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tags: micro, lecture_note
---
# Utility
## The Utility Maximization Problem

Suppose the consumer has a rational, continuous, and locally nonsatiated preference relation over $\mathbb{R}^n_+$, and $u(\cdot)$ is a continuous utility function representing her preference.

Given price vector $p \gg 0$ and wealth $w>0$, the consumer's problem of choosing her most preferred consumption bundle is the following **utility maximization problem (UMP)**:

$$\max_{x \ge 0} u(x) \\ \text{such that } p \cdot x \le w$$

### Existence of Solution
If $p \gg 0$ and $u(x)$ is continuous, then there exists a solution for the utility maximization problem.

**proof** If $p \gg 0$, then the budget set $B_{p,w} = \{x \in \mathbb{R}^n_+ | p \cdot x \le w\}$ is compact since its closed and bounded. As $u(\cdot)$ is continuous, we can apply Weierstrass's theorem.

### F.O.C.

If $u(\cdot)$ is differentiable, then the solution $x^* \in x(p,w)$ of UMP must satisfied following necessary conditions:

$$ \frac{\partial u(x^*)}{\partial x_i} \le \lambda p_i, i = 1,...,n, \text{with equality if } x_i^* > 0$$

Where $\lambda \ge 0$ is the Largrange multiplier. The same condition in matrix notation,

$$ \nabla u(x^*) \le \lambda p \text{ and } x^* \cdot [\nabla u(z^*) - \lambda p] = 0$$
## The Walrasian Demand Correspondence/Function

We define the set $x(p,w)$ as the optimal consumption vectors in the UMP under the price-wealth pair. Because there may be more than one optimal vector a pair of $(p,w)$, we call $x(p,w)$ a Walrasian demand correspondence. When $x(p,w)$ is single-valued for all $(p,w)$, we call it Walrasian demand function.

### Properties
Suppose that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated preference realtion $\succeq$ on $\mathbb{R}^n_+$, then the Walrasian demand correspondence $x(p,w)$ has following properties:

1. Homogeneity of degree zero in $(p,w)$: $x(ap, aw)=x(p,w)$ for any scalar $a>0$.

2. Walras law: $p \cdot x = w$.

3. Convexity/Uniqueness: If $\succeq$ is convex, then $u(\cdot)$ is quasiconcave and $x(p,w)$ is a convex set. If $\succeq$ is strictly convex, then $u(\cdot)$ is strictly quasiconcave and $x(p,w)$ is single-valued.

### Explanations
1. H.D.0 in $(p,w)$ is an obvious result since the budget set does not change.

2. If the point is not on the budget hyperplane $\{x \in \mathbb{R}^n| p \cdot x = w\}$, the consumer can always find a better point in the budget set.

3. The quasiconvaity of $u(\cdot)$ just follows the definition. If $x$ and $x'$ are two points in $x(p,w)$, then $x'' = ax + (1-a)x, a \in (0,1)$ is also in the buget set $B_{p,w}$. Since $\succeq$ is convex, $x''$ could not worse than $x$ and $x'' \in x(p,w)$.
If $\succeq$ is strictly convex and there are two points $x$ and $x'$ in $x(p,w)$, then $x'' = ax + (1-a)x, a \in (0,1)$ is better than $x$ and $x'$. This result contracdicts the definition of $x(p,w)$, so $x(p,w)$ must be single-valued.

