Quality bias in price elasticity

Abstract

This article examines the potential quality bias in price elasticities in cross-sectional demand analysis and develops a framework that can be used to avoid this problem. Both analytical and empirical results indicate that ignoring quality adjustment in either prices or quantities can cause biased price elasticities, which may lead to erroneous marketing decisions and policy implications.

Acknowledgements

The author is grateful to Todd M. Schmit for research assistance, to Brian W. Gould for providing data, and Tracy Boyer for comments.

Notes

¹ Hick's composite commodity theorem states that if prices of individual goods move in parallel, then the corresponding group of goods can be treated as a single commodity (Deaton and Muellbauer, 1993, p. 121).

² The denominator of Equation 5 is derived by differentiating Equation 4 with respect to lnP_c , and ∂lnL_c /∂lnP_c is derived from the demand function for the elementary good x_i . By assuming weak separability, the vector of elementary good demand, x, can be written as a function of total expenditures on composite commodity c and the price vector of elementary goods that comprise the composite commodity c. Since the demand functions of these elementary goods are homogeneous of degree zero in total expenditures and prices, the vector of individual goods’ demand x can be represented by:

Then, from Equations 2 and 3, L_c is only a function of x at a given P* and in turn, is a function of E_c/P_c . That is L_c  = g(E_c/P_c ). Differentiating this relationship with respect to P_c and multiplying both sides by P_c/L_c gives:

Also, taking the natural logarithm of the relationship, E_c  = P_c  · q_c  · L_c (where Q_c  = q_c  · L_c ) and differentiating with respect to lnP_c produces:

Combining two equations above with ∂lnL_c /∂ln(E_c /P_c ) = ϕ/(ϕ +  _M ), we have:

The term ∂lnL_c /∂ln(E_c/P_c ) is derived by differentiating L_c  = g(E_c (M)/P_c ) with respect to the total income, M. A complete derivation is available from the author upon request.

³ Differentiating Equation 4 with respect to lnM results in:

With the assumption of ∂lnP_c /∂lnM = 0, one now has ϕ = dlnV_c /dlnM. The assumption above is well acceptable because price and income variables are orthogonal in the estimation of most demand functions.

⁴ Regression results of Equations 13 and 14 and detailed procedure to obtain price elasticities have been suppressed due to the space limitation. However, this information is available from the author upon request.