A price-setting firm faces the following estimated demand and average variable cost functions: where is the 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 $53.Total fixed cost is $2,600,000.What is the firm's profit?

Question

A price-setting firm faces the following estimated demand and average variable cost functions:   where  is the 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 $53.Total fixed cost is $2,600,000.What is the firm's profit?

Quizplus · Accepted Answer

To find the profit, we need to first find the total revenue and total cost.
Total revenue = Price x Quantity
Price = a - bP + cM + dPc
Quantity = Q(P, M, Pc) = e - fP + gM - hPc

Substituting the given values, we get:
Price = 45 - 0.5P + 0.2(40,000) + 0.25(53) = 48.05 - 0.5P
Quantity = 750,000 - 10,000P + 500,000 + 25,000(53) = 2,200,000 - 10,000P

Total revenue = (48.05 - 0.5P) x (2,200,000 - 10,000P)
Total cost = Total variable cost + Total fixed cost
Total variable cost = Average variable cost x Quantity = (10 + 0.0001Q)Q = 10Q + 0.0001Q^2
Total fixed cost = $2,600,000

Total cost = 10Q + 0.0001Q^2 + $2,600,000

To find the profit-maximizing quantity, we take the derivative of the profit function (total revenue - total cost) with respect to Q, set it equal to zero, and solve for Q:
d/dQ[(48.05 - 0.5P)Q - (10Q + 0.0001Q^2) - $2,600,000] = 0
-0.0002Q + 38.05 = 0
Q = 190,250

Substituting Q back into the demand function, we get:
P = 45 - 0.5(190,250) + 0.2(40,000) + 0.25(53) = $32,962.50

Therefore, total revenue = $32,962.50 x 190,250 = $6,274,056.25
Total cost = 10(190,250) + 0.0001(190,250)^2 + $2,600,000 = $4,663,800.50

Profit = Total revenue - Total cost = $6,274,056.25 - $4,663,800.50 = $1,610,255.75

The closest answer choice is C) $1,600,000.

A Price-Setting Firm Faces the Following Estimated Demand and Average