The Envelope Theorem and Comparative Statics of Nash Equilibria

GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.

13, 201–224 (1996)

0034

The Envelope Theorem and Comparative Statics of Nash Equilibria∗ Michael R. Caputo Department of Agricultural Economics, University of California, Davis, California 95616 Received March 16, 1994

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player’s indirect objective function. The fundamental comparative statics matrix of Nash equilibria for the ith player in an N -player static game includes the equilibrium response of the other N − 1 players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game. Journal of Economic Literature Classification Numbers: C72, C61. © 1996 Academic Press, Inc.

INTRODUCTION The methodology of comparative statics, first introduced systematically into the economics literature by Samuelson (1947), has a well-established history and foothold in the literature dealing with individual static models of firm and consumer behavior. In contrast, only in the past decade or so has the literature on static noncooperative game theory investigated the comparative statics properties of the models, as Dixit (1986) has noted. Moreover, the initial applications of comparative statics in noncooperative games were in the context of tightly specified models, and it was not until the Dixit (1986) and Quirmbach (1988) papers that a more general comparative statics exercise was performed. Even though Dixit (1986) and Quirmbach (1988) aimed at a more general comparative statics analysis than their predecessors, their results still do not ∗

An earlier version of this paper was presented at the Economic Theory Seminar and Natural Resources Workgroup at University of California, Davis. The author thanks Richard J. Sexton, Eugene Silberberg, and James E. Wilen for helpful suggestions, as well as an anonymous referee for clarifying insights. Giannini Foundation Paper No. 1113 (for identification purposes only). 201 0899-8256/96 $18.00 Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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capture the fundamental qualitative structure inherent in single-decision-maker (i.e., single player or agent) comparative statics results, such as symmetry and positive semidefiniteness of the comparative statics matrix. While they do treat various forms of conjectures, the heterogeneous product model studied by Dixit (1986) is a duopoly, and the N -player model studied by both authors deals only with a homogeneous good. In all cases the firm’s objective function is less general than that studied here, and no constraints are placed on any of their models, in contrast to the game presented here. One purpose of this paper, therefore, is to provide an exhaustive comparative statics analysis of Nash equilibria in N player static games that is on par with the comparative statics analysis of general models of single-decision-maker problems a` la Silberberg (1974). Given the more general model analyzed here, one may think that there are few qualitative results forthcoming from the comparative statics analysis, but such is not the case. The form or structure of the qualitative comparative statics results, however, is markedly different from the form one would expect using the primal or classical methodology (see, e.g., Silberberg, 1990, Chapter 6), and this demonstrates why simplifying mathematical assumptions (e.g., linear demand or constant marginal cost) or ad hoc modeling assumptions (e.g., local asymptotic stability of the Nash equilibrium) are often introduced into the game in an attempt to gain tighter comparative statics results for the level of the decision variables (i.e., for terms such as ∂ x/∂α, where x is a decision variable and α is a parameter). The results of this research show that all games admitting a sufficiently smooth Nash equilibrium contain qualitative comparative statics, but use of the primal comparative statics methodology obscures the nature and form of them. In contrast to the relatively tedious mathematical manipulations of the Dixit (1986) and Quirmbach (1988) papers, this paper shows that with very little mathematical manipulation and without any curvature or stability assumptions exogenously imposed on the game, the fundamental comparative statics matrix of Nash equilibria for the ith player in an N -player static game is symmetric positive semidefinite subject to constraint. In addition, the fundamental comparative statics matrix contains the Nash equilibrium response of the other N − 1 players in the game to the parameter perturbation, as well as the impact that the other N − 1 players’ decision variables have on the marginal value of the ith player’s decision variables. Thus the strategic nature of the Nash equilibrium plays a crucial role in altering the fundamental comparative statics matrix of single-decision-maker models as set forth by Silberberg (1974). The Envelope Theorem, first introduced by Samuelson (1947) in a general setting, has been used in countless single-decision-maker optimization models. A general statement and proof of the Envelope Theorem for Nash equilibria in general N -player static games, however, is lacking. Another contribution of this paper, therefore, is to provide a simple, rigorous, and elegant proof of the Envelope Theorem in the aforementioned setting. Given the vastly different qualitative comparative statics properties of single-

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agent models and N -player Nash equilibria static games, one may suspect that the envelope results are quite different too, and this is so. For N -player static games, the Envelope Theorem not only picks up the direct effect of a parameter perturbation on the ith player’s indirect objective function, as it does in the single player situation, but it also picks up the strategic effect the parameter perturbation has on the other N − 1 players’ decision variables, something which does not appear in the single-player case. The assertions of the introduction are proven via the primal–dual methodology of Silberberg (1974). This elegant methodology avoids the tedious manipulations of the primal or classical methodology and directly reveals the fundamental nature of the Envelope Theorem and comparative statics. The intrinsic symmetric positive semidefiniteness of the comparative statics matrix is a natural consequence of the second-order necessary conditions of the primal–dual optimization problem defining the Nash equilibrium. Furthermore, no ad hoc stability conditions or exogenous curvature assumptions are invoked to derive the comparative statics, and thus the qualitative results derived here are truly fundamental to Nash equilibria. A fundamental identity linking the value of the Nash equilibrium choice function to the value of the reaction function is also derived. The identity allows for a Slutsky-like decomposition of the effect of a parameter perturbation on the Nash equilibrium choice function into strategic and nonstrategic components. From the Slutsky-like decomposition and the aforementioned curvature result comes a simple explanation of why the derivation of refutable comparative statics results for the level of the decision variables is difficult via primal methodology in models of strategic behavior. Unlike single-agent models, knowledge of how a parameter affects the marginal value of the ith player’s decision variables in an N -player static game is not sufficient to determine the Nash equilibrium comparative statics for the level of the ith player’s decision variables. The ith player must also determine how a change in the other N − 1 players’ decision variables affects the marginal value of the ith player’s decision variables, as well as how the parameter perturbation affects the Nash equilibrium value of the decision variables of the other N − 1 players. Thus only when mathematically strong simplifying assumptions are imposed on the game is there any hope of obtaining refutable comparative statics results for the level of each player’s decision variables. The use of special cases is therefore necessary in order to obtain tight qualitative results in game theoretic models of behavior when a primal methodology is employed to conduct the comparative statics analysis, as this methodology focuses the researchers’ attention on the level of the decision variables rather than on the function of the decision variables that will necessarily possess refutable results, unlike the primal–dual methodology used here. In a tangentially related piece, Slade (1994) derived a necessary and sufficient condition for a Nash equilibrium to be observationally equivalent to a single

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optimization problem. In other words, her Proposition 1 gives a necessary and sufficient condition under which the computation of Nash equilibria, which is a fixed-point calculation, can be reduced to solving a single optimization problem with a fictitious-objective function. Slade (1994), however, did not discuss the resulting comparative statics properties of the optimal solution of such reducible games, nor did she have too. The reason is that the comparative statics properties of the class of games satisfying Slade’s (1994) Proposition 1 are identical to the comparative statics properties of its equivalent fictitious-objective function optimization problem, the latter of which is a special case of Silberberg’s (1974) theorems. In contrast to the class of games satisfying Slade’s (1994) Proposition 1, the comparative statics theorems derived here are a generalization of Silberberg’s (1974). Finally, before moving on to the formal analysis, it is important to point out what is new and what is not new in the paper, so as to make its contribution as clear as possible. Theorem 1 on the generalization of the Envelope Theorem, Theorem 2 and its associated corollary on the comparative statics of Nash equilibria, and Theorem 3 on the comparative statics of reaction functions are new results for Nash equilibria, heretofore undiscovered. The methodology used to derive the results, the primal–dual methodology of Silberberg (1974), is not new, but is used here for the first time in a Nash equilibria setting. Also, the Cournot heterogeneous product duopoly model used to demonstrate the abstract theorems in a more concrete setting is not new, but the application of the aforementioned theorems to the model leads to many new qualitative results that previous research was unable to uncover.

