The Biases in Applying Static Demand Models Under Dynamic Demand

Abstract

This article investigates the mechanisms that underlie the biases in price elasticities of demand in applying static demand models under dynamic demand, which have been pointed out by previous empirical studies. It studies three sources of biases: disregard of state variables (affecting short-run elasticity); inconsistent utility parameter estimates; and changing expectations of consumers (affecting long-run elasticity). Disregard of state variables, such as durable product holdings, which is not negligible but not paid much attention to in the literature, leads to an overestimate of short-run own elasticities. Inconsistent utility parameter estimates arises due to the failure to account for consumers’ future expectations and unobserved state variables. Changing expectations of consumers are not explicitly specified in the static model, and this also leads to biased results when applying static models. Regarding the magnitude of the biases, the first and the third sources of biases might induce large biases in price elasticities, especially when the focus is on the large conditional choice probability products. Possible remedies for the use of static demand models are also discussed.

1 Introduction

Many markets are characterized by dynamic demand structures, wherein future demand has connections with current demand. Examples of goods with dynamic demand include durable goods, storable goods, goods with switching costs, and experience goods. Given the significant presence of these goods in many industries, empirical analysis of these goods is essential for a deeper understanding of these markets.

So far, to analyze these goods, many studies have applied static demand models, in which the connections between current and future demand are not explicitly specified. For example, Berry et al. (1995) (henceforth BLP) applied a static demand model to automobiles—a typical example of durables. Nevertheless, previous empirical studies that specify dynamic demand models have found that applying static demand models yields biased estimates of price elasticities of demand, as summarized in Table 1. For example, Hendel and Nevo (2006), who studied storables, showed that static demand estimates overestimate own elasticities by 30%, and underestimate cross elasticities by up to a factor of five. Gowrisankaran and Rysman (2012), who studied new durables, showed that applying a BLP-style static demand model yields price elasticities that are very close to zero.

This paper investigates why the application of static demand models yields biased price elasticities when demand is dynamic. This understanding is important for the appreciation of static models. Though the estimation and simulation of dynamic demand models are getting easier due to the accumulated knowledge and advancement of computational power, analysis with dynamic models requires greater effort. In addition, for researchers who are interested in the supply side, the introduction of dynamic demand structures might not be attractive, since they need to consider firms’ dynamic price setting behaviors so as to make the models consistent.¹ If we can expect that the biases are small for the goods of interest, this can justify the studies that use static demand models. If not, researchers should be careful with respect to the use of these static models.

In this study, we investigate three sources of biases: disregard of state variables, inconsistent utility parameter estimates; and changing expectations of consumers. We consider two elasticities: short-run elasticity (the response of current period’s demand to the current period’s temporary price change, given consumers’ fixed future expectations), and long-run elasticity (the response of the current period’s demand to the permanent or long-term price change, allowing the changing expectations of consumers). The main results are as follows:

• Disregard of state variables: Applying a static model that ignores states—e.g., durable goods holdings—leads to the overestimation of short-run own price elasticity.

• Inconsistent utility parameter estimates: Some remedies—such as introducing time and group dummy variables—mitigate the biases under some conditions.

• Changing expectations of consumers: Show the sign of the differences between short-run and long-run elasticities.

• The first and third biases are large when the focus is on the product with large conditional choice probabilities (CCPs). Hence, researchers should be careful with respect to the use of static demand models when the focus is on the products with large CCPs or market shares, which are the main focus of competition policies.

Note that this paper does not cover all of the dynamic components. There are many types of dynamics, including consumer learning, that are not discussed in this paper; enumerating all of the possible cases would take so much space. Hence, this is beyond the scope of this paper. Nevertheless, the framework or insights I provide in this paper are fundamental, and would be helpful for assessing the biases in applying static models that not directly discussed in this paper—though slight modifications might be necessary.

Table 1 Literature on the biases in static demand models

The rest of this article is organized as follows: In Section 2, I describe the relationship between this study and previous studies. In Section 3, we develop a dynamic demand model and discuss the three biases. In Section 4, we extend the base model and discuss storable goods. In Section 5 we discuss how the results in this article provide insight into the previous studies’ findings summarized in Table 1. Section 6 concludes. All the proofs of the statements are shown in Appendix A.

2 Literature

First, this study closely relates to the recent literature on dynamic demand and dynamic discrete choice (DDC) models. The literature on DDC models has investigated the identification of dynamic structural parameters, including the discount factor [e.g., Magnac and Thesmar (2002), Abbring and Daljord (2020)]. Also, numerous empirical studies have found the biases in applying static models as is summarized in Table 1.

However, it is not clear whether these results—such as the sign and magnitude of the biases—are the same in other markets. In fact, some of the studies that analyze the same market showed contradicting results. In this study, we attempt to explain the mechanism that underlies the results. This study contributes to the literature by stressing the importance and investigating the magnitude of the biases that are due to disregarding states, to which previous studies have paid less attention.

Gowrisankaran and Rysman (2020) also developed a general model of dynamic demand, and the framework in this article is similar to theirs. Their focus is surveying previous studies and clarifying the implicit assumptions that are imposed in previous empirical studies, which alleviate the computational burden of solving dynamic models. In contrast to their article, we focus on the difference between dynamic and static demand models.

Fukasawa (2022) also investigates dynamic demand using a similar framework, but the focus differs: Fukasawa (2022) focuses on the supply side, and examines the cases in which a firm’s dynamic pricing behavior can be approximated as static under dynamic demand. In contrast, the current paper focuses on the demand side, and investigates whether applying a static demand model yields a good approximation of the true dynamic demand model. Note that Fukasawa (2022) shows that a firm’s dynamic pricing behavior under dynamic demand can be approximated as static when CCPs of the firm’s product are small for all consumer types and states. The condition is the same as the one for small biases in price elasticities that are shown in the current article, and we can see that the condition is important for the appreciation of static models under dynamic demand.

This article also contributes to many studies that apply static demand models to goods with dynamic demand. To justify the use of static models, several articles have discussed some remedies. For instance, Goldberg and Verboven (2001), who studied automobiles, argued that adding time and brand dummy variables mitigates the problem. Nevertheless, the discussion has been limited to informal approaches. In this study, I formally show the extent to which these remedies alleviate the issues.

3 Model

3.1 Model Setup

3.1.1 State Variables

Let $x_{t}\in X_{t}$ be a consumer’s individual-level states at the beginning of time t. $X_{t}$ denotes the set of individual-level states. For instance, $x_{t}$ indicates whether the consumer already owns durable products, in the case of durables. In this study, we assume that $X_{t}$ is a discrete set, for notational simplicity. Nevertheless, we can easily extend this to the case where $X_{t}$ is a continuous set. Let $Pr_{lt}(x_{t})$ be the ratio of type l consumers at state $x_{t}$ among type l consumers at time t. Note that $\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})=1$ holds by definition.

3.1.2 Utility Function

In this study, we assume that each consumer purchases at most one product in each period. Let $\mathcal {J}_{t}$ denotes the set of products that are available at time t. “0” represents the outside option: the option of not buying any product. Further, let $a_{t}\in \mathcal {J}_{t}\cup \{0\}$ be the choice of a consumer at time t. $a_{t}=j$ means that the consumer purchases product j, and $a_{t}=0$ means that the consumer does not purchase any product.

Let the expected discounted utility of type-l consumer i—whose state is $x_{t}$ and choice is $a_{t}$, given product prices $p_{t}\equiv (p_{jt})_{j\in \mathcal {J}_{t}}$, continuation values $g_{t}\equiv (g_{ljt}(x_{t}))_{l,x_{t},j\in \mathcal {J}_{t}\cup \{0\}}$, market information $\Omega _{t}$ (including observed and unobserved product characteristics $(X_{jt})_{j\in \mathcal {J}_{t}},(\xi _{jt})_{j\in \mathcal {J}_{t}}$ but excluding product prices), and random individual-level random preference shock $\epsilon _{it}\equiv (\epsilon _{ijt})_{j\in \mathcal {J}_{t}\cup \{0\}}$—be $v_{ilt}\left( x_{t},p_{t},g_{t},\Omega _{t},\epsilon _{t},a_{t}\right)$. Here, ${{p_{jt}}}$ denotes product j’s price at time t. Type l consumer i maximizes utility $v_{ilt}(x_{t},p_{t},g_{t},\Omega _{t},\epsilon _{t},a_{t})$ with respect to $a_{t}\in \mathcal {J}_{t}\cup \{0\}.$

Utility $v_{ilt}(x_{t},p_{t},g_{t},\Omega _{t},\epsilon _{t},a_{t})$ is in the following form:

$$\begin{aligned} v_{ilt}\left( x_{t},p_{t},g_{t},\Omega _{t},\epsilon _{t},a_{t}\right) ={\left\{ \begin{array}{ll} -\alpha _{l}p_{jt}+f_{ljt}(\Omega _{t})+\phi _{ljt}(x_{t},\Omega _{t})+\beta _{C}g_{ljt}(x_{t},\Omega _{t})+\epsilon _{ijt} &{} \text {if }a_{t}=j\\ f_{l0t}(x_{t},\Omega _{t})+\beta _{C}g_{l0t}(x_{t},\Omega _{t})+\epsilon _{i0t} &{} \text {if}\ a_{t}=0 \end{array}\right. }, \end{aligned}$$

(1)

where $f_{ljt}$ denotes the flow utility that type l consumers gain when they buy product j; that utility depends on the observed and unobserved product characteristics $X_{jt}$ and $\xi _{jt}$. $f_{l0t}(x_{t})$ denotes the flow utility that type l consumers at state $x_{t}$ gain when not buying anything. For instance, it represents the utility from continuing the use of previous durable product $x_{t}$ in the case of durables. $\phi _{ljt}(x_{t},\Omega _{t})$ denotes the flow utility for consumers at state $x_{t}$ when they buy product j, which is not captured in $f_{ljt}$. For instance, it represents the resale value of old durable products $x_{t}$. $(\epsilon _{ijt})_{j\in \mathcal {J}_{t}\cup \{0\}}$ denotes the individual-level random preference shock; we assume that these shocks follow an i.i.d. mean-zero type-I extreme value distribution. $\beta _{C}$ represents the consumers’ discount factor, and E represents the expectation operator. $\alpha _{l}$ represents type l consumers’ marginal utility of money; we assume $\alpha _{l}>0$ holds for all l.

In addition, the continuation value $g_{ljt}(x_{t})$ is in the following form:

$$\begin{aligned} g_{ljt}(x_{t},\Omega _{t})=E\left[ V_{lt+1}^{C}(x_{t+1},p_{t+1},g_{t+1},\Omega _{t+1})|x_{t},p_{t},\Omega _{t},a_{t}=j\right] . \end{aligned}$$

(2)

Note that $g_{t}$ depends on the expected path of future product prices $\{p_{t+\tau }\}_{\tau \ge 1}$.

$V_{lt}^{C}(x_{t},p_{t},g_{t},\Omega _{t})$ is the value function of type l consumers given states $x_{t}$ at time t given product prices $p_{t}$ and continuation values $g_{t}$; and this is defined as follows:

$$\begin{aligned} V_{lt}^{C}(x_{t},p_{t},g_{t},\Omega _{t})\equiv E_{\epsilon _{it}}\left[ \max _{a_{t}\in \mathcal {J}_{t}\cup \{0\}}v_{ilt}\left( x_{t},p_{t},g_{t},\Omega _{t},\epsilon _{it},a_{t}\right) \right] , \end{aligned}$$

where $E_{\epsilon _{it}}$ denotes the expectation operator with respect to random i.i.d. shocks $\epsilon _{it}\equiv \{\epsilon _{ijt}\}_{j\in \mathcal {J}_{t}\cup \{0\}}$. Under the assumption that $\epsilon _{ijt}$ follows i.i.d. type-I extreme value distribution, the following formula holds:

$$\begin{aligned} V_{lt}^{C}(x_{t},p_{t},g_{t},\Omega _{t})=\log \left( \sum _{j\in \mathcal {J}_{t}}\exp \left( -\alpha _{l}p_{jt}+f_{ljt}(\Omega _{t})+\phi _{ljt}(x_{t},\Omega _{t})+\beta _{C}g_{ljt}(x_{t},\Omega _{t})\right) +\exp \left( f_{l0t}(x_{t},\Omega _{t})+\beta _{C}g_{l0t}(x_{t},\Omega _{t})\right) \right) . \end{aligned}$$

(3)

Hereinafter, we omit the term $\Omega _{t}$ to simplify the notation.

