Demand Curve

Demand Curve for Health Care

Health economists use the demand curve for two separate jobs. The first is to tell a response to price apart from everything else. A co-payment or coinsurance change moves use along the curve, while changes in illness, income or medical practice move the whole curve, and confusing the two biases estimated price responses. The second is valuation, since the height of the curve is the usual starting point for putting a money value on care and on the efficiency cost of insurance. Its height can be read as marginal willingness to pay only under certain conditions, and estimates from the RAND Health Insurance Experiment and a natural experiment, together with an illustrative co-payment change, show both jobs in practice, while a behavioural caveat limits its welfare uses in insurance. The wider determinants of demand have their own page.

The price-quantity schedule and its inverse

Marshall described a person's demand as a list of the prices at which that person would buy different amounts, which he called a demand schedule, and drew it as a curve with quantity on the horizontal axis and price on the vertical axis. For a market, he took total demand to be the sum of the demands of all the individuals in it. He also observed that irregular needs, such as for the services of an expert surgeon, leave no individual demand schedule, but that the aggregate behaviour of many people is regular enough to study. Demand curves for care are therefore usually market or population curves.

In health care the price on the vertical axis is the price the patient pays at the point of use, whatever the service costs the insurer or health system. The schedule can be written as a demand function with the other determinants held fixed, as follows.

$$Q = D(k;\, Y, p_o, H)$$

where $Q$ is the quantity of care used in a period (visits, prescriptions or spending), $k$ is the out-of-pocket price per unit, $Y$ is income, $p_o$ is the prices of other goods and services, including substitutes and complements, and $H$ is health status and the other characteristics of the population. The demand curve traces $Q$ as $k$ varies, with $Y$, $p_o$ and $H$ fixed.

Turning the schedule round gives the inverse demand curve, which reads the price at which each unit would just be bought. With a linear schedule both forms are simple, as shown below.

$$Q = a - b\,k, \qquad P(Q) = \frac{a - Q}{b}$$

where $a$ is the quantity used at a zero price, $b$ is the fall in quantity for each £1 rise in the out-of-pocket price, $P(Q)$ is the inverse demand curve, giving the highest price at which the $Q$-th unit would be used, and $a/b$ is the choke price at which use falls to zero.

Movement along the curve and shifts of the curve

A change in the out-of-pocket price moves use along a given curve. A change in anything held fixed in the demand function moves the whole curve. Marshall defined an increase in a person's demand as a willingness to buy more at the same price, and as much at a higher price; a general increase is one that runs throughout the whole list of prices, beyond the current price alone.

For care, a new prescription charge or a higher coinsurance rate is a movement. An ageing population, an epidemic, new clinical guidelines or a dearer substitute service shifts the curve. The supplier-induced demand hypothesis adds a provider-driven shift, and because demand for care is a derived demand from the demand for health, any change in health status shifts it too. As Aron-Dine, Einav and Finkelstein note, moral hazard in health insurance has come to mean the price sensitivity of demand for care conditional on underlying health status. On that definition it is a response along the out-of-pocket-price curve, governed by its slope, and not a shift in it.

Insurance can be drawn either way, so the price axis needs to be stated. Against the out-of-pocket price, a lower coinsurance rate is a movement down the curve. Against the full price of care, the same curve rotates, as below.

$$Q = D(c\,P)$$

where $c$ is the coinsurance rate, the share of the full price paid by the patient, and $P$ is the full price of a unit of care. With linear demand, use at full price $P$ becomes $a - b\,c\,P$, so a lower $c$ makes the curve steeper when price is on the vertical axis, pivoting it about the point where it meets the quantity axis; with free care ($c = 0$) it becomes vertical.

Reading the height as marginal willingness to pay

Marshall called the price a person is just willing to pay for a further unit the demand price, and noted that the price measures the marginal utility of the good to each purchaser individually. In welfare analysis the inverse demand curve is therefore read as marginal willingness to pay for each unit of care, and the area under it as the gross value of the care used. That reading needs three conditions.

Income effects are small. The ordinary curve holds income fixed rather than utility, and Willig's result, discussed on the consumer surplus page, makes its area a close money measure only when the income elasticity of demand or the price change is small.

Patients judge the value of care correctly. If patients misjudge benefits, through present bias, low salience of symptoms or false beliefs about treatment, the height of the curve measures what they believe care is worth, which may differ from what it does for their health.

