6  The Welfare Theorems of General Equilibrium

6.1 Introduction

This section studies the welfare properties of competitive equilibrium. We state and prove the two fundamental theorems of welfare economics: that every price equilibrium with transfers is Pareto optimal (the first theorem), and that under convexity assumptions every Pareto optimal allocation can be supported as a price (quasi)equilibrium with transfers (the second theorem). We then examine when a quasiequilibrium is a genuine equilibrium.

  • Reading material: MWG Ch. 16

Definition 6.1 (General Equilibrium) A private ownership economy specified by \((\{X_{i},\succsim_{i}\}_{i \in \mathcal{I}},\{Y_{j}\}_{j \in \mathcal{J}}, \{\omega_{i},\theta_{i}\}_{i \in \mathcal{I}})\) has a competitive (or Walrasian) equilibrium if there exists an allocation \((x^{*},y^{*})\) and price vector \(p\), such that

  1. For every \(j\), \(y^{*}_{j}\) satisfies \(p\cdot y_{j} \le p \cdot y_{j}^{*}\) for all \(y_{j} \in Y_{j}\).
  2. For every \(i\), \(x^{*}_{i} \succsim_{i} x\) for all \(x \in \{x \in X_{i}: p\cdot x \le p\cdot \omega_{i} + \sum_{j}\theta_{ij} (p\cdot y^{*}_{j}) \}\)
  3. \(\sum_{i} x^{*}_{i} = \sum_{i} \omega_{i} + \sum_{j}y^{*}_{j}\)

Definition 6.2 (Price Equilibrium with Transfers) A private ownership economy specified by \((\{X_{i},\succsim_{i}\}_{i \in \mathcal{I}},\{Y_{j}\}_{j \in \mathcal{J}}, \{\omega_{i},\theta_{i}\}_{i \in \mathcal{I}})\) has a price equilibrium with transfers if there is an allocation \((x^{*},y^{*})\), a price vector \(p\) and a collection of wealth levels \(\{w_{i}\}_{i \in \mathcal{I}}\) where \(\sum_{i}w_{i} = p\cdot\sum_{i}\omega_{i} + \sum_{j}p\cdot y^{*}_{j}\) for which

  1. For every \(j\), \(y^{*}_{j}\) satisfies \(p\cdot y_{j} \le p \cdot y_{j}^{*}\) for all \(y_{j} \in Y_{j}\).
  2. For every \(i\), \(x^{*}_{i} \succsim x\) for all \(x \in \{x \in X_{i}:p \cdot x \le w_{i} \}\)
  3. \(\sum_{i} x^{*}_{i} = \sum_{i} \omega_{i} + \sum_{j}y^{*}_{j}\)

Definition 6.3 (Pareto optimality) A feasible allocation \((x,y)\) is Pareto optimal if there is no other feasible allocation \((x',y')\) that Pareto dominates it. That is, for which \(x'_{i} \succsim x_{i}\) for all \(i\) and \(x'_{i}\succ x_{i}\) for at least one \(i\).

First fundamental theorem of welfare economics.

If preferences are locally non-satiated, 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 PEwT with associated wealth levels \(\{w_{i} \}_{i \in \mathcal{I}}\). Then if \(x_i \succ x_{i}^{*}\) then \(p\cdot x_{i} > w_{i}\). (Anything that is strictly preferred must not be affordable.) Under local non-satiation however, if \(x_i \succsim x_{i}^{*}\) then \(p\cdot x_{i} \ge w_{i}\). [Proof by contradiction. Assume that \(p\cdot x_{i} < w_{i}\), then by LNS, there exists an \(x'_{i}\) that is preferred to \(x_{i}\) and is affordable.]

Suppose that \((x,y)\) Pareto dominates \((x^{*},y^{*})\). Then by the previous property \(p\cdot x_{i} \ge w_{i}\) and by the property before that \(p\cdot x_{i} > w_i\) for at least one \(i\). Hence, \(\sum_{i}p x_{i} > \sum_{i}w_{i} = p\cdot\sum_{i}\omega_{i} + \sum_{j}p\cdot y^{*}_{j}\). But since \(y^{*}_{j}\) is profit maximizing for firm \(j\) at prices \(p\), \(p\cdot\sum_{i}\omega_{i} + \sum_{j}p\cdot y^{*}_{j} \ge p\cdot\sum_{i}\omega_{i} + \sum_{j}p\cdot y_{j}\) which implies that \((x,y)\) is not feasible.

