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

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1 Chapter 0 Strategic Behaviour Exercise 0. Table 0. is the strategic form representation of a simultaneous move game in which strategies are actions. s b s b s b 3 s a 0; 3; 4; 3 s a ; 4 0; 3 3; s a 3 ; ; 0 ; Table 0.: Elimination and equilibrium. Is there a dominant strategy for either of the two agents?. 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:. No player has a dominant strategy.. Both s a 3 and s b can be eliminated as individually irrational. 3. With common knowledge of rationality we can eliminate the dominated strategies: s a 3 and s b : 4. The Nash Equilibria in pure strategies are (s a ; s b ) and (s a ; s b 3) 53

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

3 Microeconomics Exercise 0.3 A taxpayer has income y that should be reported in full to the tax authority. There is a at (proportional) tax rate 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 ' and if the audit uncovers under-reporting then the taxpayer is required to pay the full amount of tax owed plus a ne F.. Set the problem out as a game in strategic form where each agent (taxpayer, tax-authority) has two pure strategies.. 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, ' and F? Outline Answer. See Table 0.3. Taxpayer Tax-Authority s b s b Audit Not audit s a conceal [ ] y F; y + F ' y; 0 s a report [ ] y; y ' [ ] y; y Table 0.3: The tax audit game. 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 a and the tax authority audits with probability b. (a) Expected payo to the taxpayer is a = a b [[ ] y F ] + b y which, on simplifying, gives + [ a ] b [ ] y + b [ ] y ; a = [ ] y + a b y a b F: So we have d a d a = b y b F cfrank Cowell

4 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR π b π *b taxpayer reaction tax authority reaction 0 π *a π a Figure 0.: The tax audit game It is clear that da d T 0 as b S b where a b := y y + F : (0.) So the taxpayer s optimal strategy is to conceal with probability if the probability of audit is too low ( b < b ) and to conceal with probability zero if the probability of audit is high. (b) Expected payo to the tax-authority is b = b [ a [y + F '] + [ a ] [y ']] which, on simplifying, gives So we have + b [ a [0] + [ a ] [y]] b = [ a ] y + a b [y + F ] b ' d b d b = a [y + F ] It is clear that db T 0 as a T a where d b a := ' ' y + F : (0.) So the tax authority s optimal strategy is to audit with probability 0 if the probability of the taxpayer concealing is low ( a < a ) and to audit with probability if the probability of concealment is high. a ; b as illus- (c) This yields a unique mixed-strategy equilibrium trated in Figure 0.. cfrank Cowell

5 Microeconomics (d) The e ect of a change in any of the model parameters on the equilibrium can be found by di erentiating the expressions (0.) and (0.). we = 'y [y + F ] > = F y [y + F ] > = y + F > = ' [y + F ] < = = y [y + F ] < 0: cfrank Cowell

6 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Exercise 0.4 Take the battle-of-the-sexes game in Table 0.4. s b s b [West] [East] s a [West], 0,0 s a [East] 0,0, Table 0.4: Battle of the sexes strategic form. Show that, in addition to the pure strategy Nash equilibria there is also a mixed strategy equilibrium.. Construct the payo -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 speci ed random variable takes the value with probability and with probability ; they agree to play s a ; s b with probability and s a ; s b with probability ; show that this correlated mixed strategy always produces a payo on the frontier. Outline Answer. Suppose a plays [West] with probability a and b plays [West] with probability b. The expected payo to a is a = a b [] + b [0] + [ a ] b [0] + b [] So we have = a b + [ a ] b = a b + 3 a b (0.3) d a d a = + 3b It is clear that da d T 0 as b T a 3. The expected payo to b is And so b = b [ a [] + [ a ] [0]] + b [ a [0] + [ a ] []] = a b + [ a ] b = a b + 3 a b : (0.4) d b d b = + 3a It is clear that db T 0 as a T d b 3. So there is a mixed-strategy equilibrium where a ; b = 3 ; 3. cfrank Cowell

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

8 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Exercise 0.5 Rework Exercise 0.4 for the case of the Chicken game in Table 0.5. s b s b s a ; ; 3 s a 3; 0; 0 Table 0.5: Chicken strategic form Outline Answer υ b 3 (½, ½) 0 3 υ a Figure 0.3: Chicken: payo s. Suppose a plays s a with probability a and b plays s b with probability b. The expected payo to a is a = a b [] + b [] + [ a ] b [3] + b [0] So we have = a + 3 b a b (0.5) d a d a = b It is clear that da d T 0 as b S a. The expected payo to b is b = b [ a [] + [ a ] []] + b [ a [3] + [ a ] [0]] = b + 3 a a b (0.6) And so d b d b = a cfrank Cowell

