The Decline of Drudgery

Transcription

1 The Decline of Drudgery by Brendan Epstein and Miles Kimball April 2010 Abstract Over the last centuries there has been a dramatic world-wide increase in real output. In particular, US data shows that since 1960 aggregate consumption has roughly doubled while the average number of work-hours per person has stayed more or less constant. In spite of large secular increases in the real wage and consumption, contrary to what is expected given plausible income and substitution e ects, work hours have remained trendless. This paper seeks to establish a theoretical explanation for this paradox of hard work. We argue that secular increases in job utility stemming from decreases in drudgery and increases in amenities can help cancel out strong income effects that would otherwise reduce work hours. We show that at the rm level there are strong incentives for decreases in drudgery since rms with lower drudgery have a competitive advantage by being able to attain relatively lower production costs. Moreover, as economies become richer, increases in amenities are endogenously optimal. We also show that the Frisch elasticity of labor supply is decreasing in job utility. Therefore, the higher job utility is the lower the volatility of work hours attributable to labor supply given temporary changes in the real wage. Overall, the welfare e ects associated with increases in job utility can be substantial. In addition, the e ects of declines in drudgery can be very similar, and actually identical under certain circumstances, to those of increases in technology. 1 Introduction Over the last centuries there has been a dramatic world-wide increase in real output. In particular, US data shows that since 1960 aggregate consumption has roughly doubled while Epstein: Ph.D.(C), Department of Economics, University of Michigan, Ann Arbor, MI ( epsteinb@umich.edu). Kimball: Professor, Department of Economics, University of Michigan, Ann Arbor, MI 48109, and NBER ( mkimball@umich.edu). Note: please do not cite or circulate without permission. 1

2 the average number of work hours per person has stayed more or less constant. Income e ects on labor supply are substantial. 1 Given a low elasticity of intertemporal substitution, 2 a considerable reduction in work hours should have taken place. This, however, has ultimately not occurred. Why, in light of the previous, are people still working so hard? Moreover, what are the welfare implications of this paradox of hard work? An ideal explanation to the hard-work paradox involves a theory that is applicable crossperson and cross-nationally for shocks to individuals as well as secularly. Given increases in the real wage, work hours will remain constant if either the marginal utility of consumption remains high or the marginal disutility of work remains low. Things that may aid the former are habit formation, keeping up with the Joneses, or the introduction of new goods. In terms of the latter, one can consider the consumption-work-hours cross-partial derivative in individuals utility function being positive 3 or jobs getting nicer. Given substantial income e ects one possible explanation for the lack of a strong trend in labor hours in the face of signi cant trends in wages and consumption is that the income and substitution e ects on labor supply come close to cancelling each other out (Kimball and Shapiro 2008). Empirical research in (Coulibaly 2006) shows that ceteris paribus individuals are willing to supply more work hours in the service sector than in the manufacturing one, suggesting that the utility associated with work in the former is higher than with that in the latter. This implies that in addition to substitution e ects, increases in on-the-job utility can also help o set income e ects. Such explanation is in line with earlier mentioned 1 See, for example, (Kimball and Shapiro 2008). 2 See, for example, (Hall 1988), (Barsky, Juster, Kimball, and Shapiro 1997), and (Basu and Kimball 2002). 3 See (King, Plosser, and Rebelo 1988). 2

3 properties of the utility function that would account for trendless labor hours by capturing the e ects of jobs getting more pleasant. The focus of this paper is precisely on this. In particular, we develop and analyze a theory that centers on the supply of labor hours by explicitly incorporating on-the-job utility in order to formalize and further understand the implications of changes in the nature of work on macroeconomic behavior. As our focus is in regards to on-the-job utility, a natural point of departure for our analysis is the theory of compensating di erentials, which originates in the rst ten chapters of Book I of The Wealth of Nations, (Smith 1776). The modern reference on this topic is (Rosen 1986). Figures 1 and 2 show two standard results from equalizing di erences, where W denotes the real wage, J on-the-job utility, and Y output. In particular, the solid line in Figure 1 is a wage/job-utility frontier showing that jobs that o er lower on-the-job utility should be expected to be associated with higher real wages as a means of compensating individuals for such lower on-the-job utility: individuals face a tradeo between these two variables. As the solid line in Figure 2 shows, a similar tradeo is faced by rms in terms of a job-utility/output frontier: o ering higher on-the-job utility is costly in terms of output. This is because, as explained in (Rosen 1986), rms can divert part of their productive resources towards making the quality of their jobs better. Given workers individual preferences and rms idiosyncratic costs of on-the-job utility in terms of output, each economic actor respectively optimizes by choosing a feasible point on the (J; W ) plane and the (Y; J) plane. Of course, the secular increases in the real wage and consumption referred to earlier in this introduction are the result of increases in output. In Figures 1 and 2, all else equal, increases in output and the real wage are consistent with movements along the plotted curves 3

