近畿大学学術情報リポジトリ

全文

(1)Robustness. of equilibrium. yardstick. competition. model*. Fujimoto. Masaki Abstract. in a two-firm. It is shown that, in a two-firm yardstick. competition. game, a Nash equilib-. rium outcome is robust under changes in the objectives of the firms (i.e. maximization of own profit or minimization firms (i.e. follower-follower. of opponent's. profit) and changes in the relations between. or leader-follower,. or one-shot, short-term,. Specifically, it is shown that, in the model, (a minimax point) (a Stackelberg. equilibrium). = (a Nash equilibrium).. tions of the one-shot Nash equilibrium. JEL classification: C72;. The. point is globally stable under the best response. Yardstick. L51. competition;. Nash. equilibrium;. Minimax. point;. Collusion;. equilibrium. 21, 2010 accepted. September. making. repeated. rule implies its robustness.. C73. Stackelberg. *. Also, collusion other than repeti-. These results suggest that the price rule implies efficiency of the equilibrium. outcome, and the transfer. Key words:. = (a maxmini point) —. is impossible in a (finitely or infinitely). game setting, and the Nash equilibrium dynamics.. or long-term).. author. is grateful. invaluable. to Hideo. comments. Nakai. and seminar. and suggestions.. participants. However,. remaining. at Kinki errors. University are my own.. for.

(2) V 88. 1. V 1 • 2. Introduction. This paper aims to clarify the structure sent its theoretical real economy. since been. tribution. The yardstick. applied. Applications. implications,. regulation. reimbursement. motor-vehicle. railway industry. implications. pretations tion.. are then consistent. Specifically,. collusion). the. However, the structure. outcome.. realized. of. One implication of this. A variety of distinct hypotheses. with the observed outcome of the yardstick outcome. has the appearance. and intercompeti-. of cooperation. implication of the present results is robustness. rium outcome to changes in firms' This robustness. tory price and transfer tions),. Medicare's. (or. as well as of various types of competition.. Another significant. firms.. 1998), and. competition model, a variety of be-. can generate the same equilibrium. fact is the difficulty of hypothesis testing.. 1997), the op-. have scarcely been explored.. This paper shows that, in a two-firm yardstick havior patterns. countries.. electricity dis-. (Mizutani,. (Ylvinger,. of hospital costs in the US (Shleifer, 1985).. the model and its theoretical. in many. (Cowan, 1997), Norwegian. inspections. in the. by Shleifer (1985), and has. of local monopolists. include the UK water industry. of Swedish. for policy-making. was introduced. (Dalen et al., 1998), the Japanese. eration. competition model and pre-. which could be important. competition. to the. of a yardstick. patterns. of behavior and the relations. implies that, once the regulator. determines. can be achieved, even if a firm's. relation between firms differ from those expected by the regulator,. that the discriminatory. between. the discrimina-. rules (which require knowledge of the demand and cost func-. the social optimum. lator did not understand. of the equilib-. how long competition transfer. omitted, even if introduction. objective and/or. or even if the regu-. between firms continues.. rule is the source of the robustness. of the discriminatory. It appears. and cannot be. price rule and the price-taking. havior of firms seem to be sufficient for the realization. maximization. about. collusion (Col-i, i = 1, 2), and results. problems (Max-i, i = 1, 2).. Specifically, it is shown that. 72 ( 72 )—. be-. of the social optimum.. These results are classified into three categories: results about minimization lems (Min-i, i = 1, 2), results. the. probabout.

(3) Robustness of equilibrium in a two-firm yardstick competition model (Fujimoto) 1.. ((Min-1) Proposition. 1) the Nash equilibrium outcome can be attained even if. each firm simultaneously instead of maximizing 2.. ((Min-2). Proposition. and independently. minimizes its opponent's. profit,. its own profit. 4) The Nash equilibrium. outcome can be attained. even. if one firm is a leader and the other firm is a follower, and the leader minimizes its opponent's. profit,. given a profit-maximizing. behavior. of the fol-. lower. 3.. ((Col-1) Proposition. 2) The Nash equilibrium outcome becomes a solution of. the joint profit maximization 4.. ((Col-2) Proposition. problem for the firms.. 5) If relations. then collusion that can be sustained. between firms continue for a long period, in the long run involves repetitions. of the. one-shot Nash equilibrium. 5.. ((Max-1). Proposition. 3) The Nash equilibrium. if each firm simultaneously. and independently. outcome can be attained maximizes. even. the difference be-. tween its profit and that of its opponent. 6.. ((Max-2). Proposition. 6) The Nash equilibrium. outcome can be attained. even. if one firm is a leader and the other firm is a follower, and the leader maximizes its own profit, given a profit-maximizing. In this paper, Section 2 presents Shleifer's. This is a simplified. version. of. (1985) original model, which involves only two firms, and the demand func-. tion is linear and the cost function stated above.. Generalizations. We now describe the model. There are two symmetric. facing a downward-sloping for both firms.. is quadratic.. Section 3 presents. the results as. of these results are also discussed.. 2. model.. the model.. behavior of the follower.. The. model. We use a two-firm firms, 1 and 2.. version of Shleifer's. (1985). Each firm acts as a local monopolist. local demand function q (p), which is taken to be identical. We use a linear demand function. of the form q (p) = a—/3p, defined. over an interval p E [0, a/,3] with a, ,3 > 0 and q' (p) = —,3< 0. 73 ( 73 )—. Each firm has an.