9 Microeconomics It is clear that db T 0 as a S d b. So there is a mixed-strategy equilibrium where a ; b = ;.. See Figure 0.3. 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 ; and clearly lies inside the frontier. 4. Once again a correlated strategy would produce an outcome on the broken line. cfrank Cowell 006 6

10 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR [LEFT] Alf [RIGHT] Bill [left] [right] [left] [right] Charlie [L] [M] [R] [L] [M] [R] [L] [M] [R] [L] [M] [R] Figure 0.4: Bene ts of restricting information Exercise 0.6 Consider the three-person game depicted in Figure 0.4 where strategies are actions. For each strategy combination, the column of gures in parentheses denotes the payo s to Alf, Bill and Charlie, respectively.. For the simultaneous-move game shown in Figure 0.4 show that there is a unique pure-strategy Nash equilibrium.. 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. 3. Now take the version of part 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 and. Does this mean that restricting information can be socially bene cial? Outline Answer. The strategic form of the game can be represented as in Table 0.6 from cfrank Cowell 006 6

11 Microeconomics s a :[LEFT] s a :[RIGHT] s c s c s c 3 s c s c s c 3 L M R L M R s b [left] 0; ; 3 ; ; 0; :0 ; ; ; ; 0 ; ; 0 s b [right] 0; 0; 0 0; 0; 0 0; 0; 0 ; 0; 0 ; ; ; 0; 3 Table 0.6: Alf, Bill, Charlie Simultaneous move s c s c s c 3 L M R Heads [left,left] 0; ; 3 ; ; 0; :0 Tails [right,right] ; 0; 0 ; ; ; 0; 3 Table 0.7: Alf, Bill correlate their play which it is clear that the best responses for the three players are as follows: BR a (left; L) = RIGHT BR a (left; M) = fleft; RIGHTg BR a (left; R) = fleft; RIGHTg 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) = fleft; rightg BR b (RIGHT; R) = left BR c (LEFT; left) = L BR c (LEFT; right) = fl; M; Rg 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 payo of at the Nash equilibrium: total payo is 3.. Charlie knows the coordination rule but not the outcome of the coin toss. The payo s are now as in Table 0.7. Note that neither of the possible action combinations by Alf and Bill would have emerged under the Nash equilibrium in part. It is clear that now the expected payo to Charlie of playing L is :5; the expected payo of playing R is also :5. But the expected payo of playing M is. So Charlie s best response is M Everybody gets a payo of with certainty: total payo is 6. cfrank Cowell

12 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR 3. Charlie now knows both the coordination rule and the outcome of the coin toss. From Table 0.7 it is clear that his best response is L if it is heads and R if it is tails. Now he gets a payo of 3 and the others get an equal chance of 0 or : total payo is 4, less than that under part but more than under part. cfrank Cowell

13 Microeconomics Exercise 0.7 Consider a duopoly with identical rms. The cost function for rm f is C 0 + cq f ; f = ; : The inverse demand function is 0 q where C 0, c, 0 and are all positive numbers and total output is given by q = q + q.. Find the isopro t contour and the reaction function for rm.. Find the Cournot-Nash equilibrium for the industry and illustrate it in q ; q -space. 3. Find the joint-pro t maximising solution for the industry and illustrate it on the same diagram. 4. If rm acts as leader and rm as a follower nd the Stackelberg solution. 5. Draw the set of payo possibilities and plot the payo s for cases -4 and for the case where there is a monopoly. Outline Answer. Firm s pro ts are given by = pq C0 + cq = 0 q + q q C0 + cq So it is clear that a typical isopro t contour is given by the locus of q ; q satisfying 0 c q + q q = constant see Figure 0.5. The FOC for a maximum of with respect to q keeping q constant is 0 q + q c = 0 which yields the Cournot reaction function for rm q = q = 0 c q (0.7) a straight line. Note that this relationship holds wherever rm can make positive pro ts. See Figure 0.6 which shows the locus of points that maximise for various given values of q.. By symmetry the reaction function for rm is q = 0 cfrank Cowell c q (0.8)

14 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR q profit q Figure 0.5: Iso-pro t curves for rm q χ ( ) q Figure 0.6: Reaction function for rm cfrank Cowell

