Competitive Equilibrium

Competitive Equilibrium in Health Economics

A competitive equilibrium is a type of equilibrium for a whole economy, made up of a price for every good and an allocation of goods, at which consumers and firms take the prices as given, each makes its best choice at those prices, and supply equals demand in every market at once. Kenneth Arrow's 1963 paper on the welfare economics of medical care used it as the norm against which the medical care industry was compared, and most later arguments about why health care markets are different start from the same comparison. The idea matters for appraisal too, because a market price measures opportunity cost only when the market behind it is close to a competitive equilibrium.

Price-taking, optimising and market clearing

Arrow described the competitive model as the flows of services and prices that would arise if each person traded at the going prices as if their decisions had no influence over them, with prices at which the services available equalled the amounts others wished to buy. In his Nobel lecture he reduced competitive analysis to two principles, optimising behaviour by agents who take prices as given, and prices set so that supply equals demand on each market. Debreu's Nobel lecture adds that producers maximise profit and consumers meet a budget set by the value of their endowment and their share of profits.

For an exchange economy without production the two conditions can be written as follows.

$$x_i^{*} \in \arg\max_{x_i} \left\{ u_i(x_i) \mid p^{*} \cdot x_i \le p^{*} \cdot \omega_i \right\} \ \text{for every } i, \qquad \sum_{i} x_i^{*} = \sum_{i} \omega_i$$

where $i$ indexes consumers, $x_i$ is a bundle that consumer $i$ could choose, $u_i$ is that consumer's utility function, $\omega_i$ is the bundle the consumer owns at the start (the endowment), $p^{*}$ is the vector of equilibrium prices and $x_i^{*}$ is the bundle chosen at those prices. The first condition is utility maximisation within a budget, and the second says that the goods demanded add up to the goods available.

Price-taking is plausible in a market with many small traders and hard to defend for a dominant hospital group or a single national buyer of medicines. The one-market version, with everything else held fixed, is the market equilibrium. The economy-wide version is also called a general or Walrasian equilibrium.

Existence and uniqueness of equilibrium prices

Arrow's lecture defines the general equilibrium of the economy as the set of prices that make every excess demand zero. He also states Walras' law, under which the value of planned spending by all households equals the value of endowments plus profits at any prices.

$$z(p) = \sum_{i} \left[ x_i(p) - \omega_i \right], \qquad p \cdot z(p) = 0, \qquad z(p^{*}) = 0$$

where $z(p)$ is the vector of aggregate excess demands at prices $p$, $x_i(p)$ is consumer $i$'s demand at those prices, the middle expression is Walras' law and the last one defines the equilibrium price vector $p^{*}$.

Walras and his successors argued for existence by counting equations and unknowns, an argument Debreu says cannot convince a mathematician. Arrow and Debreu proved existence in their joint 1954 paper with Kakutani's fixed point theorem. Debreu adds that global uniqueness proved too demanding a requirement, so theorists settled for local uniqueness. The result has practical weight, since computable general equilibrium models, such as those used to cost pandemics, solve numerically for this price vector.

Why the competitive outcome is the efficiency benchmark

The normative weight of the concept comes from the theorems of welfare economics. In Arrow's lecture, each optimising consumer sets the marginal rate of substitution between two goods equal to their price ratio, and because everyone faces the same prices, the rates are equal across people, so no group can gain by trading further.

$$MRS^{A}_{HX} = MRS^{B}_{HX} = \frac{p_H}{p_X}$$

where $MRS^{A}_{HX}$ and $MRS^{B}_{HX}$ are the amounts of good $X$ that consumers $A$ and $B$ would each give up for one more unit of good $H$, and $p_H$ and $p_X$ are the prices of the two goods.

Arrow's 1963 paper gives both results. By his First Optimality Theorem, a competitive equilibrium in which every commodity relevant to costs or utilities is priced is optimal in the sense of Pareto efficiency. By his Second, with no increasing returns and some minor conditions, any optimal state can be reached as a competitive equilibrium once purchasing power is suitably redistributed. He noted that shifting purchasing power towards the ill would raise the demand for medical services and, in the short run, their price.