(4) X88 initial constant. marginal. spending R(c).. 1 • 2. cost co, and can reduce co to a constant. We use a quadratic. cost function. marginal. of the form R(c). cost c by. = r(c— co)2, de-. fined over an interval c E CO,co] , where r > 0. It can be verified that all assumptions imposed on R(c) and R"(c). by Shleifer (1985) are satisfied: R(co) = 0,. (c) = 2y(c— co) < 0,. = 2y > 0.. The profits of firm i(i = 1, 2) are given by:. V' = (pi-ci)q. (pi) —R(ci)+Ti,. where T is a lump-sum transfer. (1). to the firm.. Social welfare in the local market. of the firm i is defined as the sum of the con-. sumer surplus and the firm's profit:. tip,. q(x)dx. + (pi- ci)q(Pi)— R (ci).. In the command optimum,. the regulator. (2) subject to the breakeven. constraint. (2). determines. V'. ci, pi, and Ti so as to maximize. 0.. The social optimum in each market i is characterized. by the first-order. conditions:. R(c*) = T*. (3). p* = c*. (4). —Ri(c*) = q(p*) .. (5). sumptions. a ya,3 that2< Assume co <—and< imposed by Shleifer. 2y. Then it is rapidly verified that all of the as(1985) are satisfied: q(0) = a < 2yco = — (0),. q(co) = a—/3c0 > 0 = —/r(co) , and —q' (c)— R"(c). = /3-2y < 0 (the second-order. condition for (2)). Assume also that the regulator transfer. commits himself to the discriminatory. rules given by T = R (c,) and pi =. mitment and choose costs accordingly,. a. j), and that firms believe this com-. as in Shleifer (1985). 74 ( 74 )—. price and. The total profit of firm.

(5) Robustness i, expression. Vi(ci,. of equilibrium. (1), is then. in a two-firm yardstick. rewritten. competition model (Fujimoto). as:. (c;—ci)q(cd—R(ci)+R(cd = —yci+ [(2yco—a)+,3c11ci+(2--13)0—(22-co—a)c; Cr. (6). —(22-co —a)]. Define 74+ ,1 = {(ci, < ci < r—ScJ+2rco—a} and 7/!F+,2 = {(Ci,cd r—s c r 1 +2rco—a < ci < ci}. Then the set of marginal costs that attain a positive level of profit for firm i is defined as 74+ = 74 +,1U74 +,2. We write 74 when the strict inequalitiesare replacedby weak inequalities, and write the boundary of the set. (. 74Nas V.. ).. It can be verified from (6) that 74(1 7/0=2rco2 —a 2rco—aAs will be r— 2y—a shown below, the intersection coincides with a unique symmetric Nash equilibrium point (6, c2*)of the model. Thus, it is possible to identify the Nash equilibrium point without solving the profit maximization problem for the firms.. Figure 1 shows the. profit surface of firm 1 when the demand function is q(p1) = 34-0.5p1 and the cost function is R (ci). (c1-25)2.(1). 20•*". Figure. 1.. (1) The demand function used here is the same as that used in Potters et al. (2004), and the value of y is selected such that the slope of a firm's best response curve, dc? /dc;, becomes the same in both models. For further details of the relation between their model and the present one, see Remark 5.2. 75 ( 75 )—.

