Difficulty: Intermediate | Prerequisites: Consumer theory (indifference curves, utility maximisation, budget constraints), familiarity with perfect substitutes, perfect complements and Cobb-Douglas utility functions.
This material sits in the general equilibrium and welfare economics portion of intermediate micro. It builds on everything you learnt about individual consumer choice and asks: what happens when two consumers trade with each other, and when is the result efficient? If you skipped the unit on utility function types (linear, Leontief, Cobb-Douglas), go back to that first.
An Edgeworth Box is a diagram that shows every possible way to divide fixed quantities of two goods between two consumers. The marginal rate of substitution (MRS) tells you each consumer's willingness to trade one good for the other. An allocation is Pareto efficient when no reallocation can make one consumer better off without hurting the other, and the set of Pareto-efficient allocations that also leave both consumers at least as well off as their endowment is the Core.
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. 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 two goods between two people.
Utility function
A mathematical representation of a consumer's preferences, assigning a number to each bundle so that preferred bundles get higher numbers.
Think of it as a scoring system for how much a consumer likes a particular combination of goods.
Perfect substitutes (linear utility)
A utility function of the form U(X, Y) = aX + bY. Indifference curves are straight lines with constant slope. The MRS is constant at a/b everywhere.
In simple terms, the consumer is always willing to swap X for Y at the same fixed rate, regardless of how much of each they hold.
Perfect complements (Leontief / min utility)
A utility function of the form U(X, Y) = min(aX, bY). Indifference curves are L-shaped with a kink along the line where aX = bY. The consumer gets no benefit from having more of one good unless the other good increases proportionally.
Think of it as left shoes and right shoes: an extra left shoe is worthless without a matching right shoe.
Cobb-Douglas utility
A utility function of the form U(X, Y) = X^α · Y^β. Indifference curves are smooth, convex, and never touch either axis. The MRS varies with the bundle.
In simple terms, the consumer always wants some of both goods and their willingness to trade changes depending on how much of each they currently have.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to give up good Y to obtain one more unit of good X while staying on the same indifference curve. Formally, MRS = MU_X / MU_Y (the ratio of marginal utilities).
Think of it as the consumer's personal exchange rate between the two goods at a given bundle.
Pareto efficiency (Pareto optimality)
An allocation is Pareto efficient if there is no other feasible allocation that makes at least one consumer strictly better off without making anyone worse off. At a Pareto-efficient interior allocation with smooth preferences, MRS_A = MRS_B.
In simple terms, you cannot reshuffle the goods to help someone without hurting someone else.
Contract curve
The set of all Pareto-efficient allocations inside the Edgeworth Box. At every point on the contract curve (for smooth, convex preferences), the two consumers' indifference curves are tangent to each other.
Think of it as the "best possible deals" line: any trade that lands on this curve cannot be improved for both parties.
The Core
The subset of Pareto-efficient allocations where both consumers are at least as well off as at their initial endowment. An allocation is in the Core if it is Pareto efficient AND neither consumer would prefer to walk away and keep their endowment instead.
In simple terms, it is the set of deals both consumers would voluntarily agree to.
Endowment (initial allocation)
The bundle of goods each consumer starts with before any trade occurs. Labelled "W" in Edgeworth Box diagrams.
Dimensions of the box. The width equals the total supply of good X across both consumers. The height equals the total supply of good Y. For example, if A has X = 2, Y = 8 and B has X = 6, Y = 8, the box is 8 units wide and 16 units tall.
Origins. Consumer A's origin is the bottom-left corner (0, 0). Consumer B's origin is the top-right corner. B reads the box "upside down and backwards": moving right from B's origin means B gets less X, and moving down means B gets less Y.
Plotting an allocation. Any point in the box simultaneously defines A's bundle (read from A's origin) and B's bundle (read from B's origin). The initial endowment is labelled W.
Indifference curves. A's indifference curves are drawn in the usual orientation (from A's origin). B's indifference curves are drawn from B's origin, so they appear rotated 180 degrees relative to A's. Where an A-curve and a B-curve cross, there is room for mutually beneficial trade. Where they are tangent, the allocation is Pareto efficient.
