---
tags: micro, lecture_note, book
---

# Welfare

In the following models, we assume there are $I$ consumers, $J$ producers, and $L$ commodities.
## Economy
### Economy
**Definition** An economy is a set of consumers' preferences, producers' technology, and the initial endowments. We use the notation $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J,\omega)$ to represent an economy.

- $X_i \subset \mathbb{R}^L$ : consumer $i$'s consumption set, $i=1,...,I$.
- $\succeq_i$ : consumer $i$'s preference, which is assumed to be complete and transitive, $i=1,...,I$.
- $Y_j \subset \mathbb{R}^L$: producer $j$'s production set, $j=1,...,J$.
- $\omega \subset \mathbb{R}^L$: the initial endowment.

In a pure exchange economy, we define the technology as free disposal technology, i.e., $Y_j = - \mathbb{R}^L_+$.

### Allocation
**Definition** An allocation $(x,y)=(x_1,...,x_I, y_1,...,y_J)$ is a set of consumption vectors $x_i \in X_i$ for each consumer $i=1,...,I$ and production vectors $y_j \in Y_j$ for each producer $j=1,...,J$. An allocation is feasible if

$$\sum_i^I x_i = \omega + \sum_j^J y_j$$

We denote the set of feasible allocations as

$$A = \{(x_1,...,x_I, y_1,...,y_J) \in X_1 \times ... X_I \times Y_1 \times ... \times Y_J| \sum_i^I x_i = \omega + \sum_j^J y_j \}$$

### Pareto Optimal
**Definition** An allocation $(x,y)$ **Pareto dominates** $(x',y')$ if $x_i \succeq_i x'$ for all $i$ and $x_i \succ_i x'$ for some $i$.

**Definition** A feasible allcation $(x,y)$ is **Pareto optimal/Pareto efficient** if there is no other feasible allocation Pareto dominates $(x,y)$.

**Note:** Some textbooks (such as Varian(1992)) define *weakly* Pareto improvement as improving all consumers' utility and *strongly* Pareto improvement as improving some consumers' utility and preserving others utility levels. MWG only discusses the case of strongly Pareto improvement.

### Private Ownership
**Definition** A private ownership economy is an economy with the distribution of endoment and firm profits. We use the notation $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J, \{(\omega_i), \theta_{i1},...,\theta_{iJ}\}_{i=1}^I)$ to represent a private ownership economy.

- $\omega_i \in \mathbb{R}^L$ : the initial endowment belongs to consumer $i$, $i=1,...,I$.

- $\theta_{i1},...,\theta_{iJ}$ : consumer $i$'s share of firm $j$'s profit, $i=1,...,I$. Since all consumers own all firms, it must satisfy $\sum_{i=1}^I \theta_{ij} =1, j=1,...,J$.

### Walrasian/Competitive Equilibrium
**Definition** Given a private ownership economy $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J, \{(\omega_i), \theta_{i1},...,\theta_{iJ}\}_{i=1}^I)$, an allocation $(x^*,y^*)$ and a price vector $p^*=(p_1^*,...,p_L^*)$ constitute a Walrasian/competitive equilibrium if

1. (Profit maximization) For each firm $j$, $y^*_j$ solves

$$\max_{y^* \in Y_J} p^* \cdot y^*_j .$$

2. (Utility maximization) For each consumer $i$, $x^*_i$ is maximal for $\succeq_i$ under the budget set

$$B_i = \{x_i \in X_i | p^* \cdot x^* \le p^* \cdot \omega_i + \sum_{j=1}^J \theta_{ij} (p^* \cdot y^*_j)\}.$$

3. (Market cleaning)

$$\sum_{i=1}^I x^*_{i} = \omega + \sum_{j=1}^J y^*_{j}.$$

### Price Equilibrium with Transfers
**Definition** Given an economy $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J,\omega)$ , an allocation $(x^*,y^*)$ and a price vector $p^*=(p_1^*,...,p_L^*)$ constitute a price equilibrium with transfers if there is an assignment of wealth levels $(w_1, ..., w_I)$ with $\sum_{i=1}^I w_i = p \cdot \omega + \sum_{j=1}^J p \cdot y_j^*$ such that

1. (Profit maximization) For each firm $j$, $y^*_j$ solves

$$\max_{y^* \in Y_J} p^* \cdot y^*_j .$$

2. (Utility maximization) For each consumer $i$, $x^*_i$ is maximal for $\succeq_i$ under the budget set

$$B_i = \{x_i \in X_i | p^* \cdot x^*_i \le w_i\}.$$

3. (Market cleaning)

$$\sum_{i=1}^I x^*_{i} = \omega + \sum_{j=1}^J y^*_{j}.$$

**Note:** We assume that government can induce a transfer/wealth redistribution that would not affect productions and preferences.

