ANSWER KEY GAME THEORY, ECON 95 PROFESSOR A. JOSEPH GUSE (1) (Gibbons 1.6) Consider again the Cournot duopoly model with demand given by the marginal willingness to pay function: P(Q) = a Q, but this time where the two firms have different marginal costs, c 1, and c. What is the NE if < c i < a/ for each firm? ANSWER. Again the general method is to use the information given to write down profit functions for each firm... π 1 (q 1,q ) = q 1 (a q 1 q c 1 ) π (q 1,q ) = q (a q 1 q c ) Note how we using the fact that each firm has its own cost function; c 1 appears in the expression for π 1. Setting the derviative of π 1 w.r.t. q 1 equal to zero, we can find firm 1 s best response function π 1 (q 1,q ) q 1 = a q 1 q c 1 = (1) Similarly we have for firm q 1 (q ) = a q c 1 () q (q 1 ) = a q 1 c Substituting the expression for q from equation () into equation (1) we will find the equilibrium level of firm 1 s output, q 1. () Similarly for firm we have a q1 = ( ) a q 1 c c 1 q 1 = a c 1 +c (4) q = a c +c 1 Note that if the restriction given in the problem holds and < c i < a/ for each firm, i = 1 and i =, then the right-hand side expressions for each firm s equilibrium output 1
PROFESSOR A. JOSEPH GUSE Firm s Output Level a c 1 q 1 (q ) a c q (q 1 ) = max { }, a c q1 a c 1 a c Firm 1 s Output Level Figure 1. levels in equations () and () are both guaranteed to be positive. The second part of the question asks, what if this restriction is not satisfied. In particular, what if c 1 < c and c > a+c 1. In this case, it would appear as though the RHS of equation (4) has firm making a negative quantity. One way to think about this is to just trust the math and say OK firm is maximizing profit but making a negative quantity - which could be interpretted as buying output, making them a consumer instead of a producer. Would they be getting positive consumer surplus? Let s check. Suppose firm 1 makes the output specified by equation (). The price is a a c 1+c = a+c 1 c. If we interpret firm s cost c as their marginal willingness to pay as a consumer then the question is a+c 1 c < c? If so, then firm will be getting positive consumer surplus from buying output. And indeed using the assumption that c > a+c 1, this is the case. Another way to think about this is to assumption that neither firm would never produce a negative quantity (even if it could find a price lower than its own marginal cost). In this case we have to be more precise about the firms best response functions. Instead of the expression we wrote down in (1) and (), we must write { a q c 1 q 1 (q ) = max }, { a q1 c q (q 1 ) = max, Now under the assumption that c > a + c 1, we know from above that q will be. Hence firm 1 will produce the monopoly output level q c 1. Figure 1 shows the equilibrium output levels for the case where the original restriction hold. Figure shows the equilibrium output levels for the case where c > a+c 1. }
ANSWER KEY GAME THEORY, ECON 95 Firm s Output Level q 1 (q ) a c q (q 1 ) a c a c 1 Firm 1 s Output Level Figure. () (Gibbons 1.7) Bertrand homogenous products. Demand for firm i s output is given by q i (p i,p j ) = Firm i s profit when p i < p j is given by a p i when p i < p j a p i when p i = p j when p i > p j π i (p i,p j ) = (a p i )(p i c) Remember in the Bertrand model, firms choose price, not quantity, so we much differentiate w.r.t firm i s price and set it equal to zero to find profit maximizing price. π i p i = a p i = p i c p i = a+c But this is only the profit maximizing choice if the other firm has set a higher price. What should firm i do if j has set a price lower than a+c? The only answer is that firm i must undercut j in order to sell anything at all! This undercutting will go on until both firms have set price equal to c, the marginal cost. Figure shows this - sort of. 1 1 The difference between the true best response functions and the lines drawn in Figure is that in truth a best response by firm i to the other firm, j, setting a price above c and below a+c is to set ones price below theirs, but only just below. That is p i = p j ǫ where ǫ > but is as small as possible. This is not possible to draw accurately. Therefore what I ve drawn is a situation where a firm must either match the other firm s price exactly or set it lower by an appreciable percentage difference. Since matching results in splitting the market with the other firm, the best
4 PROFESSOR A. JOSEPH GUSE p 1 (p ) Firm s Price a+c p (p 1 ) 45 line c c a+c Firm 1 s Price Figure. Firm 1 s BR function in green. Firm s best response function in blue. Note that when Firm 1 s price is below c, Firm s BR is make sure to set his price high enough so that he gets none of the market share. This explains the shaded blue area in the diagram; as long as firm is setting her price above firm 1 s price, its a best response. The green shaded areas is similarly explained by swaping Firms 1 and in this logic. () (Gibbons 1.8) This is the Hotelling model of spatial competition. (The second problem analyzed in Hotelling s 199 paper). We are thinking here about an election with or more candidates who run for office by competing on a single issue. Each candidate can take a position anywhere on the unit interval and votes are distributed uniformly along the unit interval. The assumption is that voters will vote for the candidate whose position is closest to their own. (a) candidates. When there are only two candidates the only equilibrium in this game is (.5,.5). That is both candidates take the same position and that position is the median of the distribution of voters. In this case the median is.5 because we are assuming a uniform distribution. I will prove this in two steps. First I will show that in any equilibrium both candidates must be taking the same position. Second I will show that the only position both can take in equilibrium is the median,.5. To see that both must take the same position, suppose that they did not. Then there is a candidate i taking position x i and a candidate j taking position x j with x i < x j. thing is to undercut by that fixed percentage amount. This is why the best response curves depicted are just off the 45 line.