(6) X88. 1 • 2 '-4-. Since 0 <r—Sr < 2y, we have< <. < 2y.. = 0,74+,2 n. 1 <if< i 7,and< 1 <if r—SSrr iS—r In both cases, 74+ , i n 74+, 2 = 0. By the definitions, 74+, i n 74+,1 +,2 = 0, so that 24+ n 7/_2 = 0. Obviously, 71!.kC174= 0 holds (see. Figure 1) . These results imply the uniqueness of the subgame-perfect equilibrium of an infinitely repeated version of the game (Section 3.2.2) and the equivalence of Nash and Stackelberg equilibria (Section 3.2.3). We now define the Nash equilibrium of the model.. Definition. 1. (Nash equilibrium). . An action profile (cc, cc) is a Nash equilibrium of. the model, if ci is a solution to the maximization. problem:. max Vi(ci, c;). (7). for each i = 1, 2. By differentiating (6) with respect to ci, we obtain the first-order condition:. —2yci+ (2yco—a)+/3c; = 0.. (8). Thus, the best response function of firm i is c (cd = (27co—a)-H3c;, and the unique 2y. 2yco —a27co—a). symmetric. Nash equilibrium. is (cC, c2) =.Obviously, 2 y-3'. and-order. condition for the maximization. problem. the sec-. 2y—,(3 is satisfied:. 62vi(ci, c) 2I uci. 27. < 0. From the first line of (6),a more general version of the first-order condition is written as q(c;) = —1r(ci). Thus, the Nash equilibrium cc = cz = c* coincideswith the social optimum that maximizes (2).(2)Since co < 73a , the realizedprice p' = c* is lower. than. the initial. marginal. cost. co, and. neither. firm. gains. a positive. level. of profit.. (2) Potters et al. (2004) used a benefit of slack function B (ci) instead of the cost function R(ci) usedby Shleifer (1985),and showedthat the symmetricNash equilibriumdoesnot coincide with the social optimum if the discriminatory schemefii(ci, c1) = c;(i # j) is replacedby the uniform schemepi(ci, c2) = (ci +c2)/2 for i = 1, 2. This is becausea firm behavesas a price-takerunder the discriminatoryschemebut as a price-makerunderthe uniform scheme. 76 ( 76 )—.

(7) Robustnessof equilibriumin a two-firmyardstickcompetition model(Fujimoto). 3. Results. 3.1 Changes in the firms' objectives The definitionof Nash equilibriumimpliesthat if firms simultaneouslyand independently maximizetheir own profit for a given choiceof the opponent,then the realized outcome is (c*,c*) . In two-firm competition models, such as the Cournot duopolyand the Bertrand duopoly,the realization of a Nash equilibriumoutcomealso impliesthat firms simultaneouslyand independentlymaximizetheir ownprofit, given the opponent's choice. In a two-firm yardstick competition model, however,realization of the outcome (c*,c*) does not necessarily imply that firms maximizetheirown profit,given the opponent's choice. 3.1.1 Results about minimization problems The followingresult states that, under a certain condition,even if firms change their objectivesfrom maximization of their ownprofit to minimization of their opponent's profit,the realizedoutcomeis still (c*,c*). Proposition 1 ( (Min-1) Thwarting Behavior under Relative Performance Evaluation).. Supposethat theparametersof the demandandcostfunctionssatisfy13< r. Then 2rco—a theNashequilibrium outcome(c*, c*) withc = 2 is a uniquesolutionto the minimizar— tionproblem:. min Vi(c;, c;),. where Vi(ci, c1) =. (9). Eq(c;) — (ci)] — i(ci, c1) and Cc;—cilEq(ci)—q(ci)] 0. for anyci and Proof From (6),the profit function of firm i is given by:. c;) = (r—$)c;—[(2rco—a) — ci]. 77 ( 77 )—. —(2yc0—a) ci]. (10).

(8) X88 When )3 < y, the first-order. 1 • 2. condition for the above minimization. problem is both nec-. essary and sufficient. By differentiating. (10)with respect to c1, we obtain the first-order. 2 (7—'3) c1—(2rco—a)+,3ci. for j = 1, 2.. Remark 1.. = 0. (11). By solving this pair of equations we obtain a solution (c*, c*).. ^. 1. This result cannot be obtained in the Cournot duopoly or the Bertrand. duopoly.(3) To see this, consider a simple version of Cournot's firms. conditions:. (1 and 2) produce a homogeneous. good, and the quantity. duced by each firm i(i = 1, 2) is determined maximized,. given the quantity. q1(j. model in which two of good qi pro-. so that its own profit 7-ci(qi,q1) is. i) produced by the other firm.. In this. model, the profit function of firm i is given by:. qj). La— (qi+ qi) —c] qi,. where the market marginal. (12). demand function is p (qi, q1) = a— (qi+q") , and c is a constant. cost of the firm such that c < a.. Nash equilibrium. point (q1*,q2) =. Then there is a unique symmetric. 3 3 a = (a— c a— c). Since 07-ci (qi, qi qi). < 0,. each firm i will choose a maximum possible level qi = a when aiming to minimize its opponent's. profit ri-j. In this situation. cause of excess supply,. so that. the market-clearing. both firms. earn negative. price is p = 0 belevels of profits,. rci(a, a) = —ca. We now consider Bertrand's goods, and simultaneously Tli(pi, pi) is maximized.. model, in which firms 1 and 2 produce differentiated choose their prices pi(i = 1, 2) so that their own profit. The profit function. of firm i is given by:. p,) = [pi— clIa —pi+ bpi], (3). For further (1992).. (13). details about the Cournot duopoly and Bertrand. 78 ( 78 )—. duopoly models, see. Gibb. ons.

