Dr. Jeraisy, a well-known plastic surgeon, has a reputation for being one of the best surgeons for reconstructive nose surgery. Dr. Jeraisy enjoys a rather substantial degree of market power in this market. She has estimated demand for her work to be:
Q = 480 – 0.2P
Where Q is the number of nose operations performed monthly and P the price of a nose operation.
• What is the inverse demand function for Dr. Jeraisy’s services?
• What is the marginal revenue function?
The average variable cost function for reconstructive nose surgery is estimated to be: AVC = 2Q2 – 15Q + 400
Where AVC is average variable cost (measured in dollars) and Q is the number of operations per month. The doctor’s fixed costs each month are US$8000
• If the doctor wishes to maximize her profit, how many nose operations should she perform each month?
• What price should Dr. Jeraisy charge to perform a nose operation?
• How much profit does she earn each month?
To find the inverse demand function, we need to solve the demand equation for P.
Given: Q = 480 - 0.2P
Rearranging the equation, we get:
0.2P = 480 - Q
Dividing both sides by 0.2, we have:
P = (480 - Q)/0.2
So the inverse demand function for Dr. Jeraisy's services is:
P = 2400 - 5Q
To find the marginal revenue function, we differentiate the inverse demand function with respect to Q.
dP/dQ = -5
Therefore, the marginal revenue function is:
MR = -5
To maximize profit, a firm should produce where marginal revenue (MR) is equal to marginal cost (MC). In this case, the marginal cost is the derivative of the average variable cost function.
AVC = 2Q^2 - 15Q + 400
Taking the derivative of the average variable cost function:
dAVC/dQ = 4Q - 15
To find the profit-maximizing quantity, set MR equal to MC:
-5 = 4Q - 15
Solving for Q, we get:
4Q = 10
Q = 2.5
Since we cannot perform fractional operations, Dr. Jeraisy should perform 2 nose operations per month to maximize profit.
To determine the price she should charge, substitute the optimal quantity (Q = 2) into the inverse demand function:
P = 2400 - 5Q
P = 2400 - 5(2)
P = 2400 - 10
P = 2390
Therefore, Dr. Jeraisy should charge a price of $2390 per nose operation.
To calculate the monthly profit, we need to subtract the total cost from the total revenue.
Total revenue = price x quantity
Total revenue = $2390 x 2
Total revenue = $4780
Total cost consists of fixed costs and variable costs.
Fixed costs = $8000
Variable costs can be determined by substituting the quantity (Q = 2) into the average variable cost function:
AVC = 2Q^2 - 15Q + 400
AVC = 2(2)^2 - 15(2) + 400
AVC = 8 - 30 + 400
AVC = 378
Total cost = fixed costs + variable costs
Total cost = $8000 + $378
Total cost = $8378
Profit = Total revenue - Total cost
Profit = $4780 - $8378
Profit = -$3598
Hence, Dr. Jeraisy incurs a loss of $3598 per month.