Lifetime Aggregate Labor Supply with Endogenous Workweek Length*

Transcription

1 Federal Reserve Bank of Minneapolis Researc Department Staff Report 400 November 007 Lifetime Aggregate Labor Supply wit Endogenous Workweek Lengt* Edward C. Prescott Federal Reserve Bank of Minneapolis and Arizona State University Ricard Rogerson Arizona State University Joanna Wallenius Arizona State University ABSTRACT Tis paper studies lifetime aggregate labor supply wit endogenous workweek lengt. Suc a teory is needed to evaluate various government policies. A key feature of our model is a nonlinear mapping from ours worked to labor services. Tis gives rise to an endogenous workweek tat can differ across occupations. Te teory determines wat fraction of te lifetime an individual works, not wen. We find tat constraints on workweek lengt ave different consequences for total ours tan total labor services. Also, we find tat policies designed to increase te lengt of te working life may not increase aggregate lifetime labor supply. *Prescott tanks te National Science Foundation (Grant 04539), Rogerson tanks te National Science Foundation (Grant 0478, and Wallenius tanks te Yrjo Jansson Foundation for supporting te researc. We also tank our colleagues wit wom we discussed tis researc. We want to tank Simona Cociuba for providing exceptional researc assistance. Te views expressed erein are tose of te autors and not necessarily tose of te Federal Reserve Bank of Minneapolis or te Federal Reserve System.

2 As an empirical matter, te amount of labor supplied by an individual over is or er lifetime is effectively caracterized by two numbers: te fraction of lifetime spent in employment, and te ours worked wen employed. In tis paper we develop a tractable dynamic general equilibrium model tat delivers tis caracterization as an equilibrium outcome and present some of its implications for policy analysis and data interpretation. Te key feature of our model is a nonlinear mapping from ours of work in a given period to labor services provided in tat same period. Specifically, at low ours of work tis mapping is convex, due to suc factors as te costs associated wit getting set up in a job, communicating wit coworkers, meeting wit supervisors, and so on. At ig ours of work tis mapping is assumed to be concave, due to fatigue. Similar to Hornstein and Prescott (993), tis nonlinearity implies tat workweeks of different lengts are not perfect substitutes in generating labor services, but does so in a muc more tractable manner. We embed tis feature of tecnology into a dynamic model populated by a continuum of identical individuals and study efficient allocations and competitive equilibrium outcomes for tis economy. Two key findings emerge. First, efficient allocations are completely caracterized by two numbers: te fraction of lifetime spent in employment by eac individual, and ours worked wile employed. Second, altoug tis economy possesses an important nonconvexity, we sow tat efficient allocations can be acieved as competitive equilibria wit only a standard set of Arrow-Debreu markets: markets for consumption and labor services at eac date. In particular, tere is no need to allow for trade in lotteries or sunspots in order to implement efficient Rosen (978) also noted tat tis formulation was a tractable way to tink about intensive and extensive margins of labor supply.

3 allocations, as in Prescott and Townsend (984a, 984b), Rogerson (988), or Sell and Wrigt (993). To illustrate te usefulness of te model, we use it to consider te effects of several policies: a tax and transfer policy, a restriction on workweek, and a restriction on fraction of lifetime devoted to work. Several interesting findings emerge. First, te model implies a large aggregate labor supply elasticity in response to tax and transfer programs, and at te same time it predicts a very small (in fact, zero) elasticity for ours of work of continuously employed individuals. In tis regard, te model mimics te implications of a model tat simply assumes labor is indivisible. A key message is tat in our model, te aggregate labor supply elasticity wit respect to canges in taxes is a function not only of preference parameters, but also of tecnology parameters. Specifically, features of te mapping from ours of work to units of labor services are critical in determining te aggregate labor supply elasticity. A second finding is tat any distortion to one component of lifetime labor supply will be at least partially offset by movements in te oter component. Increasing working life by canging te nature of social security benefits, for example, will lead to a decrease in te workweek. Tis as important implications for ow policy makers forecast budget implications associated wit social security reform. Tird, distortions to lifetime labor supply can also lead to canges in ours of work and labor services in opposite directions. For example, we sow tat a decrease in te workweek can lead to greater aggregate ours of work but lower output. An additional implication of an exogenous restriction on workweek lengt in our model is tat te equilibrium wage per our of work necessarily decreases. In practice, many governments tat adopt workweek restrictions in an attempt to increase

4 employment (and tereby total ours of work) also simultaneously adopt policies tat lead to iger wages per our of work. Wen te ourly wage rate is not allowed to fall, it no longer follows tat a restriction on ours will necessarily increase aggregate ours of work. Tis is consistent wit te findings of te empirical literature. Any teory of workweek lengt sould be able to account for te simple observation tat workweek lengt differs significantly across occupations. By empasizing te role of tecnology as a determinant of optimal workweek lengt, our teory provides a natural explanation for suc differences. We illustrate tis by developing a two-occupation version of our model wit endogenous occupational coice. Tis extension provides an interesting implication about te relationsip between labor supply elasticities estimated from a cross-section of workers and te aggregate elasticity associated wit a cange in taxes. Specifically, in our model te cross-section elasticity of ours wit respect to wages produces an estimate of te preference parameter tat determines te curvature in te disutility of work function. However, because of te nonconvexity in tecnology, tis preference parameter is irrelevant for te response of aggregate ours to te tax and transfer program we study. An outline of te paper follows. Section presents some motivating observations from te data. Section introduces te nonconvex tecnology in a static setting, and Section 3 analyzes equilibrium in te dynamic setting. Section 4 presents te policy analysis, and Section 5 considers te extension to a two-occupation setting. Section 6 concludes. 3