### Cournot and Engel Aggregation
Given Walras law, we can derive two equations.
#### Cournot Aggregation
Differentiating $p\cdot x(p, w) = w$ with respect to $p_k$, we can get

$$\sum_{i=1}^n p_i \frac{\partial x_i(p,w)}{\partial p_k} +x_k(p,w) = 0$$

Multiplying above equation by $\frac{p_k}{w}$, it becomes

$$
\sum_{i=1}^n \frac{p_k}{w} p_i \frac{\partial x_i(p,w)}{\partial p_k} +\frac{p_k}{w} x_k(p,w)
$$

Multiplyin the first term with $\frac{x_i}{x_i}=1$ and rearrange,

\begin{align*}
&= \sum_{i=1}^n \frac{p_i x_i}{w} \frac{\partial x_i(p,w)}{\partial p_k}\frac{p_k}{x_i} +x_k(p,w)\frac{p_k}{w}\\
&= \sum_{i=1}^n B_i(p,w) \varepsilon_{ik} + B_k(p,w) = 0,
\end{align*}

where $B_i(p,w)$ is the budget share spended on good $i$ and $\varepsilon_{ik}$ is the price elasticity between good $i$ and good $k$. This equation is called the **Cournot Aggregation**.

#### Engel Aggregation

Differentiating $p\cdot x(p, w) = w$ with respect to $w$, we will get

$$\sum_{i=1}^n p_i \frac{\partial x_i(p,w)}{\partial w} = 1$$

Multiplying each term with $\frac{w x_i}{w x_i}$, it becomes

\begin{align*}
&\sum_{i=1}^n p_i \frac{\partial x_i(p,w)}{\partial w} \frac{w x_i}{w x_i}\\
&=\sum_{i=1}^n \frac{p_i x_i}{w}\frac{\partial x_i(p,w)}{\partial w} \frac{w}{x_i}\\
&= \sum_{i=1}^n B_i(p,w) \varepsilon_{iw}=1,
\end{align*}

where $\varepsilon_{iw}$ is the income elasticity of good $i$. This equation is called the **Engel Aggregation**.

## The Indirect Utility Function

If we can find the optimal $x(p,w)$ in the UMP, we can define the indirect utility function
$$v(p,w) = u(x(p,w))$$, which is the value function of the utility maximazation problem.

### Properties
Suppose that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated preference realtion $\succeq$ on $\mathbb{R}^n_+$, then the indirect utility function $v(p,w)$ has following properties:

1. Homogeneous of degree zero in $(p,w)$.

2. Strictly increasing in $w$ and non-increasing in $p_i$ for any $i$.

3. Quasiconvex in $(p,w)$: the set $\{(p,w) | v(p,w) \le \bar{v}\}$ is convex for any $\bar{v}$.

4. Continuous in $p$ and $w$.

### Explanations
1. H.D.0 in $(p,w)$ is an obvious result since the budget set does not change.

2. Following the change of budget set.

3. Suppose that $v(p,w) \le \bar{v}$ and $v(p',w') \le \bar{v}$. For any $a \in [0,1]$, let $p'' = ap + (1-a)p'$ and $w'' = a w + (1-a) w'$. The budget set $B_{p'',w''} = \{x \in \mathbb{R}^n_+ | p'' \cdot x \le w''\}$ is the subset of $B_{p,w} + B_{p',w'}$ since
$$ap \cdot x + (1-a) p' \cdot x \le a w + (1-a) w'$$
hence $v(p'',w'') \le \bar{v}$ and $v(p,w)$ is quasiconvex in $(p,w)$.

4. Who knows?

## Roy's Identity

$$x(p,w) = - \frac{\nabla_p v(p,w)}{\nabla_w v(p,w)}$$

**Proof**

$$v(p,w) = u(x(p,w))$$

$$\nabla_p v(p,w) = \nabla_p (u(x(p,w))) = D_x u(\cdot) D_p x(p,w)$$

$$\nabla_w v(p,w) = \nabla_w (u(x(p,w))) = D_x u(\cdot) D_w x(p,w)$$

Since $x$ is the optimal solution, we can apply the envelope theorem,

$$D_p x(p,w) = -\lambda x$$

$$D_w x(p,w) = \lambda$$

as the desired result.

This complicated form is connected to the invariant property of the ordinal utility function. Since the optimal solution will not change under different utility/indirect utility functions based on the same preference, we must normalize it.