Most of intermediate micro studies one market at a time. General equilibrium asks the harder question: can every market clear at once, and is the outcome any good? This page answers it in the cleanest setting there is — two people, two goods, a fixed pile to divide — through the Edgeworth box, which folds a whole exchange economy into one picture. It is the topic that most often sends Cambridge University undergraduates to an economics tutor.
1 · The pure-exchange economy and the box
Take the smallest economy that still teaches something: two consumers, Ana and Ben, and two goods, coffee (x) and muffins (y). Between them they hold a fixed 12 coffees and 12 muffins, with no production — all they can do is trade their starting bundles, their endowments.
The box shows this at once. Its width is the total coffee, its height the total muffins. Ana’s origin is the bottom-left corner, read the usual way: coffee rightward, muffins upward. Ben’s is the top-right corner, flipped — coffee leftward, muffins downward.
Slow down on that flip; it is the step students trip on. Its reward: every point in the box is a complete allocation, since Ana’s bundle read from the bottom-left and Ben’s from the top-right always sum to 12 and 12.
2 · Gains from trade: the lens
Give each consumer indifference curves — Ana’s bowing around her origin, Ben’s around his. A curve’s slope is the marginal rate of substitution (MRS): the muffins you would trade for one more coffee while staying equally happy. The single-consumer toolkit behind these curves has its own page; the new thing here is that two people share one box.
Take an allocation where the curves cross rather than touch. Between them lies a lens, and every point inside sits on a higher curve for both consumers — so a trade into the lens helps each, which is possible exactly when the two MRSs differ. An allocation is Pareto efficient when no such lens remains: no one can gain without the other losing, and the curves are tangent. (For the wider welfare theory — the utility possibility frontier, Arrow’s theorem — see the welfare economics page.)
3 · The contract curve and the core
Collect every tangency, MRSA = MRSB. That locus is the contract curve, running corner to corner. With Cobb–Douglas tastes it is exact: writing MRS for u = xay1−a as [a/(1−a)]·(y/x) and equating Ana’s to Ben’s gives
yA = b·H·xA / (a·W + (b − a)·xA).
When Ana leans to coffee and Ben to muffins the curve bows off the diagonal; identical tastes flatten it onto the diagonal.
The contract curve is pure efficiency. The core narrows it to the stretch both consumers prefer to their own endowment, bounded where the endowment indifference curves cross it. Nobody trades to somewhere worse than staying put, so voluntary, efficient trade lands in the core.
4 · Prices and the competitive equilibrium
Now let a market post a price. Let a coffee cost p, a muffin the unit of account. Each consumer buys the best bundle their endowment’s value can afford — their demand, which for Cobb–Douglas spends a fixed income share on each good (the Lagrange exercise on its own page).
A competitive (Walrasian) equilibrium is a price p* at which everyone optimises and every market clears. Its budget line runs through the endowment, and both consumers’ choices meet at one tangency point.
The First Welfare Theorem says every competitive equilibrium is Pareto efficient — given price-taking, no market power, no externalities and complete markets — because a common tangent forces MRSA = MRSB = p*, putting the equilibrium on the contract curve. The Second Welfare Theorem reverses it: any efficient allocation is reachable as an equilibrium after a lump-sum transfer of endowments, given convex preferences — a proof beyond this page.
Worked example — Ana and Ben trade coffee and muffins
Setup. Box 12 × 12. Ana’s tastes are uA = x0.6y0.4 (coffee-leaning), Ben’s uB = x0.4y0.6 (muffin-leaning). Ana starts at (2, 10), Ben at (10, 2).
Step 1 — Why they trade. At the endowment MRSA = (0.6/0.4)(10/2) = 7.5 and MRSB = (0.4/0.6)(2/10) = 0.13 — wildly apart, so it is inefficient. Ana would swap 7.5 muffins for a coffee; Ben barely values it. So Ana buys coffee, Ben buys muffins.
Step 2 — Demands. At price p, Ana’s income is 2p + 10 and she spends 0.6 on coffee: xA = 0.6(2p + 10)/p, yA = 0.4(2p + 10). Ben’s income is 10p + 2: xB = 0.4(10p + 2)/p, yB = 0.6(10p + 2).
Step 3 — Clearing. Set xA + xB = 12. It solves to p* = 1 — one coffee per muffin. Back-substituting: Ana (7.2, 4.8), Ben (4.8, 7.2), and muffins clear, 4.8 + 7.2 = 12.