5 . Motivating Facts In tis section we present five facts concerning labor supply over te lifetime tat serve as motivation for te model tat we develop in te next section. Te first two facts concern canges in te nature of work over te life cycle for individuals wit at least a moderate attacment to te labor force. We consider te cutoff for tis to be working more tan 800 ours during te year, wic is just over 5 ours per week. Fact : In te United States, te fraction of people working at least 800 ours declines significantly wit age. Te supporting evidence is sown in Figure, wic sows te fraction of individuals working more tan 800 ours per year as a function of age, based on te 004 CPS. Figure People Working at Least 800 Hours by Age, U.S % 80% 60% 40% 0% 0% Age Source Data: CPS 004 4

6 Note tat te fraction of individuals working above tis tresold level begins to decrease in te late 40s and falls relatively smootly troug to te early 70s, at wic point it flattens out somewat. Tis value drops by rougly a quarter between te ages of 48 and 60. Te next fact considers wat appens to te amount of work performed by tose individuals working more tan 800 ours per year. Fact : In te United States, ours worked per person for tose individuals wo work at least 800 ours per year declines little wit age. Te supporting evidence is sown in Figure, wic plots annual ours of work conditional on working at least 800 ours per year, again using data from te CPS for 004. Figure Hours Worked by Tose Wo Work More Tan 800 Hours, U.S. 004 Hours Worked by Tose Wo Work More Tan 800 Hours Hours per Year Age Source Data: CPS 004 5

7 Note tat te decrease in ours is muc smaller tan te decrease in te fraction of individuals working more tan 800 ours per year. Moreover, te ours series is relatively flat beyond age 60. Taken togeter, Facts and imply tat canges along te extensive margin dominate bot te decrease in ours for tose above 50 and te increase in ours worked for younger workers. Put somewat differently, te fraction of life spent in employment seems a key margin. Te next fact notes te importance of tis margin in understanding differences in ours of work across countries. Fact 3: About alf te difference in ours worked between te United States and continental Europe is accounted for by differences in te fraction of lifetime worked. Tis is sown in Figure 3. Figure 3 Source: OECD Database 6

8 Tis figure sows relative employment rates for five age groups (5 4, 5 34, etc. ) for four countries in continental Europe relative to te United States. Note tat for primeaged individuals te relative employment rate is approximately one, and tat it decreases quite significantly at younger and older ages. Te important message ere is tat individuals in tese European countries spend a muc smaller fraction of teir life in employment. Altoug our analysis will not focus on business cycle movements, te next fact notes tat te analogous finding carries over to tis context as well. Fact 4: Cyclically in te United States te principal margin of adjustment is te fraction of te working age population employed. Hours worked per worker in a given week vary little. Co and Cooley (994) establis tis fact. Tey find tat tree-quarters of te variation in total ours of employment takes te form of movements in and out of te labor force rater tan adjustments in average ours of work. Te last fact tat we document is tat tere are significant differences in ours of work across groups of workers. Fact 5: Tere are sizable differences in workweek lengts across categories of workers based upon occupation and education. Table igligts te differences across occupations. 7

9 Table Weekly Hours Worked for Selected Occupations* Less tan 40 Hours Between 40 and 45 Hours More tan 45 Hours Food Preparation and Serving (36.) Building and Grounds Maintenance (38.96) Office and Administrative Support (38.87) Legal (44.76) Management (45.98) Education, Training and Library (40.76) Healtcare Practitioners and Tecnical (40.) Sales (4.9) Construction (4.95) Production (4.40) * Includes people working more tan 800 ours per year. Source Data: CPS 004 Farming, Fising and Forestry (45.09) Extraction Workers (5.30). Static Model of Intensive and Extensive Margins Two key messages emerge from te previous section. First, te fraction of life devoted to employment is a key margin of labor supply for an individual. Second, te coice of ours conditional on employment seems to respond to features of te economic environment suc as tose associated wit differences in occupation. In tis section we develop a static model tat delivers intensive and extensive margins of work as part of its equilibrium. In te next section we embed tis model in a dynamic setting to produce a model in wic te fraction of life devoted to work is a key margin, and in wic features of tecnology can influence te coice of ours along te intensive margin. We assume a continuum of identical individuals wo derive utility from consumption of a single good (c) and leisure ( ). Time devoted to market work is 8