THE GAME AND ASSUMPTIONS Consider a noncooperative game in normal form given by the triplet 0 = (N , S , P ), where N = {1, 2, . . . , N } is the set of players and N , the number of players, is finite, Si ⊂ Rni is the strategy set of player i ∈ N , si ∈ Si is N S j is the feasible strategy of player i ∈ N , S = S1 × S2 × · · · × S N = × j=1 the strategy space of the game, s = (s1 , . . . , s N ) ∈ S is a strategy combination specifying the actions taken by every player in the game, Pi (s) is the realvalued payoff function for player i ∈ N , giving the payoff to player i when the strategy combination s ∈ S is chosen by the N players in the game, and P (s) = (P1 (s), . . . , P N (s)) ∈ R N is the payoff vector giving the payoff to every player when the N players elect the strategy combination s ∈ S . This is the prototype definition of a normal form game as, say, given by Jehle (1991, p. 264) or Nishimura and Friedman (1981). It can be shown that if N is finite, Si ⊂ Rni is compact and convex ∀ i ∈ N , Pi ∈ C (0) , ∀ s ∈ S and ∀ i ∈ N , and Pi is quasi-concave in si , ∀ s ∈ S and ∀ i ∈ N , then the aforementioned game has a Nash (1951) equilibrium point. Nishimura and Friedman (1981) were able

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to relax the quasi-concavity assumption, something that had not previously been possible in proving existence. More recently, Baye et al. (1993) have provided a substantial generalization of earlier existence theorems by relaxing (i) the continuity and quasi-concavity assumptions on the payoff functions and (ii) the compactness and convexity of the strategy spaces. Now consider the static game given by the N simultaneous constrained optimization problems φi (α) ≡ max { f i (x1 , . . . , x N ; α) s.t. g(x1 , . . . , x N ; α) = 0} , xi

i = 1, . . . , N , (1)

whose Lagrangian functions are defined by G i (x1 , . . . , x N , λi ; α) ≡ f i (x1 , . . . , x N ; α) + λi0 g(x1 , . . . , x N ; α), i = 1, . . . , N ,

(2)

where λi ∈ Rm i = 1, . . . , N is the ith player’s vector of Lagrange multipliers, and 0 denotes transposition. The Lagrangian functions G i , i = 1, . . . , N , will be referred to as the “Game Lagrangians.” Before moving on, it is important to see how the definition of a game as embodied in the N simultaneous constrained optimization problems in (1) follows from the definition of the normal form game 0. The number of players N in the set of players N in 0 corresponds exactly to the number of constrained optimization problems in (1), there being a unique constrained optimization problem in (1) for every player in the set N . The strategy set Si of player i in 0 can then be defined in the context of (1) as Si ≡ {xi ∈ Rni | g(x1 , . . . , x N ; α) = 0},

i ∈ N,

and thus the decision vector xi of optimization problem i in (1) can be identified as the strategy si of player i ∈ N in 0. Finally, the payoff function Pi (s) of player i in 0 is simply the objective function f i (x1 , . . . , x N ; α) of optimization problem i in (1), ∀ i ∈ N . The nature and origin of the parameter vector α is clear in the game (1), as it is a vector of exogenous variables that parametrically shift the constraint functions (i.e., strategy set) and the objective function (i.e., payoff function) of each player in the game. Thus, in the context of the normal form game 0, the parameter vector α can be introduced by making explicit (i) the dependence of the strategy set of each player in the game on α, say Si (α), ∀ i ∈ N , and (ii) the dependence of the payoff function of each player in the game on α, say Pi (s; α), ∀ i ∈ N . In order to make the mapping from the normal form game 0 to the system of simultaneous constrained optimization problems (1) as clear and as well motivated as possible, consider the following heterogeneous product duopoly Cournot model. Let xi ∈ Rn+ , i = 1, 2, be the vector of inputs (i.e., strategies)

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chosen by each firm (i.e., player) at the market determined unit prices of w ∈ Rn++ . The output yi , i = 1, 2, of each firm is given by the firm specific production function yi = f i (xi ), i = 1, 2, while the inverse demand function for the product of each firm is given by pi = Di (y1 , y2 ; βi ), i = 1, 2 where βi > 0 is a firm specific demand shifting parameter. The revenue for each firm is thus Ri (x1 , x2 ; βi ) ≡ Di ( f 1 (x1 ), f 2 (x2 ); βi ) f i (xi ),

i = 1, 2.

The firms are asserted to maximize their profits, but they are assumed to face a cash flow or expenditure constraint in that firm i has ki > 0, i = 1, 2, dollars to spend on the inputs. Hence the simultaneous constrained optimization problems analogous to game (1) are given by φi (α) ≡ max{πi = Ri (x1 , x2 ; βi ) − w 0 xi s.t. w 0 xi = ki }, xi

i = 1, 2,

where α ≡ (β1 , β2 , k1 , k2 , w) is the parameter vector of the problem. The parameter vector α thus arises in the game due to firm specific effects, as in βi and ki , and from common effects, as in the vector w. The normal form representation of the above game 0 = (N , S , P ) can now be given. The set of players N = {1, 2} is simply the pair of firms engaged in the interaction, with N = 2 being the finite number of players. The strategy set of player i is then Si = {xi ∈ Rn+ | w 0 xi = ki }, ∀ i ∈ N . Notice that the strategy set of player i depends on a player specific parameter ki and on a common parameter vector w. The strategy of firm i is its choice of its input vector xi , ∀ i ∈ N . Finally, the payoff function Pi for player i is the profit of firm i, πi = Ri (x1 , x2 ; βi ) − w 0 xi , ∀ i ∈ N , and it too depends explicitly on a player specific parameter βi and on a common parameter vector w. Thus, in this example, the constraints in the game arise due to an expenditure constraint on each firm, and the parameters come from the price taking behavior of the firms in the input market, firm specific demand shift parameters, and the firm specific constraint parameters. Return now to the game (1). The following assumptions are imposed on the game (1): (A.1) f i : Rn 1 × · · · × Rn N × R p → R, f i ∈ C (2) , i = 1, . . . , N . (A.2) g: Rn 1 × · · · × Rn N × R p → Rm , g ∈ C (2) , m < n i , i = 1, . . . , N . (A.3) ∃ unique reaction functions (i.e., best-reply functions) to (1) for each ∗ (α ◦ ); ε), denoted by xi = ri (x−i , α), player, ∀ α ∈ B(α ◦ ; δ) and ∀ x−i ∈ B(x−i where x−i ≡ (x1 , . . . , xi−1 , xi+1 , . . . , x N ), i = 1, . . . N , and where ri (x−i , α) ≡ arg max{ f i (x1 , . . . , x N ; α) s.t. g(x1 , . . . , x N ; α) = 0}, xi

i = 1, . . . , N . (A.4) ∃ a unique Nash equilibrium to (1), ∀ α ∈ B(α ◦ ; δ), denoted by

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x = x ∗ (α) ≡ (x1∗ (α), . . . , x N∗ (α)), where ∗ ∗ xi∗ (α) ≡ arg max{ f i (xi ↑ x−i (α); α) s.t. g(xi ↑ x−i (α); α) = 0}, xi

i = 1, . . . , N , ∗ ∗ ∗ (α) ≡ x1∗ (α), . . . , xi−1 (α), xi , xi+1 (α), . . . , x N∗ (α). and where xi ↑ x−i ∗ (2) ◦ (A.5) xi ∈ C , ∀ α ∈ B(α ; δ), i = 1, . . . , N .

