Table 10.1: Elimination and equilibrium. 1. Is there a dominant strategy for either of the two agents?

Chapter 10 Strategic Behaviour Exercise 10.1 Table 10.1 is the strategic form representation of a simultaneous move game in which strategies are actions. s b 1 s b 2 s b 3 s a 1 0, 2 3, 1 4, 3 s a 2 2, 4 0, 3 3, 2 s a 3 1, 1 2, 0 2, 1 Table 10.1: Elimination and equilibrium 1. Is there a dominant strategy for either of the two agents? 2. Which strategies can always be eliminated because they are dominated? 3. Which strategies can be eliminated if it is common knowledge that both players are rational? 4. What are the Nash equilibria in pure strategies? Outline Answer: 1. No player has a dominant strategy. 2. Both s a 3 and s b 2 can be eliminated as individually irrational. 3. With common knowledge of rationality we can eliminate the dominated strategies: s a 3 and s b 2. 4. The Nash Equilibria in pure strategies are (s a 2, s b 1) and (s a 1, s b 3) 153

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.2 Table 10.2 again represents a simultaneous move game in which strategies are actions. s b 1 s b 2 s b 3 s a 1 0, 2 2, 0 3, 1 s a 2 2, 0 0, 2 3, 1 s a 3 1, 3 1, 3 4, 4 Table 10.2: Pure-strategy Nash equilibria 1. Identify the best responses for each of the players a, b. 2. Is there a Nash equilibrium in pure strategies? Outline Answer 1. For player A the best reply is s a 2 if player B plays s b 1, s a 1if B plays s b 2, s a 3 if B plays s b 3.For player B the best reply is s b 1 if A plays s a 1, s b 2 if A plays s a 2, s b 3 if A plays s a 3 2. The unique Nash Equilibrium is (s a 3, s b 3) c Frank Cowell 2006 154

Microeconomics Exercise 10.3 A taxpayer has income y that should be reported in full to the tax authority. There is a flat (proportional) tax rate γ on income. The reporting technology means that that taxpayer must report income in full or zero income. The tax authority can choose whether or not to audit the taxpayer. Each audit costs an amount ϕ and if the audit uncovers under-reporting then the taxpayer is required to pay the full amount of tax owed plus a fine F. 1. Set the problem out as a game in strategic form where each agent (taxpayer, tax-authority) has two pure strategies. 2. Explain why there is no simultaneous-move equilibrium in pure strategies. 3. Find the mixed-strategy equilibrium. How will the equilibrium respond to changes in the parameters γ, ϕ and F? Outline Answer 1. See Table 10.3. Taxpayer Tax-Authority s b 1 s b 2 Audit Not audit s a 1 conceal [1 γ] y F, γy + F ϕ y, 0 s a 2 report [1 γ] y, γy ϕ [1 γ] y, γy Table 10.3: The tax audit game 2. Consider the best responses: Tax-Authority s best response to conceal is audit Taxpayer s best response to audit is report Tax-Authority s best response to report is not audit Taxpayer s best response to not audit is conceal 3. Suppose the taxpayer conceals with probability π a and the tax authority audits with probability π b. (a) Expected payoff to the taxpayer is υ a = π a [ π b [[1 γ] y F ] + [ 1 π b] y ] which, on simplifying, gives + [1 π a ] [ π b [1 γ] y + [ 1 π b] [1 γ] y ], υ a = [1 γ] y + π a [ 1 π b] γy π a π b F. So we have dυ a dπ a = [ 1 π b] γy π b F c Frank Cowell 2006 155

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR π b 1 π *b taxpayer reaction tax authority reaction 0 π *a 1 π a Figure 10.1: The tax audit game It is clear that dυa dπ 0 as π b π b where a π b := γy γy + F. (10.1) So the taxpayer s optimal strategy is to conceal with probability 1 if the probability of audit is too low (π b < π b ) and to conceal with probability zero if the probability of audit is high. (b) Expected payoff to the tax-authority is υ b = π b [π a [γy + F ϕ] + [1 π a ] [γy ϕ]] which, on simplifying, gives So we have + [ 1 π b] [π a [0] + [1 π a ] [γy]] υ b = [1 π a ] γy + π a π b [γy + F ] π b ϕ dυ b dπ b = πa [γy + F ] ϕ It is clear that dυb 0 as π a π a where dπ b π a := ϕ γy + F. (10.2) So the tax authority s optimal strategy is to audit with probability 0 if the probability of the taxpayer concealing is low (π a < π a ) and to audit with probability 1 if the probability of concealment is high. (c) This yields a unique mixed-strategy equilibrium ( π a, π b) as illustrated in Figure 10.1. c Frank Cowell 2006 156

Microeconomics (d) The effect of a change in any of the model parameters on the equilibrium can be found by differentiating the expressions (10.1) and (10.2). we have π a γ = ϕy [γy + F ] 2 > 0; π b ϕ = F y [γy + F ] 2 > 0. π a ϕ = 1 γy + F > 0; π a F = ϕ [γy + F ] 2 < 0; π b ϕ = 0. π b F = γy [γy + F ] 2 < 0. c Frank Cowell 2006 157

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.4 Take the battle-of-the-sexes game in Table 10.4. s b 1 s b 2 [West] [East] s a 1 [West] 2,1 0,0 s a 2 [East] 0,0 1,2 Table 10.4: Battle of the sexes strategic form 1. Show that, in addition to the pure strategy Nash equilibria there is also a mixed strategy equilibrium. 2. Construct the payoff-possibility frontier. Why is the interpretation of this frontier in the battle-of-the-sexes context rather unusual in comparison with the Cournot-oligopoly case? 3. Show that the mixed-strategy equilibrium lies strictly inside the frontier. 4. Suppose the two players adopt the same randomisation device, observable by both of them: they know that the specified random variable takes the value [ 1] with probability π and 2[ with probability ] 1 π; they agree to play s a 1, s b 1 with probability π and s a 2, s b 2 with probability 1 π; show that this correlated mixed strategy always produces a payoff on the frontier. Outline Answer 1. Suppose a plays [West] with probability π a and b plays [West] with probability π b. The expected payoff to a is υ a = π a [ π b [2] + [ 1 π b] [0] ] + [1 π a ] [ π b [0] + [ 1 π b] [1] ] So we have = 2π a π b + [1 π a ] [ 1 π b] = 1 π a π b + 3π a π b (10.3) dυ a = 1 + 3πb dπa It is clear that dυa dπ 0 as π b 1 a 3. The expected payoff to b is And so υ b = π b [π a [1] + [1 π a ] [0]] + [ 1 π b] [π a [0] + [1 π a ] [2]] = π a π b + 2 [1 π a ] [ 1 π b] = 2 2π a 2π b + 3π a π b. (10.4) dυ b = 2 + 3πa dπb It is clear that dυb 0 as π a 2 dπ b 3. So there is a mixed-strategy equilibrium where ( π a, π b) = ( 2 3, 3) 1. c Frank Cowell 2006 158