3.1.3 Choice Probability

The CCP that type l consumer buys product j at time t conditional on being at states $x_{t}$ is:

$$\begin{aligned} s_{ljt}^{(ccp)}(x_{t},p_{t},g_{t})= & {} Pr\left( v_{ilt}(x_{t},p_{t},g_{t}, \epsilon _{it},a_{t}=j)>v_{ilt}(x_{t},p_{t},g_{t},\epsilon _{it},a_{t}=k)\ \forall k\in \mathcal {J}_{t}\cup \{0\}-\{j\}\right) \nonumber \\= & {} \frac{\exp \left( -\alpha _{l}p_{jt}+f_{ljt}+\phi _{ljt}(x_{t})+\beta _{C}g_{ljt}(x_{t})\right) }{\exp \left( V_{lt}^{C}(x_{t},p_{t},g_{t})\right) }. \end{aligned}$$

(4)

The CCP that type l consumer does not buy any product at time t conditional on being at states $x_{t}$ is:

$$\begin{aligned} s_{l0t}^{(ccp)}(x_{t},p_{t},g_{t})=\frac{\exp \left( f_{l0t}(x_{t})+\beta _{C}g_{l0t}(x_{t})\right) }{\exp \left( V_{lt}^{C}(x_{t},p_{t},g_{t})\right) }. \end{aligned}$$

(5)

The probability that type l consumer buys product j at time t is:

$$\begin{aligned} s_{ljt}(p_{t},g_{t})=\sum _{x_{t}\in X_{t}}s_{ljt}^{(ccp)}(x_{t},p_{t},g_{t})\cdot Pr_{lt}(x_{t}). \end{aligned}$$

(6)

The probability that type l consumer does not buy any product at time t is:

$$\begin{aligned} s_{l0t}(p_{t},g_{t})=\sum _{x_{t}\in X_{t}}s_{l0t}^{(ccp)}(x_{t},p_{t},g_{t})\cdot Pr_{lt}(x_{t}). \end{aligned}$$

(7)

The market share of product j at time t—the fraction of consumers who buy product j at time t—is:

$$\begin{aligned} s_{jt}(p_{t},g_{t})=\int s_{ljt}(p_{t},g_{t})dP(l), \end{aligned}$$

(8)

where dP(l) denotes the measure of type l consumers.

The fraction of consumers not purchasing any product at time t is:

$$\begin{aligned} s_{0t}(p_{t},g_{t})=\int s_{l0t}(p_{t},g_{t})dP(l). \end{aligned}$$

(9)

3.1.4 State Transition

The transition probability of consumer-level states $x_{t}$ is given by $\psi (x_{t+1}|x_{t},a_{t})$. It depends on the previous period’s states $x_{t}$ and choices $a_{t}$. For instance, in the case of durables that depreciate over time, $x_{t+1}$, product holding at time $t+1$, depends on the previous state $x_{t}$ and the product choice $a_{t}$ at time t. The transition process depends on the depreciation rate of the durables.

Note that $Pr_{lt}(x_{t})$ satisfies the following state transition formula:

$$\begin{aligned} Pr_{lt+1}(x_{t+1})=\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})\cdot \sum _{j\in \mathcal {J}_{t}\cup \{0\}}s_{ljt}^{(ccp)}(x_{t},p_{t},g_{t})\cdot \psi (x_{t+1}|x_{t},a_{t}=j). \end{aligned}$$

(10)

The dynamic demand system is composed of equations (2)-(10).

3.1.5 Price Elasticity of Demand

In this study, short-run price elasticity is defined as the elasticity of the current period’s demand in response to the current period’s temporary price change, given fixed consumers’ future expectations. Long-run price elasticity is defined as the elasticity of the current period’s demand change in response to the permanent or long-term price change, allowing changing expectations of consumers.² The difference between long-run and short-run elasticities is whether the continuation value $g_{t}$ changes or not.

Under the specifications above, short-run own price elasticity of product j at time t given product prices $p_{t}^{0}=(p_{jt}^{0})_{j\in \mathcal {J}_{t}}$ and continuation values $g_{t}^{0}$ is:

$$\begin{aligned} \eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})\equiv -\frac{\partial s_{jt}(p_{t}^{0},g_{t}^{0})}{\partial p_{jt}}\frac{p_{jt}^{0}}{s_{jt}^{0}}=\left[ \int \alpha _{l}\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{ljt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})(1-s_{ljt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0}))dP(l)\right] \frac{p_{jt}^{0}}{s_{jt}^{0}}. \end{aligned}$$

(11)

Short-run cross price elasticity of product k with respect to product $j(\ne k)$ at time t given product prices $p_{t}^{0}$ and continuation values $g_{t}^{0}$ is:

$$\begin{aligned} \eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\equiv \frac{\partial s_{kt}(p_{t}^{0},g_{t}^{0})}{\partial p_{jt}}\frac{p_{jt}^{0}}{s_{kt}^{0}}=\left[ \int \alpha _{l}\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{ljt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})s_{lkt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})dP(l)\right] \frac{p_{jt}^{0}}{s_{kt}^{0}}. \end{aligned}$$

(12)

Here, we define the term $s_{jt}^{0}\equiv s_{jt}(p_{t}^{0},g_{t}^{0})$.

3.2 First Bias: Disregard of State Variables and Short-run Price Elasticity

In this subsection, we focus on the bias that is associated with the disregard of state variables. Here, we consider the case where no persistent consumer heterogeneity exists in the dynamic model so as to clarify the point. First, we can derive a static representation of the dynamic demand model with no random coefficients with the use of equations (4) - (7):³

$$\begin{aligned} s_{jt}(p_{t},g_{t})= & {} \frac{\exp (-\alpha p_{jt}+f_{jt}+\widehat{c_{jt}}(p_{t},g_{t}))}{1+\sum _{k\in \mathcal {J}_{t}}\exp (-\alpha p_{kt}+f_{kt}+\widehat{c_{kt}}(p_{t},g_{t}))};\text { and} \end{aligned}$$

(13)

$$\begin{aligned} s_{0t}(p_{t},g_{t})= & {} \frac{1}{1+\sum _{k\in \mathcal {J}_{t}}\exp (-\alpha p_{kt}+f_{kt}+\widehat{c_{kt}}(p_{t},g_{t}))}, \end{aligned}$$

(14)

where

$$\begin{aligned} \widehat{c_{jt}}(p_{t},g_{t})\equiv \log \left( \frac{\sum _{x_{t}\in X_{t}}\frac{\exp \left( \phi _{jt}(x_{t})+\beta _{C}g_{jt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }\cdot Pr_{t}(x_{t})}{\sum _{x_{t}\in X_{t}}\frac{\exp \left( f_{0t}(x_{t})+\beta _{C}g_{0t}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }\cdot Pr_{t}(x_{t})}\right) . \end{aligned}$$

(15)

Next, suppose that the value of $\widehat{c_{jt}}$ is available at product prices $p_{t}^{0}$ and continuation values $g_{t}^{0}$. Then, we can construct the following static demand model with market share $\widehat{s_{jt}}$⁴:

$$\begin{aligned} \widehat{s_{jt}}(p_{t};p_{t}^{0},g_{t}^{0})= & {} \frac{\exp (-\alpha p_{jt}+f_{jt} +\widehat{c_{jt}}(p_{t}^{0},g_{t}^{0}))}{1+\sum _{k\in \mathcal {J}_{t}}\exp (-\alpha p_{kt} +f_{kt}+\widehat{c_{kt}}(p_{t}^{0},g_{t}^{0}))};\text { and} \end{aligned}$$

(16)

$$\begin{aligned} \widehat{s_{0t}}(p_{t};p_{t}^{0},g_{t}^{0})= & {} \frac{1}{1+\sum _{k\in \mathcal {J}_{t}}\exp (-\alpha p_{kt}+f_{kt}+\widehat{c_{kt}}(p_{t}^{0},g_{t}^{0}))}. \end{aligned}$$

(17)

Note that $\widehat{s_{jt}}(p_{t}=p_{t}^{0};p_{t}^{0},g_{t}^{0})=s_{jt}(p_{t}^{0},g_{t}^{0})=s_{jt}^{0}$ holds by construction.

Under the static model, the own price elasticity of product j at time t is:

$$\begin{aligned} \widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})\equiv -\frac{\partial \widehat{s_{jt}}(p_{t}=p_{t}^{0};p_{t}^{0},g_{t}^{0})}{\partial p_{jt}}\frac{p_{jt}^{0}}{\widehat{s_{jt}}(p_{t}^{0};p_{t}^{0},g_{t}^{0})}=\alpha s_{jt}^{0}(1-s_{jt}^{0})\cdot \frac{p_{jt}^{0}}{s_{jt}^{0}}. \end{aligned}$$

(18)

Similarly, the cross price elasticity of product k with respect to product $j(\ne k)$ at time t is:

$$\begin{aligned} \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})\equiv & {} \frac{\partial \widehat{s_{kt}}(p_{t}=p_{t}^{0};p_{t}^{0},g_{t}^{0})}{\partial p_{jt}}\frac{p_{jt}^{0}}{\widehat{s_{kt}}(p_{t}^{0};p_{t}^{0},g_{t}^{0})}=\alpha s_{jt}^{0}s_{kt}^{0}\cdot \frac{p_{jt}^{0}}{s_{kt}^{0}}. \end{aligned}$$

(19)

Then, static elasticities $\widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0}),\widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})$ and dynamic short-run elasticities $\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0}),\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})$ satisfy the following propositions:

Proposition 1

$\widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})\ge \eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})$.

Furthermore, equality holds only when $s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})=s_{jt}(p_{t}^{0},g_{t}^{0})\ \forall x_{t}\in X_{t}.$

Proposition 2

The following inequalities hold:

$$\begin{aligned} \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})\left\{ \begin{array}{c}>\\ =\\< \end{array}\right\} \eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\text { if }Cov_{jkt}(p_{t}^{0},g_{t}^{0})\left\{ \begin{array}{c} <\\ =\\ > \end{array}\right\} 0. \end{aligned}$$

where

$$\begin{aligned} E_{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t},g_{t})\equiv & {} \sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})s_{jt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t});\text {and}\\ Cov_{jkt}(p_{t},g_{t})\equiv & {} \sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})\left( s_{jt}^{(ccp)} (\widetilde{x_{t}},p_{t},g_{t})-E_{x_{t}}s_{jt}^{(ccp)} (x_{t},p_{t},g_{t})\right) \\{} & {} \left( s_{kt}^{(ccp)} (\widetilde{x_{t}},p_{t},g_{t})-E_{x_{t}}s_{kt}^{(ccp)}(x_{t},p_{t},g_{t})\right) . \end{aligned}$$

The first proposition implies that the short-run price elasticity is overestimated when applying the static model. With regard to cross elasticities, the relative size of the cross elasticities is unclear, since we cannot determine the sign of $Cov_{jkt}$ in general. Nevertheless, we can derive a stronger result for durables with unit stock where consumers own at most one durable product:

Corollary 1

Suppose that $\phi _{jt}(x_{t})=\phi _{t}(x_{t})\ \forall j\in \mathcal {J}_{t}$. Then, for durables with unit stock (inventory), $Cov_{jkt}(p_{t}^{0},g_{t}^{0})\ge 0$ and $\widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})\le \eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})$.