15 Microeconomics The Cournot-Nash solution is where (0.7) and (0.8) hold simultaneously, i.e. where q = 0 c 0 c q (0.9) The solution is at q = q = q C where q C = 0 3 see Figure 0.7. The price is c. c (0.0) q χ ( ) q C χ ( ) q Figure 0.7: Cournot-Nash equilibrium 3. Writing q = q + q, the two rms joint pro ts are given by = pq [C 0 + cq] = [ 0 q] q [C 0 + cq] The FOC for a maximum is 0 c q = 0 which gives the collusive monopoly solution as q M = 0 c : (0.) with the corresponding price [ 0 + c]. However, the break-down into outputs q and q is in principle unde ned. Examine Figure 0.8. The points (q M ; 0) and (0; q M ) are the endpoints of the two reaction functions cfrank Cowell

16 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR (each indicates the amount that one rm would produce if it knew that the other was producing zero). The solution lies somewhere on the line joining these two points. In particular the symmetric joint-pro t maximising outcome (q J ; q J ) lies exactly at the midpoint where the isopro t contour of rm is tangent to the isopro t contour of rm. q χ ( ) (0,q M ) q J q C χ ( ) (q M,0) q Figure 0.8: Joint-pro t maximisation 4. If rm is the leader and rm is the follower then rm can predict rm s output using the reaction function (0.7) and build this into its optimisation problem. The leader s pro ts are therefore given as 0 q + q q C0 + cq which, using (0.7), becomes 0 q + 0 c q q C0 + cq = The FOC for the leader s problem is 0 c q q C 0 (0.) [ 0 c] q = 0 so that the leader s output is qs = 0 c and, using (0.7), the follower s output must be cfrank Cowell qs = 0 c 4

17 Microeconomics Cournot Joint pro t max Stackelberg leader Stackelberg follower output price pro t 0 c c. [ 0 c] 0 c 4 [ 0 + c] 0 c c [ 0 c] 0 c 4 9 C 0 [ 0 c] 8 C 0 8 C c [ 0 c] 6 C 0 Table 0.8: Outcomes of quantity competition linear model see Figure 0.9. The price is c. q q C q S χ ( ) (q M,0) q Figure 0.9: Firm as Stackelberg leader 5. The outcomes of the various models are given in Table 0.8.and the possible payo s are illustrated in Figure 0.0. Note that maximum total pro t on the boundary of the triangle is exactly twice the entry in the Joint pro t max row, namely 4 [ 0 c] = C 0. This holds as long as there are two rms present i.e. right up to a point arbitraily close to either of the end-points. But if one rm is closed down (so that the other becomes a monopolist) then its xed costs are no longer incurred and the monopolist makes pro t M := 4 [ 0 c] = C 0. In Figure 0.0 the point marked is where both rms are in operation but rm is getting all of the joint pro t and the point ( M ; 0) is the situation where rm is operating on its own. cfrank Cowell

18 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Π (0, Π M ) (Π J,Π J ) (Π C,Π C ) (Π S,Π S ) 0 C 0 Π { (Π M,0) Figure 0.0: Possible payo s cfrank Cowell

19 Microeconomics Exercise 0.8 An oligopoly contains N identical rms. The cost function is convex in output. Show that if the rms 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 rm. Pro ts for a typical rm are given by where p q f + K q f C q f (0.3) K := NX j= j6=f is the total output of all the other rms, which of course rm 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 (0.4) Given that all the rms are identical we may rewrite condition (0.4) as p q (q) q N + p (q) C q = 0 (0.5) where q is industry output. This in turn can be rewritten as p (q) = C q + N (0.6) where := p (q) qp q (q) is the elasticity of demand. The result follows immediately: as N becomes large (0.6) approaches p (q) = C q (0.7). cfrank Cowell 006 7

20 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Exercise 0.9 Two identical rms 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 rms enter the market Bertrand competition then takes place afterwards. If the rms make their entry decision sequentially, what is the equilibrium? Outline Answer. The rms rst decide whether to enter (and hence incur the xed 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 rms decide to enter, the unique Nash equilibrium of the Bertrand pricing game is to set prices equal to marginal cost, (p ; p ) = (c; c). This yields overall pro ts for the two rms ( ; ) = ( K; K).. The extensive form is shown in Figure 0.. Figure 0.: The entry game 3. To nd the Subgame Perfect Nash Equilibrium, solve the game by backwards induction. If rm decides to enter, rm s optimal strategy is not to enter (pro t of 0 compared to K). If rm one decides not to enter, then rm should enter. Firm two can observe the action of rm, thus it can form history-dependent strategies. The optimal strategy is not to enter if rm entered, and to enter if rm did not enter the market, (not enter;enter). Thus, the decision for rm is between entering and receiving pro ts of M K, or not entering and receiving 0. For K small, rm will decide to enter. The unique Subgame Perfect Nash Equilibrium in strategies is thus (enter; (not enter,enter)); (0.8) yielding the equilibrium outcome (enter;not enter). cfrank Cowell 006 7

