Chapter 17 Vertical price restraint Suppose that a car dealer has a local monopo

Question

Chapter 17 Vertical price restraint Suppose that a car dealer has a local monopoly in selling Volvos. It pays to Volvo for each car that it sells, and charges each customer . The demand curve is best described by the linear function , where Q is the number of cars sold. (a) What is the profit-maximizing price for the dealer to set? At this price, how many Volvos will the dealer sell and what will the dealer's profit be? (Hint: the results won't be exact numbers, but will be functions of the wholesale price ) (b) Now let's think about how the situation looks from the car manufacturer's point of view. What is the demand curve facing Volvo? Suppose that it costs Volvo $5 to produce each car. What is the profit-maximizing choice of ? What will Volvo's profit be? What would be the retail price ? What profit will the dealer earn at Volvo's profit-maximizing choice of wholesale price ? (c) Suppose that Volvo operates the dealership itself and sells directly to its customers. What would be the profit-maximizing price ? What is Volvo's profit? (d) Suppose instead that Volvo can impose an RPM agreement on its independent retailers. What price will Volvo actually set?

Explanation / Answer

a.

Let P be the price charged by the car manufacturer to the dealer. And p is the price charged by the dealer to the consumers. The marginal cost of the dealer is P.

Dealer’s demand function is

            p = 30 – Q

            TR = pQ = (30 - Q)Q

            TR = 30Q – Q2

            MR = dTR/dQ
            MR = 30 - 2Q

Find the profit maximizing quantity of the consumer as follows.

            MC = MR

            P = 30 – 2Q

            Q = 15 – P/2                …(1)

which is the profit maximizing quantity of the dealer.

The price charged by the dealer is

            p = 30 – Q

            p = 30 – (15 – P/2)

            p = 15 + P/2

Profit of the dealer      = pQ – PQ

                                    = Q(p – P)

                                    = (15 – P/2 )(15 + P/2 – P)

                                    = (15 – P/2 )2

b.

Equation (1) is the car manufacturer’s demand function. This may be written in the inverse form as

            P = 30 – 2Q

            TRm = PQ = (30 – 2Q)Q = 30Q – 2Q2

            MRm = dTRm/dQ = 30 – 4Q

Find the profit-maximizing quantity of the car manufacturer as follows:

            MRm = 5

            30 – 4Q = 5

            Q = 6.25 units.

Price charged by the car manufacturer:

            P = 30 – 2Q = 30 – 2(6.25) = $17.50

Profit of the car manufacturer = PQ – 5Q = 17.50(6.25) – 5(6.25) = $78.125

Price charged by the dealer:

            p = 15 + (17.50/2) = $23.75

c.

If Volvo operated dealership itself, then Volvo’s demand function is

            P = 30 – Q

            TR = PQ = (30 - Q)Q

            TR = 30Q – Q2

            MR = dTR/dQ
            MR = 30 - 2Q

Find the profit maximizing quantity of the consumer as follows.

            MC = MR

            5 = 30 – 2Q

            Q = 12.50

which is the profit maximizing quantity of Volvo.

The price charged by the volvo is

            P = 30 – Q

            p = 30 – 12.50

            p = 17.50

Profit of the dealer      = PQ – 5Q

                                    = Q(P – 5)

                                    = $12.50(17.50 – 2)

                                    = $156.25

d.

Volvo will set a price equal to the marginal cost $5 and a fixed fee of $156.25.

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