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Comparative statics for integrable Nash equilibria
Economics Letters North-Holland
23 (1987) 19-21
COMPARATIVE
STATICS
19
FOR INTEGRABLE
NASH EQUILIBRIA
*
Michael J. STUTZER Federal Reserve Bank of Minneapolis, Minneapolis, MN 55480, USA Received
18 September
1986
This paper derives comparative statics results for stable Nash equilibria in integrable, concave orthogonal games. Application of these results to cost curve shifts in the asymmetric Cournot oligopoly immediately uncovers some apparently new comparative statics results.
1. Two comparative statics results I generalize the single agent results of Pauwels (1979) to derive results for a subclass of the class of concave orthogonal games defined by Rosen (1965) whose Nash equilibria are defined by maxf’(x’; XI
x)‘(, (Y),
i=l
,.‘.>
N,
subject to
g’(x’;
a) = 0,
(1)
where xl E R”, a E R”, g’isavectorofr,constraintfunctions,andx)’(=(x’,...,x’~‘,~’~’,...,x~). Throughout, a subscripted function denotes its gradient with respect to the subscripted vector, i.e., f, = (ag/ax,), and a subscripted vector of functions denotes its Jacobian matrix with respect to the subscript, i.e., g, = (ag’/axj). Hessian matrices are denoted by double subscripts. An apostrophe denotes the transpose operator. A Nash equilibrium for (Y is a vector R = (a’, . . . , ZN) solving (1) for i = 1,. . . , N. Subscripting to denote partial derivatives, we assume the following for i = 1,. . . , N: A. 1.
The maps f’
and g’ are twice differentiable.
There exist _? and fi = (A’, . . . , fi”) A.,?.
such that
,$(a’,
2;
2)‘(,
a) =f;s(c?;
J?(,
a) + g$?
L’,2(2’,
2; PC,
a) = g’(?;
a) = 0.
= 0,
A.3.
d’L;>,r d’ < 0
A.4.
2 satisfying A.2’ and A.3’ is a Nash equilibrium for CX.
for all u’ E R”, u’ # 0,
which satisfy g:,(.Y;
(Y)u’ = 0.
* The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. The material contained is of a preliminary nature, is circulated to stimulate discussion, and is not to be quoted without permission of the author.
0165-1765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
M.J. Stutzer
20
statics for integrable Nash equilibria
/ Comparative
The rank of g$ = r; < n.
A.5.
Define the Nn-order partitioned
square matrix L,,,
whose k th diagonal block is the Hessian
LtkXk, and whose (k, 1) block is the square matrix fsX1 = (a’f k/ax,k ax:), k f 1.
Assume furthermore that A.6.
The Nn-order square matrix L,,
is symmetric
and A.7.
The matrix L,,
is negative definite on the null space of g,.
Because A.1 guarantees the symmetry of L:,,,, A.6 reduces to the assumption that, for all i #j, the matrix fd!,, equals the matrix fiyxI.This assumption permits some asymmetries, and is weaker than the assumption of identical agents - symmetric Nash equilibrium - that is used in many studies. This assumption guarantees that the Jacobian of the first order conditions A.2 is symmetric, which is the Frobenius condition for mathematical integrability [Hurwicz (1971, p. 189)], guaranteeing the existence of a Lagrangian having critical points (2, fi) at solutions to A.2. Condition A.7 can be motivated in two ways. First, it guarantees that the aforementioned Lagrangian has a local constrained maximum in x at 2. Alternatively, one can consider the dynamic adjustment mechanism, dx’,‘dt
= L)r(x’;
di(,
a);
i=l,...,
N;
x(0) given,
(2)
which is related to that used by Rosen (1965, p. 529). Stutzer (1984, p. 8) shows that the addition of A.7 guarantees local stability of a Nash equilibrium 2. Denote matrices L$, = (C32Lk/13x~3aj) and L,, = ( L:I,, . . . , LEN,)‘. Stutzer used the method of Pauwels to show the following comparative statics results. (CSI )
fi, 1
IS positive semidefinite on the null space of g,, and is [ %X positive definite off its subspace of vectors u for which Lxau belongs to the space spanned by the row vectors of g,.