ANSWER KEY GAME THEORY, ECON 95 5 Note that candidate i will get all the votes to her left and half the votes lying between her and j for a total of x i + x j x i or x i+x j votes. But this cannot be a best response for candidate i, since moving closer to j can only increase her votes. For example she could move to position x i = x i+x j - half way between i and j - which would increase her votes to x i+x j + x j x i which is obviously more than x i+x j. Therefore we have shown that if there are any pure strategy equilibria in this game, they cannot involve i taking a different position from j, since i could always improve her vote count by moving to the position half-way between. Let x = x i = x j be the common position taken by candidates i and j in some equilibrium. To see that this has to be the median of the distribution (i.e..5), suppose that it is not. Then x =.5 + δ. For concreteness suppose that δ > (a similar argument will work for δ < ). Note that both candidates are getting exactly half the vote. If this is the case, then x i = x cannot be a best response for candidate i. In particular, i could increase her vote count to more than half by moving closer to.5. Hence all values of x.5 cannot represent pure-strategy equilibria. To see that.5 is an equilibrium, suppose x j were.5. If i adopts the same position, they will split the vote each getting.5. Can i do any better? No. If she moves to the left then j will win all votes to the right of.5 outright and half the votes bretween.5 and i s new position - leaving i with strictly less than half the vote. Simlilarly if she move to the right. Hence.5 is a best response for i to j. (b) Candidates. There are no PSNE in the game with candidates. To see this first note that there can be no equlibrium in this game where all three candidates share the same position. If they did, they would each get exactly 1 of the vote. Suppose the common position they share is to the right of the median voter, then any one of the three could do strictly better by moving a bit to the left. So it cannot be a best response for any player to remain there. Similarly for the case where the common position lie on the median or to the left. Next it can be shown that there can be no equilibrium in which two of the candidates share one position and the third occupies a different one. Suppose they did. Then let x i < x j = x k so that i sits to the left of j and k who share a position. i can always get more votes by moving closer to j and k, so x i cannot be a best response. Next it can be show that there can be no equilibrium in which all three candidates take unique positions. Suppose they did. Let x i < x j < x k. Again i can always improve her situation by moving to the right so this cannot be a best response for her.
6 PROFESSOR A. JOSEPH GUSE These arguments cover all possibles cases. QED. Note that a PSNE is possible with candidates, if we distribute the votes on a ring instead of along a line. (For example imagine an electorate all who live along the shores of a lake and the issue is where to build a sewage plant. Everyone s preferences are such that each voter wants the sewage plant on the opposite side of the lake from where they live.) Assume again that the distribution is uniform. Now any set of positions {x 1,x,x } where each position is measured in radians and x i x j π is a positive integer for any i j is an equilibrium. In other words any set of positions which divide the ring into exact thirds. Geometry matters! What implications does this have if the population is not uniformly distributed around the lake. Is the sewage plant more likely to be built ialong the densely populated shores or among the grand estates where their are few voters? Note: another interpretation of this game is that candidates are not vote-maximizers per se. They only care about being the candidates with the most votes, but the margin of victory is irrelevent to them. As some of you ponited out, in this case, there may be equilibria. Such equilibria can come about because in certain configurations, each losing candidates could change the identity of the winner but not in such a way to be the winner himself.