Production Games Monopolistic Competition Monopolistic competition arises when there are a large number of price-setting firms in an industry with free entry. Suppose there are n firms. In the short run, each firm faces an nth of the market demand (y) Short run analysis is not very different to that of the monopolist. The firm charges pfor an output of y"and makes T profit. The only difference demand is given by pn(y), an nth of total demand. However, in the long run other firms are free to enter if incumbents are making a profit or leave if they make a loss The Long run Firms will continue to enter until there is no incentive for further entry. That is, when profits are driven to zero. As more firms enter demand is reduced to each of the incumbent firms. Suppose once m firms are in the industry profits for each firm They will produce y each at a price of p'in the long run. Each firm is profit maximising, but receiving zero profit. They are average cost pricing. This is still inefficient. A perfectly competitive firm would produce at p= MC, which is a lower price and higher output
Production — Games 1 Monopolistic Competition • Monopolistic competition arises when there are a large number of price-setting firms in an industry with free entry. • Suppose there are n firms. In the short run, each firm faces an nth of the market demand curve. .................. ...... ............................ ................................................................................................................................................................................................................................................................................ . . . . . . . . . . . . ............ ............. ............. ............. ............ ............. ............. ............. 0 p y MC AC p ∗ y ∗ pn(y) MRn π • Short run analysis is not very different to that of the monopolist. The firm charges p ∗ for an output of y ∗ and makes π profit. The only difference is that demand is given by pn(y), an nth of total demand. • However, in the long run other firms are free to enter if incumbents are making a profit or leave if they make a loss. Production — Games 2 The Long Run • Firms will continue to enter until there is no incentive for further entry. That is, when profits are driven to zero. .................... ............................ ................................................................................................................................................................................................................................................................................ . . . . . . . ......... ............. ............. ............. ............. 0 p y AC MC p ∗ y ∗ pm(y) MRm • As more firms enter demand is reduced to each of the incumbent firms. Suppose once m firms are in the industry profits for each firm are zero. They will produce y ∗ each at a price of p ∗ in the long run. • Each firm is profit maximising, but receiving zero profit. They are average cost pricing. This is still inefficient. A perfectly competitive firm would produce at p = MC, which is a lower price and higher output
Production Games Strategic Behaviour The problem with this analysis is that firms ignore the behaviour of their competitors Consider the short run and suppose there were just two firms. Suppose D(p)= l-p and AC=0 for simplicity. If firm A operates in monopolistic competition, they think they face half the demand: Da(p)=(1-p)/2.They roduce at MR=MC. The inverse demand curve is p= 1-2yA. So MR=1-4yA. They produce at 1-4yi=0 This gives: y*= 1/4. Suppose firm B takes this into consideration. Then the actual inverse demand B faces is AyB=1-1/4-yB and not 1-2yB. Profit maximisation will yield a different quantit In fact, firm B would set MR= MC which yields: 3/4-2yg=0 giving yi= 3/8 Notice if firm B took monopolistic competition profits they would get g= 1/8 With strategic behaviour however, firm B gets TB=(1-1/4-3/8)3/8=9/61. This is better Consider the long run. Is there a way to prevent other firms from entering and hence maintaining positive profits? Game theory is the study of strategic behaviour. A game is three things: 1. Players: The individuals(consumers or firms ete. )involved. 2. Strategies: Each players choice set- what they can do 3. Payoffs: The utilities or profits which depend on the strategies all of the players take Anything which has these three features can be described as a game. Game theory is a useful language for describing all kinds of situations- but that is not all
Production — Games 3 Strategic Behaviour • The problem with this analysis is that firms ignore the behaviour of their competitors. • Consider the short run and suppose there were just two firms. Suppose D(p) = 1 − p and MC = 0 for simplicity. • If firm A operates in monopolistic competition, they think they face half the demand: DA(p) = (1 − p)/2. They produce at MR = MC. The inverse demand curve is p = 1 − 2yA. So MR = 1 − 4yA. They produce at 1 − 4y ∗ A = 0. • This gives: y ∗ A = 1/4. Suppose firm B takes this into consideration. Then the actual inverse demand B faces is 1 − y = 1 − yA − yB = 1 − 1/4 − yB and not 1 − 2yB. Profit maximisation will yield a different quantity. • In fact, firm B would set MR = MC which yields: 3/4 − 2yB = 0 giving y ∗ B = 3/8. • Notice if firm B took monopolistic competition profits they would get πB = 1/8. • With strategic behaviour however, firm B gets πB = (1 − 1/4 − 3/8)3/8 = 9/64. This is better. • Consider the long run. Is there a way to prevent other firms from entering and hence maintaining positive profits? Production — Games 4 Games • Game theory is the study of strategic behaviour. A game is three things: 1. Players: The individuals (consumers or firms etc.) involved. 2. Strategies: Each player’s choice set — what they can do. 3. Payoffs: The utilities or profits which depend on the strategies all of the players take. • Anything which has these three features can be described as a game. • Game theory is a useful language for describing all kinds of situations — but that is not all