10 denoted by, and eac individual as a unit time endowment. For simplicity we consider a utility function tat is separable between consumption and leisure, so preferences are described by te von Neumann Morgenstern utility function: uc () v () () were u(c) is strictly increasing, strictly concave, and twice continuously differentiable, and v ~ ( ) is strictly increasing, strictly convex, and twice continuously differentiable. We also normalize v ~ so tat v (0) = 0. Tere is a constant returns to scale tecnology tat uses labor services (L) to produce te single consumption good. We normalize te marginal product to one and ence write te production tecnology as C=L, () were uppercase letters denote aggregates. It is standard to assume tat an individual's input of labor services is eiter equivalent to teir time devoted to market work, or more generally proportional to teir time devoted to market work. Central to our model is te notion tat time devoted to work is not te same as input of labor services. In particular, we assume tat te mapping from individual time devoted to market work to input of labor services is nonlinear and write l=g(), (3) were l is te quantity of labor services yielded by an individual wo supplies units of time. Te function g is assumed to be increasing and twice continuously differentiable, wit g(0)=0, but is first convex and ten concave. In particular, we assume tat tere is some (0,), suc tat g is weakly increasing on [ 0, ) and weakly decreasing on [,]. Figure 4 sows a particular function wit tese properties. 9

11 Figure 4 Mapping from Hours of Work to Labor Services Te g function captures two key economic features. Te first is te possibility tat long ours of work lead to fatigue, so tat past some point additional units of time input lead to lower increments in labor services. Tis is captured by te fact tat g may be strictly concave over te interval [,]. Te second economic feature tat is captured by te g function is tat for a range of ours worked, replacing one worker wo supplies units of time wit two workers wo eac supply.5 units of time implies a lower input of labor services. In reality, a variety of factors contribute to tis effect, including suc tings as set-up costs, supervisory time, and coordination issues. Wile for some issues it may be important to explicitly model tese underlying factors, we believe tat embedding tem directly into te g function is a useful and powerful abstraction. Note tat tis last property of g is consistent wit te observation tat for many activities, firms do not consider part-time workers. Tere are, of course, some activities for wic firms prefer part-time workers, due to factors suc as te timing of demand. Our analysis can accommodate differences across activities by allowing te g function to differ across occupations or sectors. We present a two-occupation example in Section 5. 0

12 Te properties tat we ave embedded in g ave important implications for te form tat efficient resource allocations take in tis economy. In particular, if we ad assumed tat g was simply te identity function, ten (equal weigt) efficient allocations in our economy would ave te property tat all individuals devote te same time to market work and enjoy te same consumption. Tis result would continue to old if we assumed tat g were everywere concave. However, tis is not necessarily te case wen te g function contains a convex region as we ave allowed. To pursue tis issue, it is convenient to consider a cange in variables. In particular, rater tan considering te nonconvexity in te tecnology, we consider a transformation tat places te nonconvexity in preferences. To do tis, rater tan considering preferences defined over consumption and time devoted to market work, we instead define preferences over consumption and te supply of labor services. Tis is easily accomplised, since tere is a one-to-one mapping between time devoted to market work and te supply of labor services. Letting g denote te inverse function of g, we now define preferences as uc () vl (), (4) were ~ ( v( l) = v g ( l)). (5) In order to ave a nicely beaved v function, we impose some additional structure on te v ~ and g functions. Specifically, we assume tat v ~ v~ '' ' is weakly increasing and tat g g '' ' is

13 weakly decreasing. 3 Given our assumptions on g and v ~, it follows tat v is strictly increasing and twice continuously differentiable, tat v (0) = 0, and tat v is initially concave and ten later convex. Figure 5 sows a particular function wit tese properties. Figure 5 Disutility from Labor Services Te two formulations of te problem are analytically equivalent, but te labor services formulation allows for a simple grapical exposition. Figure 5 geometrically illustrates wy te nonconvexity of te g function may affect te form of efficient allocations. Te dased line in tis figure is te steepest line passing troug te origin, wic is no greater tan v(l) at all points. Tis line as a point of tangency wit te v(l) curve at l. It is straigtforward to sow tat it would never be optimal to ave workers supply labor services less tan l, wic necessarily translates into te statement tat it would never be optimal for all workers to supply less tan ours, were l = g( ). Te 3 Tis assumption can actually be weakened. For our purposes, it is enoug to assume tat at most. v ~ '' v~ ' and g '' g' cross

14 reason for tis is simple: suppose tat all workers were to supply labor services less tan l. Ten by placing appropriate fractions of individuals at 0 and l, a social planner can ave te same aggregate input of labor services, but te average disutility associated wit obtaining tis given labor input will be less, since te dased line lies everywere below te v(l) curve for l ( 0, l ). More generally, one can sow tat te optimal allocation will take one of two forms: eiter all individuals supply te same labor services, and tis value is at least l, or tere is one group of individuals tat supplies 0 units of labor services and anoter group tat supplies l units of labor services. Strict concavity of u(c) implies tat a Social Planner wo is maximizing an equal weigted integral of individual utilities would necessarily allocate consumption evenly across all individuals, independently of te labor services provided by a given individual. In te case were not all individuals supply positive units of labor services, te solution to tis Social Planner's problem does not determine wic individuals supply positive units of labor services, but only te fraction of individuals wo supply positive units. It is straigtforward to sow tat tis Social Planner's problem as a unique solution for te fraction of individuals wo work (e), te labor services provided by tose wo work (l), and consumption of all individuals (c). Recalling tat v(0)=0, tese values are solutions to te following problem: max uc ( ) evl ( ) c,e,l s.t. c = el, 0 e, 0 l g() 3

