Welfare costs of shopping trips

Abstract

Using data on the number of visitors at the store level, this paper attempts to measure the welfare costs of traditional shopping trips for the U.S. census blocks. The investigation is based on an economic model, where individuals living in census blocks decide on which store to shop from based on the shopping-trip costs and idiosyncratic benefits. The welfare gains from removing shopping-trip costs in percentage terms are shown to depend on the weighted average of log distance measures between shopping stores and census blocks. The results show that the welfare gains from removing shopping-trip costs is about 4% for the average census block, with a range between 0.021 and 18% across census blocks that is further connected to their demographic or socioeconomic characteristics, especially their population density. Several practical policy implications follow regarding how shopping-trip costs can be reduced to achieve higher welfare gains.

Appendix

This section contains the technical derivations of certain results in the main text.

1.1 A1 Derivation of the expected utility

The number of individuals living in each location is fixed. Each individual living in any location chooses the shopping store that offers the maximum utility. Since the maximum of a sequence of Fréchet distributed random variables is itself Fréchet distributed, the distribution of utility for individuals living in census block b across all possible shopping stores is as follows:

$$\begin{aligned} 1-G_{b}\left( u\right) =1- \prod \limits _{s} e^{-\Psi _{bs}u^{-\theta }} \end{aligned}$$

(25)

where the left-hand side is the probability that an individual of census block b gets a utility higher than u, and the right hand side is one minus the probability that the individual of census block b has utility less than u for all possible shopping locations. It is implied that:

$$\begin{aligned} G_{b}\left( u\right) =e^{-\Psi _{b}u^{-\theta }} \end{aligned}$$

(26)

where

$$\begin{aligned} \Psi _{b}=\sum _{s}\Psi _{bs} \end{aligned}$$

(27)

Given this Fréchet distribution for utility in census block b, the expected utility ${\overline{U}}_{b}$ in census block b is implied as:

$$\begin{aligned} {\overline{U}}_{b}= {\displaystyle \int \limits _{0}^{\infty }} \theta \Psi _{b}u^{-\theta }e^{-\Psi _{b}u^{-\theta }}dy \end{aligned}$$

(28)

Defining the following change of variables:

$$\begin{aligned} y=\Psi _{b}u^{-\theta } \end{aligned}$$

(29)

and

$$\begin{aligned} dy=\theta e^{-\Psi _{b}u^{-\left( \theta +1\right) }} \end{aligned}$$

(30)

the expected utility can be rewritten as follows:

$$\begin{aligned} {\overline{U}}_{b}= {\int \limits _{0}^{\infty }} \left( \Psi _{b}\right) ^{\frac{1}{\theta }}y^{-\frac{1}{\theta }}e^{-y}du \end{aligned}$$

(31)

which is:

$$\begin{aligned} {\overline{U}}_{i}=\left( \Psi _{b}\right) ^{\frac{1}{\theta }}\Gamma \left( 1-\frac{1}{\theta }\right) \end{aligned}$$

(32)

Using $\Psi _{b}=\sum _{s}\Psi _{bs}$, it is finally implied that:

$$\begin{aligned} {\overline{U}}_{b}=\left( \sum _{s}\Psi _{bs}\right) ^{\frac{1}{\theta }} \Gamma \left( 1-\frac{1}{\theta }\right) \end{aligned}$$

(33)

which is the expression for expected utility in the main text.

1.2 A2 Derivation of shopping store probabilities

The probability that an individual of census block b chooses to shop at store s out of all possible shopping locations (represented by r) is as follows:

$$\begin{aligned} \lambda _{bs}&=\Pr \left[ u_{bs}\ge \max \left\{ u_{br}\right\} ;\forall r\right] \nonumber \\&= {\int \limits _{0}^{\infty }} {\prod \limits _{r\ne s}} G_{br}\left( u\right) dG_{bs}\left( u\right) \nonumber \\&= {\int \limits _{0}^{\infty }} {\prod \limits _{r}} \theta \Psi _{bs}u^{-\left( \theta +1\right) }e^{-\Psi _{br}u^{-\theta } }du\nonumber \\&= {\int \limits _{0}^{\infty }} \theta \Psi _{bs}u^{-\left( \theta +1\right) }e^{-\Psi _{b}u^{-\theta } }du \end{aligned}$$

(34)

Since we have:

$$\begin{aligned} \frac{d}{du}\left[ -\frac{1}{\Psi _{b}}e^{-\Psi _{b}u^{-\theta }}\right] =\theta u^{-\left( \theta +1\right) }e^{-\Psi _{b}u^{-\theta }} \end{aligned}$$

