1. Welfare Theorems

Chapter theme: the welfare theorems. The two fundamental theorems linking "competitive equilibrium" and "Pareto optimality" in the general-equilibrium framework. §1.1 General equilibrium: the components of an economy (commodity space \(\mathbf L\), households set \(\mathbf I\) with endowment \(\mathbf e^i\) / utility \(u^i\) / consumption possibility set \(\mathbf X^i\), firms set \(\mathbf J\) with production possibility set \(\mathbf Y^j\), ownership \(\theta^i_j\), prices \(\mathbf p\)); Definition 1.1 feasible allocation (feasible + market clears \(\sum\mathbf x^i=\sum\mathbf y^j+\sum\mathbf e^i\)); Definition 1.2 competitive equilibrium (household utility max s.t. \(\mathbf p\cdot\mathbf x\le\mathbf p\cdot\mathbf e^i+\sum\theta^i_j\pi^j\), firm profit max); Definition 1.3 Pareto optimality; Definition 1.4 local non-satiation. §1.2 First welfare theorem: Theorem 1.1 under local non-satiation, competitive equilibrium \(\Rightarrow\) Pareto optimal (by contradiction). §1.3 Second welfare theorem: Definitions 1.5–1.7 set-sum, (strict) concavity, (strict) quasi-concavity; Theorem 1.2 the separating hyperplane theorem; Assumptions HH/FF (\(\mathbf X^i\) convex & \(u^i\) continuous strictly quasi-concave; aggregate production set-sum \(\mathbf Y\) convex); Theorem 1.3 Pareto optimal \(\Rightarrow\) there exists a price under which firms maximize profit and households minimize expenditure (separating-hyperplane proof); Proposition 1.1 (Arrow's Remark) if the Pareto allocation is not the "cheapest" point in the budget set, then expenditure minimization = utility maximization, so the Pareto optimum is a competitive equilibrium. Remarks 1.1–1.3: the significance (solve the social planner's problem instead of the competitive equilibrium; the justification for a representative agent) and the prerequisites (no externalities / taxes / market power / real role of money).

Definition 1.1 (Feasible allocation) An allocation \(\{\mathbf x^i,\mathbf y^j\}\) is a feasible allocation if it satisfies: - For each household \(i\in\mathbf I\), the consumption \(\mathbf x^i\) is possible, i.e. \(\mathbf x^i\in\mathbf X^i\). - For each firm \(j\in\mathbf J\), the production \(\mathbf y^j\) is possible, i.e. \(\mathbf y^j\in\mathbf Y^j\). - Market clears: demand equals supply: $$\sum_{i\in\mathbf I}\mathbf x^i=\sum_{j\in\mathbf J}\mathbf y^j+\sum_{i\in\mathbf I}\mathbf e^i$$

Definition 1.2 (Competitive equilibrium) A competitive equilibrium is a feasible allocation \(\{\mathbf x^i,\mathbf y^j\}\) and a price vector \(\mathbf p\) such that: - Every household \(i\), for \(\forall i\in\mathbf I\), solves the utility maximization problem $$\mathbf x^i\in\arg\max_{\mathbf x\in\mathbf X^i}u^i(\mathbf x)$$ subject to $$\mathbf p\cdot\mathbf x\le\mathbf p\cdot\mathbf e^i+\sum_{j\in\mathbf J}\theta^i_j\pi^j$$ - Every firm \(j\), for \(\forall j\in\mathbf J\), solves the profit (denoted by \(\pi^j=\mathbf p\cdot\mathbf y\)) maximization problem: $$\mathbf y^j\in\arg\max_{\mathbf y\in\mathbf Y^j}\mathbf p\cdot\mathbf y$$

