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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

Electronically published December 11, 2019 [ Journal of Political Economy, 2020, vol. 128, no. 1] © 2019 by The University of Chicago. All rights reserved. 0022-3808/2020/12801-0001$10.00

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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.

labor supply in the past, present, and future 119

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

labor supply in the past, present, and future 121

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.

labor supply in the past, present, and future 123

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.

labor supply in the past, present, and future 125

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.

labor supply in the past, present, and future 127

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).

labor supply in the past, present, and future 129

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

,

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