A small firm faces an inverse demand function of P = 100 - Q. Its total cost fun
ID: 1174410 • Letter: A
Question
A small firm faces an inverse demand function of P = 100 - Q. Its total cost function is given by TC = .5Q2. (You should see right away that marginal revenue is thus MR = 100 – 2Q, and it also happens that marginal cost is just MC = Q. Both MR and MC are the first derivatives of total revenue and total cost. And a quick comment on MC: unlike some marginal cost functions we’ve seen, this one is not constant, because marginal cost is getting $1 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.
(Hint: basing compensation on revenue will motivate revenue maximization rather than profit maximization!)
Hint: since the plan 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.)
Owners’ proposal: CEO keeps 10% of TR.
Firm price:
Firm output:
Total revenue:
Firm profit:
CEO compensation:
Remaining profit for owners:
Firm price:
Firm output:
Total revenue:
Firm profit:
CEO compensation:
Remaining profit for owners:
Explanation / Answer
Given P = 100 - Q, TC = 0.5Q2 ,
MR = 100 - 2Q, MC = Q.
Here, CEO is having a proposal to maximize revenue.
MR = 0
100 - 2Q = 0
2Q = 100
Q* = 50 units
P* = $ 50/unit (=100 - 50)
Total revenue = PQ = $ 2500 (= 50*50)
Firms profit = TR - TC = $ 2500 - 0.5*502 = $ 2500 - 1250
Firms profit = $ 1250
CEO compensation = 10% of TR = $ 250 (=10% of $ 2500)
Remaining profit for owners = $ 1000 (=$2500 - 1250 -250)
(Note: TR - TC - CEO compensation )