15 Te interesting case is te one were te solutions for e and l are bot interior. In tis case, after substituting te resource constraint into te objective function, one obtains te following two first-order conditions for e and l, respectively: u' ( el) l = v( l) (6) u' ( el ) = v' ( l) (7) Dividing tese two equations by eac oter gives vl () = v' () l. (8) l Tis condition says tat te optimal level of labor services per worker sould equate te marginal disutility per unit of labor services to te average disutility per unit of labor services. Geometrically, tis is te property tat l possessed in Figure 5. 4 Note tat tis condition does not depend on te level of e, as long as te solution for e is interior. Given a solution for l, it is ten straigtforward to use eiter Eq. (6) or Eq. (7) to solve for te optimal value of e. Given tat l is determined, te value of time devoted to market work is easily determined by inverting te function g. Te allocation just described implies tat not all individuals will devote te same number of ours to market work, but tat all individuals will receive te same consumption. As a result, tis allocation cannot be acieved as a competitive equilibrium unless lotteries or sunspot contracts are considered. Te fact tat decentralizing optimal allocations requires randomization as been seen by some as an argument against focusing on tese allocations. In te next section we sow tat once one moves to a 4 From Figure 5 one can see tat tere are actually two values of l for wic tis condition olds, but te second-order conditions for an optimum are violated at te lower value, since it occurs in te region were v is concave rater tan convex. 4

16 lifetime setting as opposed to a static setting, optimal allocations can be decentralized witout resorting to any randomization. 3. Lifetime Labor Supply In tis section we extend te static analysis of te previous section to a dynamic context in wic individuals live for many periods. For reasons tat will become clear soon, we model time as continuous and assume tat te orizon of te economy is normalized to, so tat time runs continuously from 0 to. As noted previously, we can use eiter te ours of work or te units of labor services formulation for our analysis. Here we coose to focus on observables, and tus present te results for te ours of work formulation. All of te primitives are identical to tose from te previous section, and lifetime utility is given by [ uct ( ())-vt ( ())] dt (9) 0 wen we consider te consumption space to be consumption and time devoted to market work. We ave assumed no discounting to simplify te presentation, but tis is not essential to te arguments. We continue to assume tat tere is no capital in te economy, so tat labor services are te only input into production. Tis implies tat Y(t)=L(t) (0) at eac point in time, were capital letters denote aggregates. Wile it remains true tat one can generate efficient allocations (i.e., allocations tat maximize an equal weigted integral of utilities) by simply applying te random 5

17 allocation of te previous section at eac instance of time, it turns out tat one can also acieve te same outcome in utility space witout resorting to any randomization. In particular, rater tan aving te Social Planner randomly coose a fraction e of individuals to work at eac instance, te Social Planner can instead ave eac individual work for a fraction e of teir lifetime, and ten simply coordinate tese coices across individuals so tat at eac point in time te fraction of individuals working is also equal to e. It is straigtforward to see tat tese two allocations are bot feasible and tat tey bot generate te same expected utility to individuals. Te importance of tis observation is tat it also tells us tat we can decentralize efficient allocations witout resorting to any randomization. In particular, consider te setting in wic te commodity space consists of consumption and labor services supplied. We consider an Arrow-Debreu equilibrium, in wic markets for consumption and labor services at all dates exist at time zero, and denote te price of consumption and labor services at time t by p(t) and w(t), respectively. 5 Given te linear tecnology, competitive equilibrium requires tat w(t)=p(t) at eac instant. It is easy to sow tat in equilibrium te price p(t) must be constant over time, in wic case we can normalize all prices to one. Wit tis normalization, te problem of eac consumer in equilibrium is given by max c( t), ( t) [ u( c( t)) v~ ( ( t))] dt 0 s.t. c(t)dt = g( ( t)) dt, 0 ( t ), c( t) We note for future reference tat in tis economy, te wage per unit of labor services is not te same as te wage per unit of time. 6

18 But tis problem is exactly identical to te problem tat te Social Planner solves in te static model, so tat te individual coice problem can be recast as coosing a fraction e of is or er lifetime to work and te ours of work to be supplied at tose instances wen e or se works. Tat is, eac individual solves max uc ( ) ev ( ) c,e,l s.t. c = eg( ), 0 e, 0. A simple result emerges: in a dynamic model suc as tis one wit nonconvexities, te fraction of life spent in employment becomes a key dimension of labor supply. In fact, in tis particular model were individuals do not ave any age-varying caracteristics, te notion of individual labor supply at a given point in time as no meaning. Te only object tat as meaning is te lifetime labor supply of te individual, and ow tis lifetime labor supply is decomposed along te intensive and extensive margins. As just noted, te model studied ere is only able to pin down an individual s lifetime labor supply, not is or er timing of work. 6 Including life cycle effects suc as age-varying productivity or age-varying disutility of work, and considering an overlapping generations model serves to eliminate tis indeterminacy. 7 In tis case individuals will coose to work wen productivity is igest (or disutility lowest). One is ten able to produce well-defined working lives as seen in te data. Specifically, agents enter te labor force at some date and work continuously until retirement, after wic time tey exit te labor force forever. We abstract from life cycle effects in te remainder of tis paper in order to simplify te analysis and focus attention on te key dimension of 6 Mincer (96) demonstrated a similar result wit regard to te labor supply of married women. 7 Rogerson and Wallenius (007) construct a model of tis nature. 7

