Consumer Surplus

Consumer Surplus as the Marshallian Area Under the Demand Curve in Health Programme Appraisal

Consumer surplus turns an ordinary demand curve into a money measure of gain or loss, which is why it appears whenever a health programme changes what people pay, in cash or in time, to use a service. As the Marshallian measure, it is read from observed use at different charges or travel distances, with no hypothetical questions. The area can be calculated as a level or as the change when a price moves, and it enters cost-benefit analysis of user charges, access and insured demand for care. It stands in a defined relation to compensating variation and equivalent variation, and an illustrative rise in a clinic charge shows the calculation, but the measure cannot value a change in a person's health.

From Dupuit's bridges to Marshall's tea

The modern idea comes from Jules Dupuit, a French civil engineer whose papers from 1844 onwards sought a money measure of the benefit of public works. He measured what he called relative utility as the area under the demand curve less what is paid, and judged bridges by how use, and surplus, responded to rising tolls. He also identified an area under the demand curve beyond the quantity sold that neither consumers nor producers could claim, which he called "lost utility"; the same reasoning underlies the modern idea of deadweight loss.

Marshall developed the concept in Book III, Chapter VI of his Principles of Economics, where he calls it consumer's surplus. His illustration is a man who would just buy one pound of tea a year at 20 shillings, a second at 14s., and further pounds at 10s., 6s., 4s. and 3s., and who buys seven pounds when the price is 2s. Each pound is worth at least the price at which he would just buy it, so the seven pounds are worth at least 59 shillings (20 + 14 + 10 + 6 + 4 + 3 + 2). He pays 7 × 2 = 14 shillings, which leaves a surplus of at least 45 shillings. For a whole market, Marshall showed the total surplus as the area between the demand curve and the price line.

Two of Marshall's caveats still matter for health appraisal. He accepted that the area adds up satisfactions on one scale only if a shilling is taken to mean the same to every buyer, and he noted that a pound's worth of satisfaction means much more to a poor person than to a rich one. He also observed that demand prices far from the customary price are largely conjecture, so the main practical use of the concept is to value changes in surplus caused by price changes near the current price, not the whole area.

Measuring the area as a level and as a change

The level of consumer surplus at a price is the area under the inverse demand curve up to the quantity bought, less what is spent. The same area can be read sideways, as the area to the left of the demand curve between the price paid and the price at which demand falls to zero. Both readings give the same number, as follows.

$$CS(p) = \int_{0}^{x(p, Y)} P(q)\,dq - p\,x(p, Y) = \int_{p}^{\bar{p}} x(s, Y)\,ds$$

where $CS(p)$ is consumer surplus at price $p$, $x(p, Y)$ is the ordinary, or Marshallian, demand at that price with income $Y$ held fixed, $P(q)$ is the inverse demand curve giving the most a marginal buyer would pay for the $q$-th unit, $s$ is a variable of integration over price, and $\bar{p}$ is the choke price at which demand falls to zero. Holding income fixed, not utility, is what separates this measure from the Hicksian ones. For a market, demand is the sum of all individual demands at each price.

Appraisal usually needs the change in surplus when a price moves from $p^0$ to $p^1$, not the level. The change is the area to the left of the demand curve between the two prices, written as follows.

$$\Delta CS = CS(p^1) - CS(p^0) = -\int_{p^0}^{p^1} x(s, Y)\,ds$$

where $\Delta CS$ is the change in consumer surplus, $p^0$ and $p^1$ are the prices before and after the change, and the other terms are as defined above. Only the stretch of the demand curve between the two prices is needed, which is the part that data on use near current charges can support.

With linear demand both quantities take simple forms, writing $x(p)$ for $x(p, Y)$, as shown below.

$$x(p) = \alpha - \beta p, \qquad CS(p) = \frac{x(p)^2}{2\beta}, \qquad \Delta CS = -(p^1 - p^0)\,\frac{x^0 + x^1}{2}$$

where $\alpha$ is the quantity demanded at a zero price, $\beta$ is the fall in quantity for each unit rise in price, $\bar{p} = \alpha / \beta$ is the choke price, and $x^0$ and $x^1$ are the quantities at the old and new prices. For a price rise the loss divides into a rectangle, $(p^1 - p^0)x^1$, which is the extra amount paid by those who keep using the service, and a triangle, $(p^1 - p^0)(x^0 - x^1)/2$, which is the surplus lost on the use that stops.