4 as indicated by the accompanying arrows, moving the economy from points a to b and c to d. Such movements are associated with decreases in on-the-job utility. Therefore, if on-thejob utility is positively related to the amount of hours individuals desire to spend at work, trendless labor hours in the face of strong income e ects consistent with increases in output and wages require shifts in the wage/job-utility and job-utility/output frontiers to the dashed lines shown in Figures 1 and 2. Therefore, the economy s choice set must be expanding and optimal choices must be moving in the northeast direction as exempli ed by points a 0 and d 0. Our principal analysis focuses on the macroeconomic implications of factors that shift the economy s output/on-the-job-utility possibility frontier. In particular, we develop a model that allows us to study the interaction of work hours (which stands in for all aspects of the job that interfere with leisure and home production), e ort (which stands in for all aspects of a job whose cost is in terms or proportionate changes in e ective productive input from labor), amenities (which we de ne to be job characteristics whose cost is in terms of goods), and drudgery (which is a variable capturing everything else that matters for job utility). We argue that secular increases in job utility stemming from decreases in drudgery and increases in amenities can help cancel out strong income e ects that would otherwise reduce work hours. We show that at the rm level, there are strong incentives for decreases in drudgery since rms with lower drudgery have a competitive advantage by being able to attain relatively lower production costs. Moreover, as economies become richer, increases in amenities are endogenously optimal. We also show that the Frisch elasticity of labor supply is decreasing in job utility. Therefore, the higher job utility is, the lower the volatility of work hours attributable to labor supply given temporary changes in the real wage. Overall, 4

5 the welfare e ects associated with increases in job utility can be substantial. In addition, the e ects of declines in drudgery can be very similar, and actually identical under certain circumstances, to those of increases in technology. This paper proceeds as follows. Section 2 establishes our general theoretical framework. Section 3 examines general equilibrium results and explores the e ects of changes in labor-augmenting technology and drudgery, as well as the implications of heterogeneity in production and the labor force. Then, Section 4 shows that there are strong rm-level microeconomic incentives to foster decreases in drudgery. In turn, Section 5 examines the role of amenities. Section 6 addresses welfare e ects and the long-run behavior of equilibrium work hours, and Section 7 concludes. 2 The General Framework The model is cast in continuous time. Throughout the paper though, we omit time indexes in order to avoid notational clutter. As our focus is on the labor market, we assume the context of a small open economy in which agents can freely borrow and lend at the exogenously determined real interest rate r (equal to, the the rate at which all economic agents discount the future, in steady state). 5

6 2.1 Households For simplicity, we begin our analysis by focusing on e ort and drudgery, and defer the treatment of amenities until further in the paper. 4 Consider a representative worker. Let e i denote the intensity devoted by a such individual to on-the-job task i and E = (e 1 ; e 2 ; e 3 ; :::) the vector of all such possible intensities per tasks. We assume these intensities are determined optimally by rms, and for simplicity focus on perfect monitoring so that moral hazard problems are not an issue. Let D denote the drudgery level associated with work and J = J (E; D) the function that maps E and D into hourly utility associated with being at work. The maximized value of this function can in principle take on any sign. Let J (E; D) = maxfj (E; D)g such that (E) = E, E where is a function mapping the vector E into the number E. E gives e ective productive input from an hour of labor before multiplication by labor-augmenting technology. henceforth refer to E as e ort per worker and J (E; D) as the job utility function. We We assume that J T 0 has the following properties: J EE < 0, J D < 0, and J DD < 0. Therefore, our framework allows for the possibility of job utility being increasing in e ort over some ranges and decreasing over others. Moreover, J D < 0 implies that in (E; J) space a decrease in drudgery causes an upward shift in the job utility function. That is, lower drudgery results in higher job utility at any given e ort level. As an example of how to interpret J, consider two production techniques: 1 and 2. 4 As will be shown, understanding the role of amenities is straightforward once the implications of drudgery are clear. 6

7 Suppose that production technique 1, J 1, results in relatively higher job-utility levels at lower e ort levels, and production technique 2, J 2, results in relatively higher job-utility levels at higher e ort levels. Then, as shown in Figure 3, in (E; J) space the job-utility function J is the upper envelope (bold) of these two techniques. We return to this issue when we analyze the cross-industry implications of our theory. Let a representative household s utility be a function of consumption of the nal good C - which throughout our analysis is the numeraire -, work hours H, e ort E, and drudgery D. We assume that households are in nitely lived, consist of a representative worker, and seek to maximize Z Z e t Udt = e t (U (C) + (T H) + H J (E; D))dt. (1) Above, is the rate at which all economic agents discount the future, t denotes time, T is an individuals s total per-period time endowment, U represents consumption utility and is such that U 0 > 0 and U 00 < 0, and denotes utility from o -the-job leisure, satisfying 0 > 0 and 00 < 0. Note that the functional form in (1) implies that our model nests the standard formulation under which the contribution to utility of an extra hour spent at work is 0, which in this case holds whenever J = 0. Additive separability simpli es our exposition and does not alter our main messages. Consider a worker employed in a job that demands e ort E and is characterized by drudgery D. The individual s utility maximization problem is, taking the hourly real wage rate per worker W paid by the rm as given, to choose a path for consumption, assets M, 7

