Wages, Business Cycles, and Comparative ... - Semantic Scholar
Wages, Business Cycles, and Comparative Advantage# (Forthcoming in the Journal of Monetary Economics, Volume 46:1; 2000)
Yongsung Chang* Department of Economics, University of Pennsylvania, Philadelphia, PA 19104, USA
Abstract The standard equilibrium models of business cycles face a puzzling fact that total hours vary greatly over the business cycle without much variation in aggregate wages. The model augments the standard RBC model to include Lucas span of control. Distinction between market and non-market and managerial and non-managerial work makes aggregate wages far less cyclical than individual wages. Cross-sectional comparative advantage between market and non-market sector in the workforce substantially increases the response of aggregate hours to shifts in relative productivity. As a result, the model provides a reconciliation between data and equilibrium macroeconomics.
Key Words: Wages; Business Cycles; Comparative Advantage JEL Classification: E32; J31; J62
#
The previous version of the paper has been circulated under the title “Cyclical Behavior of
Occupational Choices and Wages.” I thank Mark Bils for advice and encouragement. I also thank Jeff Campbell, Pat Kehoe, Bob King, Monika Merz, Sergio Rebelo, Richard Rogerson, Alan Stockman, Randy Wright, an anonymous referee, and workshop participants at Chicago, Penn, Penn State, Rice, Rochester, and Texas A&M for helpful comments. All remaining errors are mine. *
Email: [email protected]
1. Introduction One of the big puzzles in macroeconomics has been that total hours vary greatly over the business cycle without much variation in wages. The equilibrium approach to economic fluctuation pioneered by Lucas and Rapping (1969) and put forward by Kydland and Prescott (1982) and Long and Plosser (1983) views the variation of total hours of work as people’s willingness to intertemporally substitute leisure and work over the business cycle. As Barro and King (1984) illustrate, under standard preferences the equilibrium business-cycle models require a strongly pro-cyclical real wages to be consistent with the movement in hours and consumption in the data.1 The model in this paper augments the standard real-business-cycle (RBC) model to include Lucas’ (1978) span of control and Rosen’s (1982) hierarchy, where workers are assigned to managerial, production, and non-market tasks based on comparative advantage. It provides a partial resolution to the movement of hours and wages in the data. The model shows that heterogeneity of workers significantly decreases the movement of aggregate wages over the business cycle and that the comparative advantage between market and non-market activities may substantially increase the response of aggregate hours to shifts in relative productivity. Previous studies have shown that entry and exit of low-wage workers create a countercyclical composition bias in aggregate wages (e.g., Stockman, 1983; Bils, 1985; Solon, Barsky,
1
The implicit labor contract can generate acyclical wages (e.g., Azariadis, 1974; Gomme and
Greenwood, 1995; Boldrin and Horvarth, 1995). Yet it is still incapable of explaining why labor productivity is not highly correlated with hours over the business cycle.
1
and Parker, 1994). Aggregate wages constructed by the Bureau of Labor Statistics (BLS) are based on the wages of non-supervisory workers at a point in time. The transition matrix of the occupational change constructed from the Panel Study of Income Dynamics (PSID) for 19711992 indicates that there is a significant cyclical movement between non-supervisory and supervisory occupations (managers and self-employed). For example, the likelihood of moving from a non-supervisory occupation to a supervisory occupation increases by 16.2% during expansions relative to recession while the likelihood of moving in the opposite direction decreases by 26.9%. This implies that aggregate wages are subject to composition effect at both ends of the wage distribution. Not only do less-skilled workers enter the workforce, biasing down wages and productivity in expansion, but higher-wage earners become managers and self-employed, lowering aggregate wages further.2
In fact, when the cyclicality is
measured as a percentage response to output growth, the composition effect reduces the cyclicality of aggregate wages by one-third relative to individual wages both in the model and in the data. In order to match the observed movement in total hours worked in the data, the representative-agent models often rely on labor-supply elasticity that is beyond the admissible estimates based on micro data.3 The model in this paper explores the relationship between the cross-sectional comparative advantage and aggregate labor-supply elasticity. A worker who has a very strong and clear comparative advantage would not exhibit frequent movement
2
See also Kydland (1984), Cho and Rogerson (1988), and Cho (1995) for business-cycle models that
incorporate the first composition bias. 3
For an empirical estimate of labor-supply elasticity, see Ghez and Becker (1975) or Altonji (1986).
2
between sectors in response to shifts in relative productivity. However, a weak comparative advantage implies frequent movement between sectors. Home production is introduced to represent the activity in the non-market sector, and the notion of comparative advantage is formalized by the cross-sectional correlation of productivity between the market and home production in the workforce. The quantitative analysis shows that it is capable of generating a realistic movement of aggregate hours in response to shifts in total factor productivity (TFP) under the cross-sectional comparative advantage observed in the PSID.4 When the skill distribution is calibrated by the wage distribution of the PSID, the model is very successful in matching the cyclical behavior of aggregate wages, labor productivity, and employment in the data. Employment in the model is nearly as volatile as that in the data, and yet it is not highly correlated with labor productivity.5 Aggregate wages and labor productivity are mildly pro-cyclical, as in the data. The model also produces interesting dynamics for relative wages and employment in the data. The skill-premium is known to be counter-cyclical. Based on the occupational classification, the PSID data are in accord with this view in general: wages of workers in lowgrade occupations are more pro-cyclical. However, contrary to the conventional wisdom, I
4
A lottery economy such as Rogerson (1988) and Hansen (1985) generates a high aggregate labor-
supply elasticity as well. However, a lottery economy has a counter-intuitive implication that people who draw bad lots are hired. Yet job assignmentincluding market participationis determined based on comparative advantage in our model. 5
Alternatively, one can reduce this correlation by introducing an additional disturbance that shifts the
labor-supply curve (e.g., Benhabib, Rogerson, and Wright, 1991; Greenwood and Hercowitz 1991; Bencivenga, 1992; and Christiano and Eichenbaum, 1992).
3
find that wages of workers in managerial jobs, who are among the highest paid, show the most pro-cyclical pattern over the business cycle. According to the model, the managerial wage depends on the span of control, and the span of control exhibits a strongly pro-cyclical pattern. Under capital-skill complementarity the demand for production labor increases sharply at the beginning of an expansion before capital is accumulated. This is consistent with the empirical observation in the U.S. quarterly data, as the employment of production workers tends to lead that of non-production workers over the business cycle. This paper is organized as follows. Section 2 provides empirical evidence of the hierarchical structure of occupations and the cyclical movement of labor and wages that are consistent with the prediction of the model. Section 3 develops the general-equilibrium model with job assignment. Section 4 presents the static model to illustrate its important features. In Section 5, the dynamic model is calibrated, and its response to stochastic variations in TFP is analyzed. Section 6 is the conclusion.
2. Some Evidence of Labor-Market Fluctuations The model in this paper assumes three occupations: managers, production workers, and home workers. These occupations are hierarchically ordered in terms of required skill. The model predicts a cyclical movement of workers between low-grade and high-grade occupations in response to wage differentials over the business cycle. For example, workers move from non-market to market sectors and non-managerial to managerial positions in expansions in response to pro-cyclical wages and managerial-wage premiums. Thus, the aggregate wage based on non-supervisory workers can be less cyclical than individual wages due to the composition effect at both ends of the wage distribution. 4
To determine the empirical relevance of these features, based on the annual PSID data for 1971-1992, the following aspects of the labor-market fluctuation are examined. First, evidence of a hierarchical structure and comparative advantage among the three occupations is presented.
Second, wage response over the business cycle is estimated across these
occupations. Third, based on the transition matrix, the cyclical movement of workers among the three occupations is analyzed. Finally, the composition effects in aggregate wages are measured.
Table 1 shows the summary statistics, and the Appendix contains a detailed
explanation of the data.
Hierarchical Structure of Occupations If occupations are ordered hierarchically in terms of required skill, and if workers are assigned to these occupations according to their comparative advantage, workers who change their occupations from lower-grade jobs to higher-grade jobs must be relatively more-skilled and high-wage workers in their former occupations. At the same time, these workers must be relatively less-skilled and low-wage workers in their new occupations. In other words, new managers are poor managers compared to existing ones, even though they were previously relatively good workers. Table 2A shows average wages at time t in each cell of the transition matrix among managers (self-employed + not self-employed managers), non-managers (not self-employed and non-managerial workers), and non-market workers (not employed). The numbers in parentheses are average wages relative to those of existing workers in the target occupation. For example, the number 7.71 (-3.12) in the (1,2)th cell in Table 2A implies that the average wage of new workers from the non-market sector in time t is $7.71 (in 1984 dollars), and it is 5
lower than the average wage of existing workers by $3.12. The hierarchical structure of occupations and assignment of workers according to comparative advantage imply that the numbers in parentheses in the upper-diagonal terms must be negative, and the lower-diagonal terms must be positive. Table 2B compares the wages of movers and stayers at time t-1. The numbers in parentheses are those relative to the stayers in the previous occupation. For the same reasons, the upper-diagonal terms must be positive, and the lower-diagonal terms must be negative. There is no exception in the sign of these relative wages of movers and stayers.
Cyclical Behavior of Wages across Occupations Table 3 shows the cyclicality of wages, hours, and incomes across occupations. The cyclicality is measured as the percentage response to real GDP growth, the coefficient b1 of the regression 6 ∆ log( X it ) = b0 + b1 ∆ log(real GDPt ) + eit .
(1)
Consistent with previous studies by Bils and Solon et al., individual wages have been quite procyclical during the sample period. However, impacts of business cycles are not uniform across occupations. To avoid the composition effect due to changes of occupation, the sample consists of workers who were in the same occupation during two consecutive periods. According to the previous studies on skill-premium, workers with low skill tend to show more cyclical wages (e.g., Dunlop 1939; Reder, 1955), and this view seems to apply to occupational
6
The advantage of first-difference estimation is twofold. It both ensures stationarity and eliminates any
fixed effects in the panel data.
6
categories as well. 7 Table 2 shows that the workers in the relatively low-wage group, such as operatives and laborers, show more cyclical wages and hours than do others. However, managerial workers exhibit most cyclical wages, although they are the most-skilled workers in terms of average wages.8 Self-employed workers, who are also involved in managerial tasks, show highly cyclical hourly earnings as well. In sum, both individual wages and managerialwage premiums are pro-cyclical during 1971-1992, which provides incentive for workers to move from non-market to market and non-managerial to managerial positions during expansions.
Transition Matrix of Occupations In order to examine the cyclical movement of workers among three occupations, the transition matrix is constructed from the same PSID data. The sample periods are divided into expansion, normal, and recession periods based on the real GDP growth rates. The expansion period is defined as the period of GDP growth rate above 4% (Table 4A), and the recession period is defined as the period of GDP growth rate less than 1% (Table 4B). For instance, the likelihood of moving from the non-market sector to non-managerial positions in the market is 0.261 during expansions. This likelihood drops to 0.193 during recessions. The likelihood of
7
The conventional wisdom of counter-cyclical skill-premium is based mainly on educational attachment
and labor-market experience. Raisian (1983) and Keane and Prasad (1993) challenged this view by reporting that they did not find a significant cyclical pattern in such skill-premium in the PSID. 8
One might view this pro-cyclical managerial premium in favor of the quasi-fixed labor theory,
presented in Oi (1962). This theory predicts a higher utilization of skilled workers during booms. However, according to Table 3, less-skilled workers show more pro-cyclical hours.
7
moving from workers to managers during expansions is 0.059, and this likelihood drops to 0.049 during recessions. Table 4C shows the difference of transition matrix between expansion and recession in terms of percentage: 50 × (Table 4A − Table 4B)/(Table 4A + Table 4B) . The likelihood of moving from workers to managers increases by 16.2% in expansions while the likelihood of moving from managers to workers decreases by 26.9%. Overall, the movement of labor from the lower-grade occupations to the higher-grade occupations seems much stronger in booms, and much weaker in recessions. In fact, in Table 4C all upper-diagonal terms are positive and all lower diagonal terms are negative.9
Composition Effects in Wages Finally, I examine the composition effects of wages in the data. Since the aggregate wage constructed by the BLS is based on non-supervisory workers only, aggregate wages are subject to composition effects at both ends of the wage distribution during expansions: the entry of low-wage workers from the non-market sector and the exit of high-wage workers to self-employed and managers. To distinguish these two effects, I construct both average wages (average wages of the entire workforce) and aggregate wages (average wages of non-
9
There exists a significant movement of workers between the non-market sector and managerial
positions as well. For example, movement from non-market to managers is 0.053 in expansions and 0.038 in recessions. Yet the model in this paper does not allow such jumps due to its strict hierarchy structure.
