Dynamic Consumer Demand

Abstract

In this chapter, alternative dynamic models of consumer behavior are presented and evaluated. I first review historical models, including the partial adjustment model and state adjustment model, which are mainly empirical in nature. Next, models based on the theory of the consumer are presented. These modeling approaches are both myopic models (i.e., models that ignore consumer’s choices on future marginal utility), and fully intertemporal models. The intertemporal models discussed include both single-equation models and multivariate models. The rational addiction model is an important single-equation model. Multivariate generalizations of intertemporal models include the Multivariate Rational Addiction model (MRA), the SNAP model, and Habits-as-Durables model. To show how to empirically implement the system of dynamic consumer demand functions approach, I present a data set for alcoholic beverages and estimate demand parameters for the MRA model. Results indicate close conformity with theory.

Appendix: Proof of Theorem 4

Following Wohlgenant (2012), Eq. (9.21) can be written as a matrix polynomial in

Lags as follows:

$$\left( {\beta^{ - 1} \varvec{I}L^{2} + \beta^{ - 1} \varvec{B}^{ - 1} \varvec{A}L + \varvec{I}} \right)L^{ - 1} E_{\varvec{t}} \varvec{q}_{t} = \lambda \beta^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t}$$

Because $\varvec{B}$ is a positive definite matrix, we know there exists an orthogonal matrix $\varvec{P}$ such that $\varvec{PBP^{\prime}} = \varvec{N}$, a diagonal matrix with all positive diagonal elements. Define $\varvec{S} = \varvec{N}^{0.5} \varvec{P}$, where $\varvec{N}^{0.5}$ is a diagonal matrix whose elements are positive square roots. Because $\varvec{B} = \varvec{P^{\prime}NP} = \varvec{S^{\prime}S}$, we see that $\varvec{B}$ can be written as the product of two square root matrices. Because ${-}\varvec{A}$ is a positive definite matrix, $- \varvec{S^{\prime}}^{ - 1} \varvec{AS}^{ - 1}$ is also positive definite. Therefore, there exists an orthogonal matrix $\varvec{R}$ such that $- \varvec{RS^{\prime}}^{ - 1} \varvec{AS}^{ - 1} \varvec{R^{\prime}} = \varvec{M}$, a diagonal matrix whose diagonal elements are positive. Define $\varvec{Q} = \varvec{S}^{ - 1} \varvec{R^{\prime}}$, then $- \varvec{Q}^{ - 1} \varvec{B}^{ - 1} \varvec{AQ} = - \varvec{RS^{\prime}}^{ - 1} \varvec{A} \varvec{S}^{ - 1} \varvec{R^{\prime}} = \varvec{M}$. Thus, we have proved $- \varvec{B}^{ - 1} \varvec{A}$ is similar to $- \varvec{A}$, implying it has the same positive roots. The rest of the steps are identical to Wohlgenant (2012), where we multiply both sides of the matrix polynomial by $\varvec{Q}^{ - 1}$ and post-multiply by $\varvec{Q}$ to obtain

$$\varvec{Q}^{ - 1} \left( {\beta^{ - 1} \varvec{Q}L^{2} - \beta^{ - 1} \varvec{QM}L + \varvec{Q}} \right)\varvec{Q}^{ - 1} L^{ - 1} E_{\varvec{t}} \varvec{q}_{t} = \lambda \beta^{ - 1} \varvec{Q}^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t} \mathop \Rightarrow \limits_{{}}$$

$$\left( {\beta^{ - 1} \varvec{I}L^{2} - \beta^{ - 1} \varvec{M}L + \varvec{I}} \right)\varvec{Q}^{ - 1} L^{ - 1} E_{\varvec{t}} \varvec{q}_{t} = \lambda \beta^{ - 1} \varvec{Q}^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t} \mathop \Rightarrow \limits_{{}}$$