8 and work-hours to maximize (1) subject to M _ = rm + + W H C, where the price of consumption has been normalized to 1, represents non-labor, non-interest income, and for any variable X, _X refers to its change over time. The current-value Hamiltonian associated with the household s problem is given by H = U (C) + ( (T H) + H J (E; D)) + (rm + + W H C), (2) where is the costate variable giving the marginal value of real wealth in the household s dynamic control problem. The rst-order condition for consumption implies that U 0 (C) =. Substituting the underlying expression for optimized consumption into the Hamiltonian we can state the Hamiltonian maximized over C as H = U U 0 1 () (rm + C) + [ (T H) + H (W + J (E; D))]. (3) Maximizing H over both C and H is the same as maximizing H over H. Note that only the second term on the right-hand-side of (3) depends on H. Therefore, to study the household s labor supply decision we can focus on the optimization subproblem max H (T H) + H B, (4) where B = W + J (E; D) (5) 8

9 represents hourly (marginal) net job bene ts. 5 Note that B captures the utils per hour that an individual derives from on-the-job activities. The individuals s optimization subproblem implies that for any H > 0 the rst-order necessary condition for optimal per worker labor-hours satis es 0 (T H) = B. (6) Therefore, at the optimal level of hours per worker the marginal utility from o -the-job leisure is set equal to hourly net job bene ts. It follows that, as shown in Figure 4, 0 (T H) is the labor-hours supply function. Moreover, the market clearing device for work hours is in fact marginal net job bene ts. Given the worker s objective function, the solution to his or her problem is fully characterized by the rst-order condition for consumption, equations (5) and (6), and the Euler condition _ = ( r). In steady state _ so = r. In addition, combining equations (5) and (6) yields W = 0 (T H) J (E; D). (7) This means that if a household supplies positive labor hours, then the household s utility maximizing choice of hours per worker equates the product of the wage and the marginal value of real wealth to the di erence between the marginal utility of an hour o the job and that of an hour on the job. In the additively separable case of U, we normalize J and so that 0 (T ) = 0. To show 5 This solution method is similar to the one used in (Kimball and Shapiro 2008). 9

10 this, consider U + ~ +H ~ J with ~ 0 (T ) =, where is a constant. De ne (X) = ~ (X) X and J (E; D) = ~ J (E; D) H. Then, 0 (T ) = 0, and U = U + ~ (T H) + H ~ J (E; D) (T H) H =) U = U + ~ (T H) + H ~ J (E; D) T, which is equivalent to U + +H ~ J. ~ Given this normalization, J > 0 means that if the worker has no other job options, then the worker would be willing to spend some time on the job even if unpaid, while J < 0 means that the worker would never do such job unless paid. Proposition 1. The Frisch elasticity of labor supply is decreasing in job utility. Proof. Consider once more the solution to the worker s optimization subproblem. Since work hours are a direct function of marginal net job bene ts, we can write d log H = d log B. Given B = W + J, holding everything constant except wages d log B = dw= (W + J). Manipulating, it follows that d log B is equal to d log W= (1 ), where = J=W. Therefore, d log H = d log B =) d log H=d log W = = (1 ) (8) is the Frisch elasticity of labor supply. Proposition 1 implies that the higher job utility is, the lower the volatility of work hours attributable to labor supply given temporary changes in the real wage. Moreover, note that B = W + J implies that B = (W (1 )). Therefore, can be interpreted as the fraction of the wage that is a compensating di erential. 10

11 2.2 Firms Consider a representative rm whose jobs are characterized by drudgery D. Firms produce output Y by means of a function that takes as inputs capital K, workers N, hours per worker H, e ort per worker E, and is subject to an exogenous labor-augmenting technology parameter Z. In particular, let a rm s production function be given by Y = (K) (ZEHN) 1, where 2 (0; 1). We refer to ZEHN as total e ective labor productivity and ZE as hourly e ective labor productivity. Let R denote the rental rate of capital. 6 It follows that for any output level Y a rm s cost minimization problem involves choosing capital K and total work hours HN to minimize RK + W (HN) such that (K) (ZEHN) 1 = Y. Letting denote the Lagrange multiplier associated with this problem, it is straightforward to show that the rst-order necessary conditions for cost minimization are given by K : R = K 1 (ZEHN) 1 (9) and HN : W = (1 ) K (ZEHN) ZE. (10) Combining the previous two equations yields the rm s optimal relationship between capital and total e ective labor, K = [(= (1 )) (W=ZE) =R] ZEHN. (11) 6 We assume no adjustment costs, so that R = r +, where is the capital depreciation rate. 11

12 We henceforth refer to! = W=ZE as the e ective wage. Substitution of (11) into the rm s constraint implies after slight rearrangement that ZEHN = (= (1 )) Y R!. (12) Now, consider the objective function RK + W (HN), which can be restated as RK +! (ZE) HN, and substitute out K using (11). This yields! ZEHN= (1 ), which upon substituting out ZEHN by use of (12) and rearranging implies that the rm s cost function is given by C (!; R; Y ) = R = (1 ) 1! 1 Y. (13) Since R is exogenous to the rm, given (13) the remaining issue in solving the rm s optimization subproblem is minimizing the e ective wage. In solving this optimization subproblem we assume that rms take the marginal value of real wealth as given, as they do the rental rate of capital R and equilibrium hourly net job bene ts B. Of course, and B may di er for di erent types of workers. For now, we assume the existence of a representative household, and return to this issue later in the paper. Then, the rm s optimization subproblem amounts to min! = W=ZE W, E such that W + J (E; D) = B. (14) Combining these implies that W=ZE = (B J (E; D)) =ZE, which after rearranging 12

