## Abstract

This paper elucidates the role played by the heterogeneity of interactions between the endogenous variables of a model in determining the model’s behavior. It is known that comparative statics are well-behaved if these interactions are relatively small, but the formal condition imposed on the Jacobian which typically captures this idea–diagonal dominance–ignores the distribution of the interaction terms. I provide a new condition on the Jacobian—*mean positive dominance*—which better captures a trade-off between the size and heterogeneity of interaction terms. In accord with Samuelson’s (Foundations of economic analysis, Oxford University Press, London, 1947) correspondence principle, I also show that mean positive dominance yields stability and uniqueness results. I apply the results to provide new, or to generalize known, comparative statics results in the following settings: optimization problems, platform monopoly, normality, differentiable games including Cournot oligopoly, and competitive exchange economies.

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## Notes

- 1.
Early contributions to the monotone comparative statics literature include Topkis (1998), Vives (1990), Milgrom and Roberts (1990), and Milgrom and Shannon (1994). More recent developments and applications include Amir and Lambson (2000), Echenique (2002), Quah and Strulovici (2009), Jensen (2010), Amir and Lazzati (2016), Reynolds and Rietzke (2016), Cosandier et al. (2017), and others.

- 2.
- 3.
- 4.
Although Samuelson (1947) coined the term, he never provided a formal definition, and therefore different authors define the correspondence principle differently. However, Samuelson did illustrate this principle through a series of examples, and it is clear from these examples that he thought of the correspondence principle as referring to a large intersection between the set of conditions which imply well-behaved comparative statics and the set of conditions which imply stable dynamics.

- 5.
Applied to a game-theoretic context where each equation represents a player’s best response function, there is a clear connection to the social interactions literature. In this setting say that the interactions effect

*reinforces*the partial (or private) effect if \(\hbox {sgn}(\hbox {PE})=\hbox {sgn}(\hbox {IE})\ne 0;\) the interactions effect*counteracts*the private effect if \( \hbox {sgn}(\hbox {PE})=-\hbox {sgn}(\hbox {IE})\ne 0\). The*social multiplier*is \(\frac{\hbox {TE}}{\hbox {PE}},\) and this is greater than one if and only if the interactions effect has the same sign as the partial effect. - 6.
To use alternative terminology, this is a model of interdependent preferences (e.g., Pollak 1976).

- 7.
I use element-wise notation throughout the paper, so \(b\ge 0\) means \( b_{i}\ge 0\) for \(i=1,\ldots ,n.\)

- 8.
A

*P*-matrix is a matrix with positive principal minors. - 9.
A \(P_{0}\)-matrix is a matrix with nonnegative principal minors.

- 10.
*A*is positive (semi)definite if \(z^{\prime }Az(\ge )>0\) for every nonzero vector \(z\in {\mathbb {R}}.\)*A*does not need to be symmetric. - 11.
A matrix is

*positive stable*if all of its eigenvalues have positive real parts. - 12.
If \(f_{kk}=0\) for some

*k*and the Jacobian or its negation is a \(B_{0}\)-matrix, then the*k*th row contains only zeros, so the Jacobian is not invertible. - 13.
The matrix \(A=\left( a_{ij}\right) \) is

*strictly diagonally dominant*if, for \(i=1,\ldots ,n,\)\(\left| a_{ii}\right| >\sum _{j\ne i,j=1}^{n}\left| a_{ij}\right| .\) The off-diagonal terms are “ identical by rows” if, for all*i*, \( a_{ij}={\bar{a}}_{i},\) for all \(j\ne i,\)\(j\in \left\{ 1,\ldots ,n\right\} \). - 14.
One exception is Simon (1989), but the analysis is limited to the context described two paragraphs above.

- 15.
In Theorem 6w in Gale and Nikaido (1965), univalence follows if

*X*is an open convex set and \(D_{x}f\) is*weakly positive quasi-definite,*meaning that \(\det D_{x}f\)\(>0\) and \(\frac{1}{2}\left( D_{x}f+\left[ D_{x}f\right] ^{T}\right) \) is positive semidefinite. By Lemma 3, a sufficient condition for \(D_{x}f\) to be weakly positive quasi-definite is for \(\frac{1}{2}\left( D_{x}f+\left[ D_{x}f\right] ^{T}\right) \) to be an invertible \(B_{0}\)-matrix. - 16.
We can allow for positive externalities from companions as long as some strangers generate negative externalities. This case would be situated in the upper left or lower right quadrants of Fig. 1. The pure congestion case is represented in the lower left quadrant.