Microeconomics 2. See Figure 10.2. Note that, unlike oligopoly where the payoff (profit) is transferable, in this interpretation the payoff (utility) is not so the frontier has not been extended beyond the points (2,1) and (1,2). The lightly shaded area depicts all the points in the attainable set of utility can be thrown away. The heavily shaded area in Figure 10.2 shows the expected-utility outcomes achievable by randomisation. The frontier is given by the broken line joining the points (2,1) and (1,2). Figure 10.2: Battle-of-sexes: payoffs 3. The utility associated with the mixed-strategy equilibrium is ( 2 3, 2 3) and clearly lies inside the frontier in Figure 10.2. 4. Given that the probability of playing [West] is π, the expected utility for each player is υ a = 2π + [1 π] = 1 + π υ b = π + 2 [1 π] = 2 π If we allow π to take any value in [0, 1] this picks out the points on the broken line in Figure 10.2. c Frank Cowell 2006 159

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.5 Rework Exercise 10.4 for the case of the Chicken game in Table 10.5. s b 1 s b 2 s a 1 2, 2 1, 3 s a 2 3, 1 0, 0 Table 10.5: Chicken strategic form Outline Answer υ b 3 2 1 (1½, 1½) 0 1 2 3 υ a Figure 10.3: Chicken: payoffs 1. Suppose a plays s a 1 with probability π a and b plays s b 1 with probability π b. The expected payoff to a is υ a = π a [ π b [2] + [ 1 π b] [1] ] + [1 π a ] [ π b [3] + [ 1 π b] [0] ] So we have = π a + 3π b 2π a π b (10.5) dυ a dπ a = 1 2πb It is clear that dυa dπ 0 as π b 1 a 2. The expected payoff to b is And so υ b = π b [π a [2] + [1 π a ] [1]] + [ 1 π b] [π a [3] + [1 π a ] [0]] = π b + 3π a 2π a π b (10.6) dυ b dπ b = 1 2πa It is clear that dυb 0 as π a 1 dπ b 2. So there is a mixed-strategy equilibrium where ( π a, π b) = ( 1 2, 2) 1. c Frank Cowell 2006 160

Microeconomics 2. See Figure 10.3. The lightly shaded area depicts all the points in the attainable set of utility can be thrown away. The heavily shaded area shows the expected-utility outcomes achievable by randomisation 3. The utility associated with the mixed-strategy equilibrium is ( 1 1 2, 1 1 2) and clearly lies inside the frontier. 4. Once again a correlated strategy would produce an outcome on the broken line. c Frank Cowell 2006 161

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR [LEFT] Alf [RIGHT] Bill [left] [right] [left] [right] Charlie [L] [M] [R] [L] [M] [R] [L] [M] [R] [L] [M] [R] 0 1 3 2 2 2 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 2 2 0 1 1 0 1 0 0 2 2 2 1 0 3 Figure 10.4: Benefits of restricting information s a 1 :[LEFT] s a 2 :[RIGHT] s c 1 s c 2 s c 3 s c 1 s c 2 s c 3 L M R L M R s b 1 [left] 0, 1, 3 2, 2, 2 0, 1.0 1, 1, 1 2, 2, 0 1, 1, 0 s b 2 [right] 0, 0, 0 0, 0, 0 0, 0, 0 1, 0, 0 2, 2, 2 1, 0, 3 Table 10.6: Alf, Bill, Charlie Simultaneous move Exercise 10.6 Consider the three-person game depicted in Figure 10.4 where strategies are actions. For each strategy combination, the column of figures in parentheses denotes the payoff s to Alf, Bill and Charlie, respectively. 1. For the simultaneous-move game shown in Figure 10.4 show that there is a unique pure-strategy Nash equilibrium. 2. Suppose the game is changed. Alf and Bill agree to coordinate their actions by tossing a coin and playing [LEFT],[left] if heads comes up and [RIGHT],[right] if tails comes up. Charlie is not told the outcome of the spin of the coin before making his move. What is Charlie s best response? Compare your answer to part 1. 3. Now take the version of part 2 but suppose that Charlie knows the outcome of the coin toss before making his choice. What is his best response? Compare your answer to parts 1 and 2. Does this mean that restricting information can be socially beneficial? Outline Answer c Frank Cowell 2006 162

Microeconomics s c 1 s c 2 s c 3 L M R Heads [left,left] 0, 1, 3 2, 2, 2 0, 1.0 Tails [right,right] 1, 0, 0 2, 2, 2 1, 0, 3 Table 10.7: Alf, Bill correlate their play 1. The strategic form of the game can be represented as in Table 10.6 from which it is clear that the best responses for the three players are as follows: BR a (left, L) = RIGHT BR a (left, M) = {LEFT, RIGHT} BR a (left, R) = {LEFT, RIGHT} BR a (right, L) = RIGHT BR a (right, M) = RIGHT BR a (right, R) = RIGHT BR b (LEFT, L) = left BR b (LEFT, M) = left BR b (LEFT, R) = left BR b (RIGHT, L) = left BR b (RIGHT, M) = {left, right} BR b (RIGHT, R) = left BR c (LEFT, left) = L BR c (LEFT, right) = {L, M, R} BR c (RIGHT, left) = L BR c (RIGHT, right) = R it is clear that (RIGHT, left, L) is the unique Nash equilibrium. Everyone gets a payoff of 1 at the Nash equilibrium: total payoff is 3. 2. Charlie knows the coordination rule but not the outcome of the coin toss. The payoffs are now as in Table 10.7. Note that neither of the possible action combinations by Alf and Bill would have emerged under the Nash equilibrium in part 1. It is clear that now the expected payoff to Charlie of playing L is 1.5; the expected payoff of playing R is also 1.5. But the expected payoff of playing M is 2. So Charlie s best response is M Everybody gets a payoff of 2 with certainty: total payoff is 6. 3. Charlie now knows both the coordination rule and the outcome of the coin toss. From Table 10.7 it is clear that his best response is L if it is heads and R if it is tails. Now he gets a payoff of 3 and the others get an equal chance of 0 or 1: total payoff is 4, less than that under part 2 but more than under part 1. c Frank Cowell 2006 163

