Chapter 31: Duopoly
Chapter 31: Duopoly 31.1: Introduction We now apply the game-theoretic ideas that we developed in chapter 30 to the study of duopoly. This is a market in which there are just two sellers. It is a case between competition and monopoly. In the former the firms take the price as given; in the latter the monopolist can set the price. In duopoly each firm has an influence on the price but must take into account that the other firm also has an influence on the price. The profits of each firm depend on the decisions of both firms – and they must both take that into account when taking their decisions. As we will see the outcome of the duopoly game depends upon its rules. In this chapter we consider two different sets of rules – the first we call the quantity-setting game (which is referred to as the Cournot model after the economist Cournot), and the second the price-setting game (which is referred to as the Bertrand model after the economist Bertrand). In each of these we keep things as simple as possible by assuming that the firms are identical in their technology and in the good that they produce. We shall assume that both firms have constant returns to scale, with a cost function given by C(q) = cq, where q is the quantity produced and c is the constant marginal (and average) cost. Let us begin by considering the quantity-setting game (the Cournot game).
31.2: The Quantity-Setting Game In this game, the two duopolists, simultaneously and independently, choose the level of their output. Firm 1 chooses q1 and firm 2 chooses q2. Then the market decides the price of the product, p, through the aggregate demand curve: p = a – b(q1 + q2)
(31.1)
Note the implicit assumption here – that the good that the two firms produce is identical and that the aggregate quantity sold determines the price at which it can be sold, through the aggregate demand curve. We shall see if there is a Nash Equilibrium in this game. That is, we will see if there is a pair (q1, q2) for which each firm is happy with its decision (output) given the decision (output) of the other. As we know from chapter 30, if such an equilibrium exists it must lie on the intersection of the two firms’ reaction functions. We must therefore start by finding these two reaction functions. That is, we must find the optimal decision of each firm – as a function of the decision of the other firm. We work in (q1, q2) space, and we start with finding the reaction function of firm 1. To find this we introduce a new concept – that of an iso-profit curve. This, as you might anticipate, is a locus of points for which the profit is the same. Let us take firm 1 and write down an expression for its profits. Denoting the profits of firm 1 by π1 we have that profits are revenue minus costs, so if the firm produces q1 and sells this quantity at a price of p then the profits are given by π1 = p q1 – c q1
Now we must remember that the price is determined by the aggregate demand function (31.1). Substituting this is in the above expression we get π1 = [a – b(q1 + q2)] q1 – c q1 from which it follows that the profits of firm 1 are given by π1 = (a – c - b q2) q1 – b q12
(31.2)
Note very importantly that this is a function of both q1 and q2. It is linear in q2. In fact, note that as q2 increases the profits of firm 1 decrease – since the coefficient on q2 is negative. It is quadratic in q1 with a negative coefficient on q1 - so as q1 increases then first the profits of firm 1 increase and then they decrease. An iso-profit curve for firm 1 is defined by π1 = constant, that is by (a – c - b q2) q1 – b q12 = constant Plotted in (q1, q2) space an iso-profit curve is quadratic. We give a particular example (in which a = 110, b = 1 and c = 10) in figure 31.1.
We have already noted that the profits of firm 1 fall as q2 rises. It follows from this that physically higher iso-profit curves correspond to lower profits. In fact in figure 31.1, the profits attached to the 10 iso-profit curves pictured – starting at the physically lowest – are 2300, 2000, 1700, 1400, 1100, 800, 500, 200 and –100 (yes the top curve has a loss of 100). Now to find the reaction function of firm 1 we need to find, for each level of output of firm 2, the optimal level of output of firm 1 – by this we mean the profit-maximising level of output. So given some level of output of firm 2, that is, given some horizontal line in the above figure, we want to find the point on it which lies on the iso-profit curve with the greatest level of output – that is we want to find the physically-lowest iso-profit curve. It should be clear that this point must be at the summit of an iso-profit curve. Joining all these points together we find the reaction function of firm 1. It is shown in figure 31.3. We can also find it mathematically. The proof is found in the Mathematical Appendix and leads to the following reaction curve for firm 1: q1 = (a – c – b q2)/(2b)
(31.3)
In the case we are considering (where a = 110, b =1 and c = 10) this gives q1 = (100 - q2)/2 which is the straight line in figure 31.3 joining the peaks of the iso-profit curves – as we have already shown verbally1. We have found the reaction function of firm 1. Let us just look at it a little. One point on it is where q2 = 0. The optimal response is q1 = 50. So if firm 2 produces nothing the best thing that firm 1 can do is to produce 50. A moment’s thought should convince you that 50 must therefore be the monopoly output. A rather longer thought may be needed to interpret the other extreme of the reaction function – at which firm 2 produces 100 and the optimal response of firm 1 is to produce nothing. We shall later show that the output of 100 is what would be produced if the firms behaved competitively. In the meantime you might like to ponder why. Because we have identical firms we can easily work out the iso-profit map for firm 2 and its reaction function. We simply interchange q1 and q2 in what we have done. We thus get figure 31.4.
