Two identically able agents are competing for a promotion. The promotion is awarded on the basis of output (whomever has the highest output, gets the promotion). Because there are only two workers competing for one prize, the losing prize=0 and the winning prize =P. The output for each agent is equal to his or her effort level times a productivity parameter (d). (i.e. Q2=dE1 , Q2=dE2). If the distribution of “relative luck” is uniform, the probability of winning the promotion for agent 1 will be a function of his effort (E1) and the effort level of Agent 2 (E2). The formula is given by...

Prob(win)=0.5 + α(E1-E2), where α is a parameter that reflects uncertainty and errors in measurement. High measurement errors are associated with small values of α (think about this: if there are high measurement errors, then the level of an agent’s effort will have a smaller effect on his/her chances of winning). Using this information, please answer the following questions. Both workers have a disutility of

effort C(E)=E2 .

a) What is the formula for the expected utility of agent 1? What is the expected utility of agent 2?

b) What is the optimal level of effort (E1) for agent 1? Does the size of the prize matter in determining his optimal amount of effort? What about the value of α? Explain. Does agent 1’s decision on how hard to work depend on E2, i.e. on how hard the other agent works? Why or why not?

c) What is the optimal level of effort (E2) for agent 2? Does agent 2’s optimal effort depend on agent 1’s effort choice?

d) Suppose α = 0.01 and P=1000. How hard will each agent work? What is the probability that agent 1 will win the promotion? Agent 2? How do these probabilities change when we raise P to 2000? Which prize will agents prefer to face, 1000 or 2000? Under which one will they work harder?