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

microeconomic s the analytics of constrained optimal decisions assignment 6 the oligopoly model(I): competition in quantities  2016 Kellogg School of Management assignment 6 page |1 the federal funds market time line FED commits funds F G with resulting market demand r ( F ) = [90 – 0.5 F G ] – 0.5( F 1 + F 2 ) ● this initial commitment of funds by FED basically “reduces” the market size left for the two banks market demand for funds is initially r ( F ) = 90 – 0.5 F ● where F is the total funds supplied to the market by the players F = F 1 + F 2 + F G the two banks form a cartel and supply together an amount F B r ( F G,F B ) = [90 – 0.5 F G ] – 0.5 F B ● the solution to this game follows the same logic as the solution without the FED ► What is cartel’s offer?  Marginal revenue. The cartel behaves as a monopolist (choosing F B ) that faces the demand r ( F ) = [90 – 0.5 F G ] – 0.5 F B. The marginal revenue is easy to find: MR B ( F B ) = [90 – 0.5 F G ] – F B  Profit maximizing condition. The condition for monopolist MR B ( F B ) = MC B ( F B ) with a zero marginal cost gives [90 – 0.5 F G ] – F B = 0  Cartel’s offer. From the profit maximizing condition: F B = 90 – 0.5 F G

microeconomic s the analytics of constrained optimal decisions assignment 6 the oligopoly model(I): competition in quantities  2016 Kellogg School of Management assignment 6 page |2 the federal funds market ► What is FED’s offer?  There are two cases we consider - uninformed FED : the FED believes the two banks compete Cournot, thus the FED expects that, given FED’s supply F G the two banks will supply F 1 ( F G ) = 60 – F G /3 and F 2 ( F G ) = 60 – F G /3 - informed FED : the FED believes the two banks cooperate into a cartel, thus the FED expects that, given FED’s supply F G the two banks will supply together F B = 90 – 0.5 F G ► Uninformed FED  We saw in this case that the FED will supply F G = 120. However, the actual (and definitely not expected by FED) reaction from the banks is to supply F B = 90 – 0.5 F G = 30 with a resulting FFR of r ( F ) = 90 – 0.5( F G + F 1 + F 2 ) = 90 – 0.5( ) = 15.  Each bank’s profit is half of the cartel’s profit:  1 =  2 = 1/2  [ r(F B )  F B ] = 1/2  [15  30] = 225  Obviously the FED misses the target of 10 for the FFR because it does not anticipate correctly banks’ behavior.

microeconomic s the analytics of constrained optimal decisions assignment 6 the oligopoly model(I): competition in quantities  2016 Kellogg School of Management assignment 6 page |3 the federal funds market ► Informed FED  The FED believes the two banks cooperate into a cartel, thus the FED expects that, given FED’s supply F G the two banks will supply together F B = 90 – 0.5 F G with a resulting FFR of r ( F ) = 90 – 0.5( F G + F 1 + F 2 ) = 90 – 0.5( F G + 90 – 0.5 F G ) = 45 – 0.25 F G  With a target of FFR * = 10 the FED solves the equation 45 – 0.25 F G = 10 with solution F G = 140  Each bank supplies: F 1 = F 2 = 1/2  [90 – 0.5 F G ] = 1/2  [90 – 0.5  140] = 10  Each bank’s profit is  1 =  2 = r(F)  F 1 = r(F)  F 2 = 10  10 = 100  Obviously the FED gets the target of 10 for the FFR because it does anticipate correctly banks’ behavior.

microeconomic s the analytics of constrained optimal decisions assignment 6 the oligopoly model(I): competition in quantities  2016 Kellogg School of Management assignment 6 page |4 sustainable cartels ► TASK I: Cournot non-Cooperative Solution. Given the market demand P ( Q ) = 12 – Q, with Q = q 1 + q 2, the residual demands, marginal revenues and reaction functions are:  firm 1: residual demand P 1 ( q 1 | q 2 ) = (12 – q 2 ) – q 1 marginal revenue MR 1 ( q 1 | q 2 ) = (12 – q 2 ) – 2 q 1 reaction function (12 – q 2 ) – 2 q 1 = 0 gives q 1 ( q 2 ) = 6 – 0.5 q 2  firm 2: residual demand P 2 ( q 2 | q 1 ) = (12 – q 1 ) – q 2 marginal revenue MR 2 ( q 2 | q 1 ) = (12 – q 1 ) – 2 q 2 reaction function (12 – q 1 ) – 2 q 2 = 0 gives q 2 ( q 1 ) = 6 – 0.5 q 1  The two reaction functions are q 1 ( q 2 ) = 6 – 0.5 q 2 q 2 ( q 1 ) = 6 – 0.5 q 1 with Cournot non-cooperative solution q 1 ( nc ) = 4, q 2 ( nc ) = 4 P ( nc ) = 4,  1 ( nc ) = 16,  2 ( nc ) = 16

microeconomic s the analytics of constrained optimal decisions assignment 6 the oligopoly model(I): competition in quantities  2016 Kellogg School of Management assignment 6 page |5 sustainable cartels ► TASK I: Cournot Cooperative (Cartel) Solution. Given the market demand P ( Q ) = 12 – Q, with Q = q 1 + q 2, the marginal revenue for a monopolist serving this market is: marginal revenue MR ( Q ) = 12 – 2 Q with Cournot cooperative solution Q ( c ) = 6, q 1 ( c ) = 3, q 2 ( c ) = 3 P ( c ) = 6,  1 ( c ) = 18,  2 ( c ) = 18 ► TASK I: Best Deviation to Cartel Solution. If firm 1 plays the cooperative (cartel) solution, i.e. q 1 ( c ) = 3, then firm 2’s best reaction to this is given by q 1 ( q 2 ( c )) = 6 – 0.5 q 2 ( c ) = 4.5 The results of the deviation are: q 1 ( c ) = 3, q 2 ( d ) = 4.5 P ( d ) = 4.5,  1 ( d ) = 13.5,  2 ( d ) = 20.25

microeconomic s the analytics of constrained optimal decisions assignment 6 the oligopoly model(I): competition in quantities  2016 Kellogg School of Management assignment 6 page |6 sustainable cartels ► TASK II:  Alternative 1. Given a discount rate r, firm 2 makes a profit (from deviating in the first period and then playing the non-cooperative Cournot solution) of  Alternative 2. Given a discount rate r, firm 2 makes a profit (from never deviating) of Firm 2 will prefer Alternative 2 to Alternative 1 if: that is ► Solution : whenever r < 0.89 firm 2 will not deviate (the discount rate is ”too small” which means “the future counts” and it’s better to be patient and wait for getting higher profits in the future).