Assumptions (A.1) and (A.2) are the prototype continuity assumptions made in comparative statics analysis when dealing with generic objective functions and constraint functions. The restriction m < n i , i = 1, . . . , N , guarantees that for the ith player there are more choice variables than there are constraints, ensuring that the constraints do not determine the Nash equilibrium of the game independent of the objective function. Because the focus of the paper is on a qualitative characterization of the Nash equilibrium and reaction functions of (1), this is the motivation for assumption (A.3), for without existence there is no qualitative analysis to perform. Assuming the second-order sufficient conditions hold for (1) for each individual player i, i = 1, . . . , N , it follows from (A.1), (A.2), and the Implicit Function Theorem that ri ∈ C (1) locally, i = 1, . . . , N . It is important to note, however, that if f i is a strictly concave function of xi , i = 1, . . . , N , then there exists a globally unique reaction function for each i = 1, . . . , N . Hence this sufficient condition implies (A.3), so (A.3) can be seen as a consequence of such a curvature assumption. The main thrust of the paper, however, is on deriving the fundamental comparative statics properties of the Nash equilibrium of the game (1) without imposing any additional structure on the game other than assuming existence and enough smoothness so as to be able to characterize the Nash equilibrium using the differential calculus. In this way, the resulting theorems are independent of any curvature or stability assumptions imposed on the game and thus truly represent the intrinsic comparative statics properties of the Nash equilibrium of the game. Hence, the aforementioned sufficient conditions are not maintained in the paper. Assumption (A.4) asserts the local existence of a unique Nash equilibrium, and its justification rests on the same motivation as that for assumption (A.3). Again, one may impose sufficient conditions on the game (1) so as to ensure existence of a Nash equilibrium as in, say, Theorem 1 of Nishimura and Friedman (1981). Such sufficient conditions, however, are not imposed on the game for the aforementioned reasons. In contrast to a single-player form of (1), (A.5) must be made in order to characterize the comparative statics matrix of the Nash equilibrium. This will become apparent in the proof of Theorem 2 in Eqs. (1.A) and (2.A) of the Appendix. Given (A.4), the definition of a Nash equilibrium implies that each optimal choice xi = xi∗ (α), i = 1, . . . , N , is an optimal solution to the following con-

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strained single-player static optimization problem ∀ α ∈ B(α ◦ ; δ): ª © ∗ ∗ (α); α) s.t. g(xi ↑ x−i (α); α) = 0 , φi (α) ≡ max f i (xi ↑ x−i xi

(3)

i = 1, . . . , N . The optimal value function for the ith player, φi , can also be defined by the constructive method, since by (A.4), a unique Nash equilibrium exists: φi (α) ≡ f i (x ∗ (α); α),

i = 1, . . . , N .

(4)

Definition (3), it turns out, is more useful for the purposes of envelope results and comparative statics from a primal–dual point of view. Two more points must be made before closing out this section. The first is that the Nash equilibrium x = x ∗ (α) of the game (1) can be defined as the fixed point of the map N N S j → × j=1 Sj , r : × j=1

(5)

where r = (r1 , . . . , r N )0 is the vector of the reaction functions of the game (1). Alternatively, and equivalently, the Nash equilibrium x = x ∗ (α) of the game (1) may be seen as the simultaneous solution of the N sets of first-order necessary conditions of the individual constrained optimization problems (1). Second, while either of these two views of a Nash equilibrium is equivalent to the definition of a Nash equilibrium as given by (A.4) (or equivalently, as given by the solution to the constrained optimization problem (3)), it is the view of the Nash equilibrium of the game (1) as given by (A.4) (or the solution to (3)) that allows a dual vista of the Nash equilibrium via the primal–dual methodology of Silberberg (1974) and thus allows a comprehensive qualitative characterization of the Nash equilibrium of the game. Hence, it is by viewing the Nash equilibrium as embodied in (A.4) that the dual analysis which follows is allowed.

FUNDAMENTAL QUALITATIVE PROPERTIES OF NASH EQUILIBRIA Consider the primal–dual version of (3) © ª ∗ ∗ (α); α) − φi (α) s.t. g(xi◦ ↑ x−i (α); α) = 0 , (6) max Di (α) ≡ f i (xi◦ ↑ x−i α

i = 1, . . . , N ,

where xi◦ = xi∗ (α ◦ ), i = 1, . . . , N , is the value of the Nash equilibrium function that solves (3) given α = α ◦ . Given this fact, Di (α) ≤ 0, ∀ α ∈ B(α ◦ ; δ), by construction of (6). Hence Di (α) attains its maximum value of zero at α = α ◦ by definition. The Lagrangian functions for the primal–dual problem (6) are defined

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by ∗ ∗ Li ≡ f i (xi◦ ↑ x−i (α); α) − φi (α) + λi0 g(xi◦ ↑ x−i (α); α),

i = 1, . . . , N . (7) Silberberg (1974) has pointed out that the vectors of Lagrange multipliers for the primal and primal–dual single-player problems are identical. This can be seen in the more general Nash equilibria setting under investigation here by (i) setting ∗ ∗ (α) = xi ↑ x−i (α ◦ ) in (6), (iii) noting that φi α = α ◦ in (6), (ii) letting xi◦ ↑ x−i is independent of xi , and (iv) letting xi be the choice vector in (6) rather than α. The following first-order necessary conditions hold at α = α ◦ and λi = λi◦ = ∗ λi (α ◦ ) by construction of (6), i = 1, . . . , N : ¸ N · ∗ X ∂ f i (xi◦ ↑ x−i (α); α) ∂ x j∗ (α) ∂ Li ≡ ∂α ∂ xj ∂α j=1 j6=i

∗ (α); α) ∂φi (α) ∂ f i (xi◦ ↑ x−i − (8a) ∂α ∂α ¸ N · ∗ ∗ X ∂g(xi◦ ↑ x−i (α); α) ∂ x j∗ (α) ∂g(xi◦ ↑ x−i (α); α) + λi0 =0 λi0 + ∂ xj ∂α ∂α j=1

+

j6=i

∂ Li ∗ = g(xi◦ ↑ x−i (α); α) = 0. ∂λi

(8b)

Given (8), a straightforward argument establishes the following theorem. THEOREM 1 (Envelope Theorem for Nash Equilibria). Under assumptions (A.1)–(A.5) on the game (1), the following envelope result holds: ¸ ∗ N · ∗ X ∂ x j (α) ∂ f i (x ∗ (α); α) ∂φi (α) ∗ 0 ∂g(x (α); α) ≡ (9a) + λi (α) ∂α ∂ xj ∂ xj ∂α j=1 j6=i

∂g(x ∗ (α); α) ∂ f i (x ∗ (α); α) + λi∗ (α)0 ∂α ∂α ¯ ¸ N · X ∂ x j∗ (α) ∂G i (x, λi ; α) ¯¯ ∗ (α) (9b) ≡ ¯ λx=x ∗ ∂ xj ∂α i =λi (α) j=1 j6=i ¯ ∂G i (x, λi ; α) ¯¯ ∗ (α) , i = 1, . . . , N , α ∈ B(α ◦ , δ). + ¯ λx=x ∗ ∂α i =λi (α) +

Proof. By (A.4) the first-order necessary conditions (8) hold ∀ α ∈ B(α ◦ ; δ) when all the functions are evaluated at the same point in B(α ◦ ; δ). Solve (8a) for ∂φi (α)/∂α using the above observation to get (9a), and then use the definition of the Game Lagrangians (2) and the definition of the partial derivative to derive (9b).

