Difficulty: Intermediate
Prerequisites: Consumer theory basics (utility functions, indifference curves, budget constraints), basic calculus (partial derivatives). Chapters on consumer optimisation should be reviewed first.
General equilibrium analysis asks what happens when all markets in an economy clear at once, rather than studying one market in isolation. The Edgeworth box is the workhorse diagram for a two-person, two-good exchange economy: it lets you visualise every possible allocation of the total endowment between two consumers, overlay their indifference curves, and identify which allocations are efficient.
This material builds directly on individual consumer optimisation (maximising utility subject to a budget constraint) and extends it to settings where two consumers trade with each other. If you are comfortable finding an optimal bundle for one consumer given prices and income, you already have most of the tools you need here. The new question is: at what prices will both consumers' choices be mutually compatible, so that every unit of every good is accounted for?
An Edgeworth box puts two consumers' choice problems into one diagram, with the total endowment of two goods as the box dimensions. Each consumer maximises utility given their budget; an allocation is Pareto efficient when the consumers' marginal rates of substitution are equal (or when one consumer has a corner solution where no further mutually beneficial trade exists). The set of all Pareto-efficient allocations inside the box is the contract curve.
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 at the bottom-left; Consumer B's origin is at the top-right. Every point inside the box represents a feasible allocation of both goods between the two 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 starting allocation, the bundle each consumer holds before any trade takes place. In the Edgeworth box it is a single labelled point, because specifying what A holds automatically determines what B holds (the remainder).
Think of it as each person's "starting hand" before trading begins.
Budget constraint (in an exchange economy)
The set of bundles a consumer can afford when they sell their entire endowment at market prices and buy back goods. For consumer i with endowment (Wᵢˣ, Wᵢʸ), income is M = Pₓ·Wᵢˣ + Pʸ·Wᵢʸ. The constraint is Pₓ·X + Pʸ·Y = M.
In simple terms, you cash in everything you own at today's prices, then go shopping.
Optimal bundle
The utility-maximising bundle on the budget constraint. Found by setting MRS equal to the price ratio (for interior solutions) or by applying the specific optimality condition for the utility function type.
Indifference curve
A curve connecting all bundles that give a consumer the same level of utility. In the Edgeworth box, A's curves bow toward A's origin (bottom-left) and B's curves bow toward B's origin (top-right).
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for the other while staying on the same indifference curve. Formally, MRS = -(MUₓ / MUʸ). For a Cobb-Douglas utility U = Xᵃ·Yᵇ, MRS = -(a/b)·(Y/X).
Think of it as "how much Y would I give up for one more unit of X without feeling worse off?"
Pareto efficiency (Pareto optimality)
An allocation is Pareto efficient if there is no way to make one consumer better off without making the other worse off. At an interior Pareto-efficient allocation, MRSₐ = MRSᵦ. Graphically, the two consumers' indifference curves are tangent.
Contract curve
The locus of all Pareto-efficient allocations inside the Edgeworth box. It connects the two origins and traces every point where indifference curves are tangent.
Perfect complements utility
U(X, Y) = min(X, Y) or more generally min(aX, bY). The consumer always wants goods in a fixed ratio. Indifference curves are L-shaped. The optimal bundle sits at the kink of the L, where aX = bY.
Think of it as left shoes and right shoes: having extra of just one does nothing for you.
Cobb-Douglas utility
U(X, Y) = Xᵃ·Yᵇ. Indifference curves are smooth and convex. The consumer spends a fixed fraction of income on each good. Demand for X: X* = (a/(a+b))·(M/Pₓ). Demand for Y: Y* = (b/(a+b))·(M/Pʸ).
Excess demand and excess supply
Excess demand for a good = total quantity demanded minus total quantity available (the total endowment). If positive, there is excess demand; if negative, there is excess supply. In equilibrium, excess demand is zero for every good.
Add the two consumers' endowments of each good to get total X and total Y. These totals set the box dimensions.
Consumer A's origin is at the bottom-left corner. Consumer B's origin is at the top-right corner, with axes running leftward and downward.
Plot the initial endowment point W. From A's perspective, read W's coordinates normally. From B's perspective, measure from the top-right corner.
Example: if A has (1, 7) and B has (7, 1), total X = 8 and total Y = 8. W sits at (1, 7) from A's origin, which is (7, 1) from B's origin.
Each consumer's income equals the value of their endowment at the prevailing prices: M = Pₓ·Wˣ + Pʸ·Wʸ.
Both consumers face the same prices. The budget line passes through the endowment point W, because at those prices the consumer can always afford exactly what they started with.