The symmetric matrix [ g:L:,]
Suppose the constraints in (1) are augmented by s,-vectors of additional ‘just binding’ constraints g’+(Z’; a) = 0; i = 1,. . . , N. Assuming the analogous conditions A.l-A.7 for (1) with the augmented set of constraints, we also have: (CS2)
The matrix
is symmetric and negative semidefinite, where the Jacobian of the additional Lagrange multipliers flz =
with fii”,’ =
M.J. Stutzer
/ Comparative
stqtics for integrable Nash equilibrra
21
2. An application: Cost curve shifts in integrable Coumot oligopoly Let (Y, denote maxf’(x’; xl
a positive
cost curve shift parameter
x)‘(, u)=x~p(x~+,r,x~)
-(Y;cI(xI);
for firm i. Then,
(1) becomes
i=l,...,N.
Stutzer (1984, pp. 9-10) shows that relatively innocuous conditions on demand and cost functions guarantee that A.l-A.5 are satisfied. The integrability condition A.6 will be satisfied when fi,,, = f,“i,,. This occurs either when all firms are identical, i.e., the equilibrium is symmetric, or when pXX= 0, i.e., the inverse market demand curve is locally linear at a Nash equilibrium 2. Bergstrom and Varian (1985, p. 7) did not recognize the latter. The latter, assumed herein, admits asymmetric costs. The stronger global linearity condition is also frequently used in oligopoly models [Dixit (1986, p. IlO)]. A.7 can be satisfied by assuming, for all i, that there are no increasing returns to scale left unexploited in equilibrium [Stutzer (1984, p. ll)]. Positive definiteness in (CSl) implies that z?:, < 0; i=l >. . >N. A proportional, upward shift in a firm’s cost curve lowers its output. The symmetry in (CSl) implies the reciprocity relations
c:z;, = c;,a; ;
i,
j=l
,...> N.
(4)
If firm i has higher marginal cost than firm j, an upward shift in firm i’s costs has a greater impact on firm j’s output than vice versa. Positive definiteness implies that all principal minors in (CSl) are positive. For example, evaluating the upper left, second-order principal minor easily yields
The ‘own effects’ of firm cost curve shifts on their own outputs ‘dominate’ the ‘cross effects’ the shifts have on the outputs of other firms. Readers can derive and interpret the other higher-order inequalities. Suppose that government regulations raise the marginal costs of firms. Worried about the possible output reductions, the government also enacts a regulation forcing firm i to maintain its former output level. Adding the ‘just binding’ constraint g’+(x’; a) = 2’ - x’ = 0, Stutzer (1984, p. 13) shows that the negative semidefiniteness in (CS2) implies
,:,a;I < ,;,a;,+;
allj=
l,...,
N.
The negative response of firm j to an upward shift on its cost curve will be no more negative if one constrains any firm i’s output from falling. In fact, (6) is still true if any number of firms are subject to this constraint. References Bergstrom, T.C. and H.R. Varian, 1985, Two remarks on Coumot equilibria, Economics Letters 19, 5-8. Dixit, A., 1986, Comparative statics for oligopoly, International Economic Review 27, 1077121. Hurwicz, L., 1971, On the problem of integrability of demand functions, in: J. Chipman et al., eds., Preferences utility, and demand (Harcourt Brace, New York) 174-214. Pauwels, W., 1979, On some results in comparative statics analysis, Journal of Economic Theory 21, 4833490. Rosen, J.B., 1965, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica 33, 520-534. Stutzer, M.J., 1984, Correspondence principles for concave orthogonal games, Staff report 90 (Research Department, Federal Reserve Bank of Minneapolis, Minneapolis, MN).
