37 In sum, our present analysis should be viewed as one way to look at the long-run data, and it should carefully be compared to others, especially since their implica- tions for the future differ markedly. For example, the Stone-Geary for- mulation implies that the future will see flat hours, independently of how future productivity evolves, whereas the implications of the prefer- ences we propose here suggest a tight hours-productivity link. So on this issue we side with Keynes.

36 See, e.g., Wachter and Yogo (2010). 37 For a model of structural change between home and market production that gener-

ates long-run changes in hours worked in the market place, see Ngai and Pissarides (2008).

150 journal of political economy

Finally, it should be emphasized that the preference class we propose is consistent with other features of preferences introduced in applied work, such as habits of a variety of forms.38 It is also possible to consider an Epstein-Zin version of our preferences.39 The purpose here is thus not to take issue with preference formulations that depart from time invari- ance or time separability but rather to show that these can be combined with the present setting.

Appendix

A. Proofs

We now present the proofs of lemma 1, lemma 2, and theorem 1. We start by proving the two lemmata, characterizing the MRS function between c and h and the curvature with respect to consumption: the RRAc function. The proof of the- orem 1 then uses these lemmata to derive the final characterization. Because the proofs will involve a large number of auxiliary functions that are functions of ei- ther hc n=ð12nÞ or h, we economize somewhat on notation by sometimes denoting hc n=ð12nÞ by x and by systematically letting fi be a function of x whereas mj is a func- tion of h (where i and j are indices for the different functions we define). A sequence of constants will also appear; they are denoted Ak, accordingly, from k 5 1 and on.

1. Proof of Lemma 1

Because l is arbitrary, we can set it in (5) so that cl12n 5 1. This delivers

2 u2 1, hc

n= 12nð Þð Þ u1 1, hc

n= 12nð Þ� � 5 wc21= 12nð Þ:

Evaluating (5) at l 5 1, we obtain 2u2ðc, hÞ=u1ðc, hÞ5w. Inserting this expres- sion yields

u2 c, hð Þ u1 c, hð Þ

5 c1= 12nð Þ u2 1, hc

n= 12nð Þð Þ u1 1, hc

n= 12nð Þ� � : (A1)

Now identifying q(x) as u2ð1, xÞ=u1ð1, xÞ, where x 5 hcn=ð12nÞ, gives the result in lemma 1. QED

It follows from lemma 1 and u being twice continuously differentiable that q is continuously differentiable.

38 One can, e.g., add a term Xt 5 c r t X

12r t21 in front of h

121=n to the GHH case we look at, hence obtaining the preferences studied in Jaimovich and Rebelo (2009).

39 Details are available from the authors upon request.

labor supply in the past, present, and future 151

2. Proof of Lemma 2

The second first-order condition, (6), holds for all l so that it can be differentiated with respect to l and then evaluated at l 5 1 and divided by (6) again to yield

1 2 nð Þcg12n u11 cg 12n, hg2nð Þ

u1 cg 12n, hg2n

� � 2 nhg2n u12 cg12n, hg2nð Þ u1 cg

12n, hg2n � �

5 1 2 nð Þc u11 c, hð Þ u1 c, hð Þ

2 nh u12 c, hð Þ u1 c, hð Þ

: (A2)

This equation has to hold for all g (and consequently one must adjust R, but R does not appear in the equation). Moreover, it has to hold for all c and h; it has to hold for all h because assumption 1 allows any w and hence any h (given an ar- bitrary c). Given this, by setting g so that cg12n 5 1 we can state (A2) as

1 2 nð Þ u11 1, hc n= 12nð Þð Þ

u1 1, hc n= 12nð Þ� � 2 nhcn= 12nð Þ u12 1, hc

n= 12nð Þð Þ u1 1, hc

n= 12nð Þ� � 5 1 2 nð Þc u11 c, hð Þ

u1 c, hð Þ 2 nh

u12 c, hð Þ u1 c, hð Þ

which holds for all c and h. We conclude that the right-hand side of equation (A2) depends only on hcn=ð12nÞ; that is, we can write

