### Stephan disutility of effort curve

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1. Stefan has just been hired as a consultant to advise the campus construction company on how to cut its costs. The compensation scheme offered by CC Inc. is such that Stephan's pay depends on his output, i.e. on the total dollars saved by CC after following Stephan's advice. Stephan's output, Q, (in dollars) depends on the amount of effort he exerts, E, in a very simple way: Q = E. Stephan's pay, Y, is a linear function of his output, i.e. Y = a +bE. In other words, Stephan gets paid 'a' just for taking the assignment, and in addition is paid 'b' dollars for every dollar he saves CC Inc. Thus, b is his commission rate.

Stephan's utility (U) depends on two things: income and effort. In particular, U = Y - C(E), where C(E) is the disutility of effort. For this question assume C(E) = E3/3.

A) Graph Stephan's disutility of effort curve, (C(E). Graph his marginal disutility of effort curve, dC/dE. Does he exhibit increasing or decreasing marginal disutility of effort?

B) Assuming Stephan is a utility maximizer, derive his utility maximizing effort (and output) level as a function of the parameters of his pay scheme, a and b. How much effort will he choose if:

b = .36? If b = 1? Does Stephan's choice depend on the level of 'a'.

C) Illustrate your answer graphically. In one diagram, show Stephan's total level of income (Y) and disutility of effort (C(E)). Directly below this, show his optimal choice using marginal benefit and cost curves.

2. Now assume that CC Inc. is hiring Marie as a consultant to computerize its office operations. Everything is the same in Marie's situation as in Stephan's except that her disutility of effort is different. Specifically C(E) = E2 / 10.

A) How much effort will Marie exert if her commission rate is 20%? 100%? Does effort depend on 'a'?

B) Now suppose that Marie and CC are negotiating the terms of her contract (i.e. the level of 'a' and 'b') before she starts.  Before they settle on who gets exactly what in the end, they agree that, whatever contract they ultimately choose, it must be efficient, i.e. it must maximize the sum of CC's profits and Marie's utility. If the contract is efficient, what will be the level of 'b'? Comment.

C) Suppose now that to retain Marie, CC Inc. must offer her a utility level of at least 1 unit (because this is what she can get working for another firm). What must be 'a'? If Marie's alternative utility is 5 units, will she work for CC Inc.?

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