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Unformatted text preview: Class 17: Complementary Monopolists Remark . This class is an application of reaction functions and Nash Equilibrium with continuous choice. Remark . This class also demonstrates the concept of pricing externalities : A company&s pricing may a/ect another company&s pro¡ts even when the two companies do not compete. Q . Microsoft is a (near) monopolist in the operating system market and a (near) monopolist in the o¢ ce applications (word processing, spreadsheets) market. As most consumers buy both the operating system and o¢ ce applications these are (near) perfect complements. Would consumers bene¡t if the these perfect complements OS and APPLICATIONS were sold by separate monopolists (so that one company would be an OS monopolist and another company would be an APPLICATIONS monopolist) instead of a single integrated monopolist (that sells both OS and APPLICATIONS) ? A . Solution Strategy: STEP 1. Solve for the optimal total price of OS and APPLICATIONS when one monopolist sells the two complementary products. STEP 2. Solve for the equilibrium total price of OS and APPLICATIONS when two separate monopolists sell the two complementary products. STEP 3. Compare the total price in the former case and the total price in the latter case. We assume that the demand curve is D ( p ) = 1000 & p: We assume that the total cost function is TC ( q ) = 0 : 1 The marginal costs are thus zero in this example. Denote the price of OS by p 1 , the price of APPLICATIONS by p 2 , and the total price by p: Naturally, p = p 1 + p 2 must hold. Step 1. The monopolist solves max p 1 ;p 2 f ( p 1 + p 2 ) (1000 & p 1 & p 2 ) g : Assuming that the monopolist sets p 1 = p 2 ; we can substitute p 2 for p 1 in this expression and rewrite it as max p 1 f 2 p 1 (1000 & 2 p 1 ) g : To solve for the optimal price p 1 we again solve for p 1 that satis&es MR = MC: The change in TR is now 2 ( p 1 + & p 1 ) (1000 & 2 ( p 1 + & p 1 )) & 2 (1000 & 2 p 1 ) which can be rewritten as 2& p 1 (1000 & 4 p 1 & & p 1 ) The rate of change of TR is therefore 2& p 1 (1000 & 4 p 1 & & p 1 ) & p 1 which can be rewritten as 2 (1000 & 4 p 1 & & p 1 ) : The MR is therefore (set & p 1 to zero in the above expression) 2 (1000 & 4 p 1 ) : 2 In this example MC = 0 by assumption. Let p & 1 denote the optimal price. At the optimum MR = MC , and therefore in this example at the optimum 2 (1000 & 4 p & 1 ) = 0 : Solving this equation for p & 1 yields p & 1 = 250 : This is the equilibrium price charged by the monopolist for one of the goods. Because we assumed that p 1 = p 2 ; the total price ( p & = p & 1 + p & 2 ) for OS and APPLICATIONS is p & = 250 + 250 when one monopolist sells both products. Step 2. Now the monopolist 1 solves max p 1 f p 1 (1000 & p 1 & p 2 ) g and the monopolist 2 solves max p 2 f p 2 (1000 & p 1 & p 2 ) g : We &rst solve for the reaction function p & 1 ( p 2 ) of monopolist 1....
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 Winter '10
 Mikko
 Equilibrium, Externalities, Supply And Demand, p1

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