Edgeworth Box

Edgeworth Box in Welfare Economics and Health Care

An Edgeworth box is a diagram that sets the indifference maps of two people inside one rectangle whose sides measure the total amounts of two goods, so that each point is one complete division of those goods between the two. For health economists it pictures the Pareto test and the welfare theorems that underlie the welfarist case for markets and insurance. It shows in a single figure which reallocations leave both people better off, which allocations are Pareto efficient, and why efficiency alone cannot choose between them. The box also carries the welfarist assumptions, judging outcomes only by each person's own preferences, that extra-welfarist health economics later questioned.

How the box is built from two indifference maps

The box describes a pure exchange economy with two people, here A and B, two goods and fixed totals of each. Osborne and Rubinstein build it by taking each person's diagram of indifference curves and budget line, rotating person B's diagram by 180 degrees and sliding the two together until the initial bundles coincide. A point inside the rectangle then gives A's bundle measured from the bottom left origin and B's bundle measured from the top right origin.

$$x_A + x_B = \bar{X}, \qquad y_A + y_B = \bar{Y}$$

Here $x_A$ and $x_B$ are the amounts of good X held by A and B, $y_A$ and $y_B$ are their amounts of good Y, and $\bar{X}$ and $\bar{Y}$ are the fixed totals, which set the width and height of the box. Every point satisfies both equations, so one person gains a good only if the other loses it. The endowment is the starting point, what each person owns before any trade.

The name honours Francis Ysidro Edgeworth, although historians disagree over who drew the first box. McLure and Montesano conclude that Edgeworth introduced indifference curves and the contract curve, drawing them in his Mathematical Psychics of 1881, while Pareto, tentatively in 1902 and definitively in 1906, introduced the box diagram itself.

Mutual gains and the tangency condition

The indifference curves of A and B that pass through the endowment usually enclose a lens-shaped area. The MIT OpenCourseWare notes by Chen describe this area as containing all beneficial trades, because any allocation inside it puts both people on higher indifference curves. A move into the lens is therefore a Pareto improvement. Trade can continue until no further move helps one person without hurting the other, which happens where the two indifference curves touch without crossing.

$$MRS_A = \frac{\partial u_A / \partial x_A}{\partial u_A / \partial y_A} = \frac{\partial u_B / \partial x_B}{\partial u_B / \partial y_B} = MRS_B$$

Here $u_A$ and $u_B$ are the two utility functions, the partial derivatives are each person's marginal utilities of the two goods, and $MRS_A$ and $MRS_B$ are their marginal rates of substitution, the amount of good Y each would give up for one more unit of good X. Where the two rates differ, each person values the goods differently at the margin, and a small swap at a rate between the two leaves both better off. Where they are equal, the allocation is Pareto efficient.

Joining every such tangency gives the set of efficient allocations, which is the contract curve. The Saylor Academy text adds that the efficient set includes a point at which one person gets everything and the other nothing, and that an efficient allocation can be wildly unfair. Osborne and Rubinstein also define the core, which in the two-person box is the part of the efficient set that both people like at least as much as their endowments. Which efficient point is reached therefore depends on where the endowment sits.

The two welfare theorems in one diagram

A price ratio appears in the box as a straight line through the endowment, and the line is the budget line of both people at once. A competitive equilibrium is a price ratio at which the bundle A chooses and the bundle B chooses are the same point, so that demand equals supply for both goods. At that point each indifference curve is tangent to the price line and therefore to the other, so the rates of substitution are equal.

$$MRS_A = MRS_B = \frac{p_x}{p_y}$$

Here $p_x$ and $p_y$ are the prices of goods X and Y, so the ratio is minus the slope of the price line. In the Saylor Academy text's wording, the first welfare theorem states that any equilibrium of the price system is Pareto efficient. The second states that any Pareto-efficient point is an equilibrium of the price system for some endowment, and the same text notes that Arrow and Debreu demonstrated both for any number of people and goods.