### Walras Law

When $l-1$ markets are clearing, then $l$ markets are clearing.

## Partial Equilibrium

### Setting
**Definition** In the partial equilibrium model, we assume there are only $2$ goods, $m$ and $l$, and each agent's has a quasi-linear utility function. We use $x_i$ to represent the consumption of $l$ for agent $i$, and their utility function is:

$$u_i(m_i,x_i)=m_i + \phi_i(x_i).$$

$$ \text{ such that } \phi'_i(x_i)>0, \phi_i''(x_i) <0 \text{ for all } x_i \ge 0$$

Initially, agents only have endowments for good $m$, which is $\omega_i$.

We also assume that each firm can use good $m$ to produce good $x$ with a marginal cost increasing cost function. We use $q_i$ to reprecent the output of firm $i$, and their cost function is $c_i(q_i)$:

$$ \text{ such that } c'_i(q_i)>0, c''_i(q_i) >0 \text{ for all } q_i \ge 0$$

### Competitive Equilibrium
In this model, the conditions of competitive equilibrium are

1. (Profit maximization) For each firm $j$, $y^*_j$ solves

$$\max_{q_j \ge 0} p^* q_j - c_j(q_j) .$$

2. (Utility maximization) For each consumer $i$,

$$\max_{m_i \in \mathbb{R}, x_i \in \mathbb{R}_+ } m_i + \phi_i(x_i) \\ \text{ such that } m_i + p^* x_i \le \omega_{mi} + \sum_{j=1}^J \theta_{ij}(p^*q^*_j - c_j(q^*_j)).$$

it is equivalent to

$$\max_{ x_i \in \mathbb{R}_+ } \phi_i(x_i) - p^* x_i + \omega_{mi} + \sum_{j=1}^J \theta_{ij}(p^*q^*_j - c_j(q^*_j)).$$

3. (Market cleaning)

$$\sum_{i=1}^I x^*_{i} = \omega + \sum_{j=1}^J y^*_{j}.$$

Since the utility functions and the cost functions are well-behaviored, we hava the following necessary and sufficient conditions:

1. (Profit maximization) For each firm $j$,

$$p^* \le c'_j(q_j^*), \text{ with equality if } q_j^* >0.$$

2. (Utility maximization) For each consumer $i$,

$$\phi'(x_i^*) \le p^*, \text{ with equality if } x_i^* >0.$$

3. (Market cleaning)

$$\sum_{i=1}^I x^*_{i} = \sum_{j=1}^J q^*_{j}.$$

### Welfare

The utility possibiliy set under a fixed consumption and production of good $l$ at $(\bar{x}_1, ..., \bar{x}_I, \bar{q}_1, ..., \bar{q}_J)$ is

$$\{(u_1, ..., u_I): \sum_{i=1}^I u_i \le \sum_{i=1}^I \phi_i(\bar{x}_i) + \omega_m - \sum_{j=1}^J c_j(\bar{q}_j)\}$$

The above expression implies that the boundary of utility possibility set is linear.

Consider a optimization problem of Utilitarian social welfare function:

$$\max_{x_i, q_j} \sum_{i=1}^I u_i(x_i, m_i) =\sum_{i=1}^I \phi_i(x_i) - \sum_{j=1}^J c_j(q_j)+ \omega_m \\ \text{ such that } \sum_{i=1}^I x_i - \sum_{j=1}^J q_j =0.$$

Let $\mu$ be the Largrange multiplier, the necessary and sufficient conditions of the above problem are:

1. For each $j$,

$$\mu \le c'_j(q_j^*), \text{ with equality if } q_j^* >0.$$

2. For each $i$,

$$\phi'(x_i^*) \le \mu, \text{ with equality if } x_i^* >0.$$

3.
$$\sum_{i=1}^I x^*_{i} = \sum_{j=1}^J q^*_{j}.$$

These conditions are exactly the same conditions for the competitive equilibrium. This coincidence implies the first theorem of welfare economics.

