Structural Models in Marketing: Consumer Demand and Search

Abstract

As marketers move away from being focused only on “local” effects of marketing activities, e.g., what happens when I change price by 1%, in order to better understand the consequences of broader shifts in policy, the need for structural models has also grown. In this chapter, I will focus on a small subset of such “structural models” and provide brief discussions of what we mean by structural models, why we need them, the typical classes of structural models that we see being used by marketers these days, along with some examples of these models. My objective is not to provide a comprehensive review. Such an endeavor is far beyond my current purview. Rather, I would like to provide a basic discussion of structural models in the context of the marketing literature. In particular, to keep the discussion focused, I will limit myself largely to models of demand rather than models of firm behavior.

I thank Anita Rao and S Sriram for their useful comments on an earlier version. My thanks to the Kilts Center at the University of Chicago for financial support. Note that parts of this chapter appear elsewhere in “Handbook of Marketing Analytics with Applications in Marketing, Public Policy, and Litigation”: (Edward Elgar; Natalie Mizik & Dominique M. Hanssens, Editors).

Appendix: Deriving the Indirect Utility Function (Equation (6.15))

In the marketing literature, it is typical to characterize a consumer’s utility function as follows

$$ u_{ijt} = \alpha_{ij} + x_{jt} \beta_{i} + \epsilon_{ijt} $$

(6.14)

i: consumer, j: brand, t: time period. And one of the $x_{jt}$’s was price, $p_{jt}$. Strictly speaking, however, since the above equation has prices embedded in it, it is better referred to as an “indirect utility” function that is obtained from a direct utility function that is being maximized subject to a budget constraint. Here we show how the direct utility function as in Eq. (6.14) of the chapter may lead to an indirect utility function like Eq. (6.14) above. Consider a world where there are only 2 “goods”—Strawberry yogurt and raspberry yogurt, represented by their respective quantity $q_{1}$ and $q_{2}$. The consumer derives utility from these two goods according to the relationship:

$$ u (q_{1}, q_{2}) = \psi_{1} q_{1} + \psi_{2} q_{2} $$

where $\psi_{1}$, $\psi_{2} > 0$, $\psi_{1}$ and $\psi_{1}$ are the “quality” indices of the 2 flavors. Then the consumers’ indifference curves would look like the dotted lines in the figure here.

Now the consumer’s budget constraint for these goods can be written as $p_{1} q_{1} + p_{2} q_{2} \leq B$ where p ₁ and p ₂ are the prices of the two goods. We represent the budget set by the continuous line in the figure.

Given the linear indifference curves and budget set, the utility maximizing condition for the consumer lies in a “corner” i.e. to spend all the money (budget) on either strawberry or on raspberry. In the above case spending all the money on raspberry (good2) yields lower utility to the consumer than the alternative. So the consumer makes the “discrete” choice of picking only strawberry yogurt. This is because $u_{D} > u_{A}$ in the figure. At the $q_{1^{*}}$ “corner”, $q_{2^{*}} = 0$ and the consumer obtains utility $u_{D}$. At the $q_{2^{*}}$ “corner”, $q_{1^{*}} = 0$ and the consumer obtains utility $u_{A}$. So $u_{D} = \psi_{1} q_{1}$ (i.e. utility when $q_{2} = 0$) and $u_{A} = \psi_{2} q_{2}$ (i.e. when $q_{1} = 0$). From the budget constraint we know that when, $q_{2} = 0$; $q_{1} = (B/{p_{1}})$, when $q_{1} = 0$; $q_{2} = (B/{p_{2}})$. So the “indirect utility” $V_{D} = (\psi_{1} B)/(p_{1})$ and $V_{A} = (\psi_{2} B)/(p_{2})$.

Since we know $V_{D} > V_{A}$ in the above case, we can write $(\psi_{1} B)/(p_{1}) > (\psi_{2} B)/(p_{2})$ and since $B > 0$ then $(\psi_{1})/(p_{1}) > (\psi_{2})/(p_{2})$. We can now characterize $\psi_{1}$ and $\psi_{2} > 0$ by writing them as:

$$ \psi_{1} = \exp(x_{1} {\tilde \beta} + e_{1}) \quad \psi_{2} = \exp(x_{2} {\tilde \beta} + e_{2}) $$

where $x_{1}$ and $x_{2}$ are observable attributes (to the researcher) and $e_{1}$ is unobservable as is $e_{2}$. So,

$$ (\exp(x_{1} {\tilde \beta} + e_{1}))/(p_{1}) = (\exp(x_{2} {\tilde \beta} + e_{2}))/(p_{2}) $$

Taking logs on both sides:

$$ x_{1} {\tilde \beta} + e_{1} - \ln p_{1} {\text{ > }}x_{2} {\tilde \beta} + e_{2} - \ln p_{2} $$

(6.15)