Supermarket responses to Wal-Mart Supercenter expansion: a structural approach

Abstract

The impact of Wal-Mart in lowering incumbents’ retail prices has been well documented by previous studies using reduced form models. This article uses a structural model to examine the pricing behavior and promotion responses of incumbent supermarkets to a rapid expansion of Wal-Mart Supercenters (WMS) using the Dallas–Fort Worth milk market as a case study. Empirical results verify that WMS expansion disciplines incumbent supermarkets by decreasing oligopoly power and numbing consumer responsiveness to promotion. In addition, WMS expansion lures away price-sensitive consumers, leaving incumbent supermarkets to face more price-inelastic but lower demands for milk.

Appendix: Deriving the estimating equations

This appendix gives the derivation of the equations to be estimated. Each supermarket, $i=1,...,N$, chooses quantity, $q_{i},$ and promotion expenditures, $m_{i},$ to maximize

$$\begin{aligned} Max_{q_{i},a_{i}}\left\{ \pi _{i}=P(Q)q_{i}-c(q_{i})-m_{i}:Q=f(m,S)\right\} \end{aligned}$$

(10)

which has as necessary conditions for maximization,

$$\begin{aligned} \frac{\partial \pi _{i}}{\partial q_{i}}=\frac{\partial P}{\partial Q}\frac{\partial Q}{\partial q_{i}}q_{i}+P-\frac{\partial c}{\partial q_{i}}=0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial \pi _{i}}{\partial m_{i}}=\frac{\partial P}{\partial Q}\frac{\partial Q}{\partial m}\frac{\partial m}{\partial m_{i}}q_{i}+\frac{\partial q_{i}}{\partial Q}\frac{\partial Q}{\partial m}\frac{\partial m}{\partial m_{i}}P-\frac{\partial g}{\partial m_{i}}=0. \end{aligned}$$

The derivation for the estimating equation of price is given by the first necessary condition. Multiplying the first term by $\frac{Q}{Q}$ yields

$$\begin{aligned} P=-\frac{\partial P}{\partial Q}Q\,\frac{\partial Q}{\partial q_{i}}\frac{q_{i}}{Q}+\frac{\partial c}{\partial q_{i}} \end{aligned}$$

letting $\eta =\frac{\partial Q}{\partial P}\frac{1}{Q}$ and $\theta _{j}=\theta _{i}=\frac{\partial Q}{\partial q_{i}}\frac{q_{i}}{Q}$, and aggregating across firms yields the following equation to be estimated:

$$\begin{aligned} P=-\frac{\Theta }{\eta }+\frac{\partial c}{\partial Q}. \end{aligned}$$

(11)

The derivation for the estimating equation of marketing mix is given by the second first-order condtion. Multiplying the first term by $\frac{Q}{Q}\frac{m_{i}}{m_{i}}$ and the second term by $\frac{m_{i}}{m_{i}}\frac{q_{i}}{q_{i}}$ yields

$$\begin{aligned} \frac{\partial P}{\partial Q}Q\,\frac{\partial Q}{\partial a}\frac{1}{Q}\,\frac{\partial m}{\partial m_{i}}m_{i}\,\frac{q_{i}}{m_{i}}+\frac{\partial q_{i}}{\partial Q}\frac{Q}{q_{i}}\,\frac{\partial Q}{\partial m}\,\frac{\partial m}{\partial m_{i}}m_{i}\,\frac{q_{i}}{m_{i}}P=\frac{\partial g}{\partial m_{i}} \end{aligned}$$

letting $\varepsilon =\frac{\partial Q}{\partial m}\frac{1}{Q}$ and $\psi =\frac{\partial m}{\partial m_{i}}m_{i}$, aggregating across firms, inverting, and normalizing marginal cost of marketing to one yields the following equation to be estimated

$$\begin{aligned} \frac{m}{Q}=\psi \varepsilon \left( \frac{1}{\eta }+\frac{P}{\Theta }\right) . \end{aligned}$$

(12)