General equilibrium and welfare · Lesson 4 of 6
The two welfare theorems
Lessons 1 and 2 built two ideas: the efficient allocations, and the allocation a competitive market delivers. The welfare theorems join them. The first says the market’s outcome is always among the efficient ones. The second says that any efficient outcome you like can be the market’s outcome, if you first move the endowments. Between them, they separate the question of efficiency from the question of fairness.
A question to hold on to
Which of the following is not true in the case of the second theorem of welfare economics? (A) If consumers exhibit convex preferences, every Pareto efficient allocation is a possible competitive equilibrium in a pure exchange economy. (B) In an economy involving production, the convexity of production sets ensures that a Pareto efficient allocation can be achieved as a market equilibrium. (C) It holds when there are increasing returns to scale. (D) It implies that in the market system the allocative role and the distributive role can be separated from each other.
UGC NET June 2023Solved at the end ↓
The questions
Two questions about markets
The welfare theorems answer two questions that run in opposite directions.
- Is the market’s outcome efficient? (First theorem.) Start with prices; end with efficiency.
- Can the market reach any efficient outcome? (Second theorem.) Start with an efficient allocation you want; find the prices.
They are the modern, precise form of Adam Smith’s invisible hand: people pursuing their own interests, trading at common prices, end up on the contract curve. Arrow and Debreu gave the rigorous proofs, independently, in 1951.
Competition is efficient
The first theorem: competition is efficient
First fundamental theorem. Every competitive equilibrium is Pareto efficient. The assumptions behind it are about the market, not about the shape of tastes: people are price-takers who pursue their own self-interest (Hyderabad 2011), there are no externalities, property rights are well defined, and preferences satisfy local nonsatiation (a nearby bundle is always better). Convexity is not needed.
The proof is short enough to remember, and it is the heart of the theorem:
- Suppose a competitive equilibrium were not efficient. Then some feasible allocation x′ makes someone better off and no one worse off.
- The person who is better off at x′ must not have been able to afford it at the equilibrium prices, or she would have chosen it. So x′ costs her more than her wealth.
- Everyone else is no worse off. If x′ cost her less than her wealth, she could buy x′ and still have money left to buy something better, which is impossible if she had already picked her best bundle. So x′ costs her at least her wealth.
- Add up. The total cost of x′ is strictly more than the total wealth, which is the value of the total endowment. But x′ is feasible: it uses no more than the total endowment, so its value cannot exceed that. Contradiction.
Pause and predict
Which of these does the first welfare theorem need?
Any efficient point is reachable
The second theorem: any efficient point is reachable
Second fundamental theorem. If preferences are convex (and continuous and satisfy nonsatiation), then every Pareto efficient allocation is a competitive equilibrium for some redistribution of the endowments (UGC NET 2023). The idea is a geometric one. At an efficient allocation, the two indifference curves are tangent, so there is a common tangent line: the price line. Redistribute endowments so that both budget lines lie along it, and each person’s best choice is the allocation you wanted.
What the redistribution looks like matters: it must be a lump-sum transfer, one that does not depend on anything the person can change by her behaviour. The theorem is the one that speaks to redistribution (JNU 2016; Hyderabad 2019), and it is why economists say the allocative role and the distributive role of the market can be separated: let prices do the first, and move the endowments to do the second.
Interactive
Try it: the transfer
Pick a point on the contract curve with the slider. The solid gold line is the price line through it. The dotted line has the same slope but passes through the endowment ω: the vertical gap between the two lines is the lump-sum transfer needed. The green dot is where the market takes the economy from ω alone: it lands on the contract curve, as the first theorem says. Then switch to the non-convex preset and watch the second theorem fail.
Both people are Cobb–Douglas, whose indifference curves bulge toward the origin. Person 1 is endowed with a lot of x.
Price line through the target┄ the same slope through ω● the market from ω
First theorem
Left to the market from ω, the economy settles at (3.8, 6.82) at p_x/p_y = 1.2. That allocation is on the contract curve: it is Pareto efficient.
Second theorem
At p_x/p_y = 1.26 each person's best choice, with just enough wealth to afford the target, is the target itself. At these prices person 1's endowment is worth 5.23 and the target costs 4.76, so she must hand person 2 a lump-sum transfer worth 1.08 units of y. With that transfer both budget lines lie along the gold line, and the market delivers the target.
The gold dot is your target; slide it along the contract curve. Convex tastes always let a single price line touch both indifference curves at once. Without convexity the line can cut through one of them.
The fine print
What breaks them
| Assumption | First theorem | Second theorem | What goes wrong without it |
|---|---|---|---|
| Price-taking, complete markets | Needed | Needed | Market power and missing markets move prices away from marginal cost and marginal benefit |
| No externalities, well-defined property rights | Needed | Needed | The market ignores a cost or benefit that falls on others |
| Local nonsatiation (“more is better”) | Needed | Needed | A satiated person can sit at an equilibrium that wastes goods others could use |
| Convex preferences and production sets | Not needed | Needed | Demand jumps: no price makes both people choose the target (the lab’s second preset). Increasing returns make production sets non-convex (UGC NET 2023) |
The answer to the opening question follows. Increasing returns to scale make the production set non-convex, so the second theorem does not hold with them. The other three statements are true: convex preferences and convex production sets are exactly what the second theorem needs, and the separation of allocation from distribution is its message.
