---
tags: micro
---

# Demand

## 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 perferneces.

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)**:

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

### 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.

## The Walrasian Demand Correspondence/Function

We define the set $x(p,w)$ as the optimal consumption vectore in the UMP under the price-wealth pair. Because there maybe 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. Walra's 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.

### Explainations
1. H.D.0 in $(p,w)$ is an obvious reslut 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 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.

### F.O.C.

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

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

In matrix notation,

$$ \nabla u(x^*) \le \lambda p \text{ and } [\nabla u(z^*) - \lambda p] \cdot x^* = 0$$

## 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 $w$.

### Explainations
1. H.D.0 in $(p,w)$ is an obvious reslut 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?

## The Expenditure Minimization 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 perferneces.

Given price vector $p \gg 0$ and a scalar $u>u(0)$, the consumer's problem of choosing a consumption bundle that minimizes her expenditure and gives her utility as high as $u$ is the following **expentiture minimization problem (EMP)**:

$$\min_{x \ge 0} p \cdot x \\ \text{such that } u(x) \ge u$$

### Proposition
Suppose that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated preference realtion $\succeq$ on $\mathbb{R}^n_+$, then we following propositions :

1. If $x^*$ is optimal in the UMP when $w>0$, then $x^*$ is optimal in the EMP when the required utility level is $u(x^*)$. The minimized expenditure level in this EMP is $w$.

2. If $x^*$ is optimal in the EMP when the required utility level is $u >u(0)$, then $x^*$ is optimal in the UMP when wealth is $p \cdot x^*$. The mazimized utility level in this UMP is $u$.

*Note*: above propositions are followed by definitions and the property of locally nonsatiated.

## The Hicksian (Compensated) Demand Correspondence/ Function

We call the solution of EMP the Hicksian demand correspondence and denoted as $h(p, u)$. If $h(p, u)$ only contains one element in each $(p,u)$, it becomes Hicksian 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 Hicksian demand correspondence $h(p,u)$ has following properties:

1. Homogeneity of degree zero in $p$

2. No excess utility: For any $x \in h(p,u), u(x)=u$.

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

### Explainations

1. Obvious.

2. Suppose that $x \in h(p,u)$ and $u(x) > u$. Consider the a bundle $x' = a x'$ for $a \in (0,1)$. By continuity of $u(\cdot)$, when $a$ is very closed to $1$, $u(x')>u$ and $p \cdot x' < p \cdot x'$. Hence, $x$ colud not be a solution in the EMP.

3. Obvious.

## The Expenditure Function

If we can find the optimal solution $x^* \in h(p,u)$ in the EMP, we can define the expenditure function
$$e(p,u) = p \cdot x^*$$ which is the value function of the expenditure minimization problem.

### Properties

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

1. Homogeneous of degree one in $p$.

2. Strictly increasing in $u$ and nondecreasing in $p_i$ for any $i$.

3. Concave in $p$.

4. Continuous in $p$ and $u$.

### Explainations

1. Obvious.

2. Suppose that $e(p, u)$ is not strictly increasing in $u$, and let $x$ and $x'$ be two bundles that $u(x')>u(x)$ but $p \cdot x \ge p \cdot x'$. Consider the a bundle $x'' = a x''$ for $a \in (0,1)$. By continuity of $u(\cdot)$, when $a$ is very closed to $1$, $u(x'')>u(x)$ and $p \cdot x'' < p \cdot x''$.
The property of non-decreasing in $p_i$ is obvious.

3. Fix a utility level $\bar{u}$ and let $p'' = ap +(1-a)p$ for $a \in [0,1]$. Suppose that $x''$ is an optimal bundle in the EMP when price is $p''$.
$$e(p'', \bar{u}) = p'' \cdot x'' \\ =ap \cdot x'' +(1-a)p' \cdot x'' \\ \ge ae(p,\bar{u}) + (1-a)e(p', \bar{u})$$

### F.O.C.

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

$$ p \ge \lambda \nabla u(x^*) \text{ and } x^*[p - \lambda \nabla u(x^*) ] = 0$$ for some $\lambda > 0$.

### Identities
Following the assumptions of the preference, the Hicksian and Walasian demand correspondences have following relations:

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

### Compensated Law of Demand
Suppose $h(p,u)$ is a Hicksian demand function, then for $p'$ and $p''$:

$$(p''-p')\cdot [h(p'',u) - h(p',u)] \le 0$$

**Proof** By the construction of $h(p,u)$, we have below two inequalities:

$$p'' \cdot h(p'',u) \le p'' \cdot h(p',u) $$

$$p' \cdot h(p',u) \le p' \cdot h(p'',u) $$
Subtracting these two inequalities yields the results.

## Demand, Indirect Utility and Expenditure Functions

In the following section, we assume that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated preference realtion $\succeq$ on $\mathbb{R}^n_+$ and $p \gg 0$.

### Shephard's Lemma

Let $e(p,u)$ and $h(p,u)$ be expenditure function and the Hicksian demand function, we have

$$h_i(p,u) = \frac{\partial e(p,u)}{\partial p_i}, \forall i=1,...,n$$
or in the matrix notation:
$$h(p,u) = \nabla_p e(p,u)$$

### Properties
Suppose that $h(p,u)$ is continuously differentiable at $(p,u)$, and denot its derivative matrix as $D_p h(p,u)$. Then

1. $D_p h(p,u)=D^2_p e(p,u)$.

2. $D_p h(p,u)$ is a negative semidefinite matrix.

3. $D_p h(p,u)$ is a symmetric matrix.

4. $D_p h(p,u)p=0$.