13 yields J = B Z!E. (15) Note that the rm s constraint, as in the standard theory of compensating di erentials, implies a tradeo between the real wage and on-the-job utility. In (E; J) space (15) is the equation of a line with intercept B and slope Z!. That is, (15) represents a rm s isocost lines in (E; J) space: it traces out all the e ort and job utility combinations that are consistent with any given e ective wage. As the rm s optimization subproblem involves minimizing the e ective wage, in (E; J) space the solution to its subproblem involves being on the isocost line that has the algebraically greatest feasible slope. In (E; J) space such feasibility is implicitly determined by the rm s job utility function as it represents all the job utility and e ort combinations that a given rm is able to o er. The left panel of Figure 5 shows the solution to the rm s optimization subproblem in (E; J) space. Given equilibrium marginal net job bene ts B, lower e ective wages are consistent with counter clockwise rotations in the rm s isocost lines. As seen in the left panel of Figure 5! 00 >! >! 0 and! is the rm s optimal e ective wage: it can do better than! 00, and although! 0 is preferred to!, the former is not feasible conditional on the rm s job utility function. In turn, the right panel of Figure 5 shows the determination of the number of hours per worker given marginal net job bene ts B and hours supply H S. The relevance of job utility for the rm s optimization subproblem now becomes clear: job utility is the rm s e ective constraint in minimizing costs. As shown in Figure 5, the solution to the rm s optimization subproblem occurs at the 13

14 highest feasible point of tangency between the rm s job utility function and one of its isocost lines. This means that optimality is implicitly de ned by J E = Z! =) J E E = W. (16) Since ; E > 0, for positive wages it is an endogenous result from (16) that at the optimal choice of e ort J E < 0. In particular given ; E > 0 if (16) were satis ed on a non-decreasing portion of the job utility function, then this would imply a non-positive real wage. Figure 6 shows the solution to the rm s problem when optimality condition (16) occurs on the increasing portion of the rm s job utility function, implying a negative wage: in this case, the employee pays the rm to work for it. Mark Twain s Tom Sawyer is a literary example of such circumstance that nonetheless conveys the point well. In a wellknown section of the novel Tom is punished by his aunt and required to whitewash her fence. Instead, he cleverly persuades his friends that whitewashing the fence is great fun and in fact has them pay him for the experience of whitewashing the fence for him. Another example is Dude Ranches. These are ranches that people pay to go to and experience ranch life, which includes doing actual work for the ranch. Figure 7 shows a circumstance under which the real wage is zero - i.e. volunteer work. Interestingly, however, in recent years the internet has brought about a new category of volunteer workers emerging by way of social networking websites, which are run by rms that pro t by selling advertising spaces on their internet sites. In this case the product is the social networking website itself. Although maintenance of the website requires trained 14

15 personnel that actually do get paid for their work, the actual value of the product, and therefore of an advertising spot, depends on the number of participants, which itself depends on the input of users who e ectively receive no wage in exchange for their services. The analysis so far shows that the solution methodology employed in solving a rm s optimization subproblem is the same regardless of the sign of the wage. For ease of exposition we henceforth restrict attention to cases under which the real wages associated with any given job are positive. Moreover, note that although we focused on depicting cases in which the job utility function is concave, the tangency optimization method we use is robust to situations as shown in Figure 1. More generally, we have not needed to assume concavity of the rm s job utility function in order to solve its optimization subproblem. The analysis devoted to real wages of di erent signs suggests that our theory is in line with that of compensating di erentials. The left panel of Figure 5 is similar in spirit to the discussion in (Rosen 1986) that points to a tradeo between job attributes and productivity. In particular, Rosen explains that rms may shift resources from production over to improving job attributes, in which case there is a loss in output, but at the same time for any given level of output the associated real wage is lower. This is because higher job attributes constitute the means for compensating di erences. In that sense, Rosen s discussion can be read as one in which rms have a choice between investing in capital that can be used in producing output, or capital that is used as an input in the production of job attributes (and useless in the production of output). In terms of the left panel of Figure 5, the previous corresponds to a north-west movement along the job utility function. This relates back to our discussion in the introductory section regarding gures 1 and 2. Our research extends 15

16 the theory of compensating di erentials by focusing on the primitives that determine the position of the job utility function itself, one of which is drudgery. Moreover, in our discussion of amenities, we will show that aspects of job utility other than e ort are in the rm s control and lead to shifts in the job utility function as opposed to movements along. This results in a complementary way of thinking about the determinants of job utility relative to Rosen s discussion. Ultimately, we are able to make inferences regarding the evolution of job utility across time, and also predictions of its evolution conditional on changes in the economy s labor-augmenting technology and marginal value of real wealth. 3 Equilibrium In this section, we examine the implications of the model in a standard macroeconomic context. For now, we continue to assume a representative-agent framework with regards to both rms and individuals. Although rms set the real wage they pay their employees, we impose that rms are price-takers in the product market. Moreover, any given rm is assumed to produce the nal consumption good itself, which, as stated earlier, we treat as the numeraire. In this context pro t maximization for any rm entails a marginal cost of production equal to the price of output. Using (13) implies that under perfect competition rms with positive output must have 1 = R =( (1 ) 1 )! 1. (17) 16

