We analyze the choice of hours of work in the steady state by making use of a model that ignores intertemporal considerations, in contrast to the analysis of search intensity in chapter 5. This is not a serious restric- tion for our purposes, since the problems that we wish to highlight in the choice of hours of work do not depend on the differences between current income from work and permanent income. A dynamic analysis of the choice of hours of work along the lines of the analysis of the choice of hours of search in chapter 5 is, of course, possible. But in the steady state current income from work (the wage rate) is not very different from permanent income (rW in the notation of earlier chapters), so a model that ignores intertemporal considerations is likely to give similar results. In contrast, current income during search (unemployment income) is very different from permanent income (rU ) even in the steady state, and the individual searches in order to make the transition from the current state to another, so ignoring intertemporal considerations in the choice of search intensity would eliminate the main results.
We assume that the worker’s instantaneous utility depends on current income and current hours of work. We write instantaneous utility during employment as
(7.17) where w now denotes the hourly wage rate, h denotes hours of work, and the length of the day is normalized to unity. The utility function is linear in income (wh) and nonlinear in hours of work. If income is higher, the marginal cost of hours of work is higher, indicating complementarity between consumption and leisure. These assumptions parallel the ones that we made in chapter 5 to analyze hours of search; see (5.31).
We assume for simplicity that the intensity of search is fixed and un- employed workers derive no direct utility from leisure and have no in- come. Hence the two equations giving the worker’s present-discounted utility during unemployment and during employment in some job j are, respectively,
(7.18) (7.19) The W without a subscript denotes utility in the representative job. The hourly wage rate is determined, as before, by the Nash solution to the bargaining problem. Hours of work may be determined either by the worker in such a way as to maximize his utility or by a bargain between the firm and the worker along the lines of the wage bargain. Since the search externalities do not influence the decision with regard to hours of work, nor does the chosen number of hours influence the number of jobs and number of workers that come into the market, the efficient number of hours of work is determined by a Nash bargain between the firm and the worker. We show that if workers choose their own hours of work, they will choose to work too few hours. The reason for this inefficiency is related to the existence of trading costs, though not to the search externalities.
If hours of work are chosen by workers, hjmaximizes Wjfor given wj
and U. Hence optimum hours satisfy
(7.20) The wage rate does not affect hours because the income and substitu- tion effects offset each other, not an unreasonable property in the steady state. Labor-market tightness does not affect hours because the worker chooses his hours of work after he has found the job, and he knows he can revise his choice when he changes jobs.
If hours of work are chosen after a bargain between the firm and the worker, their determination is more complicated but still straightfor- ward. Let p be the product per hour input. Then, the present-discounted value of profit from a vacant job and from a filled job j are, respectively, (7.21) (7.22) Wages and hours of work in job j are chosen to maximize the product
(7.23) with Wjgiven by (7.19). Wj -U Jj V
(
)
b(
-)
1-b, 0<b<1, rJj =h pj -h wj j - l .Jj rV= -pc q+( )
q(
J-V)
, ¢ -(
)
-(
)
= f f 1 1 1 h h h j j j . rWj =w hj jf(
1-hj)
+l(
U-Wj)
. rU=q qq( )(
W U-)
,The wage rate that maximizes (7.23) satisfies the condition
(7.24) Hours of work also maximize (7.23), and the condition they satisfy is
(7.25) To derive a wage equation from (7.24), we make use of the property that in equilibrium wages and hours of work are the same in all jobs. Also in equilibrium V = 0, so from (7.21)
(7.26) Substituting into (7.22), we obtain
(7.27) Computing also W - U from (7.18) and (7.19), we obtain
(7.28) Hence, substituting V = 0, J from (7.26), W - U from (7.28), and r + l from (7.27) into condition (7.24), we arrive at the wage equation
(7.29) Wage equation (7.29) is of the same form as the one we had before, (1.23), except that now because w is an hourly wage rate, the cost of hiring is the hourly cost c/h and unemployment income z is zero. Equi- librium hours of work are determined from (7.25). Considering again (7.24) and (7.25) in equilibrium, when wages and hours are common to all jobs, we obtain, after substitution of W - U from (7.24) into (7.25),
(7.30) The elasticity of the utility of leisure with respect to hours of work is equal to the ratio of the marginal product of labor to the wage rate. But
¢ -
(
)
-(
)
= f f 1 1 h h h p w. w p c h = b ÊË1+ qˆ¯. W U wh h r q - =(
-)
+ +( )
f l q q 1 . h p w r q pc -(
)
- +( )
= l q 0. J pc q =( )
q . b f f f b Jj -V wj hj hj Wj U p wj(
)
(
-)
- ¢( )
( )
Ê Ë ˆ ¯+(
-)(
-)(
-)
= 1 1 . 1 0 . . b(
Jj -V)
f(
1-hj)
-(
1-b)(
Wj -U)
=0.in equilibrium the marginal product of labor must be strictly greater than the wage rate, as (7.27) implies, because firms must be compensated for their hiring costs. Hence (7.30) implies that when hours are set after a Nash bargain,
(7.31) By the concavity of f(1 - h), the elasticity expression in (7.31) increases in h, so hours are now higher than implied by the worker’s choice rule, (7.20). Now, if hours of work are determined by a bargain between the firm and the worker, they exceed the hours that maximize the worker’s utility.
