CAPITULO VI: COSTOS INDIRECTOS DE FABRICACION 6.1 Concepto y características6.1 Concepto y características
6.8 Tratamiento de los costos semifijos y semivariables
We derived the critical condition for the supply of jobs (1.30) by assum- ing that each firm has only one job. In this section we derive the same condition again by assuming that the firm employs many workers, and that on average it is large enough to eliminate all uncertainty about the flow of labor. We continue assuming that the wage rate is given by an implicit bargain at the individual level. That is, wages are fixed as if the firm engages in Nash bargains with each employee separately, by taking the wages of all other employees as given. This assumption is clearly the closest one to competitive wage determination in this market environ- ment. In deciding how many jobs to open up the firm anticipates the wage correctly but chooses the number of jobs by taking it as given. This is consistent with profit maximization when there are no long-term con- tracts and there is a perfect second-hand market for capital goods.
Let Kiand Nibe the capital and employment of firm i, and let F(Ki, pNi) be a constant returns to scale production function. The parameter
p is a labor-augmenting productivity parameter. The firm buys capital
equipment Kiat the price of its output and pays workers real wage w,
given by (1.31) and taken as given by the firm. We assume that there are no costs of adjustment for capital but adjusting employment involves some linear costs of adjustment.
The firm loses workers at the rate lNi. In order to recruit workers, it
has to open up job vacancies and advertise. Suppose that each vacancy costs the firm pc in recruitment costs and returns a worker at the rate
q(q), where q is outside the firm’s control. Let Vibe the number of the
firm’s vacancies. Then the firm’s labor force changes according to (3.1) The firm’s choice variable in (3.1) is Vi.
The present-discounted value of the firm’s expected profit is
(3.2) Pi= e-rt
[
F K pN(
i i)
-wNi-pcVi-Ki- K dti]
•Ú
, ˙ d , 0 ˙ . Ni =q( )
q Vi-lNiwhere d is the rate of depreciation of the capital stock. The firm maxi- mizes (3.2) with respect to Kiand Visubject to (3.1). Denoting by x the
co-state variable associated with (3.1), we get the following Euler con- ditions, satisfied by the optimal path of Kiand Nifor given paths of p
and q :
(3.3)
(3.4) (3.5) For constant p and q, there is a steady-state solution with constant w, Ni,
and Kisatisfying
(3.6) (3.7) The steady-state solution for the firm’s vacancies, Vi, is then obtained
from the constraint (3.1) for N.i= 0:
(3.8) Now, because F(Ki, pNi) is of constant returns to scale, we can re-
express F1(Ki, pNi) and F2(Ki, pNi) as functions of one variable, Ki/pNi.
But then all variables other than Ki/pNiin both (3.6) and (3.7) are market
variables: None is indexed by i. Hence in the steady state all firms will have the same ratio Ki/pNi, which we denote by k. We define
(3.9) where pf(k) is output per person employed. Hence
(3.10) (3.11) Substituting from (3.10) into (3.6), we get
F K pN2
(
i, i)
=f k( )
-kf k¢( )
. F K pN1(
i, i)
= ¢f k( )
, f k pN F K pN F K pN i i i i i( )
=(
)
= Ê Ë ˆ ¯ 1 1 , , , V N q i i =( )
l q . pF K pN w r q pc i i 2(
,)
- - 0. +( )
= l q F K pN1(
i, i)
- - =d r 0, -e-rtpc q+( )
q x=0. e pF K pN w x dx dt rt i i -[
(
)
-]
- + = 2 , l 0, e F K pN d dt e rt i i rt -[
(
)
-]
-(
- -)
= 1 , d 0,(3.12) which is condition (1.28). Substituting also from (3.11) into (3.7), we get (3.13) which, noting (3.12), is the job creation condition (1.30).
Condition (3.8) implies that in the steady state all firms choose the same ratio of vacancies to employment, and therefore (3.8) also gives the ratio of all vacancies to total employment. To revert to the notation of chapter 1, let SVi= quL, where L is the labor force, and SNi= (1 - u)L,
so (3.8) becomes
(3.14) Re-arrangement of (3.14) gives (1.32), the final equilibrium condition of chapter 1.