21 Microeconomics 4. Clearly, there is a rst-mover advantage, since even a small xed cost leads to the monopoly outcome and hence strictly positive pro ts for the rst mover. cfrank Cowell

22 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Exercise 0.0 Two rms have inherited capacity from the past so that production must take place subject to the constraint q f q f ; f = ; There are zero marginal costs. Let () be the Cournot (quantity-competition) reaction function for each rm. If the rms compete on prices show that the following must be true in a pure-strategy equilibrium:. Both rms will charge the same price p.. p = p q + q. 3. p p q + q. 4. q q. Outline Answer. If p < p then, if q = q rm could make higher pro ts by raising its price. Otherwise rm would be undercutting rm so that rm would be forced to reduce its price or lose all its sales to rm. Hence we must have p = p = p in equilibrium.. Consider two cases: (a) If p > p q + q.then for one or both rms it must be the case that q f < q f the price is too high to exhaust capacity. In which case one of the rms could reduce the price slightly, capture sales from the other rm and increase pro ts. (b) If p < p q + q.then both rms are producing to capacity. Each could increase pro ts just by raising prices to its (rationed) customers. Therefore p = p q + q (0.9) 3. If p < p q + q.one of the rms must be capacity constrained: (a) If rm is capacity constrained it could raise its price by p and make additional pro ts pq. (b) Otherwise, if rm is capacity constrained then rm s pro ts are q p q + q (0.0) If rm is also capacity constrained then it could increase pro ts by raising its price. So we may take rm as not being capacity constrained. Given the de nition of () as rm s best response function (0.0) can be written: cfrank Cowell q p q + q

23 Microeconomics with and the price must be q = q (0.) p = p q + q (0.) 4. Suppose q > q : If there is a pure strategy equilibrium then, by (0.9) p = p q + q < p q + q (0.3) which contradicts (0.) cfrank Cowell

24 high cost Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Exercise 0. In winter two identical ice-cream rms have to choose the capacity that they plan to use in the summer. To install capacity q costs an amount kq where k is a positive constant. Production in the summer takes place subject to q f q f ; f = ; 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 + q. 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.. Let () be the Cournot reaction function for either rm in the after problem (as in Exercise 0.0). In the context of a diagram such as Figure 0.6 explain why this must lie strictly above the Cournot reaction function for the before problem.. Let q C be the Cournot-equilibrium quantity for the before problem. Write down the de nition of this in terms of the present model. 3. Suppose that in the summer competition between the rms takes place in terms of prices (as in Exercise 0.0). Show that a pure-strategy Bertrand equilibrium for the overall problem is where both rms produce q C. Outline Answer q low cost. q Figure 0.: Low-cost and high-cost reaction functions for rm First consider how the reaction function of rm would di er if the constant marginal cost were lower than its current value. Given a linear inverse demand function q and constant marginal costs c, the reaction function for rm is 0 q = 0 c cfrank Cowell q (0.4)

25 Microeconomics q after before after before q Figure 0.3: Reaction functions before and after capacity costs are sunk see Exercise 0.7. So for high and low values of c we have the situation depicted in Figure 0.. 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 rm 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 0.3. q χ(q ) ½k/β q C χ(q ) ½k/β. q Figure 0.4: Equilibrium capacity Clearly if (0.4) is the after reaction function for rm, so that q = q once capacity cost is sunk, the before reaction functions for rms cfrank Cowell

26 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR and are, respectively q = 0 c k q (0.5) q = 0 c k so that the before reaction function for each rm is () k : q (0.6) The value q C is found from setting q = q = q C in (0.5) and (0.6) to give q C = 0 c k : (0.7) 3 This gives the solution to the amount of capacity q ; q to install in the rst period 3. If there is price competition in the second period then, from the solution to Exercise 0.0 p = p q + q ; we have the solution to the ice-cream sellers Summer price problem as p = p (q C ) : cfrank Cowell

