Q4. Dr. Shakeel, a well-known orthopedics, with a reputation of being one of the best
surgeons in the Maldives, enjoys a substantial degree of market power and has fixed costs
per month at MVR8000. He has estimated his demand curve and average variable cost
function per month to be as follow:
Q = 480 – 0.2P
AVC = 2Q2 – 15Q +400
a) If the doctor wishes to maximize his profits, how many operations should he perform each
month and at what price? How much profit does he earns in each month?
A4. Well, it sounds like Dr. Shakeel has quite the business going on! Let's crunch some numbers and find out the best course of action.
To maximize profits, we need to find the quantity and price that will give the doctor the highest profit. We can do that by finding the point where marginal revenue (MR) equals marginal cost (MC).
To find the MR, we can take the derivative of the demand curve with respect to quantity:
MR = d(480 - 0.2P)/dQ = -0.2
Now that we have the MR, let's find the MC by taking the derivative of the AVC function with respect to quantity:
MC = d(2Q^2 - 15Q + 400)/dQ = 4Q - 15
Setting MR equal to MC, we can solve for Q:
-0.2 = 4Q - 15
4Q = 14.8
Q ≈ 3.7 (let's say 4 operations)
Now, let's substitute this value of Q back into the demand curve to find the corresponding price:
Q = 480 - 0.2P
4 = 480 - 0.2P
0.2P = 476
P ≈ MVR 2380 (I hope that's not an arm and a leg!)
To find the profit, we subtract the total cost (TC) from total revenue (TR). TC is the sum of fixed costs (FC) and variable costs (VC):
TC = FC + VC
VC = AVC x Q = (2Q^2 - 15Q + 400) x Q = 8Q^2 - 60Q + 1600
FC = MVR 8000 (fixed costs per month)
TR = PQ = 2380 x 4 = MVR 9520
TC = 8000 + (8 x 4^2 - 60 x 4 + 1600) = MVR 8640
Profit = TR - TC = 9520 - 8640 = MVR 880
So, Dr. Shakeel should perform 4 operations each month and charge MVR 2380 for each operation. With that, he will earn a profit of MVR 880 per month. Quite a lucrative business, I must say!
To maximize profits, the doctor should determine the level of output where marginal revenue (MR) equals marginal cost (MC). The MR can be computed by finding the derivative of the demand equation, and the MC can be calculated by finding the derivative of the average variable cost (AVC) equation.
Given:
Demand curve: Q = 480 - 0.2P
Average variable cost: AVC = 2Q^2 - 15Q + 400
Step 1: Calculate the marginal revenue (MR)
To find MR, take the derivative of the demand curve with respect to Q:
MR = dQ / dP * P + Q
Differentiating Q = 480 - 0.2P with respect to P:
dQ / dP = -0.2
Therefore MR = -0.2P + Q
Step 2: Calculate the marginal cost (MC)
To find MC, take the derivative of the AVC equation with respect to Q:
MC = dAVC / dQ
Differentiating AVC = 2Q^2 - 15Q + 400 with respect to Q:
dAVC / dQ = 4Q - 15
Therefore MC = 4Q - 15
Step 3: Set MR equal to MC and solve for Q
-0.2P + Q = 4Q - 15
Bring all terms involving Q to one side:
0.2P - 3Q = -15
Substituting Q = 480 - 0.2P:
0.2P - 3(480 - 0.2P) = -15
Simplifying the equation:
0.2P - 1440 + 0.6P = -15
0.8P - 1440 = -15
Combine like terms:
0.8P = 1440 - 15
0.8P = 1425
Divide both sides by 0.8:
P = 1425 / 0.8
P = 1781.25 (approx.)
Step 4: Calculate the respective value of Q using the demand curve equation:
Q = 480 - 0.2P
Q = 480 - 0.2 * 1781.25
Q = 480 - 356.25
Q = 123 (approx.)
So, the doctor should perform approximately 123 operations each month and charge a price of MVR1781.25.
Step 5: Calculate the profit earned per month
To calculate profit, subtract the total cost from total revenue. The total revenue can be calculated by multiplying the quantity (Q) by the price (P). The total cost is the sum of fixed costs and total variable costs (AVC * Q):
Total revenue = Price * Quantity
Total cost = Fixed cost + (AVC * Quantity)
Total revenue = 1781.25 * 123 = MVR219,243.75
Total cost = 8000 + (2(123)^2 - 15(123) + 400) * 123 = MVR137,076
Profit = Total revenue - Total cost
Profit = 219,243.75 - 137,076
Profit = MVR82,167.75
Therefore, the doctor earns a profit of approximately MVR82,167.75 each month.