Definition 1.3 (Pareto Optimal Allocation) A feasible allocation \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is Pareto optimal if there does not exist any other feasible allocation \(\{\mathbf x^i,\mathbf y^j\}\) which is preferred by everyone, i.e. no allocation \(\{\mathbf x^i,\mathbf y^j\}\) could be found such that $$u^i(\mathbf x^i)\ge u^i(\bar{\mathbf x}^i)\ \text{ for }\forall i\in\mathbf I$$ $$u^{i'}(\mathbf x^{i'})>u^{i'}(\bar{\mathbf x}^{i'})\ \text{ for some }i'\in\mathbf I$$

Definition 1.4 (Local Non-Satiation) A utility function \(u^i:\mathbf X^i\to\mathbb R\) is said to be locally non-satiated if, for \(\forall\mathbf x\in\mathbf X^i\) and any neighborhood around \(\mathbf x\) denoted by \(B_\varepsilon(\mathbf x)\), \(\exists\tilde{\mathbf x}\in B_\varepsilon(\mathbf x)\) such that \(u^i(\tilde{\mathbf x})>u^i(\mathbf x)\).

Theorem 1.1 (First Welfare Theorem) Assume that all households have utility function satisfying the local non-satiation property. Suppose that \(\{\mathbf p,\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is a competitive equilibrium, then \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is a Pareto optimal allocation.

Proof By contradiction, assume that there is a feasible allocation \(\{\mathbf x^i,\mathbf y^j\}\) that Pareto dominates \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\), i.e. we have \(u^i(\mathbf x^i)\ge u^i(\bar{\mathbf x}^i)\) for \(\forall i\in\mathbf I\) and \(u^{i'}(\mathbf x^{i'})>u^{i'}(\bar{\mathbf x}^{i'})\) for some \(i'\in\mathbf I\). Then, it implies that $$\mathbf p\cdot\mathbf x^i\ge\mathbf p\cdot\bar{\mathbf x}^i,\quad\text{for }\forall i\in\mathbf I\tag{1.1}$$ $$\mathbf p\cdot\mathbf x^{i'}>\mathbf p\cdot\bar{\mathbf x}^{i'},\quad\text{for some }i'\in\mathbf I\tag{1.2}$$ - First, let's prove (1.1). Suppose not. There exists \(i\) such that \(u^i(\mathbf x^i)\ge u^i(\bar{\mathbf x}^i)\) and \(\mathbf p\cdot\mathbf x^i<\mathbf p\cdot\bar{\mathbf x}^i\), then by local non-satiation, we can find \(\hat{\mathbf x}^i\) in \(B_\varepsilon(\mathbf x^i)\), the neighborhood of \(\mathbf x^i\), such that \(u^i(\hat{\mathbf x}^i)>u^i(\mathbf x^i)\) and \(\mathbf p\cdot\hat{\mathbf x}^i\le\mathbf p\cdot\bar{\mathbf x}^i\). This is possible because we can set \(\varepsilon\) arbitrarily small to make total cost close enough but still has the local non-satiation condition. Since \(\hat{\mathbf x}^i\) is still budget feasible, its existence contradicts the fact that \(\bar{\mathbf x}^i\) maximizes utility. So (1.1) must be true. - Then, let's prove (1.2). Suppose not. There exists \(i'\) such that \(u^{i'}(\mathbf x^{i'})>u^{i'}(\bar{\mathbf x}^{i'})\) and \(\mathbf p\cdot\mathbf x^{i'}\le\mathbf p\cdot\bar{\mathbf x}^{i'}\). Then, \(\mathbf x^{i'}\) is in the budget feasible set, which contradicts the fact that \(\bar{\mathbf x}^{i'}\) maximizes utility. So (1.2) must be true.

