## Calculate the optimal lifetime incomes and firms, Managerial Economics

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Dan and Ann are chemical engineers working for a biotech company. Each of them would like to be promoted to a managerial position, but only one of them can get the job. Their supervisors have indicated that the one who produces more publications in scientific journals over the next three years will be promoted. Each published article increases the firm's revenue by \$10 (take the numbers to be in 1,000s).

The number of articles each can produce depends on how hard they work. If y is the number of published articles at the end of the third year, then yA = 0.5eA + _A and yD = 0.5eD + _D, where e represents effort and _ is a luck factor over which the researchers have no control. _A and _D are distributed such that (_A - _D) is uniform on [-½, ½]. Both Dan and Ann are risk-neutral and have disutility of effort given by C(e) = 2e2.

Let w0 = 0 be the wage each of the two gets during the three years before the promotion decision, W+ the lifetime income of a manager and W- the lifetime income of the employee who was not promoted. The firm wants to set the wages so as to maximize profit. Dan and Ann are willing to engage in the promotion contest if their expected lifetime utility is at least zero.

Calculate the optimal lifetime incomes W+ and W- the firm will promise the employees. What are the firm's expected profits from hiring Dan and Ann?

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