Efficiency in this sense is narrow. Arrow's lecture recalls that Pareto efficiency in no way implies distributive justice, and the OpenStax microeconomics text observes that consumers' ability to pay reflects the income distribution. In health care this is the central difficulty, because a competitive allocation gives care according to ability to pay, whatever the need.

Worked example with two people trading clinic sessions

All numbers in this example are illustrative. A small exchange economy has 20 physiotherapy sessions, written $H$, and 200 units of all other goods, written $X$, which serve as the unit of account, so $p$ is the price of a session. Person $A$ has a long-term musculoskeletal condition and owns no sessions and 40 units of $X$. Person $B$ is well and owns all 20 sessions and 160 units of $X$. Their preferences are Cobb-Douglas, $u_A = H^{0.5} X^{0.5}$ and $u_B = H^{0.25} X^{0.75}$, so demands take a simple form.

$$H_i = \frac{a_i \, m_i}{p}, \qquad X_i = (1 - a_i) \, m_i, \qquad m_i = p \, h_i + x_i$$

where $a_i$ is the share of wealth person $i$ spends on sessions (0.5 for $A$ and 0.25 for $B$), $m_i$ is the market value of their endowment, $h_i$ and $x_i$ are the sessions and other goods they own, and $H_i$ and $X_i$ are the amounts they demand.

1. Solve for the market-clearing price. Setting total demand for sessions equal to the 20 available gives the price.

$$p^{*} = \frac{a_A x_A + a_B x_B}{\bar{H} - a_A h_A - a_B h_B}$$

where $\bar{H}$ is the total number of sessions, 20. With $h_A$ equal to 0 and $h_B$ equal to 20, the numerator is 0.5 × 40 + 0.25 × 160 = 60 and the denominator is 20 minus 5, or 15, so the price is 60 / 15 = 4.

2. Find the allocation. $A$ buys 0.5 × 40 / 4 = 5 sessions and keeps 0.5 × 40 = 20 units of $X$. $B$ has wealth 4 × 20 + 160 = 240, keeps 0.25 × 240 / 4 = 15 sessions and consumes 0.75 × 240 = 180 units of $X$. Sessions add up to 5 + 15 = 20 and other goods to 20 + 180 = 200, so both markets clear.

3. Check efficiency. With these preferences the marginal rate of substitution of person $i$ is $(a_i / (1 - a_i)) \times X_i / H_i$. For $A$ it is 1 × 20 / 5 = 4 and for $B$ it is (1/3) × 180 / 15 = 4. Both equal the price, so no further trade can help one person without harming the other.

4. Transfer purchasing power to the person who is ill. A lump sum of 60 units of $X$ moves to $A$, who now owns 100, while $B$ keeps 20 sessions and 100 units of $X$. The numerator becomes 0.5 × 100 + 0.25 × 100 = 75 and the denominator is unchanged, so the new price is 75 / 15 = 5. $A$ buys 0.5 × 100 / 5 = 10 sessions and consumes 50 units of $X$. $B$ has wealth 5 × 20 + 100 = 200, keeps 0.25 × 200 / 5 = 10 sessions and consumes 150. The marginal rates of substitution are 50 / 10 = 5 and (1/3) × 150 / 10 = 5, so this allocation is efficient too.

5. Interpret the two equilibria. Both allocations are efficient, yet the person with the long-term condition receives 5 sessions in one and 10 in the other, and the price rises after the transfer as Arrow expected. Which allocation is better cannot be read off the model without a judgement about distribution, the job of a social welfare function or of an explicit health objective.

How health care departs from the competitive conditions

Of Arrow's three preconditions for the theorems, medical care most plainly fails the marketability of every good and service relevant to costs and utilities. Communicable disease was his example of a product whose intrinsic technological characteristics prevent a suitable price from being enforced, an externality case where Arrow saw subsidy, tax or compulsion as necessary. He argued that protection against many risks was simply not available to buy, and that under uncertainty information becomes a commodity, the root of the information asymmetry between doctor and patient.