27 Microeconomics Exercise 0. There is a cake of size to be divided between Alf and Bill. In period t = Alf o ers Bill a share: Bill may accept now (in which case the game ends), or reject. If Bill rejects then, in period t = Alf again makes an o er, which Bill can accept (game ends) or reject. If Bill rejects, the game ends one period later with exogenously xed payo s of to Alf and to Bill. Assume that Alf and Bill s payo s are linear in cake and that both persons have the same, time-invariant discount factor <.. What is the backwards induction outcome in the two-period model?. How does the answer change if the time horizon increases but is nite? 3. What would happen if the horizon were in nite? Alf period [offer γ ] Bill [accept] (γ, γ ) period [reject] Alf [offer γ ] [accept] Bill [reject] (δγ, δ[ γ ]) period 3 (δ γ, δ [ γ ]) Figure 0.5: One-sided bargaining game. Begin by drawing the extensive form game tree for this bargaining game. Note that payo s can accrue either in period (if Bill accepts immediately), in period (if Bill accepts the second o er), or in period 3 (Bill rejects both o ers). Using this time frame to discount all payo s back to period we nd the game tree shown in Figure 0.5. We can solve this game using backwards induction. Assume the game has reached period where Alf makes an o er of to Bill (keeping for himself). Bill can decide whether to accept or reject the o er made by Alf.: the cfrank Cowell

28 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR best-response function for Bill is [accept] if [ ] [reject] otherwise Since Alf wants to maximize his own payo, he would not o er more than [ ] to Bill, leaving him (Alf) with = +. The other option is to o er less today and receive tomorrow, discounted back to date. But since <, + > and hence Alf would o er = [ ] to Bill, which is accepted. Thus, going back to date, where Alf would o er Bill for himself), the best-response function for Bill is now [accept] if [ ] (keeping [reject] otherwise By a similar argument as before, Alf would not o er more than [ ] in period, and thus has the choice between receiving = + in period, or receiving + in period. But again, since <, we nd + > [ + ] and hence Alf will o er = [ ] to Bill in period, which is accepted; Alf s equilibirum share is = [ ].. Now consider a longer, but nite time horizon. The structure of the backwards induction solution outlined above shows that as the time horizon increases from T = bargaining rounds to T = T 0, the o er made by Alf reduces to = T 0 [ ] which is accepted immediately by Bill; Alf s share is = T 0 [ ]. 3. Now consider an in nite time horizon. The solution for the nite case would suggest that as T!, ( ; )!(; 0). However, this reasoning is inappropriate for the in nite 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 o er made by Alf, looks identical to the game just played. In both games, there is a potentially in nite number of future periods. This insight enables us to nd the equilibrium outcome of this game. Assume that the continuation game that follows if Bill rejects has a solution with allocation (; ). Then, in the current period, Bill will accept Alf s o er if [ ], as before. Thus, given a solution (; ), Alf would o er = [ ]. But if is a solution to the continuation game, it has to be a solution to the current game as well, and hence =. It follows that Alf will o er = [ ] = cfrank Cowell = 0 to Bill, and Bill accepts immediately.

29 Microeconomics Exercise 0.3 Take the game that begins at the node marked * in Figure 0.6. [NOT INVEST] [INVEST] * ** [In] [Out] [In] [Out] [FIGHT`] [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 0.6: Entry deterrence. Show that if M > J > F then the incumbent rm will always concede to a challenger.. 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 rm 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 payo s in each location are as in part and the incumbent s overall payo is the undiscounted sum of the payo s over all locations. Show that, however large N is, all the challengers will enter and the incumbent never ghts. Outline Answer. 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 0.9, where the rst part of the monopolist s strategy speci es the action after the entrant played [in], while the second speci es the action after [out]. cfrank Cowell 006 8

30 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR (Entrant) [in] [out] ([concede]; [concede]) J ; J M ; (incumbent) ([concede]; [ght]) J ; J M ; ([ght]; [concede]) F ; 0 M ; ([ght]; [ght]) F ; 0 M ; Table 0.9: Weak monopolist: strategic form We nd immediately that there are four Nash equilibria: ([concede]; [concede]); [in] ([concede]; [ght]); [in] ([ght]; [concede]); [out] ([ght]; [ght]); [out] Note that the outcome ([ght]; [in]), where the entrant enters and the incumbent ghts, is not an equilibrium outcome. E i [In] [Out] [FIGHT`] I i [CONCEDE] (Π M, Π) _ (Π F,0) (Π J, Π J ) Figure 0.7: Challenge i (b) To nd the Subgame Perfect Nash equilibrium, we have to nd the strategy combinations that are Nash equilibria in every subgame. There are two subgames here at rm s decision nodes. In the case that the entrant ( rm ) chose [in], the best response is to choose [concede], while if the entrant chose [out], the best response is to choose either [concede] or [ght]. But the best response of the entrant to those best responses is to choose [in]. Thus, the Subgame-Perfect Nash Equilibria are (([concede]; [concede]); [in]) and (([concede]; [ght]); [in]) cfrank Cowell 006 8

