Equilibrium

Equilibrium in Health Economics

In economics, an equilibrium is an outcome at which the plans of all the agents in a model are mutually consistent, so that no one has a reason to act differently given what everyone else is doing. Health economists use the idea to predict where a market for care or a contest between payers, providers and patients will settle, and how that resting point moves after a policy change. Health care is also the standard example of a sector where the competitive equilibrium of the textbook is a poor guide to what is observed, so much of the economics of health is about why. The single-market case is worked through on the market equilibrium and equilibrium price pages.

Partial, general and strategic equilibrium

Equilibrium can be analysed in a single market, across the whole economy or in a game between strategic players. A partial equilibrium looks at one market, such as that for a diagnostic test, and holds prices and incomes elsewhere fixed. It is reached when the price makes the quantity buyers want equal to the quantity sellers offer.

$$Q_D(P^*) = Q_S(P^*)$$

where $Q_D$ is the quantity demanded, $Q_S$ is the quantity supplied and $P^*$ is the price at which the two are equal. At a higher price there is excess supply, and at a lower price excess demand, or a shortage, and these pressures move a freely priced market back towards $P^*$.

Kenneth Arrow's 1963 paper on medical care described the competitive model as one in which individuals trade at going prices they cannot influence, with prices such that the services available equal the amounts others are willing to buy. A general equilibrium extends the condition to every market at once, which Arrow states as some set of prices that will clear all markets, so a change in one market feeds through to prices and incomes elsewhere.

A strategic, or Nash, equilibrium applies where each agent's payoff depends on what the others choose. Each player's strategy is a best reply to the strategies of the rest, so no single player can do better by deviating. The Nash equilibrium page treats this form in detail, and game theory covers the wider tools for strategic interaction in health care.

Competitive equilibrium as the benchmark for efficiency

Equilibrium is a descriptive idea, but it carries normative weight through the theorems of welfare economics. Arrow's statement of the first optimality theorem is that if a competitive equilibrium exists, and if all commodities relevant to costs or utilities are priced in the market, the equilibrium is optimal in the sense of Pareto efficiency, with no other allocation that makes all participants better off.

His second optimality theorem runs the other way. With no increasing returns and certain minor conditions, every optimal state is a competitive equilibrium for some initial distribution of purchasing power, so policy could leave allocation to the market and confine itself to redistributing money income. Arrow added that if the actual market differs significantly from the competitive model, separating allocation from distribution becomes, in most cases, impossible. The efficiency case for leaving health care to competitive markets therefore rests on conditions that health care often fails.

Why health care markets rarely reach the competitive equilibrium

Arrow listed three preconditions for competitive equilibria and optimal states to coincide, namely that a competitive equilibrium exists, that all goods and services relevant to costs and utilities are marketable, and that returns are not increasing. Medical care departs from them in several ways, the main forms of market failure, and each changes what kind of equilibrium, if any, the market reaches.

Unequal information is the first. Arrow observed that the physician's information about the consequences and possibilities of treatment is necessarily much greater than the patient's, and that both parties know it. Information asymmetry lets providers shape demand, the basis of supplier-induced demand, so demand in a simple equilibrium model is no longer independent of supply.

Insurance is the second. Arrow noted that under insurance the cost of medical care depends on the choice of doctor and the willingness to use services as well as on the illness, the problem of moral hazard. He also wrote that insurance removes the incentive to shop around for better prices, so market forces tend to be replaced by direct institutional control. Where buyers know more about their own risk than insurers, adverse selection limits which contracts an insurance market can sustain in equilibrium.

The third is restricted supply and spillovers. Arrow pointed out that licensing restricts entry and so supply, raising the cost of care, and he gave the spread of communicable disease as the obvious medical example of an externality, since a person who is not immunised risks the health of others as well as their own.

Regulated prices, a policy choice rather than a market failure, are a fourth departure. Many health systems set the price paid by patients at zero or below the level that would clear the market, so excess demand persists and something other than money rations care. Riganti, Siciliani and Fiorio note that economists have argued that, with excess demand, waiting times act as non-monetary prices that bring the demand for and supply of health care into equilibrium. Using Italian data for 2010 to 2014, they estimated that demand for elective surgery is inelastic to waiting time, with elasticities between minus 0.15 and minus 0.24, while supply is more elastic.