The problem with the static model is that the values of $\widehat{c_{jt}}$ is treated as fixed even when product prices $p_{t}$ change. In reality, $\widehat{c_{jt}}$ depends on prices since $V_{t}^{C}(x_{t},p_{t},g_{t})$ is a function of prices, and the static model cannot capture such an effect. Consequently, elasticities that are based on the static model and the short-run elasticities that are based on the dynamic model do not necessarily coincide. Of course, we can allow changing values of $\widehat{c_{jt}}$ if we can compute the derivatives of $\widehat{c_{jt}}$ with respect to prices. Nevertheless, to do so, we need knowledge of the distribution of consumers’ states ($Pr_{t}(x_{t})$), and this process is generally omitted in the standard static models.

Note that when $Pr_{t}(x_{t})=1$ (only one state; no role of states) and $\beta _{C}=0$ (myopic consumers), $\widehat{c_{jt}}=\phi _{jt}-f_{0t}$ holds.⁵ Then, $\widehat{c_{jt}}$ does not depend on prices $p_{t}$, and there is not a problem even when applying the static model.

3.2.1 Magnitude of the Biases in Price Elasticities

We can derive the upper bound of the biases in short-run price elasticities that are associated with disregarding states. The following proposition shows the statement:

Proposition 3

The following inequalities hold:

$$\begin{aligned} 0\le \widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})-\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})\le & {} \alpha p_{jt}^{0}\left( \max _{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) ;\ \text {and}\\ \left| \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\right|\le & {} \alpha p_{jt}^{0}\left( \max _{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) . \end{aligned}$$

These inequalities indicate that the biases in short-run elasticities are small when CCPs $s_{jt}^{(ccp)}(x_{t},p_{t}^{0})$ are sufficiently small for all of the states $x_{t}$.

To understand why the size of the CCPs affects the magnitude of the biases in short-run price elasticities, consider the simplified setting where $\phi _{jt}(x_{t})=0$ and $\beta _{C}=0$. Then, the utility function that is based on the dynamic model can be reformulated as:

$$\begin{aligned} v_{it}(x_{t},p_{t},\epsilon _{t},a_{t})= & {} {\left\{ \begin{array}{ll} -\alpha p_{jt}+f_{jt}+\epsilon _{ijt} &{} \text {if }a_{t}=j,\\ f_{0t}(x_{t})+\epsilon _{i0t} &{} \text {if}\ a_{t}=0 \end{array}\right. } \end{aligned}$$

Further, we assume that only one product j exists in the market. Then, by defining the term $\Delta v_{it}(x_{t},\epsilon _{it})\equiv v_{it}(x_{t},p_{t},\epsilon _{t},a_{t}=j)-v_{it}(x_{t},p_{t},\epsilon _{t},a_{t}=0)$ where $\epsilon _{it}=\epsilon _{ijt}-\epsilon _{i0t}$, the market share $s_{jt}$ that is based on the static model can be expressed as:

$$\begin{aligned} s_{jt}= & {} \sum _{x_{t}\in X_{t}}Pr(v_{it}(x_{t},p_{t},\epsilon _{t},a_{t}=j)>v_{it}(x_{t},p_{t},\epsilon _{t},a_{t}=0))\cdot Pr_{t}(x_{t})\\= & {} \sum _{x_{t}\in X_{t}}Pr(\Delta v_{it}(x_{t},\epsilon _{it})>0)\cdot Pr_{t}(x_{t}). \end{aligned}$$

Next, by (16) and (17), we can specify the “static” utility function that is based on the static model:

$$\begin{aligned} \widehat{v_{it}}(p_{t},\epsilon _{t},a_{t})={\left\{ \begin{array}{ll} -\alpha p_{jt}+f_{jt}+\widehat{c_{jt}}+\epsilon _{ijt} &{} \text {if }a_{t}=j,\\ \epsilon _{i0t} &{} \text {if}\ a_{t}=0 \end{array}\right. } \end{aligned}$$

By defining the term $\Delta \widehat{v_{it}}(\epsilon _{it})\equiv \widehat{v_{it}}(p_{t},\epsilon _{it},a_{t}=j)-\widehat{v_{it}}(p_{t},\epsilon _{it},a_{t}=0)$ where $\epsilon _{it}=\epsilon _{ijt}-\epsilon _{i0t}$, static market share $\widehat{s_{jt}}$ can be expressed as:

$$\begin{aligned} \widehat{s_{jt}}= & {} Pr(\widehat{v_{it}}(p_{t},\epsilon _{t},a_{t}=j)>\widehat{v_{it}}(p_{t},\epsilon _{t},a_{t}=0))\\= & {} Pr(\Delta \widehat{v_{it}}(\epsilon _{it})>0) \end{aligned}$$

Consequently, market shares and price elasticities are determined by the distributions of $\Delta v_{it}(x_{t},\epsilon _{it})$ (in the case of the dynamic model) and by $\Delta \widehat{v_{it}}(\epsilon _{it})$ (in the case of the static model).

The solid line in Figure 1 shows the shape of the density function of $\Delta v_{it}(x_{t},\epsilon _{t})$. Here, we assume that $\{\epsilon _{ijt}\}_{j\in \mathcal {J}_{t}}$ follows i.i.d. type-I extreme value distribution. Since there are multiple types of consumers with different $x_{t}$, multiple peaks exist. In contrast, in the static model, we abstract away the existence of consumer-level states and fit a single distribution of $\Delta \widehat{v_{it}}(\epsilon _{it})$ as in the dashed line in Figure 1. The two distributions take different shapes, and lead to biased estimates of price elasticities.

Generally, consumers who purchase the small CCP product are located in the right tail of the distribution. In the tail of the distribution, we can find minor differences between the solid line and the dashed line. Consequently, the difference between static and dynamic models is small when we focus on the small CCP product.

Fig. 1

Distribution of the utilities $\Delta v_{it}(x_{t},\epsilon _{it})$ and $\Delta \widehat{v_{it}}(\epsilon _{it})$. Notes: The solid line shows the density function of ${{\Delta v_{it}(x_{t},\epsilon _{it})}}$ based on the dynamic model accounting for the existence of states $x_{t}$. The dashed line shows the density function of $\Delta \widehat{v_{it}}(\epsilon _{it})$ based on the “static” model.

Example

Durable goods with exogenous replacement timing

To understand the results more clearly, consider the example of durable goods with exogenous replacement timing, where consumers consider purchases only when they do not have any product. Formally, consider the following setting: $s_{jt}^{(ccp)}(x_{t}\ne 0,p_{t}^{0},g_{t}^{0})\approx 0$. Here, $x_{t}=0$ denotes the state where the consumer does not possess any product at the beginning of time t.

Then, since $s_{jt}(p_{t},g_{t})=Pr_{t}(x_{t}=0)\cdot s_{jt}^{(ccp)}(x_{t}=0,p_{t},g_{t})$ holds, by (11), short-run own price elasticity that is based on the dynamic model is:

$$\begin{aligned} \eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})\equiv \alpha p_{jt}^{0}(1-s_{jt}^{(ccp)}(x_{t}=0,p_{t}^{0},g_{t}^{0})). \end{aligned}$$

By (18), the own price elasticity that is computed from the static model is:

$$\begin{aligned} \widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})\equiv \alpha p_{jt}^{0}(1-s_{jt}(p_{t}^{0},g_{t}^{0})). \end{aligned}$$

Using these simple formulas, we can easily compute the biases that are in short-run own price elasticities when applying the static model. Table 2 shows numerical examples.

Table 2 Bias in short-run own price elasticity (durable goods with exogenous replacement timing)

As this table shows, the bias is large when the fraction of consumers who currently possess the products is high ($Pr_{t}(x_{t}=0)$ is small), and consumer’s CCP of the product ($s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})$) is high. For instance, when $Pr_{t}(x_{t}=0)=0.2$ and $s_{jt}^{(ccp)}(x_{t}=0,p_{t}^{0},g_{t}^{0})=0.8$, the bias is 320%. This large bias comes from the implicit assumption in the static model that even the consumers who already own any product will consider a purchase as if they do not own anything. In the true dynamic model, consumers who already own products will not buy. Note that the bias is small if the value of $s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})$ is small, even when $Pr_{t}(x_{t}=0)$ is high.

3.2.2 Remedies for Static Demand Models

In this section, we have not yet introduced any random coefficients into the static model. Nevertheless, by introducing random coefficients, especially the coefficient on the constant term, we might be able to mitigate the bias. Here, to make the point clear we consider the case without persistent consumer heterogeneity in the dynamic model as in Section 3.2. Note that a similar argument holds even when persistent consumer heterogeneity exists in the dynamic model.

By introducing random coefficients in the static model, we can derive a static representation of the dynamic model in an alternative way. The static model is composed of market shares $\widetilde{s_{jt}}$, type-specific choice probabilities $\widetilde{s_{\widetilde{l}jt}}(p_{t},g_{t})$, additional terms $\widetilde{c_{\widetilde{l}jt}}$ and a mapping $\sigma$ from $x_{t}$ to $\widetilde{l}$ such that:

$$\begin{aligned} \widetilde{s_{\widetilde{l}jt}}(p_{t},g_{t})= & {} \frac{\exp (-\alpha p_{jt}+f_{jt}+\widetilde{c_{\widetilde{l}jt}}(g_{t}))}{1+\sum _{k\in \mathcal {J}_{t}}\exp (-\alpha p_{kt}+f_{kt}+\widetilde{c_{\widetilde{l}kt}}(g_{t}))};\\ \widetilde{s_{jt}}(p_{t},g_{t})= & {} \int \widetilde{s_{\widetilde{l}jt}}(p_{t},g_{t})dP(\widetilde{l});\\ P(\widetilde{l}=\sigma (x_{t}))= & {} Pr_{t}(x_{t});\ \text {and}\\ \widetilde{c_{\widetilde{l}jt}}(g_{t})\equiv & {} \phi _{jt}(x_{t})-f_{0t}(x_{t})+\beta _{C}g_{jt}(x_{t})-\beta _{C}g_{0t}(x_{t})\ \text {for}\ \widetilde{l}=\sigma (x_{t}). \end{aligned}$$

where P denotes the density of type $\widetilde{l}$ consumers. Then, we can easily show that the static and dynamic models are related in the following ways:

$$\begin{aligned} \widetilde{s_{\widetilde{l}jt}}(p_{t},g_{t})= & {} s_{jt}^{(ccp)}(x_{t},p_{t},g_{t})\ \ \ \text {for}\ \widetilde{l}=x_{t};\ \text {and}\\ \widetilde{s_{jt}}(p_{t},g_{t})= & {} s_{jt}(p_{t},g_{t}). \end{aligned}$$

Then, own price elasticity of product j is:

$$\begin{aligned} \widetilde{\eta _{jt}}(p^{0},g_{t}^{0})=-\frac{\partial \widetilde{s_{jt}}(p_{t}^{0},g_{t}^{0})}{\partial p_{jt}}\frac{p_{jt}^{0}}{\widetilde{s_{jt}}(p_{t}^{0},g_{t}^{0})}=\alpha \int \widetilde{s_{ljt}}(p_{t}^{0},g_{t}^{0})(1-\widetilde{s_{ljt}}(p_{t}^{0},g_{t}^{0}))dP(l)\frac{p_{jt}^{0}}{s_{jt}^{0}}=\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0}). \end{aligned}$$

The cross elasticity of product k with respect to product j is:

$$\begin{aligned} \widetilde{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})=\frac{\partial \widetilde{s_{kt}}(p_{t}^{0},g_{t}^{0})}{\partial p_{jt}}\frac{p_{jt}^{0}}{\widetilde{s_{kt}}(p_{t}^{0},g_{t}^{0})}=\alpha \int \widetilde{s_{ljt}}(p_{t}^{0},g_{t}^{0})\widetilde{s_{lkt}}(p_{t}^{0},g_{t}^{0})dP(l)\frac{p_{jt}^{0}}{s_{kt}(p_{t}^{0},g_{t}^{0})}=\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0}). \end{aligned}$$

These elasticities imply that we can obtain consistent estimates of the short-run elasticities, if we can fit the static model so that $\widetilde{s_{\widetilde{l}jt}}(p_{t}^{0},g_{t}^{0})\approx s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})$ for $\widetilde{l}=\sigma (x_{t})$ and $\alpha$ is consistently estimated.