- 17.
In this example system

*f*is a demand system and the interaction terms represent an individual’s marginal demand response to a one unit increase in other’s consumption of the good. - 18.
This example does not apply well to all congestion situations. Probably the most reasonable assumption for traffic congestion is anonymous effects since in the vast majority of cases each additional vehicle creates same negative externality. In this case the demand curve is likely downward sloping. That being said, it may be possible for toll operators to select for better drivers, and consequently face a less elastic demand curve, by selling passes only to those who have good driving records.

- 19.
Although it may seem counterintuitive, it follows from Eq. (3) that interaction terms have a positive influence on the total effect if \(-f_{ij}/f_{ii}\ge 0\) and a negative influence if \( -f_{ij}/f_{ii}\le 0.\)

- 20.
In both cases the row sum of the third row is 2.6. But the largest positive off-diagonal term in the first case is 0.8 while in the second it is 1.6. Since \(n=3,\) we have \(2.6>3\times 0.8\) but \(2.6<3\times 1.6.\)

- 21.
Recall that

*F*(*x*;*t*) is*supermodular**in**x*if \(f_{ij}\ge 0\) for all \(j\ne i\) and all \(i=1,\ldots ,n,\) or, equivalently, if the off-diagonals of the Hessian \(D_{x}f(x;t)\) are nonnegative. In this sense the MCS literature restricts attention to off-diagonal terms which have the same sign. - 22.
This version of the profit function can also accommodate membership fees; see Rochet and Tirole (2006).

- 23.
From the first-order conditions and the IFT we have

$$\begin{aligned} \left[ \begin{array}{cc} U &{} -p \\ -p^{T} &{} 0 \end{array} \right] \left[ \begin{array}{c} \mathrm{d}x \\ \frac{\hbox {d}\lambda }{\hbox {d}m} \end{array} \right] =\left[ \begin{array}{c} 0 \\ -1 \end{array} \right] . \end{aligned}$$Denoting \(\mathscr {H=}\left[ \begin{array}{cc} U &{} -p \\ -p^{T} &{} 0 \end{array} \right] \), applying Cramer’s rule, and using the Laplacian expansion to evaluate the numerator, we have \(\frac{\hbox {d}\lambda }{\hbox {d}m}=-\frac{\det U}{\det {\mathscr {H}}}.\) Since

*U*is nonsingular it follows that \(\frac{\hbox {d}\lambda }{\hbox {d}m}\ne 0.\) Using the Schur complement method to calculate the determinant, and noting that the Schur complement of \({\mathscr {H}}\) with respect to*U*is \({\mathscr {H}}/U\mathscr {=}-p^{T}U^{-1}p\), we have \(\det \mathscr {H=}\det U\det {\mathscr {H}}/U=-p^{T}U^{-1}p\det U.\) Hence, \(\frac{\hbox {d}\lambda }{\hbox {d}m} =(p^{T}U^{-1}p)^{-1}\). - 24.
Recall that concavity requires

*U*to be negative semidefinite for all*x*, or \(z^{T}Uz\le 0\) for all \(x,z\in {\mathbb {R}}^{n}.\) Thus, at values of*x*where*U*is invertible, all the eignevalues of*U*are strictly negative. The eigenvalues of \(U^{-1}\) are the inverses of the eigenvalues of*U*, so \(U^{-1}\) is negative definite at values of*x*(including \({\bar{x}})\) where*U*is invertible. This implies \( p^{T}U^{-1}p<0.\) - 25.
Barthel and Sabarwal ’s (2017) results extend to monotone transformations of concave objective functions.

- 26.
A Morse function is a smooth function with nondegenerate critical points. That is, given \({\bar{t}},\)\(\frac{\partial \pi _{i}\left( {\bar{x}};{\bar{t}} \right) }{\partial x_{i}}=0\) implies \(\frac{\partial ^{2}\pi _{i}\left( \bar{ x};{\bar{t}}\right) }{\partial x_{i}^{2}}\ne 0\). It is well known that Morse functions are generic in that, given \({\bar{t}},\) they form an open, dense subset of all smooth functions \(X\rightarrow {\mathbb {R}}.\)

- 27.
In fact, if \(\frac{\partial \pi _{i}\left( a_{i},x_{-i};{\bar{t}}\right) }{ \partial x_{i}}>0\) and \(\frac{\partial \pi _{i}\left( b_{i},x_{-i};{\bar{t}} \right) }{\partial x_{i}}<0\) for all \(x_{-i}\in X_{-i},\) then “\(\frac{ \partial \pi _{i}\left( {\bar{x}}_{i},{\bar{x}}_{-i};{\bar{t}}\right) }{\partial x_{i}}=0\) implies \(\frac{\partial ^{2}\pi _{i}\left( {\bar{x}}_{i},{\bar{x}} _{-i};{\bar{t}}\right) }{\partial x_{i}^{2}}<0\)” is necessary and sufficient for there to exist a unique critical point of \(\pi _{i}\) which is also an interior maximum. This characterization extends naturally to multi-dimensional optimization problems (see Christensen 2017).