17 Rearranging, W=ZE = (1 ) 1 =R 1=(1 ) =!. (18) Thus, under perfect competition the e ective wage is actually an exogenously determined constant from the rm s point of view. 7 The economy s general equilibrium can be determined by way of two graphical tools. The rst of these is shown in Figure 8 and extends the intuition developed via Figure 5. In this case, the slope of an isocost line Z! is exogenously determined. Nonetheless, equilibrium requires that cost minimization be satis ed. Therefore, optimality continues to be summarized by a point of tangency between the job utility function and an isocost line. The left panel of Figure 8 shows optimal e ort requirements E and job utility J, which implicitly de ne the optimal real wage W =!ZE and hourly net job bene ts B. The right panel of Figure 8 shows the determination of work hours H. As shown earlier, hours supply per worker H S is given by 0 (T H) = B. What remains to be determined is the economy s marginal value of real wealth. Given this, equilibrium consumption - and therefore production - are implicitly pinned down. In general equilibrium, our open-economy framework has r =. The household s budget constraint establishes a direct relationship between consumption C and labor earnings W H, which de nes an implicit relationship between labor earnings and. In equilibrium C = rm + + W H. Given the household s rst-order condition for consumption, this implies that = U 0 (rm + + W H). (19) 7 Recall that we refer to W as the real wage and to! as the e ective wage. 17

18 Since U 0 () is decreasing in C, equation (19) yields a negative relationship between and labor earnings W H, which we call the demand for labor-earnings LE D function. Now, consider the mechanics that determine the con guration shown in Figure 8, which was shown for a given. Suppose that the marginal value of real wealth increases from to 0. Then, as the left panel of Figure 9 shows, since! is xed and Z does not change, the rm s isocost lines become steeper. The tangency condition summarizing optimality is such that B and E increase, while J decreases. The right panel of Figure 9 shows that the increase in B induces an increase in H. In addition, given that! cannot change but E increases, W must increase so that! remains xed. This implies a positive relationship between and labor earnings W H, which we call the supply of labor-earnings LE S function: = (W (; Z; D) H (B (; Z; D))). (20) As shown in Figure 10, demand and supply for labor earnings, (19) and (20), jointly determine the economy s level of and W H. 3.1 Worker Heterogeneity Before proceeding to analyzing the e ects of changes in the model s parameters, a natural question of interest is: what are the e ects of worker heterogeneity? To address this, let there be a continuum of agents inhabiting the economy and indexed by m. A type m individual 18

19 maximizes Z Z e t U m dt = e t (U (C m ) + (T m H m ) + H m J (E m ; D))dt: subject to M _ m = rm m + m + W m H m C m. We allow for di erences in individual marginal values of real wealth m and idiosyncratic productivity. Clearly, appropriately reindexed the solution to an individual s utility maximization problem and optimization subproblem remain as before. For ease of exposition, consider the case in which workers are perfect substitutes in production, and assume a type-m worker is characterized by idiosyncratic productivity m. The natural extension of a rm s production function becomes Z Y = (K) Z m E m H m N m dm 1. The rm s cost minimization problem and solution simply involves appropriately reindexing the ones obtained earlier. Then, for a given worker of type-m the rm s optimization subproblem is such that it chooses the real wage it pays this worker W m and the corresponding e ort requirement E m to minimize! = W m = m ZE m such that m W m + J (E m ; D) = B m. The rm takes as given the marginal value of real wealth of type-m workers m, as well as their equilibrium marginal net job bene ts B m. As shown in Figure 11 the intuition and solution methodology developed under a representative worker carries over to the present context of worker heterogeneity. Interestingly, notice that from the rm s point of view what is relevant about worker types is the product 19

20 m m. We henceforth refer to this product as a worker s hungriness. We can class individuals into supra types-m, which are any arbitrary household types for which the product is equal to some value M. As was the case with a representative household, given perfect competition in the product market it is straightforward to show that the equilibrium e ective wage is once more determined exogenously by (18). Therefore, under these circumstances the rm is always entirely indi erent in terms of employing any given worker type. 8 The slope of isocost lines associated with type-m individuals are given by Hence, the isocost lines associated with individuals characterized by higher values of M Z!. are steeper relative to those with lower values of. Consider the left panel of Figure 12. As shown there, for workers of type-m and N such that M > N, the earlier discussion implies that individuals characterized by less hungriness are predicted to exert lower e ort, enjoy higher job utility, and receive lower hourly net job bene ts than their counterparts with greater hungriness. Moreover, as shown in the right-hand panel of Figure 12, workers with greater hungriness are predicted to work more hours than those with lower ones. Note, however, that relative real wages are in principle ambiguous and depend on the idiosyncratic productivity of the individuals under consideration. The results in this section suggest that workers with low marginal value of real wealth and high idiosyncratic productivity can actually have relatively high hungriness and therefore be found to work relatively high hours at high e ort levels. Overall, as hinted above, it is therefore natural to think of as characterizing a worker s hungriness: workers with high are expected to work harder in terms of both hours and e ort. Finally, note that 8 Please see the appendix for a full derivation of the rm s problem in the presence of worker heterogeneity, as well as further considerations should rms be non-competitive. 20