Arrow set out both theorems early in his paper on the welfare economics of medical care. Leading into the second theorem, he noted that equilibrium depends on the initial distribution of purchasing power, so that a transfer of purchasing power from the well to the ill will increase the demand for medical services. He added in a footnote that, in practice, it is virtually impossible to find taxes and subsidies that do not harm the achievement of an optimal state.

An insurance box for the risk of illness

The box can also be drawn for uncertainty. Osborne and Rubinstein treat a payment of money in each possible future state of the world as a separate good, so that a bundle is a pair of state-contingent incomes and each person ranks bundles by expected utility.

$$U_i(c_1, c_2) = \pi_1 u_i(c_1) + \pi_2 u_i(c_2)$$

Here $c_1$ and $c_2$ are person $i$'s income if state 1 or state 2 occurs, $\pi_1$ and $\pi_2$ are the probabilities of the two states, which everyone agrees on, and $u_i$ is a concave function that makes person $i$ risk-averse. Osborne and Rubinstein show that the marginal rate of substitution where $c_1 = c_2$ equals $\pi_1 / \pi_2$. They prove that when total income is the same in both states and everyone is strictly risk-averse, the competitive equilibrium has a price ratio of $\pi_1 / \pi_2$ and each person consumes, with certainty, the expected value of their endowment. Trade then amounts to full mutual insurance.

This version links the diagram to health insurance and risk pooling. Arrow wrote that illness is to a considerable extent unpredictable, that the ability to shift the risks of illness to others is worth a price many are willing to pay, and that the welfare case for insurance is overwhelming. He also observed that many risks are not covered and that markets for risk coverage are poorly developed or nonexistent.

Illustrative example of two people sharing the cost of illness

The numbers below are illustrative, with incomes in thousands of GBP for one year. A and B each have an income of 100 when well and 40 when ill, after the costs of care and lost earnings. Exactly one of them will fall ill. State 1 is A falling ill and state 2 is B falling ill, each with probability 0.5, and both have the utility function $u(c) = \ln c$.

1. Endowments. A holds 40 in state 1 and 100 in state 2, and B holds 100 in state 1 and 40 in state 2. Total income is 40 + 100 = 140 in each state, so the box is 140 wide and 140 high.
2. Price ratio. The equilibrium price ratio equals $\pi_1 / \pi_2$, which is 0.5 / 0.5 = 1.
3. Value of each endowment. At that price A's endowment is worth 0.5 x 40 + 0.5 x 100 = 70 in expected income, and B's is worth 0.5 x 100 + 0.5 x 40 = 70.
4. Equilibrium allocation. Each person consumes 70 in both states, so the allocation is on the diagonal of the box. If A falls ill, B pays A the transfer $t = 70 - 40 = 30$, and if B falls ill, A pays B the same $t = 100 - 70 = 30$. This is mutual insurance with no loading.
5. Gain to each person. Expected utility at the endowment and at the equilibrium is shown below.

$$EU_{\text{endowment}} = 0.5 \ln 40 + 0.5 \ln 100 \approx 4.147, \qquad EU_{\text{insured}} = \ln 70 \approx 4.248$$

Here $EU$ is A's expected utility, and B's is the same by symmetry. The certain income giving the same utility as the endowment is $\sqrt{40 \times 100} \approx 63.2$, so the gain from full cover is $g = 70 - 63.2 = 6.8$ thousand GBP of certain income to each person.

1. Efficiency. At 70 in both states each person's rate of substitution is $(0.5 / 0.5) \times (u'(70) / u'(70)) = 1$, the price ratio, so the indifference curves are tangent and the allocation is Pareto efficient. Because the box is square, every point on the diagonal gives both people equal income in the two states, and every such point is efficient.
2. Efficient but not in the core. A diagonal point giving A 60 and B 80 in both states is also efficient, yet A's utility there is $\ln 60 \approx 4.094$, below the 4.147 of having no contract. It lies outside the core, and only a transfer to B outside the market, in the manner of the second theorem, would reach it.
3. Unequal risks. If state 1 had probability 0.4, A's certain income would be 0.4 x 40 + 0.6 x 100 = 76 and B's would be 0.4 x 100 + 0.6 x 40 = 64, which still sum to 140. The person less likely to fall ill ends with more, a distributional result that the box records but cannot judge.