(35)

it is implied that:

$$\begin{aligned} \lambda _{bs}=\frac{\Psi _{bs}}{\Psi _{b}}=\frac{A_{bs}\left( P_{s}\tau _{bs}\right) ^{-\theta }\left( W_{b}\right) ^{\theta }}{\sum _{r}A_{br}\left( P_{r}\tau _{br}\right) ^{-\theta }\left( W_{b}\right) ^{\theta }}=\frac{A_{bs}\left( P_{s}\tau _{bs}\right) ^{-\theta }}{\sum _{r}A_{br}\left( P_{r}\tau _{br}\right) ^{-\theta }} \end{aligned}$$

(36)

where the last expression, which is the same as in in the main text, has been obtained after $\left( W_{b}\right) ^{\theta }$’s have been effectively eliminated.

1.3 A3 Derivation of welfare changes

As shown in the main text, the welfare (expected utility) ${\overline{U}}_{b}$ of an individual living in census block b is given by the following expression:

$$\begin{aligned} {\overline{U}}_{b}=\left( \sum _{s}A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }\right) ^{\frac{1}{\theta }}\Gamma \left( 1-\frac{1}{\theta }\right) \end{aligned}$$

(37)

Taking the total derivative of this expression can be achieved as follows:

$$\begin{aligned} d{\overline{U}}_{b}=\sum _{s}\frac{\partial {\overline{U}}_{b}}{\partial A_{bs} }dA_{bs}+\sum _{s}\frac{\partial {\overline{U}}_{b}}{\partial Z_{s}}dZ_{s} +\sum _{s}\frac{\partial {\overline{U}}_{b}}{\partial \tau _{bs}}d\tau _{bs} \end{aligned}$$

(38)

When average benefits $A_{bs}$’s and productivity measures $Z_{s}$’s are unchanged (i.e., $dA_{bs}=dZ_{s}=0$), this expression can be simplified as follows:

$$\begin{aligned} d{\overline{U}}_{b}=\sum _{s}\frac{\partial {\overline{U}}_{b}}{\partial \tau _{bs} }d\tau _{bs} \end{aligned}$$

(39)

which can be rewritten as follows:

$$\begin{aligned} d{\overline{U}}_{b}&=\sum _{s}\frac{1}{\theta }\frac{{\overline{U}}_{b}\left( \frac{-\theta }{\tau _{bs}}A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }\right) }{\left( \sum _{s}A_{bs}\left( \frac{\tau _{bs}}{Z_{s} }\right) ^{-\theta }\right) }d\tau _{bs}\nonumber \\&=-\sum _{s}\frac{{\overline{U}}_{b}\left( A_{bs}\left( \frac{\tau _{bs} }{Z_{s}}\right) ^{-\theta }\right) }{\left( \sum _{s}A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }\right) }\frac{d\tau _{bs}}{\tau _{bs} } \end{aligned}$$

(40)

Taking ${\overline{U}}_{b}$ to the left-hand side results in:

$$\begin{aligned} \frac{d{\overline{U}}_{b}}{{\overline{U}}_{b}}=-\sum _{s}\frac{\left( A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }\right) }{\left( \sum _{s}A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }\right) }\frac{d\tau _{bs}}{\tau _{bs}} \end{aligned}$$

(41)

which can be rewritten as follows:

$$\begin{aligned} \frac{d{\overline{U}}_{b}}{{\overline{U}}_{b}}=-\sum _{s}\lambda _{bs}\frac{d\tau _{bs}}{\tau _{bs}} \end{aligned}$$

(42)

as the expenditure share of $\lambda _{bs}=\frac{A_{bs}\left( P_{s}\tau _{bs}\right) ^{-\theta }}{\sum _{r}A_{br}\left( P_{r}\tau _{br}\right) ^{-\theta }}$ in the main text can be rewritten as $\lambda _{bs}=\frac{A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }}{\sum _{s} A_{bs}\left( \frac{\tau _{bs}}{Z_{s}}\right) ^{-\theta }}$ by using $P_{s}=\frac{W}{Z_{s}}$ in the main text. Finally, using $\frac{dx}{x}\approx d\log x$, the following expression can be obtained:

$$\begin{aligned} d\log {\overline{U}}_{b}=-\sum _{s}\lambda _{bs}d\log \tau _{bs} \end{aligned}$$

(43)

which represents welfare changes as in the main text.