21 given constant returns to scale in production and mobile capital di erent worker types can be thought of as being on their own individual islands. Therefore the comparative steady state analysis for a small open economy with one type of worker is exactly identical to the analysis of an economy with di erences across workers. Hence, for ease of exposition we revert to a representative worker framework. 3.2 Increases in Labor-Augmenting Technology Consider the e ects of an increase in labor-augmenting technology from Z to Z 0. Figure 13 shows the changes that occur holding the marginal value of real wealth constant. As shown in the gure s left panel, for given and!, rms isocost lines become steeper. The tangency condition summarizing optimality then implies higher hourly net job bene ts, lower job utility, and higher e ort requirements. Additionally, the real wage increases. This is because W =!ZE,! is xed, and ZE increases. In turn, the right panel of Figure 13 shows the relevant e ects on hours worked. Given no change in hours supply, higher hourly net job bene ts result in an increase in hours per worker. Interestingly, note that because on impact an increase in Z induces an increase in E, the response in e ective labor productivity ZE is greater than proportional to the increase in Z. Therefore, the model suggests a mechanism through which shocks to labor-augmenting technology would be ampli ed in terms of their e ect on productivity for short-run uctuations in Z that do not have much impact on. To understand the long-run implications of changes in Z given the resulting adjustment in, consider once more Figure 10. Since for given both W and H increase due to a change in the exogenous parameter Z, labor-earnings supply shifts out. The outward shift in LE D 21

22 determines a long-run decrease in equilibrium and an increase in equilibrium W H. enters the slope of isocost lines directly. Returning to Figure 13, note that a decrease in means that after all adjustments in take place rms isocost lines will be less steep than before adjustment in. Less steep isocost lines mean lower B, H, and E, the last of these translating into lower W and higher J. The extent to which isocost lines become less steep than for given ultimately depends on the magnitude of the change in. Thus, the nal level of B, H, J, W, and J relative to their values prior to the change in Z is ambiguous. However, what is unambiguous is a long-run decrease in and increase in labor earnings W H. Note that it could well be the case that in the new long-run W is higher than before the change in Z, but H is lower. This would be a re ection of the income e ect dominating the substitution e ect. 3.3 Declines in Drudgery Consider a decrease in drudgery from D to D 0. Three possibilities need to be examined: J ED = 0, J ED > 0, and J ED < 0. The rst of these means that changes in drudgery do not a ect how taxing extra e ort is, the second that less drudgery makes extra e ort more taxing, and the last that lower drudgery makes increases in e ort less taxing. Suppose rst that J ED = 0. Given a decrease in D the job utility function shifts up. However, J ED = 0 means that this shift is a simple upward translation of the curve, and changes in drudgery do not a ect the slope of the job-utility function. As shown in the left panel of Figure 14, for given, no change occurs in the slope of the isocost lines, so at D 0 the point of tangency de ning optimality must occur at the same e ort level as under 22

23 D. However, lower drudgery does mean that at any e ort level job utility is higher than before. Moreover, marginal net job bene ts increase, meaning that work hours also increase as shown in the right panel of Figure 14. Note that because no change occurs in e ort or labor-augmenting technology, the real wage remains constant. Thus, although W does not change, the decrease in D does induce an increase in H. In terms of Figure 10, this translates into an outward shift of labor-earnings supply, meaning that in long-run equilibrium will be lower and W H higher. Because the new equilibrium is lower than before, in the new equilibrium isocost lines must ultimately be less steep than for given. This translates into lower e ort and higher job utility than before the adjustment in. Because E must decrease, then so must W. However, since equilibrium W H is higher than before, H, and therefore B, must be higher in the new equilibrium than before the change in drudgery. Now, consider the case in which J ED > 0. Once more suppose that drudgery decreases from D to D 0. The job-utility function shifts up and its slope becomes more steeply downward sloping at every e ort level. As shown in the left panel of Figure 15, for given this change results in a decrease in e ort, an increase in job utility, and an increase in hourly net job bene ts. Because labor-augmenting technology remains xed as well as the equilibrium e ective wage, the real wage decreases by 1=E times the change in e ort. As the right panel of Figure 15 shows, the resulting increase in hourly net job bene ts results in higher hours per worker. Because for given the decrease in drudgery causes H to rise but W to fall, in terms of Figure 10 the change in the labor-earnings supply curve is ambiguous. This makes long-run changes in all of the model s endogenous variables ultimately ambiguous. Finally, consider the case in which J ED < 0. When drudgery decreases from D to D 0 the 23

24 job-utility function shifts up and also for given becomes less downward sloping at every e ort level. As shown in the left panel of Figure 16, for given the result of this change is an increase in e ort and an increase in hourly net job bene ts. In turn, under these circumstances an increase in e ort along with no change in technology or the equilibrium e ective wage means that W increases. As the right panel of Figure 16 shows, the increase in hourly net job bene ts induces an increase in hours per worker. All of the above results for given are unambiguous except for the e ect on job utility. As shown in Figure 16 the change in drudgery results in an increase in job utility. However, this need not always be the case. This is because for a su ciently small upward shift in the job-utility function, it could be that the level of job utility remains constant or actually decreases. In particular, note that if job utility decreases, then for given the qualitative e ects of a decrease in drudgery and an increase in labor-augmenting technology are identical. Moreover, regardless of the change in job utility, a decrease in drudgery when J ED < 0 induces an increase in optimal e ort requirements which results in an increase in hourly e ective labor productivity. Since the decrease in D induces an increase in both H and W it follows that the product W H increases for any given. In terms of labor-earnings supply and demand, this means that the labor-earnings supply function shifts out, delivering a new long-run equilibrium value of that is lower and W H that is higher than before the decline in drudgery. Returning to Figure 16, lower makes isocost lines less steep. Therefore, the nal tangency condition capturing optimality will imply lower B, H, E, and W, and higher J than before the adjustment of. This makes the nal levels of these variables relative to their original ones 24

