Question: A small firm faces an inverse demand function of P = 180 - 2Q. Its total cost function is given by TC = Q 2

A small firm faces an inverse demand function of P = 180 - 2Q. Its total cost function is given by TC = Q². (You should see right away that marginal revenue is thus MR = 180 - 4Q, and it also happens that marginal cost is MC = 2Q. MR and MC are the first derivatives of total revenue and total cost, respectively. And a quick comment on MC: unlike some marginal cost functions we've seen, this one is not constant, because marginal cost is getting $2 higher with each additional unit of output.)

The Chief Executive Officer will manage the firm, choosing output and price. Currently, the CEO is negotiating an incentive-based contract with the shareholders of the company. The CEO has proposed that she get 30% of the total revenue brought in by the firm. The shareholders' representative has counter-offered that 20% of total revenue be given to the CEO. (Hint: basing compensation on revenue will motivate revenue maximization rather than profit maximization!)

1. How much income will each plan generate for the CEO and for the shareholders, respectively? (Hint: since both plans create incentives for the CEO to maximize revenue rather than profit, you should not set MR = MC at this point. BIG hint: revenue is maximized when selling an additional unit won't increase your revenue, or in math terms, when MR = 0.)

Find each of the following if the firm maximizes its total revenue:

Firm price:
Firm output:
Total revenue:
Total cost:
Firm profit:

If the CEO's proposal is adopted and she keeps 30% of revenue, then...

CEO compensation:
Remaining profit for owners:

If the shareholders' proposal is adopted and the CEO keeps 20% of revenue, then...

CEO compensation:
Remaining profit for owners:

2. Suppose you are asked to mediate in the negotiations.Can you propose an incentive-based compensation scheme for the CEO that both parties are likely to accept, assuming everyone would like to maximize their income?

Your proposal:

Demonstration that everyone is better off than under their own proposal and thus should accept your proposal:

Firm price:
Firm output:
Total revenue:
Firm profit:
CEO compensation:
Remaining profit for owners:

3. Now thinking even more shrewdly: what's the maximum price you could charge for your consulting services and still leave everyone better off?

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