Step 4 — First Welfare Theorem, checked. Now MRSA = 1.5(4.8/7.2) = 1 = MRSB = p*, so the equilibrium is on the contract curve — efficient. Ana’s utility rises from about 3.81 to 6.12 and Ben’s likewise, so both beat their endowment: the equilibrium is in the core.
Step 5 — A perturbation. Let Ben acquire a taste for coffee, his weight rising 0.4 → 0.6. Re-clearing lifts the price to p* = 1.5: coffee is dearer now that both compete for it.
Step 6 — New allocation. Incomes become 13 and 17; demands resolve to Ana (5.2, 5.2) and Ben (6.8, 6.8), and both markets clear.
Step 7 — Interpret. Supply never moved — still 12 and 12. One taste shifted, the price rose 1 → 1.5, and both goods reallocated: dearer coffee leaves Ana with less of it (7.2 → 5.2). The outcome stays efficient (MRSA = MRSB = 1.5) and both still gain — general equilibrium in miniature.
Given two utility functions and an endowment, could you find the single price that clears both markets — and prove the split lands on the contract curve? Solving general equilibrium from the tastes up, rather than reading it off a finished diagram, is exactly what Edgeworth-box questions reward. A one-on-one economics tutor builds the box from both origins with you until the contract curve and the equilibrium price are results you derive, not a picture you memorise. Book a trial session.
Practice
Q1. Same box and endowment — Ana (2, 10), Ben (10, 2) — but give both the strong taste u = x0.75y0.25. Find p* and each bundle.
Q2. At the base equilibrium Ana holds (7.2, 4.8) and Ben (4.8, 7.2). Compute each MRS; is the allocation Pareto efficient?
Q3. At the base endowment — Ana (2, 10), Ben (10, 2), tastes uA = x0.6y0.4 and uB = x0.4y0.6 — compute both MRS. Efficient? Who buys coffee?
Answers. Q1: p* = 9/3 = 3; Ana’s income 16 → (4, 4), Ben’s 32 → (8, 8), and markets clear. Q2: MRSA = 1.5(4.8/7.2) = 1 = MRSB, equal, so Pareto efficient (on the contract curve). Q3: MRSA = 7.5, MRSB = 0.13, unequal, so not efficient; Ana values coffee far more, so Ana buys coffee, Ben the reverse.
Key takeaways
- The box is a whole economy in one picture. Ana reads from the bottom-left origin, Ben from the flipped top-right; every point splits the fixed totals.
- Crossing curves mean gains from trade. The lens between them is the set of mutual improvements; it closes at a tangency, MRSA = MRSB.
- Contract curve is efficiency; core is agreement. The core is the part both consumers prefer to their endowment.
- A competitive equilibrium clears every market at one price — its budget line runs through the endowment, and it is Pareto efficient, sitting on the contract curve (the First Welfare Theorem). Efficiency, though, is not fairness.
Why Cambridge University students choose our economics tutoring
- The flipped origin, made intuitive: sessions rebuild the box from both corners until reading Ben’s bundle from the top-right is automatic — the exact step most students stumble on.
- Derivations, not diagrams memorised: you solve the contract curve and the equilibrium price straight from the utility functions, ready to reproduce under exam pressure.
- One-on-one and matched to your course: a tutor works from your own past papers and notation, whether your module follows Varian, Jehle and Reny, or Mas-Colell.
FAQ
Q: What is the Edgeworth box actually showing?
A: A two-person, two-good economy with a fixed total to divide. Every point is a full allocation — one bundle from the bottom-left, the other from the top-right — so the diagram holds every possible split at once.
Q: Why is one origin upside down?
A: So one point can describe both people. The totals are fixed, so once you know Ana’s bundle, Ben has the rest; measuring him from the opposite corner makes “the rest” appear automatically.
Q: What is the difference between the contract curve and the core?
A: The contract curve is every efficient allocation — all the tangency points. The core is the smaller stretch of it that both consumers prefer to their starting endowment.
Q: Is a competitive equilibrium fair?
A: Not necessarily. The First Welfare Theorem promises efficiency, not equality; a lopsided endowment gives a lopsided but still efficient equilibrium. Fairness comes from redistributing endowments first.
Q: Does the First Welfare Theorem always hold?
A: Only under its assumptions — price-taking, no market power, no externalities, complete markets. Break one, say a monopolist or a pollution spillover, and the competitive outcome can be inefficient.
Book an economics tutor for Cambridge University or online
General equilibrium rewards students who can build the box, derive the contract curve, and solve the equilibrium price from first principles — not just recognise the diagram. One-on-one sessions build that fluency on your own past papers and notation. Tell us your module and exam date, and we will match you with the right tutor this week.
