ANSWER: See the separate Latex File.
(a) Prove that high effort is first-best efficient.
ANSWER. You would pay for high effort if you could guarantee it. Profit would then be .8(200) + .2 (0) - 70 = 90. Low effort only yields .6(200) + .4(0) -50 = 70.
(b) Explain why high effort would be inefficient if the probability of success when effort is low were .75.
ANSWER. Profit would then be .8(200) + .2 (0) - 70 = 90 with high effort. Low effort yields .75(200) + .4(0) -50 = 100, which is better. With low effort, the employer can be better off with the worker being no worse off.
(c) Let the probability of success with low effort go back to .6 for the remainder of the problem. If you cannot monitor the programmer and cannot pay him a wage contingent on success, what should you do?
ANSWER. Pay him 50 to get him to work for you, and expect 70 in profit.
(d) Now suppose you can make the wage contingent on success. Let the wage be S if Wizard is successful, and F if it fails. S and F will have to satisfy two conditions: a participation constraint and an effort-choice constraint. What are they?
ANSWER. The participation constraint is that .8S + .2F > = 70. (where I am using > = to mean ``greater than or equal to'') If this is true, then the programmer will be willing to participate in the development.
The effort-choice constraint is .8S + .2F > = .6S + .4F + 20, where the 20 represents the extra payoff the programmer would get by slacking off. If this inequality is true, then the programmer has no incentive to slack off, because the payoff of .8S +. 2F that results from high effort is as big as the payoff of .6S + .4F + 20 that results from low effort.
(e) What is a contract that will achieve the first best?
ANSWER.We can rewrite the effort-choice constraint as .2 S - .2F > = 20, so S -F > = 100 and S > = 100+F.
Substitute S= 100+F into the participation constraint, and we get .8 (100+F) + .2F > = 70. Making that an equality (so we pay the minimum possible), 80 +.8F + .2F = 70, so F = -10. We will give the programmer a negative wage if the product fails.
We still have to satisfy the participation constraint, though, so we need to pay a generous wage S if the product succeeds. .8S + .2 (-10) = 70, so it must be that .8S = 72, and S =90.
These two constraints show the big problem in incentive contracts: getting the other side to participate, and getting them to exert the right effort. Much of the structure of wages and salaries can be explained by these two problems.
(f) What is the optimal contract if you cannot pay a programmer a negative wage?
ANSWER. Efficiency wage. Pay him S=100 and F=0. Your profit will be 80, compared to 70 with low effort.
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