Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

f (9)

dt, q(0)

where 6 V /U

XN dt

matching function.

1negotiations take place to

d This is 4 - -)n, which turns out to be ermines the number of jobs ,n of the number of unem- s that exist (plus exogenous mt. It is assumed that only

,Js to replace existing (but unemployed, and only the arch in the model discussed activities of production of

v, and know that there is )ment in time, a proportion -m-specific shocks making sere is thus a constant inflow

rium unemployment rate that

•

)rkers, and that every agent consists of N workers, and here is no decision on hours istant.) The unemployment t a job, and is denoted by ssed as a proportion of the ment in time, there are UN

,c1 each other".

e depends on UN and VN

(9.1)

he matching rate, and G(., .)

6. > 0, Guu < 0, GIN < 0,

is that at each instant XN

'0, ch. 1).

clop a matching model with an

Chapter 9: Search in the Labour Market

meetings occur between an unemployed worker and a firm with a job vacancy. Which particular worker meets which particular job vacancy is selected randomly.

Consider a small time interval dt. During that time interval, there are matches and VN vacant jobs, so that the probability of a vacancy being filled during tit equals (XN / VN)dt . By defining g - ..XN/VN = X/V, we can use equation (9.1) to write q as:

q =	G(UN , VN) VNG(UN /VN , 1)		= G(U /V , 1) q(0),	(9.2)
	VN	VN

is the vacancy-unemployment ratio that plays a crucial role in the analysis. Obviously, since g(0)dt measures the probability that a vacancy will be filled in the time interval can be interpreted as the instantaneous probability of a vacancy being filled, and the expected duration of a job vacancy is 1/q(0). All these results are derived more formally in the Intermezzo below.

In view of the assumptions about G(., .), the following properties of the g(0) function can be demonstrated:

dq

dO = GU (9.3)

0 < ‘ j,

and

0 dq		0 < 77(6) < 1,	(9.4)
- - — =	GU
q dO

where ?AO) is the absolute value of the elasticity of the q(0) function.3 Unemployed workers also find a match in a stochastic manner. For workers, the

instantaneous probability of finding a firm with a vacancy is given by XN/UN, the number of vacancies expressed as a fraction of the number of unemployed workers. This instantaneous probability can be written in terms of 0 also:

G(UN , VN) VNG(UN /VN, 1)		= (V /U)G(U / V , 1) = 0q()0	fie)).	(9.5)
UN	UN

The f (0) function has the following elasticity:

0 df	=[q(0) +	dg] 0	0 dq		(9.6)
f (60 d9		dO 0(1(0)	= 1 + - — = 1 - ?AO)	•
			g de

Since f (0) represents the instantaneous probability of an unemployed worker finding a job, the expected duration of unemployment equals 1/f (0) = 1/(0q(0)). This is intuitive, since unemployed workers find it easier to locate a job (and hence expect a shorter duration of unemployment) if 0 is high, i.e. if there are relatively many vacancies. The definitions of q(0) and in (9.2) and (9.5) show that there is

3 The trick is to write (9.1) as XN = GuUN + GvVN, which implies q = Gu 10 + Gv. Hence, 9(0) = Gu 1(q0) =1 — Gv /q, which is between 0 and 1 because 0 < Gv < q.

215

trading external-

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

ity. 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:

s(1 — U)N dt = 0q(0)UN dt. (9.7)

By assuming that the labour force N is large, expected and actual inflows and outflows can be assumed the same, so that (9.7) can be solved for the actual equilibrium unemployment rate:

U = s + 0q(0)'

9.8)

which implies that a U/as > 0 and au/a9 < 0.

=

Intermezzo

Some statistical theory. The search-theoretic approach makes use of some statistical techniques that may not be immediately obvious. In this intermezzo some important notions are reviewed. Further details can be found in Ross (1993, ch. 5).

A very convenient probability distribution is the exponential distribution. A continuous random variable X is exponentially distributed if its probability

density function has thL

f(x) =	{ Ae- Ax x > 0
	0	x < u

which implies that the L

F (x) E--: i f (y) dy =I

The cumulative distri random variable X attain c

F(x) P {X < x} .