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.7 Consider a duopoly with identical firms. The cost function for firm f is C 0 + cq f, f = 1, 2. The inverse demand function is β 0 βq where C 0, c, β 0 and β are all positive numbers and total output is given by q = q 1 + q 2. 1. Find the isoprofit contour and the reaction function for firm 2. 2. Find the Cournot-Nash equilibrium for the industry and illustrate it in ( q 1, q 2) -space. 3. Find the joint-profit maximising solution for the industry and illustrate it on the same diagram. 4. If firm 1 acts as leader and firm 2 as a follower find the Stackelberg solution. 5. Draw the set of payoff possibilities and plot the payoffs for cases 2-4 and for the case where there is a monopoly. Outline Answer 1. Firm 2 s profits are given by Π 2 = pq 2 [ C 0 + cq 2] = [ β 0 β [ q 1 + q 2]] q 2 [ C 0 + cq 2] So it is clear that a typical isoprofit contour is given by the locus of ( q 1, q 2) satisfying [ β0 c β [ q 1 + q 2]] q 2 = constant see Figure 10.5. The FOC for a maximum of Π 2 with respect to q 2 keeping q 1 constant is β 0 β [ q 1 + 2q 2] c = 0 which yields the Cournot reaction function for firm 2 q 2 = χ 2 ( q 1) = β 0 c 2β 1 2 q1 (10.7) a straight line. Note that this relationship holds wherever firm 2 can make positive profits. See Figure 10.6 which shows the locus of points that maximise Π 2 for various given values of q 1. 2. By symmetry the reaction function for firm 1 is q 1 = β 0 c 2β c Frank Cowell 2006 164 1 2 q2 (10.8)

Microeconomics q 2 profit q 1 Figure 10.5: Iso-profit curves for firm 2 q 2 χ 2 ( ) q 1 Figure 10.6: Reaction function for firm 2 c Frank Cowell 2006 165

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR The Cournot-Nash solution is where (10.7) and (10.8) hold simultaneously, i.e. where q 1 = β 0 c 1 [ β0 c 1 ] 2β 2 2β 2 q1 (10.9) The solution is at q 1 = q 2 = q C where q C = β 0 c 3β (10.10) see Figure 10.7. The price is 2 3 β 0 + 1 3 c. q 2 χ 1 ( ) q C χ 2 ( ) q 1 Figure 10.7: Cournot-Nash equilibrium 3. Writing q = q 1 + q 2, the two firms joint profits are given by The FOC for a maximum is Π 2 = pq [2C 0 + cq] = [β 0 βq] q [2C 0 + cq] β 0 c 2βq = 0 which gives the collusive monopoly solution as q M = β 0 c 2β. (10.11) with the corresponding price 1 2 [β 0 + c]. However, the break-down into outputs q 1 and q 2 is in principle undefined. Examine Figure 10.8. The points (q M, 0) and (0, q M ) are the endpoints of the two reaction functions (each indicates the amount that one firm would produce if it knew that the c Frank Cowell 2006 166

Microeconomics other was producing zero). The solution lies somewhere on the line joining these two points. In particular the symmetric joint-profit maximising outcome (q J, q J ) lies exactly at the midpoint where the isoprofit contour of firm 1 is tangent to the isoprofit contour of firm 2. q 2 χ 1 ( ) (0,q M ) q J q C χ 2 ( ) (q M,0) q 1 Figure 10.8: Joint-profit maximisation 4. If firm 1 is the leader and firm 2 is the follower then firm 1 can predict firm 2 s output using the reaction function (10.7) and build this into its optimisation problem. The leader s profits are therefore given as [ β0 β [ q 1 + χ 2 ( q 1)]] q 1 [ C 0 + cq 1] which, using (10.7), becomes [ β 0 β [ q 1 + β 0 c 2β 1 2 q1 ]] q 1 [ C 0 + cq 1] = 1 [ β0 c βq 1] q 1 C 0 (10.12) 2 The FOC for the leader s problem is 1 2 [β 0 c] βq 1 = 0 so that the leader s output is q 1 S = β 0 c 2β and, using (10.7), the follower s output must be q 2 S = β 0 c 4β see Figure 10.9. The price is 1 4 β 0 + 3 4 c. c Frank Cowell 2006 167

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR q 2 q C q S χ 2 ( ) (q M,0) q 1 Figure 10.9: Firm 1 as Stackelberg leader Cournot Joint profit max Stackelberg leader Stackelberg follower output price profit β 0 c 1 3β 3 β 0 + 2 3 c. [β 0 c] 2 9β C 0 β 0 c 1 4β 2 [β [β 0 + c] 0 c] 2 8β C 0 β 0 c 1 2β 4 β 0 + 3 4 c [β 0 c] 2 8β C 0 β 0 c 1 4β 4 β 0 + 3 4 c [β 0 c] 2 16β C 0 Table 10.8: Outcomes of quantity competition linear model 5. The outcomes of the various models are given in Table 10.8.and the possible payoffs are illustrated in Figure 10.10. Note that maximum total profit on the boundary of the triangle is exactly twice the entry in the Joint profit max row, namely 1 4 [β 0 c] 2 /β 2C 0. This holds as long as there are two firms present i.e. right up to a point arbitraily close to either of the end-points. But if one firm is closed down (so that the other becomes a monopolist) then its fixed costs are no longer incurred and the monopolist makes profit Π M := 1 4 [β 0 c] 2 /β C 0. In Figure 10.10 the point marked is where both firms are in operation but firm 1 is getting all of the joint profit and the point (Π M, 0) is the situation where firm 1 is operating on its own. c Frank Cowell 2006 168

