Question

Dr. John Smith is a renown dental surgeon with a reputation for being one of the best in reconstructive dental surgery. Dr. Smith enjoys a rather substantial degree of market power in this market.

The estimated direct demand function for his services is: Q = 600 − 0.25P

where Q is the number of dental surgeries performed monthly and P is the price of a typical dental surgery.

The average variable cost function for reconstructive dental surgery is estimated to be: AVC = 2Q² − 11Q + 500 where AVC is average variable cost (measured in dollars), and Q is the number of operations per month. Dr. Smith's fixed costs each month are $12,000.

a. Express the demand function for Dr. Smith's services as an inverse demand function and find the associated marginal revenue (MR) function. [Hint: To find the MR function, you can either (i) derive the first derivative of the total revenue (TR) function, i.e. ??=???/??, where TR = P.Q; or (ii) use two-times the slope rule if the demand function is linear.]

b. If Dr. Smith aims to maximize his profit, how many surgeries should he perform each month? [Hints: First find the total cost (TC) function as the sum of total variable cost and total fixed cost. Then derive the marginal cost (MC) function as ??=???/??. Profit maximisation is that quantity at which MR = MC. Alternatively, you can first derive the profit function as π = (TR – TC) and evaluate the first-order condition: ??/??=0.]

c. What price should Dr. Smith charge to perform each reconstructive dental surgery?

d. How much profit does he earn each month?

Answer 1

Direct demand function (demand in terms of price) is given as :

Q=600-0.25P,

where Q is the number of dental surgeries performed monthly and P is the price of a typical dental surgery

The AVC function is given as :

2Q² -11Q+500

Fixed costs = 12000

a)

To get the inverse demand function, we have to express the direct demand function in terms of Q. Or in other words, we have to express P(Q), i.e P as a function of Q.

We can do it by making P as the subject of the equation and bringing Q on the other side.

Thus,

Q=600-0.25P,

or, 0.25P=600-Q

or, P=(600/0.25) - (Q/0.25)

or, P =2400-4Q , which is the inverse demand function.

As the inverse demand function is linear, so the associated MR curve will be the same as this demand function, except that it will have twice the slope of the demand function.

Hence, MR= 2400-8Q, where this curve has twice the slope of the inverse demand function.

b)

Now, we have to find the TC function, which is the sum of the TVC (total variable cost) and TFC ( total fixed cost).

AVC is gotten as 2Q² -11Q+500

So, the TVC is basically Q times AVC , i.e. AVC*Q

Thus, TVC = (2Q² -11Q+500)*Q

= 2Q³ - 11Q² + 500Q

And TFC = 12000

Thus, TC = 2Q³ - 11Q² + 500Q + 12000

MC is given as the first derivative of TC with respect to Q,

Hence MC = 6Q²-22Q+500

Profit is maximised at the quantity where MR=MC.

Now, MR= 2400-8Q has already been gotten.

Equating these two, we have:

6Q²-22Q+500 = 2400-8Q

or, 6Q² -14Q -1900 =0

or, 3Q² -7Q -950=0

or, (3Q+50)(Q-19)=0

Hence, either 3Q+50=0 , i.e. Q = -50/3 , which is impossible as quantity can't be negative

or, Q-19=0, i.e. Q=19.

Hence, the correct answer is Q=19, which is the profit maximising level of Q. Thus if Dr Smith wishes to maximise his profit, he should perform 19 surgeries a month.

c)

To get the profit maximising price, he should find the price corresponding to his profit maximising surgery Q from the direct demand (or if you like, from the indirect demand) function. I'm doing it with the direct demand function Q=600-0.25P,

We have already gotten Q=19 as the profit maximising level of Q. So put Q=19 in the direct demand function and derive the corresponding P. That P will be the profit maximising P.

19=600-0.25P

or, 0.25P=600-19

or, 0.25P= 581

or, P= 581/0.25

or, P= 2324 (in dollars)

Hence Dr Smith should charge a price of $2324 for each reconstructive dental surgery.

d)

Profit is defined as TR-TC

When Q=19, P=2324, we can calculate TR and TC and then find the profit per month for Dr Smith

TR = P*Q = 19*2324 = 44156 (in dollars)

And we have derived :

TC = 2Q³ - 11Q² + 500Q + 12000

When Q=19, this gives us :

TC=2 (19)³ -11 (19)² + 500(19) +12000

= 2*6859 - 11 *361 +9500 +12000

=13718 - 3971 + 21500

= 31247 (in dollars)

Hence, profit = TR-TC = 44156 - 31247 = 12909 (in dollars)

Thus Dr Smith earns $12909 profit each month.