Where the Marshallian area is used in health programme appraisal

In a cost-benefit analysis the change in consumer surplus is one entry in the welfare accounts, set beside the change in revenue and in resource costs for the provider or payer, so that the net benefit of an option can be found. The measure suits changes that work through a price, broadly defined, and health care offers three main cases.

User charges. A new or higher charge for a prescription, consultation or clinic visit moves users up the demand curve. The rectangle in the loss is a transfer from continuing users to the health service and cancels out in a social account. What is set against the resources saved on care no longer used is the gross value users placed on that care, meaning the triangle plus the old charge on each forgone unit, which is the same as counting the triangle, the revenue lost on forgone use and the resources saved. The price elasticity of demand decides how much use is forgone.

Travel and time costs. Many health services are free at the point of use, but reaching them costs travel and time, which act as a price. The travel cost method estimates demand against that cost and values a change in access as a change in surplus. Clarke applied it to mobile breast screening units in rural Australia, adapting the usual welfare analysis because a mammogram is a binary good and estimating demand with discrete choice models; in the ten towns studied, the benefits of mobile screening outweighed the costs for towns 29 km or more from a fixed screening unit.

Demand for care under insurance. When insurance lowers the price a patient faces, through cost sharing set below the cost of care, the extra use whose value to the patient is below its cost gives a deadweight-loss triangle between the cost line and the demand curve over that extra use, the standard welfare account of moral hazard. Baicker, Mullainathan and Schwartzstein note that in standard moral hazard models the demand curve alone can be used to make welfare statements, and that much empirical work relies on this. They argue that patients also misuse care through mistakes, which they call behavioural hazard, and that welfare calculations from the demand curve alone can then be wrong by orders of magnitude or even have the wrong sign.

Marshallian and Hicksian measures for a price change

The Hicksian measures fix the person's utility, before or after the change, and ask what money change would match it; they are defined on their own pages. Consumer surplus fixes income instead, so its area mixes the substitution effect with the income effect of the price change. When demand does not depend on income, the three measures coincide.

For a fall in the price of a normal good, compensating variation is smaller than the Marshallian area and equivalent variation is larger, because the gain in real income raises demand. For a price rise the same reasoning puts the Marshallian loss between the most the person would pay to avoid the rise and the least they would accept to bear it. The exact Hicksian measures need compensated demand curves, whereas the data usually available describe ordinary market demand.

Willig's result is the usual justification for using the Marshallian area anyway. As summarised by Schlee, Willig argued that if the income elasticity of demand is small, or the price change is small, the percentage error from using consumer surplus in place of the willingness to pay for the price change is small, and that the error vanishes as the price change tends to zero. The result is about price changes, and it does not extend to large changes in goods that are not bought at a price.

Why consumer surplus does not value changes in health itself

The Marshallian measure needs a price that changes and a demand curve for the thing being valued. A person's health state is not bought at a price. A programme that cures a condition or extends life changes a quantity the person cannot adjust, so there is no ordinary demand curve whose area could be measured. The money value of such a change is a Hicksian quantity, estimated as willingness to pay for the gain, from stated preference surveys or from choice models.

Demand for health care is a derived demand, coming from the demand for health, so the area under a demand curve for care values the care as patients perceive it when they decide to use it. Where patients misjudge the benefits of care, or where clinicians shape use through supplier-induced demand, the demand curve may not trace the health value of the care, and the area inherits the error. Frameworks that value outcomes in QALYs, such as cost-utility analysis, do not use consumer surplus at all.

Worked example of raising the user charge for a clinic visit

The figures below are illustrative. A health service charges £5 for a visit to a nurse-led clinic and proposes raising the charge to £10. Annual demand for visits in the district is estimated as linear, with 10,000 visits demanded at a zero charge and 400 fewer visits for each £1 of charge, so the choke price is 10,000 / 400 = £25. Each visit costs the service £12 to provide.