- 28.
Recall that the matrix \(M=(m_{ij})\) is strictly diagonally dominant if \( \left| m_{ii}\right| >\sum _{j\ne i,j=1}^{n}\left| m_{ij}\right| \) for all

*i*. - 29.
A direct comparison to Dixit’s conditions also help illustrate the trade-off between heterogeneity and magnitude captured in Proposition 1. The normalized version of the Jacobian of the Dixit pseudogradient has ones on the main diagonal and off-diagonal terms given by \(b_{i}/a_{i}.\) Strict diagonal dominance requires \(\left| b_{i}/a_{i}\right| <\frac{1}{n-1}.\) In contrast, by inequality (11), the normalized version of the Jacobian of the Dixit pseudogradient is mean positive dominant if \(\left| b_{i}/a_{i}\right| <\frac{1}{n-1}\) when \(b_{i}\ge 0\) and \(b_{i}/a_{i}<1\) when \(b_{i}<0.\) The reason that mean positive dominance is able to relax the diagonal dominance conditions (and still guarantee well-behaved comparative statics) is that diagonal dominace fails to take advantage of the limit on heterogeneity imposed by assuming interaction effects are identical by rows.

- 30.
The term \(\frac{b_{i}}{a_{i}-b_{i}}\) equals \(\frac{\frac{\partial ^{2}\pi _{i}}{\partial x_{i}\partial x_{j}}}{\frac{\hbox {d}P}{\hbox {d}x_{i}}-\frac{\hbox {d}^{2}c_{i}}{ \hbox {d}x_{i}^{2}}}.\) If \(\frac{\partial ^{2}\pi _{i}}{\partial x_{i}\partial x_{j}}<0,\) then \(\frac{b_{i}}{a_{i}-b_{i}}\) is positive by a strict inequality version of condition (26). If \(\frac{\partial ^{2}\pi _{i}}{\partial x_{i}\partial x_{j}}\ge 0,\) then \(\frac{b_{i}}{ a_{i}-b_{i}}\) is bounded below by \(-\frac{1}{n}\) by condition (25). It follows that \(1+\sum _{i}\frac{b_{i}}{a_{i}-b_{i}}>0.\) It is worth noting that in the classical Cournot setting, subject to some mild restrictions and the condition that \(\frac{\hbox {d}P}{\hbox {d}x_{i}}-\frac{\hbox {d}^{2}c_{i}}{ \hbox {d}x_{i}^{2}}<0,\) Kolstad and Mathiesen (1987) show that uniqueness obtains if and only if \(1+\sum _{i}\frac{b_{i}}{a_{i}-b_{i}}>0.\)

- 31.
Recall that the gross substitutes property requires \(\frac{\partial f_{\ell } }{\partial p_{j}}\ge 0\) for all \(j=1,\ldots ,L+1,\)\(j\ne \ell ,\) and \(\ell =1,\ldots ,L+1.\) In other words, the matrix \(D_{p}f(p;\omega )\) has nonnegative off-diagonal elements.

- 32.
\(m_{k}=p\cdot \omega _{k}\) is consumer

*k*’s wealth. - 33.
RT also show that the seller price decreases with more captive buyers while the buyer price decreases with the prevalence of marquee buyers. In the two-sided case we can generate these results via Cramer’s Rule under the additional condition that \(\frac{\partial ^{2}\pi }{ \partial p_{j}\partial t}=0.\)

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## Acknowledgements

I am grateful to the editors and two anonymous reviewers whose comments helped improve the paper. Any errors are mine.

## Author information

## Appendix

### Appendix

### Proof of Lemma 1

(1) Since \(y=A^{-1}b\) we have for each *i*, \(y_{i}=\sum _{j=1}^{n}b_{j}\delta _{ij}.\) It follows that

where the last inequality is valid for all \(b\ge 0\) iff \(A\in {\mathscr {N}}_{cs}^{-1}.\)

(2) This follows from the fact that \(y_{i}=\sum _{j=1}^{n}b_{j}\delta _{ij}=b_{i}\delta _{ii}\) when *b* is positive only in element *i*.