**Propositions** The first and the second theorem of welfare economics hold in this economy.

If $(x, q, p)$ is a competitive equilibrium, then it can also be a solution to the Utilitarian social welfare problem. However, under the optimality of the Utilitarian social welfare function, it is impossible to increase an agent's utility without decreasing others' utilities.

For the second theorem of welfare economics, if $(x, q)$ is a Pareto optimal allocation, it is also a solution to the Utilitarian social welfare problem. Hence, a $\mu$ could be the price in the competitive equilibrium conditions, and a lump sum transfer can induce the same allocation in the competitive market.

**Note:** As discussed later, these two theorems do not hold in general.

## First Fundamental Theorem of Welfare Economics

### The First Fundamental Theorem of Welfare Economics
**Proposition** If all consumers' preferences are locally nonsatiated, and if $(x^*, y^*, p)$ is a price equilibrium with transfers, then the allocation $(x^*, y^*)$ is Pareto optimal. In particular, any Walrasian equilibrium allocation is Pareto optimal.

**Proof** Suppose that $(x^*, y^*, p)$ is a price equilibrium with transfers and that associated wealth levels are $(w_1,...,w_I)$.

For each consumer $i$, following the definition of price equilibrium, we have

$$x_i \succ_i x_i^* \implies p \cdot x_i > w_i$$

Under the assumption of local nonsatiation,

suppose that there esitst $x_i \succeq x_i^*$ and $p \cdot x_i < w_i$, then

$$ \exists x'_i \succ_i x_i^* \text{ and } p \cdot x_i' < w_i$$

hence,

$$x_i \succeq_i x_i^* \implies p \cdot x_i \ge w_i$$

Suppose $(x,y)$ is an allocation Pareto dominates $(x^*, y^*)$, then for all $i$

$$p \cdot x_i \ge w_i$$

and for some $i$

$$p \cdot x_i > w_i$$

Summing up, we have

$$\sum_{i=1}^I p \cdot x_i > \sum_{i=1}^I w_i = p \cdot \omega + \sum_{j=1}^J p \cdot y_j^*$$

However, if $(x,y)$ is a feasible allocation, it must be

$$\sum_{i=1}^I x_i = \omega + \sum_{j=1}^J y_j$$

The equation holds after mulitiplying any price vector,

$$\sum_{i=1}^I p \cdot x_i = p \cdot \omega + \sum_{j=1}^J p \cdot y_j$$

Therefore, $(x,y)$ could not be a feasible allocation and $(x^*, y^*)$ is Pareto optimal.

A Walrasian equilibrium allocation is a price equilibrium with the wealth level as price times initial endowments, which is a special price equilibrium with transfers.

**Note:** If we only want to prove the version of weakly Pareto optimal, we do not need the assumption of local nonsatiation.

## Second Fundamental Theorem of Welfare Economics

### Price Quasiequilibrium with Transfers
**Definition** Given an economy $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J,\omega)$ , an allocation $(x^*,y^*)$ and a price vector $p=(p_1,...,p_L) \neq 0$ constitute a price quasiequilibrium with transfers if there is an assignment of wealth levels $(w_1, ..., w_I)$ with $\sum_{i=1}^I w_i = p \cdot \omega + \sum_{j=1}^J p \cdot y_j^*$ such that

1. (Profit maximization) For each firm $j$, $y^*_j$ solves

$$\max_{y^* \in Y_J} p^* \cdot y^*_j .$$

2. (Utility quasi-maximization) For each consumer $i$, if $x_i \succ_i x_i^*$ then

$$ p \cdot x_i \ge w_i.$$

3. (Market cleaning)

$$\sum_{i=1}^I x^*_{i} = \omega + \sum_{j=1}^J y^*_{j}.$$

**Note:** The allocation is only quasi-equilibrium because consumers may not maximize their utility.

### The Second Fundamental Theorem of Welfare Economics
**Proposition** Give an economy $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J,\omega)$. Suppose that every $Y_j$ is convex, every $\succeq_i$ is convex and locally nonsatiated. Then, for every Pareto optimal allocation $(x^*,y^*)$, there is a price vector $p=(p_1,...,p_L) \neq 0$ such that $(x^*,y^*,p)$ is a price quasiequilibrium with transfers.