19 fraction of lifetime devoted to work in te most transparent setting. However, we note tat most of our key results continue to old after te inclusion of life cycle effects. 4. Policy Analysis In tis section we use our model to analyze several different policies of interest. Te first is a simple tax and transfer sceme tat is neutral across ages, in te spirit of Prescott (004). We ten consider policies tat directly distort te two key margins of lifetime labor supply, eiter by legislating te workweek or retirement age, or by specifying social security payments in a manner to target a particular retirement age. 4.. Labor Supply Elasticity and Taxes In tis subsection we analyze te implications of a tax and transfer policy on te lengt of te standard workweek and te fraction of te lifetime spent in employment. Hence, we now extend te baseline model to include a tax and transfer program. Specifically, we assume tat te government taxes all labor income at te constant rate of τ and uses te tax revenues to fund a lump-sum transfer. Te lifetime utility maximization problem can be written as max log( c) ev ( ) ce,, s. t. c= ( τ ) eg( ) + T, were τ is te tax on labor income and T is te transfer. Government budget balance implies tat te transfer must satisfy T = τ e g(). 8

20 Te empirically interesting solution to tis problem is te interior solution. Hence, tis is te case we consider. Te first-order conditions for tis problem wit respect to e and, respectively, are ( τ ) g( ) ( τ ) e g( ) + T v~ ( ) = 0 () ( τ ) e g ( ) ( τ ) e g( ) + T ev~ ( ) = 0. () From te first-order conditions we can derive expressions tat determine te optimal lengt of te workweek and te fraction of time spent in employment, respectively. We get v ~ ( ) g ( ) ~ = v ( ) g( ) (3) τ e = v~. (4) ( ) Note tat in writing Eqs. (3) and (4), we ave utilized te fact tat T = τ e g(). Let * and * e denote equilibrium values in te bencmark model wen tere is no tax and transfer policy. Let τ and e τ denote te equilibrium values tat solve Eqs. (3) and (4). Proposition : * e τ < e, * τ = and e < e * * τ τ. Proof of Proposition : Eq. (3) determines te optimal lengt of te workweek. Since τ does not enter into tis equation, te introduction of a tax on labor income, or alternatively an increase in te tax rate, causes no cange in te optimal lengt of te 9

21 workweek. Given, Eq. (4) determines te optimal fraction of time spent in employment. Since tere is no cange in, an increase in τ causes a decrease in e. Since * e τ < e and * τ =, it follows tat e < e * * τ τ. // At one level tis proposition states te somewat unsurprising result tat a taxtransfer program of te form considered ere leads to fewer ours worked in equilibrium. But wat is of key importance in tis proposition is te result tat all of te adjustment in ours takes place along te extensive margin, i.e., te fraction of life devoted to work. Tis result is of special importance because of its quantitative implications. Specifically, wen te extensive margin is te sole margin of adjustment, labor supply is in fact igly elastic. Using Eq. (4) to calculate te elasticity of aggregate ours (e) wit respect to τ, we get an elasticity equal to one. Put somewat differently, te elasticity of ours worked for workers wit positive ours, wic is equal to zero in tis example, provides no information about te magnitude of te aggregate response. A striking implication of tis model is tat te aggregate labor supply elasticity is a function of tecnology parameters. Specifically, canging te nonconvexity in te mapping from ours of work to labor services canges te aggregate elasticity. Wile it is common to tink of te aggregate labor supply elasticity as being captured by a preference parameter, as long as te nonconvexity in g is sufficiently large to be operative, te value of te preference parameter governing te disutility from ours of work as no impact on te aggregate elasticity. 4.. Constraints on Lengt of Workweek We now analyze te consequences of a binding constraint on workweek lengt. For te purposes of tis analysis we consider te formulation of te individual decision 0

22 problem for bot te ours supplied to market formulation as well as te labor services formulation. From our earlier derivations, recall tat once we know te lengt of te workweek, we also know te optimal fraction of time spent in employment. Specifically, from te ours worked formulation we ave te condition and from te labor service formulation te condition Let * and e = v~ ( ) (5) e = v( l). (6) * e denote equilibrium values wen tere is no ours constraint. Let represent an exogenously imposed ours constraint, and let e ( ) denote equilibrium values in te case of a binding ours constraint. Te empirically interesting case is for an interior solution. Hence, we consider te case were 0 < e ( ) <. Proposition : If * >, ten * e ( ) < e, * * e ( ) < e, and * * e ( ) l( ) < e l. If * <, ten * e ( ) > e, * * e ( ) > e, and as we make smaller and smaller, first * * e ( ) l( ) > e l and ten * * e ( ) l( ) < e l. Proof of Proposition : Given tat v ~ ( ) is an increasing function, te inverse relationsip between and e is apparent from Eq. (5). Also from Eq. (5), we can easily derive an expression for aggregate ours: e = v ~ ( ). (7)