Add (1.1) and (1.2) and sum across all households: $$\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf x^i\right)>\mathbf p\cdot\left(\sum_{i\in\mathbf I}\bar{\mathbf x}^i\right)\tag{1.3}$$ Now let's consider the production side. By the definition of competitive equilibrium, \(\bar{\mathbf y}^j\) maximizes profits of firm \(j\), which implies \(\mathbf p\cdot\bar{\mathbf y}^j\ge\mathbf p\cdot\mathbf y^j\) for \(\forall j\in\mathbf J\). Then, apply the same condition to all firms and sum up to obtain $$\mathbf p\cdot\left(\sum_{j\in\mathbf J}\bar{\mathbf y}^j\right)\ge\mathbf p\cdot\left(\sum_{j\in\mathbf J}\mathbf y^j\right)\tag{1.4}$$ Feasibility of \(\{\mathbf x^i,\mathbf y^j\}\) and \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) together imply $$\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf x^i\right)=\mathbf p\cdot\left(\sum_{j\in\mathbf J}\mathbf y^j\right)+\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf e^i\right)\tag{1.5}$$ $$\mathbf p\cdot\left(\sum_{i\in\mathbf I}\bar{\mathbf x}^i\right)=\mathbf p\cdot\left(\sum_{j\in\mathbf J}\bar{\mathbf y}^j\right)+\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf e^i\right)\tag{1.6}$$ Note that the RHS of equation (1.5) is less or equal to the RHS of equation (1.6) due to (1.4), but the LHS of equation (1.5) is strictly larger than the LHS of equation (1.6) due to (1.3). So we have reached a contradiction. \(\blacksquare\)

Definition 1.5 (Set-sum) For any \(\mathbf A,\mathbf B\in\mathbf L\), the set-sum \(\mathbf A+\mathbf B\) of \(\mathbf A\) and \(\mathbf B\) is the set of the sums of each element from \(\mathbf A\) and each element from \(\mathbf B\), i.e. $$\mathbf A+\mathbf B\equiv\{\mathbf a+\mathbf b:\forall\mathbf a\in\mathbf A,\forall\mathbf b\in\mathbf B\}$$

Definition 1.6 (Concavity) A function \(f:\mathbf X\to\mathbb R\) is strictly concave if $$f(\theta\mathbf x+(1-\theta)\mathbf x')>\theta f(\mathbf x)+(1-\theta)f(\mathbf x'),\quad\text{for }\forall\theta\in(0,1)$$ for any \(\mathbf x,\mathbf x'\in\mathbf X\) such that \(\mathbf x'\ne\mathbf x\).

Definition 1.7 (Quasi-concavity) A function \(f:\mathbf X\to\mathbb R\) is strictly quasi-concave if its upper contour set \(\{\mathbf x\in\mathbf X:f(\mathbf x)\ge f(\tilde{\mathbf x})\}\) is (strictly) convex for \(\forall\tilde{\mathbf x}\in\mathbf X\).

Theorem 1.2 (Separating Hyperplane Theorem) Suppose that \(\mathbf A,\mathbf B\subseteq\mathbb R^m\) are convex and disjoint sets. Then, \(\exists\mathbf p\in\mathbb R^m\) and \(\mathbf p\ne\mathbf 0\) such that $$\mathbf p\cdot\mathbf x\ge\mathbf p\cdot\mathbf y,\quad\text{for }\forall\mathbf x\in\mathbf A\text{ and }\forall\mathbf y\in\mathbf B$$ (Here "disjoint" means that the intersection is empty, i.e. \(\mathbf A\cap\mathbf B=\varnothing\).)

Assumption HH \(\mathbf X^i\) is convex and \(u^i:\mathbf X^i\to\mathbb R\) is continuous and strictly quasi-concave for \(\forall i\in\mathbf I\). (Strict quasi-concavity of \(u^i\) also means that the upper-contour set of \(u^i\) is strictly convex.)

Assumption FF The aggregate production set-sum of the economy is convex, i.e. $$\mathbf Y\equiv\left\{\mathbf y\in\mathbf L:\mathbf y=\sum_{j=1}^J\mathbf y^j,\ \mathbf y^j\in\mathbf Y^j,\ \text{for }\forall j\in\mathbf J\right\}$$ is convex.