31 Microeconomics which both yield the backwards induction outcome. Hence, we nd that the threat to ght after entry is not a credible strategy. However, if there were a precommitment device, such that the threat of ghting became credible, then it would be better for the entrant not to enter, so the equilibrium outcome would be ([ght]; [out]) or ([concede]; [out]), which imply Subgame-Perfect Nash equilibrium strategies (([ght]; [concede]); [out]) or (([ght]; [ght]); [out]).. We may now consider an extension of essentially the same model. The incumbent has stores I ; I ; :::; I N in local markets ; ; :::; N. There is sequence of challenges in each market from potential entrants E ; E ; :::; E N A typical encounter is depicted in Figure 0.7. The outcome in this market is independent of actions in markets ; ; :::; i. 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 the outcome is ([concede]; [in]): cfrank Cowell

32 q = χ(q ;0) Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Exercise 0.4 In a monopolistic industry rm, the incumbent, is considering whether to install extra capacity in order to deter the potential entry of rm. 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 xed cost k in the event of entry.. Let qs be rm s output level at the Stackelberg solution if rm enters. Suppose qs 6= q M, where q M is rm 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.. Let q be the incumbent s output level for which the potential entrant s best response yields zero pro ts for the entrant. In the case where entry deterrence is possible but not inevitable, show that if qs > q, then it is more pro table for rm to deter entry than to accommodate the challenger. Outline Answer. We begin by modelling the use of capacity as deterrence. q q = χ(q ;0) q = χ(q ;c) q C q M, q Figure 0.8: Quantity and capacity choices (a) Suppose the two rms were to choose capacity z and z simultaneously a minor variation on the standard Cournot model. Firm s problem is max q ;z = p q + q q cz (0.8) subject to cfrank Cowell q z (0.9)

33 q = χ(q ;0) Microeconomics with solution q = z = q ; c (0.30) where the dependence of the reaction function on the marginal cost parameter is made explicit see Figure 0.8. Note that it never pays to leave capacity unused in this version of the story. Also note that if Firm s marginal cost were cut from c to 0 then the schedule would be shifted to the right in Figure 0.8 compare the schedule (; 0) with (; c) and also Figure 0. in the answer to Exercise 0.. (b) Firm s problem is similar: subject to max q ;z = p q + q cz k (0.3) q z (0.3) Note that the term k in (0.3), being a xed entry cost, will not affect the rst order condition that characterises rm s best-response output, given that it enters the market. So its behaviour conditional on entry is also given by the reaction function : q = z = q ; c. (0.33) Taking (0.30) and (0.33) together we get the symmetric Cournot- Nash solution (q C ; q C ) see point q C.in Figure 0.8. q z q Figure 0.9: Incumbent s best response with precommitted capacity (c) But if rm s capacity were xed in advance (and cannot be sold o ) then the term cz in.(0.8) would be treated as a sunk cost, irrelevant to the optimisation problem. The optimisation problem for rm would e ectively become that of (0.8, 0.9) with c = 0. So its behaviour would be characterised by cfrank Cowell q = q ; 0 : (0.34)

34 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR Suppose rm s advance capacity is xed at z. Then, there are in principle two regimes that can apply when at the output-choice stage of the game: q z. Output can be met from pre-existing capacity and so the best response of rm (the incumbent) is given by (0.34) in this region. q > z. Extra capacity will have to be installed at marginal cost c; the best response of rm is given by (0.30) in this region. So the combined best responses will look like Figure 0.9. q q z_z_ q z _ q Figure 0.0: Limits on incumbent s capacity precommitment (d) Now consider rm 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 (0.30) and (0.33). The incumbent would not precommit to an amount less than z because it is pointless both rm and rm can always install extra capacity during the output-choice stage. Let z be the capacity level corresponding to the solution to (0.34) and (0.33) see the right-hand intersection in Figure 0.8. 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 0.0. (e) Choice of z within the range [z; z] depends on the size of the entry cost for rm. Note that, in the usual interpretation of a Cournot diagram rm s pro ts rise as one moves North-west along rm.s reaction function see Figure 0.5 in the answer to Exercise 0.7. Now consider the implications of the size of q, the output level of rm for which the rm s best response yields it zero pro ts. q z. Firm can always make a pro t for any any capacity choice made by rm such that z [z; z]. The best that rm can do is to accommodate rm s entry and will act as a Stackelberg leader. cfrank Cowell