8
supervisory workers) from the PSID, and the cyclicality of these wages is compared to that of individual wages. Table 5 shows the cyclicality (the OLS estimate of b1 in equation (1)) of individual wages, the average wage of all workers, the aggregate wage based on non-supervisory workers, and finally, the BLS aggregate wage, respectively.10 The difference in the cyclicality between individual wages and the average wage represents the composition effect due to entry and exit of less-skilled workers only. This reduces the cyclicality of wages by 16% (from .597 for non-supervisory workers to .501).11 When aggregate wage is constructed based on nonsupervisory workers (not self-employed and non-managers), the cyclicality of wage is reduced by 33% (from .597 to .396). This reflects the effect of both types of composition bias. The cyclicality of our aggregate-wage measure is similar to that of the BLS aggregate wage (.434).
10
Wages of spouses have been available since the 1979 survey. For the empirical analysis of this
composition effect, I use the wages of heads of households only in order to generate a consistent time series for average wages. 11
This is consistent with the findings in Bils and Solon, et al. The compositional effect reduces the
cyclicality of aggregate wages by about 20% in Bils and more than 50% in Solon et al. The differences can be explained as follows. Bils uses the NLSY data, which has less heterogeneity than the PSID data. The average wages of new workers are lower than those of existing workers by 19% in his data, as opposed to 30% in the PSID. Solon et al. also use the PSID data, but their aggregate-wage measure is different from the one used here. They weight wages by hours as in the BLS aggregate wage. This generates another composition effect in aggregate wages by giving higher weights to low-wage workers in booms because low-wage workers work longer hours in booms. Since the model has an extensive margin only, I do not weight the wages by hours here.
9
In sum, the PSID data does show the significant role of both composition biases in aggregate wages. Furthermore, the cyclical behavior of wages and occupational changes in the PSID data seems consistent with the prediction of the model.
3. The Model There are identical families consisting of a continuum of family members with talent or skill z ∈ [ z , z ] . The measure µ(z) describes the number of members of the family of type z.12 The family maximizes lifetime utility defined over consumption of goods produced in the z
market, C t = ∫ c t ( z ) µ ( z )dz , and goods produced from non-market activity, namely, homez
z
produced goods, H t = ∫ ht ( z ) µ ( z )dz : z
∞
max {C , I ,l ( z )}∞ E0 ∑ ρ t [log C t + B log H t ] t
t t
subject to
t =0
(2)
t =0
z
∫z
[Ω t ( z )l t ( z )]µ ( z )dz = C t + I t − u t K t
K t +1 = I t + (1 − δ ) K t ,
(3) (4)
where ρ is the discount factor, and lt(z) is the market-labor supply of family member z. There are two occupations in the market: manager and production worker. A worker can earn wt(z) as a production worker, πt(z) as a manager. There is no “learning-by-doing” on the job so that the agent chooses the job that offers the highest current wage. Thus, the market earnings
12
The family assumption greatly simplifies the analysis because the allocation of the economy is
independent of income distribution, and the occupational choice depends on workers’ talent only.
10
Ωt(z) are max[wt(z),πt(z)]. The family owns the capital stock Kt and rents it at rental rate ut. Capital depreciates at rate δ, and the investment by the family is It. Each agent has a time endowment 1+θ. The agent can either supply one unit of labor inelastically to the market or spend it on home production: lt(z) is 1 if he works in the market and 0 otherwise. The other θ units of time are always used for non-market activity that includes home production such as cleaning and child care. ht ( z ) = α ( z )(θ + 1 − lt ( z ))
(5)
α ( z ) = α 0 z α1 , 0 < α 0 , 0 ≤ α1 < 1.
(6)
Productivity in home production may depend on the worker’s skill level as well.
The
important parameter regarding the comparative advantage between market and non-market activity is returns to skill in home production, α1.
According to the market-production
technology below, the wage of non-managerial worker is proportional to his skill. This implies that α1 represents the cross-sectional correlation of productivity between market and home production. For example, if α1 = 0, workers have the same home-production productivity regardless of earning ability in the market. However, if α1 is positive, for example, if α1 = 1/2, a worker A who is four times as productive as a worker B in the market is twice as productive as B in home production. When α1 is close to 1, the worker A is equally more productive than B in the market and at home. A higher value of α1 implies weaker comparative advantage between market and non-market work across workers. The value of α1 should depend on specific non-market activity. Yet given that our measure z represents general ability, including education, organizational skill, health, and so on, it seems natural to assume there exists a fairly
11
significant correlation between productivity in the market and home production. For instance, it is more likely that parents with better educational backgrounds are better at raising children. I will also provide evidence in Section 5 of positive α1 based on relative returns to skill in occupations that are comparable to home-production activities. Specifically, I estimate the returns to schooling and labor market experience of private household workers such as baby sitters and housekeepers and compare them to those of other unskilled workers in the PSID. The quantitative analysis in Section 5 shows that a high value of α1 substantially increases the response of aggregate hours to shifts in TFP. The intuition behind this result is that weak comparative advantage (high cross-sectional correlation in productivity) allows frequent movement between market and non-market sectors. The production process in the market has a hierarchical structure, so the manager commands production labor and capital. If an agent decides to be a worker, his talent z is transformed into efficiency unit of production labor linearly so that his wage as a production worker is wtz, where wt is the wage rate for efficiency unit of production labor. If an agent with skill z becomes a manager, he rents capital kt, hires production labor nt, and produces the output according to production technology: y t ( z ) = F ( z , k t , nt ) = At zψ [ g (k t , n t )] β , ψ = 1 − β + κ , 0 < β < 1, κ > 0 ,
(7)
ε
1 1− 1 1− ε −1 ε ε where g (k t , nt ) = χk t + (1 − χ )n t , 0 < χ < 1, ε > 0.
(8)
The substitution elasticity between capital and production labor is ε, and that between capital and manager is 1, given the multiplicative specification.
Production workers are perfect
substitutes for one another so that nt is measured in efficiency units. However, managerial
12
labor is assumed to be indivisible and uncombinable; two mediocre managers are not comparable to one superior manager. Managerial work consists of managerial decision making and monitoring/supervising.
There are economies of scale in managerial decisions because
improvements in upper-level decisions have an enormous influence on the organization by affecting productivity of all lesser-ranking workers. A supervisory activity congests this scale economy and determines the optimal span of control for each manager. (See Rosen, 1982, for a detailed discussion of this feature.) These considerations are reflected in the production technology F( ): F(z,n,k) is increasing returns to scale in z, n, and k, and strictly concave in n and k. Given the wage rate for production labor in efficiency unit wt and the rental rate of the capital ut, the manager receives the residual π t ( z ) = F ( z , k t , nt ) − ut k t − wt nt .
(9)
The first-order conditions for this maximum problem include Fk ( z , k t , nt ) = ut
(10)
Fn ( z , k t , nt ) = wt .
(11)
Demand for production labor n ( z , wt , ut ) and capital k ( z , wt , ut ) by the manager z can be obtained from (10) and (11).
Economies of scale imply managerial wage convex in z:
π ' ( z ) > 0, π " ( z ) > 0 . For efficient allocation, only the most talented will become managers.
The least
talented will become home-production workers, given the one-dimensional specification of talent. Critical levels of talent, zmt, zwt > 0, exist such that if z ≥ zmt, then the agent is a manager. If zmt > z ≥ zwt, the agent is a production worker. If z < zwt, the agent works at home. An
13
allocation in this economy means two critical values of talent zmt, zwt, and the demand functions for labor and capital by a manager with talent z, n ( z , wt , ut ) , k ( z , wt , ut ) . In equilibrium, the demands for factors are equal to their supplies: z
∫z
mt
z
∫z
mt
n ( z , wt , ut ) µ ( z )dz = ∫
zmt
z wt
zµ ( z )dz
(12)
k ( z , wt , u t ) µ ( z )dz = K t .
(13)
4. Qualitative Analysis This section presents the static version of the model to illustrate its key elements. Utility is linear and capital is dismissed. The manager with talent z receives π ( z , w) = max n F ( z , n ) − wn . The first-order condition, Fn ( z , n ) = w , provides the implicit demand function for production labor n ( z; w) .
Because l(z) = 0 for z < zw, the total
consumption of both market and home-produced goods is z
∫z
z
z
z
zm
c( z ) µ ( z )dz + ∫ [α ( z )(θ + 1 − l ( z ))]µ ( z )dz = ∫
F ( z , n ( z ))µ ( z )dz + ∫
zw
z
α ( z ) µ ( z )dz .
Without loss of generality, the constant is dropped after the equality. The maximization problem is now to choose the two critical skill levels for job assignment, zm and zw, given the labor-market constraint (12). In the Lagrangian with w as the multiplier associated with the labor-market constraint, this allocation is the competitive equilibrium with w being the wage rate of production labor: max zm , zw
z
∫z
m
F ( z , n ( z ))µ ( z )dz + ∫
zw
z
α ( z )µ ( z )dz + w[∫
zm
zw
The first-order conditions for critical level zm and zw are
14
zµ ( z )dz − ∫
z
zm
n( z ) µ ( z )dz ] .
F ( z m , n( z m )) − wn ( z m ) = wz m
(14)
wz w = α ( z w ) .
(15)
Equation (14) states that the marginal manager, whose talent is zm, faces a break-even point. The marginal manager’s income will be the same as his wage as a production worker. The second condition states that the marginal market participant’s wage is the same as the value of his home production. Figure 1 illustrates the labor-market equilibrium when α1 = 0. With market technology F ( z , n ) = Azψ n β the optimal span of control, the employment of production labor under the manager z is ψ
βA 1− β 1− β n ( z , w) = ( ) z . w 1
(16)
The optimal span of control is more than proportional to the skill of the manager; d ln( n ( z )) d ln( z )
κ > = 1 + 1−β 1 as κ captures economies of scale in managerial labor. Except for the
marginal manager, managers receive more than proportional economic rent for their superior talent.
This skews the wage distribution to the right, relative to the underlying skill
distribution. Inserting (16) into the labor-market equilibrium condition (12) reveals the wage rate of production labor as a function of critical levels zm, and zw: z
w( z m , z w ) = βA(
∫z
z
ψ 1− β
µ ( z )dz
m
zm
∫z
zµ ( z )dz
)1− β .
(17)
w
It is very useful to define the aggregate index of managerial labor and production labor in efficiency units as follows: Z = [∫
z
zm
z
ψ 1− β
µ ( z )dz ]1− β and N = ∫
zm
zw
15
zµ ( z )dz .
(18)
With these aggregate indices one can recover the aggregate market-production function as Y ( zm , z w ) = ∫
z
zm
Azψ n( z ) β µ ( z )dz = AZN β .
(19)
The wage rate of production labor in (17) is equivalent to the marginal product of production labor in the aggregate production function in (19), βAZN β −1 . The aggregate home production is H (z w ) = ∫
zw
z
α ( z ) µ ( z )dz .
(20)
Using aggregate production functions (19) and (20), the problem is max zm , zw Y ( z m , z w ) + H ( z w ) . The first-order conditions for zm and zw are equivalent to (14) and (15). I examine the response of economy to shifts in TFP in the market. Suppose that working in the market becomes more profitable due to an increase in A. The critical level of α
skill that determines market participation zw is, from (6) and (14), z w = ( w0 )1−α1 . As an increase in TFP increases the wage rate in the market, more people are drawn to the market sector. (Figure 2 illustrates this.) From (16), the span of control n(z) increases as the productionwage (w) increase is smaller than the TFP increase.13 An increase in the span of control makes the employment of managers less cyclical than that of production workers because existing managers can absorb new production workers. Since the managerial wage for agent z is
13
The supply curve of production labor is upward sloping because of the existence of the non-market
sector, and the demand curve is downward sloping because β < 1 . This implies that the increase of production-wage rate will be smaller than the increase of TFP.
16
π ( z ) = ( β1 − 1) wn ( z ) , the relative wage of manager to production worker also increases as the span of control increases:
π (z) w( z )
1
κ
= ( β1 − 1)( βwA ) 1− β z 1− β .
The cross-sectional comparative advantage between the market and home in the labor force has an interesting implication on the aggregate labor-supply elasticity as weak comparative advantage implies frequent movement of workers between sectors. As Figure 3 illustrates, when α1 > 0, the same increase in TFP draws more people to the market.
The
output from home production represents the opportunity cost of labor-market participation. When productivity between market and home is correlated, it is less costly to draw less-skilled workers into the market because they have a lower opportunity cost as well.