Microeconomics Π 2 (0, Π M ) (Π J,Π J ) (Π C,Π C ) 0 1 2 (Π S,Π S ) C 0 Π 1 { (Π M,0) Figure 10.10: Possible payoffs c Frank Cowell 2006 169

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.8 An oligopoly contains N identical firms. The cost function is convex in output. Show that if the firms act as Cournot competitors then as N increases the market price will approach the competitive price. Outline Answer The assumption of convex costs will ensure that there is no minimum viable size of firm. Profits for a typical firm are given by where p ( q f + K ) q f C ( q f ) (10.13) K := N j=1 j f is the total output of all the other firms, which of course firm f takes to be constant under the Cournot assumption. Maximising this by choice of q f gives the FOC for an interior solution q j p q ( q f + K ) q f + p ( q f + K ) C q ( q f ) = 0 (10.14) Given that all the firms are identical we may rewrite condition (10.14) as p q (q) q N + p (q) C q = 0 (10.15) where q is industry output. This in turn can be rewritten as p (q) = C q 1 + 1 ηn (10.16) where η := p (q) qp q (q) is the elasticity of demand. The result follows immediately: as N becomes large (10.16) approaches p (q) = C q (10.17). c Frank Cowell 2006 170

Microeconomics Exercise 10.9 Two identical firms consider entering a new market; setting up in the new market incurs a once-for-all cost K > 0; production involves constant marginal cost c. If both firms enter the market Bertrand competition then takes place afterwards. If the firms make their entry decision sequentially, what is the equilibrium? Outline Answer 1. The firms first decide whether to enter (and hence incur the fixed cost K), then they play the Bertrand pricing game. K is thus considered a sunk cost when the second-stage game is played. When both firms decide to enter, the unique Nash equilibrium of the Bertrand pricing game is to set prices equal to marginal cost, (p 1, p 2 ) = (c, c). This yields overall profits for the two firms (Π 1, Π 2 ) = ( K, K). 2. The extensive form is shown in Figure 10.11. Figure 10.11: The entry game 3. To find the Subgame Perfect Nash Equilibrium, solve the game by backwards induction. If firm 1 decides to enter, firm 2 s optimal strategy is not to enter (profit of 0 compared to K). If firm one decides not to enter, then firm 2 should enter. Firm two can observe the action of firm 1, thus it can form history-dependent strategies. The optimal strategy is not to enter if firm 1 entered, and to enter if firm 1 did not enter the market, (,enter). Thus, the decision for firm 1 is between entering and receiving profits of Π M K, or not entering and receiving 0. For K small, firm 1 will decide to enter. The unique Subgame Perfect Nash Equilibrium in strategies is thus (, (not enter,enter)), (10.18) yielding the equilibrium outcome (,not enter). c Frank Cowell 2006 171

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR 4. Clearly, there is a first-mover advantage, since even a small fixed cost leads to the monopoly outcome and hence strictly positive profits for the first mover. c Frank Cowell 2006 172

Microeconomics Exercise 10.10 Two firms have inherited capacity from the past so that production must take place subject to the constraint q f q f, f = 1, 2 There are zero marginal costs. Let χ ( ) be the Cournot (quantity-competition) reaction function for each firm. If the firms compete on prices show that the following must be true in a pure-strategy equilibrium: 1. Both firms will charge the same price p. 2. p = p ( q 1 + q 2). 3. p p ( χ 1 ( q 2) + q 2). 4. q 1 χ 1 ( q 2). Outline Answer 1. If p 1 < p 2 then, if q 1 = q 1 firm 1 could make higher profits by raising its price. Otherwise firm 1 would be undercutting firm 2 so that firm 2 would be forced to reduce its price or lose all its sales to firm 1. Hence we must have p 1 = p 2 = p in equilibrium. 2. Consider two cases: (a) If p > p ( q 1 + q 2).then for one or both firms it must be the case that q f < q f the price is too high to exhaust capacity. In which case one of the firms could reduce the price slightly, capture sales from the other firm and increase profits. (b) If p < p ( q 1 + q 2).then both firms are producing to capacity. Each could increase profits just by raising prices to its (rationed) customers. Therefore p = p ( q 1 + q 2) (10.19) 3. If p < p ( χ 1 ( q 2) + q 2).one of the firms must be capacity constrained: (a) If firm 1 is capacity constrained it could raise its price by p and make additional profits p q 1. (b) Otherwise, if firm 2 is capacity constrained then firm 1 s profits are q 1 p ( q 1 + q 2) (10.20) If firm 1 is also capacity constrained then it could increase profits by raising its price. So we may take firm 1 as not being capacity constrained. Given the definition of χ 1 ( ) as firm 1 s best response function (10.20) can be written: c Frank Cowell 2006 173 χ 1 ( q 2) p ( χ 1 ( q 2) + q 2)

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR with and the price must be q 1 = χ 1 ( q 2) (10.21) p = p ( χ 1 ( q 2) + q 2) (10.22) 4. Suppose q 1 > χ 1 ( q 2). If there is a pure strategy equilibrium then, by (10.19) p = p ( q 1 + q 2) < p ( χ 1 ( q 2) + q 2) (10.23) which contradicts (10.22) c Frank Cowell 2006 174