The exponential distribu expected value of X is 1/ 1/A2 . Third, the random of some light bulb. Then distribution of the ren ...

light is the same as the ( bulb does not "remembe - random variable is mei.—

P{X>s+tiX>o=

The memoryless propert function (often called the represents the conditio al light bulb or a human beu

	r(t)	f (t)
		1 — F(t)

For the exponential di distribution of remaining item. As a result, the fails. find that this is indeed thL

r(t) =	f(t)	=	Ae-
	1 — F(t)		e—

We shall have the opix interesting applications in i

MUM

216

hazard rate

I

workers to jobs, and the oat: and vacancies meet in r tbour market pressure to be )lays a crucial role because existence of a trading exten.

Jr market (firms with unfillec - ed by the price mechanic"- theprice mechanism ependent on the situation it s, the probability of ration. worker. The particular exter- - Ntion or search externality

ously given job destruction s disappears at each insta

f firm-specific shocks making !I time interval dt, the proboecomes unemployed is given ). Hence, the average numb-- al dt equals s(1 - U)N dt a: given by 0q(0)UN dt. In the at the expected inflow and

I

(9.7)

1 and actual inflows and out- d for the actual equilibrium

(9.8)

-, ach makes use of some obvious. In this intermezzo Is can be found in Ross

exponential distribution. tributed if its probability

Chapter 9: Search in the Labour Market

omit

density function has the form:

	f (x) -=	{),.e-hXx > 0		(a)
		0	x < 0

which implies that the cumulative distribution function is given by:

F(x)	iX	f(y) dy =	1	- e	x> 0	(b)
			0		x 0

The cumulative distribution function F(x) measures the probability that the random variable X attains a value less than or equal to x, or in symbols:

F(x) P {X < x} . (c)

The exponential distribution has the following properties. First, E(X) = 1/A, the expected value of X is 1/A. Second, the variance of X is V(X) E(X2 )- [E(X)] 2 = 1/A2 . Third, the random variable X is memoryless. Suppose that X is the lifetime of some light bulb. Then, if the light bulb is still working at some time t, the distribution of the remaining amount of time that it will continue to shine light is the same as the original distribution. Colloquially speaking, the light bulb does not "remember" that it has already shone for t periods. Formally, a random variable is memoryless if the following holds:

P {X s-kt IX > = PVC > (d)

The memoryless property implies a very simple expression for the failure rate function (often called the function). The failure rate function r(t) represents the conditional probability density that a t-year old item (such as a light bulb or a human being) fails. It is defined as:

r(t) =	f(t)	(e)
	1 - F(t) .

For the exponential distribution, the memoryless property implies that the distribution of remaining life for a t-year old item is the same as for a new item. As a result, the failure rate function should be constant. Using (a)-(c), we find that this is indeed the case:

f(t) Xe- A t	(f)
(t)=
1 - F(t) e-At

We shall have the opportunity to use this property in economically very interesting applications in the present chapter and in Chapter 16.

217

parts. The first part

cha

job, t

1

— w

I

is prepare~

condition detci

The Foundation of Modern Macroeconomics

The search-theoretic approach also makes extensive use of the notion of a Poisson process. A Poisson process is a counting process with a number of properties. A stochastic process {M(t), t > 0} is called a counting process if M(t) represents the number of "events" that have occurred up to time t. For example, if M(t) represents the number of goals scored by one's favourite soccer star by time t, an "event" consists of your star hitting the back of the net once more. In the context of matching, M(t) represents the number of all matches that have occurred by time t. The counting process M(t) must satisfy: (i) Ai(t) > 0; (ii) M(t) is integer valued; (iii) if s < t, then M(t) M(s) > 0; and (iv) for s < t, M(t)—M(s) equals the number of events that have occurred in the interval (s, t)

(Ross, 1993, p. 208).