Example: A has (1, 7), prices Pₓ = Pʸ = 1. Income Mₐ = 1·1 + 1·7 = 8.
Example: B has (7, 1), same prices. Income M_b = 1·7 + 1·1 = 8.
Cobb-Douglas U = Xᵃ·Yᵇ
The consumer spends fraction a/(a+b) of income on X and fraction b/(a+b) on Y.
X* = [a/(a+b)]·(M/Pₓ) and Y* = [b/(a+b)]·(M/Pʸ).
Example: B has U = X^(1/4)·Y^(3/4), M = 8, Pₓ = Pʸ = 1. Then X* = (1/4)·(8/1) = 2, Y* = (3/4)·(8/1) = 6.
Perfect complements U = min(X, Y)
The consumer always chooses X = Y (for the basic min(X,Y) form). Substitute into the budget constraint: Pₓ·X + Pʸ·X = M, so X = Y = M/(Pₓ + Pʸ).
Example: A has U = min(X,Y), M = 8, Pₓ = Pʸ = 1. Then X* = Y* = 8/2 = 4.
Linear utility U = aX + bY (perfect substitutes)
MRS is constant at -a/b. The consumer compares this to the price ratio -Pₓ/Pʸ.
If a/b > Pₓ/Pʸ, the consumer spends all income on X. If a/b < Pₓ/Pʸ, all income goes to Y. If equal, any bundle on the budget line is optimal.
Example: U = 4X + Y, Pₓ/Pʸ = some ratio. MRS = -4. If Pₓ/Pʸ < 4, consumer buys only X.
MRS = -(MUₓ / MUʸ). Take the partial derivative of U with respect to X and with respect to Y, then form the ratio.
For U = X^(1/2)·Y^(1/2): MUₓ = (1/2)·X^(-1/2)·Y^(1/2), MUʸ = (1/2)·X^(1/2)·Y^(-1/2). MRS = -Y/X.
At (4, 4): MRS = -4/4 = -1.
For U = 4X + Y: MUₓ = 4, MUʸ = 1. MRS = -4 everywhere (constant).
At (4, 12): MRS = -4.
Compute MRSₐ and MRSᵦ at the given allocation.
If MRSₐ = MRSᵦ, the allocation is Pareto efficient (for interior points with smooth preferences).
If MRSₐ ≠ MRSᵦ, the allocation is not Pareto efficient, because there exist mutually beneficial trades.
Example: A has MRS = -1, B has MRS = -3 at the initial endowment. These differ, so the allocation is not Pareto efficient.
Special case: when one consumer has linear utility (constant MRS), check whether the other consumer's MRS equals that constant at the allocation in question.
For each good, sum total demand (what both consumers want at their optimum) and compare to total supply (the total endowment).
Excess demand = total demand - total endowment. If positive, there is excess demand for that good. If negative, that negative value is excess supply.
By Walras' Law, if one market clears (or has excess supply), the other market's excess demand is determined. If X has excess supply of 2, Y has excess demand of 2 (when prices are both 1).
Example: total endowment of X = 8, A demands 4 and B demands 2, so total demand = 6. Excess supply of X = 8 - 6 = 2.
\text{Income (exchange economy): } M_i = P_X \cdot W_i^X + P_Y \cdot W_i^Y\text{Cobb-Douglas demand: } X^* = \frac{a}{a+b} \cdot \frac{M}{P_X}, \quad Y^* = \frac{b}{a+b} \cdot \frac{M}{P_Y}\text{Perfect complements } U = \min(X,Y): \quad X^* = Y^* = \frac{M}{P_X + P_Y}\text{MRS (Cobb-Douglas } U = X^a Y^b\text{): } MRS = -\frac{a}{b} \cdot \frac{Y}{X}\text{Pareto efficiency condition (interior): } MRS_A = MRS_B\text{Excess demand for good } j: \quad ED_j = \sum_i D_i^j - \sum_i W_i^jEdgeworth box diagram checklist:
Box dimensions = total endowments (total X on horizontal axis, total Y on vertical).
A's origin at bottom-left, B's origin at top-right.
Endowment point W plotted and labelled.
Budget line passes through W (slope = -Pₓ/Pʸ).
Optimal bundles for each consumer plotted and labelled (A and B).
Indifference curves through W for each consumer, bowing toward their respective origins.
The Edgeworth box framework is the foundation of international trade theory. Two countries with different endowments of goods (or factors of production) can both gain from trade, and the set of mutually beneficial trades maps directly onto the region between the two indifference curves through the endowment point. Pareto efficiency is the benchmark policymakers use when evaluating whether a market outcome can be improved: if an allocation is not Pareto efficient, there exists some reallocation (or policy change) that could make someone better off without hurting anyone else.