23 (1987) 19-21
COMPARATIVE
STATICS
19
FOR INTEGRABLE
NASH EQUILIBRIA
*
Michael J. STUTZER Federal Reserve Bank of Minneapolis, Minneapolis, MN 55480, USA Received
18 September
1986
This paper derives comparative statics results for stable Nash equilibria in integrable, concave orthogonal games. Application of these results to cost curve shifts in the asymmetric Cournot oligopoly immediately uncovers some apparently new comparative statics results.
1. Two comparative statics results I generalize the single agent results of Pauwels (1979) to derive results for a subclass of the class of concave orthogonal games defined by Rosen (1965) whose Nash equilibria are defined by maxf’(x’; XI
x)‘(, (Y),
i=l
,.‘.>
N,
subject to
g’(x’;
a) = 0,
(1)
where xl E R”, a E R”, g’isavectorofr,constraintfunctions,andx)’(=(x’,...,x’~‘,~’~’,...,x~). Throughout, a subscripted function denotes its gradient with respect to the subscripted vector, i.e., f, = (ag/ax,), and a subscripted vector of functions denotes its Jacobian matrix with respect to the subscript, i.e., g, = (ag’/axj). Hessian matrices are denoted by double subscripts. An apostrophe denotes the transpose operator. A Nash equilibrium for (Y is a vector R = (a’, . . . , ZN) solving (1) for i = 1,. . . , N. Subscripting to denote partial derivatives, we assume the following for i = 1,. . . , N: A. 1.
The maps f’
and g’ are twice differentiable.
There exist _? and fi = (A’, . . . , fi”) A.,?.
such that
,$(a’,
2;
2)‘(,
a) =f;s(c?;
J?(,
a) + g$?
L’,2(2’,
2; PC,
a) = g’(?;
a) = 0.
= 0,
A.3.
d’L;>,r d’ < 0
A.4.
2 satisfying A.2’ and A.3’ is a Nash equilibrium for CX.
for all u’ E R”, u’ # 0,
which satisfy g:,(.Y;
(Y)u’ = 0.
* The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. The material contained is of a preliminary nature, is circulated to stimulate discussion, and is not to be quoted without permission of the author.
0165-1765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
M.J. Stutzer
20
statics for integrable Nash equilibria
/ Comparative
The rank of g$ = r; < n.
A.5.
Define the Nn-order partitioned
square matrix L,,,
whose k th diagonal block is the Hessian
LtkXk, and whose (k, 1) block is the square matrix fsX1 = (a’f k/ax,k ax:), k f 1.
Assume furthermore that A.6.
The Nn-order square matrix L,,
is symmetric
and A.7.
The matrix L,,
is negative definite on the null space of g,.
Because A.1 guarantees the symmetry of L:,,,, A.6 reduces to the assumption that, for all i #j, the matrix fd!,, equals the matrix fiyxI.This assumption permits some asymmetries, and is weaker than the assumption of identical agents - symmetric Nash equilibrium - that is used in many studies. This assumption guarantees that the Jacobian of the first order conditions A.2 is symmetric, which is the Frobenius condition for mathematical integrability [Hurwicz (1971, p. 189)], guaranteeing the existence of a Lagrangian having critical points (2, fi) at solutions to A.2. Condition A.7 can be motivated in two ways. First, it guarantees that the aforementioned Lagrangian has a local constrained maximum in x at 2. Alternatively, one can consider the dynamic adjustment mechanism, dx’,‘dt
= L)r(x’;
di(,
a);
i=l,...,
N;
x(0) given,
(2)
which is related to that used by Rosen (1965, p. 529). Stutzer (1984, p. 8) shows that the addition of A.7 guarantees local stability of a Nash equilibrium 2. Denote matrices L$, = (C32Lk/13x~3aj) and L,, = ( L:I,, . . . , LEN,)‘. Stutzer used the method of Pauwels to show the following comparative statics results. (CSI )
fi, 1
IS positive semidefinite on the null space of g,, and is [ %X positive definite off its subspace of vectors u for which Lxau belongs to the space spanned by the row vectors of g,.