Microeconomics Exercise 10.11 In winter two identical ice-cream firms have to choose the capacity that they plan to use in the summer. To install capacity q costs an amount k q where k is a positive constant. Production in the summer takes place subject to q f q f, f = 1, 2 where q f is the capacity that was chosen in the previous winter. Once capacity is installed there is zero marginal cost. The market for ice-cream is characterised by the inverse demand function p ( q 1 + q 2). There are thus two views: the before problem when the decision on capacity has not yet been taken; the after problem (in the summer) once capacity has been installed. 1. Let χ ( ) be the Cournot reaction function for either firm in the after problem (as in Exercise 10.10). In the context of a diagram such as Figure 10.6 explain why this must lie strictly above the Cournot reaction function for the before problem. 2. Let q C be the Cournot-equilibrium quantity for the before problem. Write down the definition of this in terms of the present model. 3. Suppose that in the summer competition between the firms takes place in terms of prices (as in Exercise 10.10). Show that a pure-strategy Bertrand equilibrium for the overall problem is where both firms produce q C. Outline Answer q 2 low cost high cost q 1 Figure 10.12: Low-cost and high-cost reaction functions for firm 2 1. First consider how the reaction function of firm 2 would differ if the constant marginal cost were lower than its current value. Given a linear inverse demand function β 0 βq and constant marginal costs c, the reaction function for firm 2 is q 2 = β 0 c 2β c Frank Cowell 2006 175 1 2 q1 (10.24)

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR q 2 after before after before q 1 Figure 10.13: Reaction functions before and after capacity costs are sunk see Exercise 10.7. So for high and low values of c we have the situation depicted in Figure 10.12. Before installation capacity costs are proportional to the amount of capacity (marginal cost of capacity is k). But once the capacity has been installed it represents sunk costs so, from the firm s point of view it is as though the marginal cost of production has been cut. So the after-installation reaction functions must be as in Figure 10.13. q 2 χ(q 2 ) ½k/β q C χ(q 1 ) ½k/β q 1 Figure 10.14: Equilibrium capacity 2. Clearly if (10.24) is the after reaction function for firm 2, so that q 2 = χ ( q 1) once capacity cost is sunk, the before reaction functions for firms c Frank Cowell 2006 176

Microeconomics 1 and 2 are, respectively q 2 = β 0 c k 2β q 1 = β 0 c k 2β so that the before reaction function for each firm is χ ( ) k 2β. 1 2 q1 (10.25) 1 2 q2 (10.26) The value q C is found from setting q 1 = q 2 = q C in (10.25) and (10.26) to give q C = β 0 c k. (10.27) 3β This gives the solution to the amount of capacity q 1, q 2 to install in the first period 3. If there is price competition in the second period then, from the solution to Exercise 10.10 p = p ( q 1 + q 2), we have the solution to the ice-cream sellers Summer price problem as p = p (2q C ). c Frank Cowell 2006 177

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.12 There is a cake of size 1 to be divided between Alf and Bill. In period t = 1 Alf offers Bill a share: Bill may accept now (in which case the game ends), or reject. If Bill rejects then, in period t = 2 Alf again makes an offer, which Bill can accept (game ends) or reject. If Bill rejects, the game ends one period later with exogenously fixed payoffs of γ to Alf and 1 γ to Bill. Assume that Alf and Bill s payoffs are linear in cake and that both persons have the same, time-invariant discount factor δ < 1. 1. What is the backwards induction outcome in the two-period model? 2. How does the answer change if the time horizon increases but is finite? 3. What would happen if the horizon were infinite? Alf period 1 [offer 1 γ 1 ] Bill [accept] (γ 1, 1 γ 1 ) period 2 [reject] Alf [offer 1 γ 2 ] [accept] Bill [reject] (δγ 2, δ[1 γ 2 ]) period 3 (δ 2 γ, δ 2 [1 γ ]) Figure 10.15: One-sided bargaining game 1. Begin by drawing the extensive form game tree for this bargaining game. Note that payoffs can accrue either in period 1 (if Bill accepts immediately), in period 2 (if Bill accepts the second offer), or in period 3 (Bill rejects both offers). Using this time frame to discount all payoffs back to period 1 we find the game tree shown in Figure 10.15. We can solve this game using backwards induction. Assume the game has reached period 2 where Alf makes an offer of 1 γ 2 to Bill (keeping γ 2 for himself). Bill can decide whether to accept or reject the offer made by Alf.: the best-response function for Bill is { [accept] if 1 γ2 δ [1 γ] [reject] otherwise c Frank Cowell 2006 178

Microeconomics Since Alf wants to maximize his own payoff, he would not offer more than δ [1 γ] to Bill, leaving him (Alf) with γ 2 =1 δ + δγ. The other option is to offer less today and receive γ tomorrow, discounted back to date 2. But since δ < 1, 1 δ + δγ > δγ and hence Alf would offer 1 γ 2 = δ [1 γ] to Bill, which is accepted. Thus, going back to date 1, where Alf would offer Bill 1 γ 1 (keeping γ 1 for himself), the best-response function for Bill is now { [accept] if 1 γ1 δ 2 [1 γ] [reject] otherwise By a similar argument as before, Alf would not offer more than δ 2 [1 γ] in period 1, and thus has the choice between receiving γ 1 = 1 δ 2 + δ 2 γ in period 1, or receiving 1 δ + δγ in period 2. But again, since δ < 1, we find 1 δ 2 + δ 2 γ > δ [1 δ + δγ] and hence Alf will offer 1 γ 1 = δ 2 [1 γ] to Bill in period 1, which is accepted; Alf s equilibirum share is γ 1 = 1 δ 2 [1 γ]. 2. Now consider a longer, but finite time horizon. The structure of the backwards induction solution outlined above shows that as the time horizon increases from T = 2 bargaining rounds to T = T, the offer made by Alf reduces to 1 γ 1 = δ T [1 γ] which is accepted immediately by Bill; Alf s share is γ 1 = 1 δ T [1 γ]. 3. Now consider an infinite time horizon. The solution for the finite case would suggest that as T, (γ 1, 1 γ 1 ) (1, 0). However, this reasoning is inappropriate for the infinite case, since there is no last period from which a backwards induction outcome can be obtained. Instead, we use the crucial insight that the continuation game after each period, i.e. the game played if Bill rejects the offer made by Alf, looks identical to the game just played. In both games, there is a potentially infinite number of future periods. This insight enables us to find the equilibrium outcome of this game. Assume that the continuation game that follows if Bill rejects has a solution with allocation (γ, 1 γ). Then, in the current period, Bill will accept Alf s offer 1 γ 1 if 1 γ 1 δ [1 γ], as before. Thus, given a solution (γ, 1 γ), Alf would offer 1 γ 1 = δ [1 γ]. But if 1 γ is a solution to the continuation game, it has to be a solution to the current game as well, and hence 1 γ 1 = 1 γ. It follows that 1 γ = δ [1 γ] γ = 1 Alf will offer 1 γ = 0 to Bill, and Bill accepts immediately. c Frank Cowell 2006 179

