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.
1-What is the inverse demand function for Dr. Jeraisy’s services?
2-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
3-If the doctor wishes to maximize her profit, how many nose operations should she perform each month?
4-What price should Dr. Jeraisy charge to perform a nose operation?
5-How much profit does she earn each month?
Q = 20
1- To find the inverse demand function, we need to solve the demand equation for P, the price of a nose operation. The given demand equation is:
Q = 480 – 0.2P
Let's solve it for P:
Q = 480 – 0.2P
Rearranging the equation:
0.2P = 480 – Q
Divide both sides by 0.2:
P = (480 – Q) / 0.2
Therefore, the inverse demand function for Dr. Jeraisy's services is:
P = 2400 – 5Q
2- The marginal revenue (MR) function is given by the derivative of the total revenue (TR) function with respect to quantity (Q). Since total revenue is the product of quantity and price (TR = Q * P), we can find the MR function by finding the derivative of the total revenue function with respect to Q.
TR = Q * P
Taking the derivative of TR with respect to Q:
MR = d(TR) / d(Q)
MR = d(Q * P) / d(Q)
MR = P + Q * d(P) / d(Q)
Since we have the inverse demand function P = 2400 - 5Q, we can substitute it into the MR equation:
MR = (2400 - 5Q) + Q * (-5)
MR = 2400 - 5Q - 5Q
MR = 2400 - 10Q
Therefore, the marginal revenue function is:
MR = 2400 - 10Q
3- To maximize profit, the doctor should produce where marginal revenue (MR) equals marginal cost (MC).
We can find the marginal cost function by taking the derivative of the average variable cost (AVC) function:
AVC = 2Q^2 – 15Q + 400
MC = d(AVC) / d(Q)
MC = d(2Q^2 – 15Q + 400) / d(Q)
MC = 4Q - 15
Setting MR equal to MC:
2400 - 10Q = 4Q - 15
10Q + 4Q = 2400 + 15
14Q = 2415
Q = 2415 / 14
Hence, the doctor should perform approximately 172.50 nose operations each month.
4- To find the price (P), we can substitute the quantity (Q) we found in step 3 into the inverse demand function:
P = 2400 - 5Q
P = 2400 - 5 * 172.50
P = 2400 - 862.50
Therefore, the price Dr. Jeraisy should charge per nose operation is approximately $1,537.50.
5- To calculate the profit, we need to subtract the total cost from the total revenue.
Total revenue = Q * P
Total cost = AVC * Q + fixed cost
Total revenue = (172.50) * (1537.50)
Total cost = (2 * (172.50)^2) - (15 * 172.50) + 400 * 172.50 + 8000
Profit = Total revenue - Total cost
Calculate the values:
Total revenue = 265,062.50
Total cost = 426,787.50
Profit = -161,725.00
Therefore, Dr. Jeraisy earns a negative profit of $161,725.00 each month.