The straight line in this figure is firm 2’s reaction function. For example, if firm 1 produces nothing firm 2 should produce 50; if firm 1 produces 100, firm 2 should produce nothing; if firm 1 produces 50 firm 2 should produce 25. And so on. We can now answer the question as to whether there is a Nash Equilibrium in the game. Let us put the two reaction curves together. We get figure 31.6. There is a unique intersection at the point (33⅓, 33⅓) and so this must be the unique Nash Equilibrium of the game. Game theory predicts that each firm will produce 33⅓ and the total output will be 66⅔ giving a price of 43⅓. You should be clear why this is a Nash equilibrium: if firm 2 produces 33⅓ then the optimal output for firm 1 (given by its reaction function) is 33⅓; and if firm 1 produces 33⅓ then the optimal output for firm 2 (given by its reaction function) is 33⅓. So in this Nash equilibrium both firms are happy with 1
Be careful when trying to interpret this as it gives q1 as a function of q2,
their decision given the decision of the other. You should note that at no other point in the graph is this true.
31.3: Collusion? You might have realised that it might be better for the two firms to try and collude – rather than choose their outputs independently of each other. We can see the possibilities for collusion if we put together figures 31.3 and 31.4. This gives us figure 31.7, into which I have inserted a curve which one might tentatively call the ‘contract curve’ between the two duopolists,
Why might we call this line the ‘contract curve’? Because it is the locus of points efficient in the sense of the profits of the two firms: given any level of profit of firm 1 then firm 2’s profit is highest on the contract curve; given any level of profit of firm 2 then firm 1’s profit is highest on the contract curve. You might find it interesting to note that the equation of this contract curve is given by q + q2 = 50 that is the joint output of the two firms is 50. Does this 50 remind you of anything – yes the monopoly output. So we get the conclusion that the two firms are behaving efficiently if they are acting as if they are a monopolist. This is a nice conclusion – though a rather obvious one. However to be on the contract curve they have to collude. Moreover each firm has a natural incentive to cheat on any collusive agreement. Why? Simply because none of the points on the contract curve are on either firm’s reaction function – so neither firm is behaving optimally given the output decision of the other. This tells us that there are natural incentives why any collusive agreement may break down and we end up at the Nash Equilibrium. We can see this more clearly by working out the profits of the firms at the Nash Equilibrium and at the symmetrical joint profit maximising situation. We have, where we have rounded the profits to the nearest whole number. q1 = 33⅓ q1 = 25
q2 = 33⅓ 1111, 1111 1041, 1389
q2 = 25 1389, 1041 1250, 1250
You might recognise the structure of this game – it is that of a classic prisoner’s dilemma. Note that there is a unique Nash Equilibrium (which we have already identified) and there is another outcome – the joint profit maximising outcome – which is better for both of them but which is not a Nash Equilibrium. Both firms have an incentive to renege on an agreement to produce 25 units. Collusion is not a Nash Equilibrium in this game.
