Section 7

First and Second Welfare Theorems

Why these theorems matter

The welfare theorems are the bridge from 'prices clear markets' to 'competitive markets are good.' Without them, a Walrasian equilibrium is just a fixed point — a description of where supply equals demand with no claim that the outcome is desirable in any sense. The First Welfare Theorem supplies the efficiency claim; the Second Welfare Theorem supplies the policy implication. Together they define what the price mechanism can and cannot do.

First Welfare Theorem (FWT)

Every Walrasian equilibrium allocation is Pareto efficient. That is: if $(\mathbf{x}^*, p^*)$ is a Walrasian equilibrium, there is no feasible allocation that makes every agent at least as well off and at least one agent strictly better off.

The intuition behind the proof is sometimes called the 'no $5 bills on the sidewalk' argument. At the equilibrium price $p^*$ , every agent has spent their entire wealth on the bundle they prefer most — by construction of Walrasian demand. Now suppose some other feasible allocation $\mathbf{x}'$ Pareto-dominated the equilibrium allocation $\mathbf{x}^*$ . Then for each agent $i$ , the bundle $x'^i$ is at least as good as $x^{*i}$ , and strictly better for at least one agent. Local non-satiation implies that if $x'^i$ is at least as good as $x^{*i}$ , it cannot cost strictly less at $p^*$ — otherwise the agent would have chosen it. And if $x'^i$ is strictly preferred, it must cost strictly more. Summing the budget inequalities across all agents, the total expenditure on $\mathbf{x}'$ exceeds the total wealth in the economy. But $\mathbf{x}'$ is feasible, so its total value equals the value of the aggregate endowment — a contradiction.

\sum_i p \cdot x'^{\,i} > \sum_i p \cdot \omega^i = \sum_i p \cdot x^{*i}

The cost argument formalised: a Pareto-dominating allocation would have to cost more in aggregate than the total endowment, which is impossible if the allocation is feasible.

Second Welfare Theorem (SWT)

Under convexity and continuity assumptions on preferences, every Pareto-efficient allocation can be supported as a Walrasian equilibrium for some redistribution of initial endowments. Formally: if $\hat{\mathbf{x}}$ is a Pareto-efficient allocation and preferences are convex, there exists a price vector $\hat p$ and an assignment of endowments $\hat\omega$ such that $(\hat{\mathbf{x}}, \hat p)$ is a Walrasian equilibrium of the economy with endowments $\hat\omega$ .

The implication for policy is direct. If a planner has views about distribution — if they want agents to end up at a particular point on the contract curve rather than wherever the initial endowments take them — the correct instrument is a lump-sum transfer of endowments, not a price intervention. Taxes on goods, price controls, or quantity restrictions all distort the margin between MRS and the price ratio, producing inefficiency. SWT says you can get to any Pareto-efficient target by simply redistributing wealth and then letting the market find the equilibrium price on its own.

How SWT decentralises an allocation

1
Pick the target
Choose any Pareto-efficient allocation $\hat{\mathbf{x}} = (\hat x^A, \hat x^B)$ — any point on the contract curve where $\text{MRS}^A = \text{MRS}^B$ .
2
Find the supporting price
Compute the common MRS at $\hat{\mathbf{x}}$ . That ratio is the equilibrium price: $p_1 / p_2 = \text{MRS}^A(\hat x^A) = \text{MRS}^B(\hat x^B)$ . At this price, the budget line is tangent to both agents' indifference curves at the target allocation.
3
Redistribute endowments
Transfer endowments so each agent's wealth at the new prices is exactly enough to afford their target bundle: $m^i = \hat p \cdot \hat x^i$ . With those endowments in place, each agent's Walrasian demand at $\hat p$ is exactly $\hat x^i$ , and markets clear.