Worked example with waiting time as the rationing price

All numbers in this example are illustrative. A publicly funded service offers an elective procedure free at the point of use. Monthly demand falls as the expected wait lengthens, because some patients go private or decide not to proceed, and monthly activity rises with the wait, because providers respond to pressure with extra sessions.

$$D(w) = a - b\,w, \qquad S(w) = c + d\,w, \qquad w^* = \frac{a - c}{b + d}$$

where $w$ is the expected waiting time in weeks, $D(w)$ is procedures demanded per month, $S(w)$ is procedures supplied per month, $a$ and $c$ are demand and supply at a zero wait, $b$ is the fall in demand per extra week, $d$ is the rise in supply per extra week, and $w^*$ is the equilibrium wait at which demand equals supply.

1. Set the parameters. Take $a = 1{,}000$, $b = 18$, $c = 460$ and $d = 36$. With no wait, demand of 1,000 procedures a month meets supply of 460, an excess demand of 540 that the money price, fixed at zero, cannot remove.

2. Find the equilibrium wait. $w^* = (1{,}000 - 460)/(18 + 36) = 540/54 = 10$ weeks.

3. Find the equilibrium activity. $D(10) = 1{,}000 - 18 \times 10 = 820$ and $S(10) = 460 + 36 \times 10 = 820$ procedures a month, so patients' and providers' plans agree at a ten-week wait.

4. Check the elasticities. The waiting-time elasticity of demand at the equilibrium is $e_D = -18 \times 10/820 = -0.22$, inside the range Riganti and colleagues estimated, and the elasticity of supply is $e_S = 36 \times 10/820 = 0.44$, so supply is the more responsive side, as in their findings.

5. Add capacity. A funding round adds 108 procedures a month at every waiting time, so $c$ rises to 568. The new wait is $w^* = 432/54 = 8$ weeks, and activity is $D(8) = 1{,}000 - 18 \times 8 = 856$, equal to $S(8) = 568 + 36 \times 8 = 856$.

6. Interpret the result. The extra capacity cuts the wait by two weeks but raises activity by only 36 procedures, from 820 to 856. The other 72 are offset because, at the shorter wait, the pressure-driven extra sessions fall by 36 × 2 = 72. Treating waiting time as a price predicts that added capacity is partly absorbed in this way. Demand also rises as the wait shortens, which limits the fall to two weeks, where demand fixed at 820 a month would give $w^* = (820 - 568)/36 = 7$ weeks, a fall of three.

Strategic equilibrium in vaccination decisions

Voluntary vaccination is the classic case of health choices that interact through a shared outcome. Bauch and Earn model a parent who compares the morbidity risk of vaccination with the risk from infection, where the chance of infection depends on how many others vaccinate.

$$E(P, p) = -r_v\,P - r_i\,\pi_p\,(1 - P)$$

where $E$ is the expected payoff of a parent who vaccinates with probability $P$, $p$ is vaccine coverage in the population, $r_v$ is the morbidity risk from vaccination, $r_i$ is the morbidity risk from infection, and $\pi_p$ is the probability that an unvaccinated child is eventually infected at coverage $p$. Because $\pi_p$ falls as coverage rises, vaccinating becomes less attractive the more others do it.

In their mixed equilibrium, parents are indifferent between vaccinating and not, which happens when the relative risk $r = r_v/r_i$ equals $\pi_{P^*}$, with coverage settling at $P^*$. For any perceived relative risk above zero, equilibrium uptake falls short of the eradication threshold. An equilibrium that is rational for each parent can still leave the disease endemic, which is the case for public vaccination policy and for counting the vaccination externalities of herd protection in evaluations.