25 in principle ambiguous. However, ambiguity only exists if the peak of the J curve after the decrease in drudgery is less than or equal to the original level of B. If after the change in D the new peak of the J curve lies above the original level of B, then any new tangency condition consistent with positive wages will necessarily deliver a new equilibrium value of B, and therefore H, that lies above the original one. This highlights that given positive wages, ongoing decreases in drudgery, regardless of the sign of J ED, will unambiguously eventually lead to increases in work hours. This is the result of decreases in D inducing upward shifts in the J curve, and stands in contrast to changes in labor-augmenting technology in which the ultimate change in work hours is always in principle ambiguous. 3.4 Heterogeneity in Production So far, we have assumed a representative rm context. Although a convenient speci cation, this is not a necessary condition for our main results to hold. Here, we show the implications of di erences in rms conditional on all being producers of the nal good or there being di erent types of goods produced across industries. In addition, we show an interesting result stemming from cross-country analysis. In all cases, we continue to assume rms are price takers Di erences in Final-Good Producers Let there be a continuum of rms indexed by i and suppose that each rm is a producer of the nal consumption good. We allow rms to di er in their labor-augmenting technology, drudgery levels, or job utility functions. Regardless of the object over which rms are 25

26 di erent it is still the case that marginal net job bene ts are the market clearing device for labor hours. In addition, the solution to rms cost minimization problem remains as earlier. With generality in mind, let a rm s labor-augmenting technology now be equal to the product of an economy-wide productivity parameter P and a rm-speci c one Z i. Therefore, a rm s labor-augmenting technology is P Z i. In this case, all results from our earlier development carry over noting that the slope of rm i s isocost lines are given by P Z i!. To make our analysis tractable, consider two rms: 1 and 2. First, assume that these rms have the same functional form for J, but rm 1 s jobs are characterized by lower drudgery than rm 2 s. As before, there is an economy-wide and exogenously set!. Suppose that Z 1 = Z 2. Then, as shown in Figure 17, rm 1 s tangency condition suggests higher marginal net job bene ts than rm 2 s. Under these circumstances, rm 1 would set the economy s level of marginal net job bene ts. Since workers take the jobs with the highest B, then rm 2 would be unable to operate. Now, consider a case with di erences in technology countervailing di erences in drudgery. Figure 18 shows that if rm 2 was endowed with technology Z 0 2 > Z 2 then both rms would o er the same marginal net job bene ts and the household would be indi erent between them. Additionally, if rm 2 s technology were given by Z 00 2 > Z 0 2, then it would actually be able to o er higher marginal net job bene ts than rm 1. In this case individuals would strictly prefer being employed at rm 2 and rm 1 would not be able to operate. Therefore, an illuminating way to view the e ects of technology and drudgery is to see drudgery as a component of an expanded concept of technology. 26

27 Now, focus on the impact of di erent marginal values of real wealth or economy-wide productivity P as captured by the product (P ) that enters the slope of rms isocost lines. In the cases considered above, the results were robust to di erent values of (P ). However, a di erent situation emerges when rms also di er in the actual functional form of their J curves. Consider, for example, Figure 19, where rms di er in their J curves, potentially in ways that cannot be described by di erences in their drudgery levels. Without loss of generality, assume Z 1 = Z 2 = Z. Given (P ), then as pictured rm 1 is able to o er the highest marginal net job bene ts and therefore rm 2 would not be able to attract any employees. As Figure 20 shows, the previous need not always be the case. Indeed, given (P ) 0 > (P ) then both rms are able to o er the same marginal net job bene ts in which case workers are indi erent between them. Moreover, for (P ) 00 > (P ) 0 the initial situation is now reversed meaning that rm 1 is unable to attract employees. In equilibrium there is a unique value of marginal net job bene ts, and any operating rm must o er these bene ts. However, as seen above such situation does not exclude the possibility that operating rms di er in terms of their idiosyncratic technology, drudgery, or job utility function. Since B is the same across all employment opportunities, individuals are willing to supply work hours to all rms. However, although individuals determine their total work hours as shown earlier in the paper, they need not be willing to spend the same amount of time working at each type of job. Without loss of generality, focus once more on two operating rms, and suppose they di er in their job utility o erings. Firm 1 o ers job utility J 1 and rm 2 o ers job utility J 2. Proposition 2 establishes exactly how a worker will choose to allocate his or her time between these two rms. In addition, this proposition 27

