Contract Curve

Contract Curve in Welfare Economics and Health Care

The contract curve is the Pareto frontier of a two-party exchange, the locus of allocations at which the indifference curves of the two parties touch. Along it the two marginal rates of substitution are equal, so every point on the curve is Pareto efficient and every point off it leaves room for a swap that helps both sides. F. Y. Edgeworth named the curve in 1881 while asking how far a bargain between two contractors is settled by the bargain itself, and CORE Econ also calls it the Pareto efficiency curve. For health economists the curve gives the efficiency side of welfare economics a definite shape, whether the parties are two clinical teams sharing staff, a payer and a manufacturer settling a price and a volume, or two households pooling the cost of illness. Choosing one efficient point over another remains a judgement about distribution that the Pareto rule cannot make.

Edgeworth's locus of settlements that consent cannot improve

Edgeworth began with almost the simplest contract, two individuals whose interest depends on two variable quantities that they have agreed not to change without mutual consent. Exchange of two goods is one case. He looked for the points from which no small step raises both parties' utility together, and proposed to call the locus of such points the contract-curve.

$$\frac{\partial P}{\partial x}\frac{\partial \Pi}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial \Pi}{\partial x} = 0$$

where $P$ and $\Pi$ are the utilities of the two parties, written as functions of the two quantities $x$ and $y$ that the contract settles. Rearranged, the condition says the ratio of the two partial derivatives is the same for both parties, so their indifference curves through the point share a tangent.

Edgeworth reached the same locus by holding the utility of one party constant and maximising the utility of the other. In health terms, fixing the health of one group at a stated level and seeking the best attainable outcome for the other traces the curve one point at a time.

The tangency condition in a two-good allocation

In the box diagram one party's bundle is measured from the bottom-left corner and the other's from the top-right corner, so every point in the rectangle splits fixed totals of two goods between them. Osborne and Rubinstein, who use the term Pareto stable for Pareto efficient, show that every allocation on the line joining the two origins is Pareto stable and that any allocation off it is not, because another allocation then makes both individuals better off. With smooth convex preferences, more of each good always preferred and positive amounts of both goods, the curve is the set of allocations satisfying the following conditions.

$$MRS_A(x_A, y_A) = MRS_B(x_B, y_B), \qquad x_A + x_B = \bar{X}, \qquad y_A + y_B = \bar{Y}$$

where $x_A$ and $y_A$ are the quantities of the two goods held by party A, $x_B$ and $y_B$ those held by party B, $\bar{X}$ and $\bar{Y}$ are the fixed totals, and $MRS_A$ and $MRS_B$ are the rates at which each party would give up the second good for one more unit of the first, each computed from that party's utility function.

When the two rates differ, the party that values the first good more highly can buy a little of it from the other at a rate between the two, and both gain. Osborne and Rubinstein define the core of the two-person economy as the allocations that are Pareto stable and that each person likes at least as much as the initial bundle, so the core is the stretch of the contract curve inside the lens formed by the two indifference curves through the starting point.

Bargaining, competition and the point that is reached

Edgeworth's central answer was that contract without competition is indeterminate, contract with perfect competition is perfectly determinate, and contract with more or less perfect competition is less or more indeterminate. In CORE Econ's farmer and landowner model, every allocation on the Pareto efficiency curve yields the same surplus of 4.5 bushels but shares it differently, and how the two share the surplus depends on their bargaining power.

Edgeworth then added traders of each type and argued that, short of a practically infinite number, a finite length of the contract-curve remains from which no contract or recontract can move the system, a length that keeps diminishing as the market approaches perfect competition. Osborne and Rubinstein prove that every competitive equilibrium allocation is Pareto stable and lies in the core, the first being what is conventionally called the first fundamental theorem of welfare economics.

A national payer negotiating with the only maker of a patented medicine is in Edgeworth's two-party case, settling a price and a volume. Every deal from which neither side could gain without the other losing lies on the contract curve, and which deal is struck depends on walk-away positions and bargaining power, which bargaining theory models with the Nash solution. A cost-effectiveness threshold that sets the payer's walk-away price bounds the stretch of the curve the payer will accept without picking a point within it.

Mutual insurance against illness on the contract curve

Osborne and Rubinstein treat money in each possible future, or state, as a separate good, so that people can insure each other with contracts that pay out in one state in return for payments in the other. Each person maximises expected utility, and risk aversion makes the Bernoulli utility function concave.

For a strictly risk-averse person, the rate of substitution equals the ratio of the agreed state probabilities, $\pi_1/\pi_2$, where that person's income is the same in both states and differs from it elsewhere. When total income is the same in both states, an allocation is therefore on the contract curve only if it gives every person the same income in both states, so the curve is the line of full mutual insurance. Osborne and Rubinstein show that the competitive equilibrium then gives each person the expected value of the initial bundle with certainty.

The same model marks the limit of risk pooling. When state 1 is a disaster that reduces total wealth, such as an epidemic that cuts everyone's earnings at once, Osborne and Rubinstein show that every person consumes less in state 1 than in state 2 in equilibrium. With unequal totals no allocation gives everyone the same income in both states, so no point on the contract curve is full insurance for all. Adding a risk-neutral insurer, they show the risk-averse members are fully insured when the insurer's resources are large enough and only partly insured otherwise, a simple model of what an insurer adds to health insurance.

Worked example with two clinical teams sharing staff hours

All numbers in this example are illustrative. Two teams in one hospital are each given 50 nurse hours and 50 therapy hours a week, 100 of each in total. Team A treats patients who gain most from nursing and team B patients who gain most from therapy, with health produced in QALYs per year as $h_A = x_A^{0.75} y_A^{0.25}$ and $h_B = x_B^{0.25} y_B^{0.75}$, where $x$ is nurse hours and $y$ is therapy hours.