Price-taking is the other weak point. Debreu recalls Edgeworth's argument that markets become more competitive as the number of agents grows and each becomes negligible. A local market with a few hospitals or a single public purchaser is far from that limit, the territory of competition theory. A market that misses the benchmark shows market failure, and what policy should do when several conditions fail at once is the subject of the theory of the second best.

Competitive prices in cost-benefit and cost-effectiveness analysis

Appraisal relies on the competitive benchmark whenever it treats a market price as a value. HM Treasury's Green Book (2026) asks for goods and services to be valued at their opportunity cost and makes market prices the starting point, excluding taxes and subsidies as transfers. It warns that market prices may fail to represent social costs and benefits when markets do not function efficiently, naming market power, externalities and imperfect information. The OpenStax text explains why an undistorted price is informative, since in a perfectly competitive market price equals marginal cost. In cost-benefit analysis a shadow price replaces a distorted price, such as one set with the market power of a patent, and willingness to pay values outcomes that have no market.

Most NHS care has no market price. The NICE manual (PMG36, section 6.3.1) reasons that, given the fixed budget of the NHS, the maximum acceptable ICER is the opportunity cost of programmes displaced by new, more costly technologies, and section 6.3.2 asks for net health benefit to be presented, when decision-making modifiers are applied, at GBP 25,000 and GBP 35,000 per QALY gained. These values stand for health forgone elsewhere in a fixed budget, so they behave more like a shadow price of the NHS budget than like a market price.

Welfarist and extra-welfarist readings of the benchmark

The competitive benchmark belongs to the welfarist tradition. Brouwer and colleagues describe Paretian economics as treating individual utilities as the outcome, holding them interpersonally incomparable and taking the initial income distribution as the starting point, and note that the Pareto improvement normally has little cutting power beyond voluntary trading in well-regulated markets. Extra-welfarist analysis, seen archetypically in cost-utility analysis, uses something like need in place of effective demand and willingness to pay, and they describe need as unrelated to ability to pay. In the worked example a welfarist sees two efficient equilibria and a separate question of distribution, whereas an extra-welfarist health service would ask how many sessions the person with the long-term condition needs.

Neighbouring equilibrium and efficiency ideas

A competitive equilibrium differs from a Nash equilibrium, in which strategic players each choose a best response to the others and prices need play no part. It is also distinct from allocative efficiency and Pareto efficiency, which are properties an allocation may have, whereas a competitive equilibrium is a price vector together with the allocation it supports.

Common misreadings of competitive equilibrium

Most errors treat the concept as a description of real health care markets or as a verdict on fairness. The four below recur in debates about markets for care.

1. Treating it as a forecast. Arrow used the competitive model as the norm against which the medical care market was compared.
2. Reading efficiency as fairness. Both allocations in the worked example are efficient, and they give the person with the long-term condition very different amounts of care.
3. Assuming a single equilibrium. Global uniqueness proved too demanding a requirement, so a model can have more than one equilibrium.
4. Using any observed price as an opportunity cost. The Green Book turns to non-market valuation where prices do not reflect social value.

Sources

• Arrow KJ. General economic equilibrium: purpose, analytic techniques, collective choice. Nobel Memorial Lecture, 12 December 1972. https://www.nobelprize.org/uploads/2018/06/arrow-lecture.pdf
• 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. https://doi.org/10.1016/j.jhealeco.2007.07.003
• Debreu G. Economic theory in the mathematical mode. Nobel Memorial Lecture, 8 December 1983. https://www.nobelprize.org/uploads/2018/06/debreu-lecture.pdf
• HM Treasury. The Green Book (2026). Updated 28 October 2026. Chapter on valuation of social costs and social benefits. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 6.3.1 and 6.3.2. https://www.nice.org.uk/process/pmg36/chapter/committee-recommendations
• Shapiro D, MacDonald D, Greenlaw SA. Principles of Microeconomics. 3rd ed. OpenStax; 2022. Section 8.4, Efficiency in perfectly competitive markets. https://openstax.org/books/principles-microeconomics-3e/pages/8-4-efficiency-in-perfectly-competitive-markets