A company is a monopolist.
The demand function for its product is as follows:
Q = 60 – 0.4P + 6Y + 2A
Where Q = quantity sold in units
P = Price per unit
Y = per capita disposal income (thousands of dollars)
A = hundreds of dollars of advertising expenditures
The firm’s average variable cost function is
AVC = Q2 – 10Q + 60
Y = 3 (thousand) and
A = 3 (hundred) for the period being analyzed.
A. If fixed costs are equal to $1,000, derive the firm’s total cost function and marginal cost function.
C(q) = fixed costs + variable cost
B. Derive a total revenue function and marginal revenue function for the firm.
C. Calculate the profit maximizing level of output and price for the firm.
D. What will the profit be?
A. To derive the firm's total cost function, we need to add the fixed costs to the variable cost function.
Total Cost (TC) = Fixed Costs + AVC
Given that fixed costs are equal to $1,000, and AVC = Q^2 - 10Q + 60, the total cost function can be derived as:
TC = $1,000 + Q^2 - 10Q + 60
To find the marginal cost (MC), we differentiate the total cost function with respect to quantity (Q).
MC = dTC/dQ
Differentiating the total cost function, we get:
MC = 2Q - 10
Therefore, the firm's total cost function is TC = $1,000 + Q^2 - 10Q + 60, and the marginal cost function is MC = 2Q - 10.
B. To derive the firm's total revenue function, we multiply the quantity sold (Q) by the price per unit (P). Since we are looking for the revenue function and its derivative, we need to isolate the price variable.
Given the demand function Q = 60 - 0.4P + 6Y + 2A, we rearrange it to solve for P:
Q - 60 = -0.4P + 6Y + 2A
-0.4P = Q - 60 - 6Y - 2A
P = (Q - 60 - 6Y - 2A) / -0.4
Now we can express total revenue (TR) as Q * P:
TR = Q * P
Substituting the expression for P, we get:
TR = Q * (Q - 60 - 6Y - 2A) / -0.4
To find the marginal revenue (MR), we differentiate the total revenue function with respect to quantity (Q).
MR = dTR/dQ
Differentiating the total revenue function, we get:
MR = -0.8Q + 60 + 6Y + 2A
Therefore, the firm's total revenue function is TR = Q * (Q - 60 - 6Y - 2A) / -0.4, and the marginal revenue function is MR = -0.8Q + 60 + 6Y + 2A.
C. The profit-maximizing level of output and price can be determined by setting marginal cost (MC) equal to marginal revenue (MR) and solving for Q.
MC = MR
2Q - 10 = -0.8Q + 60 + 6Y + 2A
Simplifying the equation, we have:
2Q + 0.8Q = 70 + 6Y + 2A
2.8Q = 70 + 6Y + 2A
Q = (70 + 6Y + 2A) / 2.8
To find the corresponding price (P), substitute the value of Q into the demand function:
P = 60 - 0.4Q + 6Y + 2A
P = 60 - 0.4 * [(70 + 6Y + 2A) / 2.8] + 6Y + 2A
Simplifying the equation will give you the price.
D. To calculate the profit, we subtract the total cost (TC) from the total revenue (TR) at the profit-maximizing level of output (Q).
Profit = TR - TC
Substituting the profit-maximizing level of output Q into both the total revenue and total cost functions, we can calculate the profit.