1. The starting split. Each team produces $h = 50^{0.75} \times 50^{0.25} = 50$ QALYs. Team A's rate of substitution is $MRS_A = 3 y_A / x_A = 3$ therapy hours per nurse hour and team B's is $MRS_B = y_B / (3 x_B) = 0.333$, so the starting split is off the curve.

2. The curve. Setting the two rates equal with the totals fixed gives the contract curve for this case.

$$3 \frac{y_A}{x_A} = \frac{100 - y_A}{3(100 - x_A)} \quad \Rightarrow \quad y_A = \frac{100 x_A}{900 - 8 x_A}$$

where $x_A$ and $y_A$ are team A's nurse and therapy hours and team B holds the remaining $x_B = 100 - x_A$ and $y_B = 100 - y_A$. The curve runs between the two origins below the diagonal, passing through 50 nurse and 10 therapy hours for team A and through 75 and 25.

3. The core. Searching along the curve for the points where each team keeps at least 50 QALYs gives team A between 68.5 and 80.5 nurse hours. At the first end team A stays at 50 and team B reaches 63.7, and at the other end the gains are reversed.

4. A market between the teams. If the teams may trade hours one for one, team A spends three quarters of the value of its hours on nursing and team B one quarter, so a price ratio of 1 clears both markets. Team A ends with 75 nurse and 25 therapy hours and team B with 25 and 75. The point lies on the curve, since 100 × 75 / (900 − 600) = 25, and both rates equal 1. Each team now produces $h = 75^{0.75} \times 25^{0.25} = 56.99$ QALYs, so total health reaches 113.98 against 100 at the start with nobody worse off, and no point on the curve has a larger total.

5. A different starting split. If team A had started with 80 of each and team B with 20, the market clears with a therapy hour priced at 0.538 nurse hours and team A ends with 92.3 nurse and 57.1 therapy hours. That point is also on the curve, with 81.9 QALYs for team A and 27.9 for team B, a total of 109.77. Both teams gain from trade, but the total is 4.20 QALYs below the maximum.

6. Interpretation. Moving onto the curve inside the core is a Pareto gain. Here the symmetric point both maximises total QALYs and gives the teams equal health, while a market starting from unequal budgets settles at a point that favours the team that started with more.

Welfare judgements along the contract curve

Each point on the contract curve gives a pair of utility or health levels, and plotting those pairs gives the frontier of what the two parties can jointly attain. Brouwer and colleagues describe the Paretian optimum as a position where no reallocation can raise one person's utility without reducing another's, adding that there may be many such optima, between which the Pareto criterion alone cannot choose. A Bergson and Samuelson social welfare function makes that choice explicit. Edgeworth himself proposed the greatest possible utility of all concerned as the basis of arbitration between contractors, a utilitarian rule for picking the point.

The welfarist and extra-welfarist schools differ over what is measured along the axes. Brouwer and colleagues note that the Pareto criterion could be applied to non-utility characteristics such as health, which is how the worked example reads team output in QALYs. They also observe that the Pareto improvement has little cutting power beyond voluntary trading in well-regulated markets. Where health care is allocated by budgets and referral, the curve works better as a test that no one loses from a reallocation than as a forecast of market outcomes. Welfarist cost-benefit analysis values outcomes by willingness to pay, and Brouwer and colleagues describe the potential Pareto criterion, under which money gains must outweigh money losses, as one that may be viewed as a serious compromise of the Pareto principle.

Contract curve beside nearby ideas

The Edgeworth box, which Osborne and Rubinstein trace to the same 1881 book, is the diagram in which the curve is drawn. The core is the part of the curve that both parties prefer to their starting bundles, and a competitive equilibrium is one point in the core. The utility possibility frontier maps the curve into the space of utilities. The production possibility frontier lists efficient combinations of outputs rather than efficient splits of given goods, and allocative efficiency asks whether the best mix of outputs has been chosen, which the contract curve alone cannot decide.

Misreadings and limits of the contract curve

Most errors arise when the curve is asked to settle fairness or to predict an outcome. The common ones in health follow.

1. Treating a point on the curve as fair. Osborne and Rubinstein state that a Pareto stable allocation might be good or bad, fair or unfair, and CORE Econ states that Pareto efficiency has nothing to do with fairness. The allocation that gives one team all the staff is on the curve.
2. Expecting a contract or a market to exist. CORE Econ notes that the name is used even where there is no contract, which is why it prefers the term Pareto efficiency curve. In health care the curve usually describes reallocations a planner could make.
3. Ignoring effects on others. Osborne and Rubinstein show that the efficiency of competitive equilibrium depends critically on each individual caring only about the bundle he consumes. Vaccination externalities and concern for other people's health break that assumption, and the equilibrium can then lie off the true curve.
4. Reading the tangency equation as sufficient. Edgeworth called coincident lines of indifference a necessary but not sufficient condition. His own example with two spheres has an impure stretch of tangencies from which the parties would still move.
5. Assuming the curve picks the outcome. With two parties Edgeworth's indeterminacy remains, and only competition among many traders shrinks the set of final settlements towards a single point.

Sources

• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• CORE Team. The Economy 1.0. CORE Econ; 2017. Unit 5, Property and power: mutual gains and conflict. https://books.core-econ.org/the-economy-v1/book/text/05.html
• Edgeworth FY. Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. London: C. Kegan Paul; 1881. https://archive.org/details/mathematicalpsyc00edgeuoft
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded 2nd ed. Open Book Publishers; 2023. Chapters 8, 10 and 11. https://doi.org/10.11647/OBP.0362