31.4: Sequential Play You may well be asking whether the outcome may be different or better if we changed the rules of the game – for example, letting the players play sequentially rather than simultaneously. This situation is referred to as the Stackelberg model after the economist Stackelberg. Let us see what happens. Consider first the situation where firm 1 chooses first and then announces the decision to firm 2 who then decides. What does firm 1 do? We have already considered this type of situation in chapter 8. Firm 1 will reason that whatever output firm 1 chooses then firm 2 will respond with the appropriate output from its (that is, firm 2’s) reaction function – after all, that gives the optimal response by firm 2 to any output decision of firm 1. Now we know the reaction function of firm 1 – it is the line shown in figure 31.8. Firm 1 knows that whatever value of q1 it chooses then firm 2 will choose the corresponding point on the reaction function. So essentially firm 1 is choosing a point on the reaction function. Which is the best point on it from firm 1’s point of view?
Presumably the point on it on the highest (in terms of profit) iso-profit curve – that is, on the physically lowest curve in the figure. This optimal point is labelled ‘1’ in the figure. It is where q1 = 50 and q2 = 25 and where the profits of firm 1 are 1250. The profits of firm 2 in this situation are just 625. Using symmetry, the outcome when firm 2 moves first is q1 = 25 and q2 = 50 with respective profits of 625 and 1250. Clearly firm 1 prefers to move first, then prefers simultaneous play and least prefers firm 2 to move first. For firm 2 the opposite is true – it prefers to move first, then prefers simultaneous play and least prefers firm 1 to move first. In this case, it is clear that the first mover has an advantage. Do you think that this is always the case?
31.5: Monopoly, Duopoly and Competition Compared We have already worked out what will happen in this specific example: under monopoly we have an output of 50, under duopoly a total output of 66⅔ and under competition a total output of 100. Correspondingly the price under monopoly is 50, under duopoly is 46⅓ and under competition is 10 (the marginal cost). We can generalise these results. We know the reaction functions of the firms. That for firm 1 is q1 = (a – c – b q2)/(2b) , so we can conclude that the monopoly output is found by putting q2 = 0: monopoly output = (a – c)/(2b) The total duopoly output is given by the sum of the values of q1 and q2 at the intersection of the two reaction functions q1 = (a – c – b q2)/(2b) and q2 = (a – c – b q1)/(2b). Hence q1 = q2 = (a – c)/(3b) and so total duopoly output = 2(a – c)/(3b) The total competitive output can be found in two ways. First it is where the price is equal to the marginal cost, that is where p = a – b(q1 + q2) (the demand curve) = c (the marginal cost). This solves to give total competitive output = (a – c)/(b) The second way is by arguing that if one firm produces so much that the other firm’s optimal output is zero then the first firm is producing the competitive outcome. This gives the same output as that stated above (put q1 = 0 in firm 1’s reaction function and solve for q2).
It is interesting to note that the duopoly output is between the monopoly output and the competitive output. Even more interesting is a generalised result that you could try to prove (though the proof requires the generalisation of the concept of a Nash Equilibrium to a game with n players). This generalised result says that the total output in an oligopoly situation with n identical firms playing the same rules is given by total oligopoly output = n (a – c)/[(n+1)b] You will see that as n approaches infinity this approaches the competitive output. This is a reassuring result. You may like to work out yourself the implication for the price of the product in the various situations. For the record the price is lowest under competition, is highest under monopoly and is in-between in an oligopoly with n firms (with the price approaching the competitive price as n approaches infinity).
31.6: A Price-Setting Game A completely different scenario emerges when we change the rules of the game. Suppose we now make it a price-setting game. That is, each firm, independently and simultaneously chooses the price for its product. If we continue to assume that the firms produce an identical product, and if we make the rather key assumption that both firms can satisfy any demand that is forthcoming, then it is clear that if one firm is charging less than the other then it will get all the demand. So the reaction functions of the two firms are easy to work out: given any price charged by the other firm the optimal price for a firm is a fraction less than the other firm – as long as the price is not lower than the marginal (average) cost. So both firms will undercut each other until they both have prices equal to the marginal cost. This is the unique Nash Equilibrium. This is an interesting case as the outcome is exactly that of competition – the price is equal to the marginal cost. If this is what the world looks like we do not need regulatory bodies to control duopolies or oligopolies.