Students often think the Edgeworth box has separate budget lines for each consumer. It does not. Both consumers face the same budget line (it passes through W with slope -Pₓ/Pʸ), but they optimise from opposite corners.
Students sometimes forget that B's axes run in opposite directions. When reading B's allocation, you measure from the top-right corner, not the bottom-left. Mixing this up leads to plotting B's optimal bundle at the wrong point.
A common error is to assume that equal utility levels mean Pareto efficiency. Pareto efficiency has nothing to do with whether the two consumers are equally happy. It is about whether the MRS values match (for interior solutions), meaning no further mutually beneficial trades exist.
Students often confuse excess demand and excess supply. If total demand for X is less than total endowment, that is excess supply of X, not excess demand. The sign matters.
⚠️ Drawing the Edgeworth box is a standard exam question. You will be expected to plot the endowment point, the budget constraint, both consumers' optimal bundles, and indifference curves through W. Label everything.
⚠️ Computing the MRS for different utility function types (Cobb-Douglas, perfect complements, perfect substitutes) and evaluating it at a specific allocation is heavily tested.
⚠️ The Pareto efficiency test (MRS_A = MRS_B?) appears frequently. Know how to state the condition and apply it to a given allocation.
⚠️ Excess demand / excess supply calculations: be prepared to compute total demand at given prices, compare to total endowment, and state whether each market clears.
True or false: In an Edgeworth box, Consumer B's origin is at the bottom-left corner. (False, B's origin is at the top-right.)
Fill in the blank: For Cobb-Douglas utility U = X^(1/3)·Y^(2/3) with income M and prices Pₓ and Pʸ, the demand for X is X* = ____. (Answer: (1/3)·(M/Pₓ))
True or false: An allocation where MRS_A = -2 and MRS_B = -2 is Pareto efficient (assuming interior solutions). (True.)
True or false: If total demand for good X at current prices exceeds total endowment, there is excess supply of X. (False, that is excess demand.)
Fill in the blank: A consumer with perfect complements utility U = min(X, Y) always chooses bundles where X = ____. (Answer: Y)
Q: Consumer A has U = min(X, Y), endowment (1, 7), and faces prices Pₓ = Pʸ = 1. What is A's optimal bundle?
A: Income = 1 + 7 = 8. With min(X,Y), the consumer sets X = Y. Budget constraint: X + Y = 8, so 2X = 8, giving X* = Y* = 4.
Q: Consumer B has U = X^(1/4)·Y^(3/4), endowment (7, 1), prices Pₓ = Pʸ = 1. What is B's optimal bundle?
A: Income = 7 + 1 = 8. Cobb-Douglas demand: X* = (1/4)·(8/1) = 2, Y* = (3/4)·(8/1) = 6.
Q: Given the allocations above, is there excess demand or excess supply in the market for X?
A: Total endowment of X = 8. Total demand for X = 4 + 2 = 6. Since demand (6) < endowment (8), there is excess supply of 2 units.
Q: Consumer A has U = X^(1/2)·Y^(1/2) and holds (4, 4). Consumer B has U = 4X + Y and holds (4, 12). Compute MRS for each at their initial allocation.
A: For A, MRS = -Y/X = -4/4 = -1. For B, MRS = -MUₓ/MUʸ = -4/1 = -4. Since -1 ≠ -4, the allocation is not Pareto efficient.
Q: In the previous question, at a Pareto-efficient allocation where both consumers hold positive amounts, what must be true about A's bundle?
A: At a Pareto-efficient interior allocation, MRS_A = MRS_B. Since B's MRS is always -4, A's MRS must also equal -4: -Yₐ/Xₐ = -4, so Yₐ = 4Xₐ.
This material connects directly to competitive equilibrium (covered in the companion study notes), where you solve for the prices that make all markets clear simultaneously. The First Welfare Theorem states that any competitive equilibrium allocation is Pareto efficient, tying together the concepts of market-clearing and efficiency from these notes.
The MRS calculations here also connect back to individual consumer theory: the tangency condition MRS = -Pₓ/Pʸ for a single consumer's optimum is the building block for the Pareto efficiency condition MRS_A = MRS_B in the two-consumer setting.
Edgeworth box, exchange economy, two-person two-good economy, initial endowment, endowment point, budget constraint in general equilibrium, optimal bundle, utility maximisation, Cobb-Douglas utility, perfect complements, Leontief preferences, min utility function, marginal rate of substitution, MRS, Pareto efficiency, Pareto optimality, contract curve, indifference curve tangency, excess demand, excess supply, Walras' Law, ECON 302, intermediate microeconomics, general equilibrium