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.13 Take the game that begins at the node marked * in Figure 10.16. 1 [NOT INVEST] [INVEST] * 2 2 ** [In] [Out] [In] [Out] [FIGHT`] 1 1 [CONCEDE] (Π M, Π) _ [FIGHT`] [CONCEDE] (Π M k, Π) _ (Π F,0) (Π J, Π J ) (Π F,0) (Π J k, Π J ) Figure 10.16: Entry deterrence 1. Show that if Π M > Π J > Π F then the incumbent firm will always concede to a challenger. 2. Now suppose that the incumbent operates a chain of N stores, each in a separate location. It faces a challenge to each of the N stores: in each location there is a firm that would like to enter the local market). The challenges take place sequentially, location by location; at each point the potential entrant knows the outcomes of all previous challenges. The payoffs in each location are as in part 1 and the incumbent s overall payoff is the undiscounted sum of the payoff s over all locations. Show that, however large N is, all the challengers will enter and the incumbent never fights. Outline Answer 1. Consider the concept of an equilibrium here. (a) First note that there are several Nash equilibria as we can see from the strategic form in Table 10.9, where the first part of the monopolist s strategy specifies the action after the entrant played [in], while the second specifies the action after [out]. We find immediately that there are four Nash equilibria: c Frank Cowell 2006 180 ([concede], [concede]), [in] ([concede], [fight]), [in] ([fight], [concede]), [out] ([fight], [fight]), [out]

Microeconomics 2 (Entrant) [in] [out] ([concede], [concede]) Π J, Π J Π M, Π 1 (incumbent) ([concede], [fight]) Π J, Π J Π M, Π ([fight], [concede]) Π F, 0 Π M, Π ([fight], [fight]) Π F, 0 Π M, Π Table 10.9: Weak monopolist: strategic form Note that the outcome ([fight], [in]), where the entrant enters and the incumbent fights, is not an equilibrium outcome. E i [In] [Out] [FIGHT`] I i [CONCEDE] (Π M, Π) _ (Π F,0) (Π J, Π J ) Figure 10.17: Challenge i (b) To find the Subgame Perfect Nash equilibrium, we have to find the strategy combinations that are Nash equilibria in every subgame. There are two subgames here at firm 1 s decision nodes. In the case that the entrant (firm 2) chose [in], the best response is to choose [concede], while if the entrant chose [out], the best response is to choose either [concede] or [fight]. But the best response of the entrant to those best responses is to choose [in]. Thus, the Subgame-Perfect Nash Equilibria are and (([concede], [concede]), [in]) (([concede], [fight]), [in]) which both yield the backwards induction outcome. Hence, we find that the threat to fight after entry is not a credible strategy. However, if there were a precommitment device, such that the threat of fighting became credible, then it would be better for the entrant not to enter, so the equilibrium outcome would be ([fight], [out]) or ([concede], [out]), which imply Subgame-Perfect Nash equilibrium strategies (([fight], [concede]), [out]) or (([fight], [fight]), [out]). c Frank Cowell 2006 181

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR 2. We may now consider an extension of essentially the same model. The incumbent has stores I 1, I 2,..., I N in local markets 1, 2,..., N. There is sequence of challenges in each market from potential entrants E 1, E 2,..., E N A typical encounter is depicted in Figure 10.17. The outcome in this market is independent of actions in markets 1, 2,..., i 1. So the equilibrium behaviour of I i and E i is determined by the situation in local market i alone. By the result in part 1 the outcome is ([concede], [in]). c Frank Cowell 2006 182

q 2 = χ(q 1 ;0) Microeconomics Exercise 10.14 In a monopolistic industry firm 1, the incumbent, is considering whether to install extra capacity in order to deter the potential entry of firm 2. Marginal capacity installation costs, and marginal production costs (for production in excess of capacity) are equal and constant. Excess capacity cannot be sold. The potential entrant incurs a fixed cost k in the event of entry. 1. Let qs 1 be firm 1 s output level at the Stackelberg solution if firm 2 enters. Suppose qs 1 q M, where q M is firm 1 s output if its monopolistic position is unassailable (i.e. if entry-deterrence is inevitable). Show that this implies that market demand must be nonlinear. 2. Let q 1 be the incumbent s output level for which the potential entrant s best response yields zero profits for the entrant. In the case where entry deterrence is possible but not inevitable, show that if qs 1 > q1, then it is more profitable for firm 1 to deter entry than to accommodate the challenger. Outline Answer 1. We begin by modelling the use of capacity as deterrence. q 2 q 1 = χ(q 2 ;0) q 1 = χ(q 2 ;c) q C q M, q 1 Figure 10.18: Quantity and capacity choices (a) Suppose the two firms were to choose capacity z 1 and z 2 simultaneously a minor variation on the standard Cournot model. Firm 1 s problem is max q 1,z Π1 = p ( q 1 + q 2) q 1 cz 1 (10.28) 1 subject to c Frank Cowell 2006 183 q 1 z 1 (10.29)

q 2 = χ(q 1 ;0) Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR with solution q 1 = z 1 = χ ( q 2 ; c ) (10.30) where the dependence of the reaction function χ on the marginal cost parameter is made explicit see Figure 10.18. Note that it never pays to leave capacity unused in this version of the story. Also note that if Firm 1 s marginal cost were cut from c to 0 then the χ schedule would be shifted to the right in Figure 10.18 compare the schedule χ ( ; 0) with χ ( ; c) and also Figure 10.12 in the answer to Exercise 10.11. (b) Firm 2 s problem is similar: subject to max = p ( q 1 + q 2) cz 2 k (10.31) q 2,z 2Π1 q 2 z 2 (10.32) Note that the term k in (10.31), being a fixed entry cost, will not affect the first order condition that characterises firm 2 s best-response output, given that it enters the market. So its behaviour conditional on entry is also given by the reaction function χ: q 2 = z 2 = χ ( q 1 ; c ). (10.33) Taking (10.30) and (10.33) together we get the symmetric Cournot- Nash solution (q C, q C ) see point q C.in Figure 10.18. q 2 z q 1 Figure 10.19: Incumbent s best response with precommitted capacity (c) But if firm 1 s capacity were fixed in advance (and cannot be sold off) then the term cz 1 in.(10.28) would be treated as a sunk cost, irrelevant to the optimisation problem. The optimisation problem for firm 1 would effectively become that of (10.28, 10.29) with c = 0. So its behaviour would be characterised by c Frank Cowell 2006 184 q 1 = χ 1 ( q 2 ; 0 ). (10.34)

