A market (or industry) demand curve is described by
Q = 50 – 0.5 P
The firm’s cost function is
TC = 10 + 2 Q
a. Find the profit-maximizing quantity and price.
b. If the industry is regulated in a way that requires it to set P = AC, how much will be sold and what will the price be?
c. If the industry is regulated in a way that requires it to set P = MC, how much will be sold and what will the price be?
To find the profit-maximizing quantity and price, we need to determine the point where the firm's profit is maximized. This occurs when the firm's total revenue (TR) minus its total cost (TC) is highest.
a. Profit is calculated as: Profit = TR - TC
Revenue is calculated as: TR = P * Q
Cost is given by the function: TC = 10 + 2Q
Substituting the given demand curve into the revenue equation, we get: TR = P * Q = (50 - 0.5P) * Q = 50Q - 0.5PQ
Combining the equations for revenue and cost, we have: Profit = TR - TC = (50Q - 0.5PQ) - (10 + 2Q)
Simplifying the expression: Profit = 50Q - 0.5PQ - 10 - 2Q = 48Q - 0.5PQ - 10
To maximize profit, we need to find the values of Q and P that maximize this equation. To do this, we take the partial derivative of the profit function with respect to Q and set it equal to zero:
∂Profit/∂Q = 48 - 0.5P = 0
Solving for P, we find: P = 48 / 0.5 = 96
Then, substitute this value of P back into the demand equation to find Q:
Q = 50 - 0.5P = 50 - 0.5 * 96 = 50 - 48 = 2
Therefore, the profit-maximizing quantity is Q = 2, and the price is P = 96.
b. If the industry is regulated in a way that requires P = AC (average cost), we need to set the price equal to the average cost function to find the quantity and price.
The average cost (AC) is calculated as: AC = TC / Q = (10 + 2Q) / Q = 10/Q + 2
Setting P equal to AC, we have: 10/Q + 2 = P = 96
Solving for Q, we find: 10/Q = 96 - 2 = 94, or Q = 10/94
Substituting this value of Q back into the demand equation to find the price:
P = 50 - 0.5(10/94) = 50 - 0.053 = 49.947
Therefore, if the industry is regulated to set P = AC, the quantity sold would be Q = 10/94 and the price would be P = 49.947.
c. If the industry is regulated to set P = MC (marginal cost), we need to find the quantity and price where the marginal cost equals the price.
The marginal cost (MC) is given by the derivative of the total cost function with respect to quantity: MC = dTC/dQ = 2
Setting MC equal to P, we have: 2 = P = 96
Substituting this value of P into the demand equation to find the quantity:
Q = 50 - 0.5P = 50 - 0.5 * 96 = 50 - 48 = 2
Therefore, if the industry is regulated to set P = MC, the quantity sold would be Q = 2 and the price would be P = 96.