Assume a monopolist with the following demand and cost relationships.
Q = 400 - 20p
TC = 10 + 5q + q2
Calculate the following:
Profit max price
Profit max quantity
TR, TC, Profit, and the elasticity at profit max q and p.
To calculate the profit-maximizing price and quantity for a monopolist, we need to find the point where marginal revenue (MR) equals marginal cost (MC).
Step 1: Find the marginal revenue (MR).
The marginal revenue is the additional revenue earned from selling one more unit of the product. In this case, the monopolist's demand function is given as Q = 400 - 20p. To find MR, we need to differentiate the demand function with respect to quantity (Q) to obtain the inverse demand function in terms of price (p).
dQ/dp = -20
Solving for dp, we get dp = -1/20 * dQ
Since MR = dTR/dQ, we can substitute -1/20 * dQ for dp in the MR equation:
MR = dTR/dQ = (-1/20 * dQ) * p = -(1/20) * Q
Step 2: Find the marginal cost (MC).
The marginal cost is the additional cost incurred from producing one more unit of the product. The cost function is TC = 10 + 5Q + Q^2. To find MC, we differentiate the cost function with respect to quantity (Q).
dTC/dQ = 5 + 2Q
Step 3: Set MR equal to MC and solve for quantity.
MR = MC
-(1/20) * Q = 5 + 2Q
Multiply both sides by 20 to eliminate the fraction:
-Q = 100 + 40Q
Combine like terms:
41Q = -100
Divide by 41:
Q ≈ -2.44 (ignore negative)
The profit-maximizing quantity is approximately 2.44 units.
Step 4: Substitute the quantity obtained into the demand function to find the price.
Q = 400 - 20p
2.44 = 400 - 20p
20p = 400 - 2.44
20p ≈ 397.56
p ≈ 19.88
The profit-maximizing price is approximately $19.88.
Step 5: Calculate Total Revenue (TR), Total Cost (TC), and Profit.
TR = p * Q
TR = 19.88 * 2.44
To calculate TC, substitute the profit-maximizing quantity (Q ≈ 2.44) into the cost function (TC = 10 + 5Q + Q^2):
TC = 10 + 5(2.44) + (2.44)^2
Profit = TR - TC
To calculate the elasticity at the profit-maximizing quantity (Q ≈ 2.44) and price (p ≈ $19.88), we need the formula for elasticity:
Elasticity = (dQ / Q) / (dp / p)
To calculate dQ / Q, differentiate the demand function with respect to Q (dQ/dQ = 1):
dQ / Q ≈ 1
To calculate dp / p, differentiate the price equation with respect to p (dp/dp = 1):
dp / p ≈ 1 / 19.88
Plug in the values into the elasticity formula to get the numerical value.