(9) Robustness of equilibrium in a two-firm yardstick competition model (Fujimoto) where the market. demand. function. 0 < b < 2, and c is a constant In this case there. S. 2.. marginal. is a unique. with c < a and. cost of the firm.. symmetric. Nash equilibrium. point. (pi, P2*)=. Tc(a +ca+c b(pi—c), the possible outcomes are (0, pm" aPi. aqpi,p.) ince=. 2— b' 2— b), (c, c) (i.e., price= marginal resents a maximum. is qi(pi, pi) = a—pi+bp;. cost !), and (pm" 0), where the superscript. max rep-. possible level.. This result does not hold if the transfer. rule Ti= R (c;) does not apply.. The. profit function of firm i is then given by:. Vi(ci,. = (c;—ci)(a-13cd—r(ci—c0)2.. (14). Thus, we obtain: 6171(ci,c.) = ac ;. 213ci+a+,3ci. (15). 62vi(ci, c.) 2 c1). 2$ < O.. (16). 3.1.2 Results for collusion The following result implies that the symmetric Nash equilibrium looks like a result of collusion only if the realized outcome (c*, c*) and the profit levels Vi(c*, c*) = 0 are observed. (Collusion in a long-term relationship will be discussed later in (Col-2) and Remark 5.2.). Proposition. 2 ( (Col-1) Collusion in a One-shot Game).. The symmetric Nash. equilibriumoutcome (c`, c*) coincideswith a solution (c., cD of the joint profit maximization problem:. max [ V1(ci, c2)+ V2(ci, c2)]. (c1,co Moreover,Vi(c,. (17). = Vi(c*, c*) = 0, where. is the marginal cost offirm i under collusion.. 79 ( 79 )—. ..

(10) X88 Proof.. 1 • 2. This result follows from:. V1(ci, c2) + V2(ci, c2) = —$(c2— c1)2.. Remark 2.. (18). 1. This result does not follow in the case of Cournot duopoly or Bertrand. duopoly.. In Cournot's. tion—2problem tion. =. is. model a necessary condition for the joint profit maximiza-. qi + q2 =,. c. which. obviously. is. not. satisfied. by. 3. If we consider a symmetric solution, then (a—c a—c) Consequently, both firms earn higher levels of profits under. 3 < (qi*,q;).. collusion than under competition, 7z-i(q`',. > 7z-1(qi% qn.. On the other hand, in Bertrand's model, a unique solution to the problem is. a+(1—b)c a+(1—b)c ) (a+ c a+c)=(pi , P2*). 2(1 —b) ' 2(1—b) 2—b' 2—b. 2. This result does not hold if the transfer rule Ti = R(cd is inapplicable. The joint profits [ Vi+ V-i]are then given by: ci)2±2rCoCi—rc-2rcoci+rcY.. (19). Thus, a unique solution to the joint profit maximization problem is c`' = c, = co and Vi(co,co) > Vi(c*,c*). These results imply the possibility of inefficientcollusion (seealso Remark 5.2 and Figure 2).. 3.1.3 Results for maximization problems The followingresult states that, evenif both firms havestronger profit incentives than conventionalprofit-maximization,the realized outcomedoes not change.. Proposition 3 ( (Max-1) Rat Race under Relative Performance Evaluation). The Nashequilibrium outcome(c*, c*) is a uniquesolutionof the maximization problem:. max [ Vi(ci, cd-17-1(ci,c1)]. (20) 80 ( 80 )—.

(11) Robustnessof equilibriumin a two-firm yardstick competitionmodel (Fujimoto) fori-1,2,i. j.. Proof The demand function q(p) is linear, so that (c1—ci)qi(ci)—q(ci) = By differentiating. q(ci).. Vi- Vi] with respect to ci and using this relation, we obtain:. 6[VI—Vi] = i(ci— c i)qi (ci) —q(ci)] —q(ci) —21r(ci) Oc i(21) = —2[q(cd+ (ci)] . The Nash equilibrium,. which satisfies. both q(c*) = —Ri(c*) and. — (c*). —R"(c*) < 0, is therefore a unique solution to the problem presented above. ^. Remark 3.. 1.. Bertrand. The same duopoly.. result. is not obtained. In Cournot's. model,. in the case. a unique. solution. problem setoutinProposition 3is(c,a2c)*> (q, market-clearing profits. of 0.. price is p = c (price= marginal In Bertrand's. If the transfer lem becomes. 3.2. duopoly. or. to the maximization. q2*). As a result, the. cost !), so that both firms show. model, a solution to the maximization. (a+(1+b)c a+(l+b)c) ,P2*). 2 2. 2.. of Cournot. problem is. rule T = R(c1) is lifted, the solution to the maximization. prob-. # (c *,c*). ( 'No—arco—a\ r—S r—S. Changes in the relationship. between firms. This section shows that, in a two-firm yardstick (c*, c*) can be realized even if the relationship Specifically, the realization. competition model, the outcome. between the firms changes radically.. of (c*, c*) does not necessarily. imply that firms simultane-. ously and independently maximize their own profit for a given choice by the opponent, as prescribed by the concept of Nash equilibrium.. 3.2.1. Results for the minimization. problem. The following result states that the same outcome (c* , c*) can be realized even if the relationship. Proposition. between firms changes and the objective of one firm changes.. 4 ((Min-2). Minimax =Nash).. The Nash equilibrium outcome (c* , c *) with. 81 ( 81 )—.

