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Unformatted text preview: Monopoly April 12, 2009 1 Monopoly Exercise 11.17 The demand is P = 100 √ Q . Find the optimal markup. This is a constant elasticity demand curve: the elasticity is =- . 5. The optimal markup is P- MC P =- 1 = 2 Notice that when demand has constant elasticity the monopolist cannot choose to operate on the elastic portion of the demand: if the demand is inelastic he is stuck with an inelastic production. Exercise 11.4 The demand is P = 9- Q . Can a monopolist ever produce 7 unit? The marginal revenue is MR = 9- 2 Q . When 7 units are produced we have MR = 9- 2(7) < On that region the demand is inelastic, therefore the monopolist will never operate there: it is more profitable to increase the price and reduce the quantity. Exercise 11.15 The demand is P = a- bQ and MC = c + eQ . Assume that a > c, 2 b + e > 0. a) Find optimal quantity and price. The marginal revenue is MR = a- 2 bQ . The optimal condition requires MR ( Q * ) = MC ( Q * ) a- 2 bQ = c + eQ = ⇒ Q * = a- c 2 b + e P * = ( a + c ) b + ac 2 b + e 1 b) What happens when c increase and when a decreases? We need to find the partial derivatives of price and quantity with respect to a and c (we assume that e > 0 ) ∂Q * ∂a = 1 2 b + e > 0 = ⇒ if a decreases, Q * decreases ∂Q * ∂c =- 1 2 b + e < = ⇒ if c increases, Q * decreases ∂P * ∂a = b + e 2 b + e > 0 = ⇒ if a decreases, P * decreases ∂P * ∂c = b + a 2 b + e > = ⇒ if c increases, P * increases Exercise 11.22 A monopolist is serving two separate markets with de- mands P 1 = 200- 2 Q 1 P 2 = 140- Q 2 The marginal cost is MC = 20 + Q 1 + Q 2 . The firm is forced to charge a single price in both markets: find the equilibrium price and quantity sold in both markets. We need to find the total demand: first, let’s invert the demands Q 1 = 100- P 2 Q 2 = 140- P and sum them Q = Q 1 + Q 2 = 240- 3 2 P . The new demand is P = 160- 2 3 P With this demand the marginal revenue is MR = 160- 4 / 3 Q . Now, the optimal condition MR = MC = ⇒ 160- 4 3 Q = 20 + Q = ⇒ Q * = 60 ,P * = 120 At this price the quantities demanded in each market are...
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This note was uploaded on 05/09/2009 for the course ECON 200 taught by Professor Junnie during the Spring '08 term at NYU.
- Spring '08