**Proof**
For each consumer $i$, define $V_i = \{x_i \in X_i | x_i \succ_i x_i^*\}$. We can difine following sets:

$$V = \sum_{i=1}^I V_i=\{\sum_{i=1}^I x_i | x_1 \in V_1,..., x_I \in V_I\}$$

$$Y = \sum_{j=1}^J y_j=\{\sum_{j=1}^J y_j | y_1 \in Y_1,..., y_J \in Y_J\}$$

Every point in $V$ is an allocation Pareto dominates $x^*$, $Y$ is the set of all possible production plans, and $(Y + \{\omega\})$ is the set of all possible goods consumed by the consumers.

**Claim 1:** Every $V_i$ is convex

**Proof:** Suppose $x_i \succ_i x_i^*$ and $x_i' \succ_i x_i^*$. W.L.O.G, we can assume $x_i \succeq_i x_i'$. Following the convexity of $\succeq_i$, we have $a x_i + (1-a) x_i' \succeq_i x_i' \succ_i x_i^*$ for any $a \in [0,1]$. Therefore, $a x_i + (1-a) x_i' \in V_i$ and $V_i$ is convex.

**Claim 2:** $V$ and $Y + \{\omega\}$ are convex

**Proof:** The sum of finite convex sets are convex.

**Claim 3:** $V \cap (Y + \{\omega\}) = \emptyset$

**Proof:** It is the direct result of Pareto optimal.

**Claim 4:** There exists $p=(p_1,...,p_L)$ and a number $r$ such that $p \cdot z \ge r$ for every $z \in V$ and $p \cdot z \le r$ for every $z \in (Y + \{\omega\})$.

**Proof:** It is the direct result of separating hyperplane theorem since both $V$ and $(Y + \{\omega\})$ are convex, and their intersection is the empty set.

**Claim 5:** If $x_i \succeq_i x_i^*$ for every $i$, then $p \cdot (\sum_{i=1}^I x_i)\ge r$.

**Proof:** By local nonsatiation, we can construct a sequence $\{x_{in} \in X_i | x_{in} \succ_i x_i \succeq_i x_i^* \}$ such that $\lim_{n \to \infty} x_{in} = x_i$. Since each $x_{in} \in V_i$, we have $p \cdot \sum_{i=1}^I x_{in} \ge r$.

$$\lim_{n \to \infty} p \cdot \sum_{i=1}^I x_{in} = p \cdot \sum_{i=1}^I x_i \ge r.$$

**Claim 6:** $p \cdot \sum_{i=1}^I x_i^* = p \cdot (\omega + \sum_{j=1}^J y_j^*)= r$

**Proof:** Since $(x^*,y^*)$ is a feasible allocation, $\sum_{i=1}^I x_i^* =\omega + \sum_{j=1}^J y_j^* \in (Y + \omega)$. Hence,$p \cdot \sum_{i=1}^I x_i^* = p \cdot (\omega + \sum_{j=1}^J y_j^*) \le r$. By claim 5, $x_i^* \succeq_i x_i^*$ and $p \cdot (\sum_{i=1}^I x_i^*)\ge r$. Therefore, $p \cdot \sum_{i=1}^I x_i^* = p \cdot (\omega + \sum_{j=1}^J y_j^*)= r$.

**Claim 7:** $p \cdot y_j \le p \cdot y_j^*$ for each $j$ and $y_j \in Y_j$.

**Proof:** By claim 4 and claim 6, we have
$$p \cdot (\omega + \sum_{j=1}^J y_j) \le r =p \cdot (\omega + \sum_{j=1}^J y_j^*).$$

If we restirced on only one firm $j$, above equation still holds:

$$p \cdot (\omega + y_j + \sum_{k\neq j} y_k^*) \le r =p \cdot (\omega + y_j^* + \sum_{k \neq j} y_k^*).$$

Hence, $p \cdot y_j \le p \cdot y_j^*$ for each $j$ and $y_j \in Y_j$.

