bE (10.84)

{bE/lK(L-EW]}

= Ky{bEp~4L-Ef,

where the second line uses the matching function, (10.68), to substitute for V.

Equations (10.83) and (10.84) imply that a is increasing in E and that a is decreasing. Thus (10.82) implies that Vv is a decreasing function of E. As E approaches L, a approaches inhnity and a approaches zero; hence VV approaches -C jr. Similarly, as E approaches zero, a approaches zero and a approaches inhnity. Thus in this case Vv approaches (A - C)/r, which we have assumed to be positive. This information is summarized in Figure 10.6.

The equilibrium level of employment is determined by the intersection of the VV locus with the free-entry condition, Vy = 0. Imposing this condition on (10.82) and using (10.83) and (10.84) to substitute for a and a yields

- - - = . (10.85)

[b£/(I - £)] + /i/y(b£)(T-i)/y(I - E)P!y + 2b+2r

This expression implicitly dehnes E, and thus completes the solution of the model.

The Impact of a Shift in Labor Demand

We now want to ask our usual question of whether the imperfection we are considering-in this case, the absence of a centralized market-affects

rVv = -C +-4--. (10.82)

a + a + 2b + 2r

Equation (10.82) expresses VV in terms of C, A, r, b, a, and a. a and a, however, are endogenous. Thus the next step is to express them in terms of E. The facts that a = M(U,y)/U (equation [10.70]), that M = bE (equation [10.69]), and that E + U = L imply

a = J. (10.83)

Similarly, (10.71) implies

M(U,V)

(A-C)/r

FIGURE 10.6 The determination of equilibrium employment in the search and matching model

the cyclical behavior of the labor market. Specifically, we are imerested in whether it causes a shift in labor demand to have a larger impact on employment and a smaller impact on the wage than it does in a Walrasian market.

To address this question, begin by considering the steady-state effects of a faU in A From (10.82) or (10.85), this shifts the Vy locus down. Thus, as Figure 10.7 shows, employment falls. In a Walrasian market, in contrast, employment is unchanged at L. Intuitively, in the absence of a frictionless market, workers are not costlessly available at the prevailing wage. The decline in A, with fixed, raises firms costs of searching for workers relative to the profits they obtain when they find one. Thus the number of firms- and hence employment-falls.

In addition, the matching function (10.68), together with the fact that ( V) equals bE is steady state, implies that steady-state vacancies are (bE/K)/(L-E)l. Thus the decline in A and the resulting decrease in the number of firms reduce vacancies. The model therefore implies a negative relation between unemployment and vacancies-a Beveridge curve.

The model does not imply substantial wage rigidity, however. From (10.83) and (10.84), the fall in E causes a to fall and a to rise: when unemployment is higher, workers cannot find jobs as easily as before, and firms can fill positions more rapidly. From (10.80), this implies that the wage falls more than proportionately with A.

The dynamics of the transition between the two steady states are also of interest. Since there is no reason for firms whose positions are filled to

Since w = A - in the Walrasian mar]<et, the same result holds there. Thus it is not clear which case exhibits greater wage adjustment. Nonetheless, simply adding heterogeneity and matchmg does not appear to generate strong wage rigidity.

FIGURE 10.7 The effects of a fall in labor demand in the search and matching model

discharge their workers, employment and unemployment do not change discontinuously at the time of the shock. The reduced attractiveness of hiring, in contrast, causes VV to fall unless some firms exit. Thus there is exit, and hence a discontinuous drop m V. In practice, this could take the form of some firms with openings stopping their attempts to fill them.

With employment and unemployment the same as before but vacancies lower, the flows into employment exceed the outflows, and so unemployment rises. Thus the fall in A leads only to a gradual rise in unemployment. Finally, as unemployment rises, the value of a vacancy would rise if vacancies did not change; thus vacancies must rise as unemployment rises. This implies that the initial drop in V exceeds its steady-state response-that is, that there is overshooting.

A temporary change in A leads to smaller employment responses. The value of a fiUed job is clearly higher when A is temporarUy low than when it is permanently low. Thus there is a smaller fall in the number of vacancies, and hence a smaller rise in unemployment. In the extreme case of an infinitesimally brief decline in A, Vv and Vu are unaffected. In this case, firms and workers simply share the loss equally by reducing the wage by half the amount that A falls, and there is no impact on employment or unemployment.

In short, although search and matching considerations have interesting implications for the functioning of labor markets, this model of them does

"In addition, as first pointed out by Oi (1962), the fact that a hrm cannot costlessly replace its workers makes it more reluctant to discharge workers m response to a temporary downturn if the marginal product of labor falls below the disutility of working. Thus m this case frictions and heterogeneity dampen the response of employment to shocks.