To fit the static model so that $\widetilde{s_{\widetilde{l}jt}}(p_{t}^{0},g_{t}^{0})\approx s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})$ for $\widetilde{l}=\sigma (x_{t})$, we should well approximate the distribution of the term $\widetilde{c_{\widetilde{l}jt}}(g_{t})=\phi _{jt}(x_{t})-f_{0t}(x_{t})+\beta _{C}g_{ljt}(x_{t})-\beta _{C}g_{l0t}(x_{t})$ by random coefficients. For example, in order to approximate the distribution of the term $f_{0t}(x_{t})$, which does not depend on product characteristics, the most straightforward way is to introduce a random coefficient on the constant term. Note that such a “reduced-form” approach might not work in all cases. The distribution of $f_{0t}(x_{t})$ changes over time based on the values of $Pr_{t}(x_{t})$, and it is not clear whether this type of strategy works well.

3.3 Second Bias: Inconsistent Utility Parameter Estimates

The discussion above implies that static models yield overestimated short-run own price elasticities, and yield underestimated short-run cross elasticities for durables with unit stock. However, this discussion hinges on the condition that the static and dynamic models share the same price coefficient $\alpha$. However, there is no guarantee that the estimated $\alpha$ that is based on the static model coincides with $\alpha$ in the dynamic model. Here, we discuss the second bias: inconsistent utility parameter estimates.

We consider the estimation process with the use of aggregate data. As in the previous subsection, we consider the static model without random coefficients. Let $f_{jt}=X_{jt}\theta +c_{0}+\xi _{jt}$, where $X_{jt},\xi _{jt}$ denote product j’s observed and unobserved characteristics. Let $Z_{jt}$ be instrumental variables that satisfy $E[\xi _{jt}|Z_{jt}]=0$. Additionally, we impose the following condition as in the static BLP model: $S_{jt}=s_{jt}\ \ \ j\in \mathcal {J}_{t}\cup \{0\}$. $S_{jt}$ denotes the market share data of product j at time t. Then, by (13) and (14) we obtain the following linear equation:

$$\begin{aligned} \log S_{jt}-\log S_{0t}=-\alpha p_{jt}+X_{jt}\theta +\widehat{c_{0}}+\zeta _{jt}, \end{aligned}$$

where $\zeta _{jt}\equiv \xi _{jt}+\widehat{c_{jt}}-E[\widehat{c_{jt}}]$ and $\widehat{c_{0}}=c_{0}+E[\widehat{c_{jt}}]$.

In general, estimating the linear equation above by treating $\zeta _{jt}$ as the error term and applying GMM does not yield consistent estimates of $\alpha$ and $\theta$. Since $\widehat{c_{jt}}$ is a function of the value function $V_{t}^{C}(x_{t},p_{t},g_{t})$ as shown in (15), it depends on the current product characteristics and prices. Consequently, $E[\widehat{c_{jt}}|Z_{jt}]\ne 0$, and it implies $E[\zeta _{jt}|Z_{jt}]\ne 0$. In addition, correlation between instrumentals $Z_{jt}$ and consumers’ expectations $g_{jt}(x_{t})=EV_{t+1}^{C}(x_{t},a_{t}=j)$, or the correlations between IVs $Z_{jt}$ and $Pr_{t}(x_{t})$ also lead to inconsistent estimates.

Note that when $Pr_{t}(x_{t})=1$ (only one state) and $\beta _{C}=0$ (myopic consumers), $\widehat{c_{jt}}=\phi _{jt}-f_{0t}$ holds.⁶ Further, if $\phi _{jt}=\phi _{t}$ for all $j\in \mathcal {J}_{t}$ and if $\phi _{t}$ and $f_{0t}$ do not change over time, we can treat $\widehat{c_{jt}}$ as a constant term that does not depend on product j, and we can consistently estimate $\alpha$ and $\theta$. If neither of these conditions holds, applying the static model leads to the inconsistent parameter estimates.

3.3.1 Remedies for Static Demand Models

The Case of Static Models Without Random Coefficients

We continue the discussion in this section: We abstract away the existence of persistent consumer heterogeneity in the dynamic model, and we consider the case where random coefficients are not introduced in the static model. The problem is that the term $\widehat{c_{jt}}(p_{t},g_{t})=\log \left( \frac{\sum _{x_{t}\in X_{t}}\frac{\exp \left( \phi _{jt}(x_{t})+\beta _{C}g_{jt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }\cdot Pr_{t}(x_{t})}{\sum _{x_{t}\in X_{t}}\frac{\exp \left( f_{0t}(x_{t})+\beta _{C}g_{0t}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }\cdot Pr_{t}(x_{t})}\right)$ is not controlled in the estimation process:

$$\begin{aligned} \log S_{jt}-\log S_{0t}=-\alpha p_{jt}+X_{jt}\theta +\widehat{c_{jt}}(p_{t},g_{t})+c_{0}+\xi _{jt}. \end{aligned}$$

If we can well approximate the term $\widehat{c_{jt}}$ with the variables other than $p_{jt}$ and $X_{jt}$, we can obtain consistent estimates of parameter estimates $\alpha$ and $\theta$.

One strategy for approximating $c_{0}+\widehat{c_{jt}}$ is the introduction of time-/-group dummy variables.⁷ Suppose that the set of products $\mathcal {J}_{t}$ can be divided into mutually exclusive groups $\mathcal {J}_{gt}$ ($g=1,\cdots ,G$), and products in the same group share the same values of $c_{0}+\widehat{c_{jt}}$ ($c_{0}+\widehat{c_{jt}}=\widehat{c_{gt}}$ $\forall j\in \mathcal {J}_{gt}$). Then, we can consistently estimate $\alpha$ and $\theta$ by treating $\widehat{c_{gt}}$ as fixed effect dummy variables:

$$\begin{aligned} \log S_{jt}-\log S_{0t}=-\alpha p_{jt}+X_{jt}\theta +\widehat{c_{gt}}+\xi _{jt}. \end{aligned}$$

Note that products in the same group share the same values of $\widehat{c_{jt}}$ only when products in the same group share the same values of $g_{jt}(x_{t})$ (continuation value given purchasing product j) and of $\phi _{jt}(x_{t})$ (flow utility from purchasing product j given state $x_{t}$ other than $-\alpha p_{jt}+f_{jt}$).⁸

The introduction of time and group dummy variables are informally proposed in Goldberg and Verboven (2001), where automobile brand is the group. If it is plausible to assume that consumers perceive that their future utility from purchasing the same brand automobiles will be the same as their current utility, the strategy works well and we can obtain the consistent estimate of $\alpha$ and $\theta$.

Note that when the market environment is stable over time, the values of $Pr_{t}(x_{t})$ and $\beta _{C}g_{jt}(x_{t})$ are mostly stable, and $\widehat{c_{jt}}$ also take mostly stable values over time. In this case, we can treat the term $\widehat{c_{jt}}$ as a constant term, and we can obtain mostly precise estimates of the parameters unless cross-sectional correlations between $\beta _{C}g_{jt}(x_{t})$ and $p_{jt}$ or $Z_{jt}$ exists.

The Case of Static Models with Random Coefficients

We abstract away the existence of persistent consumer heterogeneity in the dynamic model. Note that a similar argument holds even when persistent consumer heterogeneity exists in the dynamic model.

If we can well approximate the term $\widetilde{c_{\widetilde{l}jt}}(g_{t})=\phi _{jt}(x_{t})-f_{0t}(x_{t})+\beta _{C}g_{jt}(x_{t})-\beta _{C}g_{0t}(x_{t})$ with random coefficients and with variables other than $p_{jt}$ and $X_{jt}$, we can obtain consistent estimates of $\alpha$ and $\theta$. As in static models without random coefficients, introducing time-/-group dummy variables would mitigate the problems in some cases, because it approximates continuation values $g_{jt}(x_{t})$ to some extent. Nevertheless, the value also depends on the states $x_{t}$, and the interactions with random coefficients would be necessary. There is no guarantee that the estimated parameters are consistent, when introducing random coefficients in the static models.

3.4 Third Bias: Changing Expectations of Consumers and Long-run Price Elasticity

Next, we consider the third bias: changing expectations of consumers, which affects long-run price elasticities defined in Section 3.1. In the static model, there is no counterpart of long-run elasticities, and we compare long-run elasticity based on the dynamic model and short-run elasticities based on the dynamic model. As was mentioned in Section 3.1, the difference is whether the continuation values $g_{t}$ changes or not.

To clarify the point, we assume that consumers have perfect foresight with respect to the future price path.⁹ We consider the case where the price of product j is expected to increase between time t and time $t+T$. We assume that the price change is not expected by the consumers before time t, and the increments of the price increases are the same for all of the periods. Here, we allow the existence of persistent consumer heterogeneity. To derive statements, we define:

$$\begin{aligned} \lambda _{ljt}(x_{t},\{p_{t+\tau }\}_{\tau \ge 0})\equiv & {} \sum _{\tau =1}^{T}\beta _{C}^{\tau -1}\left[ Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=j,\{p_{t+\tau }\}_{\tau \ge 0})-Pr(l\ choose\ j\ at\ t+\tau |x_{t},\{p_{t+\tau }\}_{\tau \ge 0})\right] ;\text { and}\\ \lambda _{ljkt}(x_{t},\{p_{t+\tau }\}_{\tau \ge 0})\equiv & {} \sum _{\tau =1}^{T}\beta _{C}^{\tau -1}\left[ Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k,\{p_{t+\tau }\}_{\tau \ge 0})-Pr(l\ choose\ j\ at\ t+\tau |x_{t},\{p_{t+\tau }\}_{\tau \ge 0})\right] . \end{aligned}$$

Here, $Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k,\{p_{t+\tau }\}_{\tau \ge 0})$ denotes the probability that type l consumers with state $x_{t}$ and choice k at time t choose product j at time $t+\tau$ given future prices $\{p_{t+\tau }\}_{\tau \ge 0}$. $Pr(l\ choose\ j\ at\ t+\tau |x_{t},\{p_{t+\tau }\}_{\tau \ge 0})$ is defined in a similar way. Intuitively, $\lambda _{ljkt}(x_{t})>0$ implies that type l consumers who choose product k at time t are more likely to choose product j in the future periods.

Let $\eta _{jt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})$ be the long-run own price elasticity of product j at time t, and let $\eta _{jkt}^{(long)}$ be the long-run cross elasticity of product k with respect to product j at time t given the future price path $\{p_{t+\tau }\}_{\tau \ge 0}$. We further assume that continuation value $g_{t}^{0}$ is consistent with the future price path $\{p_{t+\tau }^{0}\}_{\tau \ge 0}$. Then, short-run and long-run price elasticities satisfy the following claim:

Proposition 4

The following inequalities hold:

$$\begin{aligned} \eta _{jt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})\left\{ \begin{array}{c}<\\ =\\> \end{array}\right\} \eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})\text { if }\lambda _{ljt}(x_{t},\{p_{t+\tau }^{0}\}_{\tau \ge 0})\left\{ \begin{array}{c}<\\ =\\> \end{array}\right\} 0\ \forall l,x_{t}\in X_{t};\text { and}\\ \eta _{jkt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})\left\{ \begin{array}{c}>\\ =\\< \end{array}\right\} \eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\text { if }\lambda _{ljkt}(x_{t},\{p_{t+\tau }^{0}\}_{\tau \ge 0})\left\{ \begin{array}{c} <\\ =\\ > \end{array}\right\} 0\ \forall l,x_{t}\in X_{t}. \end{aligned}$$

In the durable goods case, generally the current demand for a product implies less future demand for any product due to the durability of products ($\lambda _{ljt}(x_{t})<0,\lambda _{ljkt}(x_{t})<0\ j,k\in \mathcal {J}_{t}$). Then, Proposition 4 implies that the long-run own price elasticity is smaller than the short-run own price elasticity, and the long-run cross elasticity is larger than the short-run cross elasticity.¹⁰ Note that the two elasticities coincide when the consumers are myopic ($\beta _{C}=0$).