35 Microeconomics q [z; z]. Firm will make a loss for any capacity choice by rm such that q < z z. Here rm may either (a) be assured of deterring rm s entry and and choose monopoly output or (b) choose capacity z so as to deter entry or (c) accommodate entry as above q q S χ ( ) z_z_ q M _q z _ z _ q Figure 0.: Capacity choice of incumbent (f) Compare the Stackelberg and monopoly solutions in the case where there is a linear demand curve 0 In the case where rm can blockade entry and act as a monopoly case pro ts are 0 q q cq q and the solution as q = q M = 0 c (0.35) see equation (0.) in the outline answer to Exercise 0.7. Take the case where rm accommodates and acts as a Stackelberg leader. Firm s reaction as follower is given by q = q = 0 c q (0.36) see equation (0.7) in Exercise 0.7. Firm s pro ts as leader are 0 c q q (0.37) cfrank Cowell

36 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR see equation (0.) in Exercise 0.7 and the pro t-maximising output is q = qs = 0 c : So, if the demand function were linear we must have q S = q M which contradicts what is stated in the question. Therefore the demand function must be nonlinear.. See Figure 0.. If q q M then rm sets q = 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 > q M. If q > z then q = q is not credible, it is outside the range [z; z] of credible precommitments. Since the pro ts of rm increase as one moves Northwest along rm s reaction function it follows that it is not possible to deter entry. So, by assumption, q z. Also, qs > q by assumption. Hence qs ; z > q > q M. In Figure 0. it has been assumed that qs < z. We have entry accommodation if q < q and entry deterrence if q q. In this case set q = q to get on to the lowest (and so higher pro ts) isopro t contour possible. Of course this isopro t contour is below the one through qs ; S q which, in turn, is below the one which is applicable if q < q. This establishes the conclusion. Note that if qs > z then q 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 = q is outside the range of credible pre-commitments. So now rm sets q = z and concedes entry. cfrank Cowell

37 Microeconomics Exercise 0.5 Two rms 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... What would be the solution to the Bertrand price setting game?. Compute the joint-pro t maximising solution for this industry. 3. Consider an in nitely repeated game based on the Bertrand stage game when both rms have the discount factor <. What trigger strategy, based on punishment levels p = c; will generate the outcome in part? For what values of do these trigger strategies constitute a subgame-perfect Nash equilibrium? Outline Answer. Suppose rm sets a price p > c. Firm then has three options: It can set a price p >p, it can match the price of rm, p = p, or it can undercut, since there exists an > 0 such that p = p > c. The pro ts for rm in the three cases are: 8 < 0 if p > p = p c k p : if p = p p c k p if p = p For su ciently small pro ts in the last case exceed those in the other two, and hence rm will choose to undercut rm two by a small and capture the whole market. The rms will choose to share the market if they are playing a one-shot simultaneous move game, where they will set p = p = c. If the rms maximise joint pro ts the problem becomes choose k to max [k q] q cq The FOC is k q c = 0 which implies that pro t-maximising output is q M = k c so that the price and the (joint) pro t are, respectively p M = k + c M = [k c] 4 cfrank Cowell

38 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR 3. The trigger strategy is to play p = p M at each stage of the game if the other rm does not deviate before this stage. But if the other rm does deviate then in all subsequent stages set p = c. In the accompanying table rm deviates at t = 3 by setting p = p M " < p M so that rm responds with p = c to which the best response by rm is also p = c :::t rm : p M p M p M c c rm : p M p M p c c If " is small then for that one period rm would get the whole market so = M. Thereafter = 0. If the rm had cooperated it would have got = M. The present discounted value of the net gain from defecting is M M ::: = M 0 if and only if. cfrank Cowell

