Overview of the Equilibrium Model

We start by considering the basic equilibrium model of search to be detailed in this chapter. The labour market is assumed to consist of n firms and N individuals.

Each firm produces a homogenous output using labour as an input and subject to a fixed 'capital' cost. Output is sold on a perfectly competitive auction market at some price p which is considered fixed. There are two alternative simplifying assumptions regarding the generation of vacancies.

In the null offer model each firm employs at most a single individual who in conjunction with capital (the rental on which will be denoted by k) produces an output Q.

In the unlimited vacancies model each firm corresponds to the simple infinite vacancy firm of chapter 4. All randomly contacting individuals are employed by such firm and each in conjunction with capital produce Q output (so that there are constant returns to labour) .

Both types of firm face job turnover as described in Chapter 4 so that employees leave the firm over time. The probability that any individual separates from the firm in a time interval 6t is denoted by pit.

Individuals who have separated from firms search for alternative employment. With identical individuals the only equilibrium that can be sustained is one with a unique wage offer made by all firms so that all firms pay a wage w. Individuals are assumed to know of the location of firms and to contact a firm at a deterministic rate Y£ (so that y^St firms are contacted in time interval St). Where null offers are a feature of the market individuals do not know which firms currently have a vacancy therefore individuals always select a firm to contact randomly (i.e. by drawing from a hat).

The process of contacting firms (search) is assumed to involve a flow search cost c (search for an interval St costs cSt). Where offers are unlimited the decision to participate or not simply

depends upon whether the wage once obtained compensates for the period of time required to contact the firm. Where null offers are sometimes the outcome of search it also matters to the individual that some

contacts may be unsuccessful. We shall see that it is in this aspect of search decisions that positive feedback effects suggest multiple equilibria as a possibility.

Stochastic equilibrium in a market with firms and individuals as described above occurs where expected separations equal expected matching, there is no incentive to entry or exit and where the wage satisfies either the firm's monopsony offer or some Nash bargaining outcome. The possibility of a bargain over wages will be examined in detail later. It is important to bear in mind that the wage will be endogenous to search equilibrium. Associated with steady state equilibrium will be steady state unemployment.

We start our detailed exposition of the above model by reconsidering the simple vacancy search model (with turnover) in continuous time. 6.3. Unemployed Vacancy Search

As noted above, we wish to reformulate the basic vacancy search model of Chapter 2 in continuous time. For simplicity discounting will be ignored throughout this chapter.

It is easiest to start by considering the proportion of time that an individual spends unemployed when the rate of contact at firms is y^. Considering expected per period values is natural when discounting is ignored and an infinite horizon is assumed.

The infinite lifetime of an individual is spent alternating between just two states, employment and unemployment. We again have a simple stochastic process to solve for (see Figure 1).

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Employed

Unemployed

(l-uôt)

Figure 1.

The probability x6t depends on whether offers are rationed or not. In the first case xit is equal to qy^6t where q is the probability of a vacancy ((1-q)- probability of a null) in the latter case x6t is simply y^6t.

The steady state probabilities of this process can be solved for using the same techniques as in chapter 4 to yield

(6.1) p ( U ) --- y— (y+x) P ( E ) --- —

(y+x)

p(U) and p(E) are respectively the proportion of time spent in the unemployed and employed states.

When in the first of these states the individual is assumed to receive (b-c) (unemployment utility minus search cost) whilst

in the second a wage of w. Expected per period income given participation is thus simply:

V8 - (b-c)p(U) + wp(E)

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The participation wage w* can be calculated by equating Vg to the expected per period value of non participation which by definition is b. Hence the participation wage w* satisfies

(6.3) b - (b-c)p(U) - w*p(E) « 0

Provided the market wage is greater than w* all individuals will prefer participation to pure leisure.

It is important to notice that in the case of unlimited offers w* is entirely determined by exogenous parameters whilst in the case of null offers w* in part depends upon (1-q) the null offer probability that will be endogenous to our model.

One interesting result should be noted, if c * 0 then w* = b.

This follows directly from equation (6.3) using the fact that probabilities sum to one, we shall have cause to use this special case in which (even with rationed offers) the participation wage is exogenous. On

reflection the exogeneity of the participation wage in the case of zero search costs is obvious - in the absence of such costs and discounting - the probability of state occupancy is irrelevant. In such circumstances any wage greater than the value of leisure will induce participation.

When money search costs are allowed individuals must receive a wage sufficient to compensate them for unsuccessful but costly search.

The model of individual decisions outlined here is the simplest possible, allowing only one endogenous parameter ((1-q) the null

offer probability) to determine behaviour (participation), y ^ , u, b and c

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In the next section we consider in detail two alternative models of firms upon which individual search decisions will feed back and hence generate a market equilibrium.