The symmetric matrix [ g:L:,]
Suppose the constraints in (1) are augmented by s,-vectors of additional ‘just binding’ constraints g’+(Z’; a) = 0; i = 1,. . . , N. Assuming the analogous conditions A.l-A.7 for (1) with the augmented set of constraints, we also have: (CS2)
The matrix
is symmetric and negative semidefinite, where the Jacobian of the additional Lagrange multipliers flz =
with fii”,’ =
M.J. Stutzer
/ Comparative
stqtics for integrable Nash equilibrra
21
2. An application: Cost curve shifts in integrable Coumot oligopoly Let (Y, denote maxf’(x’; xl
a positive
cost curve shift parameter
x)‘(, u)=x~p(x~+,r,x~)
-(Y;cI(xI);
for firm i. Then,
(1) becomes
i=l,...,N.
Stutzer (1984, pp. 9-10) shows that relatively innocuous conditions on demand and cost functions guarantee that A.l-A.5 are satisfied. The integrability condition A.6 will be satisfied when fi,,, = f,“i,,. This occurs either when all firms are identical, i.e., the equilibrium is symmetric, or when pXX= 0, i.e., the inverse market demand curve is locally linear at a Nash equilibrium 2. Bergstrom and Varian (1985, p. 7) did not recognize the latter. The latter, assumed herein, admits asymmetric costs. The stronger global linearity condition is also frequently used in oligopoly models [Dixit (1986, p. IlO)]. A.7 can be satisfied by assuming, for all i, that there are no increasing returns to scale left unexploited in equilibrium [Stutzer (1984, p. ll)]. Positive definiteness in (CSl) implies that z?:, < 0; i=l >. . >N. A proportional, upward shift in a firm’s cost curve lowers its output. The symmetry in (CSl) implies the reciprocity relations
c:z;, = c;,a; ;
i,
j=l
,...> N.
(4)
If firm i has higher marginal cost than firm j, an upward shift in firm i’s costs has a greater impact on firm j’s output than vice versa. Positive definiteness implies that all principal minors in (CSl) are positive. For example, evaluating the upper left, second-order principal minor easily yields
The ‘own effects’ of firm cost curve shifts on their own outputs ‘dominate’ the ‘cross effects’ the shifts have on the outputs of other firms. Readers can derive and interpret the other higher-order inequalities. Suppose that government regulations raise the marginal costs of firms. Worried about the possible output reductions, the government also enacts a regulation forcing firm i to maintain its former output level. Adding the ‘just binding’ constraint g’+(x’; a) = 2’ - x’ = 0, Stutzer (1984, p. 13) shows that the negative semidefiniteness in (CS2) implies
,:,a;I < ,;,a;,+;
allj=
l,...,
N.
The negative response of firm j to an upward shift on its cost curve will be no more negative if one constrains any firm i’s output from falling. In fact, (6) is still true if any number of firms are subject to this constraint. References Bergstrom, T.C. and H.R. Varian, 1985, Two remarks on Coumot equilibria, Economics Letters 19, 5-8. Dixit, A., 1986, Comparative statics for oligopoly, International Economic Review 27, 1077121. Hurwicz, L., 1971, On the problem of integrability of demand functions, in: J. Chipman et al., eds., Preferences utility, and demand (Harcourt Brace, New York) 174-214. Pauwels, W., 1979, On some results in comparative statics analysis, Journal of Economic Theory 21, 4833490. Rosen, J.B., 1965, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica 33, 520-534. Stutzer, M.J., 1984, Correspondence principles for concave orthogonal games, Staff report 90 (Research Department, Federal Reserve Bank of Minneapolis, Minneapolis, MN).