Definition 6.4 (Price quasiequilibrium with transfers.) Given an economy, an allocation \((x^{*},y^{*})\) and a price vector \(p \ne 0\) constitute a price quasiequilibrium with transfers if there is an assignment of wealth levels \(\{w_{i}\}_{i \in \mathcal{I}}\) with \(\sum_{i}w_{i} = p\cdot\sum_{i}\omega_{i} + \sum_{j}p\cdot y^{*}_{j}\), such that

  1. For every \(j\), \(y^{*}_{j}\) satisfies \(p\cdot y_{j} \le p \cdot y_{j}^{*}\) for all \(y_{j} \in Y_{j}\).
  2. For every \(i\), if \(x_{i} \succ x^{*}_{i}\) then \(p\cdot x_{i} \ge w_{i}\).
  3. \(\sum_{i} x^{*}_{i} = \sum_{i} \omega_{i} + \sum_{j}y^{*}_{j}\)
Second fundamental theorem of welfare economics.

Consider an economy for which \(Y_{j}\) is convex and every preference relation \(\succsim_{i}\) is convex and locally non-satiated. Then, for every Pareto optimal allocation \((x^{*},y^{*})\), there is a price vector \(p\) such that \((x^{*},y^{*},p)\) is a price quasiequilibrium with transfers.

Proof. Separating hyperplanes! For each \(i\), let \(V_{i} = \{x_{i} \in X_{i}: x_{i} \succ x_{i}^{*} \}\). Define \(V = \sum V_{i} = \{ \sum_{i} x_{i}: x_{1} \in V_{1},\ldots,x_{I} \in V_{I}\}\). \(Y = \sum_{j} Y_{j}\).

Claim 1. Every \(V_{i}\) is convex.

Claim 2. The sets \(V\) and \(Y + \sum_{i} \omega_{i}\) are convex. (The sum of any two convex sets is convex).

Claim 3. \(V \cap (Y + \sum_{i} \omega_{i}) = \emptyset\), since \((x^{*},y^{*})\) is Pareto optimal. Assume the opposite. Then \(V \cap (Y + \sum_{i} \omega_{i})\) is non-empty, so there exists a feasible allocation that makes everyone strictly better off, which is a Pareto improvement — contradicting the Pareto optimality of \((x^{*},y^{*})\).

Claim 4. There exists a \(p \ne 0\) and a number \(r\) such that \(p\cdot z \ge r\) for every \(z \in V\) and \(p \cdot z \le r\) for every \(z \in Y + \sum_{i}\omega_{i}\). This is the separating hyperplane theorem.

Claim 5. If \(x_{i} \succsim x_{i}^{*}\) for every \(i\), then \(p\cdot\left(\sum_{i}x_{i}\right) \ge r\). Proof: If \(x_{i} \succsim x^{*}_{i}\) for every \(i\), by local nonsatiation, for each \(i\) there is an \(\hat x_{i}\) close to \(x_{i}\) for which \(\hat x_{i} \succ x^{*}_{i}\) so \(\hat x_{i} \in V_{i}\). Hence, \(\sum_{i}\hat x_{i} \in V\) so \(p\cdot(\sum_{i}\hat x_{i}) \ge r\), which taking limits of both sides says that \(p \cdot \sum_{i} x_{i} \ge r\).

Claim 6. \(p\cdot\left(\sum_{i}x^{*}_{i} \right) = p \cdot \left ( \sum_{i} \omega_{i} + \sum_{j}y^{*}_{j} \right ) = r\). Proof: By step 5, \(p\cdot \left ( \sum_{i} x^{*}_{i} \right) \ge r\). But since feasibility requires that \(\sum_i x^{*}_{i} = \sum_{i} \omega_{i} + \sum_{j}y^{*}_{j}\), and \(p\cdot (\sum_{i} \omega_{i} + \sum_{j}y^{*}_{j}) \le r\), \(p \cdot \left (\sum_{i}x^{*}_{i} \right) \le r\).

