Price Endogeneity in Differentiated-Product Demand

Why Prices Are Not Automatically Exogenous

In differentiated-product demand, prices are often chosen by firms rather than assigned independently of demand. A firm that knows its product is especially appealing to consumers may set a higher price. That appeal can include characteristics observed by the analyst, such as size or performance, and product quality that the analyst does not observe. When price responds to unobserved quality, a demand regression can mistake some of quality’s effect for a price effect.

The issue is not simply that price and demand move together. Demand is supposed to respond to price. The econometric concern is that the price also moves with an omitted component of demand. The resulting price coefficient may therefore fail to measure the effect of changing price while holding product quality fixed.

Definition: For the price coefficient in a linear regression controlling for $x_{jt}$, let $\widetilde p_{jt}$ be the residual from projecting price on an intercept and those controls. Price is exogenous for this coefficient when $E[\widetilde p_{jt}\xi_{jt}]=0$, and its endogeneity component is nonzero when $E[\widetilde p_{jt}\xi_{jt}]\ne 0$.

Logit Demand and the Unobserved Quality Term

Let $j$ index products and $t$ index markets. A consumer’s utility from product $j$ in market $t$ is

u_{ijt}=x_{jt}'\beta-\alpha p_{jt}+\xi_{jt}+\varepsilon_{ijt}, \qquad \alpha>0.

Here $x_{jt}$ is a vector of product characteristics observed by the analyst, $p_{jt}$ is price, $\beta$ is the vector of coefficients on observed characteristics, and $\xi_{jt}$ is unobserved product quality common to consumers in that market. The idiosyncratic utility terms $\varepsilon_{ijt}$ follow the type-I extreme-value distribution for the standard logit model. Normalize the outside option’s mean utility to zero. The logit share inversion established in the earlier material gives

y_{jt}\equiv\log(s_{jt})-\log(s_{0t}) =x_{jt}'\beta-\alpha p_{jt}+\xi_{jt},

where $s_{jt}$ is product $j$’s market share and $s_{0t}$ is the outside-option share. The inversion is useful because it turns observed shares into a demand index. It does not remove $\xi_{jt}$: that term remains in the equation and represents the part of mean utility not explained by observed characteristics and price.

Worked Example: Inverting a Product’s Logit Share

Consider a hypothetical market with two inside products and an outside option. The mean utilities of the two inside products are both $-2$. Under logit choice, each inside product has share proportional to $e^{-2}$, while the outside option has weight 1.

The share denominator is

D=1+2e^{-2}=1+2(0.135335\ldots)=1.270671\ldots\approx1.2707.

Thus each inside product has share $s_j=e^{-2}/D=0.1065$, and the outside share is $s_0=1/D=0.7870$. Inverting the share for either product gives

y_j=\log\left(\frac{s_j}{s_0}\right) =\log\left(\frac{0.1065}{0.7870}\right) =-2.000.

This recovers mean utility relative to the outside option. If the analyst knows $x_j'\beta$ and $\alpha p_j$, the remaining difference is $\xi_j$. Share inversion itself does not establish that price is uncorrelated with this remaining quality term.

How Equilibrium Pricing Creates Correlation

A firm’s price choice reflects the demand it faces. If a product has higher unobserved quality, consumers value it more at a given price. A firm setting price in response to demand conditions may then charge more for it. In that case, across product-market observations, price tends to be high precisely where $\xi_{jt}$ is high. The price is correlated with a determinant of demand that is absent from the analyst’s observed-characteristics regression.

This is an equilibrium relationship: demand conditions affect firms’ pricing choices, and prices in turn affect demand. Its exact strength and even its sign depend on the market environment and pricing behavior. The common differentiated-product case is a positive conditional relationship between price and unobserved quality, but it is not a mathematical necessity in every possible market.

Suppose, for illustration, an analyst regresses the inverted demand index $y_{jt}$ on price alone, including an intercept. The population slope is

\frac{\operatorname{Cov}(p,y)}{\operatorname{Var}(p)} =-\alpha+ \frac{\operatorname{Cov}(p,x'\beta)}{\operatorname{Var}(p)} +\frac{\operatorname{Cov}(p,\xi)}{\operatorname{Var}(p)}, \qquad \operatorname{Var}(p)>0.