(12) X88 c =. 2rco—a. 1 • 2. is a minimaxpoint of the profit function Vi of firm i(i = 1, 2), so that it is a so-. lution to the following problem:. min max Vi(ci, c;).. (22). cjci. Wethereforehavemine max, Vi(ci, ci) = Vi(c*,c*) = 0. A +13c; P y, where A (2yco—a) > 0, so that the maximizedprofit function of firm i is given by: roof.The. best response function of firm i iscz (c1) =2. Vi*(c;(c;), c;) = (A +$c)2 + (r—)690—Aci.. (23). The desired result now follows from the equations:. dVi* (c; (c;) ,c;) dc ;. (2r-13) L(2 7—,3)c;—A 1,. d2171*(ci(ci),ci) d c,?. (24). (2T-13)2 > 0 . 2y. (25). (See also Figure 1).. ^. This result implies that the same outcome (c*, c*) arises in the following situation.. One of the firms (firm i) is a follower who chooses its marginal. serving its opponent's. choice. cost ci after ob-. so that its profit can be maximized:. max Vi(ci, c;).. eo. ci. The other firm (firm j) is a leader who chooses its marginal ponent's. profit. min Vi*(c;(c;),. Remark 4.. given the opponent's. best response c; (c;):. c;).. (27). This result does not hold for the Cournot. The Nash equilibrium. in Cournot's. duopoly or Bertrand. model is (qi*, q2*) =3. 82 ( 82 )—. ). cost c; to minimize its op-. duopoly.. 3' (a— c a— c and the.

(13) Robustness of equilibrium in a two-firm yardstick competition model (Fujimoto) minimax point of 7-c1is (0, a—c) and of 7i-2is (a—c, 0).. This is because the maxi-. mized profit rci*(qi*(q;), q1) of firm i decreases monotonically On. the. other. hand,. the. Nash. equilibrium. of. with respect to q1.. Bertrand's. model. is. i I a+ca+c) 2—b'2—b' while theminimax point of7-t-1is (a+c 2' 0)and of itis(0,a+c) . The reason isthat themaximized profit 7i-i*(/); (P1), pi)offirm (pi*,p2*) =. i increases monotonically with respect to pi. Generalization: We now consider a generalization of this result.. From the enve-. lope theorem and (6), we obtain:(4) dV'(ci* (c;), c;) = (c1—c;(cd)q.' (c;) + q(ci) + dc;. (ci). (28). dVi*(c*,c*) Th d c;= 0. Thus, a sufficient condition for (c*, c*) to be a minimax point of Vi is that the maximized e profile (c*, c*) is a symmetric Nash equilibrium, so that. profit of firm i, Vi*, is convex in cf. q (cd=. —Ri(ci) and —(c). From the Nash equilibrium conditions,. < R"(c) , we obtaindc;(c') dc ;. R"(ci). < 1. The suffi-. cient condition now becomes:. (c;—c:(cd)q"(ci) + R" (c;(c;)) •R" (ci) +2R" (c;(c;)) •q' (c;) + (q' (c;))2 R"(c:(cd) whereq' (c3) < 0 and R"(c;) > 0. This conditionis always satisfied when the demand function is linear, i.e. q"(c) = 0, and the cost function is quadratic so that R"(c) is constant.. 3.2.2 Results for collusion It followsfrom Propositions 2 and 4 that, in an infinitely repeated yardstick competition, each firm can gain from a unilateral deviation from collusive action c`.(= c;) without any loss arising from being punished, if c; * c*. Collusion can therefore be sustained in the long run only when both firms repeatedly choose c*.. (4) For the envelope theorem, see Varian. (1992). 83 ( 83 )—. 2.