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Before moving to the second-order necessary conditions of the primal–dual problem (6), an interpretation of Theorem 1 and its departure from the Envelope Theorem of single-player models is necessary. Theorem 1 asserts that two separate effects result from a perturbation in a parameter vector on the ith player’s indirect objective function: (i) a direct or explicit effect (the second line of (9a) and (9b)), resulting from the explicit appearance of α in the ith player’s objective function f i and the constraint functions g j , j = 1, . . . , m, and (ii) a strategic effect (the first line of (9a) and (9b) involving the summation) resulting from the response of the other N − 1 players’ Nash equilibrium functions to the perturbation in α. The direct or explicit effect is identical to the prototype Envelope Theorem of single-decision-maker constrained static optimization models, and thus the difference between Theorem 1 and the archetype Envelope Theorem resides in the strategic effect, which itself depends in part on the number of players in the game. Hence, the strategic nature of the Nash equilibrium is crucial in determining the effect that parameter perturbations have on a given player’s indirect objective function. Notice that in using Theorem 1 in particular modeling applications, the correct envelope expression can be obtained by two equivalent methods. One way to compute the effect that a perturbation in a parameter vector has on the ith player’s indirect objective function is to simply apply the formula in (9a), remembering, of course, that the partial derivatives are taken before the objective function and constraint functions are evaluated at the Nash equilibrium of the game. According to Eq. (9b), one could follow the alternative recipe to arrive at the same result: (i) form the N − 1 vectors comprised of the partial derivatives of the ith player’s Game Lagrangian function (2) with respect to the other N − 1 players’ decision variables, (ii) compute the N − 1 comparative static matrices composed of the partial derivatives of the other N − 1 players’ Nash equilibrium choice functions with respect to the parameter of interest, (iii) compute the respective matrix product of the corresponding results in (i) and (ii) and sum up the N − 1 vectors, (iv) compute the partial derivative of the Game Lagrangian function (2) with respect to the explicit appearance of the parameter of interest, (v) evaluate the vectors in (iii) and (iv) at the Nash equilibrium x = x ∗ (α) and λi = λi∗ (α), and finally (vi) add the resulting vectors from (v) together. Continuing on with the primal–dual problem (6), the second-order necessary conditions are given by h0  ∀ h ∈ Rp 3 

∂ 2 Li h ≤ 0, ∂α 2

¸ N · ∗ X ∂g(xi◦ ↑ x−i (α); α) ∂ x j∗ (α) j=1 j6=i

∂ xj

∂α

 ∗ ∂g(xi◦ ↑ x−i (α); α)  h = 0, (10) + ∂α

i = 1, . . . , N ,

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which necessarily holds at α = α ◦ by construction of (6). In words this says that at α = α ◦ , the Hessian matrix of the Lagrangian function of the primal–dual problem (6) with respect to the parameters is negative semidefinite subject to constraint. It is important to note that this curvature restriction is independent of any curvature assumptions made about the functions f i , i = 1, . . . , N and g j , j = 1, . . . , m, as none were imposed on these functions to arrive at (10); the assertion of maximization alone produces the above curvature restruction. The following theorem expresses the curvature restriction in (10) in a more usable form, and its proof is contained in the Appendix. THEOREM 2 (Curvature Theorem for Nash Equilibria). Under assumptions (A.1)–(A.5) on the game (1), the following curvature property holds for Nash equilibria: (" # m ∂ 2 gk (x ∗ (α); α) ∂ xi∗ (α) ∂ 2 f i (x ∗ (α); α) X 0 ∗ + λik (α) (11) h ∂α∂ xi ∂α∂ xi ∂α k=1 # " N ∂ x ∗ (α)0 m 2 ∗ X ∂ ∂ xi∗ (α) ∂ 2 f i (x ∗ (α); α) X g (x (α); α) k j ∗ + + λik (α) ∂α ∂ x j ∂ xi ∂ x j ∂ xi ∂α j=1 k=1 j6=i    N ∂ x ∗ (α)0 ∂g(x ∗ (α); α)0  ∂λi∗ (α)  ∂g(x ∗ (α); α)0 X j + h ≥ 0, + ∂α ∂α ∂ xj ∂α  j=1 j6=i

  ¸ ∗ N · ∗ ∗ X ∂ x (α) ∂g(x (α); α) (α); α) ∂g(x j  h = 0, + ∀ h ∈ Rp 3  ∂ xj ∂α ∂α j=1 j6=i

i = 1, . . . , N ,

α ∈ B(α ◦ ; δ).

Rather than try to interpret Theorem 2 and contrast it with its single player counterpart, Eq. (10) of Silberberg (1974), it will be much easier to interpret and contrast the unconstrained version of Theorem 2 and Eq. (10) of Silberberg (1974). Setting g ≡ 0 in Theorem 2 yields the proof of the following unconstrained comparative statics result for Nash equilibria. COROLLARY 2 (Unconstrained Curvature Theorem). Under assumptions (A.1)–(A.5) on the game (1), if g ≡ 0, then the following curvature property holds for Nash equilibria:   N ∂ x ∗ (α)0 2  ∂ 2 f (x ∗ (α); α) ∂ x ∗ (α) X ∗ ∗ ∂ f (x (α); α) ∂ x (α) i i j i i + h ≥ 0, (12) h0  ∂α∂ xi ∂α ∂α ∂ x j ∂ xi ∂α  j=1 j6=i

∀ h ∈ R p , i = 1, . . . , N , α ∈ B(α ◦ ; δ).

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Corollary 2 has wide applicability because most oligopoly models used in the literature, including the general models of Dixit (1986) and Quirmbach (1988), either do not have any constraints or have constraints that are simple enough to substitute out of the problem, rendering it unconstrained. Moreover, many of the oligopoly models have only one decision variable per economic unit, and thus Corollary 2 reduces the comparative statics exercise down to the computation of cross partial derivatives of the economic units’ objective functions. A detailed example on the use of Corollary 2 will be presented in a following section. By setting g ≡ 0 in Eq. (10) of Silberberg (1974), it is seen that the first matrix in (12) of Corollary 2 constitutes the fundamental comparative statics matrix of single-decision-maker optimization problems. Thus the N −1 matrices in the second term in (12) constitute the difference between the fundamental comparative statics matrices of N -player games employing the Nash equilibrium solution and single-decision-maker optimization models. It is exactly these N −1 strategic matrices in (12) that create the difficulty in getting refutable comparative statics results for the level of the decision variables in games, for one must know (i) the effect that a parameter perturbation has on the Nash equilibrium value of the other N − 1 players’ decision variables and (ii) the effect that the N − 1 other players’ decision variables has on the marginal value of the ith player’s decision variables, before the ith player’s comparative statics response can be deduced. Because so much information is required to determine the comparative statics properties of the level of the decision variables of Nash equilibria relative to single-agent models, the extensive use of simplifying mathematical assumptions and special cases is to be expected in applied game theoretic modeling using the Nash equilibrium solution. An important aspect of Theorem 2 and Corollary 2 is that the comparative statics properties of Nash equilibria of N -player static games are contained in a symmetric positive semidefinite matrix subject to constraint. The conclusion is independent of any curvature properties imposed on the game (since none were imposed on the game to derive the results) and is a necessary consequence of the optimization assertion alone. Moreover, the result is not dependent upon any ad hoc stability conditions which are logically inconsistent with the static nature of the game. This puts the qualitative properties of static games as exemplified by Theorem 2 and Corollary 2 on par with those of single-decision-maker models as exemplified by Silberberg (1974). In addition, Theorems 1 and 2 and Corollary 2 show that there is a fundamental qualitative structure inherent in all static games that met assumptions (A.1)–(A.5), and thus in principle, the implications of the game are subject to refutation. Even the rather general models of Dixit (1986) and Quirmbach (1988) fall short in this regard, since by working exclusively with primal or classical methodology they were led down a path that inhibited the recovery of the fundamental symmetric positive semidefinite comparative statics matrix that underlies the Nash equilibrium of every sufficiently smooth N -player static game.