28 shows that as the household s non-labor income rm + or total time endowment T increase, its members will devote more work hours to the job that o ers the higher job utility. Proposition 2. Consider rms 1 and 2. Suppose these rms di er in their job utility and real wage o ers. In particular rm i o ers the job-utility/real wage bundle (J i ; W i ). Moreover, assume the economy s marginal value of real wealth is given by the endogenously determined, and without loss of generality suppose J 2 > J 1. Let be the fraction of total work hours an individual devotes to working in rm 1 and (1 ) that which an individual devotes to working in rm 2. Then, = (J 2 J 1 ) = (W 1 W 2 ), B = (J 2 J 1 ) = (W 1 W 2 ), and = 1 U 0 1 () rm W 1 W 2 T 0 1 (B) W 2. Proof. Since in equilibrium both rms o er the same B, then B = W 1 + J 1 and B = W 2 + J 2. Note that J 2 > J 1 implies that W 1 > W 2. Combining the previous equations yields = (J 2 J 1 ) = (W 1 W 2 ). Substituting this into rm 1 s constraint and rearranging implies that B = (J 2 J 1 ) = (W 1 W 2 ). In equilibrium from the household s budget constraint C = H (W 1 + (1 ) W 2 ) + rm +. Moreover, the household s choice of total work-hours supply satis es 0 (T H) = B which implies that H = T 0 1 (B). Substituting this into the previous yields C = (T 0 1 (B)) (W 1 + (1 ) W 2 ) + rm +. Given the household s condition for optimal consumption, U 0 1 () = C. Combining these two nal equations and rearranging yields the desired result. Note that J 2 > J 1 can be interpreted as job 2 being a dream job and job 1 being a day job. Therefore, Proposition 2 is quite intuitive: the more time an individual has to 28

29 spend on overall work activities, or the more exogenous wealth he or she is endowed with, the more time said individual will devote engaged in the dream job. Interestingly enough, note that the fraction of time devoted to the day job is also decreasing in the di erence W 1 W 2. To the extent that this di erence implicitly captures how much lower job utility the day job o ers relative to the dream job, this highlights that higher real wages o ered as compensation for relatively lower job utility are not a su cient factor to induce individuals to devote more hours to such job. Below, we consider the applicable versions of laborearning supply and demand, which are analogously derived in the present context and for industry-level di erences Industry-Level Di erences Now, suppose instead that there is a continuum of industries indexed by i. Each industry produces a di erent type of good, but rms within industries are perfectly competitive. For ease of exposition, let there be a representative rm per industry. In analogous fashion to earlier analysis, let P be an economy-wide productivity parameter and Z i the labor-augmenting technology characterizing industry i so that an industry s e ective labor productivity is P Z i E i. Let p i be the relative price of good produced by industry i. Workerside optimization is just as before, and appropriately reindexed the same is true of a rm s cost minimization problem. The industry-level optimization subproblem is therefore min W i, E i! i = W i =P Z i E i 29

30 subject to W i + J i (E i ; D i ) = B i, where B i are equilibrium marginal net job bene ts in industry i. As before this implies that W i =P Z i E i = (B i J i (E i ; D i )) =P Z i E i, which after rearranging yields J i = B i P Z i! i E i. In (E i ; J i ) space the solution to an industry s optimization subproblem again involves being on the isocost line that has the algebraically greatest feasible slope. However, in this case pro t maximization implies that for industries with positive output p i = R =( (1 ) 1 )! 1 i. Rearranging, (1 ) =(1 ) =R =(1 ) = W i p 1=(1 ) i =P Z i E i =!, where we have used the de nition of! i. Therefore, in this case, from an industry s point of view! is an exogenously determined constant. To fully appreciate the solution to a rm s optimization subproblem, note that an isocost line can be stated as J i = B i! P Z i E i =p 1=(1 ) i, and furthermore J i (E i ; D i ) = J i xp Z i E i =p 1=(1 ) i ; D i, where x = pi in P Z i E i =p 1=(1 ) 1=(1 ) i =P Z i! i. Figure 21 shows the solution to a rm s optimization subproblem ; J i space. Since the slope of isocost lines,!, is the same across industries, then industry-level optimal operations and marginal net job bene ts are determined by the point of tangency between a representative rm s isocost line and job utility function. 30

31 Given Figure 21, note that in any industry the vertical distance between B i and J i is equal 1=(1 ) to W i. Moreover, the horizontal distance between P Z i E i =pi and the origin is equal to W i =!. Now, consider two industries, i = 1; 2 with job utility functions given by J 1 = J 1 and J 2 = J 2 as depicted in Figure 2. As explained earlier, what is now relevant is the upper envelope of these job utility functions. To see this, consider Figure 22. Note that for a low marginal value of real wealth such as 0 industry 1 is able to o er the highest marginal net job bene ts in which case industry 2 does not operate. For a higher marginal value of real wealth such as 00 > 0 both industry 1 and 2 are able to o er the same marginal net job bene ts in which case the worker chooses hours allocation across industries according to Proposition 2. Finally for even higher marginal values of real wealth such as 000 > 00 industry 2 is able to o er the highest marginal net job bene ts, in which case industry 1 is unable to operate. It is interesting to consider the determination of labor-earnings supply and demand under these circumstances. LE D is a simple extension of that derived earlier. In particular, this function satis es = U 0 (rm + + H (W 1 + (1 ) W 2 )), where is the fraction of total work hours devoted to industry 1. The appropriate version of LE S is slightly di erent than earlier considered. Note that given Figure 22, for low values of only industry 1 operates, and the associated real wages, marginal net job bene ts, and work hours are relatively low. Therefore, in terms of labor-earnings supply, low values of 31