Production Games One of the most famous games is the prisoners'dilemma. This has two players with two strategies each and four associated payoffs for each player. It can be represented by the two-by-two matrix below. 5 0 This is called the normal form or strategic form of the game. All discrete action space games can be represented in such matrices. Action spaces need not be discrete-for cample, firms set quanti d consumers choose bundles of goods ed will be simultaneous moue games. All players make their decision without observing the strategies played by other agents. In many interesting games players do observe some of the actions of others. Firms may make decisions sequentially. Such dynamic games will be considered very briefly Best Responses Suppose a player's opponent in a game plays a particular strategy. What would the player do? The strategy that ises the player's payoff given their opponent's play is a best respon The collection of best responses for each possible action of their opponent is called a best response function-or a reaction function(or even a best reply function). Plotting this gives the reaction What are the best responses in the prisoners'dilemma Common practice is to underline the payoffs corresponding to best responses in such a normal form game
Production — Games 5 Some Examples • One of the most famous games is the prisoners’ dilemma. This has two players with two strategies each and four associated payoffs for each player. It can be represented by the two-by-two matrix below. C D C 3 3 5 0 D 0 5 1 1 • This is called the normal form or strategic form of the game. • All discrete action space games can be represented in such matrices. Action spaces need not be discrete — for example, firms set quantities and consumers choose bundles of goods. • Most of the games presented will be simultaneous move games. All players make their decision without observing the strategies played by other agents. • In many interesting games players do observe some of the actions of others. Firms may make decisions sequentially. Such dynamic games will be considered very briefly. Production — Games 6 Best Responses • Suppose a player’s opponent in a game plays a particular strategy. What would the player do? The strategy that maximises the player’s payoff given their opponent’s play is a best response. • The collection of best responses for each possible action of their opponent is called a best response function — or a reaction function (or even a best reply function). Plotting this gives the reaction curve. • What are the best responses in the prisoners’ dilemma? C D C 3 3 5 0 D 0 5 1 1 • Common practice is to underline the payoffs corresponding to best responses in such a normal form game
Production Games A best response to a best response to a best response to a.. is called a Nash If all the payoffs in a particular cell are underlined then the strategies that correspond to that cell are a Na rategies. Sometimes this can be very complicated. Chess In the prisoners dilemma the Nash equilibrium is (D, D] Player 2 BRI Player 1 In the above graph the best responses are plotted for each player. These are rea rves. The Nash equilibrium is where the wes intersect. This is a useful method for finding equilibria. Along the axis are the possible actions taken by players 1 and 2. BR, is the best reply function of player 1 and BR2 is player 2s. They intersect at ( D, DI, the equilibrium. What are the dotted lines for? Mixed Strategies Suppose a player could "toss a coin"to decide which action to take. What would their opponent de Such an action is called a mired strategy. For example. in the prisoners'dilemma, with a fair coin, the opponent would expect to see C half the time and D half the time. They calculate a best response If they choose to play C, half the time they would get 3, and half the time they would get 0. If they choose to play D half the time they would get 5, and half the time they would get 1. Their expected payoffs 32+0=2<3=57+1 2 mixes half-half BRI Therefore they would play D in response to such a mixture. In fact, they would play D in response to any mixture. Hence the dotted lines can be filled in as above. Mixed strategies are beyond the level required here
Production — Games 7 Nash Equilibrium • A best response to a best response to a best response to a ... is called a Nash equilibrium. • If all the payoffs in a particular cell are underlined then the strategies that correspond to that cell are a Nash equilibrium. Notice equilibrium is a collection of strategies. Sometimes this can be very complicated. Chess? • In the prisoners’ dilemma the Nash equilibrium is {D, D}. ................................................................................................................................................................................................................................................................................ ............. ............. ............. ............. ............. ............. ............. ............. ............. . . . . . . . . . Player 2 Player 1 BR2 BR1 C D C D • • • • In the above graph the best responses are plotted for each player. These are reaction curves. The Nash equilibrium is where the two reaction curves intersect. This is a useful method for finding equilibria. • Along the axis are the possible actions taken by players 1 and 2. BR1 is the best reply function of player 1 and BR2 is player 2s. They intersect at {D, D}, the equilibrium. What are the dotted lines for? Production — Games 8 Mixed Strategies • Suppose a player could “toss a coin” to decide which action to take. What would their opponent do? • Such an action is called a mixed strategy. For example, in the prisoners’ dilemma, with a fair coin, the opponent would expect to see C half the time and D half the time. They calculate a best response. • If they choose to play C, half the time they would get 3, and half the time they would get 0. If they choose to play D half the time they would get 5, and half the time they would get 1. Their expected payoffs are: 3 1 2 + 0 1 2 = 3 2 < 3 = 5 1 2 + 1 1 2 ................................................................................................................................................................................................................................................................................ ......................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . 2 mixes half-half Player 2 Player 1 BR2 BR1 C D C D • • • • Therefore they would play D in response to such a mixture. In fact, they would play D in response to any mixture. Hence the dotted lines can be filled in as above. Mixed strategies are beyond the level required here
