Question 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 + (E1-E2), where is a parameter that reflects uncertainty and errors in measurement. High measurement errors are associated with small values of (think about this: if there are high measurement errors, then the level of an agents 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)=E 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 = 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 firms 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 doesnt the firm prefer to use this prize spread?

g. Now change the firms measurement technology to = 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 firms profit now to those when was half as large, at .025. What has happened to the base salary? Explain.