Product Similarity and Cross-Price Elasticity

Abstract

We use a spatial competition model to show how an increasing similarity between two products can monotonically increase, monotonically decrease, or have a non-monotonic effect on cross-price elasticity. We relate these results to prior research that links cross-price elasticity to product similarity, and the literature on market structure, merger analysis, and price discrimination.

Appendix

Signing $\frac{\partial \eta _{12}}{\partial y}$ in the general marginal cost case. A necessary condition for the inequalities in footnote (4) to hold is:

$$\begin{aligned} c(\alpha ) < \alpha. \end{aligned}$$

(12)

From (9), we have $sign\left[ \frac{\partial \eta _{12}}{\partial y}\right] \lesseqqgtr 0$ if

$$\begin{aligned}&c(\alpha )+\frac{4}{3}\alpha \,\gtreqqless\, \frac{4}{3}\left( 1-c(1)\right) , \end{aligned}$$

(13a)

$$\begin{aligned}&\text {or, }4(1-c(1))-4\alpha -3c(\alpha)\, \lesseqgtr\, 0 \end{aligned}$$

(13b)

Consider the case when c(1) is sufficiently low: arbitrarily close to zero. Since c(.) is an increasing function, this implies $c(\alpha )$ is arbitrarily close to zero as well. The LHS of (13b) reduces to $4-4\alpha >0$ for $\alpha \in (0,1).$ Here, $sign[\frac{\partial \eta _{12}}{\partial y}]>0.$

Consider the case when c(1) is sufficiently high: arbitrarily close to 1. Then the LHS of (13b) reduces to $-4\alpha -3c(\alpha )<0$. Here, $sign[ \frac{\partial \eta _{12}}{\partial y}]<0.$

For intermediate values of c(1),  the sign of LHS of (13b) depends on the value of $\alpha$. Define the function $f(\alpha )=4(1-c(1))-4\alpha -3c(\alpha ).$ At $\alpha$ arbitrarily close to zero, $f(\alpha )$ approaches $4-4c(1)>0$ using (12). At $\alpha$ close to 1, $f(\alpha )$ is close to $-7c(1)\,<\,0.$ Then, given that f(.) is continuous and decreasing in $\alpha$, the Intermediate Value Theorem tells us that there must exist $\alpha$—say, $\overline{\alpha }$—such that $f(\overline{\alpha } )=0,$ and for $\alpha <\overline{\alpha },$ $f(\alpha )>0$ and hence $sign[ \frac{\partial \eta _{12}}{\partial y}]>0,$ and for $\alpha >\overline{ \alpha },$ $f(\alpha )<0$ and hence $sign[\frac{\partial \eta _{12}}{ \partial y}]<0.$