the analytics of constrained optimal decisions microeco nomics spring 2016 the oligopoly model (II): competition in prices ………….1the federal funds market ………….4 sustainable cartels assignment seven

microeconomic s the analytics of constrained optimal decisions assignment 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management assignment 7 page |1 mergers of complements ► Demands are different from what we studied so far: Q 1 = 96 – 2 P 1 – 2 P 2 Q 2 = 96 – 2 P 2 – 2 P 1 The difference is now that the two products are no longer substitutes but rather complements: ● the two products have to be used together ● this implies that an increase in price of one product will decrease the demand for both products ● there are no switchers in this case (the market size for both products goes up/down simultaneously ► However the derivation of the reaction functions follows the same “procedure”: ● write the profit functions for each firm ● maximize each profit function with respect to own price for each firm ● this will give you the reaction function ● in fact we can use directly the previous result since now the demands are Q 1 = a 1 – b 1 P 1 – d 1 P 2 Q 2 = a 2 – b 2 P 2 – d 2 P 1 ● which implies that the reaction functions have the same form as before but: ● we replace d 1 with – d 1 and d 2 with – d 2 P 1 = 0.5∙ a 1 / b ∙ MC 1 – 0.5∙ d 1 ∙ P 2 / b 1 P 2 = 0.5∙ a 2 / b ∙ MC 2 – 0.5∙ d 2 ∙ P 1 / b 2

microeconomic s the analytics of constrained optimal decisions assignment 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management assignment 7 page |2 mergers of complements ► Using the general reaction functions P 1 = 0.5∙ a 1 / b ∙ MC 1 – 0.5∙ d 1 ∙ P 2 / b 1 P 2 = 0.5∙ a 2 / b ∙ MC 2 – 0.5∙ d 2 ∙ P 1 / b 2 we get for our demand functions: P 1 = 0.5∙96/ ∙6 – 0.5∙2∙ P 2 /2 P 2 = 0.5∙96/ ∙6 – 0.5∙2∙ P 1 /2 that is P 1 = 27 – P 2 /2 P 2 = 27 – P 1 /2 ► This is a system of two equations with two unknowns. Since the system is symmetric we’ll solve for P 1 = P 2 : P = 27 – P /2 with P 1 = P 2 = reaction function firm reaction function firm 2 P2P2 P1P1 18 ► When the products are complements the reaction function for the Bertrand model are downward slopping.

microeconomic s the analytics of constrained optimal decisions assignment 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management assignment 7 page |3 mergers of complements ► Using the general reaction functions Q 1 = a 1 – b 1 P 1 – d 1 P 2 Q 2 = a 2 – b 2 P 2 – d 2 P 1 we get the cross elasticities as: firm 1: e 1,2 = (∆Q 1 /∆P 2 )∙P 2 /Q 1 = – d 1 ∙P 2 /Q 1 firm 2: e 2,1 = (∆Q 2 /∆P 1 )∙P 1 /Q 2 = – d 2 ∙P 1 /Q 2 For prices P 1 = 20 and P 2 = 10 we get: Q 1 = 96 – 2 P 1 – 2 P 2 = 96 – 2∙20 – 2∙10 = 36 Q 2 = 96 – 2 P 2 – 2 P 1 = 96 – 2∙10 – 2∙20 = 36 For elasticities we get firm 1: e 1,2 = – d 1 ∙P 2 /Q 1 = – 2 ∙ 10/36 = – 0.55 firm 2: e 2,1 = – d 2 ∙P 1 /Q 2 = – 2 ∙ 20/36 = – 1.11

microeconomic s the analytics of constrained optimal decisions assignment 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management assignment 7 page |4 mergers of complements ► General reaction functions P 1 = 0.5∙ a 1 / b ∙ MC 1 – 0.5∙ d 1 ∙ P 2 / b 1 P 2 = 0.5∙ a 2 / b ∙ MC 2 – 0.5∙ d 2 ∙ P 1 / b 2 If the marginal cost for Firm 2 decreases from the reaction function we see immediately that this implies: - price P 2 decreases for each price P 1 - reaction function of Firm 2 shifts left As a result the price for Firm 2 decreases and price for Firm 1 increases reaction function firm reaction function firm 2 P2P2 P1P1 (0) (1) ► The profit function of the resulting company is simply the sum of the individual profits:  =  1 +  2 = ( P 1  Q 1 ) + ( P 2  Q 2 ) – TC ( Q 1, Q 2 ) = = P 1  (96 – 2 P 1 – 2 P 2 ) + P 2  (96 – 2 P 2 – 2 P 1 ) – [100 – 6  (96 – 2 P 1 – 2 P 2 ) – 6  (96 – 2 P 2 – 2 P 1 )] = = – 2  ( P 1 + P 2 – 30) 2 + constant ► The above profit is maximized for any pair of prices ( P 1, P 2 ) such that P 1 + P 2 = 30. Symmetric solution P 1 = P 2 = 15. changes in equilibrium (Bertrand - complements) merger analysis

microeconomic s the analytics of constrained optimal decisions assignment 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management assignment 7 page |5 limited capacities ► The “negotiation” process described in the problem, in which both parts can revisit their offers until agreement is reached, points to a standard (simultaneous) Cournot solution:  contractor one residual demand P 1 = (6,300 – L 2 ) – L 1 marginal revenue MR 1 = (6,300 – L 2 ) – 2 L 1 profit maximization MR 1 = MC 1 gives (6,300 – L 2 ) – 2 L 1 = 6,000 reaction function L 1 = 150 – L 2 /2  contractor two residual demand P 2 = (6,300 – L 1 ) – L 2 marginal revenue MR 2 = (6,300 – L 1 ) – 2 L 2 profit maximization MR 2 = MC 2 gives (6,300 – L 1 ) – 2 L 2 = 6,000 reaction function L 2 = 150 – L 1 /2  Cournot equationsL 1 = 150 – L 2 /2 L 2 = 150 – L 1 /2 ► Thus both contractors will settle for 100 levels ( L 1 = L 2 = 100 ). This is shown in the diagram as the intersection of the two reaction functions. unconstrained solution L1L1 L2L Cournot Solution reaction function contractor 2 reaction function contractor 1

microeconomic s the analytics of constrained optimal decisions assignment 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management assignment 7 page |6 limited capacities ► The constraint solution is based on the same reactions functions with additional restrictions on the maximum allowed number of levels:  Cournot equationsL 1 = 150 – L 2 /2 OR L 1  50 L 2 = 150 – L 1 /2 OR L 2  50 ► Thus both contractors will settle for 50 levels ( L 1 = L 2 = 50 ). This is shown in the diagram as the intersection of the two “constrained” reaction functions. constrained solution L1L1 L2L constrained Cournot Solution constrained reaction function contractor 1 constrained reaction function contractor 2