Linear (perfect substitutes): U = aX + bY. MU_X = a, MU_Y = b, so MRS = a/b everywhere. The indifference curves are straight lines with slope -a/b. Example: U_A = 2X + Y gives MRS_A = 2/1 = 2 at every bundle. Example: U_A = X + 2Y gives MRS_A = 1/2 at every bundle. Example: U_B = 4X + Y gives MRS_B = 4/1 = 4 at every bundle.
Leontief (perfect complements): U = min(aX, bY). The MRS is not defined at the kink (where aX = bY). Off the kink, one of the goods is in excess and has zero marginal utility. Along the vertical portion of the L (where aX < bY, so Y is in excess), MRS = 0 because extra Y adds nothing. Along the horizontal portion (where aX > bY, so X is in excess), MRS is undefined (infinite) because extra X adds nothing but extra Y is infinitely valuable relative to X. Example: U_B = min(2X, Y). The kink line is 2X = Y. At (6, 8), we have 2(6) = 12 > 8 = Y, so X is in excess. The consumer is on the horizontal portion, meaning MRS is undefined (or can be said to approach infinity).
Cobb-Douglas: U = X^α · Y^β. MU_X = αX^(α-1) · Y^β, MU_Y = βX^α · Y^(β-1), so MRS = (αY) / (βX). This varies with the bundle. Example: U_B = X^(1/4) · Y^(3/4) gives MRS_B = (1/4 · Y) / (3/4 · X) = Y / (3X).
The tangency condition. For two consumers with smooth, strictly convex preferences, Pareto efficiency requires MRS_A = MRS_B. When this holds, the indifference curves are tangent and there is no lens-shaped region between them where a mutually beneficial trade exists.
When both consumers have linear utility. If MRS_A ≠ MRS_B (the indifference curves have different slopes), the Pareto-efficient allocations are corner solutions. All goods should go to the consumer who values the relatively scarce good more. For example, if U_A = X + 2Y (MRS_A = 1/2) and U_B = 4X + Y (MRS_B = 4), A values Y more and B values X more, so efficiency requires giving all X to B and all Y to A, or a mix along the edges of the box.
Specifically, with two linear consumers whose MRS differs, the contract curve runs along the edges of the Edgeworth Box (the bottom edge and the left edge, or the top edge and the right edge, depending on who values what more).
When one consumer is linear and the other is Cobb-Douglas. The linear consumer has a constant MRS. Pareto efficiency requires setting the Cobb-Douglas consumer's MRS equal to that constant. For example, if U_A = 2X + Y (MRS_A = 2) and U_B = X^(1/4) · Y^(3/4) (MRS_B = Y_B / (3X_B)), setting MRS_B = 2 gives Y_B = 6X_B. Since X_B = 12 - X_A and Y_B = 12 - Y_A, the contract curve can be written as Y_A = 12 - 6(12 - X_A) = 6X_A - 60, but only where this falls within the box (both consumers' bundles are non-negative). In this example the contract curve is Y_B = 6X_B traced through the Edgeworth Box.
Checking Pareto efficiency at a given allocation. Compare MRS_A and MRS_B. If they differ (and at least one consumer has strictly convex preferences in the relevant region), the allocation is not Pareto efficient because the indifference curves cross and there is a lens of mutually improving trades.
The Core adds a participation constraint to Pareto efficiency. An allocation is in the Core if and only if two conditions hold:
The allocation is Pareto efficient (no mutually improving reallocation exists).
Both consumers are at least as well off as at their initial endowment (neither consumer would refuse and walk away with what they started with).
Testing whether an allocation is in the Core. Compute each consumer's utility at the proposed allocation and at the endowment. If either consumer's utility falls below their endowment utility, the allocation is not in the Core (it fails the participation constraint). If both are at least as well off, then also check Pareto efficiency.
