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Consider a worker with utility function over wages and eort,




Problem 1 Eciency Wages;Consider a worker with utility function over wages and eort, u(w, e) = w e. A rm is considering;how to optimally set wages so as to induce the worker to deliver a given eort level e = e. Eort;is normally unobserved but with monitoring probability (0, 1), the rm can directly observe;the workers eort. If the rm monitors the worker and it nds that the worker is shirking, it;is committed to ring the worker. The rms revenue is directly tied to the workers eort in a;non-veriable way (that is, it is deterministic and observable to the rm, but it cannot use revenue;observations to make assertions about the workers eort in a court of law). Specically, R() = R;e;and R(0) = 0. The rms prots are given by (, w|e) = R(e) w c(), where c() is the;monitoring cost. Dene c() =.;Dene w as the market clearing wage for which unemployment is zero. Unemployment in the;economy is going to be a positive function of the dierence between the actual market wage w and;w, u (w) = w w.;w;The workers expected utility can be written up contingent on the eort choice and the rms;wage oer w;u(w, e) = w e;u(w, 0) = (1)w + u(w)b + 1 u(w) w;where the last equation reects the probability of being red 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 benets b, or nd a new job with probability;(1 u) and then receive the market wage w.;1. Determine the minimum rm wage requirement w such that the workers incentive compati;bility constraint (IC) is satised, u(w, e) u(w, 0).;2. Determine the optimal monitoring choice given that the rm sets the wage equal to w, (w).;3. Now, impose the condition that in equilibrium it must be that all rms set the same wage.;Consequently, the market wage must equal the rms optimal wage choice, w = w. Use the;binding (IC) constraint imposing w = w and the optimal monitoring choice to characterize;the equilibrium eciency wage w.;4. Determine a basic condition on R so that the above solution is indeed an equilibrium (that;(w), w |e) (0, 0|0).);is;5. How is unemployment aected by an increase in the benet level, b?;1;Problem 2 Search;Jane is searching for a summer job. She has surveyed the market and has determined that the;oer distribution for summer jobs is uniform with a lower bound of $5, 000 and an upper bound of;$20, 000. Hence, any oer in the range between $5, 000 and $20, 000 is equally likely, so Pr(of f er =;x) = 1/15000, for any x [5000, 20000]. Obtaining oers requires resources and eort, and Jane;has determined that the cost of obtaining an oer is equivalent to a monetary payment of c = 500;dollars. Jane can draw as many oers as she pleases. Each one costs her 500 dollars. After each;draw she decides to accept or reject the current oer. If she rejects the oer, she draws again. If;she accepts, she is done searching. Dene the reservation level, R, such that all oers below R are;rejected and all oers at or above R are accepted.;1. Determine the average value of a draw from the oer distribution.;2. For a given reservation level, R, determine the expected search cost.;3. For a given reservation level, R, determine the expected value of a job conditional on it being;accepted.;4. The value of search for a given reservation level strategy is the sum of expected search costs;and the expected value of an acceptable job. Determine the value of searching as a function;of the given reservation level, U (R). Draw U (R) in a gure with R on the horizontal axis;and U (R) on the vertical axis. Include the 45 line in the graph.;5. Determine the optimal choice of R.;6. What is the average value of a job that Jane accepts? What is the average number of oers;that Jane will draw before an acceptable oer is made?;7. Suppose that the oer distribution changes so that the lower bound is 0 dollars and the upper;bound is $25, 000. The distribution remains uniform so all oers between 0 and 25000 are;equally likely. What is the value of the average realization from the distribution? What is;Janes new optimal choice of R? What is the average number of oers Jane will draw before;accepting? Explain why R changes and the direction of change.;Problem 3 Unemployment;Consider the following wage search model: During a period an unemployed worker receives one;wage oer with probability. With probability 1, the unemployed worker does not receive;an oer. The wage oer is drawn from the wage oer distribution F (w) which is assumed to be;uniform over the interval [0,100]. Specically this implies that the probability of receiving an oer;w less than or equal to some level w is Pr(w w) = w/100. During any period, an employed;worker loses her job with probability. For any given reservation wage R such that unemployed;workers reject oers below R and accept oers above R, determine the steady state unemployment;rate using the logic of the bathtub model. Use your answer to calculate the unemployment rate for;= 0.017, = 0.5, and R = 35.;2


Paper#27106 | Written in 18-Jul-2015

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