1. the analytics of constrained optimal decisions microeco nomics spring 2016 the oligopoly model(I): competition in quantities ………….1the federal funds market setup ………….2 player’s profit functions session six ………….4residual demand ………….6 reaction function..……….10the cournot solution..……….14cournot model key points …..…….15 the stackelberg model..………21the cartel model …..……24 final key points

2. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |1 the federal funds market ► Several of the largest banks, together with the FED, are the participants in the interbank loan market ► Some of the banks have overnight surpluses of funds while others have overnight deficits of funds ► The interbank loan market is designed to allocate the available funds among the banks that have surplus and those who need funds. The price (interest rate) that “clears” this market is called the Federal Funds Rate ( FFR ) - this is the rate the FED announces periodically Bank 1 (surplus) Bank 1 (surplus) Bank 2 (surplus) Bank 2 (surplus) FED (manages reserves) FED (manages reserves) Bank A (deficit) Bank A (deficit) Bank B (deficit) Bank B (deficit) Interbank Loan Market clearing interest rate: FFR Interbank Loan Market clearing interest rate: FFR supply funds receive funds

3. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |2 the federal funds market ► Each of the bank with funds surplus will supply a certain amount, say F 1 and F 2 (for simplicity let’s say there are only two banks with funds surplus), therefore the total supply of funds is F 1 + F 2. ► The demand for funds depends on the resulting interest rate: r ( F ) = a – b∙F where F = F 1 + F 2. Remark. Notice the particular way in which this type of market operates: the interest rate (price of funds) adjusts such that the whole supply is transacted ! ► What are the revenues for the banks that supplied the funds? They receive the same interest rate on the funds they lend, thus: Bank 1:  1 = r ( F 1 + F 2 )∙ F 1 = [ a – b ∙( F 1 + F 2 )]∙ F 1 Bank 2:  2 = r ( F 1 + F 2 )∙ F 2 = [ a – b ∙( F 1 + F 2 )]∙ F 2 Remark. Notice the strategic feature of this market : Bank 1’s decision ( F 1 ) affects the interest rate (through the total funds supplied) which affects Bank 2’s revenue – and the same is true about Bank 2’s decision ( F 2 ) affecting Bank 1’s revenue. total funds supplied

4. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |3 the federal funds market ► Let’s take a simple example and consider the following demand function: r ( F ) = 90 – 0.5∙ F ► If Bank 1 offers F 1 = 20 mill while Bank 2 offers F 2 = 10 mill, total funds supplied are: F = F 1 + F 2 = 30 mill with FFR equal to r (30) = 75 bps (0.75%). ► The corresponding revenues are  1 = r ( F )∙ F 1 = 75∙20 = 1,500  2 = r ( F )∙ F 2 = 75∙10 = 750 FFR F 302.25002.10001.95001.8000 201.60001.50001.40001.3000 100.85000.80000.75000.7000 00.00000.000 0102030 bank 1 bank 2 30180 90 75 revenue to Bank 1 in 000’ millions demand for funds F = 30 as a sum of F 1 = 20 and F 2 = 10 300.00000.70001.30001.8000 200.00000.75001.40001.9500 100.00000.80001.50002.1000 00.00000.85001.60002.2500 0102030 bank 1 bank 2 revenue to Bank 2 in 000’ millions

5. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |4 the federal funds market D 30 mill Bank 1: 60 mill FFR 45 D 30 9060 D 30 mill FFR 55 D 30 7040 Bank 1: 40 mill F F Bank 1’s perspective ► Suppose Bank 2 offers F 2 = 30 mill. By shifting D to the left by 30 mill at each price we obtain the “market left” for Bank 1, thus the name: the residual demand curve ► How do we find the equation of the residual demand? ► The total market demand equation is FFR = 90 – 0.5∙( F 1 + F 2 ) if Bank 2 offer F 2 = 30 mill we plug this amount in the above equation to get the residual demand curve as FFR 1 = 90 – 0.5∙( F 1 + 30) = 75 – 0.5∙ F 1 ► The residual demand curve gives the FFR for each Bank 1’s offer conditional on a certain offer from Bank 2 (in our case this offer is actually F 2 = 30 mill): ● for F 1 = 60 we get FFR = 75 – 0.5∙60 = 45 ● for F 1 = 40 we get FFR = 75 – 0.5∙40 = 55 ► A subtler implication of constructing this residual demand curve is that once we condition Bank 2’s offer to some F 2 then Bank 1 has the rest of the market to serve, i.e. can behave as a monopolist on the remaining market with demand given by the residual demand curve residual demand

6. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |5 the federal funds market Bank 1’s perspective ► Let’ s try to derive the residual demand curve in a more general case, i.e. say Bank 2’s offer is F 2 ► From total demand FFR = 90 – 0.5∙( F 1 + F 2 ) we get the residual demand curve for Bank 1 as: FFR 1 = (90 – 0.5∙ F 2 ) – 0.5∙ F 1 ► From this we can find now the optimal choice of Bank 1 given that Bank 2 chooses F 2 ( we assume that marginal cost is zero ): F 1 ( F 2 ) = 90 – 0.5∙ F 2 90 90 – 0.5∙ F 2 180180 – F 2 90 – 0.5∙ F 2 MR 1 residual demand total demand F2F2 Remark The optimal choice for Bank 1 was found as the intersection of the marginal revenue curve MR 1 = (90 – 0.5∙ F 2 ) – F 1 with the marginal cost curve (equal to zero here) F FFR residual demand

7. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |6 the federal funds market reaction function 75 30 60 F2F2 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 F1F1 … what Bank 1 conjectures about Bank 2’s offer… … how Bank 1 responds to the conjectures about Bank 2… D FFR D 60 60 60 mill MR 60 F D 30 mill FFR D 30 75 MR 30 Bank 1’s perspective ► Bank 2 offers F 2 = 60 mill ► Bank 1 choice: F 1 = 60 mill ► Bank 2 offers F 2 = 30 mill ► Bank 1 choice: F 1 = 75 mill

8. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |7 the federal funds market D 30 mill Bank 2: 60 mill FFR 45 D 30 9060 D 30 mill FFR 55 D 30 7040 Bank 2: 40 mill F F Bank 2’s perspective ► Suppose Bank 1 offers F 1 = 30 mill. By shifting D to the left by 30 mill at each price we obtain the “market left” for Bank 2, thus the name: the residual demand curve ► How do we find the equation of the residual demand? ► The total market demand equation is FFR = 90 – 0.5∙( F 1 + F 2 ) if Bank 1 offer F 1 = 30 mill we plug this amount in the above equation to get the residual demand curve as FFR 2 = 90 – 0.5∙(30 + F 2 ) = 75 – 0.5∙ F 2 ► The residual demand curve gives the FFR for each Bank 2’s offer conditional on a certain offer from Bank 1 (in our case this offer is actually F 1 = 30 mill): ● for F 2 = 60 we get FFR = 75 – 0.5∙60 = 45 ● for F 2 = 40 we get FFR = 75 – 0.5∙40 = 55 ► A subtler implication of constructing this residual demand curve is that once we condition Bank 1’s offer to some F 1 then Bank 2 has the rest of the market to serve, i.e. can behave as a monopolist on the remaining market with demand given by the residual demand curve residual demand

9. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |8 the federal funds market residual demand Bank 2’s perspective ► Let’ s try to derive the residual demand curve in a more general case, i.e. say Bank 1’s offer is F 1 : ► From total demand FFR = 90 – 0.5∙( F 1 + F 2 ) we get the residual demand curve for Bank 2 as: FFR 2 = (90 – 0.5∙ F 1 ) – 0.5∙ F 2 ► From this we can find now the optimal choice of Bank 2 given that Bank 1 chooses F 1 (we assume that marginal cost is zero): F 2 ( F 1 ) = 90 – 0.5∙ F 1 90 90 – 0.5∙ F 1 180180 – F 1 90 – 0.5∙ F 1 MR 2 residual demand total demand F1F1 Remark. The optimal choice for Bank 2 was found as the intersection of the marginal revenue curve MR 2 = (90 – 0.5∙ F 1 ) – F 2 with the marginal cost curve (equal to zero here) F FFR

10. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |9 the federal funds market 75 60 F2F2 ► Bank 2’s reaction function: F 2 (F 1 ) = 90 – 0.5∙F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 … what Bank 2 conjectures about Bank 1’s offer… … how Bank 2 responds to the conjectures about Bank 1… D FFR D 60 60 60 mill MR 60 F D 30 mill FFR D 30 75 MR 30 Bank 2’s perspective ► Bank 1 offers F 1 = 60 mill ► Bank 2 choice: F 2 = 60 mill ► Bank 1 offers F 1 = 30 mill ► Bank 2 choice: F 2 = 75 mill 3060 reaction function

11. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |10 the federal funds market 75 60 F2F2 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 3060 75 30 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 ► Cournot equilibrium occurs at a pair of amounts F 1 * and F 2 * such that : ● Bank 1’s choice is “ best response ” to Bank 2’s choice: F 1 * maximizes Bank 1’s profits given that Bank 2 offers F 2 * and ● Bank 2’s choice is “ best response ” to Bank 1’s choice: F 2 * maximizes Bank 2’s profits given that Bank 1 offers F 1 * Remark. Finding the Cournot equilibrium means to find the pair ( F 1 *, F 2 *) having the properties described in the definition above. Cournot solution the cournot solution

12. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |11 the federal funds market 60 F2F2 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 60 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 ► The graphical solution is obvious but what about the algebraic solution? ► The intersection of the two reaction functions means that we are looking for the pair ( F 1, F 2 ) that solves the system of two equations given by the reaction functions: Bank 1: F 1 = 90 – 0.5∙ F 2 Bank 2: F 2 = 90 – 0.5∙ F 1 ► Plug the first equation into the second one: F 2 = 90 – 0.5∙(90 – 0.5∙ F 2 ) that is 0.75∙ F 2 = 45 with F 2 * = 60 and using the first equation we get F 1 * = 60 ► The FFR is r ( F ) = 90 – 0.5∙( F 1 *+ F 2 *) = 30 and each bank’s profit is  1 =  2 = 30∙60 = 1,800 Cournot solution the cournot solution

13. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |12 the federal funds market F2F2 F1F1 75 30 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 with zero marginal cost 60 D FFR D 60 60 60 mill MR 60 F D 30 mill FFR D 30 75 MR 30 Bank 1’s perspective ► Bank 2 offers F 2 = 60 mill ► Bank 1 choice: F 1 = 40 mill ► Bank 2 offers F 2 = 30 mill ► Bank 1 choice: F 1 = 55 mill 40 MC = 20 55 ► Bank 1’s reaction function : F 1 ( F 2 ) = 70 – 0.5∙ F 2 with marginal cost equal to 20 4055 ► Suppose MC for Bank 1 changes from 0 to 20. How is the equilibrium changing?

14. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |13 the federal funds market ► Suppose MC for Bank 1 changes from 0 to 20. How is the equilibrium changing? F2F2 F1F1 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 with zero marginal cost 60 33.33 73.33 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 ► Bank 1’s reaction function : F 1 ( F 2 ) = 70 – 0.5∙ F 2 with marginal cost equal to 20 Cournot solution ► The marginal revenue curve for Bank 1 was derived earlier as: MR 1 = (90 – 0.5∙ F 2 ) – F 1 and since the marginal cost is now MC 1 = 20 the optimal choice (and reaction function) for Bank 1 is given by the solution to (90 – 0.5∙ F 2 ) – F 1 = 20 ► The reaction function for Bank 1 is F 1 ( F 2 ) = 70 – 0.5∙ F 2 ► The Cournot solution is found at the intersection of the two reaction functions, i.e. we have to solve the system: F 1 = 70 – 0.5∙ F 2 F 2 = 90 – 0.5∙ F 1 The solution is F 1 * = 33.33 and F 2 * = 73.33. Compared to the initial solution notice how Bank 1 lost “share” of the market: it offers less while Bank 2 is offering more.

15. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |14 ► What does the Cournot solution represent conceptually? ● notice that each company has to make its individual choice based on “conjectures” of what all other companies are going to choose… the choices are made simultaneously … ● the Cournot solution calculates the “self-fulfilling conjectures”… “you will get what you expect” ► The steps in obtaining the Cournot solution : ● for each player derive the residual demand function from the total demand function… ● each player behaves as a monopolist with respect to its residual demand function and its marginal cost therefore you are able to derive player’s optimal choice ( MR = MC ); this choice will depend on the conjecture about the action of the other player – these are the reaction functions of the two players ● the Cournot solution is found by solving the system of two equations and two unknowns as defined by the reaction functions ► Two important issues: ● the decision to produce/offer a certain output is based on optimal choice therefore “lower the price to gain more share of the market” is not a valid strategy ● a player’s reaction function is derived based on the production/offer by the other player – it does not matter for what reasons the other player is offering that output (high or low MC ) – which implies that a player’s reaction function does not change when conditions particular to the other player change key points

16. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |15 the stackelberg model ► What happens if one the banks decides to announce first its choice? In who’s advantage (if any) is this revelation? ► Say Bank 1 announces first that it will supply some amount F 1 (and to prove that it immediately deposits this amount with the FED) … ► … now Bank 2 seems to have an advantage since it observes Bank 1’s choice so it can choose its own offer already knowing what Bank 1 offers … ► … what is Bank 1’s offer? Remark The model with simultaneous offers is the standard Cournot model while the model with sequential offers is called the Stackelberg model 75 60 F2F2 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 3060 75 30 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 Cournot solution

17. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |16 the stackelberg model ► First of all Bank 1’s reaction function does not matter anymore … ► … Bank 1’s reaction function, as derived under the Cournot assumptions, is relevant only when Bank 1 reacts to Bank 2’s offer … here Bank 1 reveals its choice before Bank 2 makes its own choice ► … say Bank 1 chooses to offer F 1 = 30 we know that Bank 2 optimally, i.e. maximizing its profit, by choosing the corresponding offer according to its reaction function: F 2 ( F 1 ) = 90 – 0.5∙ F 1 = 75 ► … the FFR is: r ( F ) = 90 – 0.5∙( F 1 + F 2 ) = 37.5 ► Bank 1’s profit is  1 = r ( F )∙ F 1 = 37.5∙30 = 1,125 and Bank 2’s profit is  2 = r ( F )∙ F 2 = 37.5∙75 = 2,812.5 F 2 ( F 1 ) = 75 F2F2 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 F 1 =30 ► Suppose Bank 1 is announcing first its supply F 1. How is the equilibrium changing?

18. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |17 the stackelberg model ► Suppose Bank 1 is announcing first its supply F 1. How is the equilibrium changing? ► Notice that when Bank 1 announces its offer F 1 it can infer what Bank 2 will choose by using Bank 2’s reaction function: F 2 ( F 1 ) = 90 – 0.5∙ F 1 ► The resulting FFR is r ( F ) = 90 – 0.5∙( F 1 + F 2 ( F 1 ) ) = = 90 – 0.5∙( F 1 + 90 – 0.5∙ F 1 ) = 45 – 0.25∙ F 1 ► Bank 1’s profit is  1 = r ( F )∙ F 1 = (45 – 0.25∙ F 1 )∙ F 1 = = 45∙ F 1 – 0.25∙ F 1 2 ► Optimal choice for Bank 1 is obtained by maximizing the above function: F 1 *= 90 The resulting choice for Bank 2 is F 2 * = 45 ► Resulting profits are:  1 = 2,025 and  2 = 1,012.5 F 2 ( F 1 ) = 45 F2F2 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 F 1 =90 ► Bank 1’s optimal choice is on Bank 2’s reaction function

19. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |18 the stackelberg model ► Suppose the FED announces at the beginning of the game that will supply F G funds. The two banks observe this announcement and choose their own offers, F 1 and F 2, according to the Cournot solution. What is the amount that the FED should announce such that it is absolutely sure that the resulting FFR (after the two banks make their choice) will be exactly 10? the “manipulation” game ► Let’s look at this game from a time perspective first: 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 compete now in a Cournot model given a demand r ( F ) = [90 – 0.5 F G ] – 0.5( F 1 + F 2 ) ● the solution to this game follows the same logic as the solution without the FED ► What are Bank 1’s and Bank 2’s residual demands ? ► What are Bank 1’s and Bank 2’s reaction functions ?

20. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |19 the stackelberg model the “manipulation” game ► We have to solve now the system F 1 = 90 – 0.5∙ F G – 0.5∙ F 2 F 2 = 90 – 0.5∙ F G – 0.5∙ F 1 ► The solution is found in the usual way (plug the first equation into the second, solve for F 2, then use this back into the first equation to find F 1 ) however the algebra is slightly more cumbersome since we have to carry over the extra term for F G. Anyway: F 1 ( F G ) = 60 – F G /3 F 2 ( F G ) = 60 – F G /3 ► The resulting FFR is thus r ( F ) = 90 – 0.5∙( F 1 + F 2 + F G ) = 90 – 0.5∙(60 – F G /3 + 60 – F G /3 + F G ) = 30 – F G /6 ► The FED wants a target FFR * thus solves 30 – F G /6 = FFR * with solution F G = 180 – 6∙ FFR * ► For FFR * =10 we get F G = 180 – 6∙10 = 120 and F 1 = 60 – 120/3 = 20, F 2 = 60 – 120/3 = 20

21. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |20 the stackelberg model the “manipulation” game 60 F2F2 ► bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F G – 0.5∙ F 1 bank 2’s profit-maximizing quantity graphed against the amount offered by bank 1, here F G = 120 F1F1 60 ► bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F G – 0.5∙ F 2 bank 1’s profit-maximizing quantity graphed against the amount offered by bank 2, here F G = 120 Cournot solution with no FED ( F G = 0) 20 90180 90 180 F G = 120 Cournot solution with FED (committing F G = 120) ► Notice the change in the solution with the shift in the reaction functions (from F G = 0 for the initial case to F G = 120) ► By committing funds in the initial stage of the game the FED basically “manipulates” the two banks as the FED knows how the banks will respond when faced with a certain demand; the FED’s job is simply to “figure out” the correct size of the market to be left to the two competing banks

22. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |21 the cartel model 60 F2F2 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 F1F1 60 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 Cournot solution ► The two banks split the market in two compared to the perfect competition case in which there are many banks competing (perfect competition) ► Is there a way for these two banks to collude? That means they decide how to share the market and then split the profit in hope that they would both improve on their profit obtained by competing (as a reminder the profit obtained by each bank in case they compete is 1,800) ►The idea is first to find a way to maximize the joint profit… ► … if the banks behave just like one bank (through consolidation) the resulting bank would behave as a monopolist that chooses its offer F and faces the demand function … r ( F ) = 90 – 0.5∙ F

23. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |22 the cartel model 60 F2F2 F1F1 Cournot solution ► This is the classic monopolist problem, and so first we get the marginal revenue as MR ( F ) = 90 – F and since the marginal cost is zero the solution is F * = 90. ► The resulting FFR is r ( F ) = 90 – 0.5∙ F = 45 ► … with joint profit  = r ( F )∙ F = 45∙90 = 4,050 ► If the two banks split the profit in equal parts each bank gets  1 =  2 = 2,025 ► Obviously this is an improvement over the Cournot solution for which the profits were for each bank 1,800. Cartel solution 45 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 45

24. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |23 the cartel model 60 F2F2 F1F1 Cournot solution ► The cartel outcome depends on each bank committing to offer half the monopolist solution… ► If given the opportunity, would Bank 1 deviate from offering F 1 = 45 knowing that Bank 2 will actually be honest and stick with the cartel arrangement, i.e. F 2 = 45? ► The answer is given by the reaction function of Bank 1: F 1 ( F 2 = 45) = 90 – 0.5∙ F 2 = 67.5 ► The resulting FFR is r ( F ) = 90 – 0.5∙( F 1 + F 2 ) = 33.75 ► The profits are  1 = r ( F )∙ F 1 = 2,278.125  2 = r ( F )∙ F 2 = 1,518.750 ► Clearly Bank 1 has an incentive to deviate from the cartel arrangement… Cartel solution 45 ► Bank 2’s reaction function : F 2 ( F 1 ) = 90 – 0.5∙ F 1 Bank 2’s profit-maximizing quantity graphed against the amount offered by Bank 1 ► Bank 1’s reaction function : F 1 ( F 2 ) = 90 – 0.5∙ F 2 Bank 1’s profit-maximizing quantity graphed against the amount offered by Bank 2 67.5 optimal deviation by Bank 1

25. microeconomic s the analytics of constrained optimal decisions lecture 6 the oligopoly model (I): competition in quantities  2016 Kellogg School of Management lecture 6 page |24 ► What is the difference between the Cournot and Stackelberg models? ● for Cournot choices are made simultaneously … while for Stackelberg one player announces first its choice and the other player optimally (according to its reaction function) responds ● it is this forward thinking (or anticipation) – first player to announce knows how the second player will respond so it will strategically choose its announcement – that gives an advantage to the “first mover” ► The steps in obtaining the Stackelberg solution : ● for each player derive the residual demand function from the total demand function… ● plug the reaction function of the second player (the second to make the announcement) into the price equation and then use this equation into first player’s profit ● at this point the profit function for the first player depends only on its own choice therefore the final step is to find the choice that maximizes the profit function ● the choice for the second player is found from its reaction function ► The cartel model : ● the two players decide to collude, i.e. behave as a consolidated entity that behaves as a monopolist; the solution to such a model is the standard monopolist case; each player offers half of the monopolist solution and therefore splits the market equally ● the problem with this arrangement is that each player has a strong incentive to deviate given that the other player is honest key points