31.7: Summary We have studied duopoly behaviour under a variety of assumptions. In particular we have studied a quantity-setting game and a price-setting game. In a quantity-setting institution the Nash Equilibrium is between the competitive and monopoly output (in terms of the total quantity produced and the price charged). Collusive situation is better for both - but not a Nash Equilibrium. Both firms would prefer that they were the leader. In a price-setting institution the Nash Equilibrium implies Marginal Cost pricing (so duopoly looks like competition).
31.8: Can duopolists co-operate? This exercise is a good follow-up to the game theory experiment of the previous chapter. In essence, it is the same exercise, but the context is different. Again it can be played with two teams – who play two duopolists competing in a market for some good, which they both produce. The structure of the exercise is as follows. The two duopolists produce an identical product. Each duopolist has constant marginal and average costs of 10p per unit. The aggregate demand curve for the duopolists’ product is p = 100 - (q1 + q2 ) where p is the market price (in pence) and q1 and q2 are the outputs of Teams 1 and 2 respectively. Teams must decide on their q’s; the tutor, or the person keeping control, works out p using the formula above. So, for example, if q1 = 20 and q2 = 30 then p = 50 and the revenues to Teams 1 and 2 are respectively 50 x 20 = 1000 (= £10) and 50 x 30 = 1500 (= £15). Costs are respectively 10 x 20 = 200 (= £2) and 10 x 30 = 300 (= £3), so profits are £8 and £12 respectively. As in the game theory experiment, I suggest that the teams play this simple duopoly problem 8 times. The first 4 times, no communication will be allowed between the two Teams; for the last 4 times, communication will be allowed - but no physical threats, and no enforcement of contracts by the tutor/referee. This is a little more complicated as it is not immediately clear what are the payoffs to the various decisions that the teams may take. In fact, there are not just two choices as in the game theory experiment but a whole continuum - the two quantities can be any positive numbers (though numbers greater than 100 would be a bit silly as they would imply negative prices and negative profits). Interestingly, though, it has the same structure – there is a unique Nash equilibrium (can you find it?) that is dominated by an outcome in which the two duopolists collude and collectively produce the monopoly output. The interesting question is whether that collusive outcome can be sustained – obviously without legal support (as the law in most countries is strongly against collusive monopoly-type agreements) and usually without communication (as the law also forbids that). What do you find?
31.9: Mathematical Appendix We find the reaction function for firm 1. To do this, we need to find, for each level of output of firm 2, the optimal output of firm 1. We can find this by differentiating π1 with respect to q1 (for a given level of q2), putting this expression equal to zero and then solving for q1 as a function of q2. From (31.2) we get dπ1/dq1 = (a – c - b q2) –2 b q1 and so dπ1/dq1 = 0 implies that q1 = (a – c – b q2)/(2b) which is equation (31.3).
31.2: The Quantity-Setting Game In this game, the two duopolists, simultaneously and independently, choose the level of their output. Firm 1 chooses q1 and firm 2 chooses q2. Then the market decides the price of the product, p, through the aggregate demand curve: p = a – b(q1 + q2)
(31.1)
Note the implicit assumption here – that the good that the two firms produce is identical and that the aggregate quantity sold determines the price at which it can be sold, through the aggregate demand curve. We shall see if there is a Nash Equilibrium in this game. That is, we will see if there is a pair (q1, q2) for which each firm is happy with its decision (output) given the decision (output) of the other. As we know from chapter 30, if such an equilibrium exists it must lie on the intersection of the two firms’ reaction functions. We must therefore start by finding these two reaction functions. That is, we must find the optimal decision of each firm – as a function of the decision of the other firm. We work in (q1, q2) space, and we start with finding the reaction function of firm 1. To find this we introduce a new concept – that of an iso-profit curve. This, as you might anticipate, is a locus of points for which the profit is the same. Let us take firm 1 and write down an expression for its profits. Denoting the profits of firm 1 by π1 we have that profits are revenue minus costs, so if the firm produces q1 and sells this quantity at a price of p then the profits are given by π1 = p q1 – c q1
Now we must remember that the price is determined by the aggregate demand function (31.1). Substituting this is in the above expression we get π1 = [a – b(q1 + q2)] q1 – c q1 from which it follows that the profits of firm 1 are given by π1 = (a – c - b q2) q1 – b q12
(31.2)
Note very importantly that this is a function of both q1 and q2. It is linear in q2. In fact, note that as q2 increases the profits of firm 1 decrease – since the coefficient on q2 is negative. It is quadratic in q1 with a negative coefficient on q1 - so as q1 increases then first the profits of firm 1 increase and then they decrease. An iso-profit curve for firm 1 is defined by π1 = constant, that is by (a – c - b q2) q1 – b q12 = constant Plotted in (q1, q2) space an iso-profit curve is quadratic. We give a particular example (in which a = 110, b = 1 and c = 10) in figure 31.1.