1 2 nð Þc u11 c, hð Þ u1 c, hð Þ

2 nh u12 c, hð Þ u1 c, hð Þ

5 f1 hc n= 12nð Þ� �, (A3)

where f1 is then defined by the expression on the left-hand side of equation (A2) evaluated at cg12n 5 1. Differentiating (7) with respect to c gives

u12 c, hð Þu1 c, hð Þ 2 u11 c, hð Þu2 c, hð Þ u1ðc, hÞ2

5 cn= 12nð Þq xð Þ 1 2 n

1 nc1= 12nð Þq 0 xð Þhcn= 12nð Þ21

1 2 n ; cn= 12nð Þf2 xð Þ,

where we used the notation x 5 hcn=ð12nÞ and the last equality simply defines a new function f2. Then, again using the characterization of the MRS function to replace u2ðc, hÞ=u1ðc, hÞ 5 c1=ð12nÞqðhcn=ð12nÞÞ, we obtain

u12 c, hð Þ u1 c, hð Þ

2 u11 c, hð Þ u1 c, hð Þ

c1= 12nð Þq xð Þ 5 cn= 12nð Þf2 xð Þ,

and hence

h u12 c, hð Þ u1 c, hð Þ

5 u11 c, hð Þ u1 c, hð Þ

hc1= 12nð Þq xð Þ 1 hcn= 12nð Þf2 xð Þ 5 c u11 c, hð Þ u1 c, hð Þ

xq xð Þ 1 xf 2 xð Þ:

This expression can be combined with equation (A3) to conclude that 2cu11ðc, hÞ= u1ðc, hÞ must be a function only of x; we call this function p.40 QED

40 The function p(x) is thus defined by

2 1 2 nð Þp xð Þ 1 n p xð Þxq xð Þ 2 xf 2 xð Þ½ � 5 f1 xð Þ, which straightforwardly offers a solution (that will depend on q, f1, and f2).

152 journal of political economy

3. Proof of Theorem 1

We now combine the information in lemmata 1 and 2 to complete our proof of theorem 1. We do this in two steps. First, we analyze the case with n ≠ 0, and then we analyze the case with n 5 0. Note that the case with n 5 0 is already discussed in King, Plosser, and Rebelo (1988).

The proof strategy is very similar in the two cases. First, we integrate the RRAc function in lemma 2 with respect to c to obtain a functional form for u1. As we integrate with respect to c, an unknown function of h appears. Then, by differen- tiating this u1 function with respect to h we obtain an expression that can be com- pared to a restriction on u12=u1 found in the proof of lemma 2. This comparison gives us more restrictions on the unknown function of h. Thus, since the proof of lemma 2 uses lemma 1, we are ensured that the function we arrive at is consistent with both of our lemmata. Based on our u1, we then integrate to deliver a candi- date for u. The integration yields a new unknown function of h, but we can again restrict this function by differentiating our candidate u with respect to h and comparing the result to lemma 1. This delivers our final functional form.

Case with n ≠ 0.—Note that the characterization of the RRAc function in lemma 2 can be restated as

∂ log u1 c, hð Þ ∂ logðcÞ 5 2p exp logðhÞ 1

1 2 n logðcÞ

� � :

This equation can be integrated straightforwardly with respect to logðcÞ to arrive at

u1 c, hð Þ 5 f3 hcn= 12nð Þ � �

m1 hð Þ, (A4) where f3 is a new function of x and m1 is an arbitrary function of h.

