Economic Models of Choice

Abstract

This chapter provides an introduction to choice models based on the principle of direct utility maximization. Models of direct utility are characterized by specifications of the utility function and accompanying budget constraint that allows separation of what is gained (i.e., utility) from that which is given up in an exchange. Direct utility maximization rationalizes observed choice as arising from goal-oriented consumers who are resource constrained. Marketing data overwhelmingly reflects goal-oriented behavior on the part of consumers in the high rate of zero’s present in disaggregate data, indicating that most people choose to not purchase most products that are available. By developing alternative models of direct utility maximization, we hope to spur additional research on utility formation and a more in-depth understanding of optimal firm reaction to the demands and constraints of consumers.

Appendix

Consider the direct utility model:

$$ \max u\left( x \right) =\sum \limits _{k}^{{}}{\frac{{{\psi }_{k}}}{\gamma }}\ln \left( \gamma {{x}_{k}}+1 \right) \text { subject to } p'x \le E $$

where the $\psi 's$ are assumed to sum to one (i.e., $\sum \psi _k = 1$) and $\gamma > 0$. Solving for the utility maximizing quantities, $x^*$, by the Lagrangian method will give rise to the following objective function:

$$ L=\sum \limits _{k}^{{}}{\frac{{{\psi }_{k}}}{\gamma }}\ln \left( \gamma {{x}_{k}}+1 \right) -\lambda \left( {p}'x-E \right) $$

The FOC’s for optimal demand ($x^*$) are:

$$\begin{aligned} 0=\frac{\partial L}{\partial {{x}_{1}}}=\frac{{{\psi }_{1}}}{\gamma {{x}_{1}}+1}-\lambda {{p}_{1}} \qquad&\Leftrightarrow \qquad {{\psi }_{1}}=\lambda {{p}_{1}}\left( \gamma {{x}_{1}}+1 \right) \\ 0=\frac{\partial L}{\partial {{x}_{2}}}=\frac{{{\psi }_{2}}}{\gamma {{x}_{2}}+1}-\lambda {{p}_{2}} \qquad&\Leftrightarrow \qquad {{\psi }_{2}}=\lambda {{p}_{2}}\left( \gamma {{x}_{2}}+1 \right) \\&\vdots \\ 0=\frac{\partial L}{\partial {{x}_{k}}}=\frac{{{\psi }_{k}}}{\gamma {{x}_{k}}+1}-\lambda {{p}_{k}} \qquad&\Leftrightarrow \qquad {{\psi }_{k}}=\lambda {{p}_{k}}\left( \gamma {{x}_{k}}+1 \right) \end{aligned}$$

Using $\sum \psi _k = 1$, we solve for $\lambda$:

$$ 1=\sum \limits _{k=1}^{K}{{{\psi }_{k}}}=\sum \limits _{k=1}^{K}{\lambda {{p}_{k}}\left( \gamma {{x}_{k}}+1 \right) } $$

or

$$ \lambda =\frac{1}{\sum \limits _{k=1}^{K}{{{p}_{k}}\left( \gamma {{x}_{k}}+1 \right) }}=\frac{1}{\gamma E+\sum \limits _{k=1}^{K}{{{p}_{k}}}} $$

Substituting $\lambda$ back into FOCs yields optimal demand equations:

$$\begin{aligned} x_{k}^{*}&=\frac{1}{\gamma }\left( \frac{{{\psi }_{k}}}{\lambda {{p}_{k}}}-1 \right) \\&=\frac{1}{\gamma }\left( \frac{{{\psi }_{k}}}{{{p}_{k}}}\left( \gamma E+\sum \limits _{k}^{{}}{{{p}_{k}}} \right) -1 \right) \end{aligned}$$

Substituting $x^*$ into the direct utility function allows us to obtain the expression for indirect utility (V):

$$\begin{aligned} V&\equiv u\left( {{x}^{*}} \right) \\&=\sum \limits _{k}^{{}}{\frac{{{\psi }_{k}}}{\gamma }}\ln \left( \gamma x_{k}^{*}+1 \right) \\&=\sum \limits _{k}^{{}}{\frac{{{\psi }_{k}}}{\gamma }}\ln \left( \gamma \left( \frac{1}{\gamma }\left( \frac{{{\psi }_{k}}}{{{p}_{k}}}\left( \gamma E+\sum \limits _{k}^{{}}{{{p}_{k}}} \right) -1 \right) \right) +1 \right) \\&=\sum \limits _{k}^{{}}{\frac{{{\psi }_{k}}}{\gamma }}\ln \left( \frac{{{\psi }_{k}}}{{{p}_{k}}}\left( \gamma E+\sum \limits _{k}^{{}}{{{p}_{k}}} \right) \right) \\&=\sum \limits _{k}^{{}}{\frac{{{\psi }_{k}}}{\gamma }}\left( \ln {{\psi }_{k}}-\ln {{p}_{k}}+\ln \left( \gamma E+\sum \limits _{k}^{{}}{{{p}_{k}}} \right) \right) \end{aligned}$$