A simple test for distinguishing between internal reference price theories

Abstract

A large literature demonstrates the empirical importance of internal reference price effects. There are several theories regarding how and why these effects arise. We offer a simple test that distinguishes between the two leading theories based on economically rational behavior: price as a signal of quality and price as a predictor of future prices. Our test builds on differences in how past consumer purchases interact with internal reference prices. We first validate the reliability of our test by applying it to synthetic data. We then apply our test to purchases of ketchup and diapers and find: (1) quality signaling is the dominant mechanism behind reference price effects in both categories; (2) consistent with the quality-signaling theory, reference price effects diminish as various measures of consumer experience increase; but (3) in both categories there are many individuals for whom price-prediction effects dominate quality-signaling effects.

Appendix: additional data preparation

Diaper price index

We aggregated each brand’s diaper sizes into five categories: (a) 0 and 1; (b) 2; (c) 3; (d) 4; and (e) 5 and 6. We lumped sizes 0 and 1, as well as 5 and 6, together because the size differences between 0 and 1 and 5 and 6 are very small and there were fewer purchases of these sizes individually compared to other sizes. We then created a distinct price index for each: brand × size category × week × store. Where data were available, we took a quantity-weighted average of the prices paid at that store that week for that brand and size category. We calculated the sample-period weights by city. Where no data were available for a given store/week for that diaper size/brand, we constructed a price index based on the relevant data for all stores in the same city that week. If no data were available at any store in a given week, then we used the process described earlier in the text to interpolate data from other weeks:

Household-level inventory variable

Consider a household with purchase dates t _1, t ₂,...,t _N , and let ${x_{{t_1}}},{x_{{t_2}}},...,{x_t}_N$ denote the corresponding purchase quantities on those dates. Notice that the purchase quantities are 0 in weeks that are in our sample period but not on these dates; that is, X _t=0 for $t \notin \left\{ {{t_1},{t_2},...,{t_N}} \right\}.$

We first construct a consumption rate variable:

$$ r \equiv \frac{{\,\sum\limits_{j = 1}^{N - 1} {{x_{{t_j}}}} }}{{{t_N} - {t_1}}} $$

Notice that the summation of the quantities does not include the final purchase.

We next divide households into two categories. The first category comprises households that appear to make the first purchases of their lifetimes during the sample period. These households are identified as those that took a “long time” before making a purchase in the sample period. Specifically, a household’s purchase at time t ₁ is considered to be its first ever if ${t_1} > {\max_{j = 1,2...N - 1}}\left\{ {{t_{j + 1}} - {t_j}} \right\}$. For these households, we assume that they were not active purchasers until date t ₁. That is, we exclude them from the sample for t < t ₁. From date t ₁ onward, we define their inventory at date t as follows:

$$ \begin{array}{*{20}{c}} {I{V_t}_{_1} = 0} \hfill \\{I{V_t} = \max \left\{ {I{V_{t - 1}} + {x_{t - 1}} - r,\;0} \right\}\;{\hbox{for}}\;t > {t_1}.} \hfill \\\end{array} $$

The second category of households comprises those that made initial purchases prior to the start of our sample period. These are households for whom ${t_1} \leqslant {\max_{j = 1,2...N - 1}}\left\{ {{t_{j + 1}} - {t_j}} \right\}$. For these households, we assume that

$$ \begin{array}{*{20}{c}} {I{V_t} = r\left( {{t_1} - t} \right)\;{\hbox{for}}\;t < {t_1}} \hfill \\{I{V_{{t_1}}} = 0} \hfill \\{I{V_t} = \max \left\{ {{I_{t - 1}} + {x_{t - 1}} - r,0} \right\}\;{\hbox{for}}\;t > {t_1}.} \hfill \\\end{array} $$