A Poisson process is a specific kind of counting process. Formally, the counting process {M(t), t > 0} is called a Poisson process with rate X( > 0) if:

(i) M(0) = 0; (ii) the process has independent increments; (iii) the number of events in any interval length t is Poisson distributed with mean At. Hence,

P tM(t s) M(s) = rn} e-At(Xtr	(g)
m! '

for in = 0, 1, 2, 3, ... For our purposes it is important to know something about interarrival times. Suppose that we have a Poisson process M(t), and that the first event has occurred at time We define Tr, as the elapsed time between the (n — 1)st and the nth event (for n > 1), and refer to Tr, as the interarrival time. Of course, Tr, is stochastic. A very useful property of the Poisson process is that Ti , (n = 1, 2, 3, . .) are independent identically distributed exponential random variables with parameter X and hence have a mean of 1/X (Ross, 1993, p. 214).

Within the context of the matching model this is a very handy property. Since interarrival times are distributed exponentially, the hazard rate r(t) ,--- A is constant and A dt represents the probability that a failure will take place in the time interval dt. Note that a "failure" implies that a match has occurred in this context. Hence, A can be interpreted as the instantaneous probability of a match occurring.

Firms

Each firm is extremely small, has a risk-neutral owner, and has only one job, which is either filled or vacant. If the job is filled, the firm hires physical capital K at a given interest rate r, and produces output F(K, 1). The production function is constant returns to scale and satisfies FK > 0 > FKK and FL > 0 > Fa. If the job is vacant, on the other hand, the firm is actively searching for a worker and incurs a constant search cost of yo per time unit. As was pointed out above, the probability that the firm finds a worker in time interval dt is given by q(9) dt. Since each firm

only has one job, the numb, free entry/exit

Let Jo denote the present va an occupied job, and let I t di

perfect capital market the firm following steady-state arbitr,,,.

riv = — yo + q(8) Lk —

In words, equation (9.9) says

the value of this asset must be return from the asset. The rt that must be incurred each tim fact that the vacant job can be q(9)). The capital gain is the 4

to Iv.

Since anyone who

set up a firm (with a vacancy a occur until the value of a vacar job is worth a negative amours

This implies the following expo

jv = 0

= — Yo

The final expression is intuitive which the search cost yo mus: must be such that the expected cost of the vacancy.

For a firm with a filled derived:

/Jo = F(K,1) — (r

where (r 8) is the rental Equation (9.11) says that the equals 'Jo. This must equal till

is the surpi that remains after the produL equals F(K, 1) — (r 8)K — w). TI

destruction (sfo).

The size of each firm with a flu chooses the amount of capital

218

lye use of the notion of a c with a number of propcounting process if M (- _A up to time t. For exam-

-one's favourite soccer sta - e back of the net once more. 'timber of all matches that must satisfy: (i) M(t) > e

(s)> 0; and (iv) for s < t, xcurred in the interval (s, t

rocess. Formally, the countKess with rate A( 0) if: crements; (iii) the number uted with mean At. Hence,

( g)

t to know something about nrocess M(t), and that the elapsed time between the T„ as the interarrival time. 'he Poisson process is that ibuted exponential random

,f 1/A (Ross, 1993, p. 214). s a very handy property.

Ilv, the hazard rate r(t) = failure will take place in t a match has occurred in

* , ntaneous probability of a

and has only one job, which n hires physical capital K at The production function is d FL > 0 > Fa . If the job is g for a worker and incurs a ed out above, the probability •1 by q(9) dt. Since each firm

Chapter 9: Search in the Labour Market

only has one job, the number of jobs and firms in the economy coincide, and the free entry/exit condition determines the number of jobs/firms.