Microeconomics Suppose firm 1 s advance capacity is fixed at z. Then, there are in principle two regimes that can apply when at the output-choice stage of the game: q 1 z. Output can be met from pre-existing capacity and so the best response of firm 1 (the incumbent) is given by (10.34) in this region. q 1 > z. Extra capacity will have to be installed at marginal cost c; the best response of firm 1 is given by (10.30) in this region. So the combined best responses will look like Figure 10.19. q 2 q 2 z_z_ q 1 z _ q 1 Figure 10.20: Limits on incumbent s capacity precommitment (d) Now consider firm 1 s choice when determining advance capacity to be installed in advance as a possible deterrent. Let z be the capacity level corresponding to q C the solution to (10.30) and (10.33). The incumbent would not precommit to an amount less than z because it is pointless both firm 1 and firm 2 can always install extra capacity during the output-choice stage. Let z be the capacity level corresponding to the solution to (10.34) and (10.33) see the right-hand intersection in Figure 10.18. The incumbent could not credibly precommit to an amount greater than z because this would imply that it would have capacity that would never be used in the output-choice stage. So the capacity precommitment must be in the range [z, z] see Figure 10.20. (e) Choice of z within the range [z, z] depends on the size of the entry cost for firm 2. Note that, in the usual interpretation of a Cournot diagram firm 2 s profits rise as one moves North-west along firm 2.s reaction function see Figure 10.5 in the answer to Exercise 10.7. Now consider the implications of the size of q 1, the output level of firm1 for which the firm 2 s best response yields it zero profits. q 1 z. Firm 2 can always make a profit for any any capacity choice made by firm 1 such that z [z, z]. The best that firm 1 can do is to accommodate firm 1 s entry and will act as a Stackelberg leader. c Frank Cowell 2006 185

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR q 1 [z, z]. Firm 2 will make a loss for any capacity choice by firm 1 such that q 1 < z z. Here firm 1 may either (a) be assured of deterring firm 2 s entry and and choose monopoly output or (b) choose capacity z so as to deter entry or (c) accommodate entry as above q 2 q S χ 2 ( ) z_z_ q M _q 1 z _ z _ q 1 Figure 10.21: Capacity choice of incumbent (f) Compare the Stackelberg and monopoly solutions in the case where there is a linear demand curve β 0 βq In the case where firm 1 can blockade entry and act as a monopoly case profits are [ β0 βq 1] q 1 cq 1 and the solution as q 1 = q M = β 0 c 2β (10.35) see equation (10.11) in the outline answer to Exercise 10.7. Take the case where firm 1 accommodates and acts as a Stackelberg leader. Firm 2 s reaction as follower is given by q 2 = χ 2 ( q 1) = β 0 c 2β 1 2 q1 (10.36) see equation (10.7) in Exercise 10.7. Firm 1 s profits as leader are 1 [ β0 c βq 1] q 1 (10.37) 2 c Frank Cowell 2006 186

Microeconomics see equation (10.12) in Exercise 10.7 and the profit-maximising output is q 1 = q 1 S = β 0 c 2β. So, if the demand function were linear we must have q 1 S = q M which contradicts what is stated in the question. Therefore the demand function must be nonlinear. 2. See Figure 10.21. If q 1 q M then firm 1 sets q 1 = q M sells q M and the potential entrant does not enter: blockading of entry is inevitable. By assumption entry deterrence is not inevitable so q 1 > q M. If q 1 > z then q 1 = q 1 is not credible, it is outside the range [z, z] of credible precommitments. Since the profits of firm 2 increase as one moves Northwest along firm 2 s reaction function it follows that it is not possible to deter entry. So, by assumption, q 1 z. Also, qs 1 > q1 by assumption. Hence qs 1, z > q1 > q M. In Figure 10.21 it has been assumed that qs 1 < z. We have entry accommodation if q 1 < q 1 and entry deterrence if q 1 q 1. In this case set q 1 = q 1 to get on to the lowest (and so higher profits) isoprofit contour possible. Of course this isoprofit contour is below the one through ( qs 1, S) q2 which, in turn, is below the one which is applicable if q 1 < q 1. This establishes the conclusion. Note that if qs 1 > z then q1 can be larger or smaller than z. If it is smaller, we get the same solution as above. If it is larger then entry deterrence is not possible for q 1 = q 1 is outside the range of credible pre-commitments. So now firm 1 sets q 1 = z and concedes entry. c Frank Cowell 2006 187

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.15 Two firms in a duopolistic industry have constant and equal marginal costs c and face market demand schedule given by p = k q where k > c and q is total output.. 1. What would be the solution to the Bertrand price setting game? 2. Compute the joint-profit maximising solution for this industry. 3. Consider an infinitely repeated game based on the Bertrand stage game when both firms have the discount factor δ < 1. What trigger strategy, based on punishment levels p = c, will generate the outcome in part 2? For what values of δ do these trigger strategies constitute a subgame-perfect Nash equilibrium? Outline Answer 1. Suppose firm 2 sets a price p 2 > c. Firm 1 then has three options: It can set a price p 1 >p 2, it can match the price of firm 2, p 1 = p 2, or it can undercut, since there exists an ɛ > 0 such that p 1 = p 2 ɛ > c. The profits for firm 1 in the three cases are: 0 if p 1 > p Π 1 [ 2 = p 2 c ] k p 2 [ 2 if p 1 = p 2 p 2 ɛ c ] [ k p 2] if p 1 = p 2 ɛ For ɛ suffi ciently small profits in the last case exceed those in the other two, and hence firm 1 will choose to undercut firm two by a small ɛ and capture the whole market. The firms will choose to share the market if they are playing a one-shot simultaneous move game, where they will set p 1 = p 2 = c 2. If the firms maximise joint profits the problem becomes choose k to max [k q] q cq The FOC is k 2q c = 0 which implies that profit-maximising output is q M = k c 2 so that the price and the (joint) profit are, respectively p M = k + c 2 Π M = 1 [k c]2 4 c Frank Cowell 2006 188

