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# unconstrained profit-maximizing level of price and output

Specific Motors Corporation is one of the Big Three auto manufacturers in Transylvania. Specific's share of the domestic auto market is 55 percent. The next two closest competitors control 25 and 15 percent of the market, respectively, and the rest may be accounted for by two small, specialized firms. Specific has been under pressure from Transylvania's Justice Department and the State Trade Commission for monopolistic practices. To discourage any attempts to break up Specific, management has decided to maintain its market share below 55 percent of the total domestic automobile sales revenues.

Specific estimates that to stay within its constraint sales of 55 percent of the market, its total shares should not exceed \$2.8 billion.

The firm faces the following demand and cost functions:
P=16,000 - .02Q
TC=850,000,000 + 4,000 Q

a. Calculate the unconstrained profit-maximizing level of price and output for Specific.
b. At this level, what will total sales revenue be? Total Profits?
c. If the firm constrains its sales revenue to \$2.8 billion, calculate price, output, and profit levels under the constraint.
d. What is the cost to the firm of this market-share constraint?

#### Solution Preview

Hello!

Question a
First of all, let's write the profit function for Specific:

Profit = P*Q - TC = (16000 - 0.02Q)*Q - 850000000 - 4000Q
Profit = 16000Q - 0.02Q^2 - 850000000 - 4000Q

We now have to find the Q that maximizes this equation. Here, we sue the regular procedure: we find the first derivative of Profit with respect to Q and equate it to zero:

Profit' = 16000 - 0.04Q - 4000 = 0

Isolating Q, we get:

0.04Q = ...

#### Solution Summary

This job determines unconstrained profit-maximizing level of price and output.

\$2.19