5. Quantitative Analysis The family assumption greatly simplifies the analysis of the model economy.14
It
separates the static problem of resource allocation at time t from the dynamic capitalaccumulation problem over time so that the model can be solved recursively. First, the allocation of capital and production labor across managers at time t is solved, given Kt, zmt, and zwt. Second, the paths of investment, It, consumption, Ct, and labor-supply decisions, zmt and zwt, are determined by the intertemporal consumption theory.
14
The family assumption dismisses possible income effects on labor supply across agents. However,
relaxing the family assumption along with allowing divisible labor will not affect the result on the cyclical behavior of relative wages. Given that wages are measured by total earnings divided by hours, the income effect on hours will strengthen the result of this model by reinforcing the pro-cyclical managerial wage because it will reduce the hours of high-wage earners who are managers.
17
Given the constant returns to scale in g(k,n), it is useful to write the output under manager z as y t ( z ) = At zψ (nt f ( rt )) β , where rt = kt/nt is the capital-labor ratio. The CRS of g ( ) also implies that the capital-labor ratio rt is common across managers so that rt = K t / N t . The first-order conditions (10) and (11) imply f (rt ) − rt f ' (rt ) wt = . f ' (rt ) ut
(21)
From (10) and (7), the demand for production labor by the manager z is 1
βAt zψ f ' ( rt ) 1− β 1 ) . n ( z , wt , u t ) = ( ut f ( rt )
(22)
Inserting (22) into the labor-market-equilibrium condition (12) and using the aggregate index in (18), one can write the rental rate as ut = βAt Z t ( N t f ( rt )) β −1 f ' ( rt ) .
(23)
Using (21) and (23) the wage rate for production labor is wt = βAt Z t ( N t f ( rt )) β −1 ( f ( rt ) − rt f ' ( rt )) .
(24)
The wage of manager z is, from (9), (21), and (22), π t ( z ) = (1 − β ) At Z t ( N t f ( rt )) β −1 f ( rt )n( z ) .
(25)
Then the growth rate of the relative wage of managers to production labor is •
•
•
•
π t ( z ) wt ( z ) n t ( z ) 1 r − = + χ (1 − ) t , π t ( z ) wt ( z ) n t ( z ) ε rt •
where x t =
dx ( t ) dt
(26)
. The relative wage depends on the span of control, substitution elasticity, and
capital-labor ratio. For instance, under capital-skill complementarity (ε > 1), an increase in
18
capital-labor ratio favors managers relative to workers. Again, using the aggregate index in (18), the aggregate market-production function is Y ( zmt , z wt , K t ) = ∫
z
[ At zψ ( nt ( z ) f ( rt )) β ]µ ( z )dz = At Z t ( f ( rt ) N t ) β .
z mt
(27)
The aggregate home-production function is z
z wt
z
z
H ( z wt ) = θ ∫ α 0 z α1 µ ( z )dz + ∫
α 0 z α1 µ ( z )dz .
(28)
Using (27) and (28) one can rewrite the maximization problem as ∞
max {Ct , zmt , zwt , K t +1} E 0 [∑ ρ t [log C t + B log H ( z wt )] t =0
subject to Y ( z mt , z wt , K t ) − C t − K t +1 + (1 − δ ) K t = 0 . With Lagrange multiplier λt for the resource constraint, the first-order conditions are C t −1 = λt
(29) −β
ψ
1− β (1 − β ) At Z t1− β ( N t f ( rt )) β z mt = βAt Z t ( N t f ( rt )) β −1 ( f ( rt ) − rt f ' ( rt )) z mt
(30)
Bα 0 z wt α1 H t ( z wt ) −1 = λt βAt Z t ( N t f ( rt )) β −1 ( f ( rt ) − rt f ' ( rt )) z wt
(31)
λt = ρEt [ λt +1 ( βAt +1Z t +1 ( N t +1 f ( rt +1 )) β −1 f ' ( rt +1 ) + 1 − δ )]
(32)
At Z t ( N t f ( rt )) β = C t + K t +1 − (1 − δ ) K t .
(33)
Equation (30) states that, for the marginal manager, the marginal product as a manager is equal to the marginal product as a production worker. Equation (31) states that, for the marginal market participant, the marginal product of labor is the same as the value of marginal product in home production. Equation (32) is the Euler equation. Equation (33) is the resource constraint.
19
Calibration of the Model According to the model, the wage of a non-supervisory worker is linear in worker’s talent z. This allows us to calibrate the talent distribution µ(z) directly from the cross-sectional wage distribution of the non-supervisory workers from PSID (wages of heads of households and wives). I use the wage distribution of 1983 and 1984 because it is the mid-point of the sample period and the base year of the deflator for nominal earnings. The mean and the standard deviation of the log-wage distribution are 2.17 and 0.53, respectively. This wage distribution should be viewed as a doubly truncated representation of skill distribution µ(z) for two reasons. First, workers in the non-market sector are not included. Second, as usual in the analysis with micro data, to avoid outliers due to measurement errors in reported hours and income, wages below $3 or above $100 are eliminated. This is consistent with the top-coding practice at wage $100 in the PSID. Then, the mean µz and variance σz of the underlying skill distribution µ(z) are searched among lognormal distributions to match the mean and standard deviation of the truncated wage distribution.
Specifically, when Φ( ) is the cumulative
standard normal distribution, µz and σz are chosen to satisfy the following two conditions (See Maddala, 1983.): E[ln( w) | 3 ≤ w ≤ 100) =
Φ ' [c1 ] − Φ ' [c2 ] 2.17 − µ z = Φ [c2 ] − Φ [c1 ] σz
V [ln( w) | 3 ≤ w ≤ 100] = 1 − E[ln( w) | 3 ≤ w ≤ 100]2 +
where c1 =
c1Φ ' [c1 ] − c 2Φ ' [c 2 ] (0.53) 2 = Φ [c1 ] − Φ [c2 ] σ z2
ln( 3) − µ z ln(100) − µ z and c 2 = . σz σz
20
The calibrated values are µz = 2.11 and σz = 0.58. Figure 4 shows the actual wage distribution and the calibrated skill distribution µ(z). Given the calibrated skill distribution µ(z), the steadystate critical values are chosen to match the occupational breakdown of the PSID: 12% managers, 63% non-managerial workers, and 25% non-market workers (zm =16.4 and zw =5.5). For production technology in the market we need to specify ε, χ, β, and κ. As a benchmark, the elasticity of substitution between production labor and capital (ε) is 1. Given the steady-state values of zm and zw, the first-order condition (30) imposes one constraint among χ, β, and κ. I calibrate χ and β based on disaggregate manufacturing-industry data, and equation (30) will provide κ. Using the four-digit-industry-level NBER productivity data constructed by Bartelsman and Gray (1996) from the Annual Survey of Manufacturing for 1954-1996, I calculate the shares of payment to employed labor in value-added output. The average of this share is .45. Since their data do not include some fringe-benefit payments, I use a slightly higher value: (1-χ) = .5.15 Based on three-digit-industry data, Burnside, Eichenbaum, and Rebelo (1995) report returns to scale of between .8 and .92. (See Table 5 of their paper.) According to Basu and Fernald (1996), aggregation tends to produce a higher estimate for the returns to scale. Since β represents the firm or plan-level decreasing returns to scale in our model, I use the smallest value of the estimates in Burnside et al.: β =.8. The parameter κ is set to 0.075 to satisfy equation (30). This leads to this production function for the benchmark case: y t ( z ) = At z 0.275 ( k t 0.5 nt 0.5 ) 0.8 . The capital share in total output is 0.372, which is close to
15
In fact, when the shares are adjusted for fringe benefits at the two-digit level the averages of shares
are .474 (value-added weighted) and .525 (non-weighted).
21
the values commonly used in the literature. In addition, the calibrated values of κ and β imply a relative standard deviation of log wages of managers to workers of 1.375, which is very close to the standard deviation of self-employed workers to non-self-employed workers in the PSID (1.39). According to the model, the returns to skill is 1 for non-managerial workers. Since α1 represents the returns to skill in home production, the relative returns to skill in occupations that are comparable to home production to those of other unskilled workers would provide some information on the size of α1. In the PSID, for female workers the relative returns to skill of private household workers such as baby sitters and housekeepers to those of other female unskilled workers are .62 and .70 when the skill is measured by the years of schooling and the labor-market experience (age-schooling-4), respectively. For male workers, they are .29 and .45, respectively, for schooling and labor-market experience.16 As a benchmark case, the returns to skill in home production is zero (α1 = 0). I use other values such as ½and ¾in the quantitative analysis as well. Values of other parameters are fairly standard. The discount factor is chosen to match the annual steady-state real interest rate of 6.5%. The annual depreciation rate of capital is 10%. The ratio of non-working time to working time is 3 (θ = 3). Temporary shifts in productivity At follow the first-order autocorrelation in logs as in King et al.: ln At = (1 − ρ A ) ln A + 0.9 ln At −1 + e At .
16
The standard deviation of e At is set to match the
Specifically, for female unskilled workers, the returns to schooling and labor-market experience are
4.24 and .55 respectively. For female private-household workers, they are 2.64 and .38. For male unskilled workers, they are 5.43 and 1.63. For male private-household workers, they are 1.56 and .73.
22
standard deviation of real GDP in the data in the benchmark case. The values of the integrals are approximated by a linear quadrature. Skill levels are confined within the range z ∈ [0.5, 300]. This range covers almost 100% of the calibrated skill distribution µ(z). B and α0 are free parameters. Table 6 summarizes the parameter values for the benchmark case. The model is solved numerically by a log-linear approximation of the first-order conditions (29)-(33) around the steady states of the economy with a deterministic trend as in King, Plosser, and Rebelo (1988). 17
Cross-Sectional Comparative Advantage and Aggregate Labor Supply Figure 5 shows the impulse responses of the benchmark economy to a one percent increase in TFP, At. All variables are percentage deviations from the steady state. Panel (1) shows the exogenous TFP shock and the capital stock.
Panel (2) shows aggregate
employment and average labor productivity. Aggregate employment and labor productivity increase about .4% and .7%, respectively, in the first period. Panel (3) shows aggregate consumption, investment, and output. The responses are similar to those from the standard RBC model.
17
Panel (4) shows the employment of production workers and managers.
As is common in the literature, a deterministic trend in productivity is incorporated to allow the
output, consumption, investment, and capital grow over time as in the data. When the productivity of 1−β
production labor in g(k,n), home production α(z), and managerial labor grow at rates γ, γ, and γ 1− β + κ , respectively, the model has a balanced growth path where output, capital, investment, and consumption of both market goods and home produced goods grow at a common growth rate γ. See King et al. (1987) for a detailed discussion of how to obtain a stationary economy from one with balanced growth.
23
Employment of production workers is more volatile because of the pro-cyclical span of control. Panel (5) shows the average span of control, that is, the average employment of production labor per managers. Span of control increases initially, and then goes below the steady state in the later stage of an expansion as market participation decreases. In the later stage of expansion, the demand for home-produced goods increases along with the consumption of market goods.
Panel (6) shows the wages of production workers and
managers. Since capital accumulation is neutral (ε = 1), the wage gap between managers and workers represents the span-of-control effect only. Table 7 reports the population moments of the model economy. The first column is the benchmark case. Like other RBC models with productivity shocks, labor productivity is too pro-cyclical (0.98 relative to 0.753 in the data) and the employment is not so volatile as in the data (0.246 relative to 0.7 in the data). As is explained in the static case, the response of aggregate employment depends on the cross-sectional comparative advantage. As the cross-sectional correlation of productivity in the market and at home increases, employment becomes more volatile. For example, when α1 = 1/2, the relative volatility of employment increases from 0.246 to 0.387. When α1 = 3/4, the relative volatility of employment increases to 78% of the data (0.544 relative to 0.7). As employment becomes more volatile, the composition effect due to entry and exit of less-skilled workers reduces the labor productivity-employment correlation significantly (0.314 when α1 = 3/4). As a result, the volatility of output increases as well. Figure 6 shows the impulse response for the case of α1 = 3/4. Aggregate employment is much more volatile relative to labor productivity as it increases almost 1% initially, yet labor productivity increases only .3%.
24
While the relative volatility of employment to labor productivity in the benchmark case (.295) is far less than that in the data (1.062), the volatility is now quite close to data (.796).
Capital-Skill Complementarity Capital-skill complementarity has been emphasized in the literature because it creates an interesting short-run dynamics for relative employment (e.g., Rosen, 1968; Griliches, 1969). In this case, the short-run production function is not homothetic, and the relative demand for labor depends on the stage of business cycles.