The discrepancy between utility-maximizing and Nash hours is not due to the search externalities; it is related to costly exchange. Workers choose the number of hours of work by comparing their marginal cost (loss of leisure) with the marginal benefit to themselves (the hourly wage rate). If the number of hours is determined by the Nash bargain, a similar marginal comparison is made but now the costs and benefits to the firm are also taken into account. The joint net marginal cost of one more hour to the firm and the worker is still the loss of leisure to the worker, but the joint gain is the product from one more hour. Thus the two methods of determining hours give the same result only when the wage rate is equal to the marginal product of labor. This cannot happen in equilib- rium in this model because the firm’s hiring costs drive a wedge between marginal product and wages.
In equilibrium (7.30) simplifies further by making use of (7.29). Sub- stituting (7.29) into (7.30), we derive
(7.32) Labor-market tightness and the worker’s share in the wage bargain influence optimal hours because they influence the wedge between the marginal product of labor and the equilibrium wage rate. Higher tight- ness or share of labor reduce the wedge, so they also reduce hours toward the level chosen by the utility-maximizing worker: The inefficiency that arises when workers choose their own hours is less in tight markets. (The inverse relation between tightness and hours of work is a feature of the steady state, and it does not necessarily hold over the cycle.)
¢ -
(
)
-(
)
=(
+)
f f b q 1 1 1 1 h h h c h . ¢ -(
)
-(
)
> f f 1 1 1 h h h .Market equilibrium with variable hours of work is determined as before, except for small and obvious changes because of the reinterpre- tation of the wage rate and the product of labor in per-hour units. Thus in the steady state the Beveridge curve and the job creation curve, respectively, (1.21) and (1.24), are given by the same equations as before, and the hourly wage rate is also given by the same equation. Hours of work in the case where the worker chooses his own hours after he finds the job are given by the elasticity condition (7.20). In this case equilib- rium hours depend only on the utility function, and they do not interact with the rest of the model. But in the efficient bargain case the hours condition changes to (7.32), so hours are influenced by all variables that influence equilibrium tightness. Any parameter that increases equilib- rium tightness reduces equilibrium hours. Equation (7.32) is very much in the spirit of the other equilibrium relations that we have found in the labor market analysis of part II of the book, such as the intensity condi- tion (5.29) and the participation condition (7.4). Equilibrium labor- market tightness (the ratio of vacancies to unemployment) is critical in all of them because it is the variable that transmits the effects of para- meter changes to them.
7.4 Notes on the Literature
Labor force participation in the context of a job-matching model was considered by McKenna (1987). A partial treatment of added- and discouraged-worker effects in a model of search appeared in Pissarides (1976a, b). Other equilibrium matching models that considered chang- ing labor force participation include Bowden (1980) and Blanchard and Diamond (1989).
The topics discussed in this chapter have received extensive treatment in the labor economics literature but not in the context of job-matching models. For example, partial models of added-worker and discouraged- worker effects were extensively discussed by Cain (1966) and Bowen and Finegan (1969). Generally, the early studies found that the discouraged- worker effect dominates the added-worker effect. More recently Lund- berg (1985) has used a transitions approach to estimate a small but statistically significant added-worker effect for married women.
Issues in the supply of hours of work in dynamic models are discussed in several papers in the new classical macroeconomics, following the lead