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FUNDAMENTAL QUALITATIVE PROPERTIES OF REACTION FUNCTIONS Recall that from (A.3) the ith player’s reaction function or best-reply function xi = ri (x−i , α), i = 1, . . . , N , is defined as the solution to the ith player’s optimization problem alone, treating the decision variables of the other players x−i as fixed or parametric, just like the parameter vector α. Given the definition of the reaction function, it immediately follows that the qualitative comparative statics properties of the reaction functions are completely determined by Eq. (10) of Silberberg (1974). Thus the fundamental comparative statics matrix with respect to α is qualitatively identical for reaction functions of N -player static games and single-player constrained optimization problems. The above observation results in the proof of the following theorem. THEOREM 3 (Reaction Function/Single-Player Equivalence Theorem). Under assumptions (A.1)–(A.5) on the game (1), the following curvature result holds for the reaction functions: (" # m ∂ 2 gk (ri (x−i , α), x−i ; α) ∂ 2 f i (ri (x−i , α), x−i ; α) X 0 λˆ ik (x−i , α) + h ∂α∂ xi ∂α∂ xi k=1 ) ∂ri (x−i , α) ∂g(ri (x−i , α), x−i ; α)0 ∂ λˆ i (x−i , α) × + h ≥ 0, (13) ∂α ∂α ∂α ,α),x−i ;α) , h = 0, ∀ h ∈ R p 3 ∂g(ri (x−i∂α ∗ ◦ i = 1, . . . , N , α ∈ B(α ; δ), x−i ∈ B(x−i (α ◦ ); ε);

here λi = λˆ i (x−i , α) is the optimal solution for the ith player’s Lagrange multiplier vector corresponding to the reaction function solution. Moreover, if in addition g ≡ 0, then the curvature property reduces to ¸ · 2 0 ∂ f i (ri (x −i , α), x −i ; α) ∂ri (x −i , α) h ≥ 0, (14) h ∂α∂ xi ∂α ∗ ∀ h ∈ R p , i = 1, . . . , N , α ∈ B(α ◦ ; δ), x−i ∈ B(x−i (α ◦ ); ε). The mathematical link between the Nash equilibrium solution xi = xi∗ (α), i = 1, . . . , N , and the reaction function solution xi = ri (x−i , α), i = 1, . . . , N , will now be elucidated. If in solving for the ith player’s reaction function, the ith player takes the choice variables of the other N − 1 players as fixed at their Nash equilibrium value, then the value of the ith player’s reaction function is identically equal to the value of the ith player’s Nash equilibrium function. More formally, using the definitions of the Nash equilibrium functions and reaction functions in (A.4) and (A.3), respectively, one has ∗ (α), α), xi∗ (α) ≡ ri (x−i

α ∈ B(α ◦ ; δ), i = 1, . . . , N .

(15)

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Differentiating the fundamental identity (15) with respect to α yields the relationship between the comparative statics matrix of the Nash equilibrium solution and that of the reaction function solution for the ith player N X ∂ri ∂ri ∂ x j∗ ∂ xi∗ ≡ + , ∂α ∂ x j ∂α ∂α j=1

α ∈ B(α ◦ ; δ), i = 1, . . . , N .

(16)

j6=i

Equation (16) asserts that the effect of a change in the parameter vector α on the ith player’s Nash equilibrium function xi∗ , represented by the n i × p matrix ∂ xi∗ /∂α, can be broken up into two matrix effects: (i) an n i × p nonstrategic or reaction function effect matrix ∂ri /∂α, which measures the direct effect of a change in the parameter vector α on the ith player’s reaction function, holding the decision variables of the other N − 1 players Pfixed at their Nash equilibrium value, and (ii) an n i × p strategic effect matrix j6N=i=1 (∂ri /∂ x j )(∂ x j∗ /∂α), which measures the indirect effect that a change in the parameter vector α has on the ith player’s reaction function through the other N − 1 players’ Nash equilibrium response to the parameter perturbation. The fundamental identity (15) and derivative decomposition (16) will be exploited in the next section, where a specific economic game is investigated for its qualitative properties. It will also be shown that it is the strategic effect that is responsible for the difficulty in signing the comparative statics of the level of the decision variables of Nash equilibria via primal methodology.

THE COURNOT HETEROGENEOUS PRODUCT DUOPOLY REVISITED Consider the case of two firms, i = 1, 2, producing a single related (i.e., substitute) but not identical (or homogeneous) product yi , i = 1, 2. Let pi = Di (y1 , y2 ) be the inverse demand function facing firm i, where pi is the price of output for firm i, i = 1, 2. Denote the total production cost function for firm i as Ci (yi ) i = 1, 2. It is assumed that the output of each firm is subject to its own specific unit tax ti > 0, i = 1, 2. Defining Ri (y1 , y2 ) ≡ Di (y1 , y2 )yi − Ci (yi ), i = 1, 2, as ith firm’s net of production cost revenue function, the heterogeneous product duopoly model can be stated as φi (t1 , t2 ) ≡ max {πi (y1 , y2 ; ti ) ≡ Ri (y1 , y2 ) − ti yi } , yi

i = 1, 2.

(17)

Finally, let yi = yi∗ (t1 , t2 ), i = 1, 2, be the Nash equilibrium output “supply” functions, and let yi = ri (yj , ti ), i 6= j = 1, 2, be the reaction function equilibrium output “supply” functions for the game (17). First consider the qualitative comparative statics properties of the reaction functions. Recall that the reaction function for firm i is defined as the solution to

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the ith firm’s first-order necessary condition, treating the other firm’s decision variable as given or fixed. That is, yi = ri (yj , ti ), i 6= j = 1, 2, is the solution to ∂ Ri (y1 , y2 ) ∂πi ≡ − ti = 0, ∂ yi ∂ yi

i = 1, 2,

(18)

assuming, of course, that the second-order sufficient condition ∂ 2 πi /∂ yi2 ≡ ∂ 2 Ri /∂ yi2 < 0, i = 1, 2, holds for each firm, so that the use of the Implicit Function Theorem on (18) is justified. The first thing to notice from (18) is that the reaction function for firm i does not depend directly or explicitly on the unit tax of firm j. In other words, a change in the unit tax of firm j does not affect the reaction function of firm i. A straightforward application of the Implicit Function Theorem to (18) yields the comparative statics of the reaction functions: −∂ 2 Ri /∂ yi ∂ yj ∂ri = ≷ 0, i 6= j = 1, 2, ∂ yj ∂ 2 Ri /∂ yi2 ( 0 if i 6= j = 1, 2 ∂ri = 1 < 0 if i = j = 1, 2; ∂t j ∂ 2 Ri /∂ yi2

(19a) (19b)

here the derivatives in (19) for firm i are evaluated at yi = ri (yj , ti ), i 6= j = 1, 2. By (19a), the reaction function for each firm can be upward or downward sloping in general, for ∂ 2 Ri /∂ yi ∂ yj ≷ 0, i 6= j = 1, 2, in general. The reaction functions slope down if and only if ∂ 2 Ri /∂ yi ∂ yj < 0, i 6= j = 1, 2, that is, if and only if the marginal revenue of firm i falls as the output of firm j increases. Equivalently, using the terminology introduced by Bulow et al. (1985), the reaction functions slope down if and only if the output of each firm is a strategic substitute for the output of the other firm. The assumption that the two outputs are substitutes, however, is neither necessary nor sufficient for this to hold, since ∂ Di ∂ 2 Di ∂ 2 Ri ≡ + yi ≷ 0, ∂ yi ∂ yj ∂ yi ∂ yi ∂ yj

i 6= j = 1, 2,

(20)

holds in general, even when the substitute product assumption is imposed, i.e., when ∂ Di /∂ yj < 0, i 6= j = 1, 2, holds. Thus, in general, even in this simple model, empirical information and/or further assumptions on the mathematical structure of the model are needed to get the so-called “normal” result of downward sloping reaction functions, a point made earlier by Bulow et al. (1985, p. 490). The assertions of maximization and Nash behavior, and the assumption of substitute products, are not strong enough to generate the popular result. An obvious sufficient condition for ∂ 2 Ri /∂ yi ∂ yj < 0, i 6= j = 1, 2, to hold is that the products are substitutes and ∂ 2 Di /∂ yi ∂ yj ≤ 0, i 6= j = 1, 2.1 1 An example where ∂ D /∂ y < 0 and ∂ 2 R /∂ y ∂ y > 0 holds is p = D (y , y ) ≡ y −2 y −1 . 1 2 1 1 2 1 1 1 2 1 2 Simply compute ∂ D1 /∂ y2 ≡ −y1−2 y2−2 < 0 and ∂ 2 R1 /∂ y1 ∂ y2 ≡ y1−2 y2−2 > 0.