Microeconomics 3. The trigger strategy is to play p = p M at each stage of the game if the other firm does not deviate before this stage. But if the other firm does deviate then in all subsequent stages set p = c. In the accompanying table firm 2 deviates at t = 3 by setting p = p M ε < p M so that firm 1 responds with p = c to which the best response by firm 2 is also p = c. 1 2 3 4 5...t firm 1: p M p M p M c c firm 2: p M p M p c c If ε is small then for that one period firm 2 would get the whole market so Π 2 = Π M. Thereafter Π 2 = 0. If the firm had cooperated it would have got Π 2 = 1 2 Π M. The present discounted value of the net gain from defecting is 1 2 Π M 1 2 Π [ M δ + δ 2 + δ 3 +... ] = 1 2 Π 1 2δ M 1 δ 0 if and only if 1 2 δ 1. c Frank Cowell 2006 189

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.16 Consider a market with a very large number of consumers in which a firm faces a fixed cost of entry F. In period 0, N firms enter and in period 1 each firm chooses the quality of its product to be H, which costs c > 0, or L, which costs 0. Consumers choose which firms to buy from, choosing randomly if they are indiff erent. Only after purchasing the commodity can consumers observe the quality. In subsequent time periods the stage game just described is repeated indefinitely. The market demand function is given by ϕ(p) if quality is believed to be H q = 0 otherwise where ϕ ( ) is a strictly decreasing function and p is the price of the commodity. The discount rate is zero. 1. Specify a trigger strategy for consumers which induces firms always to choose H quality. Hence determine the subgame-perfect equilibrium. What price will be charged in equilibrium? 2. What is the equilibrium number of firms, and each firm s output level in a long-run equilibrium with free entry and exit? 3. What would happen if F = 0? Outline Answer 1. For a given price p profits for i are given by [ ] 1 Π = q i [p c] 1 + r + 1 [1 + r] 2 +... F = q i [p c] F (10.38) r if it produces H quality for ever and q i p 1 + r F if it produces low quality in one period and then never sells any product thereafter. The trigger strategy is to stay with the firm unless it is observed to have changed its quality to L, in which case the firm will never again have customers. This punishment ensures SPNE (Note the zero demand for L quality.) To ensure that this strategy is incentive-compatible, and thus consistent with SPNE, we need Hence q i p q i [p c] [ 1 + 1 1 + r + 1 [1 + r] 2 +... = 1 + r q i [p c] (10.39) r p [1 + r] c (10.40) in equilibrium. The equilibrium price is suffi ciently high that the firm is unwilling to sacrifice its future profits for a one-off gain from producing L quality and selling it at a high price. c Frank Cowell 2006 190 ]

Microeconomics 2. From (10.38) the long-run zero-profit condition implies q(p) [p c] r We have the market-clearing condition F (10.41) Nq i = q(p) (10.42) This (10.42) determines the equilibrium number of firms ( N = int q(p) p c ) rf (10.43) From (10.40) and (10.41) we get q i = 1 N rf p c (10.44) 3. Without entry cost, but with demand elastic, q(p) is determined, but q i and N are indeterminate. c Frank Cowell 2006 191

Microeconomics CHAPTER 10. STRATEGIC BEHAVIOUR Exercise 10.17 In a duopoly both firms have constant marginal cost. It is common knowledge that this is 1 for firm 1 and that for firm 2 it is either 3 4 or 1 1 4. It is common knowledge that firm 1 believes that firm 2 is low cost with probability 1 2. The inverse demand function is 2 q where q is total output. The firms choose output simultaneously. What is the equilibrium in pure strategies? Outline Answer Let marginal cost be denoted by c. Firm 1 s and firm 2 s profits are given by, respectively: Π 1 := [ 2 q 1 q 2] q 1 C 1 q 1 (10.45) Π 2 := [ 2 q 1 q 2] q 2 C 2 cq 2 (10.46) Maximising (10.46) with respect to q 2, given q 1 we have the FOC 2 q 1 2q 2 c = 0 which yields the Cournot reaction function for firm 2 q 2 = χ 2 ( q 1 ; c ) = 1 1 2 c 1 2 q1 (10.47) (compare the answer for Exercise 10.7). The two cases (low-cost and high-cost) for firm 2 are therefore q 2 L = 5 8 1 2 q1 (10.48) q 2 H = 3 8 1 2 q1 (10.49) compare this with Figure 10.12 in Exercise 10.11. In the light of the above the expected profits for firm 1 are 1 [[ 2 q 1 q 2 ] L q 1 C 1 q 1] + 1 [[ 2 q 1 q 2 2 2 H] q 1 C 1 q 1] [ 2 q 1 1 2 q2 L 1 ] 2 q2 H q 1 C 1 q 1 Under the Cournot assumption firm 1 takes ql 2 and q2 H for firm 1 is [ 2 2q 1 1 2 q2 L 1 ] 2 q2 H 1 = 0 as given. So the FOC which implies q 1 = 1 2 1 4 [ q 2 L + q 2 H] (10.50) as the best response of firm 1 to firm 2. Substituting from (10.48) and (10.49) into (10.50) we have q 1 = 1 2 1 [ 5 4 8 1 2 q1 + 3 8 1 ] 2 q1 (10.51) c Frank Cowell 2006 192

Microeconomics which simplifies to q 1 = 1 2 1 [ 1 q 1 ] 4 (10.52) = 1 4 + 1 4 q1 (10.53) From which we find the solution as q 1 = 1 3 q 2 L = 5 8 1 6 = 29 48 q 2 H = 3 8 1 6 = 17 48 c Frank Cowell 2006 193