6  The Welfare Theorems of General Equilibrium

6.1 Introduction

This section introduces the classical theory of individual preference and the connection between this and the observable choices that individuals make. To do this, we derive the demand that is implied by the preferences of an individual and the budget set available to them.

  • Reading material: MWG Ch. 15

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 \(i\), \(x^{*}_{i} \succsim x\) for all \(x \in \{x \in X_{i}: p\cdot x \le p\cdot \omega_{i} + \theta_{i}\cdot (p\cdot y^{*}_{j}) \}\)
  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. Since \(V \cap (Y + \sum_{i} \omega_{i}) = \emptyset\) is non-empty, there exists a feasible allocation that makes everyone better off which is a Pareto improvement.

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 \(i\). Then 7 and 8 satisfy the first two conditions, and the third follows from the feasibility of the allocation (which follows from claims 3,4,5.

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 \(w_{i}\), if \(x_{i} \succ x_{i}^{*}\) 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 cheaper than \(w_{i}\)) then \(x_{i} \succ x_{i}^{*}\) implies that \(p\cdot x_{i} > w_{i}\).

Proof. Draw both bundles on the budget line and the cheaper one inside. Then by continuity there exists an \(\alpha\) for which the convex combination of the better and worse is better than \(x^{*}\). This is a contradiction because the convex combination costs less but is preferred.

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.