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Equation (19b) asserts that a rise in the unit tax facing firm i will lower its reaction function level of output, but that a change in the unit tax facing the other firm will have no direct (or explicit) effect on its reaction function. The comparative statics ∂ri /∂ti < 0, i = 1, 2, are exactly the same results, qualitatively, for a monopolistic or competitive firm facing a unit tax on its output (see, e.g., Silberberg 1990, Chapter 1). Alternatively, the comparative statics results in (19) can be derived using Theorem 3. For example, since ∂ 2 πi /∂ yi ∂ yj ≡ ∂ 2 Ri /∂ yi ∂ yj and ∂ 2 πi /∂ yi ∂ti ≡ −1, by using Eq. (14) of Theorem 3 it follows that sign

∂ 2 πi ∂ri = sign , i 6= j = 1, 2, ∂ yj ∂ yi ∂ yj ∂ri ≤ 0, i = 1, 2, ∂ti

(21a) (21b)

which is identical to the information contained in (19). Equation (21a) says that the slope of the ith firm’s reaction function is determined by the effect that the jth firm’s output has on the ith firm’s marginal revenue function or, in other words, whether the output of firm j is a strategic substitute or strategic complement for the output of firm i. Now turn attention to the Nash equilibrium solutions yi = yi∗ (t1 , t2 ), i = 1, 2, of the two firms. Recall that the Nash equilibrium solutions are defined as the simultaneous solution of both firms’ first-order necessary conditions: ∂ R1 (y1 , y2 ) ∂π1 ≡ − t1 = 0 ∂ y1 ∂ y1

(22a)

∂ R2 (y1 , y2 ) ∂π2 ≡ − t2 = 0. ∂ y2 ∂ y2

(22b)

Assuming that the Jacobian determinant of (22) is nonzero at the Nash equilibrium solution, that is, ¯ ¯ ¯ ∂ 2 π1 ∂ 2 π1 ¯¯ ¯ ¯ ∂ y2 ∂ y1 ∂ y2 ¯¯ ¯ 1 (23) (y1 = yi∗ (t1 , t2 ), i = 1, 2), |J | = ¯ ¯ 6= 0, 2 ¯ ¯ ∂ 2 π2 ∂ π 2 ¯ ¯ ¯ ∂y ∂y ∂ y2 ¯ 2

1

2

then the Implicit Function Theorem gives yi = yi∗ (t1 , t2 ), i = 1, 2, as the welldefined simultaneous solution to (22). Note that the main diagonal entries of |J | are negative by the second-order sufficient condition for each firm, that is, ∂ 2 πi /∂ yi2 < 0, i = 1, 2, but the off diagonal entries are in general indeterminate in sign, just like single-agent optimization problems. Unlike single-agent problems, however, the sign of |J | is not given by a second-order sufficient condition, since

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the Jacobian matrix J is not identical to the Hessian matrix of a single underlying optimization problem. In other words, the Jacobian matrix of (22) results from the decisions of two strategically linked agents and thus cannot be related to the Hessian matrix of a single underlying optimization problem; hence the sign of |J | is not linked to a second-order sufficient condition of a single underlying optimization problem. In contrast, this would be the case under the necessary and sufficient condition of Proposition 1 of Slade (1994). By the Implicit Function Theorem, since |J | ≷ 0 in general, all the Nash equilibrium comparative statics of the Cournot duopoly are ambiguous, so no refutable implications emerge (or so it would seem) from even this simple model. An application of the Implicit Function Theorem to (22) yields the comparative statics of the Nash equilibrium solution and demonstrates the above assertion: ∂ 2 π2 /∂ y22 ∂ y1∗ ≡ ≷0 ∂t1 |J |

(24a)

−∂ 2 π2 /∂ y2 ∂ y1 ∂ y2∗ ≡ ≷0 ∂t1 |J |

(24b)

−∂ 2 π1 /∂ y1 ∂ y2 ∂ y1∗ ≡ ≷0 ∂t2 |J |

(24c)

∂ 2 π1 /∂ y12 ∂ y2∗ ≡ ≷ 0. ∂t2 |J |

(24d)

Thus, unlike single-agent models, the denominator of all the Nash equilibrium comparative statics in (24) is not signable, and thus all the comparative statics are ambiguous. Even if both reaction functions slope downward so that ∂ 2 πi /∂ yi ∂ yj < 0, i 6= j = 1, 2, it is still not possible to sign the comparative statics expressions in (24). This observation is the driving force behind numerous authors’ attempts at signing |J |. Dixit (1986) provides a clear and general exposition of the signing of |J |. The approach taken asserts an ad hoc dynamic adjustment process to the Nash equilibrium. Using Taylor’s Theorem to provide a linear approximation to the nonlinear dynamical system, the Jacobian matrix J rears its head again. Then using necessary and sufficient conditions for local asymptotic stability of the Nash equilibrium, the conclusion |J | > 0 is reached, and many previously ambiguous comparative statics results are then signable. Unfortunately, as most authors (including Dixit, 1986) recognize, this approach to signing |J | is wholly unsatisfactory because it lays a dynamic adjustment process on top of an inherently static model. In other words, the above approach to signing |J | is in the exact nature of Samuelson’s (1947) Correspondence Principle. As Brock (1986) and Brock and Malliaris (1989, Chapter 7) have documented, however, Samuelson’s Correspondence Principle has been the sub-

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ject of numerous criticisms over the past two decades, and thus this approach to signing |J | represents a regression rather than an advancement. In other words, as Brock (1986) and Brock and Malliaris (1989) point out, the method of imposing a dynamic adjustment process on a fundamentally static economic model is ad hoc and without logical foundation. Only by modeling the economic process as dynamic from first principles (i.e., as a differential game) does the stability analysis take on any real economic meaning and have a logical foundation, and as such, the dynamic adjustment process used over the past decades is without a logical foundation. Corollary 2 shows that no such ad hoc dynamics need be imposed on the problem for refutable comparative statics results to emerge. Thus the comparative statics that result from an application of Corollary 2 to any Nash equilibria are logically consistent and not ad hoc. For example, an application of Corollary 2 to the Cournot duopoly model implies that, for i = 1,   −1 · ∂ y ∗ ∂ y ∗ ¸ · ∂ y ∗ ∂ y ∗ ¸0 · ∂ 2 π ¸ · ∂ y ∗ ∂ y ∗ ¸ 1 1 1 2 2 1 1   + (25) ∂t ∂t ∂t ∂t ∂ y ∂ y ∂t ∂t 1 2 1 2 2 1 1 2 0  " #  ∗ 2 ∗ ∗ ∗ 2 −∂ y1∗ ∂t1

=

0

 = 

−∂ y1∗ ∂t2

0

+

∂ y2∗ ∂ 2 π1 ∂ y1∗ ∂ y∗ − ∂t11 ∂t1 ∂ y2 ∂ y1 ∂t1 ∂ y2∗ ∂ 2 π1 ∂ y1∗ ∂t2 ∂ y2 ∂ y1 ∂t1

∂ y2 ∂t1 ∂ y2∗ ∂t2

∂ π1 ∂ y2 ∂ y1 ∂ 2 π1 ∂ y2 ∂ y1

∂ y1 ∂t1 ∂ y1∗ ∂t1

∂ y2 ∂t1 ∂ y2∗ ∂t2

∂ π1 ∂ y2 ∂ y1 ∂ 2 π1 ∂ y2 ∂ y1

∂ y2∗ ∂ 2 π1 ∂ y1∗ ∂ y∗ − ∂t21 ∂t1 ∂ y2 ∂ y1 ∂t2 ∂ y2∗ ∂ 2 π1 ∂ y1∗ ∂t2 ∂ y2 ∂ y1 ∂t2