**Claim 8:** For every $i$, if $x_i \succ_i x_i^*$, then $p \cdot x_i \ge p \cdot x_i^*.$

**Proof:** By claim 5 and claim 6, we have

$$p \cdot (x_i + \sum_{k \neq i}x_k^*) \ge r =p \cdot (x_i^* + \sum_{k \neq i}x_k*) $$

**Claim 9:** The wealth levels $w_i= p \cdot x_i^*, i=1,...,I$ support $(x^*,y^*,p)$ as a price quasiequilibrium with transfers.

**Proof:** Claim 7 ensures profit maximization for each firm; Claim 8 ensures the utility quasi-maximization; the market is cleaning since $(x^*,y^*)$ is a feasible allocation.

### When Price Quasiequilibrium is Price Equilibrium

**Proposition** Given the price vector $p$ and consumption $x^*$, suppose that $X_i$ is convex and $\succeq_i$ is continuous, and $x \succ_i x^*$ implies $p \cdot x \ge p \cdot x^*$. If there exists another cheaper bundle $x_i' \in X_i$ such that $p \cdot x' < p \cdot x^*$, then $x \succ_i x^*$ implies $p \cdot x > p \cdot x^*$.

**Proof**
Suppose that $x \succ_i x^*$ and $p \cdot x = p \cdot x^*$. Consider the convex combination between $x$ and $x'$, $a x + (1-a) x'$. Since $X_i$ is convex, $a x + (1-a) x' \in X_i$ for $a \in [0,1]$. The continuity of $\succeq_i$ implies that there exists $a \in [0,1]$ such that $a x + (1-a) x' \succ_i x_i^*$. However, it is a contradiction because $p \cdot (a x + (1-a) x') < p \cdot x^*$. Therefore, if there is another cheaper bundle $x_i' \in X_i$ such that $p \cdot x' < p \cdot x^*$, then $x \succ_i x^*$ implies $p \cdot x > p \cdot x^*$.

**Proposition** Suppose that for every $i$, $0 \in X_i$, $X_i$ is convex and $\succeq_i$ is continuous. Then any price quasiequilibrium with transfers that has $(w_1 ,..., w_I) \gg 0$ is a price equilibrium with transfers.

**Proof**
The above assumptions ensure that there is a cheaper consumption bundle for every consumer.

### Second FTWE on Exchange Economy
**Proposition** Give a pure exchange economy $(\{(X_i, \succeq_i)\}_{i=1}^I,\{Y_j\}_{j=1}^J,\omega)$. Suppose that every $Y_j = -\mathbb{R}^L_+$, every $X_i = \mathbb{R}^L_+$, every $\succeq_i$ is convex , continuous, and strongly monotone, and $\omega \gg 0$. Then, for every Pareto optimal allocation $(x^*,y^*)$, there is a price vector $p=(p_1,...,p_L) \neq 0$ such that $(x^*,y^*,p)$ is a price equilibrium with transfers.

**Proof**
We already know that there is a price vector $p=(p_1,...,p_L) \neq 0$ such that $(x^*,y^*,p)$ is a price quasiequilibrium with transfers. Since every $Y_j = -\mathbb{R}^L_+$, the profit maximization condition implies that any good price could not be negative.

As $\omega \gg 0$, we have $p \cdot \omega = \sum_{i=1}^I p \cdot x_i^* =\sum_{i=1}^I w_i>0$. Hence, for some $i$, $w_i>0$. For them, there exists consumption bundles cheaper than $x_i^*$, and $x_i^*$ is the utility maximization under $p$. Hence, the strongly monotone assumption of preferences ensures that any good has positive price. Otherwise, no one can have utility maximization bundle under $p$. Since $p \gg 0$, $w_i = p \cdot x_i^* \ge 0.$ If $w_i >0$, then $x_i^*$ is the utility maximization bundle under $p$. If $w_i =0$, then $x_i^*=0$ and $x_i^*$ is also the utility maximization bundle under $p$. As a result, $(x^*,y^*,p)$ is a price equilibrium with transfers.