The Two Welfare Theorems

Walk along the utility possibility frontier. Pick a point with the slider and move it around.

The competitive equilibrium always sits on the frontier. That is the First Welfare Theorem. Other points on the frontier are reachable too. Support the point with the right price and shift income in one lump sum. Then the market lands there by itself. That is the Second Theorem, and the formula below gives the transfer.

For example, say you and a 🧑 friend split 10 🍬 candies. Every split that uses all 10 is efficient. But if your friend's share feels too small, there is a fix. Hand a few candies to your friend before trading starts. Then plain market trading reaches the split you wanted.

(u_1, u_2) \text{ on the frontier} \iff \text{Pareto efficient}

T = p \cdot (x^1 - e^1) \quad \text{(lump-sum transfer)}

Part of these learning paths

General Equilibrium and Welfare

Examples & Exercises

Verify directly via marginal rates of substitution that the competitive equilibrium allocation $x^1 = x^2 = (5, 5)$ is Pareto efficient (First Welfare Theorem).

MRS_i(x^i) = \frac{a_i}{1-a_i}\frac{x^i_2}{x^i_1}

x \text{ Pareto efficient} \iff MRS_1(x^1) = MRS_2(x^2)

Equilibrium utilities

(5, 5)

sit on the utility possibility frontier.

Show proof

$MRS_1(5, 5) = \frac{0.5}{0.5} \cdot \frac{5}{5} = 1$ , and $MRS_2(5, 5) = 1$ . Both marginal rates of substitution equal the equilibrium price $p_1/p_2 = 1$ , so the indifference curves are tangent and the allocation is on the contract curve — Pareto efficient.
In utility space, $(u_1, u_2) = (5, 5)$ , which lies exactly on the utility possibility frontier $u_2 = 10 - u_1$ traced along the contract curve (the diagonal). This is the First Welfare Theorem: competitive equilibrium is Pareto efficient.

Find the lump-sum transfer that decentralizes the Pareto optimum $x^1 = (6, 6)$ , $x^2 = (4, 4)$ as a competitive equilibrium (Second Welfare Theorem).

p_1 = MRS_1(6, 6) = 1 \quad (\text{supporting price})

T = p \cdot (x^1 - e^1) = 1 \cdot (6-7) + (6-3) = 2

Target

(6, 6)

— supporting price

1

, transfer

T = 2

Show proof

$(6, 6)$ is on the diagonal (the contract curve), hence Pareto optimal. The common marginal rate of substitution there is $MRS = 1$ , so the supporting price is $p_1 = 1$ .
Consumer 1 must afford $(6, 6)$ , requiring wealth $1 \cdot 6 + 6 = 12$ .
Original wealth is $w_1 = 10$ , so a transfer of $T = 12 - 10 = 2$ does it.
After the transfer, demands are $x^1 = (0.5 \cdot 12,\ 0.5 \cdot 12) = (6, 6)$ , $x^2 = (4, 4)$ , and markets clear.

12. Equilibrium and Welfare