Microeconomics · A module in six lessons
General equilibrium and welfare economics, from the Edgeworth box to Arrow
Can a market of selfish people produce a good outcome, and what does “good” mean? This module answers with one picture, the Edgeworth box, and builds up to the two welfare theorems, social welfare functions and Arrow’s impossibility theorem. Each lesson has a lab to move, and 128 past-paper questions sorted by level finish it.
128past-paper questions, each placed in a lesson
74of them the Delhi School’s, gathered in Lesson 6
6lessons, all live, each with a lab
8interactive labs, from a draggable box to an equilibrium solver
Start at the beginning, or check where you are.Six lessons, about 5 hours in all. You need indifference curves and the MRS; the consumer-choice page covers them. Each lesson opens with a real exam question, builds the idea with a diagram you can move, and ends with that question solved and a set of past papers.
The path
Three stages, six lessons
0 of 6 doneYour progress is saved on this device.
- Edgeworth 1881, Walras 1874Exchange
Which splits of the goods are efficient, and which one do prices pick?
- 1The Edgeworth box and the contract curveTwo people, two goods, one box. Pareto improvements, the lens of mutually beneficial trades, the contract curve for Cobb–Douglas, Leontief and linear tastes, and why efficiency says nothing about fairness.45 min · 17 past questions
- 2Competitive equilibrium and Walras' lawPrices as a line through the endowment, excess demand, why one market can be left out (Walras' law), the numeraire, free goods, and how to compute an equilibrium step by step.45 min · 15 past questions
- Pareto 1906, Arrow and Debreu 1951Efficiency and the welfare theorems
When does competition give an efficient economy, and what does fairness have to do with it?
- 3Efficiency with production: the three conditionsEfficiency in exchange, in production and in the product mix. Robinson Crusoe, the production possibilities curve, and why an indirect tax breaks the condition MRS = MRT.35 min · 7 past questions
- 4The two welfare theoremsEvery competitive equilibrium is efficient; every efficient allocation can be reached by competition after a transfer. What each needs, what breaks them, and why the second separates efficiency from distribution.40 min · 6 past questions
- Bergson 1938 to Arrow 1951Choosing and the Delhi School
How do we rank efficient outcomes, and what does the DSE entrance ask in the same topics?
- 5Social welfare, compensation and Arrow's theoremThe utility possibility frontier, Bergson–Samuelson welfare functions (utilitarian, Rawls, Nash), the Kaldor, Hicks and Scitovsky tests, no-envy, and why majority voting cycles.45 min · 9 past questions
- 6The Delhi School economiesThe DSE entrance asks the same topics with unusual tastes: Leontief, lexicographic, max and perfect substitutes. A solver for any pair, and the four patterns behind 74 questions.60 min · 74 past questions
Where to start
A two-minute check
One question from each lesson. Your first wrong answer tells you where to begin; get them all and you can go straight to the cheat sheet.
Question 1 of 6
At a point in the Edgeworth box, person 1's MRS is 3 and person 2's is 1. Then
Revise
Cheat sheet: every result on one screen
What each idea says, who gave it and when, how to use it, and the trap examiners set on it.
Pareto efficiencyPareto, 1906
It saysNo change helps someone without hurting someone else
Use it forTest an allocation; find the contract curve
The trapIt says nothing about fairness: giving everything to one person is efficient
Contract curveEdgeworth, 1881
It saysAll efficient allocations: where MRS₁ = MRS₂ for smooth tastes
Use it forEdgeworth box problems
The trapWith kinks or corners there may be no tangency; reason from the shapes
The coreEdgeworth; named by Shubik, 1959
It saysEfficient allocations inside the lens through the endowment
Use it forWhere bargaining can end
The trapA stretch of the curve, not a point
Competitive equilibriumWalras, 1874
It saysEach person picks her best affordable bundle and all markets clear
Use it forCompute prices and bundles
The trapOnly relative prices matter: fix a numeraire
Walras' lawWalras
It saysThe value of aggregate excess demand is zero at all prices
Use it forSolve one market, check the other
The trapN − 1 markets clear ⇒ the Nth clears; not 'each excess demand is zero'
Free goods
It saysA good in excess supply even at a zero price is free
Use it forLeontief economies with more of one good
The trapOnly complements can sit at a zero price; others want unlimited amounts
Three efficiency conditions
It saysMRS equal across consumers; MRTS equal across producers; MRS = MRT
Use it forTaxes, market structures, Crusoe
The trapMRS is tastes, MRT is technology; an indirect tax breaks MRS = MRT
First welfare theoremArrow and Debreu, 1951
It saysEvery competitive equilibrium is Pareto efficient
Use it forAssertion–reason questions
The trapNeeds no externalities and nonsatiation; does not need convexity
Second welfare theoremArrow and Debreu, 1951
It saysWith convexity, every efficient allocation is an equilibrium after a lump-sum transfer
Use it forRedistribution questions
The trapFails with increasing returns or non-convex tastes
Social welfare functionBergson, 1938; Samuelson, 1947
It saysW = f(U₁, …, U_m): ranks allocations
Use it forChoose a point on the utility frontier
The trapUtilitarian Σ U, Rawls min U (not max), Nash Π U
Kaldor and HicksKaldor, Hicks, 1939
It saysCould the gainers compensate the losers (Kaldor); could the losers not bribe the gainers (Hicks)?