The choice belongs to the patient and the benefits stay with the patient. Where clinicians shape use, observed demand partly reflects provider behaviour. Where care protects other people, as with vaccination, the private curve leaves out the positive externality.

When the conditions hold, the area between the curve and the price is consumer surplus.

Estimating demand curves for care from RAND and natural experiments

A scatter of observed prices and quantities rarely traces a demand curve. Prices and use are set jointly with supply, people with higher expected use choose more generous insurance, and the curve shifts over time, so estimation needs price variation unrelated to the other determinants of use.

The RAND Health Insurance Experiment supplied that variation by design. Between 1974 and 1981 it insured more than 5,800 people from about 2,000 households at six sites in the United States and randomly assigned families to plans with coinsurance rates of 95, 50, 25 or 0 per cent, plus a mixed plan and an individual deductible plan.

Aron-Dine, Einav and Finkelstein's reanalysis shows what the experiment does and does not deliver. Average annual medical spending on the 95 per cent plan was 39 per cent below that on free care. Their pairwise arc elasticities of total spending with respect to the coinsurance rate were mostly negative, ranging from about −0.1 to −0.5. The widely quoted elasticity of −0.2 came from combining the experimental data with an episode-of-care model and further assumptions, not directly from the randomised comparisons. Because the plans were non-linear, with caps on out-of-pocket spending, no single price describes each plan. In the authors' illustration, applying the −0.2 elasticity to a move from RAND's 25 per cent coinsurance plan to a constant 28 per cent coinsurance predicted spending reductions of 18, 9 or 14 per cent, depending on whether the RAND plan was summarised by a price of 10, 17 or 13 cents per dollar of spending. An experiment of this kind identifies a few points on a curve for a given population and contract, leaving the rest of the curve to assumption.

A natural experiment adds points in another setting. Chandra, Gruber and McKnight used a policy change that raised co-payments for retired public employees in California and found physician visits and prescription drug use price sensitive, with arc elasticities similar to RAND's. Unlike RAND, they found offsetting rises in hospital use, concentrated among people with chronic disease.

Worked example with a physiotherapy co-payment, a shift in need and a welfare check

The figures below are illustrative. Annual demand for physiotherapy sessions in a district is linear, with 12,000 sessions used at a zero co-payment and 300 fewer for each £1 of co-payment, so the choke price is £40. Each session costs the health service £30 to provide.

1. Movement along the curve. The co-payment rises from £10 to £20 and use changes as follows.

$$Q_0 = 12000 - 300 \times 10 = 9000, \qquad Q_1 = 12000 - 300 \times 20 = 6000$$

where $Q_0$ and $Q_1$ are sessions per year at the £10 and £20 co-payments. The price change alone removes 3,000 sessions.

2. A shift in the same year. An ageing population raises demand by 1,500 sessions at every co-payment, so the new curve is $Q = 13{,}500 - 300k$, and use at a £20 co-payment is as follows.

$$Q_1' = 13500 - 300 \times 20 = 7500$$

where $Q_1'$ is sessions per year at the £20 co-payment on the shifted curve.

3. What before-and-after data show. An analyst who sees only the two observed points calculates the following.

$$\frac{7500 - 9000}{20 - 10} = \frac{-1500}{10} = -150$$

where the result is the apparent fall in sessions per £1 of co-payment. The true slope is 300 per £1. The observed fall of 1,500 sessions is the movement of 3,000 offset by the shift of 1,500, so treating the two points as one curve halves the price response.

4. Read the height of the original curve. The inverse demand curve gives the following.

$$P(9000) = 40 - \frac{9000}{300} = 10$$

where $P(9000)$ is the most that would be paid for the 9,000th session, so the marginal session at a £10 co-payment is valued at £10. The 3,000 sessions deterred by the rise to £20 were valued at between £10 and £20 each.

5. The deadweight loss of cover read from the demand curve. At a £10 co-payment, use runs to 9,000 sessions, but sessions are worth their £30 cost only up to the level below.

$$Q^{*} = 12000 - 300 \times 30 = 3000$$

where $Q^{*}$ is use at which marginal willingness to pay equals cost. The loss on the 6,000 sessions valued below cost is a triangle.

$$DWL = \frac{1}{2} \times (30 - 10) \times (9000 - 3000) = 60000$$

where $DWL$ is the deadweight loss in pounds per year on this reading.