(3) In this case \(b_{i}={\bar{b}}\ge 0\) for all *i*, so \(y_{i}= \sum _{j=1}^{n}b_{j}\delta _{ij}={\bar{b}}\left( \sum _{j=1}^{n}\delta _{ij}\right) \ge 0,\) where the last inequality is valid for all \({\bar{b}}>0\) iff \(A\in {\mathscr {N}}_{rs}^{-1}.\)

### Proof of Lemma 3

(1) This follows from the facts that a *B*-matrix has a strictly positive determinant, the determinant is a continuous function of the matrix entries, and weak inequalities are preserved in the limit.

(2) The proof of this claim is a slight modification of the proof of Proposition 2.5 in Peña (2001). Consider the set of natural numbers \(M=\left\{ 1,\ldots ,n\right\} ;\) let \( \alpha \) be a subset of *M* with *k* elements, and let \(\alpha ^{\prime }\) be the complement of \(\alpha \) in *M*, \(\alpha ^{\prime }=M\backslash \alpha \). The elements of \(\alpha \) and \(\alpha ^{\prime }\) are understood to be arranged in increasing order. Denote by \(A[\alpha ]\) any principal submatrix of *A* with elements \(\left( a_{ij}\right) \) with \(i,j\in \alpha .\)

First we establish that \(A[\alpha ]\) has nonnegative row sums. To this end, note that \(a_{ii}\ge \sum _{h\in H}\left| a_{ih}\right| ,\) where \( H=\left\{ h|1\le h\le n\text { and }a_{ih}<0\right\} .\) This follows from the fact that rearranging (7) implies \( a_{ii}-a_{ii}^{+}\ge \sum _{j\ne i}(a_{ii}^{+}-a_{ij})\) and observing that \( a_{ii}\ge a_{ii}-a_{ii}^{+},\)\(a_{ii}^{+}-a_{ij}\ge 0\) for all \(i\ne j,\) and \(a_{ii}^{+}-a_{ij}\ge \left| a_{ij}\right| \) if \(a_{ij}<0.\)

To show that \(ka_{ij}\le \sum _{s\in \alpha }a_{is}\) for all \(i\ne j\in \alpha ,\) assume instead that for some \(i\ne j\in \alpha ,\)\( ka_{ij}>\sum _{s\in \alpha }a_{is}\) to derive a contradiction. Since \( ka_{ii}^{+}\ge ka_{ij}\) for all *i*, it follows that for some *i*,

which contradicts the assumption that *A* is a \(B_{0}\)-matrix.

(3)–(4) These are straightforward corollaries of (2).

(5) From Eq. (7) it follows that if \(a_{ii}^{+}=0,\) then \(a_{ii}\ge 0.\) If \(a_{ii}^{+}>0,\) then since \(\sum _{j=1}^{n}a_{ij}\le a_{ii}+(n-1)a_{ii}^{+},\) Eq. (7) implies \( a_{ii}+(n-1)a_{ii}^{+}\ge na_{ii}^{+},\) or \(a_{ii}\ge a_{ii}^{+}.\)

(6) \(A+A^{T}\) is a symmetric \(B(B_{0})\)-matrix by Lemma 2, so it is positive (semi)definite from Peña (2001) and part (4). *A* is positive (semi)definite if and only if \(A+A^{T}\) is positive (semi)definite, and similarly for \(A^{T}.\)

(7) Again, *A* is positive (semi)definite if and only if \(A+A^{T}\) is positive (semi)definite, and similarly for \(A^{T}.\)

(8) This follows from Proposition 2.3 and Theorem 4.3 in Peña (2001). In an effort to keep this paper reasonably self-contained, I will state the key parts of these results here. Let \(A=(a_{ik})_{1\le i,k\le n}\) be a real matrix. Define, for each \(i=1,\ldots ,n,\)

Then define, for each \(i=1,\ldots ,n,\)

Letting \(\lambda \) be an eigenvalue of *A*, the first part of Peña’s (2001) Theorem 4.3 implies

Proposition 2.3 in Peña (2001) states that the real matrix *A* is a *B*-matrix if and only if for all \( i=1,\ldots ,n,\)

It follows that \(A^{T}\) is a *B*-matrix if and only if for all \(i=1,\ldots ,n,\)

Consequently, \(\min \left\{ \theta _{1},\ldots ,\theta _{n}\right\} >0\) if *A* and \(A^{T}\) are *B*-matrices.