Let Jo denote the present value of the profit stream originating from a firm with

an occupied job, and let jv designate the same for a firm with a vacancy. With a perfect capital market the firm can borrow freely at the given interest rate, and the following steady-state arbitrage equation holds for a firm with a vacancy:

rjv = — Yo q(9) [Jo — /17	(9.9)
	] •

In words, equation (9.9) says that a vacant job is an asset of the firm. In equilibrium, the value of this asset must be such that the capital cost rJv is exactly equal to the return from the asset. The return consists of two parts, i.e. the constant search cost that must be incurred each time unit (—yo) plus the expected capital gain due to the fact that the vacant job can be filled in the future (with instantaneous probability q(9)). The capital gain is the difference in value of a filled and a vacant job, i.e.

to — /v•

Since anyone who is prepared to incur the constant search cost each time unit can set up a firm (with a vacancy) and start looking for a worker, free entry of firms will occur until the value of a vacant job is exactly equal to zero. Conversely, if a vacant job is worth a negative amount, exit of firms takes place and vacancies disappear. This implies the following expression:

Iv = 0 0 = —Yo + q(09)I0 Io = yolq(0). (9.10)

The final expression is intuitive. The expected duration of a vacancy is 1/q(6) during which the search cost yo must be incurred. In equilibrium the number of jobs/firms must be such that the expected profit of a filled job is exactly equal to the expected

cost of the vacancy.

For a firm with a filled job, the following steady-state arbitrage equation can be derived:

rjo = F (K , 1) — (r + 8)K — w —

(9.11)

where (r + 8) is the rental charge on capital goods, and w is the real wage rate. Equation (9.11) says that the asset value of a filled job is Jo and its capital cost

equals rjo. This must equal the return from the filled job, which consists of two parts. The first part is the surplus created in production, i.e. (the value of) output

that remains after the production factors capital and labour have been paid (this equals F(K, 1) — (r + 8)K — w). The second part is the expected capital loss due to job

destruction (sJo).

The size of each firm with a filled job is determined in the usual manner. The firm chooses the amount of capital it wants to rent such that the value of the firm is

219

The Foundation of Modern Macroeconomics

maximized. In terms of (9.11) we can write this problem as:

max (r + s)1 F (K , 1) — (r + 8)K — w	FK(K , 1) = r + 8 .	(9.12)
{K)

This is the usual condition equating the marginal product of capital to the rental charge on capital. By substituting (9.10) and (9.12) into (9.11), we obtain: 4

	(r + s)yo	= F (K, 1) — FK(K, 1)K — w
	q(0)
	FL(K , 1)	— w = yo	(9.13)
	r + s	q(9)•

The left-hand side of (9.13) represents the value of an occupied job, equalling the present value of rents (accruing to the firm during the job's existence) using the risk-of-job-destruction-adjusted discount rate, r + s, to discount future rents. The right-hand side of (9.13) is the expected search costs. With free exit/entry of firms, the value of an occupied job exactly equals the expected search costs (see above). 5

Workers

The worker is risk neutral and lives forever, and consequently only cares about the expected discounted value of income (Diamond, 1982, p. 219). A worker with a job earns the wage w, whilst an unemployed worker obtains the exogenously given "unemployment benefit" z. This may consist of a real transfer payment from the government but may also include the pecuniary value of leisure. Let YE denote the present value of the expected stream of income of a worker with a job, and let Yu denote the same for an unemployed worker. Then the following steady-state arbitrage equation can be derived for a worker without a job:

rYu = z + q(9) [YE — Yu] • (9.14)

In words, equation (9.14) says that the asset Yu is the human capital of the unemployed worker. The capital cost of the asset must be equal to the return, which consists of the unemployment benefit, z, plus the expected capital gain due to finding a job, i.e. YE — Yu. As Pissarides (1990, p. 10) points out, rYu can be interpreted in two ways. First, it is the yield on human capital of an unemployed worker during search. It measures the minimum amount for which the worker would be willing to stop searching for a job, and hence has the interpretation of a reservation wage. The second

4 We have used the linear homogeneity of the production function, which implies that F = FKK +

1 x FL, so that F — FKK = FL .

s If there were no search costs for the firm (yo = 0), the model would yield the standard productivity condition for labour (FL = w). With positive search costs, however, the factor labour receives less than its marginal product. This is because the marginal product of labour must be sufficiently large to cover the capital cost of the expected search costs.

interpretation is that of " unemployed worker can —

For a worker with a job I

a

rY E = w s[YE — Yid -

The permanent income of there is a non-zero probab By solving (9.14)-(9.15)

	rr u =	±s)z±0,
		r+s+0q, )

rY E = sz + [r + eq(0)11 r+s+0q(H)

where the second expresi be willing to search for a ji

Wages

What happens when a jot is a pure economic rent creal expected search costs by t between the two parties? I some going market A, v with impersonal exchan, between the two parties is bargaining. Fortunately, al in two-person bargaining s

We assume that all firr rate is the same everywh, solution of the model, wl discuss the macroeconon ically adequate description firm-worker pair that is i: such pairings as given.

