The Madison Corporation, a monopolist, receives a report from a consulting firm

Question

The Madison Corporation, a monopolist, receives a report from a consulting firm concluding that the demand function for its product is

Q = 78 - 1.1P + 2.3Y + 0.9A

where Q is the number of units sold, P is the price of its product (in dollars), Y is per capita income (in thousands of dollars), and A is the firm’s advertising expenditure (in thousands of dollars). The firm’s average variable cost function is

AVC= 42 - 8Q + 1.5Q2

where AVC is average variable cost (in dollars).

a. Can we determine the firm’s marginal cost curve?

b. Can we determine the firm’s marginal revenue curve?

c. If per capita income is $4,000 and advertising expenditure is $200,000, can we determine the price and output where marginal revenue equals marginal cost? If so, what are they?

Explanation / Answer

a) In the question we are given the average variable cost (AVC) . By multiplyin it with the the number of units produced we can find the average total cost ( here there is no fixed cost given , so we assume average total cost is average variable cost )

TVC=AVC*Q= 42Q - 8Q2 + 1.5Q3

Differentiating wrt Q ( First order condition we get )

d(TC)/dQ= 42-16Q+4.5Q2 =MC ( ie cost per additional unit produced )

b ) Marginal revenue is given by the first order condition of the total revenue

We know that revenue is simply the product of quantity and price

Also we are given the demand curve Q = 78 - 1.1P + 2.3Y + 0.9A

From here we can find the value of P in terms of Q,Y and A and multiply it by Q

1.1P = 78 + 2.3Y + 0.9A-Q

P=( 78 + 2.3Y + 0.9A - Q )/1.1

TR= P*Q =( 78Q+2.3Q+0.9AQ-Q2 )/1.1

d(TR)/dQ= (78 + 2.3Y + 0.9A - 2Q)/1.1 = MR

c)In this part we are given the values of Y and A . By substituting them in MR equation we get MR in terms of Q

Y = 4000 , A= $200000 but we are substituting Y as $4 and A as $ 200 because there values are to be given in thousand dollars . SO by taking $4 we are in as way sayinh that income is $4000 ( as the vakue of income would be in thousand dollars )

MR= (78+ 2.3 (4)+0.9(200)-2Q)/1.1

= (78+9.2+180-2Q)/1.1

= 267.2-2Q/1.1

Comparing MR and MC

MR=MC

(267.2-2Q)/1.1 = 42-16Q+4.5Q2

267.2-2Q/1.1=( 42-16Q+4.5Q2 )

242.9-1.82Q=( 42-16Q+4.5Q2 )

4.5Q2 - 14.18Q - 200.9 = 0

Solving this we get

Q=8.44 and -5.20

Quantity cannot be negative

So Q= 8.44

Price

P=(78-Q+2.3Y+0.9A)/1.1

P= 78-8.44 +2.3(4)+0.9(200)/1.1

P= (78-8.44+9.2+180)/1.1

P= 235.24

(Comment for doubts if any )

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.