### Proof of Proposition 3

The necessary first-order conditions for a maximum are

To apply Proposition 2, we need to show that the Hessian of \(-\pi \) is mean positive dominant. The first-order conditions imply that \(\frac{\partial Q}{\partial p}\equiv \frac{\partial Q}{\partial p_{i}}\) for all *i* at a maximum, so the terms along the main diagonal of the Hessian of \(-\pi \) are \(-\frac{\partial ^{2}\pi }{\partial p_{i}^{2}} =-\left( \sum p-c\right) \frac{\partial ^{2}Q}{\partial p_{i}^{2}}-2\frac{ \partial Q}{\partial p}.\) The off-diagonal terms are \(-\frac{\partial ^{2}\pi }{\partial p_{i}\partial p_{j}}=-\left( \sum p-c\right) \frac{\partial ^{2}Q}{\partial p_{i}\partial p_{j}}-2\frac{\partial Q}{\partial p}.\) The remainder of the proof exploits the fact that \(\frac{\partial ^{2}\pi }{\partial p_{i}\partial p_{j}}=\frac{\partial ^{2}\pi }{\partial p_{j}\partial p_{i}}.\)

Since \(-\frac{\partial ^{2}\pi }{\partial p_{i}\partial p_{j}}\ge 0\) for some \(j\ne i,\)the Hessian (of \(-\pi )\) is mean positive dominant if, for \( i=1,\ldots ,n,\)

which simplifies to \(\frac{\partial ^{2}Q}{\partial p_{i}^{2}}+\sum _{j\ne i} \frac{\partial ^{2}Q}{\partial p_{j}\partial p_{i}}\le n\max \left\{ \frac{ \partial ^{2}Q}{\partial p_{j}\partial p_{i}}|j\ne i\right\} ,\) as desired. We have now shown that the profit function is *B*-concave under the conditions in Proposition 3, so Proposition 2 applies.

### Proof of Proposition 3

(1) Note that for \(i,j=\left\{ S,B\right\} \) and \(j\ne i,\)

The first-order conditions are \(\left( \sum p-c\right) \frac{\partial Q}{ \partial p_{i}}+Q=0\) for all *i* which implies \(D^{i}D^{j}=-\left( \sum p-c\right) \frac{\partial D^{i}}{\partial p_{i}}D^{j}\) for \(i,j\in \left\{ B,S\right\} \) and \(i\ne j.\) Thus, \(\frac{D^{S}}{\frac{\partial D^{S}}{ \partial p_{S}}}=\frac{D^{B}}{\frac{\partial D^{B}}{\partial p_{B}}}\) in equilibrium. It follows that \(\frac{\partial ^{2}Q}{\partial p_{i}^{2}}\le \frac{\partial ^{2}Q}{\partial p^{i}\partial p^{j}}\) for \(i\ne j\in \left\{ B,S\right\} \) if and only if, for \(i\in \left\{ B,S\right\} ,\)

This last condition is precisely log-concavity of the quasi-demand functions.

(2) \(\frac{\partial \pi }{\partial p_{B}\partial p_{S}}=\frac{\partial D^{B} }{\partial p_{B}}D^{S}+D^{B}\frac{\partial D^{S}}{\partial p_{S}}+\left( \sum p-c\right) \frac{\partial D^{B}}{\partial p_{B}}\frac{\partial D^{S}}{ \partial p_{S}}=D^{B}\frac{\partial D^{S}}{\partial p_{S}}\le 0,\) where the second equality follows from the first-order conditions.

(3) With captive buyers, we have \(\frac{\partial ^{2}\pi }{\partial p_{B}\partial t}=D^{S}>0\) and \(\frac{\partial ^{2}\pi }{\partial p_{S}\partial t}=\left( \sum p-c\right) \frac{\partial D^{S}}{\partial p_{s}} +D^{S}=0\) by the first-order condition, so we conclude that \(p^{B}+p^{S}\) and \(p^{B}\) increase with more captive buyers.

With marquee buyers, \(\frac{\partial ^{2}\pi }{\partial p_{B}\partial t}=- \frac{\partial D^{S}(p_{S}-t)}{\partial p_{S}}\left( D^{B}(p_{B})+\left( \sum p-c\right) \frac{\partial D^{B}}{\partial p_{B}}\right) =0\) using the first-order conditions. Once again using the first-order conditions and the assumption that \(D^{S}\) is log concave, we have

Thus, \(p^{B}+p^{S}\) and \(p^{S}\) increase with the addition of marquee buyers.^{Footnote 33}

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### Cite this article

Christensen, F. Comparative statics and heterogeneity.
*Econ Theory* **67, **665–702 (2019) doi:10.1007/s00199-018-1116-x

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### Keywords

- Comparative statics
- Mean positive dominance
- B-matrix
- Correspondence principle
- Cournot oligopoly
- Normal goods

### JEL Classification

- C6
- C72
- D11
- D4
- D5