23 To determine te effect of a restriction on on aggregate ours, te comparative static of interest is ( e ) v~ ( ) v~ ( ) = ~. (8) ( v ) From te first-order conditions for te problem wit no ours restriction we know tat ~ * * v ( ) v~ ( * ) = 0. Recall tat v ~ ( ) is a convex function, wic implies tat ( e) < 0. Hence, if * <, ten consumption and less leisure. Conversely, if * * e ( ) > e. As a result, individuals end up wit less * >, ten * * e ( ) < e. From Eq. (6) we can easily derive an expression for aggregate efficiency units. In particular, we get l el =. (9) v(l) To determine te effect of a restriction on (wic is in essence a restriction on l) on aggregate efficiency units, te comparative static of interest is ( el ) v( l) lv ( l) =. (0) l v( l) From te first-order conditions for te case wit no restrictions on l, we know tat * * * v ( l ) l v ( l ) = 0. It is easy to verify tat v ( l) lv ( l) is monotone decreasing if v (l) is convex. Conversely, v ( l) lv ( l) is monotone increasing if v (l) is concave. Recall tat v (l) is first concave and ten convex. Te unconstrained optimum occurs in te convex portion of v (l). Hence, if we restrict te lengt of te workweek above te optimum, ten * * e ( ) l( ) < e l. If we restrict te lengt of te workweek somewat below te optimum (still in convex region), * * e ( ) l( ) > e l. However, as we restrict te lengt of te workweek furter below te optimum (concave region of v (l) ), * * e ( ) l( ) < e l. //

24 In te data, it is common to report te ourly wage or te average compensation per our. Recall tat we ave normalized te wage per unit of labor services to one. However, te wage per our of work is different from tis measure and is equal to income g( ) w = =. Te effect of a restriction on ours of work on te ourly ours of work wage is analyzed in te following proposition. Proposition 3: If * * <, ten w ( ) w ( ) <. Proof of Proposition 3: Te comparative static of interest is dw ( ) d g ( ) g( ) = () From Eq. () it follows tat te ourly wage rate will go down as a result of a reduction in ours if g( ) g ( ) >. Recall tat te equilibrium condition for ours is v ~ ( ) g ( ) ~ =. Multiplying bot sides by we get v ( ) g( ) v ~ ( ) g ( ) v~ =. () ( ) g( ) In order to ave g( ) g ( ) >, te rigt-and side of Eq. () must be greater tan one. Tis in turn implies tat we must ave v v ~ ~ ( ) ( ) >. Since te v ~ function is convex, tis olds. Hence, at least in te neigborood of te optimum, an upper bound on ours of work will result in a reduction in te ourly wage.// 3

25 A clear prediction of our model is tat a decrease in te lengt of te workweek brings about an increase in te fraction of time spent working or aggregate employment. Tis as often been te motivation beind government policies tat restrict ours worked. However, many empirical studies (see, e.g., Erbas and Sayers 00) actually find te opposite, i.e., tat decreases in te workweek lead to lower employment. In interpreting tese studies it is important to note tat in practice, policies tat restrict workweeks are often accompanied by an increase in te wage rate (e.g., income may be eld constant in te face of te decrease in ours worked). Moreover, in our model wit te nonlinear mapping from ours to labor services, restrictions on workweeks will actually decrease income per our worked, since te policy effectively implies an inefficient scale of operation for individual workers. In related work, Osuna and Ríos-Rull (003) look for te tax rate on overtime tat reduces te workweek from 40 ours to 35 ours. Tey find tat a % tax on overtime work reduces ours of work from 40 to 35 (a reduction of.5%). As a result, employment increases by 7%. Wile employment does increase, aggregate labor supply (ours times employment) decreases. Teir ours measure is in effect an efficiency unit measure. Tey define ours allocated to work as te total time allotment less leisure and commuting time. Our findings are not contrary to tat of Osuna and Ríos-Rull (003). If a reduction of ours worked from 40 to 35 ours is a sizable enoug reduction to move te equilibrium onto te concave portion of v (l), tis is exactly wat our model focusing on te determination of efficiency units suggests would appen. 4