3.4.1 Magnitude of the Biases in Price Elasticities

The next proposition shows the upper bound of the biases in long-run price elasticities:

Proposition 5

The following inequalities hold:

$$\begin{aligned} \left| \eta _{jt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})-\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})\right|\le & {} \kappa _{jt}\left( \max _{l,x_{t+\tau },\tau \ge 1}\alpha _{l}s_{ljt+\tau }^{(ccp)}(x_{t+\tau },p_{t+\tau }^{0},g_{t+\tau }^{0})\right) ;\text { and}\\ \left| \eta _{jkt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\right|\le & {} \kappa _{jt}\left( \max _{l,x_{t+\tau },\tau \ge 1}\alpha _{l}s_{ljt+\tau }^{(ccp)}(x_{t+\tau },p_{t+\tau }^{0},g_{t+\tau }^{0})\right) , \end{aligned}$$

where $\kappa _{jt}\equiv 2\beta _{C}\frac{1-\beta _{C}^{T}}{1-\beta _{C}}p_{jt}^{0}$.

The inequalities indicate that the biases in long-run elasticities are small when future CCPs are sufficiently small for all of the states and consumer types. In general, consumers expect that the probability that they will purchase the product in the future is small if the CCP of a product is small. Then, they are less likely to be affected by the future price change of the small CCP product.

4 Extension: Storable Goods

We can extend the discussion in Section 3 to storables, by introducing the choice of consumption level other than the product choice, which is also the essential component in the storable goods models.

Here, we consider the model where consumers solve the dynamic optimization problem with inventory represented by the following Bellman equation:¹¹

$$\begin{aligned} \widetilde{V_{lt}^{C}}(x_{t},p_{t})= & {} \int \max _{C_{lt},\{d_{iljt}\}_{j\in \mathcal {J}_{t}}}\Biggr [U(C_{lt})-F(x_{t}+q_{t}-C_{lt})+\sum _{j\in \mathcal {J}_{t}}d_{iljt}\left( -\alpha p_{jt}+f_{ljt}+\epsilon _{ijt}\right) \\{} & {} \ \ \ +\beta _{C}E\left[ \widetilde{V_{lt+1}^{C}}(x_{t}+q_{t}-C_{lt},p_{t+1})|x_{t},p_{t}\right] \Bigg ]p(\epsilon )d\epsilon . \end{aligned}$$

Here, $x_{t}$ denotes the inventory of the consumer at time t. $d_{iljt}$ denotes the number of purchases of product j; for instance, $d_{iljt}=1$ implies that the consumer purchases a product j at time t. $C_{lt}$ denotes the amount of consumption, and $U(C_{lt})$ represents the utility from the consumption of storables. Here, we assume that only the quantity of consumption matters (brand does not matter in the consumption stage). $q_{t}=\sum _{j\in \mathcal {J}_{t}}q_{jt}d_{iljt}$ denotes the purchased quantity at time t, where $q_{jt}$ denotes the package size of product j. $x_{t}+q_{t}-C_{lt}$ represents the quantity of the goods for storage, and $F(x_{t}+q_{t}-C_{lt})$ represents the storage cost. We assume that $x_{t+1}=x_{t}+q_{t}-C_{lt}$ holds.

In the following discussion, we assume that consumers purchase at most one product ($\sum _{j\in \mathcal {J}_{t}}d_{iljt}\le 1$) in each period. Then, the optimal consumption level given the purchase of product j at time t is:

$$\begin{aligned} C_{lt}^{*}|_{j}=\arg \max _{C_{lt}}U(C_{lt})-F(x_{t}+q_{jt}-C_{lt})+\beta _{C}E\left[ \widetilde{V_{lt+1}^{C}}(x_{t}+q_{jt}-C_{lt},p_{t+1})|x_{t},p_{t}\right] . \end{aligned}$$

The optimal consumption level given no purchase at time t is:

$$\begin{aligned} C_{lt}^{*}|_{0}=\arg \max _{C_{lt}}U(C_{lt})-F(x_{t}-C_{lt})+\beta _{C}E\left[ \widetilde{V_{lt+1}^{C}}(x_{t}-C_{lt},p_{t+1})|x_{t},p_{t}\right] . \end{aligned}$$

Note that $C_{lt}^{*}|_{j\in \mathcal {J}_{t}\cup \{0\}}$ does not depend on the current prices $p_{t}$, given the continuation values $E\left[ \widetilde{V_{lt+1}^{C}}(x_{t}+q_{jt}-C_{lt}^{*}|_{j\in \mathcal {J}_{t}\cup \{0\}},p_{t+1})|x_{t},p_{t}\right]$. In contrast, $C_{lt}^{*}|_{j\in \mathcal {J}_{t}\cup \{0\}}$ may depend on the future prices $\{p_{t+\tau }\}_{\tau \ge 1}$ through the terms on continuation values $g_{t}$.

Then, by defining $\phi _{ljt}(x_{t})\equiv U(C_{lt}^{*}|_{j})-F(x_{t}+q_{jt}-C_{lt}^{*}|_{j})\ \text {for }j\in \mathcal {J}_{t}$, $f_{l0t}(x_{t})\equiv U(C_{lt}^{*}|_{j})-F(x_{t}+q_{jt}-C_{lt}^{*}|_{j})$, and $g_{ljt}(x_{t})\equiv E\left[ \widetilde{V_{lt+1}^{C}}(x_{t}+q_{jt}-C_{lt}^{*}|_{j},p_{t+1})|x_{t},p_{t}\right]$, we can specify the utility from choosing the alternative $j\in \mathcal {J}_{t}\cup \{0\}$ as in (1):

$$\begin{aligned} v_{ilt}\left( x_{t},p_{t},g_{t},\epsilon _{it},a_{t}\right) ={\left\{ \begin{array}{ll} -\alpha _{l}p_{jt}+f_{ljt}+\phi _{ljt}(x_{t})+\beta _{C}g_{ljt}(x_{t})+\epsilon _{ijt} &{} \text {if }a_{t}=j\\ f_{l0t}(x_{t})+\beta _{C}g_{l0t}(x_{t})+\epsilon _{i0t} &{} \text {if}\ a_{t}=0 \end{array}\right. }. \end{aligned}$$

4.1 Short-run Price Elasticity

Here, we consider the setting without persistent consumer heterogeneity as in the discussion in Section 3.2. Since $\phi _{jt}(x_{t})$ and $f_{0t}(x_{t})$ does not depend on the current product prices $\{p_{kt}\}_{k\in \mathcal {J}_{t}},$ given the continuation values $g_{t}$, we can easily show that the same statements in Section 3.2 (Propositions 1 and 2) hold. In terms of short-run cross elasticities, the signs of the biases are not clear, as was shown in Proposition 2. Nevertheless, for the “same package size products,” short-run cross elasticities are underestimated:

Corollary 2

For storable goods $j,k\in \mathcal {J}_{t}$ with $q_{jt}=q_{kt}$, $Cov_{jkt}(p_{t}^{0},g_{t}^{0})\ge 0$ and $\widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})\le \eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})$.

4.2 Long-run Price Elasticity

If the future price change does not affect the current consumption level, then the terms $\phi _{ljt}(x_{t})$ and $f_{l0t}(x_{t})$ do not depend on future prices, and we can easily show that the statements in Section 3.4 (Proposition 4) also holds even for storables. In the storable goods case, in general the current demand for a product implies less future demand for any product $(\lambda _{ljt}(x_{t})<0,\lambda _{ljkt}(x_{t})<0\ j,k\in \mathcal {J}_{t})$. Then, Proposition 4 implies that the long-run own price elasticity is smaller than the short-run own price elasticity, and the long-run cross price elasticity is larger than the short-run cross price elasticity.

Nevertheless, in reality, future price change may affect the current consumption level. For instance, consumers who expect higher future prices may reduce the amount of consumption and increase inventory. Consequently, Proposition 4 cannot be directly applied to storables, and another bias—changing consumption level in response to the future price change—exists.

5 Applications of the Results to Empirical Researches

In this section, we discuss how the results so far provide insights into the previous studies’ findings with respect to the biases in applying static demand models. Here, we focus on the results of four papers in Table 1.

5.1 Automobiles ( Chen et al. (2008) and Schiraldi (2011))

Chen et al. (2008) analyzed the automobile market and showed that a static model overestimates short-run own price elasticity¹² by 14%. In contrast, Schiraldi (2011), who also analyzed the automobile market, showed that a static model underestimates short-run own price elasticity by 73%. We can guess that the difference comes from the primary sources of the biases: Chen et al. (2008)’s result is due mainly to disregarding states; and Schiraldi (2011)’s result is due primarily to the inconsistent utility parameter estimates.

Chen et al. (2008) considered the market with only one homogeneous new car model and one homogeneous used car model. Moreover, their economic model did not incorporate persistent consumer heterogeneity. Since they introduced new and used car dummy variables in the estimation, the identification mainly comes from time-series variations. Since they implicitly considered the setting where the market is stationary, $Pr_{t}(x_{t})$ and consumers’ expectations $g_{jt}(x_{t})$ are mostly stable over time. Though they used cost-shifters as instruments that may be correlated with consumers’ expectations over time, static estimates yielded only minor biases. In fact, the bias of the price coefficient was only 2%. Then, the effect of the bias that was due to the disregard of states dominated, and the static model overestimated the short-run price elasticity.

In contrast, Schiraldi (2011) considered the market with multiple products. The identification came from both time-series and cross-sectional variations. When focusing on cross-sectional variations, consumers’ continuation values $g_{ljt}(x_{t})$ are positively correlated with the product prices $p_{jt}$, since higher quality and expensive products would have higher remaining values in the next period. The positive correlation leads to a large positive correlation between price $p_{jt}$ and $\widetilde{c_{ljt}}$, and consequently, an underestimation of the price coefficient.¹³ Since the bias that is due to the underestimation of the price coefficient is so large (bias:82%), we can suspect that the effect of that bias dominated the bias that is due to the disregard of states,¹⁴ and the static model underestimated the short-run own elasticity.

5.2 New Durable Goods ( Gowrisankaran and Rysman, 2012)

Gowrisankaran and Rysman (2012) studied a new durable goods market with replacement demand. They showed that the estimated price coefficient was sufficiently close to zero when applying a static BLP model. In their model, we can expect that not only continuation values $g_{ljt}(x_{t})$ are correlated with prices $p_{jt}$ as in Schiraldi (2011), but also $Pr_{lt}(x_{t}=0)$ is correlated with prices $p_{jt}$. Here, $x_{t}=0$ denotes the state where consumers do not own any durable product. Especially in the latter periods of the diffusion process, the decreasing fraction of no-stock consumers $Pr_{lt}(x_{t}=0)$ induces lower demand $s_{jt}$. Prices $p_{jt}$ decline over time simultaneously, and a static estimate fits the model as if declining prices induces lower demand.¹⁵

5.3 Storable Goods ( Hendel and Nevo, 2006)

Hendel and Nevo (2006) argued that applying a static model leads to the overestimation of long-run own price elasticities and underestimation of long-run cross elasticities. The biases come from three sources: disregard of states (consumer inventory); inconsistent utility parameter estimates; and changing expectations of consumers. Disregard of states leads to the overestimation of short-run own elasticities for all of the products, and to the underestimation of short-run cross elasticities for the same package size products, as was discussed in Section 4. Changes in consumers’ expectations affect the current consumption level and product choice. If the current consumption level does not change in response to the future price change, it is plausible to assume that long-run elasticities are higher than short-run elasticities, as was discussed in Section 4.