Worked example (from the homework). U_A = X + 2Y, U_B = 4X + Y. Endowments: A gets (4, 2), B gets (0, 2). Endowment utilities: U_A(4, 2) = 4 + 4 = 8, U_B(0, 2) = 0 + 2 = 2.
Allocation (i): A gets (0, 2), B gets (4, 2). U_A = 0 + 4 = 4 < 8. A is worse off. Not in the Core.
Allocation (ii): A gets (0, 4), B gets (4, 0). U_A = 0 + 8 = 8 ≥ 8. U_B = 16 + 0 = 16 ≥ 2. Both pass. Check Pareto efficiency: MRS_A = 1/2, MRS_B = 4. Since these differ and both consumers have linear utility, the only Pareto-efficient allocations are corner solutions where all X goes to B and/or all Y goes to A. This allocation gives all X to B and all Y to A, so it is Pareto efficient. It is in the Core.
Allocation (iii): A gets (3, 4), B gets (1, 0). U_A = 3 + 8 = 11 ≥ 8. U_B = 4 + 0 = 4 ≥ 2. Both pass. Check Pareto efficiency: MRS_A = 1/2, MRS_B = 4. These differ, but neither consumer is at a corner where only edge-of-box allocations are efficient. This is an interior allocation that is not on the contract curve. Not Pareto efficient. Not in the Core.
Allocation (iv): A gets (1, 2), B gets (3, 2). U_A = 1 + 4 = 5 < 8. A is worse off. Not in the Core.
MRS for linear utility U = aX + bY: MRS = a / b (constant)
MRS for Cobb-Douglas U = X^α · Y^β: MRS = (α / β) × (Y / X)
MRS for Leontief U = min(aX, bY): Undefined at the kink (aX = bY). Along the vertical arm (Y in excess): MRS = 0. Along the horizontal arm (X in excess): MRS → ∞.
Edgeworth Box dimensions: Width = X_A + X_B (total X). Height = Y_A + Y_B (total Y).
Contract curve condition (smooth preferences): MRS_A(X_A, Y_A) = MRS_B(X_B, Y_B), where X_B = total X - X_A and Y_B = total Y - Y_A.
Indifference curve equation for linear utility: Set aX + bY = k (a constant), giving Y = (k - aX) / b, a straight line.
Indifference curve equation for Leontief: Set min(aX, bY) = k, giving an L-shape with the corner at X = k/a, Y = k/b.
Indifference curve equation for Cobb-Douglas: Set X^α · Y^β = k, giving Y = (k / X^α)^(1/β), a smooth convex curve.
Students often think "Pareto efficient" means "fair" or "equal." It does not. An allocation where one consumer has everything and the other has nothing can be Pareto efficient.
Students confuse the contract curve with the Core. The contract curve is all Pareto-efficient allocations. The Core is the subset of the contract curve where both consumers do at least as well as at their endowment. The Core is always a portion of the contract curve, never larger than it.
When computing MRS for a Leontief (min) function, students often try to take partial derivatives as if the function were smooth. It is not differentiable at the kink. The MRS is undefined there. Use the logic of which good is in excess instead.
Students sometimes draw B's indifference curves from A's origin. In the Edgeworth Box, B's curves must be drawn from B's origin (top-right corner), so they appear flipped relative to A's.
⚠️ You will almost certainly be asked to draw an Edgeworth Box with labelled axes, the endowment point W, and indifference curves for both consumers. Practise drawing these for all three utility types (linear, Leontief, Cobb-Douglas) until you can do it quickly.
⚠️ Computing MRS and checking whether MRS_A = MRS_B is the fastest way to test Pareto efficiency. This is a very common exam question.
⚠️ The Core question requires you to check two things: participation constraint (utility at the new allocation ≥ endowment utility for both) and Pareto efficiency. Students who check only one condition lose marks.
⚠️ Contract curve derivation for a linear + Cobb-Douglas pairing is a standard problem. Set the Cobb-Douglas MRS equal to the linear MRS (which is a constant) and solve for the relationship between X_B and Y_B.
