HELP!!!!!
One and only Inc is a monopolist. The demand function for its product is estimated to be Q=60-0.4P +6Y+2A
Y=3,000
P=Price per Unit
Y=Per capita disposable personal income (thousands of dollars)
A=hundreds of dollars of advertising expenses
The Firms average variable cost function is AVC=Q²-10Q+60
Y is equal to 3(thousand) and A is equal to 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.
b.) Derive a total revenue function and a marginal revenue function for the firm.
c.) Calculate the profit maximizing level of price and output for One and Only Inc.
d.) What profit and loss will One and Only earn?
e.) If fixed costs were $1,200, how would your answers change for parts (a) through (d)?
a.) To derive the firm's total cost function, we need to add fixed costs (FC) to average variable costs (AVC). The formula is:
Total cost (TC) = Fixed costs (FC) + Average variable costs (AVC)
Given that fixed costs are equal to $1,000, the total cost function becomes:
TC = $1,000 + (Q² - 10Q + 60)
To find the marginal cost (MC), we need to take the derivative of the total cost function with respect to quantity (Q):
MC = d(TC)/dQ = d/dQ($1,000 + Q² - 10Q + 60)
MC = 2Q - 10
b.) To derive the total revenue (TR) function, we need to multiply the quantity (Q) by the price (P). The formula is:
TR = Q * P
To find the marginal revenue (MR), we need to take the derivative of the total revenue function with respect to quantity (Q):
MR = d(TR)/dQ = d/dQ(Q * P)
MR = P
c.) To find the profit-maximizing level of price and output, we need to equate marginal cost (MC) to marginal revenue (MR) and solve for Q:
MC = MR
2Q - 10 = P
2Q = P + 10
Q = (P + 10)/2
Substituting the expression for Q into the demand function:
Q = 60 - 0.4P + 6Y + 2A
(P + 10)/2 = 60 - 0.4P + 6(3) + 2(3)
(P + 10)/2 = 60 - 0.4P + 18 + 6
(P + 10)/2 = 84 - 0.4P
P + 10 = 168 - 0.8P
1.8P = 158
P ≈ 87.78
Therefore, the profit-maximizing level of price is approximately $87.78. To find the corresponding quantity, substitute this value back into the demand function:
Q = 60 - 0.4(87.78) + 6(3) + 2(3)
Q ≈ 36.11
Thus, the profit-maximizing level of output is approximately 36.11 units.
d.) To calculate the profit, we need to subtract the total cost from the total revenue:
Profit = Total revenue - Total cost
Profit = (Q * P) - (FC + AVC)
Using the values we determined, the profit is:
Profit = (36.11 * 87.78) - (1,000 + (36.11² - 10 * 36.11 + 60))
e.) If fixed costs were $1,200, the total cost function becomes:
TC = $1,200 + (Q² - 10Q + 60)
Repeating the calculations from parts a), b), c), and d) with this new total cost function would yield the updated results.