The relative demand for unskilled labor
increases in the beginning of an expansion when capital is not yet accumulated. However, subsequent capital accumulation will substitute for the unskilled labor and will favor the skilled labor. Figure 7 shows the impulse response when production labor is a better substitute for capital than managerial labor (ε = 3/2). At the outset of an expansion, the employment of production workers increases sharply. This is reflected in the impulse response of the span of control. It sharply increases in the beginning and retreats as capital accumulates. In fact, employment of production workers tends to lead that of non-production labor in the post-war U.S. quarterly data. The relative wage of managers to workers increases in the beginning because of the spike in span of control. It is sustained even after the span of control falls below the steady-state level because capital accumulation favors managerial labor, which is relatively complementary to capital. (Recall equation (26).) Due to the higher substitution between capital and production labor, employment becomes more volatile and its volatility (0.618) is now quite close to that of data (0.7). A higher volatility of employment reinforces
25
the composition bias and the labor productivity-employment correlation is close to zero (0.052) as in the data.18 19
Two Composition Effects in Aggregate Wages The model generates less cyclical aggregate wages and labor productivity than those in the standard RBC models. This is due to the changes in skill mix of the workforce over the business cycle. Since the aggregate wage constructed by the BLS is based on non-supervisory workers only, there are two composition effects in aggregate wages during expansions: the entry of low-wage workers from the non-market sector and the exit of high-wage workers to self-employed and managers. Parallel to the empirical analysis in Section 2, I construct the average wages (average wages of the entire workforce) and the aggregate wages (average wages of production workers) from the model. The last column of Table 8 shows the cyclicality of these wages based on the PSID data in Section 2.
18
To compare the performance of the model, implied regression
For comparison, Figure 8 shows the impulse responses for the case of ε = 2/3, the opposite case to the
capital-skill complementarity. The span of control is relatively flat over the business cycle because the relative demand for production labor does not sharply increase in the beginning, and capital accumulation complements production workers in the later stage of the business cycle. The wage growth is reversed in the early stage of the business cycle, even though the span of control stays above the steady state. 19
The results based on the Hodrick-Prescott filter show a similar pattern to those in Table 7 except for
labor productivity-employment correlations. They exhibit slightly higher correlation in the model and lower correlation in the data than in Table 7.
26
coefficientthe ratio of the covariance between the output and wage measure to the variance of outputis calculated. In the benchmark case, the composition effect reduces the cyclicality of average wages and aggregate wages by 10% (from 0.911 to 0.816) and 12% (from 0.911 to 0.804), respectively. Because employment is not so volatile as in the data, the composition effect is not so great as in the data. When the employment is nearly as volatile as in the data (the model with α1 = 3/4, ε = 3/2), the cyclicality of average wage decreases by 19% (from 0.734 to 0.593), and that of aggregate wage decreases by 34% (from 0.734 to 0.483).20 The composition effect predicted by the model is very close to what we find from the PSID data in Section 2 (16% and 33%, respectively). In sum, both in the data and in the model, not only the first bias (due to entry and exit of low-wage workers between market and home) but also the second bias (due to the movement between non-supervisory workers and managerial workers) significantly reduces the cyclicality of wages.
6. Concluding Remarks Equilibrium models of the business cycle that emphasize shifts in TFP as a major source of economic fluctuation do not explain why aggregate hours vary greatly over the business cycle without much variation in aggregate wages. This paper suggests a resolution to this problem by recognizing the heterogeneity of skill possessed by workers. The model in this paper augments the standard RBC model to include distinctions between managerial and nonmanagerial employment and home and market production.
20
Due to differences in detrending method and frequency between the panel data and the model, I
compare the relative size of compositional effects instead of comparing the numbers directly.
27
Business cycles are associated with the systematic movement of workers that makes aggregate wages and labor productivity less cyclical than individual wages. Not only do lessskilled workers enter the workforce, biasing down wages and productivity in expansion, but higher-wage earners become managers and self-employed, lowering aggregate wages further. Weak comparative advantage between market and non-market work implies frequent movement between sectors in response to relative productivity shifts. Under a comparative advantage that is consistent with PSID, this model is capable of matching the key moments in the aggregate labor market. Aggregate hour is nearly as volatile as in the data, and yet it is not highly correlated with labor productivity. Aggregate wage and labor productivity exhibit mildly pro-cyclical behavior. The model produces interesting dynamics for relative wages and employment as well. Wages of managerial workers are highly pro-cyclical, as in the PSID data. As the employment of production labor tends to lead that of non-production labor in the data, under capital-skill complementarity the relative demand for production labor sharply increases at the beginning of an expansion. This model abstracts from any friction in the labor market. For instance, in an economy with a rigid labor market, the response of the economy may be quite different from those predicted by the model. Examples of such friction include search friction, specific human capital, hiring costs, and firing costs. While the model assumes a fixed one-dimensional talent distribution in the economy, allowing multi-dimensional talent or endogenous evolution of jobspecific skills such as learning-by-doing will enrich the employment dynamics across occupations.
28
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33
Appendix: Data
The PSID data consist of a random sample and a poverty sample. Only the random sample is used to represent the skill distribution of the aggregate economy. The sample period is 1971-1992. The sample consists of heads of households and wives who are 18-60 years old. Wage data for wives are available only since 1979. Wage data are annual hourly earnings (annual labor incomes divided by annual hours). The wages are for those workers who were working in the private and non-agricultural sectors for at least 100 hours per year, and whose hourly wage rate was above $3 in 1983 dollars. Wages are deflated by the Consumer Price Index. The base year is 1983. The descriptive statistics for workers in the market sector are given in Table 1. In the estimation of the cyclicality of wages in different occupations in Table 3, the sample consists of workers who were in the same occupation in two consecutive periods to capture the pure price changes of labor in the corresponding category. In Table 7, because wives’ wages are available only since 1979, data from heads of households are used in order to generate a consistent time series of average wages. Aggregate data are quarterly for 1955:I - 1994:IV from the Citibase. Output is real GDP (in 1987 constant dollars). Consumption is non-durables and services. Investment is gross fixed investment. Employment is employed man-hours based on the BLS establishment survey.
34
Table 1: Summary Statistics from the PSID, 1971-1992. Variables
Mean
Standard Deviation
Observations
Age
36.87
10.96
89867
Years of schooling
12.89
2.53
89032
Head of household
0.66
0.47
89867
Female dummy
0.44
0.49
89867
Real wages
10.88
7.24
59827
Annual working hours
2044.83
649.23
59827
Annual labor income
22499.42
17222.97
59827
35
Table 2A: Comparison of Wages of Movers and Stayers: Wages at Time t. (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
NA
7.71 (-3.12)
9.10 (-4.53)
(Non-Managers) t-1
NA
10.83 (0)
11.66 (-2.97)
(Managers) t-1
NA
11.94 (1.11)
14.63 (0)
Note: Numbers in parentheses are average wages relative to the stayers in the new occupation.
Table 2B: Comparison of Wages of Movers and Stayers: Wages at Time t-1. (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
NA
NA
NA
(Non-Managers) t-1
8.48 (-2.14)
10.62 (0)
11.59 (0.97)
(Managers) t-1
10.07 (-4.35)
11.51 (-2.91)
14.42 (0)
Note: Numbers in parentheses are average wages relative to the stayers in the old occupation.
36
Table 3: Cyclical Behavior of Wages, Hours, and Labor Income (Occupational categories are only for the non-self-employed.) Occupation
Wage
Hour
Income
Obs.
Average Wage
All
.542
.523
1.065
48456
11.46
(.063)*
(.064)*
(.071)*
.747
.565
1.312
4353
13.58
(.337)*
(.206)*
(.332)*
.828
.179
1.007
2273
14.77
(.237)*
(.198)
(.212)*
Professional/
.453
.316
.770
8438
14.05
Technical
(.136)*
(.150)*
(.151)*
Clerical/Sales
.373
.234
.607
9359
10.39
(.126)*
(.141)
(.152)*
.582
.505
1.087
5384
11.92
(.150)*
(.150)*
(.160)*
Operatives/
.637
1.014
1.652
8931
8.37
Laborer
(.131)*
(.162)*
(.112)*
Self-Employed
Managerial
Craftsmen
Note: Numbers in parentheses are standard deviations. * significant at 5%
37
Table 4A: Transition Matrix of Occupational Changes: Expansions (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
0.686
0.261
0.053
(Non-Managers) t-1
0.061
0.881
0.059
(Managers) t-1
0.040
0.176
0.783
Total
0.16
0.65
0.19
Table 4B: Transition Matrix of Occupational Change: Recessions (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
0.768
0.193
0.038
(Non-Managers) t-1
0.085
0.865
0.049
(Managers) t-1
0.066
0.231
0.704
Total
0.24
0.613
0.148
Table 4C: Difference in Percentage between Expansions and Recessions ( 50 × (Table 4A − Table 4B)/(Table 4A + Table 4B) ) (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
11.4
29.9
31.8
(Non-Managers) t-1
-33.8
1.8
16.2
(Managers) t-1
-47.8
-26.9
10.8
38
Table 5: Two Composition Effects in Aggregate Wages from the PSID Wages
Data
Obs.
Supervisory workers
0.873 (.244)
5952
Non-supervisory workers
0.597 (.071)
27667
Average wage of all workers
0.501 (.179)
20
Aggregate wage based on non-supervisory workers
0.396 (.188)
20
BLS aggregate wage
0.434 (.158)
20
39
Table 6: Parameter Values for the Benchmark Case Parameters
Description
µz = 2.11
Mean of lognormal skill distribution µ(z)
σz = 0.58
Standard deviation of lognormal skill distribution µ(z)
zm = 16.4
Steady-state critical-skill level for zmt
zw = 5.5
Steady-state critical-skill level for zwt
β = 0.8
Curvature in production function y = Az1-β+κ[g(n,k)]β
ε=1
Substitution elasticity between n and k
χ = 0.5
Capital share in g(n,k)
κ = 0.075
Economies-of-scale parameter
δ = 0.025
Quarterly depreciation rate
θ=3
Ratio of non-working time to working time
ρ = 0.988
Quarterly discount factor
α1 = 0
Home-production technology α( z ) = α 0 z α1
ρA = 0.9
Autocorrelation of productivity shock At
40
Table 7: Population Moments of the Models α1=0
α1=1/2
α1=3/4
α1=3/4
ε=1
ε=1
ε=1
ε=3/2
σY
3.98
4.167
4.388
4.591
3.98
σC / σY
0.703
0.694
0.685
0.627
0.812
σI / σY
2.369
2.43
2.5
2.666
2.07
σEmp /σY
0.246
0.387
0.544
0.618
0.7
σEmp / σY/Emp
0.295
0.514
0.796
0.819
1.062
cor(Y, Emp)
0.745
0.752
0.760
0.652
0.753
cor(Y,Y/Emp)
0.98
0.94
0.855
0.786
0.715
cor(Emp, Y/Emp)
0.599
0.484
0.314
0.052
0.079
Statistics
Data
Note: Data are linearly detrended. σC /σY: standard deviation of consumption relative to Y. Emp: total employment in the model, total employed hours in the data. cor(C,Y): correlation of C and Y.
41
Table 8: Composition Effect in Wages from the Model α1=0
α1=3/4
α1=3/4
ε=1
ε=1
ε=3/2
cov (Wman,Y) / Var (Y)
0.926
0.834
0.779
0.873
cov (Wprod,Y) / Var (Y)
0.911
0.800
0.734
0.597
cov (Waverage,Y) / Var (Y)
0.816
0.585
0.593
0.501
cov (Waggregate,Y) / Var (Y)
0.804
0.557
0.483
0.396
Statistics
Data
Note: Wman: wages of managers including self-employed. Wprod: wages of non-managerial workers. Waverage: average wage of workers in the market sector. Waggregate: average wage of non-managerial workers.
42
Figure 1: Equilibrium in the Labor Market
Wage
π( z,w) : wages as a manager
wz : wages as a production worker
α(z ) : value of home production
zw
0
zm
z
Density
home 0
µ (z) managers
production workers zw
zm
43
z
Figure 2: Productivity Increase in the Market
Wage π'(z)
π (z) w'z wz
α (z) zw' zw
0
zm' zm
z
Density
home 0
µ (z) managers
production workers
zw' zw
zm' zm
44
z
Figure 3: Productivity Increase in the Market when α′(z) > 0 and α′(z) = 0.
Wage
w'z wz α (z) α0
0
zw'
z
zw
45
Figure 4: Wage Distribution of the PSID () and the Lognormal Distribution (--).