We have already noted that the profits of firm 1 fall as q2 rises. It follows from this that physically higher iso-profit curves correspond to lower profits. In fact in figure 31.1, the profits attached to the 10 iso-profit curves pictured – starting at the physically lowest – are 2300, 2000, 1700, 1400, 1100, 800, 500, 200 and –100 (yes the top curve has a loss of 100). Now to find the reaction function of firm 1 we need to find, for each level of output of firm 2, the optimal level of output of firm 1 – by this we mean the profit-maximising level of output. So given some level of output of firm 2, that is, given some horizontal line in the above figure, we want to find the point on it which lies on the iso-profit curve with the greatest level of output – that is we want to find the physically-lowest iso-profit curve. It should be clear that this point must be at the summit of an iso-profit curve. Joining all these points together we find the reaction function of firm 1. It is shown in figure 31.3. We can also find it mathematically. The proof is found in the Mathematical Appendix and leads to the following reaction curve for firm 1: q1 = (a – c – b q2)/(2b)
(31.3)
In the case we are considering (where a = 110, b =1 and c = 10) this gives q1 = (100 - q2)/2 which is the straight line in figure 31.3 joining the peaks of the iso-profit curves – as we have already shown verbally1. We have found the reaction function of firm 1. Let us just look at it a little. One point on it is where q2 = 0. The optimal response is q1 = 50. So if firm 2 produces nothing the best thing that firm 1 can do is to produce 50. A moment’s thought should convince you that 50 must therefore be the monopoly output. A rather longer thought may be needed to interpret the other extreme of the reaction function – at which firm 2 produces 100 and the optimal response of firm 1 is to produce nothing. We shall later show that the output of 100 is what would be produced if the firms behaved competitively. In the meantime you might like to ponder why. Because we have identical firms we can easily work out the iso-profit map for firm 2 and its reaction function. We simply interchange q1 and q2 in what we have done. We thus get figure 31.4.
The straight line in this figure is firm 2’s reaction function. For example, if firm 1 produces nothing firm 2 should produce 50; if firm 1 produces 100, firm 2 should produce nothing; if firm 1 produces 50 firm 2 should produce 25. And so on. We can now answer the question as to whether there is a Nash Equilibrium in the game. Let us put the two reaction curves together. We get figure 31.6. There is a unique intersection at the point (33⅓, 33⅓) and so this must be the unique Nash Equilibrium of the game. Game theory predicts that each firm will produce 33⅓ and the total output will be 66⅔ giving a price of 43⅓. You should be clear why this is a Nash equilibrium: if firm 2 produces 33⅓ then the optimal output for firm 1 (given by its reaction function) is 33⅓; and if firm 1 produces 33⅓ then the optimal output for firm 2 (given by its reaction function) is 33⅓. So in this Nash equilibrium both firms are happy with 1
Be careful when trying to interpret this as it gives q1 as a function of q2,
their decision given the decision of the other. You should note that at no other point in the graph is this true.