Now observe that it follows from the proof of lemma 2 that hu12ðc, hÞ=u1ðc, hÞ can also be written as a function of x alone: it equals 2pðxÞxqðxÞ 1 xf 2ðxÞ. We use this fact to further restrict the function m1. In particular, by taking derivatives in equation (A4) with respect to h, multiplying by h, and dividing by u1, we obtain an expression for hu12ðc, hÞ=u1ðc, hÞ that can be written as

f4 hc n= 12nð Þ� � 1 m01 hð Þh

m1 hð Þ ,

where f4 is defined by f4ðxÞ ; f 03 ðxÞx=f3ðxÞ. For the consistency of these two ex- pressions for hu12ðc, hÞ=u1ðc, hÞ—the one just stated and the arbitrary function of x given above (2pðxÞxqðxÞ 1 xf 2ðxÞ)—it must be that m01ðhÞh=m1ðhÞ is a con- stant.42 Hence, m1ðhÞ 5 A1hk for some constants A1 and k; that is, it is isoelastic. Using this fact in (A4) gives

41 The integration delivers an expression for log u1ðc, hÞ as a function of log x plus a function of h. The latter function can be a function only of h since c was integrated over. The function of log x can be rewritten as a function of x. Equation (A4) is then obtained after raising e to the left- and right-hand sides of this equation, and f3 and m1 are defined accordingly.

42 If m01ðhÞh=m1ðhÞ would depend on h, consistency could not be fulfilled for any given combination of c and h.

labor supply in the past, present, and future 153

u1 c, hð Þ 5 f3 hcn= 12nð Þ � �

A1h k : (A5)

Since n ≠ 0, the expression on the right-hand side can equivalently be written f5ðhð12nÞ=ncÞhk, by defining f5ðxÞ 5 A1f3ðxð12nÞ=nÞ. Therefore, (A5) can be easily inte- grated with respect to c to deliver

u c, hð Þ 5 f6 hcn= 12nð Þ � �

hk2 12nð Þ=n 1 m2 hð Þ, (A6)

where f6 is the new function that results from the integration of f5 over c and m2 is an arbitrary function of h (as the integration was over c). With the aim of further restricting m2, we can express u2 as

u2 c, hð Þ 5 u1 c, hð Þc1= 12nð Þq xð Þ 5 f3 xð ÞA1hkc1= 12nð Þq xð Þ 5 f7 hcn= 12nð Þ � �

hk21=n, (A7)

where we have used the characterization of the MRS function in lemma 1, (A5), and finally the definition f7ðxÞ ; f3ðxÞA1x1=nqðxÞ. We can now check consistency by differentiating u with respect to h in (A6) and comparing with (A7). The re- sult is

k 2 1 2 n

� � f6 xð Þhk21=n 1 cn= 12nð Þf 06 xð Þhk2 12nð Þ=n 1 m02 hð Þ ; f8 xð Þhk21=n 1 m02 hð Þ,

where the equality comes from collecting terms and defining a new function f8 accordingly. For consistency, thus, this expression has to equal f7ðxÞhk21=n for all x and h. This is possible if and only if m02ðhÞ 5 A2hk21=n, where A2 is a constant. Concentrating first on the case where k 2 1=n ≠ 21, we obtain m2ðhÞ 5 ð11k 2 1=nÞ21A2h11k21=n 1 A3 ; A4h11k21=n 1 A3. The constant A3 can be set ar- bitrarily, as it does not affect choice. The second term in (A6) can thus be merged together with the first term using factorization, and we can write uðc, hÞ as f9ðxÞh11k21=n 1 A3, with f9ðxÞ ; f6ðxÞ 1 A4. Now note that h11k21=n 5 x11k21=nc2nð11k21=nÞ=ð12nÞ, so that uðc, hÞ can be written as f9ðxÞx11k21=nc2nð11k21=nÞ=ð12nÞ 1 A3. Now define vðxÞ ; ½ð1 2 jÞf9ðxÞx11k21=n�1=ð12jÞ and j ; kn=ð1 2 nÞ, and we conclude that we can write uðc, hÞ 5 ½ðc � vðxÞÞ12j 2 1�=ð1 2 jÞ (where A3 has been set to 21=ð1 2 jÞ).