46
Figure 5: Impulse Responses of the Economy When α1=0 and ε=1
47
Figure 6: Impulse Responses of the Economy When α1=3/4 and ε=1
48
Figure 7: Impulse Responses of the Economy When α1=3/4 and ε=3/2
49
Figure 8: Impulse Responses of the Economy When α1=3/4 and ε=2/3
50
Yongsung Chang* Department of Economics, University of Pennsylvania, Philadelphia, PA 19104, USA
Abstract The standard equilibrium models of business cycles face a puzzling fact that total hours vary greatly over the business cycle without much variation in aggregate wages. The model augments the standard RBC model to include Lucas span of control. Distinction between market and non-market and managerial and non-managerial work makes aggregate wages far less cyclical than individual wages. Cross-sectional comparative advantage between market and non-market sector in the workforce substantially increases the response of aggregate hours to shifts in relative productivity. As a result, the model provides a reconciliation between data and equilibrium macroeconomics.
Key Words: Wages; Business Cycles; Comparative Advantage JEL Classification: E32; J31; J62
#
The previous version of the paper has been circulated under the title “Cyclical Behavior of
Occupational Choices and Wages.” I thank Mark Bils for advice and encouragement. I also thank Jeff Campbell, Pat Kehoe, Bob King, Monika Merz, Sergio Rebelo, Richard Rogerson, Alan Stockman, Randy Wright, an anonymous referee, and workshop participants at Chicago, Penn, Penn State, Rice, Rochester, and Texas A&M for helpful comments. All remaining errors are mine. *
Email: [email protected]
1. Introduction One of the big puzzles in macroeconomics has been that total hours vary greatly over the business cycle without much variation in wages. The equilibrium approach to economic fluctuation pioneered by Lucas and Rapping (1969) and put forward by Kydland and Prescott (1982) and Long and Plosser (1983) views the variation of total hours of work as people’s willingness to intertemporally substitute leisure and work over the business cycle. As Barro and King (1984) illustrate, under standard preferences the equilibrium business-cycle models require a strongly pro-cyclical real wages to be consistent with the movement in hours and consumption in the data.1 The model in this paper augments the standard real-business-cycle (RBC) model to include Lucas’ (1978) span of control and Rosen’s (1982) hierarchy, where workers are assigned to managerial, production, and non-market tasks based on comparative advantage. It provides a partial resolution to the movement of hours and wages in the data. The model shows that heterogeneity of workers significantly decreases the movement of aggregate wages over the business cycle and that the comparative advantage between market and non-market activities may substantially increase the response of aggregate hours to shifts in relative productivity. Previous studies have shown that entry and exit of low-wage workers create a countercyclical composition bias in aggregate wages (e.g., Stockman, 1983; Bils, 1985; Solon, Barsky,
1
The implicit labor contract can generate acyclical wages (e.g., Azariadis, 1974; Gomme and
Greenwood, 1995; Boldrin and Horvarth, 1995). Yet it is still incapable of explaining why labor productivity is not highly correlated with hours over the business cycle.
1
and Parker, 1994). Aggregate wages constructed by the Bureau of Labor Statistics (BLS) are based on the wages of non-supervisory workers at a point in time. The transition matrix of the occupational change constructed from the Panel Study of Income Dynamics (PSID) for 19711992 indicates that there is a significant cyclical movement between non-supervisory and supervisory occupations (managers and self-employed). For example, the likelihood of moving from a non-supervisory occupation to a supervisory occupation increases by 16.2% during expansions relative to recession while the likelihood of moving in the opposite direction decreases by 26.9%. This implies that aggregate wages are subject to composition effect at both ends of the wage distribution. Not only do less-skilled workers enter the workforce, biasing down wages and productivity in expansion, but higher-wage earners become managers and self-employed, lowering aggregate wages further.2
In fact, when the cyclicality is
measured as a percentage response to output growth, the composition effect reduces the cyclicality of aggregate wages by one-third relative to individual wages both in the model and in the data. In order to match the observed movement in total hours worked in the data, the representative-agent models often rely on labor-supply elasticity that is beyond the admissible estimates based on micro data.3 The model in this paper explores the relationship between the cross-sectional comparative advantage and aggregate labor-supply elasticity. A worker who has a very strong and clear comparative advantage would not exhibit frequent movement
2
See also Kydland (1984), Cho and Rogerson (1988), and Cho (1995) for business-cycle models that
incorporate the first composition bias. 3
For an empirical estimate of labor-supply elasticity, see Ghez and Becker (1975) or Altonji (1986).
2
between sectors in response to shifts in relative productivity. However, a weak comparative advantage implies frequent movement between sectors. Home production is introduced to represent the activity in the non-market sector, and the notion of comparative advantage is formalized by the cross-sectional correlation of productivity between the market and home production in the workforce. The quantitative analysis shows that it is capable of generating a realistic movement of aggregate hours in response to shifts in total factor productivity (TFP) under the cross-sectional comparative advantage observed in the PSID.4 When the skill distribution is calibrated by the wage distribution of the PSID, the model is very successful in matching the cyclical behavior of aggregate wages, labor productivity, and employment in the data. Employment in the model is nearly as volatile as that in the data, and yet it is not highly correlated with labor productivity.5 Aggregate wages and labor productivity are mildly pro-cyclical, as in the data. The model also produces interesting dynamics for relative wages and employment in the data. The skill-premium is known to be counter-cyclical. Based on the occupational classification, the PSID data are in accord with this view in general: wages of workers in lowgrade occupations are more pro-cyclical. However, contrary to the conventional wisdom, I
4
A lottery economy such as Rogerson (1988) and Hansen (1985) generates a high aggregate labor-
supply elasticity as well. However, a lottery economy has a counter-intuitive implication that people who draw bad lots are hired. Yet job assignmentincluding market participationis determined based on comparative advantage in our model. 5
Alternatively, one can reduce this correlation by introducing an additional disturbance that shifts the
labor-supply curve (e.g., Benhabib, Rogerson, and Wright, 1991; Greenwood and Hercowitz 1991; Bencivenga, 1992; and Christiano and Eichenbaum, 1992).
3
find that wages of workers in managerial jobs, who are among the highest paid, show the most pro-cyclical pattern over the business cycle. According to the model, the managerial wage depends on the span of control, and the span of control exhibits a strongly pro-cyclical pattern. Under capital-skill complementarity the demand for production labor increases sharply at the beginning of an expansion before capital is accumulated. This is consistent with the empirical observation in the U.S. quarterly data, as the employment of production workers tends to lead that of non-production workers over the business cycle. This paper is organized as follows. Section 2 provides empirical evidence of the hierarchical structure of occupations and the cyclical movement of labor and wages that are consistent with the prediction of the model. Section 3 develops the general-equilibrium model with job assignment. Section 4 presents the static model to illustrate its important features. In Section 5, the dynamic model is calibrated, and its response to stochastic variations in TFP is analyzed. Section 6 is the conclusion.
2. Some Evidence of Labor-Market Fluctuations The model in this paper assumes three occupations: managers, production workers, and home workers. These occupations are hierarchically ordered in terms of required skill. The model predicts a cyclical movement of workers between low-grade and high-grade occupations in response to wage differentials over the business cycle. For example, workers move from non-market to market sectors and non-managerial to managerial positions in expansions in response to pro-cyclical wages and managerial-wage premiums. Thus, the aggregate wage based on non-supervisory workers can be less cyclical than individual wages due to the composition effect at both ends of the wage distribution. 4
To determine the empirical relevance of these features, based on the annual PSID data for 1971-1992, the following aspects of the labor-market fluctuation are examined. First, evidence of a hierarchical structure and comparative advantage among the three occupations is presented.
Second, wage response over the business cycle is estimated across these
occupations. Third, based on the transition matrix, the cyclical movement of workers among the three occupations is analyzed. Finally, the composition effects in aggregate wages are measured.
Table 1 shows the summary statistics, and the Appendix contains a detailed
explanation of the data.
Hierarchical Structure of Occupations If occupations are ordered hierarchically in terms of required skill, and if workers are assigned to these occupations according to their comparative advantage, workers who change their occupations from lower-grade jobs to higher-grade jobs must be relatively more-skilled and high-wage workers in their former occupations. At the same time, these workers must be relatively less-skilled and low-wage workers in their new occupations. In other words, new managers are poor managers compared to existing ones, even though they were previously relatively good workers. Table 2A shows average wages at time t in each cell of the transition matrix among managers (self-employed + not self-employed managers), non-managers (not self-employed and non-managerial workers), and non-market workers (not employed). The numbers in parentheses are average wages relative to those of existing workers in the target occupation. For example, the number 7.71 (-3.12) in the (1,2)th cell in Table 2A implies that the average wage of new workers from the non-market sector in time t is $7.71 (in 1984 dollars), and it is 5
lower than the average wage of existing workers by $3.12. The hierarchical structure of occupations and assignment of workers according to comparative advantage imply that the numbers in parentheses in the upper-diagonal terms must be negative, and the lower-diagonal terms must be positive. Table 2B compares the wages of movers and stayers at time t-1. The numbers in parentheses are those relative to the stayers in the previous occupation. For the same reasons, the upper-diagonal terms must be positive, and the lower-diagonal terms must be negative. There is no exception in the sign of these relative wages of movers and stayers.
Cyclical Behavior of Wages across Occupations Table 3 shows the cyclicality of wages, hours, and incomes across occupations. The cyclicality is measured as the percentage response to real GDP growth, the coefficient b1 of the regression 6 ∆ log( X it ) = b0 + b1 ∆ log(real GDPt ) + eit .
(1)
Consistent with previous studies by Bils and Solon et al., individual wages have been quite procyclical during the sample period. However, impacts of business cycles are not uniform across occupations. To avoid the composition effect due to changes of occupation, the sample consists of workers who were in the same occupation during two consecutive periods. According to the previous studies on skill-premium, workers with low skill tend to show more cyclical wages (e.g., Dunlop 1939; Reder, 1955), and this view seems to apply to occupational
6
The advantage of first-difference estimation is twofold. It both ensures stationarity and eliminates any
fixed effects in the panel data.
6
categories as well. 7 Table 2 shows that the workers in the relatively low-wage group, such as operatives and laborers, show more cyclical wages and hours than do others. However, managerial workers exhibit most cyclical wages, although they are the most-skilled workers in terms of average wages.8 Self-employed workers, who are also involved in managerial tasks, show highly cyclical hourly earnings as well. In sum, both individual wages and managerialwage premiums are pro-cyclical during 1971-1992, which provides incentive for workers to move from non-market to market and non-managerial to managerial positions during expansions.
Transition Matrix of Occupations In order to examine the cyclical movement of workers among three occupations, the transition matrix is constructed from the same PSID data. The sample periods are divided into expansion, normal, and recession periods based on the real GDP growth rates. The expansion period is defined as the period of GDP growth rate above 4% (Table 4A), and the recession period is defined as the period of GDP growth rate less than 1% (Table 4B). For instance, the likelihood of moving from the non-market sector to non-managerial positions in the market is 0.261 during expansions. This likelihood drops to 0.193 during recessions. The likelihood of
7
The conventional wisdom of counter-cyclical skill-premium is based mainly on educational attachment
and labor-market experience. Raisian (1983) and Keane and Prasad (1993) challenged this view by reporting that they did not find a significant cyclical pattern in such skill-premium in the PSID. 8
One might view this pro-cyclical managerial premium in favor of the quasi-fixed labor theory,
presented in Oi (1962). This theory predicts a higher utilization of skilled workers during booms. However, according to Table 3, less-skilled workers show more pro-cyclical hours.
7
moving from workers to managers during expansions is 0.059, and this likelihood drops to 0.049 during recessions. Table 4C shows the difference of transition matrix between expansion and recession in terms of percentage: 50 × (Table 4A − Table 4B)/(Table 4A + Table 4B) . The likelihood of moving from workers to managers increases by 16.2% in expansions while the likelihood of moving from managers to workers decreases by 26.9%. Overall, the movement of labor from the lower-grade occupations to the higher-grade occupations seems much stronger in booms, and much weaker in recessions. In fact, in Table 4C all upper-diagonal terms are positive and all lower diagonal terms are negative.9
Composition Effects in Wages Finally, I examine the composition effects of wages in the data. Since the aggregate wage constructed by the BLS is based on non-supervisory workers only, aggregate wages are subject to composition effects at both ends of the wage distribution during expansions: the entry of low-wage workers from the non-market sector and the exit of high-wage workers to self-employed and managers. To distinguish these two effects, I construct both average wages (average wages of the entire workforce) and aggregate wages (average wages of non-
9
There exists a significant movement of workers between the non-market sector and managerial
positions as well. For example, movement from non-market to managers is 0.053 in expansions and 0.038 in recessions. Yet the model in this paper does not allow such jumps due to its strict hierarchy structure.