Theorem 1.3 (Second Welfare Theorem) Suppose that \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is a Pareto optimal allocation. Then there exists a price vector \(\mathbf p\) such that: 1. All firms maximize profits with \(\bar{\mathbf y}^j\), i.e. for \(\forall j\in\mathbf J\), $$\mathbf p\cdot\mathbf y^j\le\mathbf p\cdot\bar{\mathbf y}^j,\quad\text{for }\forall\mathbf y\in\mathbf Y^j$$ 2. All households minimize expenditure with \(\bar{\mathbf x}^i\) to attain the utility at least as much as the utility attainable by \(\bar{\mathbf x}^i\), i.e. $$\bar{\mathbf x}^i\in\arg\min_{\mathbf x\in\mathbf X^i}\mathbf p\cdot\mathbf x\quad\text{subject to}\quad u^i(\mathbf x)\ge u^i(\bar{\mathbf x}^i)$$

Proof Denote \(\mathbf A\) as the set-sum of the upper contour sets of all the households: $$\mathbf A\equiv\left\{\mathbf x\in\mathbf L:\mathbf x=\sum_{i\in\mathbf I}\mathbf x^i,\ \mathbf x^i\in\mathbf X^i,\ u^i(\mathbf x^i)\ge u^i(\bar{\mathbf x}^i),\ \text{for }\forall i\in\mathbf I\right\}$$ By Assumption HH, the upper contour set for each household \(i\) is convex. Since set-sum preserves convexity, \(\mathbf A\) is also a convex set.

By Assumption FF, \(\mathbf Y\) is convex. Define \(\mathbf B\equiv\mathbf Y+\sum_{i\in\mathbf I}\mathbf e^i\), then \(\mathbf B\) is also convex.

Since \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is a Pareto optimal allocation, by definition of Pareto optimal allocation, $$\text{int}(\mathbf A)\cap\mathbf B=\varnothing\tag{1.7}$$ By the Separating Hyperplane Theorem, there exists a vector \(\mathbf p\ne\mathbf 0\) such that $$\mathbf p\cdot\mathbf x\ge\mathbf p\cdot\mathbf z,\quad\text{for }\forall\mathbf x\in\mathbf A\text{ and }\forall\mathbf z\in\mathbf B$$ which implies $$\mathbf p\cdot\mathbf x\ge\mathbf p\cdot\mathbf y+\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf e^i\right)\tag{1.8}$$ for \(\forall\mathbf x\in\mathbf A\) and \(\forall\mathbf y\in\mathbf Y\). (1.8) holds particularly for \(\mathbf x=\sum_{i\in\mathbf I}\bar{\mathbf x}^i\) and \(\mathbf y=\sum_{j\in\mathbf J}\bar{\mathbf y}^j\), i.e. $$\mathbf p\cdot\left(\sum_{i\in\mathbf I}\bar{\mathbf x}^i\right)\ge\mathbf p\cdot\left(\sum_{j\in\mathbf J}\bar{\mathbf y}^j\right)+\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf e^i\right),\quad\forall\mathbf y\in\mathbf Y\tag{1.9}$$ The feasibility of \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) implies (1.9) binding, i.e. $$\mathbf p\cdot\left(\sum_{i\in\mathbf I}\bar{\mathbf x}^i\right)=\mathbf p\cdot\left(\sum_{j\in\mathbf J}\bar{\mathbf y}^j\right)+\mathbf p\cdot\left(\sum_{i\in\mathbf I}\mathbf e^i\right)$$ So, \(\mathbf p\cdot\bar{\mathbf y}^j\ge\mathbf p\cdot\mathbf y^j\) holds for \(\forall\mathbf y^j\in\mathbf Y^j\), which means that firms are maximizing their profits.