∂ y1 ∂t2 ∂ y1∗ ∂t2





is a locally symmetric positive semidefinite matrix. For i = 2, Corollary 2 implies that   0 · ∂ y ∗ ∂ y ∗ ¸ · ∂ y ∗ ∂ y ∗ ¸0 · ∂ 2 π ¸ · ∂ y ∗ ∂ y ∗ ¸ 2 2 2 1 1 2 2   + (26) ∂t ∂t ∂t ∂t ∂ y ∂ y ∂t ∂t 1 2 1 2 1 2 1 2 −1 " #  ∂ y∗ ∂ 2 π ∂ y∗ ∂ y∗ ∂ 2 π ∂ y∗  2 2 1 2 1 2 0 0 ∂t ∂ y ∂ y ∂t ∂t ∂ y ∂ y ∂t = +  ∂ y1∗ 21 2 ∂ y1∗ ∂ y1∗ 21 2 ∂ y2∗  −∂ y2∗ −∂ y2∗  = 

∂t1

∂t2

∂ y1∗ ∂ 2 π2 ∂ y2∗ ∂t1 ∂ y1 ∂ y2 ∂t1 ∂ y1∗ ∂ 2 π2 ∂ y2∗ ∂ y∗ − ∂t12 ∂t2 ∂ y1 ∂ y2 ∂t1

1 ∂ π2 2 ∂t2 ∂ y1 ∂ y2 ∂t1

1 ∂ π2 2 ∂t2 ∂ y1 ∂ y2 ∂t2

∂ y1∗ ∂ 2 π2 ∂ y2∗ ∂t1 ∂ y1 ∂ y2 ∂t2 ∂ y1∗ ∂ 2 π2 ∂ y2∗ ∂ y∗ − ∂t22 ∂t2 ∂ y1 ∂ y2 ∂t2

 

is a locally symmetric positive semidefinite matrix too. The symmetry and curvature results in (25) and (26) are fundamental to the Cournot duopoly model and heretofore have not been discovered by other researchers. No ad hoc stability conditions need be invoked, nor do assumptions about the slopes of the reaction

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functions have to be made to reach the above conclusions, and as such, they represent the intrinsic comparative statics properties of the Cournot duopoly model. It is not too surprising that these results have been left undiscovered, for they require the computation of unconventional comparative statics expression from a classical or primal comparative statics viewpoint. What is so elegant about the primal–dual methodology of Silberberg (1974) is that it always finds the fundamental comparative statics matrix of any optimization problem. Since the matrices (25) and (26) are locally symmetric positive semidefinite, it follows that ¸ ∗ · ∗ 2 ∂ y1 ∂ y2 ∂ π1 −1 ≥0 (27a) ∂t1 ∂ y2 ∂ y1 ∂t1 ∂ y2∗ ∂ 2 π1 ∂ y1∗ ≥0 ∂t2 ∂ y2 ∂ y1 ∂t2 ∂ y1∗ ∂ 2 π2 ∂ y2∗ ≥0 ∂t1 ∂ y1 ∂ y2 ∂t1 ¸ ∗ · ∗ 2 ∂ y2 ∂ y1 ∂ π2 −1 ≥ 0, ∂t2 ∂ y1 ∂ y2 ∂t2

(27b)

(27c)

(27d)

all hold, along with the reciprocity conditions due to the symmetry of the matrices. Equations (27b) and (27c) assert that either all three terms are nonnegative, or one is nonnegative and two are nonpositive. For example, from (27b), if ∂ 2 π1 /∂ y2 ∂ y1 < 0, that is, if the output of firm two is a strategic substitute for the output of firm one, then an increase in the unit tax on firm two must affect the output levels of firms one and two in a qualitatively opposite manner. In contrast, if ∂ 2 π1 /∂ y2 ∂ y1 > 0, so that the output of firm two is a strategic complement for the output of firm one, then either (i) both firms do not produce less when the unit tax on firm two increases or (ii) both firms do not produce more when the unit tax on firm two increases. More generally, Eqs. (27) show that not only does the slope of the reaction function (and thus whether the firms’ outputs are strategic substitutes or complements) play an important role in the comparative statics of the Cournot equilibrium (recall 21a), but so does the magnitude of the slope, thus reinforcing the earlier point that a great deal more about the mathematical and economic structure of the game must be known relative to single-agent problems in order to derive refutable comparative statics results for the level of each players’ decision variables. Now consider Eq. (15), the identity linking the value of the Nash equilibrium choice functions to the value of the reaction functions. For say, firm 1 of the Cournot duopoly, Eq. (15) becomes y1∗ (t1 , t2 ) ≡ r1 (y2∗ (t1 , t2 ), t1 ).

(28)

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Differentiation of (28) with respect to t1 yields ∂r1 ∂ y2∗ ∂r1 ∂ y1∗ ≡ + ≷ 0. ∂t1 ∂ y2 ∂t1 ∂t1

(29)

The derivative decomposition in (29) highlights the important role played by the slope of the reaction function (or, equivalently, by (21a), whether the output of firm two is a strategic substitute or complement for the output of firm one) in determining the comparative statics of the Cournot duopoly. Equation (29) shows that the effect of a change in the unit tax on firm one has two effects on the output level of firm one: (i) a nonstrategic effect ∂r1 /∂t1 , measuring the effect the change in t1 has on the reaction function of firm one, holding the output of firm two fixed at its Nash equilibrium level, and (ii) a strategic effect (∂r1 /∂ y2 )(∂ y2∗ /∂t1 ), measuring the effect the change in t1 has on the reaction function of firm one via the response of firm two to the change in t1 . By (21b), ∂r1 /∂t1 < 0, so the ambiguity in signing ∂ y1∗ /∂t1 is solely the result of the strategic effect (∂r1 /∂ y2 )(∂ y2∗ /∂t1 ). Thus it should now be clear why refutable comparative statics results on the level of the decision variables are difficult to come by in strategic models: one must know how other players will react to parameter perturbations in order to determine one’s own comparative statics response. The use of simplifying or special cases is therefore a natural response by the users of game theory in their attempt to draw tight conclusions about the comparatives statics properties of the level of the decision variables using primal comparative statics methodology from a model which has fundamental but unconventional qualitative comparative statics properties. Differentiating (28) with respect to t2 yields ∂r1 ∂ y2∗ ∂ y1∗ ≡ ≷ 0. ∂t2 ∂ y2 ∂t2

(30)

Thus for the unit tax on firm two, which only enters the profit flow for firm two explicitly, only the strategic effect matters in determining the comparative statics of the Cournot equilibrium for firm one. Thus firm one changes its output level when t2 changes, but only in response to firm two changing its output level as t2 changes. Finally, consider the envelope property of Theorem 1. For firm one, say, ∂ R1 ∂ y2∗ ∂φ1 (t1 , t2 ) ≡ − y1∗ ≷ 0 ∂t1 ∂ y2 ∂t1

(31a)

∂ R1 ∂ y2∗ ∂φ1 (t1 , t2 ) ≡ ≷ 0, ∂t2 ∂ y2 ∂t2

(31b)

