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154 ) = eeP(w  e)dt + e~"4c" ( ) + d  e)Vv{At)]. (10.26) The first term of (10.26) reflects utility during the interval (0, Af). The probabihty that the worker is stiU employed at time t is e If the worker is employed, flow utility is w  e. Discounting this back to time 0 yields an expected contribution to lifetime utility of e*+4w  e)? The second term of (10.26) reflects utility after At. At time At, the worker is employed with probability e"*, and is unemployed with probability 1 e. Combining these probabilities with the Vs and discounting yields the second term. If we compute the integral in (10.26), we can rewrite the equation as Ve(At) = t[1  e (+>f](w  ¸) (10.27) + e[eVE{At) + (1  e)Vu{At)]. Solving this expression for Vfc (At) gives VEiAt) = (w ~¸) + le41  eWuiAt). (10.28) As described above, Ve equals the limit of ( ) as Af approaches zero. (Similarly, Vu equals the limit of VviAt) as t approaches zero.) To find this "If time is discrete rather than continuous, we look one period ahead. See Sargent (1987b) for an introduction to dynamic programming. Because of the steadystate assumption, if it is optimal for the worker to exert effort initially, it continues to be optimal. Thus we do not have to allow for the possibility of the worker beginning to shirk. brief interval of time and use the V; s themselves to summarize what occurs after the end of the interval. Consider first a worker who is employed and exerting effort at time 0. Suppose temporarily that time is divided into intervals of length At, and that a worker who loses his or her job during one interval cannot begin to look for a new job until the beginning of the next interval. Let ( ) and Vr/(At) denote the values of employment and unemployment as of the beginning of an interval under this assimiption. In a moment we will let At approach zero. When we do this, the constraint that a worker who loses his or her job during an interval cannot find a new job during the remainder of that interval becomes irrelevant. Thus ) will approach Ve. If a worker is employed in a job paying a wage of w, ( 1) is given by
limit, we apply IHopitals rule to (10.28). This yields •  Ve = "tKw ¸) + hVul (10.29) p + b Equation (10.29) can also be derived intuitively. Think of an asset that pays dividends at rate w  ¸ per umt time when the worker is employed and no dividends when the worker is unemployed, and assume that the asset is being priced by riskncutral investors with required rate of return p. Since the expected present value of lifetime dividends of this asset is the same as the workers expected present value of lifetime utility, the assets price must be Ve when the worker is employed and Vu when the worker is unemployed. For the asset to be held, it must provide an expected rate of return of p. That is, its dividends per unit time, plus any expected capital gains or losses per unit time, must equal pVe. When the worker is employ ed. dividends per unit time are w  e, and there is a probability b per unit time of a capital loss of VeVu Thus, pVe = (w  ¸)  HVe  Vu). (10.30) Rearranging this expression yields (10.29). If the worker is shirking, the "dividend" is w per unit time, and the expected capital loss is (b + q)(Vs  Vu) per unit time. Thus reasoning parallel to that used to derive (10.30) implies pVs=w{b + q){VsVu). (10.31" Finally, if the worker is unemployed, the dividend is zero and the expected capital gain (assuming that hrms pay sufficiently high wages that employed workers exert effort) is a(V£  Vu) per unit time. Thus, pVu = a{yEVu). (10.32 The NoShirking Condition The firm must pay enough that Ve > Vs; otherwise its workers exert no effort and produce nothing. At the same time, since effort cannot exceed ¸, there is no need to pay any excess over the minimum needed to induce effort. Thus the firm chooses w so that Ve just equals VsP . Ve = Vs. (10.331 iEquations (10.31) and (10.32) can also be derived by defining Vu(M) and Vs(Af) and proceeding along the lines used to derive (10.29). "Since all firms are the same, they choose the same wage. Thus Vf and Vs do not depend on what firm a worker is employed by.
"We are assuming that the economy is large enough that although the breakup ot any individual job is random, aggregate breakups are not. Since Ve and Vs must be equal, (10.30) and (10.31) imply (w  ¸)  HVe ~Vu) = w~{b + ){  Vv), (10.34) V£  Vu = . (10.35) Equation (10.35) implies that firms set wages high enough that workers strictly prefer employment to unemployment. Thus workers obtain rents. The size of the premmm is increasing in the cost of exerting effort, e, and decreasing in firms efficacy in detecting shirkers, q. The next step is to find what the wage must be for the rent to employment to equal elq. Rearranging (10.30) to obtain an expression for w yields w = ¸ + pVe + (  Vv) = ¸ + pVv + { + p)(Ve ~ Vv) (10.36) = e+{a + b + p){VEVu), where the last line uses (10.32) to substitute for pVu Thus for Ve  Vu to equal e/q, the wage must be w = e + (a + b + p). (10.37) This condition states that the wage needed to induce effort is increasing in the cost of effort (¸), the ease of findmg jobs (a), the rate of job breakup (b), and the discount rate (p), and is decreasing in the probability that shirkers are detected (q). It turns out to be easier to express the wage needed to prevent shirking in terms of employment per firm, L, rather than the rate at which the unemployed find jobs, a. To substitute for a, we use the fact that, since the economy is in steady state, movements into and out of unemployment must balance. The number of workers becoming unemployed per unit time is JV (the number of firms) times I (the number of workers per firm) times b (the rate of job breakup).* The number of unemployed workers finding jobs is L NL times a. Equating these two quantities yields Equation (10.38) imphes a b = Lb/(L  NL). Substituting this into (10.37) yields
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