Difficulty: Intermediate | Prerequisites: Basic utility maximisation, budget constraints, indifference curves, Lagrangian optimisation
General equilibrium theory asks what happens when every market in an economy clears at once, rather than looking at one market in isolation. The Edgeworth box is the workhorse diagram for a two-consumer, two-good exchange economy: it lets you visualise both consumers' choices, the endowment, and whether trade can make both better off, all in one picture. To follow this material you need to be comfortable solving constrained optimisation problems for Cobb-Douglas and perfect-complement (Leontief) utility functions, and you should know how to draw budget lines and indifference curves.
This topic sits at the bridge between individual consumer theory (earlier in Econ 302) and welfare economics. It is also the foundation for understanding Walrasian equilibrium and the First Welfare Theorem.
Two consumers start with fixed bundles of two goods and trade at given prices. Each sells their endowment and buys the bundle that maximises their utility subject to their budget constraint. If total demand for each good equals total supply, the market is in equilibrium. The Edgeworth box plots both consumers' choices in one diagram, making it easy to spot whether the proposed prices actually clear the market.
Edgeworth box
A rectangular diagram whose width equals the total endowment of good X and whose height equals the total endowment of good Y. Consumer A's origin is the bottom-left corner; Consumer B's origin is the top-right corner, with axes running in the opposite direction. Every point inside the box represents a feasible allocation of both goods between both consumers.
In simple terms, it is a box that shows every possible way to split a fixed pile of two goods between two people.
Initial endowment (W)
The bundle each consumer starts with before any trade takes place. In the Edgeworth box the endowment is a single point, because once you fix what A has, B gets the remainder.
Think of it as: the "starting position" before the market opens.
Budget constraint (in an exchange economy)
The line through the endowment point with slope equal to the negative price ratio, -Px/Py. Each consumer can afford any bundle on or below this line, valued at the same total as their endowment.
In simple terms, it is the set of bundles a consumer can buy after selling everything they own at market prices.
Perfect complements (Leontief) utility
A utility function of the form U(X, Y) = min(aX, bY). The consumer always wants X and Y in the fixed ratio b:a. Indifference curves are L-shaped, with the kink along the line aX = bY.
Think of it as: left shoes and right shoes. Extra left shoes are worthless without matching right shoes.
Cobb-Douglas utility
A utility function of the form U(X, Y) = X^α · Y^β. The consumer always spends fraction α/(α+β) of income on X and fraction β/(α+β) on Y. Indifference curves are smooth and convex.
In simple terms, the exponents tell you the spending shares.
Excess demand
The amount by which total quantity demanded of a good exceeds total quantity supplied (total endowment) at the current prices. Positive excess demand means buyers want more than is available.
Excess supply
The amount by which total quantity supplied exceeds total quantity demanded. Equivalently, excess supply = negative excess demand.
Walras' Law
If every consumer satisfies their budget constraint, the total value of excess demand across all markets sums to zero. A practical consequence: if one market clears, the other must clear too (in a two-good economy).
Think of it as: money spent somewhere has to come from somewhere.
Two consumers (A and B), two goods (X and Y).
Each consumer has an initial endowment of both goods.
Prices Px and Py are given (or to be solved for).
Each consumer's income = the market value of their endowment: I = Px · Xw + Py · Yw, where (Xw, Yw) is the endowment.
When U_A(X, Y) = min(2X, Y):
The consumer always chooses the kink of the L-shaped indifference curve, where 2X = Y.
Substitute Y = 2X into the budget constraint: Px · X + Py · (2X) = I.
Solve for X, then Y = 2X.
Worked example (from the homework): Endowment = (4, 4), Px = 2, Py = 1.
Income: I = 2(4) + 1(4) = 12.
Kink condition: Y = 2X.
Budget: 2X + 1(2X) = 12, so 4X = 12, so X = 3.
Y = 2(3) = 6.
A's optimal bundle: (3, 6).
When U_B(X, Y) = X^(1/3) · Y^(2/3):
The spending-share rule gives the answer directly.
Spend fraction α/(α + β) = (1/3)/1 = 1/3 of income on X.
Spend fraction β/(α + β) = (2/3)/1 = 2/3 of income on Y.
Worked example: Endowment = (4, 4), Px = 2, Py = 1.
Income: I = 2(4) + 1(4) = 12.
Spending on X: (1/3)(12) = 4, so X = 4/Px = 4/2 = 2.
Spending on Y: (2/3)(12) = 8, so Y = 8/Py = 8/1 = 8.
B's optimal bundle: (2, 8).
Total X in the economy = X_A endowment + X_B endowment = 4 + 4 = 8. This is the box width.
Total Y = 4 + 4 = 8. This is the box height.
A's origin: bottom-left. B's origin: top-right.
Plot W (the endowment) at A's coordinates (4, 4). From B's perspective this is also (4, 4).
The budget line passes through W with slope = -Px/Py = -2.
Plot A's optimal bundle at (3, 6) from A's origin.
Plot B's optimal bundle at (2, 8) from A's origin. (From B's origin this is (8-2, 8-8) = (6, 0).)
Total demand for X = X_A + X_B = 3 + 2 = 5. Total supply of X = 8.