This formula follows by taking the covariance of $p$ with each term in $y=x'\beta-\alpha p+\xi$, then dividing by $\operatorname{Var}(p)$. The observed-characteristics term cannot be dropped unless it is constant or uncorrelated with price. If the regression includes the relevant observed characteristics, the appropriate expression instead uses the variation in price left after linearly removing those controls.

Result: Let $\widetilde p$ be the residual from the population linear projection of price on an intercept and the included observed characteristics $x$. By the Frisch–Waugh–Lovell theorem, the coefficient on price in the population regression of $y$ on the intercept, $x$, and $p$ is $$ -\alpha+\frac{\operatorname{Cov}(\widetilde p,\xi)}{\operatorname{Var}(\widetilde p)}, \qquad \operatorname{Var}(\widetilde p)>0. $$ The formula requires residual price variation after controls. The second term is the price-endogeneity component of the regression slope.

Deriving the Conditional Price-Coefficient Formula

The derivation makes clear why the control variables matter. Project $p$ on an intercept and $x$, and write $p=\widehat p+\widetilde p$, where $\widehat p$ is the linear projection and $\widetilde p$ is its residual. By construction, $\widetilde p$ is uncorrelated with the intercept and each included component of $x$. The residualized outcome is therefore

\widetilde y =y-\operatorname{Proj}(y\mid 1,x) =-\alpha\widetilde p+\widetilde \xi,

where $\widetilde \xi=\xi-\operatorname{Proj}(\xi\mid 1,x)$. The regression slope of residualized $y$ on residualized price is

\frac{\operatorname{Cov}(\widetilde p,\widetilde y)} {\operatorname{Var}(\widetilde p)} = -\alpha+ \frac{\operatorname{Cov}(\widetilde p,\widetilde \xi)} {\operatorname{Var}(\widetilde p)} = -\alpha+ \frac{\operatorname{Cov}(\widetilde p,\xi)} {\operatorname{Var}(\widetilde p)}.

The last equality holds because $\widetilde p$ is uncorrelated with the projection of $\xi$ on the included controls. Thus the formula concerns the price–quality relationship after controlling for observed characteristics. If this conditional covariance is positive, the regression slope is greater (less negative) than the demand parameter $-\alpha$. The coefficient can even become positive if the endogeneity component is large enough.

Worked Example: A Two-Product Market with Endogenous Prices

Consider repeated hypothetical two-product markets. The outside option has mean utility zero, the price coefficient is $\alpha=1$, and there are no varying observed characteristics. Product 2 has mean utility $-2$ in both markets. Product 1 has unobserved quality $\xi_1$, and suppose its equilibrium prices in the two markets follow the relationship $p_1=2+2\xi_1$. This relationship represents a hypothetical pricing response to quality; it is not a universal pricing rule.

In the first market, let $\xi_1=0$, so $p_1=2$. Product 1’s mean utility is $\delta_1=-p_1+\xi_1=-2$. In the second market, let $\xi_1=1$, so $p_1=4$, and $\delta_1=-4+1=-3$. The inverted demand outcomes are $y_1=-2$ and $y_1=-3$, respectively, because $y_1=\delta_1$.

The shares are consistent with this inversion. In the first market, the denominator is $1+e^{-2}+e^{-2}=1.2707$, so $s_{01}=1/1.2707=0.7870$ and $s_{11}=e^{-2}/1.2707=0.1065$. In the second market, the denominator is $1+e^{-3}+e^{-2}=1.1851$, so $s_{01}=0.8438$ and $s_{11}=e^{-3}/1.1851=0.0420$. In each case, $\log(s_{11}/s_{01})$ equals the stated mean utility, to rounding.

The price-only regression slope across these two equally weighted observations is

\frac{-3-(-2)}{4-2}=-0.500.

The structural price coefficient is $-\alpha=-1$, so the regression slope is biased upward by $0.500$. The covariance formula gives the same result: the equally weighted population covariance of $p_1=(2,4)$ and $\xi_1=(0,1)$ is $0.500$, and the population variance of price is $1.000$. Hence

-\alpha+\frac{\operatorname{Cov}(p_1,\xi_1)} {\operatorname{Var}(p_1)} =-1+\frac{0.500}{1.000} =-0.500.