### Explanations

1. It is based on Shephard's lemma.

2. Because expenditure function is concave, its Hessian matrix must be negative semidefinite. Since the diagonal term of the matrix is non-positive, the own price net effect is also non-positive, i.e. the law of demand.

3. The Hessian matrix of a twice differentiable function is symmetric, which is called [Young's theorem or Schwarz's theorem](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives). Hence, the net cross price effects of two goods must be the same, i.e. if $X$ is a net substitute of $Y$, then $Y$ is also a net substitute of $X$.

4. It is based on Euler's formula and the fact that the Hicksian demand function is H.D.0 in $p$. Hence, one good could not be substitue or complement with all goods.

### The Slutsky Equation

If the demand function is differentiable, we have

$$\frac{\partial h_i(p,u)}{\partial p_j} = \frac{\partial x_i(p,w)}{\partial p_j} +\frac{\partial x_i(p,w)}{\partial w} x_j(p,w) , \forall i,j$$

Some textbooks exchange the terms as:

$$ \frac{\partial x_i(p,w)}{\partial p_j} = \frac{\partial h_i(p,u)}{\partial p_j} - \frac{\partial x_i(p,w)}{\partial w} x_j(p,w) , \forall i,j$$

**Proof**
Since $h_i(p,u) = x(p,e(p,u))$, taking derivative on it with $p_j$, we have

$$\frac{\partial h_i(p,u)}{\partial p_j} = \frac{\partial x_i(p,e(p,u))}{\partial p_j} +\frac{\partial x_i(p,e(p,u))}{\partial e} \frac{\partial e(p,u)}{\partial p_j}\\ = \frac{\partial x_i(p,w)}{\partial p_j} +\frac{\partial x_i(p,w)}{\partial w} h_j(p,u) \\ = \frac{\partial x_i(p,w)}{\partial p_j} +\frac{\partial x_i(p,w)}{\partial w} x_j(p,w)$$

**Explanations**

The equation tell us that the gross price effect is the sum of net price effect and the wealth effect. We can see that in general, the gross price effect of good $X$ on goog $Y$ is different from the gross price effect of good $Y$ on goog $X$. In extreme case, even the sign would not the same. Besides, although the net own price effect is non-positive, Giffen goods still exist with extreme negative wealth effect.

### Roy's Identity

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

## Welfare Evaluation

Suppose the original price vector is $p^0$ and a consumer's wealth is $w$, we want to evaluate the consumer's welfare change after the price changing to $p^1$.

## Measures

There are two method to evalute the welfare change: equivalent variation (EV) and the compensating variation (CV).

$$EV(p^0,p^1,w)=e(p^0,u^1) - e(p^1,u^1) = e(p^0,u^1) - w$$

$$CV(p^0,p^1,w)=e(p^0,u^0) - e(p^1,u^0) = w - e(p^1,u^0)$$

In one sentence, EV is "The differnce of your purchasing power based on current price"; CV is "How much the government need to compensate so you can accept the new price."

Assuming the differentiability of expenditure function, we can get following equations by Shephard's lemma:

$$EV(p^0,p^1,w)=e(p^0,u^1) - e(p^1,u^1) =\int_{p^1}^{p^0} h(p,u^1) dp$$

$$CV(p^0,p^1,w)=e(p^0,u^0) - e(p^1,u^0) = \int_{p^1}^{p^0} h(p,u^0) dp$$

Moreover, we can define the change of consumer surplus as the intergral of demand function:

$$\Delta CS(p^0,p^1,w)= \int_{p^1}^{p^0} x(p,w) dp$$

The above three formulas tell us theat the sign of three measures must be the same.

### Comparision

Since the three measures are just the integrals of three different function on the same region, we can check their relationships by comparing the three functions.

Recall the identities between Walrasian demand functions and Hicksian demand function:

$$x(p,e(p,u)) = h(p,u)$$

We have following two equations:

$$x(p,e(p,u^0)) = h(p,u^0)$$
$$x(p,e(p,u^1)) = h(p,u^1)$$

After taking derivative of $u$, we can get:

$$\frac{\partial x_i(p,e(p,u))}{\partial u} = \frac{\partial x_i(p,e(p,u))}{\partial e} \frac{\partial e}{\partial u}=\frac{\partial h_i(p,u)}{\partial u}$$

Since $\frac{\partial e}{\partial u}>0$, the sign of $\frac{\partial h_i(p,u)}{\partial u}$ is the same as $\frac{\partial x_i(p,e(p,u))}{\partial e}$. i.e. the good is normal or inferior.

Also, the Slutsky's equations tells us:

$$\frac{\partial h_i(p,u)}{\partial p_i} = \frac{\partial x_i(p,u)}{\partial p_i} +\frac{\partial x_i(p,u)}{\partial w} x_i(p,w)$$

We can use above results to compare EV, CV, and CS during the price change of only one good. Suppose the price of good $i$ change and the price of other goods remain the same. We drop the subscript $i$ for convenience.( it's a minor abuse of notation):

Suppose $p^1 < p^0$ and $i$ commodotity is a normal good, we have

$$h(p,u^0)< x(p,w)< h(p,u^1)$$

Hence,

$$CV= \int_{p^1}^{p^0} h(p,u^0) dp < \Delta CS = \int_{p^1}^{p^0} x(p,w) dp< EV= \int_{p^1}^{p^0} h(p,u^1) dp$$

Conversely, if $p^1 < p^0$ and $i$ commodotity is a inferior good, we have:

$$h(p,u^1)< x(p,w)< h(p,u^0)$$

$$EV < \Delta CS < CV$$

Moreover, if there is no wealth effect on good $i$, the three measures are the same.