31.3: Collusion? You might have realised that it might be better for the two firms to try and collude – rather than choose their outputs independently of each other. We can see the possibilities for collusion if we put together figures 31.3 and 31.4. This gives us figure 31.7, into which I have inserted a curve which one might tentatively call the ‘contract curve’ between the two duopolists,
Why might we call this line the ‘contract curve’? Because it is the locus of points efficient in the sense of the profits of the two firms: given any level of profit of firm 1 then firm 2’s profit is highest on the contract curve; given any level of profit of firm 2 then firm 1’s profit is highest on the contract curve. You might find it interesting to note that the equation of this contract curve is given by q + q2 = 50 that is the joint output of the two firms is 50. Does this 50 remind you of anything – yes the monopoly output. So we get the conclusion that the two firms are behaving efficiently if they are acting as if they are a monopolist. This is a nice conclusion – though a rather obvious one. However to be on the contract curve they have to collude. Moreover each firm has a natural incentive to cheat on any collusive agreement. Why? Simply because none of the points on the contract curve are on either firm’s reaction function – so neither firm is behaving optimally given the output decision of the other. This tells us that there are natural incentives why any collusive agreement may break down and we end up at the Nash Equilibrium. We can see this more clearly by working out the profits of the firms at the Nash Equilibrium and at the symmetrical joint profit maximising situation. We have, where we have rounded the profits to the nearest whole number. q1 = 33⅓ q1 = 25
q2 = 33⅓ 1111, 1111 1041, 1389
q2 = 25 1389, 1041 1250, 1250
You might recognise the structure of this game – it is that of a classic prisoner’s dilemma. Note that there is a unique Nash Equilibrium (which we have already identified) and there is another outcome – the joint profit maximising outcome – which is better for both of them but which is not a Nash Equilibrium. Both firms have an incentive to renege on an agreement to produce 25 units. Collusion is not a Nash Equilibrium in this game.
31.4: Sequential Play You may well be asking whether the outcome may be different or better if we changed the rules of the game – for example, letting the players play sequentially rather than simultaneously. This situation is referred to as the Stackelberg model after the economist Stackelberg. Let us see what happens. Consider first the situation where firm 1 chooses first and then announces the decision to firm 2 who then decides. What does firm 1 do? We have already considered this type of situation in chapter 8. Firm 1 will reason that whatever output firm 1 chooses then firm 2 will respond with the appropriate output from its (that is, firm 2’s) reaction function – after all, that gives the optimal response by firm 2 to any output decision of firm 1. Now we know the reaction function of firm 1 – it is the line shown in figure 31.8. Firm 1 knows that whatever value of q1 it chooses then firm 2 will choose the corresponding point on the reaction function. So essentially firm 1 is choosing a point on the reaction function. Which is the best point on it from firm 1’s point of view?
Presumably the point on it on the highest (in terms of profit) iso-profit curve – that is, on the physically lowest curve in the figure. This optimal point is labelled ‘1’ in the figure. It is where q1 = 50 and q2 = 25 and where the profits of firm 1 are 1250. The profits of firm 2 in this situation are just 625. Using symmetry, the outcome when firm 2 moves first is q1 = 25 and q2 = 50 with respective profits of 625 and 1250. Clearly firm 1 prefers to move first, then prefers simultaneous play and least prefers firm 2 to move first. For firm 2 the opposite is true – it prefers to move first, then prefers simultaneous play and least prefers firm 1 to move first. In this case, it is clear that the first mover has an advantage. Do you think that this is always the case?
31.5: Monopoly, Duopoly and Competition Compared We have already worked out what will happen in this specific example: under monopoly we have an output of 50, under duopoly a total output of 66⅔ and under competition a total output of 100. Correspondingly the price under monopoly is 50, under duopoly is 46⅓ and under competition is 10 (the marginal cost). We can generalise these results. We know the reaction functions of the firms. That for firm 1 is q1 = (a – c – b q2)/(2b) , so we can conclude that the monopoly output is found by putting q2 = 0: monopoly output = (a – c)/(2b) The total duopoly output is given by the sum of the values of q1 and q2 at the intersection of the two reaction functions q1 = (a – c – b q2)/(2b) and q2 = (a – c – b q1)/(2b). Hence q1 = q2 = (a – c)/(3b) and so total duopoly output = 2(a – c)/(3b) The total competitive output can be found in two ways. First it is where the price is equal to the marginal cost, that is where p = a – b(q1 + q2) (the demand curve) = c (the marginal cost). This solves to give total competitive output = (a – c)/(b) The second way is by arguing that if one firm produces so much that the other firm’s optimal output is zero then the first firm is producing the competitive outcome. This gives the same output as that stated above (put q1 = 0 in firm 1’s reaction function and solve for q2).