With regard to inconsistent estimates, Hendel and Nevo (2006) empirically showed that applying a static model results in the overestimation of the price coefficient. To understand this, consider a simplified model in which there are only two states: state with-/-without inventory ($x_{t}\ne 0$ and $x_{t}=0$). Further, we abstract away consumer heterogeneity and assume that consumers do not buy anything at the inventory state $(x_{t}\ne 0)$. Then, $\widehat{c_{jt}}$ can be expressed as:¹⁶

$$\begin{aligned} \widehat{c_{jt}}(p_{t},g_{t})=\log \left( \frac{\frac{\exp (\phi _{jt}(x_{t}=0)+\beta _{C}g_{jt}(x_{t}))}{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)}{1-\frac{\sum _{k\in \mathcal {J}_{t}}\exp \left( -\alpha p_{kt}+f_{kt}+\phi _{kt}(x_{t})+\beta _{C}g_{kt}(x_{t})\right) }{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)}\right) . \end{aligned}$$

(20)

The equation implies that $\widehat{c_{jt}}$ is an increasing function of $Pr_{t}(x_{t}=0)$. Generally, consumers are more likely to buy storables under temporary price reductions and when they are at the no-inventory state. Then, it is plausible to assume that $Pr_{t}(x_{t}=0)$ and prices $p_{jt}$ are negatively correlated ($Pr_{t}(x_{t}=0)$ is high when prices are low, and vice versa). Then, there is negative correlation between prices $p_{jt}$ and $\widehat{c_{jt}}$. Of course, other effects—such as the correlation between $p_{jt}$ and $g_{jt}(x_{t})$—would exist. Nevertheless, these effects might be dominated by the effect of the negative correlation between $Pr_{t}(x_{t}=0)$ and prices $p_{jt}$.¹⁷

6 Conclusion

In this article, I have investigated the mechanisms that underlie the biases in applying static demand models when the true demand structure is dynamic. I studied three biases: disregard of state variables; inconsistent utility parameter estimates; and changing expectations of consumers. The bias that is due to the disregarding of states, which has not been discussed much in the previous literature, leads to the overestimation of short-run own price elasticities. The first and third biases are small when the focus is on the small conditional choice probability (CCP) products.

In this study, we assumed that idiosyncratic preference shock $\epsilon _{ijt}$ follows type-I extreme value distribution as does most of the literature. Nevertheless, there is no guarantee that the distributional assumption is correct. Considering the results in nonparametric settings might be a promising extension of this study.

In addition, we can find an analogy with the limited consideration set models (e.g., Abaluck and Adams-Prassl (2021); Crawford et al. (2021)). For example, in the default specific models—which is one category of the consideration set models—only a fraction of consumers make purchase decisions. Similarly, in the extreme case of durables, only a fraction of consumers with no product holdings make additional purchase decisions.¹⁸ However, the fraction of consumers who make decisions is generally not observed, and this causes problems. It would be interesting to investigate how the identification strategies in the recent literature mitigate the issues that are associated with the use of static models.

Proof

1.1 Proof of the Statements in Section 3.1

1.1.1 Derivation of Equations (13) and (14)

Using equations (4) - (7),

$$\begin{aligned}{} & {} \log s_{jt}(p_{t},g_{t})-\log s_{0t}(p_{t},g_{t})\\= & {} \log \left( \sum _{x_{t}\in X_{t}}\frac{\exp \left( -\alpha p_{jt}+f_{jt} +\phi _{jt}(x_{t})+\beta _{C}g_{jt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t}) \right) }\cdot Pr_{t}(x_{t})\right) \\{} & {} -\log \left( \frac{\exp \left( f_{0t}(x_{t})+\beta _{C}g_{0t}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }\cdot Pr_{t}(x_{t})\right) \\= & {} -\alpha p_{jt}+f_{jt}+\log \left( \sum _{x_{t}\in X_{t}}\frac{\exp \left( \phi _{jt}(x_{t}) +\beta _{C}g_{jt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) } \cdot Pr_{t}(x_{t})\right) \\{} & {} -\log \left( \frac{\exp \left( f_{0t}(x_{t})+\beta _{C}g_{0t}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }\cdot Pr_{t}(x_{t})\right) \\= & {} -\alpha p_{jt}+f_{jt}+\widehat{c_{jt}}(p_{t},g_{t}). \end{aligned}$$

Consequently, we obtain the equations by defining the terms $\widehat{c_{jt}}$.

1.2 Proof of the Statements in Section 3.2

1.2.1 Proof of Proposition 1 and 2

It is sufficient to prove the following lemma:

Lemma 1

The following equations hold:

$$\begin{aligned} \widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})-\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})= & {} \alpha \sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}}) \left[ s_{jt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})- E_{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t},g_{t})\right] ^{2}\\{} & {} \cdot \frac{p_{jt}^{0}}{s_{jt}^{0}}, \text { and}\\ \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})= & {} -\alpha Cov_{jkt}(p_{t}^{0},g_{t}^{0})\cdot \frac{p_{jt}^{0}}{s_{kt}^{0}}. \end{aligned}$$

Own Price Elasticity:

Proof

By (11) and (18),

$$\begin{aligned} \widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})-\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})= & {} \left[ \alpha s_{jt}^{0}(1-s_{jt}^{0})\right. \\{} & {} \left. -\alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\cdot s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\left( 1-s_{jt}^{(ccp)}(x_{t},p_{t}^{0})\right) \right] \frac{p_{jt}^{0}}{s_{jt}^{0}}\\= & {} \alpha \left[ \left( -\left( \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) ^{2}+\sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})^{2}\right) \right] \frac{p_{jt}^{0}}{s_{jt}^{0}}\\= & {} \alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\left[ s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})-\sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})s_{jt}^{(ccp)}(\widetilde{x_{t}},p_{t}^{0},g_{t}^{0})\right] ^{2}\cdot \frac{p_{jt}^{0}}{s_{jt}^{0}}. \end{aligned}$$

$\square$

Cross Price Elasticity:

Proof

By (12) and (19),

$$\begin{aligned}{} & {} \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)} (p_{t}^{0},g_{t}^{0}) \\= & {} \left[ \alpha s_{jt}^{0}s_{kt}^{0}-\alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\cdot s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})s_{kt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right] \frac{p_{jt}^{0}}{s_{kt}^{0}}\\= & {} -\alpha \left[ \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\left( s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})-\sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})s_{jt}^{(ccp)}(\widetilde{x_{t}},p_{t}^{0},g_{t}^{0})\right) \cdot \right. \\{} & {} \left. \left( s_{kt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})-\sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})s_{kt}^{(ccp)}(\widetilde{x_{t}},p_{t}^{0},g_{t}^{0})\right) \right] \frac{p_{jt}^{0}}{s_{kt}^{0}}\\= & {} -\alpha Cov_{jkt}(p_{t}^{0},g_{t}^{0})\cdot \frac{p_{jt}^{0}}{s_{kt}^{0}}. \end{aligned}$$

$\square$

1.2.2 Proof of Corollary 1

Proof

First, continuation value $g_{jt}(x_{t})$ does not depend on the states $x_{t}$ for durables with unit stock, because current product holdings do not affect future continuation values when they sell or throw away the old product and purchase a new product.

Next, we define $\overline{s_{jt}}(p_{t},g_{t})\equiv s_{jt}^{(ccp)}(x_{t},p_{t},g_{t})/\left( \sum _{m\in \mathcal {J}_{t}}s_{mt}^{(ccp)}(x_{t},p_{t},g_{t})\right)$, which represents the conditional choice probability of product j given the purchase of any product. $\overline{s_{jt}}(p_{t},g_{t})$ does not depend on consumers’ states $x_{t}$, because $\overline{s_{jt}}(p_{t},g_{t})=\frac{\frac{\exp \left( \phi _{t}(x_{t})-\alpha p_{jt}+f_{jt}+\beta _{C}g_{jt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }}{\sum _{m\in \mathcal {J}_{t}}\frac{\exp \left( \phi _{t}(x_{t})-\alpha p_{mt}+f_{mt}+\beta _{C}g_{mt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }}=\frac{\exp \left( -\alpha p_{jt}+f_{jt}+\beta _{C}g_{jt}\right) }{\sum _{m\in \mathcal {J}_{t}}\exp \left( -\alpha p_{mt}+f_{mt}+\beta _{C}g_{mt}\right) }$. Then, we can derive:

$$\begin{aligned} Cov_{jkt}(p_{t},g_{t})\equiv & {} \sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})\left( s_{jt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})-E_{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t},g_{t})\right) \left( s_{kt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})-E_{x_{t}}s_{kt}^{(ccp)}(x_{t},p_{t},g_{t})\right) \\= & {} \sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})\left( \left( 1-s_{0t}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})\right) \overline{s_{jt}}(p_{t},g_{t})-E_{x_{t}}\left( 1-s_{0t}^{(ccp)}(x_{t},p_{t},g_{t})\right) \overline{s_{jt}}(p_{t},g_{t})\right) \cdot \\{} & {} \ \ \ \left( \left( 1-s_{0t}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})\right) \overline{s_{kt}}(p_{t},g_{t})-E_{x_{t}}\left( 1-s_{0t}^{(ccp)}(x_{t},p_{t},g_{t})\right) \overline{s_{kt}}(p_{t},g_{t})\right) \\= & {} \overline{s_{jt}}(p_{t},g_{t})\overline{s_{kt}}(p_{t},g_{t})\sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})\left( s_{0t}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})-E_{x_{t}}s_{0t}^{(ccp)}(x_{t},p_{t},g_{t})\right) ^{2}\ge 0. \end{aligned}$$

$\square$

1.2.3 Proof of Proposition 3

Own Price Elasticity:

Proof

It follows that:

$$\begin{aligned} \widehat{\eta _{jt}}(p_{t}^{0},g_{t}^{0})-\eta _{jt}^{(short)}(p_{t}^{0},g_{t}^{0})= & {} \left[ \alpha \left( -\left( \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) ^{2}+\sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})^{2}\right) \right] \frac{p_{jt}^{0}}{s_{jt}^{0}}\\\le & {} \frac{\sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\cdot \alpha s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})}{\sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})}p_{jt}^{0}\\\le & {} \frac{\left( \max _{x_{t}}\alpha s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) \cdot \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})}{\sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})}p_{jt}^{0}\\= & {} \alpha p_{jt}^{0}\left( \max _{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) . \end{aligned}$$

$\square$

Cross Price Elasticity:

Proof

When $\widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\ge 0$ holds, then:

$$\begin{aligned} \left| \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\right|= & {} \left[ \alpha s_{jt}^{0}s_{kt}^{0}-\alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\cdot s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})s_{kt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right] \frac{p_{jt}^{0}}{s_{kt}^{0}}\\\le & {} \frac{\alpha s_{jt}^{0}s_{kt}^{0}}{s_{kt}^{0}}p_{jt}^{0}\\= & {} \left( \alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) p_{jt}^{0}\\\le & {} \alpha \max _{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t}^{0})p_{jt}^{0}\ \left( \because \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})=1\right) . \end{aligned}$$

On the other hand, when $\widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\le 0$ holds, then:

$$\begin{aligned} \left| \widehat{\eta _{jkt}}(p_{t}^{0},g_{t}^{0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\right|= & {} \left[ -s_{jt}(p_{t}^{0},g_{t}^{0})s_{kt}(p_{t}^{0},g_{t}^{0})+\alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\cdot s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})s_{kt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right] \frac{p_{jt}^{0}}{s_{kt}^{0}}\\\le & {} \frac{\alpha \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})\cdot s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})s_{kt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})}{s_{kt}^{0}}p_{jt}^{0}\\\le & {} \frac{\left( \max _{x_{t}}\alpha s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})\right) \cdot \sum _{x_{t}\in X_{t}}Pr_{t}(x_{t})s_{kt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})}{s_{kt}^{0}}p_{jt}^{0}\\\le & {} \alpha \max _{x_{t}}s_{jt}^{(ccp)}(x_{t},p_{t}^{0},g_{t}^{0})p_{jt}^{0}. \end{aligned}$$

Hence, the statement holds. $\square$

1.3 Proof of the Statements in Section 3.4

As in Section 3.4, we assume that consumers have perfect foresight on the future price path.

