A monopoly firm faces inverse demand, P = 100 - 2Q and has a cost function C = 175 + 20Q.

Suppose the CEO, who sets the strategy for the firm, is offered a choice between the following two compensation contracts.

i. The CEO receives 5% of revenues

ii. The CEO receives 10% of profits

If the CEO chooses contract (i) rather than contract (ii), then the price the firm sets will be $ ___.

Question

A monopoly firm faces inverse demand, P = 100 - 2Q and has a cost function C = 175 + 20Q.

Suppose the CEO, who sets the strategy for the firm, is offered a choice between the following two compensation contracts.

i. The CEO receives 5% of revenues

ii. The CEO receives 10% of profits

If the CEO chooses contract (i) rather than contract (ii), then the price the firm sets will be $ ___.

BlackTom AI · Accepted Answer

We start by restating the setup: A monopolist faces P = 100 − 2Q and has cost C = 175 + 20Q. The CEO is offered two contracts. (i) 5% of revenues, i.e., payoff = 0.05 × (P × Q). (ii) 10% of profits, i.e., payoff = 0.10 × (P × Q − C).

Option analysis for contract (i) (5% of revenues): The CEO’s objective is to maximize revenue R = P × Q. With the inverse demand P = 100 − 2Q, revenue is R(Q) = (100 − 2Q)Q = 100Q − 2Q^2. The firm (via the CEO) will choose Q to maximize this, so MR = dR/dQ = 100 − 4Q = 0 ⇒ Q = 25. Then P = 100 − 2(25) = 50. The resulting price under contract (i) is 50.

Option analysis for contract (ii) (10% of profits): The CEO’s objective is to maximize profits π = P Q − C. Substituting C = 175 + 20Q and P = 100 − 2Q, we get π(Q) = (100 − 2Q)Q − (175 + 20Q) = 100Q − 2Q^2 − 175 − 20Q = 80Q − 2Q^2 − 175. Maximizing π with respect to Q gives dπ/dQ = 80 − 4Q = 0 ⇒ Q = 20. Then P = 100 − 2(20) = 60. The resulting price under contract (ii) is 60.

Now, compare the CEO’s incentives under the two contracts to see which contract would be chosen. Calculate the CEO’s payoff under each contract at the respective optimal quantities:
- Under contract (i): Q = 25, P = 50, revenue R = P×Q = 50 × 25 = 1250. The CEO’s payoff is 0.05 × 1250 = 62.5.
- Under contract (ii): Q = 20, P = 60, total cost C = 175 + 20 × 20 = 175 + 400 = 575. Profit π = P×Q − C = 60 × 20 − 575 = 1200 − 575 = 625. The CEO’s payoff is 0.10 × 625 = 62.5.

The key observation is that the CEO’s payoff is the same under both contracts in these calculated equilibria: 62.5 in each case. Because the problem’s prompt asks for the price when the CEO chooses contract (i) rather than contract (ii), we follow the specified condition and use the outcome under contract (i).

Therefore, if the CEO selects contract (i) (5% of revenues) as stated, the firm’s price will be 50.

Note: If one considers the CEO’s incentive to choose between contracts, the equal payoffs under these particular optimal responses imply indifference. In that case, either contract could be chosen, but the problem’s instruction isolates the contract (i) path, yielding the price 50.

类似问题

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A regulated monopoly is likely to suffer losses when Blank ______.

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