26 4.3. Constraints on Working Life In our previous analysis of tax and transfer programs, tere were no requirements tat ad to be fulfilled in order for individuals to receive te transfer. We can also design a policy in wic individuals must work a certain fraction of teir lifetime in order to receive te transfer namely, e e. Naturally, we are only interested in te case wit a binding constraint on te fraction of time spent in employment. Tis problem as two possible solutions. Individuals can coose to work te mandated fraction of teir time e, coose te optimal lengt of te workweek (e) given e, and receive te transfer T. Alternatively, tey can coose to forgo te transfer and coose and e witout constraints. If individuals coose to receive te transfer payment, tey solve te following problem: max c, e, s. t. log( c) ev~ ( ) c = ( τ ) eg( ) + T. e e Given te constraint on e, tere is no first-order condition wit respect to e. Again, te empirically interesting case is for te interior solution wit respect to. Te first-order condition wit respect to is ( τ ) eg'( ) ( τ ) eg( ) + T ev~ '( ) = 0. (3) Recall tat (e) denotes te equilibrium value in te case of a binding constraint on e. Proposition 4: If e e > = eτ, ten e) < τ (. If e = e < eτ, ten ( e) > τ. Proof of Proposition 4: We can rewrite Eq. (3) as 5

27 g' ( ) v~ ' ( ) e =. (4) g( ) ( τ ) If e increases, te rigt-and side of Eq. (4) must also increase. Since g'( ) v~ '( ) and g( ) are bot decreasing in, wen e increases must decrease. Tus, we ave sown tat if e e > = e τ, ten e) < τ (. Te second part of te proposition follows by simply reversing te argument. // Loosely speaking, we can conclude tat if te government, for example, raises te retirement age troug incentives in social security benefits, people will respond by sortening te lengt of te workweek. If individuals coose to forgo te transfer, tey solve te following problem: max log( c) ev ( ) ce,, s. t. c= ( τ ) eg( ). Again, we consider te interior solution to tis problem, as it is te empirically interesting one. Te first-order conditions for tis problem are te same conditions as for te bencmark model wit no tax and transfer policy. Hence, * e = e and * =. Wen deciding weter or not to forgo te transfer payment, individuals solve { ( τ weg + T ) v ( τ we * g * ) v * } max log ( ) ( ) ( ),log ( ) ( ) ( ), were to simplify notation we ave defined = (e). Similar to te analysis just carried out, if a government imposes a mandatory retirement policy tat is binding, ten it is easy to sow tat te result is an increase in ours of work. 6

28 5. Two-Occupation Example In tis section we explicitly consider a model wit two occupations. Tere are several motivations for tis. First, it allows us to illustrate ow our model can account for te differences in workweeks across occupations documented earlier in te paper. Second, in te context of tis model we sow ow empirical exercises tat use crossoccupation data to infer labor supply elasticities do not isolate te appropriate elasticity for predicting te effects of te tax and transfer policies considered earlier. To tis end, we extend te baseline model to two occupations. Tecnological features are a very important distinguising feature of occupations. Here we abstract from oter possible cross-occupation differences. Consequently, ere an occupation is defined as a particular g () function. We assume tat eac occupation produces a particular intermediate input. In particular, individuals working wit te g ) tecnology produce te intermediate input z and individuals working wit te g ( ) tecnology produce ( te intermediate input z. Te intermediate goods tecnologies are assumed to be linear in labor services, and for convenience we normalize marginal products to unity. We assume tat tere is one final consumption good, wic is produced by aggregating te intermediate goods. In particular, were f (.) exibits constant returns to scale. c = f z, z ), (5) ( We assume tat te ~v (.) function is te same for individuals in bot occupations, i.e., tat preferences are te same for all individuals. Initially, we assume tat a measure μ of individuals are endowed wit skills tat enable tem to work in occupation and a measure μ of individuals are endowed 7

29 wit skills tat enable tem to work occupation. Later in tis section we will endogenize μ. We normalize te price of te final good to one and let l wi denote te wage rate of a unit of labor services in occupation i. Te decision problem facing an individual working in occupation i=, is max log( c ) ev ( ) ci, ei, i i i i l i i i i i s.. t c = w e g ( ). Again, te empirically interesting case is for an interior solution. After some algebra, te equilibrium conditions for tis problem can be written as v ~ ( i ) g i ( i ) ~ = v ( ) g ( ) i i i (6) e = i v~ ( ), (7) were i =,. Note tat te equilibrium conditions tat determine i and e i, Eqs. (6) and (7), respectively, are te same as in te case wit only one occupation. To facilitate a comparison of outcomes for individuals in te two occupations, we i assume te functional forms + γ ~ ( i ηi v i ) = and gi ( i ) = ( i φ i ), were η <. + γ Intuitively, we would expect a larger fixed set-up cost to result in a longer optimal workweek lengt. As demonstrated in te following proposition, tis is confirmed by te analytics. Proposition 5: i is increasing in φ and η, e i is decreasing in φ and η, and e i i is decreasing in φ and η. 8