8
supervisory workers) from the PSID, and the cyclicality of these wages is compared to that of individual wages. Table 5 shows the cyclicality (the OLS estimate of b1 in equation (1)) of individual wages, the average wage of all workers, the aggregate wage based on non-supervisory workers, and finally, the BLS aggregate wage, respectively.10 The difference in the cyclicality between individual wages and the average wage represents the composition effect due to entry and exit of less-skilled workers only. This reduces the cyclicality of wages by 16% (from .597 for non-supervisory workers to .501).11 When aggregate wage is constructed based on nonsupervisory workers (not self-employed and non-managers), the cyclicality of wage is reduced by 33% (from .597 to .396). This reflects the effect of both types of composition bias. The cyclicality of our aggregate-wage measure is similar to that of the BLS aggregate wage (.434).
10
Wages of spouses have been available since the 1979 survey. For the empirical analysis of this
composition effect, I use the wages of heads of households only in order to generate a consistent time series for average wages. 11
This is consistent with the findings in Bils and Solon, et al. The compositional effect reduces the
cyclicality of aggregate wages by about 20% in Bils and more than 50% in Solon et al. The differences can be explained as follows. Bils uses the NLSY data, which has less heterogeneity than the PSID data. The average wages of new workers are lower than those of existing workers by 19% in his data, as opposed to 30% in the PSID. Solon et al. also use the PSID data, but their aggregate-wage measure is different from the one used here. They weight wages by hours as in the BLS aggregate wage. This generates another composition effect in aggregate wages by giving higher weights to low-wage workers in booms because low-wage workers work longer hours in booms. Since the model has an extensive margin only, I do not weight the wages by hours here.
9
In sum, the PSID data does show the significant role of both composition biases in aggregate wages. Furthermore, the cyclical behavior of wages and occupational changes in the PSID data seems consistent with the prediction of the model.
3. The Model There are identical families consisting of a continuum of family members with talent or skill z ∈ [ z , z ] . The measure µ(z) describes the number of members of the family of type z.12 The family maximizes lifetime utility defined over consumption of goods produced in the z
market, C t = ∫ c t ( z ) µ ( z )dz , and goods produced from non-market activity, namely, homez
z
produced goods, H t = ∫ ht ( z ) µ ( z )dz : z
∞
max {C , I ,l ( z )}∞ E0 ∑ ρ t [log C t + B log H t ] t
t t
subject to
t =0
(2)
t =0
z
∫z
[Ω t ( z )l t ( z )]µ ( z )dz = C t + I t − u t K t
K t +1 = I t + (1 − δ ) K t ,
(3) (4)
where ρ is the discount factor, and lt(z) is the market-labor supply of family member z. There are two occupations in the market: manager and production worker. A worker can earn wt(z) as a production worker, πt(z) as a manager. There is no “learning-by-doing” on the job so that the agent chooses the job that offers the highest current wage. Thus, the market earnings
12
The family assumption greatly simplifies the analysis because the allocation of the economy is
independent of income distribution, and the occupational choice depends on workers’ talent only.
10
Ωt(z) are max[wt(z),πt(z)]. The family owns the capital stock Kt and rents it at rental rate ut. Capital depreciates at rate δ, and the investment by the family is It. Each agent has a time endowment 1+θ. The agent can either supply one unit of labor inelastically to the market or spend it on home production: lt(z) is 1 if he works in the market and 0 otherwise. The other θ units of time are always used for non-market activity that includes home production such as cleaning and child care. ht ( z ) = α ( z )(θ + 1 − lt ( z ))
(5)
α ( z ) = α 0 z α1 , 0 < α 0 , 0 ≤ α1 < 1.
(6)
Productivity in home production may depend on the worker’s skill level as well.
The
important parameter regarding the comparative advantage between market and non-market activity is returns to skill in home production, α1.
According to the market-production
technology below, the wage of non-managerial worker is proportional to his skill. This implies that α1 represents the cross-sectional correlation of productivity between market and home production. For example, if α1 = 0, workers have the same home-production productivity regardless of earning ability in the market. However, if α1 is positive, for example, if α1 = 1/2, a worker A who is four times as productive as a worker B in the market is twice as productive as B in home production. When α1 is close to 1, the worker A is equally more productive than B in the market and at home. A higher value of α1 implies weaker comparative advantage between market and non-market work across workers. The value of α1 should depend on specific non-market activity. Yet given that our measure z represents general ability, including education, organizational skill, health, and so on, it seems natural to assume there exists a fairly
11
significant correlation between productivity in the market and home production. For instance, it is more likely that parents with better educational backgrounds are better at raising children. I will also provide evidence in Section 5 of positive α1 based on relative returns to skill in occupations that are comparable to home-production activities. Specifically, I estimate the returns to schooling and labor market experience of private household workers such as baby sitters and housekeepers and compare them to those of other unskilled workers in the PSID. The quantitative analysis in Section 5 shows that a high value of α1 substantially increases the response of aggregate hours to shifts in TFP. The intuition behind this result is that weak comparative advantage (high cross-sectional correlation in productivity) allows frequent movement between market and non-market sectors. The production process in the market has a hierarchical structure, so the manager commands production labor and capital. If an agent decides to be a worker, his talent z is transformed into efficiency unit of production labor linearly so that his wage as a production worker is wtz, where wt is the wage rate for efficiency unit of production labor. If an agent with skill z becomes a manager, he rents capital kt, hires production labor nt, and produces the output according to production technology: y t ( z ) = F ( z , k t , nt ) = At zψ [ g (k t , n t )] β , ψ = 1 − β + κ , 0 < β < 1, κ > 0 ,
(7)
ε
1 1− 1 1− ε −1 ε ε where g (k t , nt ) = χk t + (1 − χ )n t , 0 < χ < 1, ε > 0.
(8)
The substitution elasticity between capital and production labor is ε, and that between capital and manager is 1, given the multiplicative specification.
Production workers are perfect
substitutes for one another so that nt is measured in efficiency units. However, managerial
12
labor is assumed to be indivisible and uncombinable; two mediocre managers are not comparable to one superior manager. Managerial work consists of managerial decision making and monitoring/supervising.
There are economies of scale in managerial decisions because
improvements in upper-level decisions have an enormous influence on the organization by affecting productivity of all lesser-ranking workers. A supervisory activity congests this scale economy and determines the optimal span of control for each manager. (See Rosen, 1982, for a detailed discussion of this feature.) These considerations are reflected in the production technology F( ): F(z,n,k) is increasing returns to scale in z, n, and k, and strictly concave in n and k. Given the wage rate for production labor in efficiency unit wt and the rental rate of the capital ut, the manager receives the residual π t ( z ) = F ( z , k t , nt ) − ut k t − wt nt .
(9)
The first-order conditions for this maximum problem include Fk ( z , k t , nt ) = ut
(10)
Fn ( z , k t , nt ) = wt .
(11)
Demand for production labor n ( z , wt , ut ) and capital k ( z , wt , ut ) by the manager z can be obtained from (10) and (11).
Economies of scale imply managerial wage convex in z:
π ' ( z ) > 0, π " ( z ) > 0 . For efficient allocation, only the most talented will become managers.
The least
talented will become home-production workers, given the one-dimensional specification of talent. Critical levels of talent, zmt, zwt > 0, exist such that if z ≥ zmt, then the agent is a manager. If zmt > z ≥ zwt, the agent is a production worker. If z < zwt, the agent works at home. An
13
allocation in this economy means two critical values of talent zmt, zwt, and the demand functions for labor and capital by a manager with talent z, n ( z , wt , ut ) , k ( z , wt , ut ) . In equilibrium, the demands for factors are equal to their supplies: z
∫z
mt
z
∫z
mt
n ( z , wt , ut ) µ ( z )dz = ∫
zmt
z wt
zµ ( z )dz
(12)
k ( z , wt , u t ) µ ( z )dz = K t .
(13)
4. Qualitative Analysis This section presents the static version of the model to illustrate its key elements. Utility is linear and capital is dismissed. The manager with talent z receives π ( z , w) = max n F ( z , n ) − wn . The first-order condition, Fn ( z , n ) = w , provides the implicit demand function for production labor n ( z; w) .
Because l(z) = 0 for z < zw, the total
consumption of both market and home-produced goods is z
∫z
z
z
z
zm
c( z ) µ ( z )dz + ∫ [α ( z )(θ + 1 − l ( z ))]µ ( z )dz = ∫
F ( z , n ( z ))µ ( z )dz + ∫
zw
z
α ( z ) µ ( z )dz .
Without loss of generality, the constant is dropped after the equality. The maximization problem is now to choose the two critical skill levels for job assignment, zm and zw, given the labor-market constraint (12). In the Lagrangian with w as the multiplier associated with the labor-market constraint, this allocation is the competitive equilibrium with w being the wage rate of production labor: max zm , zw
z
∫z
m
F ( z , n ( z ))µ ( z )dz + ∫
zw
z
α ( z )µ ( z )dz + w[∫
zm
zw
The first-order conditions for critical level zm and zw are
14
zµ ( z )dz − ∫
z
zm
n( z ) µ ( z )dz ] .
F ( z m , n( z m )) − wn ( z m ) = wz m
(14)
wz w = α ( z w ) .
(15)
Equation (14) states that the marginal manager, whose talent is zm, faces a break-even point. The marginal manager’s income will be the same as his wage as a production worker. The second condition states that the marginal market participant’s wage is the same as the value of his home production. Figure 1 illustrates the labor-market equilibrium when α1 = 0. With market technology F ( z , n ) = Azψ n β the optimal span of control, the employment of production labor under the manager z is ψ
βA 1− β 1− β n ( z , w) = ( ) z . w 1
(16)
The optimal span of control is more than proportional to the skill of the manager; d ln( n ( z )) d ln( z )
κ > = 1 + 1−β 1 as κ captures economies of scale in managerial labor. Except for the
marginal manager, managers receive more than proportional economic rent for their superior talent.
This skews the wage distribution to the right, relative to the underlying skill
distribution. Inserting (16) into the labor-market equilibrium condition (12) reveals the wage rate of production labor as a function of critical levels zm, and zw: z
w( z m , z w ) = βA(
∫z
z
ψ 1− β
µ ( z )dz
m
zm
∫z
zµ ( z )dz
)1− β .
(17)
w
It is very useful to define the aggregate index of managerial labor and production labor in efficiency units as follows: Z = [∫
z
zm
z
ψ 1− β
µ ( z )dz ]1− β and N = ∫
zm
zw
15
zµ ( z )dz .
(18)
With these aggregate indices one can recover the aggregate market-production function as Y ( zm , z w ) = ∫
z
zm
Azψ n( z ) β µ ( z )dz = AZN β .
(19)
The wage rate of production labor in (17) is equivalent to the marginal product of production labor in the aggregate production function in (19), βAZN β −1 . The aggregate home production is H (z w ) = ∫
zw
z
α ( z ) µ ( z )dz .
(20)
Using aggregate production functions (19) and (20), the problem is max zm , zw Y ( z m , z w ) + H ( z w ) . The first-order conditions for zm and zw are equivalent to (14) and (15). I examine the response of economy to shifts in TFP in the market. Suppose that working in the market becomes more profitable due to an increase in A. The critical level of α
skill that determines market participation zw is, from (6) and (14), z w = ( w0 )1−α1 . As an increase in TFP increases the wage rate in the market, more people are drawn to the market sector. (Figure 2 illustrates this.) From (16), the span of control n(z) increases as the productionwage (w) increase is smaller than the TFP increase.13 An increase in the span of control makes the employment of managers less cyclical than that of production workers because existing managers can absorb new production workers. Since the managerial wage for agent z is
13
The supply curve of production labor is upward sloping because of the existence of the non-market
sector, and the demand curve is downward sloping because β < 1 . This implies that the increase of production-wage rate will be smaller than the increase of TFP.
16
π ( z ) = ( β1 − 1) wn ( z ) , the relative wage of manager to production worker also increases as the span of control increases:
π (z) w( z )
1
κ
= ( β1 − 1)( βwA ) 1− β z 1− β .
The cross-sectional comparative advantage between the market and home in the labor force has an interesting implication on the aggregate labor-supply elasticity as weak comparative advantage implies frequent movement of workers between sectors. As Figure 3 illustrates, when α1 > 0, the same increase in TFP draws more people to the market.
The
output from home production represents the opportunity cost of labor-market participation. When productivity between market and home is correlated, it is less costly to draw less-skilled workers into the market because they have a lower opportunity cost as well.
5. Quantitative Analysis The family assumption greatly simplifies the analysis of the model economy.14
It
separates the static problem of resource allocation at time t from the dynamic capitalaccumulation problem over time so that the model can be solved recursively. First, the allocation of capital and production labor across managers at time t is solved, given Kt, zmt, and zwt. Second, the paths of investment, It, consumption, Ct, and labor-supply decisions, zmt and zwt, are determined by the intertemporal consumption theory.