Watch out for
The mistakes examiners are fishing for
- Convexity is a condition for the second theorem, not the first. The “which is not required” question has convex preferences as its answer.
- The first theorem says competitive equilibrium is efficient, not that efficient allocations are equilibria. That is the second theorem.
- “The theorem proves markets are fair.” It does not. A competitive equilibrium from a very unequal endowment is efficient and very unequal.
- Transfers that distort. The second theorem needs lump-sum transfers. A tax on income or on goods changes behaviour and breaks the efficiency the theorem promises.
Policy
Efficiency and equity
The two theorems together give a clean division of labour. Efficiency is the market’s job. Equity is a choice about the endowments: pick the distribution of resources you consider fair, then let competition make the economy efficient from there. The theorems also show what is wrong with trying to achieve equity by distorting prices (rent control, price ceilings): you lose efficiency without needing to.
In practice, lump-sum transfers are hard to find, since almost everything the government can tax depends on what people do. That is why the second theorem is a benchmark, not a recipe.
Two true statements, no explanation
Assertion-and-reason questions pair the first theorem with Walras’ law (CUET 2023): both are true, but the law does not explain the theorem. And UGC NET Dec 2025 pairs “all mutually beneficial trades are completed, so the competitive allocation is efficient” with “a trade that makes everyone better off from an inefficient allocation need not give an efficient allocation”. Both are true, and again the second does not explain the first: a Pareto improvement need not be the last one.
Worked questions
The opening question, solved
UGC NET 2023 · advanced
Which is not true of the second welfare theorem? (A) convex preferences, pure exchange: every efficient allocation is a possible equilibrium; (B) with production, convexity of production sets gives the same; (C) it holds under increasing returns; (D) it separates the allocative and distributive roles.
- Recall what the theorem needs.Convex preferences and convex production sets, so that a price line can separate the better set from the target.
- Increasing returns.Production sets with increasing returns are not convex. The condition fails, and so does the theorem.
Answer
C: it does not hold under increasing returns to scale.
Hyderabad 2011 · standard
According to the first fundamental theorem of welfare economics, a competitive general equilibrium is Pareto optimal when: (A) economic agents pursue their own self-interest; (B) some are altruistic; (C) all are altruistic; (D) the government sets a minimum support price and a price ceiling.
- Recall the assumption.Price-taking agents, each pursuing her own interest, nothing more: no altruism needed, no government intervention.
Answer
A.
JNU 2016 · foundation
Which addresses redistribution issues to promote the equity objectives of a planner: the first theorem, the second theorem, Walras’ law or the Pareto criterion?
- Which theorem moves endowments?The second: it says any efficient allocation, so any distribution you prefer, can be reached by redistributing endowments and letting markets work.
Answer
The second fundamental theorem (option B). Hyderabad 2019 asks the same.
UGC NET 2025 · advanced
A: When everyone trades in the competitive market place, all mutually beneficial trades are completed and the resulting equilibrium allocation is efficient. R: Even if a trade from an inefficient allocation makes everyone better off, the new allocation is not necessarily efficient.
- Is A true?Yes: the first theorem.
- Is R true?Yes: a Pareto improvement can leave room for another. Efficient means no improvement is left, which is a stronger property.
- Does R explain A?No: R is about a single trade starting from an inefficient point; A is about the end state of competition.
Answer
Both A and R are correct, but R is not the correct explanation of A (option B).
Revise
The lesson on one screen
The lesson on one screen
- First theorem
- Every competitive equilibrium is Pareto efficient. Needs price-taking, no externalities, property rights, nonsatiation. Not convexity.
- Why
- If x′ were a Pareto improvement it would cost more than total wealth, but it is feasible: contradiction.
- Second theorem
- With convex preferences and production sets, any efficient allocation is an equilibrium after a lump-sum redistribution of endowments.
- Separation
- Prices do allocation; endowments do distribution.
- What breaks them
- Externalities, market power, missing markets; for the second theorem, non-convexity (increasing returns, preferences that bend away).
- Not a claim of fairness
- A competitive equilibrium can be efficient and very unequal.
- Assertion–reason
- Walras' law and the first theorem are both true; neither explains the other.
Practice
Check yourself
Quick questions
1Which assumption does the second theorem need but the first does not?
2Which theorem tells a planner how to reach a preferred distribution without losing efficiency?
3A competitive equilibrium in a market with a pollution externality is
4The first theorem says a competitive equilibrium is
Numerical problem
The size of a lump-sum transfer
Two people each have u = √(xy). Person 1 is endowed with (9, 1) and person 2 with (1, 9). A planner wants the efficient allocation (6, 6) for person 1 and (4, 4) for person 2. At the supporting prices (1, 1), how many units of wealth must person 1 receive?
Real questions
Past papers, by level
Foundation
Definitions and who-said-what: Pareto efficiency, the contract curve, Walras' law and the welfare theorems. Most CUET, UGC NET and Hyderabad questions sit here.
1 questionPractise →
Standard
Statements to judge, matching lists, and equilibrium prices and bundles to compute with Cobb–Douglas and Leontief tastes, up to JAM, GATE and JNU level.
5 questionsPractise →
Delhi School level
This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 6: Transfers, production and fairness.
7 questionsLesson 6 →
All 6 past-paper questions for this lesson, easiest first: practise the full set →
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
Read next
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