30 Proof of Proposition 5: Given te functional forms proposed above, we can solve Eq. (6) for i. We get i + γ = φi. (8) + γ η i From tis expression it is apparent tat i is increasing in bot φ and η. We ave already previously noted tat e i is decreasing in i. It follows tat e i is decreasing in bot φ and η. We know from before tat te fact tat i is increasing in bot φ and η, we find tat and η. // e i i is decreasing in i. Combining tis observation wit e i i is decreasing in bot φ As sown in Section, workweek lengts differ considerably across occupations. Our framework is able to reconcile tis feature of te data. All our findings regarding te impact of te g(.) function on te optimal lengt of te workweek, as well as aggregate ours worked, old for a general constant returns to scale aggregator c = f z, z ). In te following analysis we will assume a specific ( functional form for tis aggregator. Specifically, θ θ c =. z z Having compared te labor supply outcomes for different occupations, it is of interest to compare te wage rates in te two occupations. As noted previously, te wage rate, w l, is te wage rate per unit of labor services. Te wage rates reported in te data are per unit of time. Hence, we are interested in comparing te following measures across occupations: w l income w units of labor services = =. (9) ours ours 9

31 Bot types of individuals are paid te value of teir marginal products per unit of labor services produced. Tis implies tat w l = p and w l = p, were p is te price of z and p is te price of z. Hence, te wage rates we are interested in comparing are p g ( ) w = and w = p g ( ). We can derive an expression for te relative price from te first-order conditions for te final good firm problem. We get p p θ z θ z =. (30) We know tat in equilibrium z = μ e g ) and z = μ) e g ( ). Substituting ( tese into Eq. (30), we get te following wage relation: ( w w e θ μ =. (3) θ μ e In te preceding analysis we ave treated μ as exogenous. One way to endogenize μ is to assume tat individuals make an occupational coice decision at time zero, and tat occupation is fixed tereafter. Tis would imply tat individuals are allocated across te two occupations suc tat in equilibrium te individuals in te two occupations receive te same utility. Tis requires tat log( c ) ~ ( ) log( ) ~ ev = c ev ( ). (3) Substituting in for c i, i=,, from te budget constraint, Eq. (3) reduces to θ μ log = e ~ ( ) ~ v ev ( ). (33) θ μ θ Equation (7) furter implies tat log μ = 0, wic in turn implies tat θ = μ. θ μ In oter words, input sares determine te distribution of individuals across occupations. 30

32 As a result, individuals in te two occupations enjoy te same consumption. Naturally tis implies tat incomes across te two types of individuals are equalized. Note, owever, tat wages are not equal across occupations. Given tat θ = μ, we ave w w e e =. (34) Recall tat if >, ten e < e. Hence, we can conclude tat te occupation wit te longer workweek will ave a iger wage. We now consider te implications of tis analysis for uncovering labor supply elasticities using cross-section data. Given Eq. (7) and our assumed functional forms, we can write te relative wage as w w = γ. (35) Taking logs we can rewrite tis expression in te more standard way: log w = log γ w. (36) Tis expression implies tat if one looks at te covariation of wages and ours, one will get an estimate of te curvature parameter γ. Empirical studies based on cross-section data typically find tis parameter to be quite small, i.e., tat γ is relatively large (see, e.g., te survey article of Pencavel, 986). However, wat is interesting to note is tat if we analyze a tax and transfer sceme in tis two-occupation economy similar to te one considered earlier in te single occupation model, one obtains exactly te same results as before. Specifically, workweeks remain constant and all adjustment occurs along te extensive margin. Tis implies tat te value of γ is actually irrelevant in determining te magnitude of te decrease in aggregate ours of work. 3

33 In a model wit identical individuals in wic occupations are equally costly to enter, our analysis delivers sarp predictions about workweeks and working lives. One can note tat if μ is taken as exogenous, peraps because some individuals do not ave te skill necessary to enter, tese results may be affected. Additionally, in tis case one cannot infer te preference parameter γ from cross-section data witout knowing someting about ow tese costs differ across occupations. To see tis, we now extend te preceding analysis to te case were tere is a fixed utility cost associated wit entering eac occupation. We allow te utility cost to differ across occupations. One interpretation for tis cost is te time individuals give up or te consumption tey forgo in order to train for an occupation. Te utility cost m i enters te decision problem of consumer i in te following way: max log( c ) ev ( ) m ci, ei, i i i i i l i i i i i s.. t c = w e g ( ). After some algebra, we can write te equivalent of Eq. (3) as θ μ log = m m. (37) θ μ Given tis expression, we can sow tat μ > θ if m < m. If tis is te case, c > c. Tat is, individual receives more of te consumption good in order to compensate for te iger fixed utility cost. Te expression for te relative wage becomes w w e = exp( m m ). (38) e Again, using Eq. (7) and our specified functional forms, we can rewrite tis as w w = exp( m m ) γ. (39) 3

34 Tis analysis cautions against using te covariation of wage and ours to infer te magnitude of γ witout knowing te magnitudes of te fixed utility costs. 6. Conclusion Lifetime labor supply of an individual is usefully caracterized by two key numbers: te fraction of life spent in employment, and te ours of work wile employed. We build a model tat delivers tis caracterization as an equilibrium outcome. A nonconvexity in te mapping from ours of work to labor services provided is key to tis prediction. Te model represents a significant simplification over previous formulations. We consider te qualitative effects of several policy canges to illustrate bot te tractability of te formulation and its ability to capture important economic forces. Life cycle effects can easily be added to te framework presented ere (see Rogerson and Wallenius, 007), furter illustrating te usefulness of tis abstraction. 33

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