Use it forPotential Pareto improvements
The trapCompensation is hypothetical, not paid
Scitovsky double criterionScitovsky, 1941
It saysPass the Kaldor test forward and fail it in reverse
Use it forRule out the paradox when frontiers cross
The trapRequires the reversal test
Arrow's impossibility theoremArrow, 1950–51
It saysNo rule has unrestricted domain, Pareto, independence and no dictator (3+ alternatives)
Use it forVoting and welfare questions
The trapIt does not say Pareto efficiency is impossible
Order them
Who said what, and when
UGC NET and CUET put these in order or match them to names. The dates are from the original publications; the Sources chapter lists where we checked them.
| Year | Contribution |
|---|---|
| 1874 | Walras, Éléments d'économie politique pure: general equilibrium and the auctioneer |
| 1881 | Edgeworth, Mathematical Psychics: the contract curve and the range of final settlements |
| 1906 | Pareto, Manuale di economia politica: Pareto optimality (French edition 1909) |
| 1938 | Bergson: the social welfare function |
| 1939 | Kaldor and Hicks: the compensation criterion |
| 1941 | Scitovsky: the double criterion and the paradox |
| 1950–51 | Arrow: the impossibility theorem; Arrow and Debreu: the two welfare theorems (1951) |
| 1954 | Arrow and Debreu: existence of competitive equilibrium |
| 1971 | Rawls: A Theory of Justice and the maximin principle |
Real questions
Practise across the module
Delhi School75
JNU13
Hyderabad12
CUET-PG12
IIT JAM7
UGC NET6
GATE3
Questions placed in this module, by exam.
The Delhi School asks more here than all the other exams together, in a distinctive style: strange tastes and an economy to solve. JAM and GATE set the numerical equilibrium problems. CUET, UGC NET and Hyderabad ask definitions, Walras’ law, the welfare theorems and who-said-what.
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.
8 questionsPractise →
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.
46 questionsPractise →
Delhi School level
The Delhi School of Economics' exchange-economy questions, collected in Lesson 6.
74 questionsLesson 6 →
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.
Quick answers
Frequently asked questions
What is an Edgeworth box?
A rectangle that shows every way two people can split fixed amounts of two goods. Its width is the total of the first good and its height the total of the second. Person 1 measures her bundle from the bottom-left corner, person 2 from the top-right, so every point is one allocation of everything. It is used to find Pareto efficient allocations (the contract curve) and, with prices, competitive equilibrium.
What is the contract curve?
The set of all Pareto efficient allocations in the Edgeworth box: the points where the two people's indifference curves are tangent, so their marginal rates of substitution are equal. From any point off the curve there is a trade that helps both. With kinked or non-convex tastes the efficient set can be a band or lie on the edges of the box.
What is Walras' law?
The value of the aggregate excess demand across all markets is zero at every price vector, because each person spends exactly her wealth. So if N − 1 markets are in equilibrium the Nth is too, and only N − 1 relative prices need to be solved for.
What do the two fundamental theorems of welfare economics say?
The first: every competitive equilibrium is Pareto efficient (it needs price-taking, no externalities and nonsatiation, but not convexity). The second: with convex preferences and production sets, every Pareto efficient allocation is a competitive equilibrium after a suitable lump-sum redistribution of endowments, so efficiency and distribution can be separated.
What are the Kaldor, Hicks and Scitovsky criteria?
Compensation tests for a change that makes some people better off and others worse off. Kaldor (1939): the gainers could compensate the losers and still be better off. Hicks (1939): the losers could not profitably bribe the gainers to forgo the change. Scitovsky (1941): the move must pass the Kaldor test and the move back must fail it, which rules out the paradox where each position appears better than the other.
What does Arrow's impossibility theorem say?
With at least three alternatives, no way of combining individual rankings into a social ranking satisfies unrestricted domain, the Pareto principle, independence of irrelevant alternatives and non-dictatorship at once. Majority rule, for example, can cycle (the Condorcet paradox). Arrow published it in 1950 and in his 1951 book Social Choice and Individual Values.
Checked
Sources and what we checked
The dates and attributions above were checked against the pages below on 24 October 2026. Where historians disagree we say so: the modern Edgeworth box diagram is credited to Pareto’s Manuale (1906) in most textbooks, though Edgeworth’s 1881 analysis came first and the attribution is debated. Every answer key in the practice sets was worked by hand against the key. Questions whose keys we think are wrong, or whose stored text cannot be solved, are left out; the reasons are given on the lesson pages and in the module’s source file.
- Edgeworth box (Wikipedia)
- Thinking outside the box: a new history of Edgeworth's and Pareto's boxes (McLure, University of Western Australia)
- Edgeworth's limit theorem (Wikipedia)
- Léon Walras (Wikipedia)
- Pareto efficiency (Wikipedia)
- Fundamental theorems of welfare economics (Wikipedia)
- Existence of an equilibrium for a competitive economy (Arrow and Debreu, Econometrica, 1954)
- Social welfare function (Wikipedia)
- Kaldor–Hicks efficiency (Wikipedia)
- Scitovsky paradox (Wikipedia)
- Arrow's impossibility theorem (Wikipedia)
- Kenneth Arrow (Wikipedia)
- A Theory of Justice (Wikipedia)
Read next
All resources →MicroeconomicsThe Edgeworth box and the contract curve, explainedHow to read an Edgeworth box, Pareto improvements and the lens of mutually beneficial trades, why efficiency means equal MRS, the contract curve for Cobb–Douglas, Leontief and perfect substitutes, and why efficient is not fair.MicroeconomicsConsumer choice: budget line and indifference curvesMove a price or income and watch the budget line and indifference curve re-tangent, then split the change into its two effects.International economicsThe Heckscher–Ohlin theory of trade, explained with diagramsFactor abundance and factor intensity, the 2×2×2 assumptions, the Edgeworth box and the production frontier, and the theorem: export what uses your abundant factor.