⚠️ With two linear consumers who have different MRS values, the contract curve runs along the edges of the box, not through the interior. This is a frequent source of lost marks on graphing questions.
True or false: If MRS_A ≠ MRS_B at an interior allocation, the allocation is Pareto efficient. (Answer: False.)
Fill in the blank: The MRS for U(X, Y) = 3X + 5Y is ______ at every bundle. (Answer: 3/5.)
True or false: Every allocation on the contract curve is in the Core. (Answer: False. It must also satisfy the participation constraint.)
Fill in the blank: For U(X, Y) = min(4X, Y), the kink line where the consumer "balances" the two goods is Y = ______. (Answer: 4X.)
True or false: In the Edgeworth Box, Consumer B's origin is at the bottom-left corner. (Answer: False. B's origin is at the top-right corner.)
Q: U_A(X, Y) = 2X + Y. What is the MRS for consumer A, and what shape are A's indifference curves?
A: MRS_A = 2. The indifference curves are straight lines with slope -2 (in the Y-on-X plane).
Q: U_B(X, Y) = min(2X, Y). At the bundle (6, 8), is consumer B at the kink, or is one good in excess? Which one?
A: At (6, 8), 2X = 12 and Y = 8. Since 2X > Y, the consumer has X in excess relative to Y. B is on the horizontal arm of the L-shaped indifference curve, not at the kink.
Q: With U_A = 2X + Y (MRS_A = 2) and U_B = min(2X, Y) (MRS undefined at the kink), is the initial allocation A = (2, 8), B = (6, 8) Pareto efficient?
A: At B's bundle, 2(6) = 12 > 8 = Y, so B is not at the kink. B's MRS is undefined (approaching infinity on the horizontal arm). Since MRS_A = 2 and B's effective MRS differs, the allocation is not Pareto efficient. A could give Y to B in exchange for X and both could be made better off.
Q: U_A = X + 2Y, U_B = 4X + Y. Both are linear with MRS_A = 1/2 and MRS_B = 4. What does the contract curve look like in the Edgeworth Box?
A: Because both consumers have linear utility with different MRS values, no interior tangency is possible. The contract curve runs along the edges of the box. Efficient allocations have all X going to B (who values it more, MRS_B = 4) and all Y going to A (who values it more, MRS_A = 1/2 means A gives up X cheaply for Y). The contract curve runs along the bottom edge (Y_A = total Y, varying X) and the left edge (X_B = total X, varying Y).
Q: U_A = 2X + Y (MRS_A = 2), U_B = X^(1/4) · Y^(3/4). Derive the contract curve.
A: Set MRS_B = MRS_A. MRS_B = (1/4 · Y_B) / (3/4 · X_B) = Y_B / (3X_B). Set Y_B / (3X_B) = 2, so Y_B = 6X_B. This is the equation of the contract curve in terms of B's consumption. In the Edgeworth Box (total X = 12, total Y = 12), X_B ranges from 0 to 2 (since Y_B = 6X_B must be ≤ 12). For X_B > 2, the contract curve hits the top edge of the box.
Q: An allocation gives U_A = 10 at the proposed trade but U_A = 8 at the endowment. U_B = 5 at the proposed trade but U_B = 6 at the endowment. Is this allocation in the Core?
A: No. Consumer B is worse off (5 < 6) than at the endowment. The participation constraint fails for B, so the allocation is not in the Core, regardless of whether it is Pareto efficient.
This material connects directly to competitive (Walrasian) equilibrium, which you will likely cover next. In a competitive equilibrium, prices adjust until both consumers maximise utility on their budget constraints and markets clear; the resulting allocation lies on the contract curve (First Welfare Theorem). Understanding Pareto efficiency and the contract curve here is the foundation for seeing why competitive markets produce efficient outcomes.
The Core concept reappears in game theory and mechanism design: the idea that an outcome must survive every coalition's ability to walk away is central to cooperative game theory.
The MRS calculations you practise here carry over to producer theory (the marginal rate of technical substitution, MRTS) and to any optimisation problem where you equate ratios of marginal values.
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