(14) V8 This. result result is stated. Proposition. V 1 • 2. as follows.. 5 ( (Col-2) Collusion in a Long-term. Relationship). . Collusion can be sus-. tained in the long run only when both firms repeatedly choose the Nash equilibrium cost c; =. = c*.. Proof. From. Proposition. 2, the. profile. (c1, c2) maximizes. the. joint. V 1± v-2] if el _ c2 _ cc. Consider the case in which the firms' marginal. profits. costs under. collusion are cc > c*. Since c; = cc > c*, firm i can choose the best deviation c; (cc) E. ( y-13c`'+2yco —a ,cc)andgain a positive level ofprofit, Vi(c; (cc), cc)> 0 (Note that (ci*(cc), cc) E V_(+ ,2). We suppose that firm j has adopted the following trigger strategy:(5). Choose cc as long as firm i chooses cc. If firm i has chosen ci. cc in the pre-. vious period, then choose c* forever after, where, from Proposition 4, c = argminc[maxc , Vi(ci, c1)] is firm j's maximum punishment.. From Proposition 2, the present value of profits that firm i receives from choosing cc repeatedly is at most1 —a1Vi(cc, cc) = 0, where a E [0, 1) is the discount factor of the firm.. On the other hand, from Proposition 4, the present value of profits that. firm i receives by deviating optimally from cc is Vi(c;(cc), cc) + 1-6 Vi(c*, c*) = Vi(c; (c9 , c9 > 0. Hence, the best deviation from the collusive action cc is profitable for firm i, and collusion (cc, cc) with cc > c* cannot be supported as a subgameperfect equilibrium outcome. By similar arguments, firm i can gain a positive profit Vi(ci, c9 > 0 by increasing its marginal cost so that (ci, cc) E 74+ , i in the lower case cc < c* (see also Figure 1). Since no ci exists such that Vi(ci, c*) > 0 when firm j chooses the Nash equilibrium cost c*, collusion (c*, c*) can be sustained in the long run.. ^. (5) The idea of a trigger strategy was introduced by Friedman (1971), and has since been frequently used to analyze incentives for collusion in a repeated game setting. For the trigger strategy, see Gibbons (1992). 84 ( 84 )—.

(15) Robustnessof equilibriumin a two-firm yardstick competitionmodel (Fujimoto) Remark 5.. 1. This result does not hold in the Cournot duopoly.(6) In Cournot's (a— c)2 model, the profit for firm i under collusion is TC = 8, the profit for firm i from the Nash equilibrium is 7z-i(qi% q (a— c)2 2*)= 9, and the profit for firm ai from optimally deviating from ql isn-i(q;(q,c),9(6 = model, collusion is sustainable because rci(q;`(q,c),q,c) >. 4c) 2. In Cournot's qi) > q2*).. 2. Potters et al. (2004) also examined the possibility of collusion in a repeated game setting.. However, the payoff structure of their model is different from. here, because they omit the transfer rule, Ti = R(ci) . Figure 2 shows the profit surface of firm 1 in their model for the demand function q(p1) = 34-0.5p1 and the benefit function B(ci) = 40c1—cf (compare Figure 2 with Figure 1) . Their parameter values are a =34, /3=0.5, y =1, co = 25, and cc = 20. (The present assumptions are satisfied for 17 < co < 68 and td. < 2y.) Under these assump1 4'an Vi(c; (cc), cc) > Vi(cc, cc) = Vi(c*, c*), where Vi(c; (cc), cc) = 49and Vi(cc, cc). = Vi(c*, c*) = 0. In our model, collusion is not sustainable in the long run. 1 Potters et al. (2004) used B(ci) = 40c1—ci, and obtained c?(c;) = 3+-4cJ and the inequality. Vi(c; (cc), c9 > Vi(cc, c9 > Vi(c*, c*),. Vi(c; (cc), cc) = 544, and Vl(cc, c9 = 400, and c*. where. c; (cc) = 8. 4 and Vi(c*, c*). and. 144. This. is why collusion is sustainable in their model. Uniqueness: We now prove uniqueness of the subgame-perfect equilibrium of a repeated yardstick. competition.. As shown in Section 2, 74+1-174+ = 0 and. 74 + n voi= 0 (i = 1, 2, i # j) (See Figure 1) .. It follows from the Folk Theorem. (Fudenberg and Maskin (1986)) that the average payoff which can be achieved as a subgame-perfect equilibrium payoff (i.e., a feasible and individually rational payoff) is zero.. From the backwards-induction argument, each firm maximizes the present. value of its payoffs in every period, given a continuation payoff of 0. Therefore, the unique subgame-perfect equilibrium of an infinitely repeated yardstick competition. (6) For collusion in the Cournot duopoly, see Gibbons (1992). 85 ( 85 )—.