Consider a particular fin Obviously the firm chaL,, due to free exit/entry) to expected gain to the firm i

faits = F (Ki, 1) — (r + 6)1

F (Ki, 1) — w A = r + s

220

YE — YU.

:n as:

=r+8.	. 1 1

pc!, ct of capital to the ren ' (9.11), we obtain: 4

(9.13)

an occupied job, equallir c.- e job's existence) using t

) discount future rents. The With free exit/entry of firm< NJ search costs (see above).'

iuently only cares about tt' 2, p. 219). A worker with a tains the exogenously given transfer payment from the leisure. Let YE denote the worker with a job, and let the following steady-state

a job:

(9.14

Liman capital of the unem- ' to the return, which concapital gain due to finding a can be interpreted in two loyed worker during search. er would be willing to stop

\ervation wage. The second

m, which implies that F = FKK -

ka yield the standard productivity [te factor labour receives less than _ist be sufficiently large to cover

Chapter 9: Search in the Labour Market

interpretation is that of "normal" or "permanent" income: the amount that the unemployed worker can consume whilst still leaving his/her human capital intact.

For a worker with a job the steady-state arbitrage equation reads as follows.

rY E = — s [YE — Yu]

(9.15)

The permanent income of an employed worker differs from the wage rate because there is a non-zero probability of job destruction causing a capital loss of

By solving (9.14)-(9.15) for rYu and rY E, the following expressions are obtained:

	rY u =	(r + s)z + 0 q(6)w	(9.16)
		r +s+9q(0)

s + [r + 1(9)] w	r(w — z)	rYu,	(9.17)
r YE =	r+s+0q(0)
r+s+9q(0)

where the second expression in (9.17) shows that w > z must hold for anybody to be willing to search for a job.

Wages

What happens when a job seeker encounters a firm with a vacancy? Clearly there is a pure economic rent created by the encounter, existing of the sum of the foregone expected search costs by the firm and the worker. But how is this surplus shared between the two parties? In this search context, it is clearly not possible to refer to some going market wage rate, because the concept of an aggregate labour market with impersonal exchange has been abandoned. The exchange that takes place between the two parties is one-on-one, and the division of the rent is a matter of bargaining. Fortunately, as we saw in Chapter 8, there is a useful solution concept in two-person bargaining situations, called the generalized Nash bargaining solution.

We assume that all firm-worker pairings are equally productive, so that the wage rate is the same everywhere. This allows us to focus on the symmetric equilibrium solution of the model, which is reasonable because the aim of this chapter is to discuss the macroeconomic implications of search theory, not to develop an empirically adequate description of the labour market. We furthermore assume that each firm-worker pair that is involved in wage negotiations takes the behaviour of other

such pairings as given.

Consider a particular firm-worker pairing i. What does the firm get out of a deal? Obviously the firm changes status from a firm with a vacancy (with value n, 0, due to free exit/entry) to a firm with an occupied job (with value Hence, the expected gain to the firm is:

rjo = F (Ki , 1) — (r + 8)Ki — wi —

	= F L(Ki , 1) — wi	(9.18)
	r + s
		221

C

The Foundation of Modern Macroeconomics

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	— Yu] + (1 — 3) log — iv] , 0 < < 1,	(9.20)
two

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):

d	Q	(1	) dJO
dwi =	— Yu dwi		P) dwi = °

	)— # 1	= 0
	r +s Y
	.k—Yu r+s

- Yu =[Lia — Jv] .

