Preference-based Demand: utility maximization problem

We then assume that the consumer has rational, continuous and locally nonsatiated preferences, represented by the utility function $u(\cdot )$​.

Assumptions

1. All prices are strictly positive ($p\gg 0$​) and linear (each unit of good $k$​ costs the same).

2. Wealth is strictly positive ($w>0$).

3. Goods are divisible:$x\in {\mathbb{R}}_{+}^L$​ and consumer can consume any bundle in budget set.

4. Set of goods is finite.

5. The consumption space is $X={\mathbb{R}}_{+}^L$.

$$\begin{array}{rl}\underset{x\ge 0}{max}& u(x)\\ & \mathrm{s}.\mathrm{t}.p\cdot x\le w.\end{array}$$

Solution

Proposition 3.D.1 (Solution existence of UMP). If $p\gg 0$​ and $u(\cdot )$​​​ is continuous, then the utility maximization problem has a solution.

Proposition 3.D.2 (Walrasian demand correspondence properties). Suppose that $u(\cdot )$​ is a continuous utility function representing a locally nonsatiated preference relation $\succsim$​ defined on the consumption set $X={\mathbb{R}}^{+}$​, then the Walrasian demand correspondence $x(p,w)$​ possesses the following properties:

1. Homogeneity of degree zero in $(p,w)$: $x(\alpha p,\alpha w)=x(p,w)$​ for any $p,w$​ and scalar $\alpha >0$​.

2. Walras' law: $p\cdot x=w$​ for all $x\in x(p,w)$​.

3. Convexity/uniqueness: If $\succsim$ is convex, so that $u(\cdot )$ is quasiconcave, then $x(p,w)$ is a convex set. Moreover, if $\succsim$ is strictly convex, so that $u(\cdot )$ is strictly quasiconcave, then $x(p,w)$​ consists of a single element.

If $u(\cdot )$ is continuously differentiable, an optimal consumption bundle $x^{\ast }\in x(p,w)$​ can be characterized in a very useful manner by means of first-order conditions.

Proposition ( Kuhn-Tucker (necessary) conditions).The Kuhn-Tucker (necessary) conditions (see Section M.K of the Mathematical Appendix) say that

• If $x^{\ast }\in x(p,w)$ is a solution to the UMP, then there exists a Lagrange multiplier $\lambda >0$ such that for all $\mathcal{l}=1,\cdots ,L$:$$\begin{array}{}\text{(3.D.1)}& \frac{\partial u(x^{\ast })}{\partial x_{\ell }}\le \lambda p_{\ell },\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{i}\mathrm{f}{x}_{\ell }^{\ast }>0.\end{array}$$

Interior optimum

Thus, if we are at an interior optimum (i.e., if $x^{\ast }\gg 0$), we must have

$$\begin{array}{}\text{(3.D.4)}& \mathrm{\nabla }u(x^{\ast })=\lambda p.\end{array}$$

If $\mathrm{\nabla }u(x^{\ast })\gg 0$, this is equivalent to the requirement​ that for any two goods $\ell$​ and $k$, we have

$$\begin{array}{}\text{(3.D.5)}& \frac{\partial u(x^{\ast })/\partial x_{\ell }}{\partial u(x^{\ast })/\partial x_k}=\frac{p_{\ell }}{p_k}.\end{array}$$

• It relates the marginal rate of substitution ${\mathrm{M}\mathrm{R}\mathrm{S}}_{lk}(x^{\ast })$(the amount of good $k$​​​​ the agent must be given to compensate him/her for a one-unit reduction in his consumption of good $\ell$) to the ratio of prices (i.e., the rate of exchange).

Boundary optimum

When the consumer's optimal bundle $x^{\ast }$ lies on the boundary of the consumption set. In this case, the gradient vector need not be proportional to the price vector.

In particular, the first-order conditions tell us that

• $\partial u_{\ell }(x^{\ast })/\partial x_{\mathcal{l}}\le \lambda p_{\ell }$​​ for those $\ell$​​ with $x_{\mathcal{l}}^{\ast }=0$​​ and
• $\partial u_{\mathcal{l}}(x^{\ast })/\partial x_{\mathcal{l}}=\lambda p_{\ell }$​​ for those $\ell$​​ with $x_{\ell }^{\ast }>0$​​​​.

The indirect utility function

Definition (indirect utility function). For each $(p,w)\gg 0$​, the utility value of the UMP​ is denoted $v(p,w)\in \mathbb{R}$​. It is equal to $u(x^{\ast })$​ for any $x^{\ast }\in x(p,w)$​.

Proposition (Roy's identity). Roy's Identity states that the Walrasian demand function is equal to minus the ratio of partial derivatives w.r.t. price and wealth, i.e.,

$$x_{\ell }(p,w)=-\frac{\partial v(p,w)/\partial p_{\ell }}{\partial v(p,w)/\partial w}.$$