4/21/2016. Intermediate Microeconomics W3211. Lecture 21: Game Theory 3 Back to the Firms. The Story So Far. Today. Monopolies. Introduction.

Transcription

1 //06 Intermediate Microeconomics W3 Lecture : Game Theor 3 Back to the Firms Introduction Columbia Universit, Spring 06 Mark Dean: mark.dean@columbia.edu The Stor So Far. 3 Toda We now have a number of tools to analze strategic situations Defined a game Thought about how to solve a game Dealt with issues of Multiple equilibria (equilibrium selection) Existence (mixed strategies) Thought about the difference between sequential and simultaneous move games Go back to thinking about firms In particular, we are going to relax the assumption that firms are price takers Think about firms that are big enough to affect price Take this into account when deciding what to do First consider the case of Monopol Onl one firm Ver boring game Then consider the case of a small number of firms Oligopol More interesting game Varian Ch. 5/6/8 Feldman and Serrano Ch /3 Monopolies 6 Up until now, we have assumed that firms have been small In particular, the treat prices as given However this is clearl not alwas the case Monopol Sometimes firms are large Get to set their own prices 5

2 //06 Monopolies 7 Wh Monopolies? 8 How does this happen? Presumabl monopolies make big profits From previous lectures, we know that excess profits should encourage new firms to enter Wh does that not alwas happen? a legal fiat; e.g. US Postal Service a patent; e.g. a new drug sole ownership of a resource; e.g. a toll highwa formation of a cartel: man firms colluding to restrict output to affect prices. examples: OPEC : think of the 73 oil crisis De Beers : controls 80% of world production of diamonds large economies of scale; e.g. local utilit companies. 9 The Demand Curve 0 We will assume that monopolies act to maximize profits Remember, that for a perfectl competitive firm, the profit maximization problem was to choose output to maximize How does this change with the monopolist? Well, we said that a monopolist gets to choose prices p Demand Function But, there will still be a relationship between price the choose and the quantit the can sell D(p) Given b the demand function The Demand curve tells the firm how much people will bu given price p The Demand Curve Demand Function The latter formulation will be more convenient P() So, what should the firm do to maximize profits? Let s take first order conditions! Profit=revenue-cost First order conditions Equivalentl it tells us how much the can charge if the want to sell the amount Revenue= Cost (Remember, ou have to worr about corner solutions etc)

3 //06 3 Revenue= Cost This is the same condition as in the perfect competition case But now marginal revenue has an additional term: This is exactl the firm taking into account the effect that increased production has on prices Revenue= Cost What does Revenue look like? Well, note that is negative, so the marginal revenue at will be less than the price Side note: at the moment, we are assuming that the onl thing the monopolist can do is charge a constant price per unit output Later in the course we will consider some sneak other things that a monopolist might want to do 5 Revenue and Demand 6 For example, if the inverse demand curve is given b Then revenue is given b And so marginal revenue is given b Demand: P() p p( ) 7 Monopolistic Output 8 So what does the optimal output look like for a monopolist? Cost Demand Revenue m The will choose in order to get marginal cost to equal marginal revenue 3

4 //06 Monopolistic Output 9 Monopolistic Output 0 Cost Demand p m Cost Demand Revenue Revenue m m What price will the charge? The price that allows them to sell this amount of output Which we can read off from the demand curve Monopolistic Output How does the behavior of a monopolist compare to that of perfectl competitive firms Remember, for a perfectl competitive firm Price equals marginal cost The suppl curve is given b the marginal cost Equilibrium happens when suppl equals demand p m p e Revenue Cost Demand m e Monopolists will suppl less and charge higher prices than a perfectl competitive industr 3 Surplus Under Perfect Competition What happens to surplus in the monopol case? p m Consumer surplus Producer surplus m

5 //06 Surplus under Monopol 5 6 p m Consumer surplus Producer surplus What happens to surplus in the monopol case? Three effects Producer surplus increases Consumer surplus decreases Total surplus decreases i.e. monopoles are inefficient m Price Mark-Up and Elasticit 7 Price Mark-Up and Elasticit 8 Let s go back to the first order conditions for the monopolist Rearranging gives But is marginal cost, and is the reciprocal of the price elasticit of demand In the perfectl competitive case, price would equal marginal cost Monopolists mark up the price above marginal cost The degree of the mark up depends on the price elasticit of demand The lower the elasticit, the higher the mark up This makes sense: the less responsive demand is to prices, the higher the price that monopolists will charge Oligopol 30 We now want to move from one firm (monopol) to a small number of firms (oligopol) Concentrate on the case of two firms (duopol) Oligopol It turns out there are different was to analze this case, depending on how we assume firms interact Sometimes called the market structure We are going to think first about the case in which firms compete on quantit Sometimes called Cournot competition In order to analze the market, we will make use of game theoretic concepts 9 5

