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161 For examples of search and matching models, see Diamond (1982); Pissarides (1985); Mortenson (1986); Howitt (1988); Blanchard and Diamond (1989); and Hosios (1990). The model in this section is closest to Pissaridess. many of our usual tools, even a basic model is relatively complicated. As a result, the model here only introduces some of the issues involved. The Model The economy consists of workers and jobs. Workers can be either employed or unemployed, and jobs can be either filled or vacant. The numbers of employed and unemployed workers are denoted E and U, and the numbers of filled and vacant jobs are denoted E and V. Each job can have at most one worker. Thus F and E must be equal. The labor force is fixed at I; thus E + U = 1. Throughout, we consider only steady states. The number of jobs is endogenous. Specifically, vacancies can be created or eliminated freely; there is a fixed cost of per unit time, however, of maintaining a job (either filled or vacant). can be thought of as reflecting the cost of capital. The model is set in continuous time. When a worker is employed, he or she produces output at rate A per unit time and is paid a wage of w per unit time. A is exogenous and is assumed to be greater than C; w is determined endogenously. For simplicity, costs of effort and of job search are ignored. Thus a workers utility per unit time is w if employed, and zero if unemployed. Similarly profits per unit time from a job are A  w  if it is fiUed, and  if it is vacant. The key assumptions of the model concern how workers become employed. Positive levels of unemployment and vacancies can coexist without being immediately eliminated by hiring. Instead, unemployment and vacancies are assumed to yield a flow of new jobs at some rate per unit time: M = M{U, V) (10.68) = KUlV, 0</3<1, 0< <1. The matching function, (10.68), proxies for the complicated process of employer recruitment, worker search, and mutual evaluation. It is not assumed to exhibit constant returns to scale. When it exhibits increasing returns ((3 + > 1), there are thickmarket effects: increasing the level of search makes the matching process operate more effectively, in the sense that it yields more output (matches) per unit of input (unemployment and vacancies). When the matching function has decreasing returns {p + < 1), there are crowding effects. In addition to the flow of new matches, there is tumover in existing jobs. As in the ShapiroStiglitz model, jobs end at an exogenous rate b per unit time. Thus the dynamics of the number of employed workers are given by
See Problem 10.15 for the implications of alternative assumptions about how the surplus is divided. ¨ = M{U, V)  bE. Since we are focusing on steady states, M and E must satisfy M{U, V) = bE (10.69) Let a denote the rate per unit time that unemployed workers find jobs, and a the rate per unit time that vacant jobs are filled, a and a are given by .  (10.70) . . MM). ,10.71) As in the ShapiroStiglitz model, we use dynamic programming to describe the values of the various states. The "return" on being employed is a "dividend" of w per unit time minus the probability b per unit time of a "capital loss" of VeVu. Thus, rVE = w HVe  Vu). (10.72) where r is the interest rate (see equation [10.30] for comparison). Similar reasoning imphes rVF ={Aw  O HVe  Vv), (10.73) rVu = a{VE  Vu), (10.74) rVv = C + a(VF  Vv). (10.75) Two conditions complete the model. First, when an unemployed worker meets a firm with a vacancy, they must choose a wage. It must be high enough that the worker wants to work in the job, and low enough that the employer wants to hire the worker. Because neither party can find a replacement instantaneously, however, these requirements do not uniquely determine the wage; instead, there is a range of wages that makes both parties better off than if they had not met. We assume that the worker and the employer set the wage so that each of them gets the same gain. That is, VeVu = Vf Vv. (10.76) Second, as described above, new vacancies can be created and eliminated costlessly. Thus the value of a vacancy must be zero. Without the frictions, the model is simple. Labor supply is perfectly inelastic at I, and labor demand is perfectly elastic at A  C. Thus, since A  > 0 by assumption, there is full employment at this wage. Shifts in
Solving the Model We solve the model by focusing on two variables, employment (E) and the value of a vacancy (Vv) Our procedure will be to find the value of Vy implied by a given level of employment, and then to impose the freeentry condition that Vv must be zero. We begin by considering the determination of the wage and the value of a vacancy given a and a. Subtracting (10.74) from (10.72) and rearranging yields VeVu = (10.77) a + b + r Similarly, (10.73) and (10.75) imply VfVv = . (10.78) a + b + r Since our splittingthesurplus assumption (equation [10.76]) implies that Ve  Vu and Ve  Vy are equal, (10.77) and (10.78) imply w Aw a + b + r a + b+ r Solving this condition for w yields (10.79)  = " v. • (10.80) a + a + 2b + 2r Equation (10.80) implies that when a and a are equal, the hrm and the worlcer divide the output from the job equally. When a exceeds a, worlcers can hnd new jobs more rapidly than firms can find new employees, and so more than half of the output goes to the worker. When a exceeds a, the reverse occurs. Recall that we want to focus on the value of a vacancy. Equation (10.75) states that rVy equalsC+a{VE  Vy), and the splittingthesurplus assumption implies that Ve  Vy equals Ve  Vu Thus rVy equals  C + a(VE  Vu)Substituting expression (10.80) for w into (10.77) for Ve  Vu implies VeVu = (10.81) a + a + 2b + 2r Thus, labor demandchanges in Alead to immediate changes in the wage and leave employment unchanged.
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