General Equilibrium and the Welfare Theorems

Equilibrium in a single market is only part of the picture. A change in the price of one good sends waves through all markets. General equilibrium (GE) is a theory describing the simultaneous equilibrium in all markets of the economy. Two fundamental welfare theorems link competitive equilibrium with the optimal allocation of resources.

The Arrow-Debreu General Equilibrium Model

Exchange economy: $I$ consumers, $L$ goods. Consumer $i$ has an initial endowment $\omega_i \in \mathbb{R}^L_+$ and preferences $\succeq_i$.

Competitive equilibrium (Walrasian): price vector $p^$ and allocation $(x_1^,...,x_I^*)$ such that:

1. Optimality: $x_i^* \succeq_i x_i$ for all admissible $x_i$ with $p^* \cdot x_i \leq p^* \cdot \omega_i$
2. Markets clear: $\sum_i x_i^* = \sum_i \omega_i$ (demand = supply for all goods)

Walras' law: $\sum_l p_l(\sum_i x_{il} - \omega_{il}) = 0$ — the total excess demand in value is always $= 0$. Corollary: if $L-1$ markets are in equilibrium, the $L$th is also in equilibrium. Prices are defined only up to a scalar (normalization: $\sum_l p_l = 1$ or $p_1 = 1$).

Existence theorem (Arrow-Debreu, 1954): under convex, continuous, monotonic preferences, a competitive equilibrium exists. Proof via the Kakutani fixed point theorem.

The Two Welfare Theorems

Pareto optimality: an allocation $(x_i)$ is Pareto optimal (PO) if there is no other admissible allocation that improves at least one consumer without worsening the others.

First welfare theorem: Any competitive equilibrium is Pareto optimal.

Proof (sketch): Suppose that under $p^$, the equilibrium $(x_i^)$ is not PO. Then there exists an allocation $(x'_i)$: $x'_i \succ_i x_i^$ for some $i$, $x'_i \succeq_i x_i^$ for the rest. From rational behavior: $x'_i \succ_i x_i^* \rightarrow p^* \cdot x'_i > p^* \cdot x_i^$ (otherwise consumer $i$ would have chosen $x'_i$). Summing over $i$: $p^ \cdot \sum x'_i > p^* \cdot \sum x_i^* = p^* \cdot \sum \omega_i$. Contradicts resource feasibility.

Second theorem: Under convex preferences, any Pareto optimal allocation can be reached as a competitive equilibrium after a suitable redistribution of initial endowments.

Political meaning: The objectives of efficiency (markets) and equity (distribution) are separable: first achieve the desired distribution through lump-sum transfers, then let markets work — they will find a PO allocation on their own.

The Edgeworth Box

Two-good, two-agent economy: $\omega_A = (6, 2)$, $\omega_B = (2, 6)$. Total endowment $\Omega = (8, 8)$.

In the Edgeworth box: the horizontal axis is good 1 ($0...8$), the vertical is good 2 ($0...8$). The initial allocation is the point $\omega = (6, 2)$. Indifference curves of $A$ are standard, those of $B$ are rotated by $180^\circ$. The mutually beneficial trade area = “lens” between indifference curves passing through $\omega$.

Contract curve (Pareto set): the set of PO allocations — where indifference curves are tangent ($MRS_A = MRS_B$). For $u_A = x_1 x_2$, $u_B = x_1 x_2$: the contract curve is the diagonal of the box $x_{1A} = x_{2A}$.

Market Failures

The first theorem assumes: no externalities, no public goods, perfect information, complete markets. Violation of any of these conditions $\rightarrow$ market failure.

Negative externalities: A plant pollutes a river — fishermen bear costs without compensation. Social costs

gt;$ private costs $\rightarrow$ overproduction. Solutions: Pigovian tax ($= MSC - MPC$), tradeable emission permits (Coase).

Coase theorem (1960): With zero transaction costs, agents themselves will reach an efficient solution to externalities through bargaining — regardless of the initial allocation of rights.

Numerical Example

Two consumers: $u_A = x_{1A}^{0.5} x_{2A}^{0.5}$, $u_B = x_{1B}^{0.5} x_{2B}^{0.5}$. $\Omega = (1, 1)$. Equilibrium prices: $p_1/p_2 = 1$ (by symmetry). Equilibrium: $x_{1A}^* = x_{2A}^* = 0.5$, $x_{1B}^* = x_{2B}^* = 0.5$. $MRS_A = x_{2A}/x_{1A} = 1 = p_1/p_2$ ✓. $MRS_B = x_{2B}/x_{1B} = 1 = p_1/p_2$ ✓. Both consumers are on the contract curve ($MRS_A = MRS_B = 1$).

Exercise: Exchange economy: two consumers $A$ and $B$, two goods. $\omega_A = (4,1)$, $\omega_B=(2,3)$. $u_A = \min(x_1, x_2)$, $u_B = x_1 + x_2$. (1) Find WE (competitive equilibrium). (2) Draw the Edgeworth box. (3) Is WE Pareto optimal? (4) If the initial endowments are redistributed ($\omega'_A = (3,2)$, $\omega'_B = (3,2)$) — how does the equilibrium change? Is this the principle of the second theorem?