Production Games The Stag Hunt Best responses in the stag hunt are again underlined. Below it is a best response diagram. The dashed lines indicate BRz. Notice there are three Nash equilibria -two pure strategy Nash equilibria and one "hidden"mixed strategy equilibrium. In any discrete action space game there is at least one Nash equilibrium Road Congestion How can game theory be applied to economic problems? Consider the following road congestion game. n students go to lectures. Should they drive(D)or take the bus(B), given they want to get there as fast as possible? If they all drive, traffic congestion implies it takes 45 minutes to get to the lecture If they all take the bus, the road is clear and it only takes 30 minntes If everyone else takes the bus and a single student takes the car, the one in the car flies to work(no other he road) in 15 minutes. If everyone else takes the car and a single student takes the bus, the one on the bus takes 60 minutes- all those cars and all those bus-stops. Everyone Else Me This is a generalisation of the prisoners'dilemma. The student will drive. In fact it is dominant to do so. All the other students will arrive at the same decision. Everyone takes the car Wouldn't it be nice if everyone took the bus
Production — Games 9 The Stag Hunt • Best responses in the stag hunt are again underlined. Below it is a best response diagram. S R S 5 5 4 0 R 0 4 3 3 ................................................................................................................................................................................................................................................................................ . .......................................................................................................................................................................................................................... . . . . . . . . . ............. ............. ............. ............. ............. ............. ............. ............. ............. ..... Player 2 Player 1 BR2 BR1 S R S R • • • • The dashed lines indicate BR2. Notice there are three Nash equilibria — two pure strategy Nash equilibria and one “hidden” mixed strategy equilibrium. In any discrete action space game there is at least one Nash equilibrium. Production — Games 10 Road Congestion • How can game theory be applied to economic problems? Consider the following road congestion game. n students go to lectures. Should they drive (D) or take the bus (B), given they want to get there as fast as possible? • If they all drive, traffic congestion implies it takes 45 minutes to get to the lecture. • If they all take the bus, the road is clear and it only takes 30 minutes. • If everyone else takes the bus and a single student takes the car, the one in the car flies to work (no other cars on the road) in 15 minutes. If everyone else takes the car and a single student takes the bus, the one on the bus takes 60 minutes — all those cars and all those bus-stops. Everyone Else Me B D B 30 min 60 min D 15 min 45 min • This is a generalisation of the prisoners’ dilemma. The student will drive. In fact it is dominant to do so. All the other students will arrive at the same decision. Everyone takes the car. • Wouldn’t it be nice if everyone took the bus?
Production Games Dynamic Games Suppose an incumbent monopolist is facing a possible entrant to a market. Such an action would reduce the onopolist's profit. This is an example of a dymamic game. The entrant makes a decision whether to enter before the monopolist can react. A simple game to model this situation is the entry deterrence game below. (1,2) (0.5) The monopolist, M, makes a "threat". If the new firm, E enters, M threatens to "fight".(Perhaps flood the market with cheap goods). The entrant will get no profit and will lose the co If Mfdoesn't fight", they split the market equally. E still loses a start up cost of 1 If E stays out of the market. they make zero profit. M would make monopoly profits of 5. But this is not a credible threat. Why not? Oligopoly An industry with a few price-setting firms is an oligopoly. If there are just two firms, it is a duopoly. 1. Cournot: Firms simultaneously choose quantities to maximise prof 2. Bertrand: Firms simultaneously choose prices to maximise profits 1. Price Leadership: Firms sequentially choose prices to maximise profits Each can be represented as a game. The first two are simultaneous move games, the second two dynamic games
Production — Games 11 Dynamic Games • Suppose an incumbent monopolist is facing a possible entrant to a market. Such an action would reduce the monopolist’s profit. This is an example of a dynamic game. The entrant makes a decision whether to enter before the monopolist can react. A simple game to model this situation is the entry deterrence game below. .............................................................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................... • .............................................................................................................................................................................................................................................................................................. • • • • E M Fight Don’t Fight Enter Don’t Enter (0, 5) (1, 2) (−1, 0) • The monopolist, M, makes a “threat”. If the new firm, E enters, M threatens to “fight”. (Perhaps flood the market with cheap goods). The entrant will get no profit and will lose the cost of starting up (1). • If M “doesn’t fight”, they split the market equally. E still loses a start up cost of 1. • If E stays out of the market, they make zero profit. M would make monopoly profits of 5. • But this is not a credible threat. Why not? Production — Games 12 Oligopoly • An industry with a few price-setting firms is an oligopoly. If there are just two firms, it is a duopoly. • Four different examples will be considered (with two firms for simplicity): 1. Cournot: Firms simultaneously choose quantities to maximise profits. 2. Bertrand: Firms simultaneously choose prices to maximise profits. 3. Stackelberg: Firms sequentially choose quantities to maximise profits. 4. Price Leadership: Firms sequentially choose prices to maximise profits. • Unlike in the non-strategic models so far, the variable that is chosen can have a striking effect upon the outcome. • Each can be represented as a game. The first two are simultaneous move games, the second two dynamic games