Exercise 5Exercise 5: Promotions and Tournaments1. 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 â€œrelative luckâ€ 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 agentâ€™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 ofeffort 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 ?? Explain. Does agent 1â€™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 2â€™s optimal effort depend on agent 1â€™s effort choice?d) Suppose ? = 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 ? = 0.025 and d = 25 (these are â€œbase-caseâ€ 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 workersâ€™ 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 firmâ€™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 doesnâ€™t the firm prefer to use this prize spread?g) Now change the firmâ€™s 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 firmâ€™s profit now to those when ? was half as large, at .025. What has happened to the base salary? Explain.Exercise 5Prize Spread P Optimal Effort E* Base Salary A Expected Utility E(U) Output Q Expected Profit E(n) 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000Exercise 53. 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 1â€™s effort2 is given by E1 and agent 2â€™s by E2. Each agentâ€™s disutilityof 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 agentsâ€™ 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 agentâ€™s output be given by 10E (where E is the agentâ€™s effort). Now compute the firmâ€™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.
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