Problem 1 â€“ Efficiency Wages;Consider a worker with utility function over wages and effort, u(w, e) = w ? e. A firm is considering how to optimally set wages so as to induce the worker to deliver a given effort level e = e ?. Effort is normally unobserved but with monitoring probability ? ? (0, 1), the firm can directly observe the workerâ€™s effort. If the firm monitors the worker and it finds that the worker is shirking, it is committed to firing the worker. The firmâ€™s revenue is directly tied to the workerâ€™s effort in a non-verifiable way (that is, it is deterministic and observable to the firm, but it cannot use revenue observations to make assertions about the workerâ€™s effort in a court of law). Specifically, R(e ?) = R ?, and R(0) = 0. The firmâ€™s profits are given by ? (?, w|e) = R(e) ? w ? c(?), where c(Â·) is the monitoring cost. Define c(?) = ?.;Define w? as the market clearing wage for which unemployment is zero. Unemployment in the;economy is going to be a positive function of the difference between the actual market wage wË† and;w?, u(wË†) = wË†?w?. wË†;The workerâ€™s expected utility can be written up contingent on the effort choice and the firmâ€™s wage offer w;u (w, e ?) = w ? e ? u(w, 0) = (1 ? ?)w + ??u(wË†)b + ?1 ? u(wË†)?wË†?;where the last equation reflects the probability of being fired in case of being caught shirking in which case we are stylistically capturing the threat of unemployment by saying that the worker will then be unemployed with probability u and receive benefits b, or find a new job with probability (1 ? u) and then receive the market wage wË†.;1. Determine the minimum firm wage requirement w ? such that the workerâ€™s incentive compati- bility constraint (IC) is satisfied, u(w ?, e ?) ? u(w ?, 0).;2. Determine the optimal monitoring choice given that the firm sets the wage equal to w ?, ??(w ?).;3. Now, impose the condition that in equilibrium it must be that all firms set the same wage. Consequently, the market wage must equal the firmâ€™s optimal wage choice, wË† = w ?. Use the binding (IC) constraint imposing wË† = w ? and the optimal monitoring choice to characterize the equilibrium efficiency wage wË†.;4. Determine a basic condition on R ? so that the above solution is indeed an equilibrium (that is, ? (??(wË†), wË†|e ?) ? ? (0, 0|0).);5. How is unemployment affected by an increase in the benefit level, b?
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