6. Check for behavioural hazard. Suppose patients undervalue sessions by £8 each because the benefits of recovery come later, so the health benefit of the marginal session is $B(Q) = 48 - Q/300$. Setting this equal to the £30 cost gives efficient use of $Q = 300 \times (48 - 30) = 5400$ sessions, and the benefit of the 9,000th session is £18. The loss is then as follows.

$$DWL_B = \frac{1}{2} \times (30 - 18) \times (9000 - 5400) = 21600$$

where $DWL_B$ is the deadweight loss in pounds per year measured against the true benefit. The demand curve overstates the loss nearly threefold. A co-payment set at the £30 cost, which the demand curve alone would favour when the value of risk protection is ignored, would cut use to 3,000 sessions and forgo sessions worth more than they cost; the co-payment that brings use to 5,400 is £22.

Welfare uses in insurance analysis and their limits

Much of the welfare economics of health insurance is drawn on demand curves. When cover sets the patient's price below cost, the triangle between the cost line and the curve over the extra use is the standard measure of the efficiency loss from moral hazard, set against the value of risk protection that the cover provides, as in step 5. The same curve gives the consumer surplus lost when a charge rises; under insurance, the extra amount that continuing users pay is a transfer to the health service or insurer, offset in expectation by lower premiums, and so is not a social cost.

Demand curves are also drawn for insurance contracts. In Einav and Finkelstein's framework, the market demand curve for a contract is the cumulative distribution of willingness to pay for it, and under adverse selection competitive firms charging one price settle where that curve meets the average cost curve. Absent income effects, the welfare loss is the area between the demand and marginal cost curves for the people left inefficiently uninsured.

Baicker, Mullainathan and Schwartzstein argue that when patients also misuse care through mistakes, which they call behavioural hazard, welfare calculations from the demand curve alone can be off by orders of magnitude or have the wrong sign, a point taken up on the consumer choice page. In cost-benefit analysis the curve values care as patients perceive it. Cost-utility analysis does not use it to value health, though demand responses still enter decision models as uptake under cost sharing.

Demand curve, demand, elasticity and willingness to pay

The demand curve is one slice of demand, the price-quantity relation with every other determinant fixed. The price elasticity of demand summarises the proportional response at a point or over an arc of the curve and differs from its slope; responses to other prices belong to cross-price elasticity. Willingness to pay is the broader welfare concept, of which the height of the curve is one revealed-preference measure for units of care; willingness to pay for health itself is a Hicksian quantity, usually elicited by stated preference.

Common misreadings of demand curves for care

Three errors recur when demand curves are used in health economics. Each can change a policy conclusion.

• Treating a shift as a movement. A fall in use after a charge rises may partly reflect changes in need, supply or guidelines; in the worked example this halves the estimated response.
• Reading the height as health value. The height reflects patients' beliefs about care at the time of use.
• Using one elasticity as the curve. The −0.2 figure rests on modelling assumptions, and applying it requires reducing a non-linear contract to one price; it also says little about prices far from those observed, including where use would fall to zero.

Sources

• Aron-Dine A, Einav L, Finkelstein A. The RAND Health Insurance Experiment, three decades later. Journal of Economic Perspectives. 2013;27(1):197-222. doi:10.1257/jep.27.1.197
• Baicker K, Mullainathan S, Schwartzstein J. Behavioral hazard in health insurance. Quarterly Journal of Economics. 2015;130(4):1623-1667. Working paper version: NBER Working Paper 18468; 2012. https://www.nber.org/papers/w18468
• Chandra A, Gruber J, McKnight R. Patient cost-sharing, hospitalization offsets, and the design of optimal health insurance for the elderly. NBER Working Paper 12972. Cambridge, MA: National Bureau of Economic Research; 2007, revised June 2009. https://www.nber.org/papers/w12972 (published as: Patient cost-sharing and hospitalization offsets in the elderly. American Economic Review. 2010;100(1):193-213)
• Einav L, Finkelstein A. Selection in insurance markets: theory and empirics in pictures. Journal of Economic Perspectives. 2011;25(1):115-138. doi:10.1257/jep.25.1.115
• Marshall A. Principles of Economics. 8th ed. London: Macmillan; 1920 (first published 1890). Book III, Chapter III.
• Schlee EE. Expected consumer's surplus as an approximate welfare measure. Economic Theory. 2008;34(1):127-155.
• Willig RD. Consumer's surplus without apology. American Economic Review. 1976;66(4):589-597.