1. Find the quantities. Demand at each charge is calculated as follows.

$$x^0 = 10000 - 400 \times 5 = 8000, \qquad x^1 = 10000 - 400 \times 10 = 6000$$

where $x^0$ and $x^1$ are visits per year at the £5 and £10 charges.

2. Calculate consumer surplus at each charge. The level is the triangle between the demand curve and the charge, as follows.

$$CS^0 = \frac{(25 - 5) \times 8000}{2} = 80000, \qquad CS^1 = \frac{(25 - 10) \times 6000}{2} = 45000$$

where $CS^0$ and $CS^1$ are consumer surplus in pounds per year at the £5 and £10 charges.

3. Calculate the change and divide it. The loss to users is £35,000, the fall from £80,000 to £45,000, which matches the area to the left of the demand curve between the two charges, as shown below.

$$-\Delta CS = (10 - 5) \times \frac{8000 + 6000}{2} = 35000$$

where $-\Delta CS$ is the loss of consumer surplus in pounds per year. Of this, 5 × 6,000 = £30,000 is the extra amount paid by users who keep coming, and 5 × 2,000 / 2 = £5,000 is the surplus lost on the 2,000 visits that no longer happen.

4. Add the service's side. Charge revenue rises from 5 × 8,000 = £40,000 to 10 × 6,000 = £60,000, a gain of £20,000. The 2,000 forgone visits save 2,000 × 12 = £24,000 of resources. The service's position improves by £44,000.

5. Find the net change. The net change in welfare is the service's gain of £44,000 less the users' loss of £35,000, or £9,000 a year. As a check, users valued the forgone visits at the area under the demand curve between 6,000 and 8,000 visits, worth £7.50 each on average, or 7.5 × 2,000 = £15,000, which is the £5,000 triangle plus the £5 old charge on each of the 2,000 visits. Set against a resource cost of £24,000, that leaves the same £9,000.

6. Interpret the result. On the Marshallian account the higher charge raises net benefit, because the visits it deters were worth £7.50 each on average to users and cost £12 each to provide. The same logic would favour a charge equal to the £12 cost, at which the last visit made is worth exactly what it costs. That holds only if the demand curve measures the value of the visits. If some deterred visits would have found disease early, the later treatment costs and health losses are missing from the area, and the gain could vanish or reverse. The £30,000 transfer falls on users, including those on low incomes, and an unweighted sum treats it as neutral. The charge doubles, so the price change is not small; if the income elasticity of demand for clinic visits is small, Willig's result still suggests the Marshallian figure is close to the Hicksian ones for this change.

Assumptions behind the Marshallian measure

The area is a welfare measure only under several assumptions. Users are taken to judge their own welfare, so the demand curve is read as their marginal valuation. Income effects are taken to be small, as Marshall assumed when he judged that changes in the marginal value of money would rarely matter in practice, or the area is accepted as an approximation in the sense of Willig's result. Other prices are held constant and only one price changes; when several prices change together, Ekelund and Hébert note that the Marshallian surplus is no longer uniquely defined and depends on the order in which prices are assumed to change, a problem first recognised by Hotelling.

Common misreadings in health economics

Several errors recur when consumer surplus is used to appraise health programmes. The whole area under a demand curve for a free service is reported as the value of the service, although it depends on an extrapolation to a choke price that data rarely support. The Marshallian area is used to value health gains or risk reductions, which have no ordinary demand curve. Consumer surplus is reported for a policy that changes several charges at once without stating the order in which the price changes are taken.

Sources

• Baicker K, Mullainathan S, Schwartzstein J. Behavioral hazard in health insurance. Quarterly Journal of Economics. 2015;130(4):1623-1667.
• Clarke PM. Cost-benefit analysis and mammographic screening: a travel cost approach. Journal of Health Economics. 1998;17(6):767-787.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford: Oxford University Press; 2015.
• Ekelund RB Jr, Hébert RF. Consumer surplus: the first hundred years. History of Political Economy. 1985;17(3):419-454.
• Marshall A. Principles of Economics. 8th ed. London: Macmillan; 1920 (first published 1890). Book III, Chapter VI.
• 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.