6 //06 Quantit Competition 3 Quantit Competition 3 Assume that firms compete b choosing output levels. If firm produces units and firm produces units then total quantit supplied is +. The market price will be p( + ). This is what makes this a game Output of firm affects the price of firm, and visa versa Suppose firm takes firm s output level choice as given. Then firm sees its profit function as ( ; ) p( ) c ( ). Given, what output level maximizes firm s profit? The firms total cost functions are c ( ) and c ( ). Suppose that the market inverse demand function is p( T) 60 T and that the firms total cost functions are 33 Then, for given, firm s profit function is ( ; ) ( 60 ). 3 c( ) and c( ) 5. Then, for given, firm s profit function is ( ; ) ( 60 ). So, given, firm s profit-maximizing output level solves I.e., firm s best response to is R( )

7 //06 I.e., firm s best response to is R( ) 5. Ke point: Optimal output of firm depends on the output of firm Firm s reaction curve R( ) Similarl, given, firm s profit function is ( ; ) ( 60 ) Similarl, given, firm s profit function is ( ; ) ( 60 ) 5. So, given, firm s profit-maximizing output level solves Similarl, given, firm s profit function is ( ; ) ( 60 ) 5. So, given, firm s profit-maximizing output level solves I.e., firm s best response to is 5 R( ). Firm s reaction curve 5 R( ). 5/ 5 7

8 //06 3 What do we think will happen in this case? Well, we can think of this as a game with Two plaers The action of each plaer is to choose a quantit Paoff is given b the profit for each firm How do we solve games? An equilibrium is when each firm s output level is a best response to the other firm s output level, for then neither wants to deviate from its output level A pair of output levels (, ) is a Cournot-Nash equilibrium if and Nash equilibrium! This is an action for each plaer which is the best for them, given what the other plaer is doing R( ) 5 5 and R ( ). 5 R( ) 5 5 and R ( ). Substitute for to get R( ) 5 5 and R ( ). Substitute for to get R( ) 5 5 and R ( ). Substitute for to get Hence

9 //06 R( ) 5 5 and R ( ). Substitute for to get Hence So the Cournot-Nash equilibrium is (, ) ( 3, 8) / Firm s reaction curve R( ) 5. Firm s reaction curve 5 R( ) Firm s reaction curve R( ) 5. Firm s reaction curve R( ). Cournot-Nash equilibrium, 3, 8. 5 Quantit Competition Generall, given firm s chosen output level, firm s profit function is ( ; ) p( ) c ( ) and the profit-maximizing value of solves p ( ) p ( ) c ( ) 0. The solution, = R ( ), is firm s Cournot- Nash reaction to. 5 Quantit Competition Similarl, given firm s chosen output level, firm s profit function is ( ; ) p( ) c ( ) and the profit-maximizing value of solves p ( ) p ( ) c ( ) 0. The solution, = R ( ), is firm s Cournot- Nash reaction to. 53 Quantit Competition Firm s reaction curve R( ). Firm s reaction curve R( ). Cournot-Nash equilibrium = R ( ) and = R ( ) 5 9

10 // Q: Are the Cournot-Nash equilibrium profits the largest that the firms can earn in total? Not if the are allowed to co-operate! Suppose the two firms want to maximize their total profit and divide it between them. Their goal is to choose cooperativel output levels and that maximize m (, ) p( )( ) c( ) c( ). This is similar to the monopolist problem! In fact, if the firms have constant costs, it is the monopolist problem Tpicall, this will give the firm higher profit Sometimes called forming a cartel Suppose two firms face an inverse market demand of p( T ) = T and have total costs of c ( ) = and c ( ) =. Can the make more in a cartel or if the compete? 57 What is each firm s per period profit in the cartel? p( T ) = T, c ( ) =, c ( ) =. If the firms collude then their joint profit function is M (, ) = ( )( + ). What values of and maximize the cartel s profit? 58 M (, ) = ( )( + ). What values of and maximize the cartel s profit? Solve π π M M M (, ) = ( )( + ). What values of and maximize the cartel s profit? Solve π π M M Solution is M = M =