14
The family assumption dismisses possible income effects on labor supply across agents. However,
relaxing the family assumption along with allowing divisible labor will not affect the result on the cyclical behavior of relative wages. Given that wages are measured by total earnings divided by hours, the income effect on hours will strengthen the result of this model by reinforcing the pro-cyclical managerial wage because it will reduce the hours of high-wage earners who are managers.
17
Given the constant returns to scale in g(k,n), it is useful to write the output under manager z as y t ( z ) = At zψ (nt f ( rt )) β , where rt = kt/nt is the capital-labor ratio. The CRS of g ( ) also implies that the capital-labor ratio rt is common across managers so that rt = K t / N t . The first-order conditions (10) and (11) imply f (rt ) − rt f ' (rt ) wt = . f ' (rt ) ut
(21)
From (10) and (7), the demand for production labor by the manager z is 1
βAt zψ f ' ( rt ) 1− β 1 ) . n ( z , wt , u t ) = ( ut f ( rt )
(22)
Inserting (22) into the labor-market-equilibrium condition (12) and using the aggregate index in (18), one can write the rental rate as ut = βAt Z t ( N t f ( rt )) β −1 f ' ( rt ) .
(23)
Using (21) and (23) the wage rate for production labor is wt = βAt Z t ( N t f ( rt )) β −1 ( f ( rt ) − rt f ' ( rt )) .
(24)
The wage of manager z is, from (9), (21), and (22), π t ( z ) = (1 − β ) At Z t ( N t f ( rt )) β −1 f ( rt )n( z ) .
(25)
Then the growth rate of the relative wage of managers to production labor is •
•
•
•
π t ( z ) wt ( z ) n t ( z ) 1 r − = + χ (1 − ) t , π t ( z ) wt ( z ) n t ( z ) ε rt •
where x t =
dx ( t ) dt
(26)
. The relative wage depends on the span of control, substitution elasticity, and
capital-labor ratio. For instance, under capital-skill complementarity (ε > 1), an increase in
18
capital-labor ratio favors managers relative to workers. Again, using the aggregate index in (18), the aggregate market-production function is Y ( zmt , z wt , K t ) = ∫
z
[ At zψ ( nt ( z ) f ( rt )) β ]µ ( z )dz = At Z t ( f ( rt ) N t ) β .
z mt
(27)
The aggregate home-production function is z
z wt
z
z
H ( z wt ) = θ ∫ α 0 z α1 µ ( z )dz + ∫
α 0 z α1 µ ( z )dz .
(28)
Using (27) and (28) one can rewrite the maximization problem as ∞
max {Ct , zmt , zwt , K t +1} E 0 [∑ ρ t [log C t + B log H ( z wt )] t =0
subject to Y ( z mt , z wt , K t ) − C t − K t +1 + (1 − δ ) K t = 0 . With Lagrange multiplier λt for the resource constraint, the first-order conditions are C t −1 = λt
(29) −β
ψ
1− β (1 − β ) At Z t1− β ( N t f ( rt )) β z mt = βAt Z t ( N t f ( rt )) β −1 ( f ( rt ) − rt f ' ( rt )) z mt
(30)
Bα 0 z wt α1 H t ( z wt ) −1 = λt βAt Z t ( N t f ( rt )) β −1 ( f ( rt ) − rt f ' ( rt )) z wt
(31)
λt = ρEt [ λt +1 ( βAt +1Z t +1 ( N t +1 f ( rt +1 )) β −1 f ' ( rt +1 ) + 1 − δ )]
(32)
At Z t ( N t f ( rt )) β = C t + K t +1 − (1 − δ ) K t .
(33)
Equation (30) states that, for the marginal manager, the marginal product as a manager is equal to the marginal product as a production worker. Equation (31) states that, for the marginal market participant, the marginal product of labor is the same as the value of marginal product in home production. Equation (32) is the Euler equation. Equation (33) is the resource constraint.
19
Calibration of the Model According to the model, the wage of a non-supervisory worker is linear in worker’s talent z. This allows us to calibrate the talent distribution µ(z) directly from the cross-sectional wage distribution of the non-supervisory workers from PSID (wages of heads of households and wives). I use the wage distribution of 1983 and 1984 because it is the mid-point of the sample period and the base year of the deflator for nominal earnings. The mean and the standard deviation of the log-wage distribution are 2.17 and 0.53, respectively. This wage distribution should be viewed as a doubly truncated representation of skill distribution µ(z) for two reasons. First, workers in the non-market sector are not included. Second, as usual in the analysis with micro data, to avoid outliers due to measurement errors in reported hours and income, wages below $3 or above $100 are eliminated. This is consistent with the top-coding practice at wage $100 in the PSID. Then, the mean µz and variance σz of the underlying skill distribution µ(z) are searched among lognormal distributions to match the mean and standard deviation of the truncated wage distribution.
Specifically, when Φ( ) is the cumulative
standard normal distribution, µz and σz are chosen to satisfy the following two conditions (See Maddala, 1983.): E[ln( w) | 3 ≤ w ≤ 100) =
Φ ' [c1 ] − Φ ' [c2 ] 2.17 − µ z = Φ [c2 ] − Φ [c1 ] σz
V [ln( w) | 3 ≤ w ≤ 100] = 1 − E[ln( w) | 3 ≤ w ≤ 100]2 +
where c1 =
c1Φ ' [c1 ] − c 2Φ ' [c 2 ] (0.53) 2 = Φ [c1 ] − Φ [c2 ] σ z2
ln( 3) − µ z ln(100) − µ z and c 2 = . σz σz
20
The calibrated values are µz = 2.11 and σz = 0.58. Figure 4 shows the actual wage distribution and the calibrated skill distribution µ(z). Given the calibrated skill distribution µ(z), the steadystate critical values are chosen to match the occupational breakdown of the PSID: 12% managers, 63% non-managerial workers, and 25% non-market workers (zm =16.4 and zw =5.5). For production technology in the market we need to specify ε, χ, β, and κ. As a benchmark, the elasticity of substitution between production labor and capital (ε) is 1. Given the steady-state values of zm and zw, the first-order condition (30) imposes one constraint among χ, β, and κ. I calibrate χ and β based on disaggregate manufacturing-industry data, and equation (30) will provide κ. Using the four-digit-industry-level NBER productivity data constructed by Bartelsman and Gray (1996) from the Annual Survey of Manufacturing for 1954-1996, I calculate the shares of payment to employed labor in value-added output. The average of this share is .45. Since their data do not include some fringe-benefit payments, I use a slightly higher value: (1-χ) = .5.15 Based on three-digit-industry data, Burnside, Eichenbaum, and Rebelo (1995) report returns to scale of between .8 and .92. (See Table 5 of their paper.) According to Basu and Fernald (1996), aggregation tends to produce a higher estimate for the returns to scale. Since β represents the firm or plan-level decreasing returns to scale in our model, I use the smallest value of the estimates in Burnside et al.: β =.8. The parameter κ is set to 0.075 to satisfy equation (30). This leads to this production function for the benchmark case: y t ( z ) = At z 0.275 ( k t 0.5 nt 0.5 ) 0.8 . The capital share in total output is 0.372, which is close to
15
In fact, when the shares are adjusted for fringe benefits at the two-digit level the averages of shares
are .474 (value-added weighted) and .525 (non-weighted).
21
the values commonly used in the literature. In addition, the calibrated values of κ and β imply a relative standard deviation of log wages of managers to workers of 1.375, which is very close to the standard deviation of self-employed workers to non-self-employed workers in the PSID (1.39). According to the model, the returns to skill is 1 for non-managerial workers. Since α1 represents the returns to skill in home production, the relative returns to skill in occupations that are comparable to home production to those of other unskilled workers would provide some information on the size of α1. In the PSID, for female workers the relative returns to skill of private household workers such as baby sitters and housekeepers to those of other female unskilled workers are .62 and .70 when the skill is measured by the years of schooling and the labor-market experience (age-schooling-4), respectively. For male workers, they are .29 and .45, respectively, for schooling and labor-market experience.16 As a benchmark case, the returns to skill in home production is zero (α1 = 0). I use other values such as ½and ¾in the quantitative analysis as well. Values of other parameters are fairly standard. The discount factor is chosen to match the annual steady-state real interest rate of 6.5%. The annual depreciation rate of capital is 10%. The ratio of non-working time to working time is 3 (θ = 3). Temporary shifts in productivity At follow the first-order autocorrelation in logs as in King et al.: ln At = (1 − ρ A ) ln A + 0.9 ln At −1 + e At .
16
The standard deviation of e At is set to match the
Specifically, for female unskilled workers, the returns to schooling and labor-market experience are
4.24 and .55 respectively. For female private-household workers, they are 2.64 and .38. For male unskilled workers, they are 5.43 and 1.63. For male private-household workers, they are 1.56 and .73.
22
standard deviation of real GDP in the data in the benchmark case. The values of the integrals are approximated by a linear quadrature. Skill levels are confined within the range z ∈ [0.5, 300]. This range covers almost 100% of the calibrated skill distribution µ(z). B and α0 are free parameters. Table 6 summarizes the parameter values for the benchmark case. The model is solved numerically by a log-linear approximation of the first-order conditions (29)-(33) around the steady states of the economy with a deterministic trend as in King, Plosser, and Rebelo (1988). 17
Cross-Sectional Comparative Advantage and Aggregate Labor Supply Figure 5 shows the impulse responses of the benchmark economy to a one percent increase in TFP, At. All variables are percentage deviations from the steady state. Panel (1) shows the exogenous TFP shock and the capital stock.
Panel (2) shows aggregate
employment and average labor productivity. Aggregate employment and labor productivity increase about .4% and .7%, respectively, in the first period. Panel (3) shows aggregate consumption, investment, and output. The responses are similar to those from the standard RBC model.
17
Panel (4) shows the employment of production workers and managers.
As is common in the literature, a deterministic trend in productivity is incorporated to allow the
output, consumption, investment, and capital grow over time as in the data. When the productivity of 1−β
production labor in g(k,n), home production α(z), and managerial labor grow at rates γ, γ, and γ 1− β + κ , respectively, the model has a balanced growth path where output, capital, investment, and consumption of both market goods and home produced goods grow at a common growth rate γ. See King et al. (1987) for a detailed discussion of how to obtain a stationary economy from one with balanced growth.
23
Employment of production workers is more volatile because of the pro-cyclical span of control. Panel (5) shows the average span of control, that is, the average employment of production labor per managers. Span of control increases initially, and then goes below the steady state in the later stage of an expansion as market participation decreases. In the later stage of expansion, the demand for home-produced goods increases along with the consumption of market goods.
Panel (6) shows the wages of production workers and
managers. Since capital accumulation is neutral (ε = 1), the wage gap between managers and workers represents the span-of-control effect only. Table 7 reports the population moments of the model economy. The first column is the benchmark case. Like other RBC models with productivity shocks, labor productivity is too pro-cyclical (0.98 relative to 0.753 in the data) and the employment is not so volatile as in the data (0.246 relative to 0.7 in the data). As is explained in the static case, the response of aggregate employment depends on the cross-sectional comparative advantage. As the cross-sectional correlation of productivity in the market and at home increases, employment becomes more volatile. For example, when α1 = 1/2, the relative volatility of employment increases from 0.246 to 0.387. When α1 = 3/4, the relative volatility of employment increases to 78% of the data (0.544 relative to 0.7). As employment becomes more volatile, the composition effect due to entry and exit of less-skilled workers reduces the labor productivity-employment correlation significantly (0.314 when α1 = 3/4). As a result, the volatility of output increases as well. Figure 6 shows the impulse response for the case of α1 = 3/4. Aggregate employment is much more volatile relative to labor productivity as it increases almost 1% initially, yet labor productivity increases only .3%.
24
While the relative volatility of employment to labor productivity in the benchmark case (.295) is far less than that in the data (1.062), the volatility is now quite close to data (.796).
Capital-Skill Complementarity Capital-skill complementarity has been emphasized in the literature because it creates an interesting short-run dynamics for relative employment (e.g., Rosen, 1968; Griliches, 1969). In this case, the short-run production function is not homothetic, and the relative demand for labor depends on the stage of business cycles.