Claim 7. Firms are profit maximizing under \(p\). Proof: For any firm \(j\) and plan \(y_{j}\), we have \(y_{j} + \sum_{h \ne j} y^{*}_{h} \in Y\). Therefore,

\[p\cdot \left ( \sum_{i}\omega_{i} + y_{j} + \sum_{h \ne j}y^{*}_{h} \right) \le r = p \cdot \left ( \sum_{i}\omega_{i} + y_{j}^{*} + \sum_{h \ne j}y^{*}_{h} \right)\]

which implies that \(p\cdot y_{j} \le p\cdot y^{*}_{j}\).

Claim 8. For consumers, all better allocations are only weakly affordable. Proof: Let \(x_{i} \succ x^{*}_{i}\). By 5 and 6, (in the same way as 7),

\[p\cdot \left ( x_{i} + \sum_{k \ne i}x^{*}_{k} \right ) \ge r = p \cdot \left ( x^{*}_{i} + \sum_{k \ne i} x^{*}_{k} \right)\]

which implies that \(p\cdot x_{i} \ge p \cdot x^{*}_{i}\).

Claim 9. Let \(w_{i} = p\cdot x^{*}_{i}\) for all \(i\). Condition 1 of the quasiequilibrium is Claim 7, condition 2 is Claim 8 (with this choice of \(w_{i}\), \(p\cdot x_{i} \ge p\cdot x^{*}_{i} = w_{i}\) whenever \(x_{i} \succ x^{*}_{i}\)), and condition 3 is the feasibility of \((x^{*},y^{*})\), which holds because it is a Pareto optimal (hence feasible) allocation. Finally, the wealth levels satisfy the required adding-up condition: summing over \(i\) and using feasibility, \(\sum_{i}w_{i} = p\cdot\sum_{i}x^{*}_{i} = p\cdot\sum_{i}\omega_{i} + \sum_{j}p\cdot y^{*}_{j}\). So \((x^{*},y^{*},p)\) is a price quasiequilibrium with transfers.

Proposition 6.1 (16.D.2) Suppose that \(X_{i}\) is convex and \(\succsim_{i}\) is continuous and that for an allocation \(x_{i}^{*}\), price vector \(p\) and wealth level \(w_{i}\), \(x_{i} \succ x_{i}^{*}\) implies that \(p\cdot x_{i} \ge w_{i}\). Then, if there is a consumption vector \(x_{i}' \in X_{i}\) such that \(p\cdot x_{i}' < w_{i}\) (there is something feasible and strictly cheaper than \(w_{i}\)), then \(x_{i} \succ x_{i}^{*}\) implies that \(p\cdot x_{i} > w_{i}\).

Proof. Suppose, toward a contradiction, that \(x_{i} \succ x_{i}^{*}\) but \(p\cdot x_{i} = w_{i}\) (the case \(p\cdot x_{i} \ge w_{i}\) with equality is the only one to rule out). Consider the bundles \(x^{\alpha} = \alpha x_{i} + (1-\alpha)x_{i}'\), which lie in \(X_{i}\) by convexity. By continuity of \(\succsim_{i}\), for \(\alpha\) close enough to \(1\) we still have \(x^{\alpha} \succ x_{i}^{*}\). But \(p\cdot x^{\alpha} = \alpha w_{i} + (1-\alpha)p\cdot x_{i}' < w_{i}\), contradicting the hypothesis that anything strictly preferred to \(x_{i}^{*}\) costs at least \(w_{i}\).

Proposition 6.2 (16.D.3) If every \(X_{i}\) is convex, \(0 \in X_{i}\), and \(\succsim_{i}\) is continuous, then any price quasiequilibrium with transfers that has \((w_1,\ldots,w_{I}) >> 0\) is a price equilibrium with transfers.

Examples.
  1. Cobb-Douglas utility
  2. Linear Utility
  3. Quasilinear utility

Edgeworth box examples.

Triangle relating CEwT, to Pareto optimality to maximizing a linear SWF.

Pareto optimality and linear social welfare.