1.3.1 Proof of Proposition 4

It is sufficient to prove the following lemma:

Lemma 2

The following equations hold:

$$\begin{aligned}{} & {} \eta _{jt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})-\eta _{jt}^{(short)}(p_{t}^{0}, g_{t}^{0})\\= & {} \frac{p_{jt}^{0}}{s_{jt}^{0}}\beta _{C}\int \alpha _{l}\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{ljt}^{(ccp)}(x_{t},p_{t}^{0})\lambda _{ljt}(x_{t},\{p_{t+\tau }^{0}\}_{\tau \ge 0})dP(l);\text { and}\\{} & {} \eta _{jkt}^{(long)}(\{p_{t+\tau }^{0}\}_{\tau \ge 0})-\eta _{jkt}^{(short)}(p_{t}^{0},g_{t}^{0})\\= & {} -\frac{p_{jt}^{0}}{s_{kt}^{0}}\beta _{C}\int \alpha _{l}\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{lkt}^{(ccp)}(x_{t},p_{t}^{0})\lambda _{ljkt}(x_{t},\{p_{t+\tau }^{0}\}_{\tau \ge 0})dP(l). \end{aligned}$$

To prove the lemma, we define the term

$$\begin{aligned} \widetilde{v_{ljt}}(x_{t},p_{t},g_{t})\equiv {\left\{ \begin{array}{ll} -\alpha _{l}p_{jt}+f_{ljt}+\phi _{ljt}(x_{t})+\beta _{C}g_{ljt}(x_{t}) &{} \text {if }j\in \mathcal {J}_{t}\\ f_{l0t}(x_{t})+\beta _{C}g_{l0t}(x_{t}) &{} \text {if }j=0 \end{array}\right. }. \end{aligned}$$

Using the term, $v_{ilt}(x_{t},p_{t},g_{t},\epsilon _{it},a_{t}=j)=\widetilde{v_{ljt}}(x_{t},p_{t},g_{t})+\epsilon _{ijt}$ holds. For convenience, we further define $\mathcal {A}_{t}=\mathcal {J}_{t}\cup \{0\}$. We omit the terms $p_{t+\tau }^{0}$ and $g_{t+\tau }^{0}$ in the following discussion to simplify the notation. First, we prove the following lemmas:

Lemma 3

Define the following term:

$$\begin{aligned} \zeta _{ljt+\tau ,u}(\widetilde{x_{t+1}})\equiv \sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}Pr(l\ choose\ j\ at\ t+\tau |x_{t+\tau -1},a_{t+\tau -1}=h)\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lht+\tau -u}}(x_{t+\tau -u})} \end{aligned}$$

for $\tau =2,\cdots ,T$ and $u=1,\cdots ,\tau -1$. Then, the following formula holds:

$$\begin{aligned} \zeta _{ljt+\tau ,u}(\widetilde{x_{t+1}})=\beta _{C}^{\tau -2}Pr(l\ choose\ j\ at\ t+\tau |\widetilde{x_{t+1}}). \end{aligned}$$

Proof

First,

$$\begin{aligned} \zeta _{ljt+\tau ,1}(\widetilde{x_{t+1}})= & {} \sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}Pr(l\ choose\ j\ at\ t+\tau |x_{t+\tau -1},a_{t+\tau -1}=h)\cdot \\{} & {} \frac{\partial V_{lt+\tau -1}^{C}(x_{t+\tau -1})}{\partial \widetilde{v_{lht+\tau -1}}(x_{t+\tau -1})}\sum _{x_{t+\tau -2}}\sum _{k\in \mathcal {A}_{t+\tau -2}}\frac{\partial \widetilde{v_{lkt+\tau -2}}(x_{t+\tau -2})}{\partial V_{lt+\tau -1}^{C}(x_{t+\tau -1})}\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lkt+\tau -2}}(x_{t+\tau -2})}\\= & {} \sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}Pr(l\ choose\ j\ at\ t+\tau |x_{t+\tau -1},a_{t+\tau -1}=h)\cdot \\{} & {} s_{lht+\tau -1}^{(ccp)}(x_{t+\tau -1})\sum _{x_{t+\tau -2}}\sum _{k\in \mathcal {A}_{t+\tau -2}}\beta _{C}\psi (x_{t+\tau -1}|x_{t+\tau -2},a_{t+\tau -2}=k)\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lkt+\tau -2}}(x_{t+\tau -2})}. \end{aligned}$$

Since

$$\begin{aligned}{} & {} \sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}Pr(l\ choose\ j\ at\ t +\tau |x_{t+\tau -1},a_{t+\tau -1}=h)\\{} & {} s_{lht+\tau -1}^{(ccp)}(x_{t+\tau -1}) \psi (x_{t+\tau -1}|x_{t+\tau -2},a_{t+\tau -2}=k)\\= & {} Pr(l\ choose\ j\ at\ t+\tau |x_{t+\tau -2},a_{t+\tau -2}=k) \end{aligned}$$

holds, we obtain:

$$\begin{aligned} \zeta _{ljt+\tau ,1}(\widetilde{x_{t+1}})= & {} \beta _{C}\sum _{x_{t+\tau -2}} \sum _{k\in \mathcal {A}_{t+\tau -2}}Pr(l\ choose\ j\ at\ t+ \tau |x_{t+\tau -2},a_{t+\tau -2}=k)\\{} & {} \frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lkt+\tau -2}}(x_{t+\tau -2})}\\= & {} \beta _{C}\zeta _{ljt+\tau ,2}(\widetilde{x_{t+1}}). \end{aligned}$$

By repeatedly applying the equation, we have:

$$\begin{aligned} \zeta _{ljt+\tau ,1}(\widetilde{x_{t+1}})=\beta _{C}^{\tau -2}\zeta _{ljt+\tau ,\tau -1}(\widetilde{x_{t+1}}). \end{aligned}$$

Here,

$$\begin{aligned} \zeta _{ljt+\tau ,\tau -1}(\widetilde{x_{t+1}})\equiv & {} \sum _{x_{t+1}}\sum _{k\in \mathcal {A}_{t+1}}Pr(l\ choose\ j\ at\ t+\tau |x_{t+1},a_{t+1}=k)\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lkt+1}}(x_{t+1})}\\= & {} \sum _{k\in \mathcal {A}_{t+1}}Pr(l\ choose\ j\ at\ t+\tau |\widetilde{x_{t+1}},a_{t+1}=k)\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lkt+1}}(\widetilde{x_{t+1}})}\\= & {} \sum _{k\in \mathcal {A}_{t+1}}Pr(l\ choose\ j\ at\ t+\tau |\widetilde{x_{t+1}},a_{t+1}=k)s_{lkt+1}^{(ccp)}(\widetilde{x_{t+1}})\\= & {} Pr(l\ choose\ j\ at\ t+\tau |\widetilde{x_{t+1}}). \end{aligned}$$

Hence, we obtain the formula. $\square$

Lemma 4

The following equation holds:

$$\begin{aligned} \frac{\partial E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=k\right] }{\partial p_{jt+\tau }}= & {} -\alpha _{l}\beta _{C}^{\tau -1}Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k). \end{aligned}$$

Proof

Using Lemma 3,

$$\begin{aligned}{} & {} \frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial p_{jt+\tau }} \\= & {} \sum _{x_{t+\tau }}\frac{\partial \widetilde{v_{ljt+\tau }}(x_{t+\tau })}{\partial p_{jt+\tau }}\frac{\partial V_{lt+\tau }^{C}(x_{t+\tau })}{\partial \widetilde{v_{ljt+\tau }}(x_{t+\tau })}\sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}\frac{\partial \widetilde{v_{lht+\tau -1}}(x_{t+\tau -1})}{\partial V_{lt+\tau }^{C}(x_{t+\tau })}\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lht+\tau -1}}(x_{t+\tau -1})}\\= & {} \sum _{x_{t+\tau }}(-\alpha _{l})s_{ljt+\tau }^{(ccp)}(x_{t+\tau })\sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}\beta _{C}\psi (x_{t+\tau }|x_{t+\tau -1},a_{t+\tau -1}=h)\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lht+\tau -1}}(x_{t+\tau -1})}\\= & {} -\beta _{C}\alpha _{l}\cdot \sum _{x_{t+\tau -1}}\sum _{h\in \mathcal {A}_{t+\tau -1}}Pr(l\ choose\ j\ at\ t+\tau |x_{t+\tau -1},a_{t+\tau -1}=h)\frac{\partial V_{lt+1}^{C}(\widetilde{x_{t+1}})}{\partial \widetilde{v_{lht+\tau -1}}(x_{t+\tau -1})}\\= & {} -\beta _{C}\alpha _{l}\zeta _{ljt+\tau ,\tau -1}(\widetilde{x_{t+1}})\\= & {} -\beta _{C}^{\tau -1}\alpha _{l}Pr(l\ choose\ j\ at\ t+\tau |\widetilde{x_{t+1}}). \end{aligned}$$

Consequently,

$$\begin{aligned}{} & {} \frac{\partial E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}}) |x_{t},a_{t}=k\right] }{\partial p_{jt+\tau }} \\= & {} \sum _{x_{t+1}}\frac{\partial V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})}{\partial p_{jt+\tau }}\psi (x_{t+1}|x_{t},a_{t}=k)\\= & {} -\alpha _{l}\beta _{C}^{\tau -1}\sum _{x_{t+1}}Pr(l\ choose\ j\ at\ t+\tau |x_{t+1})\psi (x_{t+1}|x_{t},a_{t}=k)\\= & {} -\alpha _{l}\beta _{C}^{\tau -1}Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k). \end{aligned}$$

$\square$

Lemma 5

The following equation holds:

$$\begin{aligned} \sum _{m\in \mathcal {A}_{t}}\sum _{\tau =1}^{T}\frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }}\frac{\partial s_{lkt}^{(ccp)}(x_{t})}{\partial \widetilde{v_{lmt}}(x_{t})}= & {} -\alpha _{l}\beta _{C}s_{lkt}^{(ccp)}(x_{t})\lambda _{ljkt}(x_{t}). \end{aligned}$$

Proof

First,

$$\begin{aligned}{} & {} \sum _{m\in \mathcal {A}_{t}}\sum _{\tau =1}^{T}\frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }}\frac{\partial s_{lkt}^{(ccp)}(x_{t})}{\partial \widetilde{v_{lmt}}(x_{t})}\\= & {} \sum _{\tau =1}^{T}\left[ \frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1}, \{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=k\right] }{\partial p_{jt+\tau }} \frac{\partial s_{lkt}^{(ccp)}(x_{t})}{\partial \widetilde{v_{lkt}}(x_{t})}\right. \\{} & {} \left. +\sum _{m\in \mathcal {A}_{t}-\{k\}}\frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1}, \{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }} \frac{\partial s_{lkt}^{(ccp)}(x_{t})}{\partial \widetilde{v_{lmt}}(x_{t})}\right] \\= & {} \sum _{\tau =1}^{T}\left[ \frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1}, \{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=k\right] }{\partial p_{jt+\tau }}s_{lkt}^{(ccp)} (x_{t})\left( 1-s_{lkt}^{(ccp)}(x_{t})\right) \right. \\{} & {} \left. -\sum _{m\in \mathcal {A}_{t}-\{k\}}\frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1}, \{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }} s_{lmt}^{(ccp)}(x_{t})s_{lkt}^{(ccp)}(x_{t})\right] \\= & {} \beta _{C}s_{lkt}^{(ccp)}(x_{t})\sum _{\tau =1}^{T}\left[ \frac{\partial E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=k\right] }{\partial p_{jt+\tau }}\right. \\{} & {} \left. -\sum _{m\in \mathcal {A}_{t}}\frac{\partial E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }}s_{lmt}^{(ccp)}(x_{t})\right] . \end{aligned}$$