General equilibrium and the macroeconomic evaluation of health policy

Standard cost-effectiveness analysis is partial equilibrium analysis, valuing a technology's costs and health effects while assuming that prices and incomes in the rest of the economy are unaffected. Beutels, Edmunds and Smith argued that traditional health economic analysis is ill-equipped for public health emergencies of international concern such as pandemic influenza, because the implicit assumption of partial equilibrium within the health sector, and in the economy as a whole if a broader perspective is adopted, would be violated.

A computable general equilibrium model represents producers, households, government and the use of labour and capital across the whole economy. Its benchmark solution is compared with counterfactual solutions that simulate a policy or event. In a 2009 BMJ study of pandemic influenza based on the 2004 UK economy, Smith and colleagues estimated that the costs related to illness alone ranged from 0.5% to 1.0% of gross domestic product (GBP 8.4bn to GBP 16.8bn) in low-fatality scenarios. They found that school closure increases the economic impact, particularly for mild pandemics, and that large-scale prophylactic absence from work would add markedly to the impact with few health benefits, labour-market effects that a partial equilibrium analysis would not capture.

Equilibrium states inside decision models

Equilibrium also describes the long-run behaviour of models used in economic evaluation where no market is involved. A Markov chain with fixed transition probabilities can settle into a steady state, a distribution of the cohort across health states that no longer changes from cycle to cycle.

$$\pi = \pi T, \qquad \sum_{k=1}^{K} \pi_k = 1$$

where $\pi$ is the row vector of state proportions, $\pi_k$ is the share of the cohort in state $k$, $K$ is the number of states and $T$ is the matrix of transition probabilities. Where death is an absorbing state reachable from every other state, the only steady state is the whole cohort dead, so the idea mainly serves as a check on long-run behaviour.

In a dynamic transmission model the relevant equilibrium is epidemiological. The ISPOR-SMDM task force on dynamic transmission modelling explains that the force of infection depends on the number of infectious people, changes over time and feeds back into its own future value. In the Bauch and Earn model, the system converges to a disease-free state when coverage reaches the critical level and to a stable endemic state when it does not. A shift in that equilibrium after a programme is one way such models show herd effects that a static model with a fixed infection risk leaves out.

Common misreadings of equilibrium

Several errors recur when the idea is carried into health economics. Most come from reading a model's resting point as a statement about what is desirable or what is observed.

1. Treating equilibrium as optimal. The welfare theorems need conditions that health care often fails, and voluntary vaccine uptake shows a Nash equilibrium that is stable yet inefficient.
2. Assuming an observed price clears the market. Regulated tariffs and copayments can sit alongside persistent excess demand, which shows up as queues.
3. Assuming an equilibrium exists and is reached. Arrow made existence a precondition, and models with feedback can have several resting points or reach none in a relevant time frame.
4. Ignoring spillovers outside the health sector. For a shock as large as a pandemic, partial equilibrium estimates miss economy-wide effects.

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
• Bauch CT, Earn DJD. Vaccination and the theory of games. Proceedings of the National Academy of Sciences of the USA. 2004;101(36):13391-13394. https://pmc.ncbi.nlm.nih.gov/articles/PMC516577/
• Beutels P, Edmunds WJ, Smith RD. Partially wrong? Partial equilibrium and the economic analysis of public health emergencies of international concern. Health Economics. 2008;17(11):1317-1322. https://doi.org/10.1002/hec.1339
• Greenlaw SA, Shapiro D. Principles of Economics. 2nd ed. OpenStax; 2017. Section 3.1. https://openstax.org/books/principles-economics-2e/pages/3-1-demand-supply-and-equilibrium-in-markets-for-goods-and-services
• Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. Dynamic transmission modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-5. Medical Decision Making. 2012;32(5):712-721. https://doi.org/10.1177/0272989X12454578
• Riganti A, Siciliani L, Fiorio CV. The effect of waiting times on demand and supply for elective surgery: evidence from Italy. Health Economics. 2017;26(Suppl 2):92-105. https://doi.org/10.1002/hec.3545
• Smith RD, Keogh-Brown MR, Barnett T, Tait J. The economy-wide impact of pandemic influenza on the UK: a computable general equilibrium modelling experiment. BMJ. 2009;339:b4571. https://doi.org/10.1136/bmj.b4571