And similarly for \(\forall\mathbf x^i\in\mathbf X^i\), \(\mathbf p\cdot\bar{\mathbf x}^i\le\mathbf p\cdot\mathbf x^i\), which means that households are minimizing expenditure. \(\blacksquare\)

Proposition 1.1 (Arrow's Remark) If the Pareto optimal allocation consumption \(\bar{\mathbf x}^i\) is not the "cheapest" point in the budget set of each household \(i\), i.e. for \(\forall i\in\mathbf I\), \(\exists\hat{\mathbf x}^i\in\mathbf X^i\) such that \(\mathbf p\cdot\hat{\mathbf x}^i<\mathbf p\cdot\bar{\mathbf x}^i\). Then, the Pareto optimal allocation \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) and the price vector \(\mathbf p\) is a competitive equilibrium, which means that the expenditure minimization is the same as utility maximization.

Proof We want to show that minimizing expenditure subject to attaining utility no less than \(u^i(\bar{\mathbf x}^i)\) is the same as maximizing utility subject to a budget constraint equals to \(\mathbf p\cdot\bar{\mathbf x}^i\).

We will prove this proposition by way of contradiction. Suppose not. Then, there is a \(\hat{\mathbf x}\in\mathbf X^i\) such that \(\mathbf p\cdot\hat{\mathbf x}\le\mathbf p\cdot\bar{\mathbf x}^i\) and \(u^i(\hat{\mathbf x})>u^i(\bar{\mathbf x}^i)\). Denote $$\mathbf x^\theta=\theta\hat{\mathbf x}+(1-\theta)\tilde{\mathbf x},\quad\text{for }\forall\theta\in(0,1)$$ where \(\tilde{\mathbf x}\) is the cheaper point in the proposition's assumption. Since \(u^i(\hat{\mathbf x})>u^i(\bar{\mathbf x}^i)\), the continuity and strict quasi-concavity of \(u^i\) implies that \(u^i(\mathbf x^\theta)\ge u^i(\bar{\mathbf x}^i)\) for \(\theta\) sufficiently close to zero. Note that \(\tilde{\mathbf x}\) is a strictly cheaper point than \(\bar{\mathbf x}^i\), so $$\begin{aligned}\mathbf p\cdot\mathbf x^\theta&=\theta\,\mathbf p\cdot\hat{\mathbf x}+(1-\theta)\,\mathbf p\cdot\tilde{\mathbf x}\\&<\theta\,\mathbf p\cdot\bar{\mathbf x}^i+(1-\theta)\,\mathbf p\cdot\bar{\mathbf x}^i\\&=\mathbf p\cdot\bar{\mathbf x}^i\end{aligned}$$ which contradicts with the fact that \(\bar{\mathbf x}^i\) is the expenditure minimizing solution subject to attaining utility no less than \(u^i(\bar{\mathbf x}^i)\).

So we have reached contradiction and thus there is no such \(\hat{\mathbf x}\) that \(\mathbf p\cdot\hat{\mathbf x}\le\mathbf p\cdot\bar{\mathbf x}^i\) and \(u^i(\hat{\mathbf x})>u^i(\bar{\mathbf x}^i)\), which means that \(\bar{\mathbf x}^i\) is not only expenditure minimizing but also utility maximizing. \(\blacksquare\)

Remark 1.1 The Second Welfare Theorem also implies that any Pareto Optimal allocation can be achieved by perfect competition with any initial endowment as long as appropriate lump-sum transfers among agents are arranged before the start of competition such that the Pareto Optimal allocation is budget feasible for every agent in that economy.

Remark 1.2 The Second Welfare Theorem is convenient in the sense that competitive equilibrium sometimes is hard to solve but we can instead solve the social planner's problem whose solution is Pareto Optimal, then by the Second Welfare Theorem we know that solution is also a competitive equilibrium for some price vector. Conditions making the Second Welfare Theorem work justifies the use of representative agent in our modeling.

Remark 1.3 In order to have the Second Welfare Theorem work, we must assume there is no externalities, no taxes, no market power of any agent (everyone is just a price taker) and no real role played by money, and etc.