It is interesting to note that the duopoly output is between the monopoly output and the competitive output. Even more interesting is a generalised result that you could try to prove (though the proof requires the generalisation of the concept of a Nash Equilibrium to a game with n players). This generalised result says that the total output in an oligopoly situation with n identical firms playing the same rules is given by total oligopoly output = n (a – c)/[(n+1)b] You will see that as n approaches infinity this approaches the competitive output. This is a reassuring result. You may like to work out yourself the implication for the price of the product in the various situations. For the record the price is lowest under competition, is highest under monopoly and is in-between in an oligopoly with n firms (with the price approaching the competitive price as n approaches infinity).
31.6: A Price-Setting Game A completely different scenario emerges when we change the rules of the game. Suppose we now make it a price-setting game. That is, each firm, independently and simultaneously chooses the price for its product. If we continue to assume that the firms produce an identical product, and if we make the rather key assumption that both firms can satisfy any demand that is forthcoming, then it is clear that if one firm is charging less than the other then it will get all the demand. So the reaction functions of the two firms are easy to work out: given any price charged by the other firm the optimal price for a firm is a fraction less than the other firm – as long as the price is not lower than the marginal (average) cost. So both firms will undercut each other until they both have prices equal to the marginal cost. This is the unique Nash Equilibrium. This is an interesting case as the outcome is exactly that of competition – the price is equal to the marginal cost. If this is what the world looks like we do not need regulatory bodies to control duopolies or oligopolies.
31.7: Summary We have studied duopoly behaviour under a variety of assumptions. In particular we have studied a quantity-setting game and a price-setting game. In a quantity-setting institution the Nash Equilibrium is between the competitive and monopoly output (in terms of the total quantity produced and the price charged). Collusive situation is better for both - but not a Nash Equilibrium. Both firms would prefer that they were the leader. In a price-setting institution the Nash Equilibrium implies Marginal Cost pricing (so duopoly looks like competition).
31.8: Can duopolists co-operate? This exercise is a good follow-up to the game theory experiment of the previous chapter. In essence, it is the same exercise, but the context is different. Again it can be played with two teams – who play two duopolists competing in a market for some good, which they both produce. The structure of the exercise is as follows. The two duopolists produce an identical product. Each duopolist has constant marginal and average costs of 10p per unit. The aggregate demand curve for the duopolists’ product is p = 100 - (q1 + q2 ) where p is the market price (in pence) and q1 and q2 are the outputs of Teams 1 and 2 respectively. Teams must decide on their q’s; the tutor, or the person keeping control, works out p using the formula above. So, for example, if q1 = 20 and q2 = 30 then p = 50 and the revenues to Teams 1 and 2 are respectively 50 x 20 = 1000 (= £10) and 50 x 30 = 1500 (= £15). Costs are respectively 10 x 20 = 200 (= £2) and 10 x 30 = 300 (= £3), so profits are £8 and £12 respectively. As in the game theory experiment, I suggest that the teams play this simple duopoly problem 8 times. The first 4 times, no communication will be allowed between the two Teams; for the last 4 times, communication will be allowed - but no physical threats, and no enforcement of contracts by the tutor/referee. This is a little more complicated as it is not immediately clear what are the payoffs to the various decisions that the teams may take. In fact, there are not just two choices as in the game theory experiment but a whole continuum - the two quantities can be any positive numbers (though numbers greater than 100 would be a bit silly as they would imply negative prices and negative profits). Interestingly, though, it has the same structure – there is a unique Nash equilibrium (can you find it?) that is dominated by an outcome in which the two duopolists collude and collectively produce the monopoly output. The interesting question is whether that collusive outcome can be sustained – obviously without legal support (as the law in most countries is strongly against collusive monopoly-type agreements) and usually without communication (as the law also forbids that). What do you find?
31.9: Mathematical Appendix We find the reaction function for firm 1. To do this, we need to find, for each level of output of firm 2, the optimal output of firm 1. We can find this by differentiating π1 with respect to q1 (for a given level of q2), putting this expression equal to zero and then solving for q1 as a function of q2. From (31.2) we get dπ1/dq1 = (a – c - b q2) –2 b q1 and so dπ1/dq1 = 0 implies that q1 = (a – c – b q2)/(2b) which is equation (31.3).