11 //06 M (, ) = ( )( + ). M = M = maximizes the cartel s profit. The maximum profit is therefore M = ( 8)(8) = 96. Suppose the firms share the profit equall, getting 96/ = 8 each per period. What are the firms profits if the do not co-operate? p( T ) = T, c ( ) =, c ( ) =. Given, firm s profit function is ( ; ) = ( ). 6 6 What are the firms profits if the do not cooperate? p( T ) = T, c ( ) =, c ( ) =. Given, firm s profit function is ( ; ) = ( ). The value of that is firm s best response to solves π 0 R( ). 63 What are the firms profits if the do not cooperate? ( ; ) = ( ). R( ). Similarl, R( ). The C-N equilibrium (, ) solves = R ( ) and = R ( ) = = 8. 6 What are the firms profits if the do not cooperate? So both firms earn more if the form a cartel 66 ( ; ) = ( ). = = 8. But is such a cartel stable? What would firm want to do if firm is producing at the cartel level? So each firm s profit in the C-N equilibrium is = = ()(8) 8 6 each period. 65

12 //06 Let s sa that firm knows that firm will produce at the cartel level. What is their best response? Firm produces quantit CH that maximizes firm s profit given that firm continues to produce M =. What is the value of CH? CH = R ( M ) = ( M )/ = ( )/ = 5. Firm s profit in the period in which it cheats is therefore CH = ( 5 )(5) 5 = So a profit-seeking cartel in which firms cooperativel set their output levels is fundamentall unstable. This is basicall a prisoner s dilemma Both firms would do better if the collude, but this is not an equilibrium E.g., OPEC s broken agreements & Punishment Strategies 70 So a profit-seeking cartel in which firms cooperativel set their output levels is fundamentall unstable. This is basicall a prisoner s dilemma Both firms would do better if the collude, but this is not an equilibrium Imagine a firm that is deciding between. Staing in the cartel forever. Cheating on the cartel, and then being punished forever b the other cartel member E.g., OPEC s broken agreements. But is the cartel unstable if the game is repeated man times, instead of being plaed onl once? Then there is an opportunit to punish a cheater. & Punishment Strategies 7 & Punishment Strategies 7 To determine if such a cartel can be stable we need to know 3 things: (i) What is each firm s per period profit in the cartel? (ii) What is the profit a cheat earns in the first period in which it cheats? (iii) What is the profit the cheat earns in each period after it first cheats? (i) What is each firm s per period profit in the cartel? This is just the cartel profits - 8 per period

13 //06 & Punishment Strategies & Punishment Strategies (ii) What is the profit a cheat earns in the first period in which it cheats? This is what firm would get if the best responded to firm producing the cartel output The would get 50 (iii) What is the profit the cheat earns in each period after it first cheats? This depends upon the punishment inflicted upon the cheat b the other firm & Punishment Strategies & Punishment Strategies 76 (iii) What is the profit the cheat earns in each period after it first cheats? This depends upon the punishment inflicted upon the cheat b the other firm. Suppose the other firm punishes b forever after not cooperating with the cheat. Then each firm will earn the Cournot profits - 6 in each period Note that this is a credible threat 75 To determine if such a cartel can be stable we need to know 3 things: (i) What is each firm s per period profit in the cartel? 8. (ii) What is the profit a cheat earns in the first period in which it cheats? 50. (iii) What is the profit the cheat earns in each period after it first cheats? 6. & Punishment Strategies 77 & Punishment Strategies Each firm s periodic discount factor is /(+r). The present-value of firm s profits if it does not cheat is?? Each firm s periodic discount factor is /(+r). The present-value of firm s profits if it does not cheat is 78 3

14 //06 & Punishment Strategies & Punishment Strategies Each firm s periodic discount factor is /(+r). The present-value of firm s profits if it does not cheat is Each firm s periodic discount factor is /(+r). The present-value of firm s profits if it does not cheat is The present-value of firm s profit if it cheats this period is?? The present-value of firm s profit if it cheats this period is & Punishment Strategies 8 So the cartel will be stable if ( r) r r r r. Cartels can be stable if. The game is plaed an infinite number of times. The plaers are patient enough 8 Summar 8 Summar Toda we have Modelled monopolies Modelled duopolies that compete on quantit Thought about when duopolies form cartels 83