The production box and health service efficiency

The same construction applies to inputs. In the production box described in Chen's MIT notes, the sides measure fixed totals of two inputs, each corner is the origin for one industry, and the curves are isoquants. Chen calls an input allocation technically efficient if the output of one good cannot be raised without lowering the other, which requires the two marginal rates of technical substitution to be equal.

$$MRTS_H = MRTS_C$$

Here $MRTS_H$ and $MRTS_C$ are the rates at which the two inputs can be substituted at constant output, illustrated by a hip replacement service and a cataract service sharing fixed nurse hours and theatre hours. Chen's notes derive the production possibility frontier from the set of these efficient input allocations. The production box therefore gives a picture of productive efficiency across services, and the choice of a point on the frontier is a question of allocative efficiency.

Welfarism, extra-welfarism and choosing among efficient points

Every judgement the box makes is in terms of each person's own preferences. Brouwer and colleagues quote Hurley's summary that the welfarist tenets require any policy to be judged solely in terms of the utilities individuals achieve, as assessed by the individuals themselves. The box shows the consequence they describe, which is that there are many Pareto optima and choosing between them is impossible using only the Pareto criterion. Selecting one requires an explicit judgement about distribution, which in welfarist economics is expressed through a Bergson-Samuelson social welfare function applied to the utility possibility frontier that the contract curve maps out.

Extra-welfarism changes what is compared rather than the geometry. Brouwer and colleagues argue that comparing people in terms of health, usually measured in QALYs, avoids the policy paralysis of relying on the Pareto criterion alone. They see the contrast most clearly in cost-utility analysis, which uses something like need in place of the effective demand and willingness to pay of cost-benefit analysis.

Where the box misleads in health care

The efficiency results depend on each person caring only about their own bundle. Osborne and Rubinstein state that this assumption is critical and give an example in which one person is harmed by the other's consumption and the competitive outcome is not Pareto stable. Arrow identified exactly this in health, a concern of individuals for the health of others, and treated communicable diseases as a case where a suitable price cannot be enforced. These are the externality problems behind vaccination externalities, and with them the tangency of private indifference curves no longer marks an efficient allocation.

Other limits follow from the box's simplicity. The exchange box holds totals fixed, so it shows trade in what already exists and leaves out production and its opportunity cost. The insurance version assumes agreed probabilities and complete markets for state-contingent income, whereas Arrow found markets for risk-bearing poorly developed and saw moral hazard in health insurance as a practical limit on cover. A common misreading treats a point on the efficient set as a recommended outcome. The box ranks allocations only where one is better for both people, and any choice between efficient points, as in the equity-efficiency trade-off, needs a separate value judgement.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. Author version: https://eprints.whiterose.ac.uk/id/eprint/3789/
• Chen CH. Lecture 19: Efficiency in exchange, equity and efficiency, and efficiency in production. 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare; 2007. https://ocw.mit.edu/courses/14-01-principles-of-microeconomics-fall-2007/
• McLure M, Montesano A. Thinking outside the box: Edgeworth, Pareto and the early history of the box diagram. Economics Discussion Paper 18.03. University of Western Australia; 2018. https://ideas.repec.org/p/uwa/wpaper/18-03.html
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded 2nd edition. Cambridge: Open Book Publishers; 2023. Chapters 10 and 11. https://doi.org/10.11647/OBP.0362
• Saylor Academy. Introduction to Economic Analysis, v. 1.0. Chapter 14, General equilibrium. Saylor Academy; 2012. https://saylordotorg.github.io/text_introduction-to-economic-analysis/s15-general-equilibrium.html