Excess supply of X = 8 - 5 = 3.
Total demand for Y = Y_A + Y_B = 6 + 8 = 14. Total supply of Y = 8.
Excess demand for Y = 14 - 8 = 6.
The market does not clear at these prices. This is not an equilibrium.
Notice that Walras' Law holds: the value of excess demand across both markets sums to zero. Excess demand for X in value = -3 × 2 = -6. Excess demand for Y in value = 6 × 1 = 6. Sum = 0.
This framework is the simplest version of a general equilibrium model, the kind used to study international trade (each country has an endowment and preferences) or to design auction markets where multiple goods are allocated simultaneously.
\text{Income from endowment: } I = P_X \cdot X_w + P_Y \cdot Y_w\text{Cobb-Douglas demand: } X^* = \frac{\alpha}{\alpha + \beta} \cdot \frac{I}{P_X}, \quad Y^* = \frac{\beta}{\alpha + \beta} \cdot \frac{I}{P_Y}\text{Perfect complements } U = \min(aX, bY): \text{ set } aX = bY, \text{ substitute into budget}\text{Excess demand for good } j: \quad ED_j = \sum_i X_{ij}^* - \sum_i X_{ij}^w\text{Walras' Law: } P_X \cdot ED_X + P_Y \cdot ED_Y = 0Students often think the budget constraint in an exchange economy passes through the origin. It does not. It passes through the endowment point, because the consumer can always afford to keep what they already have.
Students sometimes confuse "total endowment" with "one consumer's endowment." In the Edgeworth box, the box dimensions are the sum of both consumers' endowments, not one person's.
A common error with perfect complements is using the wrong ratio. If U = min(2X, Y), the kink is where 2X = Y, meaning Y/X = 2, not X/Y = 2.
Students frequently assume that any set of prices produces an equilibrium. The homework shows that Px = 2, Py = 1 does not clear the market here. Finding equilibrium prices is a separate problem.
⚠️ You will almost certainly be asked to solve for optimal bundles with at least one Leontief and one Cobb-Douglas consumer. Know both methods cold.
⚠️ Edgeworth box labelling is a major source of lost marks. Make sure you label both origins, both axes, the endowment W, each consumer's optimal bundle, and the budget line.
⚠️ Expect a question asking whether a given set of prices clears the market. The answer is usually no, which sets up the next question: find the equilibrium price.
⚠️ Walras' Law is a useful check. If your excess demand calculations violate it, you have made an arithmetic error.
True or False: In a two-good exchange economy, if the market for X clears, the market for Y must also clear.
True (by Walras' Law).
Fill in the blank: For U = min(3X, Y), the consumer always chooses bundles where Y = ____X.
True or False: The budget line in an Edgeworth box always passes through the endowment point.
True.
Fill in the blank: A Cobb-Douglas consumer with U = X^(0.4) · Y^(0.6) spends ____% of income on Y.
60%.
True or False: Excess supply in the X market and excess demand in the Y market can occur at the same time.
True, and Walras' Law tells you their values (in dollar terms) are equal in magnitude.
Q: Consumer A has U = min(2X, Y), endowment (4, 4), and faces prices Px = 2, Py = 1. What is A's optimal bundle?
A: Income = 2(4) + 1(4) = 12. Kink condition: Y = 2X. Budget: 2X + 2X = 12, so X = 3, Y = 6. Optimal bundle is (3, 6).
Q: Consumer B has U = X^(1/3) · Y^(2/3), endowment (4, 4), Px = 2, Py = 1. What is B's optimal bundle?
A: Income = 12. Spend 1/3 of income on X: X = (1/3)(12)/2 = 2. Spend 2/3 on Y: Y = (2/3)(12)/1 = 8. Optimal bundle is (2, 8).
Q: Given the bundles above, is the market for X in equilibrium? If not, is there excess demand or excess supply, and how much?
A: Total demand for X = 3 + 2 = 5. Total supply = 8. Excess supply of X = 3.
Q: What are the dimensions of the Edgeworth box for the economy described above?
A: Width = total X = 4 + 4 = 8. Height = total Y = 4 + 4 = 8. The box is 8 by 8.
Q: Why does the budget line in the Edgeworth box pass through the endowment rather than the origin?
A: Because the consumer's income comes from selling their endowment at market prices, so the endowment is always affordable. The line of bundles with the same market value as the endowment passes through it.
This material connects directly to competitive equilibrium (covered in the second set of notes from this homework). Once you can check whether given prices clear the market, the next step is solving for the prices that do.
It also connects to the First Welfare Theorem: any competitive equilibrium allocation is Pareto efficient. The Edgeworth box is the diagram used to prove this visually, by showing that the equilibrium sits on the contract curve.
The Leontief and Cobb-Douglas utility functions reappear in producer theory (fixed-proportions and Cobb-Douglas production functions), so the optimisation techniques here transfer directly.
Edgeworth box, exchange economy, general equilibrium, Walrasian equilibrium, Leontief utility, perfect complements, Cobb-Douglas utility, excess demand, excess supply, Walras' Law, budget constraint endowment, two-consumer two-good economy, contract curve, Pareto efficiency, price ratio, optimal bundle, indifference curve, intermediate microeconomics, ECON 302, Penn State