This rent-sharing rule can be turned into a more convenient two ways.

(9.21)

wage equation in

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 model is summarized b

222

yo0 yoV/U

			Chapter 9: Search in the Labour Market
- e used (9.12) and lir-	First, by substituting (9.18)-(9.19) into (9.21) and imposing jv = 0 (due to free
he final expression invoh--	exit/entry) we obtain:
-,.ent with the worker, tt.
18) shows what the firm		(1 — p))I =	+ ( 1 —
, 1 the worker by bargainir
r		(1 P)[]wi sYu= [FL(Ki 1) - wi		]+ (1 -
		sr + s
truck, the worker changes
c that the net gain to		(1 — p) [wi + sYu] = p [FL(Ki, — wi] + (1 — P)(r + s)Yu
		wi = (1- i3)tYu + 13FL(Ki, 1).		(9.22)

9.1'

station regarding the wa, does not accept

- inue searching as one rate that the unemployed unemployed is not wi -•

Re wi is set such that Q is

1, (9.2r

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 -							p) Jo
						18
= z + eq(e)( 1	13	p	)	q)/°(9	)	=z+/1362 i°6	(9.23)
						,

points of the firm and the the worker and the firm ring rule rolls out of the

(9.21)

tvenient wage equation in

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:

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 represents the average hiring costs per unemployed worker).

9.1.2 Market equilibrium

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

The Foundation of Modern Macroeconomics

the equilibrium values for the endogenous variables, K, w, 9, and U.

	FK(K , 1) = r + 8,		(9.25)
	FL [K(r + 0,1] —w _ Yo		(9.26)
	r + s	q(0)'
	w = (1 — 13)z + [FL [K(r + 8), 1] + Yo		(9.27)
	U = s + 0q(0) •		(9.28)

Equation (9.25) is the marginal productivity condition for capital, determining the optimal capital stock (and thus the optimal size of production) of each firm with a filled job. Since the marginal productivity of capital diminishes as more capital is added (FKK < 0), (9.25) relates the optimal capital stock (K*) to the (exogenous) rental rate on capital, i.e. K* = K(r + 6) with K' < 0. By plugging this function into, respectively, (9.13) and (9.24) we obtain (9.26) and (9.27). Equation (9.26) is a form of the zero profit condition implied by the assumption of free exit/entry of firms, and (9.27) is the wage-setting equation that rolls out of the Nash bargaining between a firm with a vacancy and an unemployed job seeker. Finally, (9.28) is the expression for the equilibrium unemployment rate. This equation is also known as the Beveridge curve (Blanchard and Diamond, 1989).

The model is recursive under the assumption of a fixed real rate of interest. First, (9.25) determines the optimal size of each producing firm as a function of the interest rate. Then (9.26)—(9.27) determine equilibrium values for w and 9 as a function of that optimal capital stock. Finally, (9.28) determines the unemployment rate, U, as a function of 9. Once 9 and U are known, the number of jobs is given by (1 — U)N + 0 UN and employment equals L = (1 — U)N.

The graphical representation of the model is given in Figure 9.1. In panel (a) the ZP curve is the zero-profit condition (9.26). It is downward sloping in (w, 9) space:

	(dw) =	(r + s)yo (6) < .	(9.29)
	dO zp q(9)2

Intuitively, a reduction in the wage increases the value of an occupied job and thus raises the left-hand side of (9.26). To restore the zero-profit equilibrium the expected search cost for firms (the right-hand side of (9.26)) must also increase, i.e. q(9) must fall and 9 must rise.

Also in panel (a), the WS curve is the wage-setting curve (9.27). This curve is upward sloping in (w, 9) space:

	dw\	ws = PY° °•	(9.30)
	T

Intuitively, the wage rises with 9 because the worker receives part of the search costs that are foregone when he strikes a deal with a firm with a vacancy (see above).

Figure 9.1. Search equi

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

— dU + Udf (e) = ( 1 - U

u

sU + Uf (0) [1 - ?AO)]6;- =

[s - f(0)U(1 - 779))]

By using U = s/(s + f) in the fin

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