Exercise: Promotions and Tournaments
1. 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?
2. Given your answers to question 1, you are now ready to make a spreadsheet. Consider the same situation as in question 1 but now suppose α = 0.025 and d = 25 (these are "base-case" values; set up your spreadsheet so that you can input any value for these parameters).
a) Think of the first column in the blank spreadsheet below as alternative values of the prize spread that the firm is considering. Using the formula for the optimal effort of each agent, fill in the first column. (Recall that this is the same effort level for both agents).
b) Let both workers' alternative utility levels be equal to 100. (This is the best total utility each worker can get from another job, and the firm has decided this is the amount of utility it will provide to both workers, to keep them from quitting. Using the formula for expected utility, calculate the level of base salary, A, (i.e what you are paid whether you win the prize or not) the firm must offer at each level of P to give workers this level of utility. Put these values in column 3. Check your calculations in column 4 by plugging your calculated values of A and E into the formula for utility. It should come out to 100 in each row of the table.
c) Calculate the total expected output produced by the two workers combined in column 5 of the table.
d) Calculate the firm's total expected profits from the two workers combined in column 6 of the table, for each alternative value of the prize spread, P. Whatever the prize spread, assume the firm offers a base salary, A, just sufficient to give each worker an expected utility level of 100. What is the profit-maximizing prize spread under these circumstances?
e) Is the base salary positive or negative at the profit maximum in part (d)? Explain why it is positive or negative.
f) Of all the possible prize spreads considered in your spreadsheet (from zero up to 3000), which one makes workers work the hardest? Why doesn't the firm prefer to use this prize spread?
g) Now change the firm's measurement technology to α = 0.025. (The firm can now determine which worker had the higher output twice as accurately). What is now the optimal prize spread, P? Compare the levels of worker effort, expected utility, output and the firm's profit now to those when α was half as large, at .025. What has happened to the base salary? Explain.
Prize Spread P Optimal Effort E* Base Salary A Expected Utility E(U) Output Q Expected Profit E(n)
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3. Two agents are competing for a promotion. The winner gets S, the loser gets zero. The probability worker 1 wins the prize is given by:
Prob(1 wins)=0.9 + 0.1(E1-E2),where agent 1's effort2 is given by E1 and agent 2's by E2. Each agent's disutility of effort is given by E /2.
a) If both agents work equally hard, which one is more likely to win the promotion?
b) Write down the formula for the expected utility of agent 1. Find his/her optimal effort as a function of the prize spread, S.
c) Write down the formula for the expected utility of agent 2. Find his/her optimal effort as a function of the prize spread, S. Explain why, in this example, both agents work equally hard for any given prize spread.
d) In the blank spreadsheet that is provided for this question, think of the various values of S as different possible prize spreads the firm is considering. In column 2, fill in the (common) effort level both agents will choose.
e) Assume neither agent is paid to show up for work (a=0). Using the definition of utility (and the fact that both agents work equally hard) fill in both agents' utility in columns 3 and 4.
f) Suppose that, to get agent 1 to take the job, she must attain an expected utility level of 20. To get agent 2 to take the job, he must get an expected utility of 5. In columns 5 and 6, compute the level of a for each agent that just induces them to take the job (i.e. that gives them expected utilities of exactly 20 and 5 respectively).
g) Suppose the firm chooses the levels of a given by part f. Let the expected value of each agent's output be given by 10E (where E is the agent's effort). Now compute the firm's output in column 7, and its profits in column 8. What is the profit- maximizing prize spread? Comment on the different levels of a for the two workers at this point.