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so that both derivatives are ambiguous in sign in general. Equations (31) show that if a parameter enters a given firm’s objective function explicitly, then the effect of a change in that parameter produces two effects, in this case on the firm’s indirect profit function. Thus an increase in t1 has two effects on the profits of firm one, (i) a direct effect −y1∗ (t1 , t2 ), which results in a reduction in the profit of firm one and (ii) a strategic effect (∂ R1 /∂ y2 )(∂ y2∗ /∂t2 ), resulting from the effect a change in t1 has on the Cournot output level of firm two, and the resulting effect the change in the output level of firm two has on the net revenue of firm one, an ambiguous effect in general. For changes in t2 , (31b) shows that the ambiguous strategic effect is the only operative change in the indirect profit function of firm one. Since the outputs of the two firms are substitutes, ∂ R1 /∂ y2 ≡ y1 ∂ D1 /∂ y2 < 0 holds. Thus if ∂ y2∗ /∂t1 ≥ 0, that is, if firm two does not lower its output when the unit tax on firm one rises, then (31a) shows that the profit of firm one will fall when its unit tax rises. Recall, however, that ∂ y2∗ /∂t1 ≷ 0 in general, so that if ∂ y2∗ /∂t1 < 0 holds and is “large enough,” then the profit of firm one will rise when its unit tax rises. In this instance the reduction in profit from the direct effect of an increase in t1 is more than offset by the rise in profit due to the strategic effect, which is represented in part by the reduced level of output of firm two. Because the outputs of the two firms are substitutes (i.e., ∂ R1 /∂ y2 < 0), (31b) shows that the effect of an increase in the unit tax on firm two on the profit of firm one is the opposite of the effect the rise in the unit tax on firm two has on the output of firm two. In other words, if firm two responds to the increase in the unit tax it faces by decreasing (increasing) its output, then the profit of firm one will increase (decrease) due to the rise in the unit tax facing firm two.

CONCLUSION For Nash equilibria of N -player static games, Theorem 1 shows that the envelope property of such games differs quite substantially from the typical singledecision-maker envelope result. The discrepancy between the models results from the strategic nature of the Nash equilibrium. Theorem 1 shows that not only does the envelope result pick up the direct effect of a parameter perturbation, but it also picks up the strategic effect the parameter perturbation has on the other N − 1 players’ Nash equilibrium choice functions. Theorem 2 shows that the fundamental comparative statics matrix for the ith player in a Nash equilibrium is symmetric positive semidefinite subject to constraint and includes the equilibrium response of the other N − 1 players to the parameter perturbation as well as the effect the other N −1 players’ decision variables have on the marginal value of the ith player’s decision variables. Because no curvature assumptions were exogenously imposed on the game, nor were any ad hoc stability properties

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assumed on the inherently static game, the qualitative results of Theorem 2 are a fundamental part of all sufficiently smooth games using the Nash equilibrium solution concept. Theorem 2 and Corollary 2 also show that if in a game one is to obtain comparative statics results on the level of the decision variables that are comparable to those in single-agent models, the information required in such games is quite extensive. Thus the use of special cases is to be expected, at least for the generation of refutable comparative statics results for the level of the decision variables in game theoretic models when primal comparative statics methodology is employed.

APPENDIX Proof of Theorem 2. First differentiate (8a) with respect to α recalling (A.5) "n · ¸ 2 ∗ j N ∗ ◦ ∗ X X ∂ x js (α) ∂ f i (xi◦ ↑ x−i (α); α) ∂ 2 Li 0 ∂g(x i ↑ x −i (α); α) ≡ + λi 2 ∂α ∂ x js ∂ x js ∂α 2 j=1 s=1 j6=i

+ ×

∂ x j∗ (α)0 ∂α "

N X q=1 q6=i

× +

m ∗ ∗ ∂ 2 f i (xi◦ ↑ x−i (α); α) X ∂ 2 gk (xi◦ ↑ x−i (α); α) + λik ∂ x j ∂ xq ∂ x j ∂ xq k=1

#

∂ xq∗ (α) ∂α ∂ x j∗ (α)0 ∂α

"

m ∗ ∗ ∂ 2 f i (xi◦ ↑ x−i (α); α) X ∂ 2 gk (xi◦ ↑ x−i (α); α) λik + ∂ x j ∂α ∂ x j ∂α k=1

##

m ∗ ∗ ∂ 2 f i (xi◦ ↑ x−i (α); α) X ∂ 2 gk (xi◦ ↑ x−i (α); α) ∂ 2 φi (α) + λ − ik ∂α 2 ∂α 2 ∂α 2 k=1 " # N m ∗ ∗ X ∂ 2 f i (xi◦ ↑ x−i (α); α) X ∂ 2 gk (xi◦ ↑ x−i (α); α) + + λik ∂α∂ x j ∂α∂ x j j=1 k=1

+

j6=i

×

∂ x j∗ (α) ∂α

,

i = 1, . . . , N ,

(1.A)

where λik , i = 1, . . . , N , k = 1, . . . , m, is the kth component of the vector λi and x js , j = 1, . . . , N , s = 1, . . . , n j , is the sth component of the vector x j .

COMPARATIVE STATICS OF NASH EQUILIBRIA

223

Next, differentiate the envelope result (9a) of Theorem 1 with respect to α: "n · ¸ 2 ∗ j N ∗ X X ∂ x js (α) ∂ 2 φi (α) ∂ f i (x ∗ (α); α) ∗ 0 ∂g(x (α); α) + λi (α) ≡ ∂α ∂ x js ∂ x js ∂α 2 j=1 s=1 j6=i " # N m ∂ x j∗ (α)0 X ∂ 2 gk (x ∗ (α); α) ∂ 2 f i (x ∗ (α); α) X ∗ + + λik (α) ∂α q=1 ∂ x j ∂ xq ∂ x j ∂ xq k=1 ×

∂ xq∗ (α)

∂α " # m ∂ x j∗ (α)0 ∂ 2 f i (x ∗ (α); α) X ∂ 2 gk (x ∗ (α); α) ∗ + + λik (α) ∂α ∂ x j ∂α ∂ x j ∂α k=1 ¸ ∂ x j∗ (α)0 ∂g(x ∗ (α); α)0 ∂λi∗ (α) + ∂α ∂ xj ∂α m ∂ 2 f i (x ∗ ; α) X ∗ ∂ 2 gk (x ∗ ; α) + + λik (α) 2 ∂α ∂α 2 k=1 " # N m 2 ∗ X ∂ x j∗ (α) ∂ ∂ 2 f i (x ∗ (α); α) X g (x (α); α) k ∗ + + λik (α) ∂α∂ x j ∂α∂ x j ∂α j=1 k=1

+

∂g(x ∗ (α); α)0 ∂λi∗ (α) , ∂α ∂α

i = 1, . . . , N .

(2.A)

Evaluate (1.A) and (2.A) at α = α ◦ and λi = λi◦ = λi∗ (α ◦ ), i = 1, . . . , N , substitute (2.A) into (1.A), cancel common terms, note that the summation index in the second line of (2.A) runs from 1 to N but that the summation index in the third line of (1.A) skips the ith term, and also note that the summation index in the seventh line of (2.A) runs from 1 to N but that the summation index in the seventh line of (1.A) skips the ith term, thus reducing (1.A) to " # N ∂ x ∗ (α ◦ )0 m 2 ◦ ◦ X ∂ 2 f i (x ◦ ; α ◦ ) X g (x ; α ) ∂ ∂ 2 Li k j ∗ ≡ − + λik (α ◦ ) ∂α 2 ∂α ∂ x j ∂ xi ∂ x j ∂ xi j=1 k=1 j6=i

∂ x ∗ (α ◦ ) × i " ∂α # m ∗ 2 ◦ ◦ ◦ ∂ 2 f i (x ◦ ; α ◦ ) X ∗ ◦ ∂ gk (x ; α ) ∂ x i (α ) − + λik (α ) ∂α∂ xi ∂α∂ xi ∂α k=1   N ∂ x ∗ (α ◦ )0 ∂g(x ◦ ; α ◦ )0 X ∂g(x ◦ ; α ◦ )0  ∂λi∗ (α ◦ ) j − + , ∂α ∂α ∂ xj ∂α j=1 j6=i

i = 1, . . . , N .

(3.A)

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Recalling (A.4), it follows that (10) and (3.A) hold ∀ α ∈ B(α ◦ ; δ). Using this observation, substituting (3.A) into (10), and multiplying through by minus unity completes the proof.

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