The Foundation of Modern Macroeconomics an intricate connection between the process linking workers to jobs, and the one linking jobs to workers. This is obvious, since workers and vacancies meet in pairs.

The variable e is the relevant parameter measuring labour market pressure to both parties involved in the labour market. This parameter plays a crucial role because the dependence of the search probabilities on 9 implies the existence of a trading externality.

There is stochastic rationing occurring in the labour market (firms with unfilled vacancies, workers without a job) which cannot be solved by the price mechanism, since worker and vacancy must first get together before the price mechanism can play any role. The degree of rationing is, however, dependent on the situation in the labour market, which is summarized by 9. If 9 rises, the probability of rationing is higher for the average firm and lower for the average worker. The particular external effect that is present in the model is called the congestion or search externality by Pissarides (1990, p. 6).

For simplicity it is assumed that there is an exogenously given job destruction process that ensures that a proportion s of all filled jobs disappears at each instant.

These jobs could be destroyed, for example, because of firm-specific shocks making previously profitable jobs unprofitable. Hence, in a small time interval dt, the probability that an employed worker loses his/her job and becomes unemployed is given by sdt (with the same holding for filled jobs, of course). Hence, the average number of workers that become unemployed in a time interval dt equals s(1 — U)N dt and the average number of unemployed who find a job is given by 9q(9)UN dt. In the steady-state the unemployment rate is constant, so that the expected inflow and outflow must be equal to each other:

where Ki denotes the capital stock of firm i, and we have used (9.12) and linear homogeneity of the firm's production function to obtain the final expression involving the marginal product of labour. (Upon reaching agreement with the worker, the firm rents capital such that FK(Ki, 1) = r +8.) Equation (9.18) shows what the firm is after: it wants to squeeze as much surplus as possible out of the worker by bargaining for a wage far below the marginal product of the worker.
What does the worker get out of the deal? If a deal is struck, the worker changes status from unemployed to employed worker, which means that the net gain to the worker is:
r —Yu) =wi— s[lq— Yu] — rYu, (9.19)
where Yu does not depend on wi, but rather on the expectation regarding the wage rate in the economy as a whole (see (9.16)). If the worker does not accept this job offer (and the wage on offer wi) then he/she must continue searching as one of many in the "pool of the unemployed". The relevant wage rate that the unemployed worker takes into account to calculate the value of being unemployed is not w i but rather the expected wage rate elsewhere in the economy.
Using the generalized Nash bargaining solution, the wage wi is set such that S2 is maximized:
max S2 log two — Yu] + (1 — 3) log — iv] , 0 < < 1, (9.20)
where Jv (= 0) and Yu can be interpreted as the "threat" points of the firm and the worker, respectively. The relative bargaining strengths of the worker and the firm are given by, respectively, /3 and 1 — /3. The usual rent-sharing rule rolls out of the bargaining problem defined in (9.20):
dJO dwi = — Yu dwi P) dwi = °
d Q (1 )
First, by substituti exit/entry) we obtain:
( 1 — 13))1 = +
(1 —,8)
[wi + sYL r + s (1 — 13) [wi + sYu]
wi=(1— $)rYu+ I 4 The worker gets a we,b, product (FL). The str, and the closer is the The second expressic we know that each f.
that Ki = K. Hence, th wi = w. This implies t rYu = z + 0q(9) [1 .
= z + 90) ( -1--
This result is intuitive.
benefit, the relative L..
and the tightness in th(
the alternative wage w = (1 — /3)z + ItL I Workers get a weighted a consists of the margina saved if the deal is stru costs per unemployed 9.1.2 Market equilib We now have all the !IL = 0 r +s Y.k—Yu r+s )— # 1 - Yu =[Lia — Jv ] . (9.21)
This rent-sharing rule can be turned into a more convenient wage equation in two ways. model is summarized b 222 9.1'
station regarding the wa, does not accept - inue searching as one rate that the unemployed unemployed is not w i -•
Re wi is set such that Q is 1, (9.2r points of the firm and the the worker and the firm ring rule rolls out of the Chapter 9: Search in the Labour Market First, by substituting (9.18)-(9.19) into (9.21) and imposing jv = 0 (due to free exit/entry) we obtain:
(1 — p))I = + ( 1 —
wi sYu FL(Ki 1) - wi r s(1 P)[] = [ r + s ]+ (1 -
(1 — p) [wi + sYu] = p [FL(Ki, — wi] + (1 — P)(r + s)Yu wi = (1- i3)tYu + 13FL(Ki, 1). (9.22)
The worker gets a weighted average of his/her reservation wage (rYu) and marginal product (FL). The stronger is the bargaining position of the worker, the larger is p and the closer is the wage to the marginal product of labour.
The second expression for the wage equation is obtained as follows. From (9.12)
we know that each firm with an occupied job chooses the same capital stock, so that Ki = K. Hence, the wage rate chosen by firm i is also the same for all firms, wi = w. This implies that rYu can be written as follows:
rYu = z 0 q(9) [YE - Yu] z + 0(1(0) (1 -
18 p) Jo = z + eq(e)( 1 _13 p ) q)(/9° =z+ ) /13 62 i° (9.23) ,6 This result is intuitive. The reservation wage is increasing in the unemployment benefit, the relative bargaining strength of the worker, the employers' search cost, and the tightness in the labour market. By substituting (9.23) into (9.22) we obtain the alternative wage equation:
- e used (9.12) and lir -
he final expression invoh--
-,.ent with the worker, tt.
18) shows what the firm ,1 the worker by bargainir truck, the worker changes c that the net gain to w= (1 - 13)z + 13 [FL(K, 1) + (9.24)
Workers get a weighted average of the unemployment benefit and the surplus, which consists of the marginal product of labour plus the expected search costs that are saved if the deal is struck (recall that yo0 yoV/U represents the average hiring costs per unemployed worker).
(9.21)
9.1.2 Market equilibrium tvenient wage equation in We now have all the necessary ingredients of the model. For convenience, the full model is summarized by the following four equations which together determine 223 By combining ZP and unemployment ratio, or .
In panel

(b) of Figure 9.
cator for labour market ti and BC is the Beveridge c (9.28), the Beveridge cu 1 ( 1 —71) 3 (f(
where U dU/U, ti in (9.4) and (9.5). 6 The I Intuitively, for a given ur fall in the instantaneous curve the unemploymci the labour market (U < s rate must rise. Equation s shifts the Beveridge can 6 This expression is obtain,-
[s + f(0)] dU + Udf (0) c —u dU + Udf

(e) = ( 1 - U sU + Uf (0) [1 - ?AO)] - 6; = [s - f(0)U(1 - 779))]
By using U = s/(s +

f) in the fin The Foundation of Modern Macroeconomics the equilibrium values for the endogenous variables, K, w, 9, and U.
(9.25)
(9.26)
(9.27)
(9.28)
FK(K , 1) = r + 8, FL [K(r + 0,1] —w _ Yo r + s q(0)'
w = (1 — 13)z + [FL [K(r + 8), 1] + Yo U = s + 0q(0) •