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notes_10 - Monopoly 1 Monopoly The demand is P = 100 Q Find...

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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 = - 0 . 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) < 0 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
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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 < 0 = 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 > 0 = 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 Q 1 = 100 - 60 = 40 Q 2 = 140 - 120 = 20 (which add exactly up to 60) 2
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2 Capturing Surplus Exercise 12.2 The demand is P = 20
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