In the special case where 1 1 k 5 1=n, we find from equation (A6) that uðc, hÞ 5 f6ðhcn=ð12nÞÞ 1 m2ðhÞ, but we also see from the arguments above that m2(h) has to equal A2 log h 1 A5, where A5 is again an arbitrary constant. Since (given n ≠ 0) we can write logðhÞ 5 logðxÞ 2 n logðcÞ=ð1 2 nÞ, our candidate u can be rewritten as uðc, hÞ 5 f6ðxÞ 2 A2n logðcÞ=ð1 2 nÞ 1 A2 logðxÞ 1 A5. The constant A5 can be set to zero, and we can write uðc, hÞ 5 2A2n=ð1 2 nÞ ½logðcÞ 2 ð1 2 nÞf6ðxÞ=ðA2nÞ 2 ð1 2 nÞlogðxÞ=n�. The factorized constant can be normalized to 21 (as it does not affect choice), and we can then define log vðxÞ ; f6ðxÞ 1 ð1 2 nÞ logðxÞ=n, an arbitrary function; this concludes the case 1 1 k 5 1=n. Hence, we obtain the utility function

u c, hð Þ 5 c � v hcn= 12nð Þð Þð Þ12j 2 1

1 2 j if j ≠ 1,

logðcÞ 1 log v hcn= 12nð Þ � �

if j 5 1:

8>< >:

154 journal of political economy

Case with n 50.—In this case, we can rewrite the RRAc function in lemma 2 as

∂ log u1 c, hð Þ ∂ logðcÞ 5 2p hð Þ: (A8)

We can integrate this equation with respect to log c to obtain

log u1 c, hð Þ 5 2p hð Þ logðcÞ 1 m3 hð Þ, (A9) where m3 is an arbitrary function, given that we integrated over c. Differentiating with respect to h then gives

u12 c, hð Þ u1 c, hð Þ

5 2p0 hð Þ log c 1 m03 hð Þ: (A10)

From the proof of lemma 2 we know that u12ðc, hÞ=u1ðc, hÞ can be written as a function of h alone (recall that n 5 0). This implies that p0ðhÞ 5 0; that is, the only version of equation (A9) that is possible is log u1ðc, hÞ 5 2j logðcÞ 1 m3ðhÞ, where j is a constant. Using this fact and raising e to both sides of (A9) then delivers

u1 c, hð Þ 5 c2jm4 hð Þ, (A11) where m4ðhÞ 5 expðm3ðhÞÞ. Integrating (A11) with respect to c, we can write

u c, hð Þ 5 c � v hð Þð Þ12j 2 1

1 2 j 1 m5 hð Þ if j ≠ 1,

m4 hð Þ logðcÞ 1 log v hð Þ if j 5 1;

8>< >: (A12)

here, in the first equation 21=ð1 2 jÞ 1 m5 is another function (of h) that ap- pears because of the integration over c and v(h) is defined from ðvðhÞ12j 2 1Þ= ð1 2 jÞ 5 m4ðhÞ, whereas in the second equation log vðhÞ is the function that ap- pears due to the integration.

We now show, along the lines of the case where n ≠ 0, that m4 and m5 will have to have very specific forms. We look at each in turn. So in the case with j ≠ 1, combine (A11) with lemma 1 to write

u2 c, hð Þ 5 c12jq hð Þm3 hð Þ: (A13)

This can be contrasted with the result of differentiating (A12) with respect to h, an operation that yields

u2 c, hð Þ 5 c12jvðhÞ2jv0 hð Þ 1 m05 hð Þ:

Since these last two equations both have to hold for all c and h, it must be that m05ðhÞ 5 0—that is, that m5(h) is a constant (which can be abstracted from).

Turning to the case where j 5 1, along the same lines we again derive two ex- pressions for u2 and check consistency. Combining (A11) with lemma 1, one finds that u2 cannot depend on c. Differentiating the second line of (A12) with respect to h, however, delivers a function of c unless m4(h) is a constant; as it does not affect choice, we set this constant to 1.

labor supply in the past, present, and future 155

In sum, in the j ≠ 1 case we obtain uðc, hÞ 5 ½ðc � vðhÞÞ12j 2 1�=ð1 2 jÞ, and in the j 5 1 case we obtain logðcÞ 1 log vðhÞ. This completes the proof for the case n 5 0. We have completed the proof of the statement in our main the- orem. QED

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