$$\left( {\varvec{I} -\varvec{\varLambda}_{1} L} \right)\left( {\varvec{I} -\varvec{\varLambda}_{2} L} \right)\varvec{Q}^{ - 1} L^{ - 1} E_{\varvec{t}} \varvec{q}_{t} = \lambda \beta^{ - 1} \varvec{Q}^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t} \mathop \Rightarrow \limits_{{}}$$

$$-\varvec{\varLambda}_{2} L\left( {\varvec{I} -\varvec{\varLambda}_{2}^{ - 1} L^{ - 1} } \right)\left( {\varvec{I} -\varvec{\varLambda}_{1} L} \right)\varvec{Q}^{ - 1} L^{ - 1} E_{\varvec{t}} \varvec{q}_{t} = \lambda \beta^{ - 1} \varvec{Q}^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t} \mathop \Rightarrow \limits_{{}}$$

$$\left( {\varvec{I} -\varvec{\varLambda}_{1} L} \right)\varvec{Q}^{ - 1} E_{\varvec{t}} \varvec{q}_{t} = - \left( {\varvec{I} -\varvec{\varLambda}_{2}^{ - 1} L^{ - 1} } \right)^{ - 1} \lambda \beta^{ - 1}\varvec{\varLambda}_{2}^{ - 1} \varvec{Q}^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t} \mathop \Rightarrow \limits_{{}}$$

$$\left( {\varvec{I} - \varvec{Q\varLambda }_{1} \varvec{Q}^{ - 1} L} \right)E_{\varvec{t}} \varvec{q}_{t} = - \varvec{Q}\left( {\varvec{I} -\varvec{\varLambda}_{2}^{ - 1} L^{ - 1} } \right)^{ - 1} \lambda \beta^{ - 1} \varvec{\varLambda }_{2}^{ - 1} \varvec{Q}^{ - 1} \varvec{B}^{ - 1} E_{t} \varvec{p}_{t}$$

where we use the results that $\varvec{M} =\varvec{\varLambda}_{1} +\varvec{\varLambda}_{2}$ and $\varvec{\varLambda}_{1}\varvec{\varLambda}_{2} = \beta^{ - 1} \varvec{I}$. Wohlgenant (2012) shows that there are exactly $2n$ distinct positive roots such that $\lambda_{1i} < 1$ and $\lambda_{2i} = \beta^{ - 1} \lambda_{1i}^{ - 1} > 1$. After substituting $E_{\varvec{t}} \varvec{q}_{t} = \varvec{q}_{t} - \varvec{u}_{t}$, we obtain the desired result. QED

Problems

• 9.1 Derive the discrete version of the State Adjustment Model, Eq. (9.10), and show how the reduced-form coefficients, the $K_{i}$’s, are related to the underlying structural parameters. Show that Eq. (9.11) is the overidentifying restriction.

• 9.2 Derive Becker et al.’s (1994) rational addiction model. Discuss the issues in estimation of the model. How would you obtain short-run and long-run price elasticities of demand from this model? How do the elasticities change depending upon not only the length of time for adjustment to price changes but also the consumer’s perception of how long the price change is expected to last?

• 9.3 Given the intraperiod profit function, Eq. (9.30), derive the SNAP model, Eq. (9.31). Discuss how expectations are modeled and how the model nests the AIDS model. Discuss why this model is highly restrictive regarding intertemporal substitution between individual goods.

• 9.4 Taxes on alcoholic beverages have favored wine and beer over spirits. The federal tax shares of alcohol price are approximately 4.6 percent for beer, 2.6 percent for wine, and 9.7 percent for spirits, respectively. Taxes are levied on a per gallon basis. Using the elasticities from the estimated MRA model in Table 9.3, estimate the effects on beer, wine, and spirits consumption from increasing the tax rates for beer and wine by 30 percent, but leaving the tax rate for spirits the same. How do the short-run and long-run effects (for permanent price change) differ?

  Table 9.3 Elasticities of Dynamic Demand for Wine, Beer, and Spirits