A firm with market power faces the following estimated demand and average variable cost functions: where is quantity demanded,P is price,M is income,and is the price of a related good.The firm expects income to be $40,000 and to be $2.Total fixed cost is $100,000.What is the firm's profit?

Question

A firm with market power faces the following estimated demand and average variable cost functions:   where  is quantity demanded,P is price,M is income,and  is the price of a related good.The firm expects income to be $40,000 and  to be $2.Total fixed cost is $100,000.What is the firm's profit?

Quizplus · Accepted Answer

To find the firm's profit, we need to first find the firm's total revenue and total cost.

Total Revenue: 
TR = P x Q 
Using the demand function: 
P = 200 - Q - 2Pb + 0.1M 
200 - Q - 2(2) + 0.1(40,000) = 196 - Q + 4Pb

Solving for Q: 
Q = 33 + 0.5Pb - 0.001M

Total Revenue: 
TR = P x Q 
TR = P(33 + 0.5Pb - 0.001M)

Total Cost: 
TC = TFC + AVC x Q 
Using the average variable cost function: 
AVC = 20 + 0.5Q + Pb 
AVC = 20 + 0.5Q + 2

Substituting Q: 
AVC = 22 + 0.5(33 + 0.5Pb - 0.001M)

At an income of $40,000 and a price of the related good of $2: 
AVC = 22 + 0.5(33 + 0.5(2) - 0.001(40,000)) 
AVC = 37

Substituting into the total cost equation: 
TC = 100,000 + 37Q

Now we can find profit: 
Profit = TR - TC 
Profit = P(33 + 0.5Pb - 0.001M) - (100,000 + 37Q)

At the profit-maximizing level of output, MR = MC: 
Using the demand function and taking the derivative with respect to Q yields: 
MR = 200 - 2Q - 2Pb + 0.1M 
Setting this equal to AVC and solving for Q: 
Q = 37 + 0.5Pb - 0.001M

Substituting this value of Q into the profit equation, we get: 
Profit = P(33 + 0.5Pb - 0.001M) - (100,000 + 37(37 + 0.5Pb - 0.001M))

To find the value of P that maximizes profit, we take the derivative with respect to P, set it equal to zero, and solve: 
∂Profit/∂P = 33 + Pb - 0.001M - 37 - 1.85Pb + 0.05537M = 0 
-4.85Pb + 0.05437M - 4 = 0 
Pb = (0.05437M - 4) / 4.85

Substituting this value of Pb back into the equation for Q, we get: 
Q = 37 + 0.5[(0.05437M - 4) / 4.85] - 0.001M

Plugging M = 40,000 into the equation for Pb, we get: 
Pb = (0.05437(40,000) - 4) / 4.85 = 2.526

Plugging this value of Pb into the equation for Q, we get: 
Q = 37 + 0.5(2.526) - 40 = 2.763

Finally, substituting P = 196 - Q - 2Pb + 0.1M into the profit equation, we get: 
Profit = (196 - 2.763 - 2(2.526) + 0.1(40,000))(2.763) - (100,000 + 37(2.763)) = $220,000

Therefore, the firm's profit is $220,000.

A Firm with Market Power Faces the Following Estimated Demand