Price rises with quality, so the observed demand index falls less sharply with price than it would if quality were held fixed. The calculation illustrates the bias mechanism, not a claim that every equilibrium must produce this particular price–quality relationship.

Why Controls Change the Calculation

A price-only slope combines the price effect, the association between price and observed characteristics, and the association between price and unobserved quality. The second component is not the endogeneity term: it may arise simply because more desirable observed characteristics command higher prices. Including those characteristics changes the relevant comparison to price variation among products with the same fitted observed attributes.

Worked Example: Accounting for a Varying Observed Characteristic

Suppose two hypothetical observations have $p=(1,3)$, $x=p$, $\beta=1$, $\alpha=2$, and $\xi=0$ in both observations. The demand index is $y=x\beta-\alpha p+\xi=p-2p=-p$, so the observed pairs are $(p,y)=(1,-1)$ and $(3,-3)$. The simple-regression slope is $(-3+1)/(3-1)=-1$.

The full covariance formula explains why:

-\alpha+ \frac{\operatorname{Cov}(p,x\beta)}{\operatorname{Var}(p)} +\frac{\operatorname{Cov}(p,\xi)}{\operatorname{Var}(p)} = -2+\frac{\operatorname{Var}(p)}{\operatorname{Var}(p)}+0 =-1.

Here price and the observed characteristic move together perfectly, while quality is constant. The price-only slope is not the structural coefficient $-2$. Moreover, because $x=p$ in this example, there is no residual price variation after controlling for $x$; the conditional price coefficient cannot be separately identified from these observations. This example shows why the varying $x'\beta$ term must be included in a price-only slope calculation and why residual variation matters when controls are used.

Interpretation and a Common Pitfall

The regression coefficient is a causal price effect only under an appropriate exogeneity condition: after conditioning on the included observed characteristics, the remaining price variation must be uncorrelated with unobserved demand quality. If that condition fails, the coefficient combines the structural price response with the price–quality association. A positive conditional covariance typically makes a negative demand coefficient less negative; it does not imply that consumers prefer higher prices.

Wrong reasoning: “The demand equation contains observed characteristics, so the price coefficient is automatically free of omitted quality.”
Correction: Controls remove the variation explained by those observed characteristics. They do not remove $\xi$. The relevant condition is that residualized price be uncorrelated with unobserved quality.
Wrong reasoning: “The price-only slope is always $-\alpha+\operatorname{Cov}(p,\xi)/\operatorname{Var}(p)$.”
Correction: That expression applies only when $x'\beta$ is constant or uncorrelated with price. Otherwise include $\operatorname{Cov}(p,x'\beta)/\operatorname{Var}(p)$, or use the residualized formula for a regression that includes controls.

The practical importance is substantial: demand estimates inform predictions about how consumers respond to price changes. If the estimated price coefficient partly captures unobserved quality, those predictions can be misleading. Instrumental-variable methods address this endogeneity problem by seeking price variation unrelated to unobserved demand quality; choosing and evaluating particular instruments requires additional assumptions and analysis.

Key takeaway: Logit share inversion yields $y=x'\beta-\alpha p+\xi$, not an equation in which unobserved quality disappears. Equilibrium prices can respond to that quality. With controls, the price-slope discrepancy is governed by the covariance between residualized price and unobserved quality, divided by the residualized price variance.

Check Your Understanding

Answer each question using the demand equation and the covariance formulas in this tutorial.

In the inverted logit equation, what does $\xi_{jt}$ represent, and why can share inversion not eliminate it?
State the condition under which the price coefficient in a regression with observed-characteristic controls equals the structural coefficient $-\alpha$.
When is the price-only slope formula $-\alpha+\operatorname{Cov}(p,\xi)/\operatorname{Var}(p)$ valid without an observed-characteristics term?
If residualized price and unobserved quality have positive covariance, how does the regression slope compare with $-\alpha$?
In the two-product example, identify the structural price coefficient, the price-only slope, and the upward bias.