Module 4 — General Equilibrium and Welfare

Core question

Can decentralised choices be mutually feasible in every market, and what assumptions connect competitive equilibrium to social efficiency?

Learning outcomes

You will be able to:

• define feasibility, Pareto efficiency, and Walrasian equilibrium;
• solve a two-consumer, two-good exchange economy;
• state and qualify both fundamental welfare theorems;
• distinguish existence, uniqueness, stability, efficiency, and fairness;
• explain how production networks transmit local shocks.

1. From one budget to an economy

Consumer i has preferences u_i, endowment ω_i, and demand:

An allocation is feasible when:

A Walrasian equilibrium is a price vector p* and allocation x* such that:

1. every consumer optimises at p*;
2. every market clears:

Prices coordinate plans; they do not create resources.

2. Edgeworth-box logic

For two consumers and two goods, the Edgeworth box contains every feasible division of total endowment. An allocation is Pareto efficient if no feasible reallocation makes one consumer better off without making the other worse off.

At a regular interior efficient allocation:

Otherwise, the consumers value a marginal trade differently and both can gain. The set of efficient allocations is the contract curve. Efficiency does not select one point on it.

3. Worked competitive equilibrium

Let total resources be (10,10). Endowments are:

and both consumers have u_i(x,y)=√(xy). Normalise p_y=1 and write p_x=p.

Incomes are:

Each spends half of income on each good, so aggregate demand for x is:

Market clearing requires 5+5/p=10, hence p*=1. Each income is 10 and each demands (5,5).

Both optimise, trade balances in value, and aggregate demand equals resources.

4. Walras' law reduces—but does not solve—the system

Aggregate excess demand is:

If every budget satisfies Walras' law, then:

Therefore, if all but one market clear and the omitted good has a positive price, the last clears too. Walras' law is an accounting restriction; it does not prove that an equilibrium exists, is unique, or is stable under price adjustment.

5. The welfare theorems—and their assumptions

First welfare theorem

Under local nonsatiation and a complete competitive-market environment, a Walrasian equilibrium is Pareto efficient.

The result can fail with:

• externalities or public goods;
• market power;
• asymmetric information;
• missing state-contingent markets;
• non-convex technologies or other institutional constraints.

Second welfare theorem

With convex preferences and production sets plus suitable regularity, a Pareto-efficient allocation can be decentralised as a competitive equilibrium after appropriate lump-sum redistribution of endowments.

This separates, in theory, distribution from price-based allocation. In practice, lump-sum transfers may be unavailable because type, wealth, or earning ability is private; taxes then change incentives.

6. Existence, uniqueness, and stability are different claims

• Existence: at least one price vector clears markets.
• Uniqueness: only one relative-price equilibrium exists.
• Local stability: small deviations trigger forces that return to it.
• Global stability: adjustment converges from a wide set of starting prices.

Continuity, convexity, and bounded feasible sets help establish existence through a fixed-point argument. Gross-substitutes conditions can support uniqueness and price adjustment. Standard preference assumptions alone do not guarantee either.

7. Add firms and production

Firms choose net output y_j from production set Y_j to maximise p·y_j. Consumers own firm profits. Feasibility becomes:

At an interior efficient allocation with production:

so consumers' willingness to substitute goods equals the economy's marginal transformation rate. Distortions anywhere in the production chain can alter prices and quantities elsewhere.

8. Current extension: production networks

Baqaee and Farhi study trade economies with international production networks and wedge-like distortions such as markups, tariffs, and nominal rigidities (2024). Their framework shows why a sector's sales share alone need not measure the full welfare effect of a shock: substitution, input-output links, and distortions generate general-equilibrium propagation.

Use the paper as a conceptual warning. A supply disruption at a small upstream input can matter through network position, while a large gross-output sector need not create an equally large welfare loss. Exact effects require an estimated network and elasticities; the theorem is not a universal multiplier.

9. Partial versus general equilibrium

The choice is about the counterfactual, not prestige: use the smallest model that contains the important feedbacks.

Practice

1. Resolve the worked economy when A's endowment becomes (9,1) and B's (1,9).
2. Find the contract-curve condition when u_A=x_Ay_A and u_B=x_B²y_B.
3. Give an efficient but highly unequal allocation.
4. Identify which welfare-theorem assumption fails in pollution, monopoly, and hidden-action insurance.
5. Draw a three-sector input network and trace one upstream productivity shock.

Quick check

• Equilibrium requires optimisation and simultaneous feasibility.
• Equal MRS is an interior efficiency condition, not a fairness rule.
• Walras' law is accounting; existence and stability need more.
• The welfare theorems are conditional institutional results.
• Networks turn local shocks into economy-wide price and quantity changes.

Next: replace price-taking with strategic interaction.