(16) Figure 2. comprises repetitions of the one-shot Nash equilibrium (c*, c*). Uniqueness of the one-shot Nash equilibrium point obviously implies that the unique subgame-perfect equilibrium of the finitely repeated version of the game also comprises repetitions of the one-shot Nash equilibrium. This argument does not imply that collusion is always impossible.. This is. because the best deviation of firm i, c; (cc) , is also a solution to the joint profit maximization problem of firms in a symmetric game, i.e. c; (cc) = c; (cc) . (This contrasts with the results for the Cournot duopoly and the Prisoners' Dilemma).(7) However, if we consider the best response dynamics ci,t+i =. t) (i = 1, 2, j. i, and. t = 0, 1, ...) an action profile (ci , t, c,, t) converges to the Nash equilibrium point (c*, c*) from any initial state (c1,o,. 0) because dc; /dc, < 1. Moreover, if we use the. best response function stated in Remark 5.2, then. c*1< 0.0011ci , 0—c*1holds in. period 5.. 3.2.3. Results for maximization. problems. We now prove that the same outcome (c*, c*) is realized when the leader (firm j) maximizes its own profit V3 instead. (7) For the Prisoners'. of minimizing its opponent's profit Vi , given the. Dilemma, see Gibbons (1992). 86 ( 86 )—.

(17) Robustnessof equilibriumin a two-firm yardstick competitionmodel (Fujimoto) opponent's best response ci (c1) (Stackelberg equilibrium).. Proposition 6 ((Max-2) Staekelberg=Nash).. The Nash equilibriumoutcome (c*, c*). becomesa Stackelbergequilibriumof the model. Proof Without loss of generality, we assume that firm i is the follower and firm j is the leader. The profit function of firm j (leader) is given by:. 1P(c;(cd, ci) = —rc,?+Aci+c(ci). where AA. (2rco—a). > 0 and c;(c;). C(r—,8)c;(c;)+,3ci— A],. =is. +,3c; 2 r. the best response function. (30). of firm. i (follower). Since the right-hand. side of (30)can be rewritten. as:. 1. A-74+Ac•4 2'2+,3cilar+S)A—,3(3r—)3)1,. (31). differentiation of (30) with respect to c; yields the first-order condition: (r+,3) (2y—,3) 22-2 [(2y—,3)ci— A] = 0. nce. d2171(C;(Cj),. (32). (y+,(3)(22--,(3)2 Si 2 < 0 and (27—$)> 0, a Stackelbergequiy2. 2rco—a2rco—a) (c*,c*), where the superscript f repre-. librium is (ci, ci) == 2 r. 2r—/3. sents the followerand 1 represents the leader.. ^. This result follows from the fact that the set of marginal costs (ci, ci) which achievea non-negativeprofit for firm j (leader), Vi+,and the best response curve of firm i (follower), c;(cd, have a unique intersection (c*, c*) (see also Figure 1). Generalization: We now considera generalization of this result. By differentiating the profit function of firm j (leader) with respect to d17-1(ci*(cd, c,). av-i(ci*(cj),c)1avj(c;(cj), + acs dc; 87 ( 87 )—. we have: c1). ac;. (33).

(18) X88 a Vi(c;(c.),c.). where. '*"1 • 2. L(c;(ci)—ci)qi(c;(ci))+q(c;(0)+Ri(c;(0)]. and. uci. avi(c; (0, c1) = ac;. 1q(c; (0) +. (01 . From the first-order and second-order. dc* conditions conditions for the maximization problem of firm i , it follows that d ( o• cf) < 1 R"( . Since the symmetric Nash equilibrium satisfies both ci = c; = cand ci) dVi( c*,c*) q(c*) = — dc (c*), it always satisfies the first-order condition:= ; 0. It therefore becomes a Stackelberg equilibrium, provided that the profit function of firm is concave in-id. d2V(c(cd, c? y.< 0. This means that, if there is a unique Stackelberg equilibrium, it must be (c*, c*). Not surprisingly, we can prove the following result in a similar way.. Proposition 7 (Maxmini = Nash) . Assumethat theparameters of the demandand costfunctions satisfy /3 < y . Thenthe Nash equilibriumoutcome(c*, c*) is a maxminipoint of theprofit function Vi offirm i(i = 1, 2) ; that is, it is a solution to thefollowingproblem:. max min Vi(ci, cd.. (34). Wethereforehave maxc ,mincVi (ci, Proof. = Vi(c*, c*) = mine maxe, Vi(ci, cd.. From the argument in Proposition 1, the first-order condition for firm j's. minimization problem is both necessary and sufficient; it yields:. c"inA (ci) =(35) 2( —13c1 y—,(3)' where A = (2rco—a) > 0.. Substitution of (35)into the profit function (6) of firm i. yields:. Vi(ci, cr(ci)). = —rcY+Aci(A 4( —,3ci)2(36) r—s) •. Thus, the desired result follows from the equations:. 88 ( 88 )—.