The relative demand for unskilled labor
increases in the beginning of an expansion when capital is not yet accumulated. However, subsequent capital accumulation will substitute for the unskilled labor and will favor the skilled labor. Figure 7 shows the impulse response when production labor is a better substitute for capital than managerial labor (ε = 3/2). At the outset of an expansion, the employment of production workers increases sharply. This is reflected in the impulse response of the span of control. It sharply increases in the beginning and retreats as capital accumulates. In fact, employment of production workers tends to lead that of non-production labor in the post-war U.S. quarterly data. The relative wage of managers to workers increases in the beginning because of the spike in span of control. It is sustained even after the span of control falls below the steady-state level because capital accumulation favors managerial labor, which is relatively complementary to capital. (Recall equation (26).) Due to the higher substitution between capital and production labor, employment becomes more volatile and its volatility (0.618) is now quite close to that of data (0.7). A higher volatility of employment reinforces
25
the composition bias and the labor productivity-employment correlation is close to zero (0.052) as in the data.18 19
Two Composition Effects in Aggregate Wages The model generates less cyclical aggregate wages and labor productivity than those in the standard RBC models. This is due to the changes in skill mix of the workforce over the business cycle. Since the aggregate wage constructed by the BLS is based on non-supervisory workers only, there are two composition effects in aggregate wages during expansions: the entry of low-wage workers from the non-market sector and the exit of high-wage workers to self-employed and managers. Parallel to the empirical analysis in Section 2, I construct the average wages (average wages of the entire workforce) and the aggregate wages (average wages of production workers) from the model. The last column of Table 8 shows the cyclicality of these wages based on the PSID data in Section 2.
18
To compare the performance of the model, implied regression
For comparison, Figure 8 shows the impulse responses for the case of ε = 2/3, the opposite case to the
capital-skill complementarity. The span of control is relatively flat over the business cycle because the relative demand for production labor does not sharply increase in the beginning, and capital accumulation complements production workers in the later stage of the business cycle. The wage growth is reversed in the early stage of the business cycle, even though the span of control stays above the steady state. 19
The results based on the Hodrick-Prescott filter show a similar pattern to those in Table 7 except for
labor productivity-employment correlations. They exhibit slightly higher correlation in the model and lower correlation in the data than in Table 7.
26
coefficientthe ratio of the covariance between the output and wage measure to the variance of outputis calculated. In the benchmark case, the composition effect reduces the cyclicality of average wages and aggregate wages by 10% (from 0.911 to 0.816) and 12% (from 0.911 to 0.804), respectively. Because employment is not so volatile as in the data, the composition effect is not so great as in the data. When the employment is nearly as volatile as in the data (the model with α1 = 3/4, ε = 3/2), the cyclicality of average wage decreases by 19% (from 0.734 to 0.593), and that of aggregate wage decreases by 34% (from 0.734 to 0.483).20 The composition effect predicted by the model is very close to what we find from the PSID data in Section 2 (16% and 33%, respectively). In sum, both in the data and in the model, not only the first bias (due to entry and exit of low-wage workers between market and home) but also the second bias (due to the movement between non-supervisory workers and managerial workers) significantly reduces the cyclicality of wages.
6. Concluding Remarks Equilibrium models of the business cycle that emphasize shifts in TFP as a major source of economic fluctuation do not explain why aggregate hours vary greatly over the business cycle without much variation in aggregate wages. This paper suggests a resolution to this problem by recognizing the heterogeneity of skill possessed by workers. The model in this paper augments the standard RBC model to include distinctions between managerial and nonmanagerial employment and home and market production.
20
Due to differences in detrending method and frequency between the panel data and the model, I
compare the relative size of compositional effects instead of comparing the numbers directly.
27
Business cycles are associated with the systematic movement of workers that makes aggregate wages and labor productivity less cyclical than individual wages. Not only do lessskilled workers enter the workforce, biasing down wages and productivity in expansion, but higher-wage earners become managers and self-employed, lowering aggregate wages further. Weak comparative advantage between market and non-market work implies frequent movement between sectors in response to relative productivity shifts. Under a comparative advantage that is consistent with PSID, this model is capable of matching the key moments in the aggregate labor market. Aggregate hour is nearly as volatile as in the data, and yet it is not highly correlated with labor productivity. Aggregate wage and labor productivity exhibit mildly pro-cyclical behavior. The model produces interesting dynamics for relative wages and employment as well. Wages of managerial workers are highly pro-cyclical, as in the PSID data. As the employment of production labor tends to lead that of non-production labor in the data, under capital-skill complementarity the relative demand for production labor sharply increases at the beginning of an expansion. This model abstracts from any friction in the labor market. For instance, in an economy with a rigid labor market, the response of the economy may be quite different from those predicted by the model. Examples of such friction include search friction, specific human capital, hiring costs, and firing costs. While the model assumes a fixed one-dimensional talent distribution in the economy, allowing multi-dimensional talent or endogenous evolution of jobspecific skills such as learning-by-doing will enrich the employment dynamics across occupations.
28
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33
Appendix: Data
The PSID data consist of a random sample and a poverty sample. Only the random sample is used to represent the skill distribution of the aggregate economy. The sample period is 1971-1992. The sample consists of heads of households and wives who are 18-60 years old. Wage data for wives are available only since 1979. Wage data are annual hourly earnings (annual labor incomes divided by annual hours). The wages are for those workers who were working in the private and non-agricultural sectors for at least 100 hours per year, and whose hourly wage rate was above $3 in 1983 dollars. Wages are deflated by the Consumer Price Index. The base year is 1983. The descriptive statistics for workers in the market sector are given in Table 1. In the estimation of the cyclicality of wages in different occupations in Table 3, the sample consists of workers who were in the same occupation in two consecutive periods to capture the pure price changes of labor in the corresponding category. In Table 7, because wives’ wages are available only since 1979, data from heads of households are used in order to generate a consistent time series of average wages. Aggregate data are quarterly for 1955:I - 1994:IV from the Citibase. Output is real GDP (in 1987 constant dollars). Consumption is non-durables and services. Investment is gross fixed investment. Employment is employed man-hours based on the BLS establishment survey.
34
Table 1: Summary Statistics from the PSID, 1971-1992. Variables
Mean
Standard Deviation
Observations
Age
36.87
10.96
89867
Years of schooling
12.89
2.53
89032
Head of household
0.66
0.47
89867
Female dummy
0.44
0.49
89867
Real wages
10.88
7.24
59827
Annual working hours
2044.83
649.23
59827
Annual labor income
22499.42
17222.97
59827
35
Table 2A: Comparison of Wages of Movers and Stayers: Wages at Time t. (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
NA
7.71 (-3.12)
9.10 (-4.53)
(Non-Managers) t-1
NA
10.83 (0)
11.66 (-2.97)
(Managers) t-1
NA
11.94 (1.11)
14.63 (0)
Note: Numbers in parentheses are average wages relative to the stayers in the new occupation.
Table 2B: Comparison of Wages of Movers and Stayers: Wages at Time t-1. (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
NA
NA
NA
(Non-Managers) t-1
8.48 (-2.14)
10.62 (0)
11.59 (0.97)
(Managers) t-1
10.07 (-4.35)
11.51 (-2.91)
14.42 (0)
Note: Numbers in parentheses are average wages relative to the stayers in the old occupation.
36
Table 3: Cyclical Behavior of Wages, Hours, and Labor Income (Occupational categories are only for the non-self-employed.) Occupation
Wage
Hour
Income
Obs.
Average Wage
All
.542
.523
1.065
48456
11.46
(.063)*
(.064)*
(.071)*
.747
.565
1.312
4353
13.58
(.337)*
(.206)*
(.332)*
.828
.179
1.007
2273
14.77
(.237)*
(.198)
(.212)*
Professional/
.453
.316
.770
8438
14.05
Technical
(.136)*
(.150)*
(.151)*
Clerical/Sales
.373
.234
.607
9359
10.39
(.126)*
(.141)
(.152)*
.582
.505
1.087
5384
11.92
(.150)*
(.150)*
(.160)*
Operatives/
.637
1.014
1.652
8931
8.37
Laborer
(.131)*
(.162)*
(.112)*
Self-Employed
Managerial
Craftsmen
Note: Numbers in parentheses are standard deviations. * significant at 5%
37
Table 4A: Transition Matrix of Occupational Changes: Expansions (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
0.686
0.261
0.053
(Non-Managers) t-1
0.061
0.881
0.059
(Managers) t-1
0.040
0.176
0.783
Total
0.16
0.65
0.19
Table 4B: Transition Matrix of Occupational Change: Recessions (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
0.768
0.193
0.038
(Non-Managers) t-1
0.085
0.865
0.049
(Managers) t-1
0.066
0.231
0.704
Total
0.24
0.613
0.148
Table 4C: Difference in Percentage between Expansions and Recessions ( 50 × (Table 4A − Table 4B)/(Table 4A + Table 4B) ) (Non-Market)t
(Non-Managers)t
(Managers)t
(Non-Market)t-1
11.4
29.9
31.8
(Non-Managers) t-1
-33.8
1.8
16.2
(Managers) t-1
-47.8
-26.9
10.8
38
Table 5: Two Composition Effects in Aggregate Wages from the PSID Wages
Data
Obs.
Supervisory workers
0.873 (.244)
5952
Non-supervisory workers
0.597 (.071)
27667
Average wage of all workers
0.501 (.179)
20
Aggregate wage based on non-supervisory workers
0.396 (.188)
20
BLS aggregate wage
0.434 (.158)
20
39
Table 6: Parameter Values for the Benchmark Case Parameters
Description
µz = 2.11
Mean of lognormal skill distribution µ(z)
σz = 0.58
Standard deviation of lognormal skill distribution µ(z)
zm = 16.4
Steady-state critical-skill level for zmt
zw = 5.5
Steady-state critical-skill level for zwt
β = 0.8
Curvature in production function y = Az1-β+κ[g(n,k)]β
ε=1
Substitution elasticity between n and k
χ = 0.5
Capital share in g(n,k)
κ = 0.075
Economies-of-scale parameter
δ = 0.025
Quarterly depreciation rate
θ=3
Ratio of non-working time to working time
ρ = 0.988
Quarterly discount factor
α1 = 0
Home-production technology α( z ) = α 0 z α1
ρA = 0.9
Autocorrelation of productivity shock At
40
Table 7: Population Moments of the Models α1=0
α1=1/2
α1=3/4
α1=3/4
ε=1
ε=1
ε=1
ε=3/2
σY
3.98
4.167
4.388
4.591
3.98
σC / σY
0.703
0.694
0.685
0.627
0.812
σI / σY
2.369
2.43
2.5
2.666
2.07
σEmp /σY
0.246
0.387
0.544
0.618
0.7
σEmp / σY/Emp
0.295
0.514
0.796
0.819
1.062
cor(Y, Emp)
0.745
0.752
0.760
0.652
0.753
cor(Y,Y/Emp)
0.98
0.94
0.855
0.786
0.715
cor(Emp, Y/Emp)
0.599
0.484
0.314
0.052
0.079
Statistics
Data
Note: Data are linearly detrended. σC /σY: standard deviation of consumption relative to Y. Emp: total employment in the model, total employed hours in the data. cor(C,Y): correlation of C and Y.
41
Table 8: Composition Effect in Wages from the Model α1=0
α1=3/4
α1=3/4
ε=1
ε=1
ε=3/2
cov (Wman,Y) / Var (Y)
0.926
0.834
0.779
0.873
cov (Wprod,Y) / Var (Y)
0.911
0.800
0.734
0.597
cov (Waverage,Y) / Var (Y)
0.816
0.585
0.593
0.501
cov (Waggregate,Y) / Var (Y)
0.804
0.557
0.483
0.396
Statistics
Data
Note: Wman: wages of managers including self-employed. Wprod: wages of non-managerial workers. Waverage: average wage of workers in the market sector. Waggregate: average wage of non-managerial workers.
42
Figure 1: Equilibrium in the Labor Market
Wage
π( z,w) : wages as a manager
wz : wages as a production worker
α(z ) : value of home production
zw
0
zm
z
Density
home 0
µ (z) managers
production workers zw
zm
43
z
Figure 2: Productivity Increase in the Market
Wage π'(z)
π (z) w'z wz
α (z) zw' zw
0
zm' zm
z
Density
home 0
µ (z) managers
production workers
zw' zw
zm' zm
44
z
Figure 3: Productivity Increase in the Market when α′(z) > 0 and α′(z) = 0.
Wage
w'z wz α (z) α0
0
zw'
z
zw
45
Figure 4: Wage Distribution of the PSID () and the Lognormal Distribution (--).
46
Figure 5: Impulse Responses of the Economy When α1=0 and ε=1
47
Figure 6: Impulse Responses of the Economy When α1=3/4 and ε=1
48
Figure 7: Impulse Responses of the Economy When α1=3/4 and ε=3/2
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Figure 8: Impulse Responses of the Economy When α1=3/4 and ε=2/3
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