By Lemma 4,

$$\begin{aligned}{} & {} \frac{\partial E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=k\right] }{\partial p_{jt+\tau }}-\sum _{m\in \mathcal {A}_{t}}\frac{\partial E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }}s_{lmt}^{(ccp)}(x_{t})\\= & {} -\alpha _{l}\beta _{C}^{\tau -1}Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k)+\alpha _{l}\beta _{C}^{\tau -1}\sum _{m\in \mathcal {A}_{t}}Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=m)\cdot s_{lmt}^{(ccp)}(x_{t})\\= & {} -\alpha _{l}\beta _{C}^{\tau -1}\left[ Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k)-Pr(l\ choose\ j\ at\ t+\tau |x_{t})\right] . \end{aligned}$$

Then, we obtain:

$$\begin{aligned}{} & {} \sum _{m\in \mathcal {A}_{t}}\sum _{\tau =1}^{T}\frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1},\{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }}\frac{\partial s_{lkt}^{(ccp)}(x_{t})}{\partial \widetilde{v_{lmt}}(x_{t})}\\= & {} -\alpha _{l}\beta _{C}s_{lkt}^{(ccp)}(x_{t}) \sum _{\tau =1}^{T}\beta _{C}^{\tau -1}\left[ Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=k)\right. \\{} & {} \left. -Pr(l\ choose\ j\ at\ t+\tau |x_{t})\right] \\= & {} -\alpha _{l}\beta _{C}s_{lkt}^{(ccp)}(x_{t})\lambda _{ljkt}(x_{t}). \end{aligned}$$

$\square$

1.3.2 Proof of Lemma 2

Own Price Elasticity:

Proof

It follows that:

$$\begin{aligned}{} & {} \eta _{jt}^{(long)}-\eta _{jt}^{(short)} = -\frac{p_{jt}^{0}}{s_{jt}^{0}} \int \left( \sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})\sum _{m\in \mathcal {A}_{t}} \sum _{\tau =1}^{T}\frac{\partial \beta _{C}E\left[ V_{lt+1}^{C}(x_{t+1}, \{p_{t+s}\}_{s\in \mathbb {N}})|x_{t},a_{t}=m\right] }{\partial p_{jt+\tau }} \frac{\partial s_{ljt}^{(ccp)}(x_{t})}{\partial \widetilde{v_{lmt}}(x_{t})}\right) dP(l)\\{} & {} \quad = -\frac{p_{jt}^{0}}{s_{jt}^{0}}\int \left( \sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t}) \left( -\alpha _{l}\beta _{C}s_{ljt}^{(ccp)}(x_{t})\lambda _{ljt}(x_{t})\right) \right) dP(l)\ \ \ (\because \text {Lemma }5)\\{} & {} \quad = \beta _{C}\frac{p_{jt}^{0}}{s_{jt}^{0}}\int \alpha _{l}\left( \sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{ljt}^{(ccp)}(x_{t})\lambda _{ljt}(x_{t})\right) dP(l). \end{aligned}$$

$\square$

Cross Price Elasticity:

We can prove as in the case of own elasticity.

1.3.3 Proof of Proposition 5

Cross Price Elasticity:

Proof

In the following, we omit $p_{t+\tau }^{0}$ and $g_{t+\tau }^{0}$ to make the notation simpler. It follows that:

$$\begin{aligned}{} & {} \left| \eta _{jkt}^{(long)}-\eta _{jkt}^{(short)}\right| = \frac{p_{jt}^{0}}{s_{kt}^{0}}\beta _{C}\int \alpha _{l}\sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{lkt}^{(ccp)}(x_{t})\left| \lambda _{ljkt}(x_{t})\right| dP(l)\\{} & {} \quad \le \beta _{C}p_{jt}^{0}\frac{\int \sum _{x_{t}\in X_{t}}Pr_{lt}(x_{t})s_{lkt}^{(ccp)}(x_{t})dP(l)}{s_{kt}^{0}}\left( \max _{l,x_{t}}\alpha _{l}\left| \lambda _{ljkt}(x_{t})\right| \right) \\{} & {} \quad = \beta _{C}p_{jt}^{0}\left( \max _{l,x_{t}}\alpha _{l}\left| \sum _{\tau =1}^{T}\beta _{C}^{\tau -1}\left[ Pr(l\ choose\ j\ at\ t+\tau |x_{t},a_{t}=j)-Pr(l\ choose\ j\ at\ t+\tau |x_{t})\right] \right| \right) \\{} & {} \quad = \beta _{C}p_{jt}^{0}\left( \max _{l,x_{t}}\alpha _{l}\left| \sum _{\tau =1}^{T}\beta _{C}^{\tau -1}\sum _{x_{t+\tau }}\left[ s_{ljt+\tau }^{(ccp)}(x_{t+\tau })Pr(x_{t+\tau }|x_{t},a_{t}=j)-s_{ljt+\tau }^{(ccp)}(x_{t+\tau })Pr(x_{t+\tau }|x_{t})\right] \right| \right) \\{} & {} \quad \le \beta _{C}p_{jt}^{0}\left( \max _{l}\alpha _{l}\cdot 2\sum _{\tau =1}^{T}\beta _{C}^{\tau -1}\max _{x_{t+\tau }}s_{ljt+\tau }^{(ccp)}(x_{t+\tau })\right) \ (\because \sum _{x_{t+\tau }}Pr(x_{t+\tau }|x_{t},a_{t}=j)=\sum _{x_{t+\tau }}Pr(x_{t+\tau }|x_{t})=1)\\{} & {} \quad = 2\beta _{C}\frac{1-\beta _{C}^{T}}{1-\beta _{C}}p_{jt}^{0}\cdot \max _{x_{t+\tau }}s_{ljt+\tau }^{(ccp)}(x_{t+\tau }). \end{aligned}$$

$\square$

Own Price Elasticity:

We can prove the statement as in the case of cross price elasticity.

1.4 Proof of the Statements in Section 4

1.4.1 Proof of Corollary 2

Proof

Let $\mathcal {J}_{gt}$ be the set of products in the same package size, and let $j,k\in \mathcal {J}_{gt}$. Then, $\phi _{jt}(x_{t})=\phi _{gt}(x_{t})$ holds for all $j\in \mathcal {J}_{gt}$, because $C_{t}^{*}|_{j}=\arg \max _{C_{t}}U(C_{t})-F(x_{t}+q_{jt}-C_{t})+\beta _{C}E\left[ V_{lt+1}^{C}(x_{t}+q_{jt}-C_{t},p_{t+1})|x_{t},p_{t}\right]$ takes common values for all $j\in \mathcal {J}_{gt}$ (same package size ($q_{jt}$) products), and $\phi _{jt}(x_{t})=U(C_{t}^{*}|_{j})-F(x_{t}+q_{jt}-C_{t}^{*}|_{j})$ holds by construction. Moreover, since $x_{t+1}=x_{t}+q_{t}-C_{t}$, continuation values $g_{jt}(x_{t})$ take common values for all $j\in \mathcal {J}_{gt}$.

Then, $s_{j|g,t}(p_{t},g_{t})\equiv s_{jt}^{(ccp)}(x_{t},p_{t},g_{t})/\left( \sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)}(x_{t},p_{t},g_{t})\right)$, the probability that a consumer purchases product j conditional on purchasing products in $\mathcal {J}_{gt}$, does not depend on states $x_{t}$, because

$s_{j|g,t}(p_{t},g_{t})=\frac{\frac{\exp \left( \phi _{jt}(x_{t})-\alpha p_{jt}+f_{jt}+\beta _{C}g_{jt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }}{\sum _{m\in \mathcal {J}_{gt}}\frac{\exp \left( \phi _{mt}(x_{t})-\alpha p_{mt}+f_{mt}+\beta _{C}g_{mt}(x_{t})\right) }{\exp \left( V_{t}^{C}(x_{t},p_{t},g_{t})\right) }}=\frac{\exp (-\alpha p_{jt}+f_{jt})}{\sum _{m\in \mathcal {J}_{gt}}\exp \left( -\alpha p_{mt}+f_{mt}\right) }$ holds. Hence, we obtain:

$$\begin{aligned}{} & {} Cov_{jkt}(p_{t},g_{t}) = \sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}}) \left( \left( \sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})\right) s_{j|g,t}(p_{t},g_{t})\right. \\{} & {} \quad \left. -E_{x_{t}}\left( \sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)}(x_{t},p_{t},g_{t}) \right) s_{j|g,t}(p_{t},g_{t})\right) \cdot \\{} & {} \left( \left( \sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t}) \right) s_{k|g,t}(p_{t},g_{t})\right. \\{} & {} \quad \left. -E_{x_{t}}\left( \sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)} (x_{t},p_{t},g_{t})\right) s_{k|g,t}(p_{t},g_{t})\right) \\{} & {} \quad = s_{j|g,t}(p_{t},g_{t})s_{k|g,t}(p_{t},g_{t})\sum _{\widetilde{x_{t}}\in X_{t}}Pr_{t}(\widetilde{x_{t}})\\{} & {} \quad \left( \sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)}(\widetilde{x_{t}},p_{t},g_{t})-E_{x_{t}}\sum _{m\in \mathcal {J}_{gt}}s_{mt}^{(ccp)}(x_{t},p_{t},g_{t})\right) ^{2}\ge 0. \end{aligned}$$

$\square$

1.5 Proof of the Statements in Section 5

1.5.1 Proof of Equation (20)

Proof

Since we consider the case where consumers do not purchase under the state $x_{t}\ne 0$, let $\phi _{jt}(x_{t}\ne 0)=-\infty$ so that $s_{jt}^{(ccp)}(x_{t}\ne 0,p_{t},g_{t})=0$. Then, $\frac{\exp (\phi _{jt}(x_{t}\ne 0)+\beta _{C}g_{jt}(x_{t}\ne 0))}{\exp (V_{t}^{C}(x_{t}\ne 0,p_{t},g_{t}))}=0$ and $\frac{\exp (f_{0t}(x_{t}\ne 0)+\beta _{C}g_{0t}(x_{t}\ne 0))}{\exp (V_{t}^{C}(x_{t}\ne 0,p_{t},g_{t}))}=1$. Hence,

$$\begin{aligned} \widehat{c_{jt}}= & {} \log \left( \frac{\frac{\exp (\phi _{jt}(x_{t}=0) +\beta _{C}g_{jt}(x_{t}=0))}{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0) +\frac{\exp (\phi _{jt}(x_{t}\ne 0) +\beta _{C}g_{jt}(x_{t}\ne 0))}{\exp (V_{t}^{C}(x_{t}\ne 0,p_{t},g_{t}))} \left( 1-Pr_{t}(x_{t}=0)\right) }{\frac{\exp (f_{0t}(x_{t}=0)+\beta _{C}g_{0t}(x_{t}=0))}{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)+\frac{\exp (f_{0t}(x_{t}\ne 0)+\beta _{C}g_{0t}(x_{t}\ne 0))}{\exp (V_{t}^{C}(x_{t}\ne 0,p_{t},g_{t}))}\left( 1-Pr_{t}(x_{t}=0)\right) }\right) \\= & {} \log \left( \frac{\frac{\exp (\phi _{jt}(x_{t}=0)+\beta _{C}g_{jt}(x_{t}=0))}{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)}{\frac{\exp (f_{0t}(x_{t}=0)+\beta _{C}g_{0t}(x_{t}=0))}{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)+\left( 1-Pr_{t}(x_{t}=0)\right) }\right) \\= & {} \log \left( \frac{\frac{\exp (\phi _{jt}(x_{t}=0)+\beta _{C}g_{jt}(x_{t}=0))}{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)}{1-\frac{\sum _{k\in \mathcal {J}_{t}}\exp \left( -\alpha p_{kt}+f_{kt}+\phi _{kt}(x_{t}=0)+\beta _{C}g_{kt}(x_{t}=0)\right) }{\exp (V_{t}^{C}(x_{t}=0,p_{t},g_{t}))}Pr_{t}(x_{t}=0)}\right) . \end{aligned}$$

$\square$