(19) Robustnessof equilibriumin a two-firm yardstick competitionmodel (Fujimoto) dVi(ci, cr(ci)) dc i. (27—S) i(2 7—,3)ci—A], 2(r—S). d2Vi(ci, cr(ci)) dc2(7—l3). (2T-13)2. (37). (38). < 0.. These results also imply that the Nash equilibrium outcome is robust to strategic manipulation by firms. To see this, consider the following two-stage game. (1)In the first stage, each firm i(i = 1, 2) simultaneously and independently reports a level of marginal cost c;"to the regulator, and (2) in the second stage, each firm simultaneously and independently chooses its actual marginal cost ci to maximize its profit, given the marginal cost reported by the opponent in the previous stage, c,r(j # i). According to Proposition 4 (Minimax =Nash), if each firm i determines ci such that its opponent's profit V' is minimized, given the opponent's best response c,*(c;) in the next stage, then the costs reported in the first stage are (cd,-,. = (c*, c*), and the Nash equilib-. rium outcome (c', c') is therefore realized in the second stage. Surprisingly, it can be shown under a certain condition that if each firm aims to maximize its own profit in determining (cd:,ci), then the realized outcome is the Nash equilibrium outcome (c*, c*) .. Consider the following two-stage game.. (1) In the. first stage, each firm i simultaneously and independently determines the pair (cdi,cil*) and reports onlyc to the regulator, where the superscript 1* represents the optimal choice in the first stage.. (2)In the second stage, each firm makes a better. choice from {cil*,c.;(c;)} . This result is stated as follows.. Proposition. 8.. Assume that the parameters of the demand and cost functions satisfy. r < )3 < 27. Then if eachfirm i determinesthe pair (cT, ci) so that its ownprofit Vi is maximized,given the opponent'sbest responseto the reported cost, c; (c0 , then. ci) = (c*, c*). and consequentlyc; (c1) = c*. Proof The profit function of firm i in the first stage, Vi(ci, cj(cT)), is given by:. —7c+Ac 1 472 [2+3c/1 (r+3)A— 23rci +3()3 —r)cTi , 89 ( 89 )—. (39).

(20) X88. 1 • 2. where A = (27co—a) > 0. By differentiating the first-order. this with respect to ci and cT,we obtain. conditions:. —4r2ci + (22,+ ,3) A +/32cT = 0,. (40). rci— A —(i3-7)cT = 0.. (41). Upon solving this pair of equations, we obtain cr =2 ,. rA. = c , ci=. c , and. ci(c1) = c* (see also (33)). Supposing that y < /3, we have: 62V1(ci, c (cli)) acs. /32(a ,),) 2,1,2<0.. (42). Note that all of the assumptions in Shleifer (1985) are satisfied and therefore Propositions 2-6 are obtained if 0 < /3 < 2y, while the assumption in Propositions 1 (Min-1) and 7 (Maxmini = Nash) is not satisfied if y <. A consequence of the con-. dition y < /3is as follows:. Proposition 9. Since the Nash equilibriumis c; = —q(c) =< 2q(co) 2(c o—c*). 2rco—a= c2 = r—/3. , wehave:. 0,(43). where q (c) = a—$c is a demandfunction with a, )3 > 0 and c* < co.. References. 1). Cowan, S., 1997. Competition in the water industry. Oxford Review of Economic Policy 13, 83-92. 2) Dalen, D. M., Moen, E., and Riis, C., 1998. Innteksregulering - gir det mer effektiv kontroll av nettmonopolene enn avkastningsregulering ? (Income regulation - does it provide a more efficient control over network monopolies than rateof-return regulation). Report prepared for the Norwegian Water Resources and 90 ( 90 )—.

(21) Robustness of equilibrium in a two-firm yardstick competition model (Fujimoto) Energy Directorate (NVE). Friedman, J. W., 1971. A noncooperative view of oligopoly. International Economic Review 12, 106-122. C4 ] Fudenberg, D., Maskin, E., 1986. The folk theorem in repeated games with discounting or with incomplete information. Econometrica 54, 533-554. Harvester Wheatsheaf, New C5 ] Gibbons, R., 1992. A Primer in Game Theory. York. C6 ] Mizutani, F., 1997. Empirical analysis of yardstick competition in the Japanese railway industry. International Journal of Transport Economics 24, 367-392. C7 ] Potters, J., Rockenbach, B., Sadrieh, A., van Damme, E., 2004. Collusion under yardstick competition: An experimental study. International Journal of Industrial Organization 22, 1017-1038. RAND Journal of C8 ] Shleifer, A., 1985. A theory of yardstick competition. Economics 16, 319-327. C9 ] Varian, Hal R., 1992. Microeconomic Analysis, 3rd ed. Norton, New York. C10] Ylvinger, S., 1998. The operation of Swedish motor-vehicle inspections: Efficiency and some problems concerning regulation. Transportation 25, 23-36. C3 ]. 91 ( 91 )—.

(22)