Inspection of the firm’s present-value expression (3.2) shows that the critical new element in this theory of the demand for factors of produc- tion is the cost of adjustment for employment [ pc/q(q)](N.i+ lNi). This
cost is linear in N.i, and it has some effect on the level of employment in
the steady state. Moreover, since q stands for labor-market tightness, the cost of adjustment depends on tightness: The firm can adjust employ- ment more cheaply when the market is less tight (lower q). The conges- tion externality that we discussed in chapter 1 is due to this new element of the theory. If all firms try to expand employment together, they compete for the pool of unemployed workers by simultaneously opening up more vacancies. This increases market tightness and increases the length of time that a given firm has to wait before a suitable employee arrives. But, if a firm tries to expand employment alone, the waiting time is shorter. Firms follow the same employment policy irrespective of what other firms do, thus ignoring the congestion that their policies create for other firms in the market.
3.2 Unemployment Income
When we analyzed the effects of productivity changes in chapter 1, we saw that the general level of productivity had an influence on employ-
q l q u u q 1 - =
( )
. p f k kf k w r q pc( )
- ¢( )
[
]
- - +( )
= l q 0, ¢( )
= + f k r d ,ment because of the fixed actual or imputed income during unemploy- ment. The Nash wage equation is a linear combination of unemployment income and labor productivity, and when the former is fixed, productiv- ity changes lead to changes in profitability and job creation. We will now argue that in long-run equilibrium it is more reasonable to allow unem- ployment income to respond to changes in some of the variables of the model, leading to a proportional relation between wages and the general level of productivity. The responses that we will describe are not derived from explicit maximizing models, but it is easy to see that commonly used models are consistent with them. We omit the derivation to avoid too many digressions from the central theme of the book. Also it will become apparent that the assumption of fixed unemployment income is innocu- ous in most applications of the model, but it makes a big difference to the analysis of the effects of permanent productivity changes.
Unemployment income consists of actual income received during unemployment and the imputed value of time to unemployed workers. If actual income consists of transfer payments, it is reasonable to assume that it is fixed in terms of the prevailing wage rate, rather than the pre- vailing price level. For example, unemployment insurance benefits may be indexed to the average wage rate, just as the taxes used to finance them are generally proportional to wage earnings and not lump sum (see also chapter 9). Any income during unemployment other than a trans- fer payment, like income earned doing odd jobs in a secondary sector of the economy, should also be in fixed proportion to income from work in the primary sector along a steady-state path. Thus the actual-income component of z does not pose any serious problems for the existence of a balanced-growth path with unemployment. It may reasonably be assumed to be proportional to average wages.
This leaves imputed income from leisure activities, which is also part of z. The value of leisure to the worker is computed as the real com- pensation that the worker requires in order to give up his time for work. If leisure time is a consumption good, the value that a worker puts on it is not independent of market returns. In general, if there is a perfect capital market and workers have a long horizon, the minimum compen- sation that a worker requires in order to give up a consumption good that he possesses is a function of his wealth. Thus, in a general utility- maximization approach to job search, z is likely to depend on both human and nonhuman wealth.
Human wealth for unemployed workers is equal to U, the “asset value” of the worker during search. Nonhuman wealth has not played a role in our model. Let it be denoted by A. Then, in a general utility- maximization framework, imputed income during unemployment can be approximated by a function of permanent income, the average yield on human and nonhuman wealth:
(3.15) with z assumed constant.
Suppose first that we ignore nonhuman wealth; for example, let A = 0. Then substitution of z from (3.15) into one of the expressions that we derived for U, such as (1.19), gives
(3.16) so in steady-state
(3.17) Further substitution of z from (3.17) into the wage equation with capital, (1.31), gives
(3.18) Thus our assumptions about the worker’s imputed unemployment income make wages proportional to the labor-augmenting productivity parameter p. The factor of proportionality depends on the worker’s share in the wage bargain, the valuation that the worker places on his leisure time (z ), the firm’s recruitment cost, and market tightness.
