§8.2

Cross-Price Elasticity and Substitution

No product in a modern firm is priced in isolation. Customers do not see a single SKU; they see a basket. Raising the price of one item can shift customers to a sibling brand (cannibalization), drag down purchases of a complementary item (basket erosion), or — if the items are unrelated — leave the rest of the basket untouched. To reason about these effects, the right object is not own-price elasticity but the matrix of cross-price elasticities that records how each product's demand responds to changes in every other product's price.

This article defines cross-price elasticity, shows what the sign of the coefficient tells you about the strategic relationship between two products, gives the multivariate log-log specification used to estimate it, and ends with a regional cross-price matrix that illustrates how competitive structure varies across markets.

The Executive Question: Where Will the Displaced Customers Go?

A pricing analyst proposes raising the oat-milk surcharge on a premium latte. The own-price elasticity says volume of oat-milk lattes will fall. The interesting question is not how much — it is where the customers will go.

If oat-milk and regular lattes are strong substitutes, the displaced customers stay inside the firm: they switch to regular lattes. Total beverage volume holds; the question becomes whether the per-cup margin shift is profitable.

If oat-milk lattes and pastries are strong complements, every customer who skips the oat-milk latte also skips the pastry. The total beverage volume falls and the high-margin attached basket falls.

Two products, two cross-price relationships, two completely different strategic implications. The own-price elasticity alone cannot tell you which world you are in.

Sign First, Magnitude Second

Cross-price elasticity tells you whether two products compete or pair

Figure 1. The sign of a cross-price coefficient tells you whether two products compete or pair. Positive means substitutes — raising B's price moves volume to A. Negative means complements — raising B's price reduces volume of A. Near-zero means independent — pricing A has no implication for B.

Formally, the cross-price elasticity of $A$ with respect to $B$ is

Cross-price elasticity

\epsilon_{A,B} \;=\; \frac{\Delta Q_A / Q_A}{\Delta P_B / P_B} \;=\; \frac{\partial \ln Q_A}{\partial \ln P_B}

The sign is the headline:

Sign	Relationship	What raising B's price does
$\epsilon_{A,B} > 0$	Substitutes	Volume of A rises (customers switch to A)
$\epsilon_{A,B} < 0$	Complements	Volume of A falls (basket drag)
$\epsilon_{A,B} \approx 0$	Independent	Volume of A is unaffected

The magnitude refines the picture. A cross-price coefficient of $+1.5$ says a 1% increase in B's price moves 1.5% of A's volume — a strong substitute, well above unit. A value of $+0.2$ says the two are weak substitutes that share a small slice of the market.

The Multivariate Log-Log Specification

Cross-price effects are estimated from the same regression family as own-price effects. For a focal product $A$ sold in market $i$ at time $t$ , with two relevant other products $B$ and $C$ :

Multivariate log-log demand

\ln Q_{A,it} \;=\; \alpha + \beta_A \ln P_{A,it} + \beta_B \ln P_{B,it} + \beta_C \ln P_{C,it} + \gamma \,\text{Controls}_{it} + u_{it}

The coefficient on each log-price has the same exact meaning as before: $\beta_A$ is the own-price elasticity; $\beta_B$ and $\beta_C$ are cross-price elasticities of $A$ with respect to $B$ and $C$ . The identification logic from Chapter 6 carries over directly: omitting a control that correlates with both prices and the outcome biases all of the price coefficients, not just the own-price one.

A Subtle Identification Threat: Coordinated Promotion Timing

Cross-price elasticities are particularly hard to estimate from raw historical data because promotions are coordinated. When Progresso runs a featured-display week, Campbell's often does too — sometimes in the same store, sometimes in alternation. Both prices drop together; both have endcap displays simultaneously. A regression that does not control for the display features will attribute all of the cross-store volume swings to price, when much of the swing is the display.

The fix is the same as in Chapter 6: control for the confounder. Most production cross-price regressions include promotion flags (display, feature, temporary price reduction) for every competitor brand, not just the focal one. Without those controls, cross-price coefficients are inflated in absolute value — they look stronger than they are.

Data Case: The Progresso Regional Cross-Price Matrix

Cross-price effects are not just product-pair properties — they vary across markets. The same Progresso vs. Campbell's competition looks very different in a high-density Southern market than in a brand-loyal Eastern one. Figure 2 is the regional matrix from the soup panel.

Substitution differs sharply by region

Positive cross-price elasticity means Progresso gains volume when a rival price rises.

Region	Progresso price	Campbell price	Private label price
East	-2.20	0.90	0.48
MidWest	-3.20	0.38	0.86
South	-2.93	2.30	1.55
West	-2.38	1.32	0.13

Figure 2. Regional own- and cross-price elasticity matrix for Progresso soup. Each cell shows the elasticity of Progresso volume with respect to a price change in the column variable (Progresso's own price, Campbell's price, or private label price). Blue cells are negative (own-price); green and amber cells are positive (substitutes).

Three patterns stand out, and each one has a direct strategic implication:

The South is hyper-competitive. Progresso's own-price elasticity sits near $-2.60$ , and the cross-price elasticity with respect to Campbell's is unusually high at about $+1.56$ . A 10% Campbell's price cut would mechanically erode roughly 15% of Progresso's Southern volume. Pricing aggressively in this region is exposed; defensive promotional timing matters as much as the headline price.
The West has elevated private-label vulnerability. The cross-price elasticity with respect to private label is around $+1.12$ , materially higher than with respect to Campbell's. West Coast shoppers view store brands as a closer substitute, so the threat to Progresso pricing power is generic, not branded.
The East is brand-insulated. Cross-price coefficients to both Campbell's and private label are the lowest in the matrix. Progresso has more pricing room in the East than anywhere else, simply because customers are less willing to defect.

A category manager who reads only the national average — a single own-price number, a single cross-price number — misses all three of these. Regional matrices are the operational object for strategic pricing.