39 Microeconomics Exercise 0.6 Consider a market with a very large number of consumers in which a rm faces a xed cost of entry F. In period 0, N rms enter and in period each rm chooses the quality of its product to be High, which costs c > 0, or Low, which costs 0. Consumers choose which rms to buy from, choosing randomly if they are indi erent. Only after purchasing the commodity can consumers observe the quality. In subsequent time periods the stage game just described is repeated inde nitely. The market demand function is given by 8 < '(p) if quality is believed to be High q = : 0 otherwise where ' () is a strictly decreasing function and p is the price of the commodity. The discount rate is zero.. Specify a trigger strategy for consumers which induces rms always to choose High quality. Hence determine the subgame-perfect equilibrium. What price will be charged in equilibrium?. What is the equilibrium number of rms, and each rm s output level in a long-run equilibrium with free entry and exit? 3. What would happen if F = 0? Outline Answer. For a given price p pro ts for i are given by " # = q i [p c] + r + [ + r] + ::: = q i [p c] F (0.38) r if it produces High quality for ever and q i p F + r if it produces low quality in one period and then never sells any product thereafter. The trigger strategy is to stay with the rm unless it is observed to have changed its quality to Low, in which case the rm will never again have customers. This punishment ensures SPNE (Note the zero demand for Low quality.) To ensure that this strategy is incentive-compatible, and thus consistent with SPNE, we need Hence q i p q i [p c] " + + r + [ + r] + ::: = + r q i [p c] (0.39) r p [ + r] c (0.40) in equilibrium. The equilibrium price is su ciently high that the rm is unwilling to sacri ce its future pro ts for a one-o gain from producing Low quality and selling it at a high price. cfrank Cowell F #

40 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR. From (0.38) the long-run zero-pro t condition implies q(p) [p c] r We have the market-clearing condition F (0.4) Nq i = q(p) (0.4) This (0.4) determines the equilibrium number of rms N = int q(p) p c rf (0.43) From (0.40) and (0.4) we get q i = N rf p c (0.44) 3. Without entry cost, but with demand elastic, q(p) is determined, but q i and N are indeterminate. cfrank Cowell 006 9

41 Microeconomics Exercise 0.7 In a duopoly both rms have constant marginal cost. It is common knowledge that this is for rm and that for rm it is either 3 4 or 4. It is common knowledge that rm believes that rm is low cost with probability. The inverse demand function is q where q is total output. The rms choose output simultaneously. What is the equilibrium in pure strategies? Outline Answer Let marginal cost be denoted by c. Firm s and rm s pro ts are given by, respectively: := q q q C q (0.45) := q q q C cq (0.46) Maximising (0.46) with respect to q, given q we have the FOC q q c = 0 which yields the Cournot reaction function for rm q = q ; c = c q (0.47) (compare the answer for Exercise 0.7). The two cases (low-cost and high-cost) for rm are therefore q L = 5 8 q H = 3 8 q (0.48) q (0.49) compare this with Figure 0. in Exercise 0.. In the light of the above the expected pro ts for rm are q ql q C q + q q L q qh q q H q C q Under the Cournot assumption rm takes ql for rm is q q L q H and q H = 0 which implies q = q 4 L + qh as the best response of rm to rm. Substituting from (0.48) and (0.49) into (0.50) we have q = q q cfrank Cowell C q as given. So the FOC (0.50) (0.5)

42 Microeconomics CHAPTER 0. STRATEGIC BEHAVIOUR which simpli es to q = From which we nd the solution as q (0.5) 4 = q (0.53) q = 3 q L = 5 8 q H = = = 7 48 cfrank Cowell

43 Chapter Information Exercise. A rm sells a single good to a group of customers. Each customer either buys zero or exactly one unit of the good; the good cannot be divided or resold. However, it can be delivered as either a high-quality or a low-quality good. The quality is characterised by a non-negative number q; the cost of producing one unit of good at quality q is C(q) where C is an increasing and strictly convex function. The taste of customer h is h the marginal willingness to pay for quality. Utility for h is U h (q; x) = h q + x where h is a positive taste parameter and x is the quantity consumed of all other goods.. If F is the fee required as payment for the good write down the budget constraint for the individual customer.. If there are two types of customer show that the single-crossing condition is satis ed and establish the conditions for a full-information solution. 3. Show that the second-best solution must satisfy the no-distortion-at-the-top principle. 4. Derive the second-best optimum. Outline Answer. If the consumer has income y then the budget constraint is x + F (q) y where is a variable taking the values 0 or, representing the cases not buy and buy.. Assume that each person s type is common knowledge. (a) If there are two taste types a ; b with The preferences are as shown in Figure.. 95 a > b (.)