Consumer Choice Problem

A consumer (purchaser of priced quantifiable goods in a market) is often modeled as facing a problem of utility maximization given a budget constraint, or alternately, a problem of expenditure minimization given a desired level of utility.

A typical formalization

In formalizing the consumer's constrained optimization problem from both sides, we will consider the "primal" problem of utility maximization and its "dual" problem of expenditure minimization.

The symbols used (with underlining indicating vectors) are:

$U\equiv$ utility

$I\equiv$ income

${\underline {x}}\equiv$ (units of) goods

${\underline {p_{x}}}\equiv$ prices (per unit)

${\underline {\alpha }}\equiv$ parameters

$V\equiv$ utility (indirect)

$E\equiv$ expenditure

Primal problem: utility maximization

The consumer's objective function, $U({\underline {x}},{\underline {\alpha }})$, is maximized subject to the budget constraint $I-{\underline {p_{x}}}\cdot \;{\underline {x}}\geq 0$. The Lagrangean function for this optimization is thus:

${\mathcal {L}}=U({\underline {x}},{\underline {\alpha }})+\lambda (I-{\underline {p_{x}}}\cdot \;{\underline {x}})$

The optimal choices ${\underline {x}}^{*}$ are Marshallian demand functions of ${\underline {p_{x}}}$, ${\underline {\alpha }}$, and $I$.

The value function $V$, with arguments (i.e., independent variables) ${\underline {p_{x}}}$, ${\underline {\alpha }}$, and $I$, is equal to the objective function evaluated at the optimal choices:

$V({\underline {p_{x}}},{\underline {\alpha }},I)=U({\underline {x}}^{*})$

Dual problem: expenditure minimization

The consumer's objective function, ${\underline {p_{x}}}\cdot \;{\underline {x}}$, is minimized subject to the constraint $U({\underline {x}},{\underline {\alpha }})\geq {\bar {U}}$. The Lagrangean function for this optimization is thus:

${\mathcal {M}}={\underline {p_{x}}}\cdot \;{\underline {x}}+\mu [{\bar {U}}-U({\underline {x}},{\underline {\alpha }})]$

The optimal choices ${\underline {x}}^{0}$ are Hicksian demand functions of ${\underline {p_{x}}}$, ${\underline {\alpha }}$, and ${\bar {U}}$.

The value function $E$, with arguments (i.e., independent variables) ${\underline {p_{x}}}$, ${\underline {\alpha }}$, and ${\bar {U}}$, is equal to the objective function evaluated at the optimal choices:

$E({\underline {p_{x}}},{\underline {\alpha }},{\bar {U}})={\underline {p_{x}}}\cdot \;{\underline {x}}^{0}.$

Duality relationships

Let $V({\underline {P_{x}}},{\underline {\alpha }},I)=U^{*}$ be the maximal level of utility attainable in the primal problem (given the prices and other parameters), and then let that $U^{*}$ be the fixed level of utility, ${\bar {U}}$, for the related dual problem. Then we have the following utility relationships:

${\underline {x}}^{*}({\underline {P_{x}}},{\underline {\alpha }},I)={\underline {x}}^{0}({\underline {P_{x}}},{\underline {\alpha }},{\bar {U}})$

$E({\underline {P_{x}}},{\underline {\alpha }},U^{*})={\underline {p_{x}}}\cdot \;{\underline {x}}^{0}=I$

$V^{*}({\underline {P_{x}}},{\underline {\alpha }},I)=U({\underline {x}}^{*},{\underline {\alpha }})=U^{*}$

$\mu ^{0}={\frac {1}{\lambda ^{*}}}$

Sources

• Lectures by Laurence Iannaccone, fall 2005.
• Eugene Silberberg, The Structure of Economics, Chapter 14.