Nonhuman wealth makes a difference to this analysis because, if we write A > 0 and use (3.15), we get, in place of (3.17),
(3.19) The wage equation then becomes
(3.20) w=
(
-)
rA f k r k c p - +( )
-(
+)
+ - È ÎÍ ˘ ˚˙ 1 1 1 b z z b d z z q . z= rA pc - + - È ÎÍ ˘ ˚˙ z z b b q 1 1 . w= f k( )
-(
r+)
k+ c p - È ÎÍ ˘ ˚˙ b d z z q 1 . z= pc - - z z b b q 1 1 . rU= pc -(
)(
-)
b z b q 1 1 , z=zr A U(
+)
, 0< <z 1,Thus wages are again a linear combination of a term that is pro- portional to productivity and one that is apparently independent of productivity.
As Phelps (1994) argues, however, although it may be reasonable to assume that nonhuman wealth is independent of market outcomes in the short run, in the longer run it adapts to labor-market earnings. A tem- porary increase in earnings may not have much influence on wealth, but a permanent increase leads to more savings and eventually raises wealth by an amount that reflects the rise in earnings.
If wealth plays an important role in determining reservation wages and the bargaining stand of workers, the slow response of it to long-term changes in labor market conditions could explain persistent effects of productivity changes on unemployment. For example, if wages obey equation (3.20) and z is not small, a permanent fall in productivity p (or of its rate of growth) can increase unemployment above its steady state for a long time (for as long as is required for nonhuman wealth to fall to a level consistent with the lower level of productivity and so allow wages to fall).
Such temporary (but potentially long-lived) unemployment responses to long-term productivity changes could be important in the empirical analysis of unemployment, in the light of the long-term productivity changes that took place in industrial countries in the 1970s. In the analy- sis that follows, however, we will concentrate on the longer-run steady- state properties of equilibrium, when nonhuman wealth, if it matters, has had time to adjust to the labor market equilibrium. Under these cir- cumstances there is no loss of generality if we ignore nonhuman wealth, avoid the modeling of consumption and savings choices, and write the value of imputed income during unemployment as in (3.15), with A = 0 and z a small positive number. The wage equation is (3.18), so higher valuation of leisure (higher z) implies higher wages. Through the higher wages it also implies lower labor market tightness (e.g., see equation (3.13)) and so lower job creation and higher unemployment. These are all intuitive results that need no further discussion.
The key property of the extended model of this section is that wages are proportional to the general productivity parameter p, with the factor of proportionality depending positively on labor-market tightness and the valuation of leisure. Substitution of (3.18) into the job creation condition, (3.13), gives, after making use of (3.12),
(3.21) Labor-market tightness in the steady state is independent of produc- tivity p.
We note in passing two points about the extended model of this section. First, a very similar but simpler wage equation can be derived by writing z = zp. What matters in (3.21) is that the imputed value of leisure is proportional to productivity. Although the simpler restriction
z = zp cannot be derived from a utility maximizing model, in many long-
run applications of the model it may be a reasonable simplification to make. The approach that we have followed here, of making the imputed value of time depend on permanent income, makes a difference when the shocks to productivity are temporary. For example, a temporary rise in p has only a temporary effect on wages, so the worker’s permanent income increases by less than the rise in productivity. This increases expected profits for as long as the productivity remains high, inducing more job creation. If the rise in productivity is perceived to be perma- nent, this does not happen.
Second, a related point can be made about the firm’s recruitment cost, which was assumed from the start to be proportional to general pro- ductivity, pc. Intuitively it may be more reasonable to write the recruit- ment cost as a function of wages, on the ground that recruitment is a labor-intensive activity. Of course, in the latter case, wages would also be proportional to productivity, but in addition they would be proportional to the marginal product of labor, f(k) - kf ¢(k). Replacing pc by wc, however, is an unnecessary complication for our purposes, and we will not take it up.
The wage equation (3.18) replaces (1.31) in the equilibrium set of equations of chapter 1, (1.29) to (1.32). The unknowns remain the same as before, k, w, q, and u. The properties of the new system, comprising (1.29), (1.30), (3.18), and (1.32) are qualitatively the same as the prop- erties of (1.29) to (1.32), with one important exception. Now general labor-augmenting productivity shocks are fully absorbed by wages, so equilibrium unemployment does not respond to them. This property makes the new model a more suitable tool for long-run analysis than the model of chapter 1. 1 1 0 -