Marshallian Demand Curve

Marshallian Demand Curve in Health Care and Health Tax Analysis

A Marshallian demand curve is the ordinary demand curve of consumer theory, traced out when a person with a fixed money budget picks the best bundle they can afford as the price of one good changes. Because money income stays the same while the price moves, a price rise leaves the person worse off, and the change in quantity mixes a substitution effect with an income effect. Health economists meet this curve whenever they estimate how use of care, medicines or taxed foods responds to the price people pay. It is the right curve for predicting uptake, purchases and health effects after a charge or a health tax, while its area is only an approximation to the money measures of welfare used in cost-benefit analysis.

Demand derived from a fixed money budget

The curve comes from utility maximisation subject to a budget constraint. With two goods, such as physiotherapy sessions and all other spending, the problem is as follows.

$$\max_{x_1, x_2} \; u(x_1, x_2) \quad \text{subject to} \quad p_1 x_1 + p_2 x_2 \le m$$

where $u$ is the person's utility function, $x_1$ is the quantity of the health good (sessions, prescription items or a taxed food), $x_2$ is a composite of all other goods, $p_1$ is the price the person pays for the health good at the point of use, $p_2$ is the price of the composite good and $m$ is money income.

The solution gives a demand function $x_1(p_1, p_2, m)$. Osborne and Rubinstein call the relation between the price of a good and its demand, with the other price and wealth fixed, the consumer's regular, or Marshallian, demand function. In health care the price on the axis is what the patient pays, such as a prescription charge, a co-payment or a self-funded fee, which can differ greatly from what the care costs the health system.

Marshall stated one general law of demand, that the amount demanded increases with a fall in price. He treated the marginal utility of money as fixed while a person's material resources are unchanged, whereas the modern curve holds money income fixed and lets its purchasing power change as the price moves.

Substitution and income effects in the slope

The slope of the Marshallian curve can be split into two parts with the Slutsky equation. The first part is the response along an indifference curve, holding utility fixed, which is the slope of the compensated (Hicksian) demand curve. The second part is the income effect of the price change, as shown below.

$$\frac{\partial x_1(p_1, p_2, m)}{\partial p_1} = \frac{\partial h_1(p_1, p_2, u)}{\partial p_1} - x_1 \frac{\partial x_1(p_1, p_2, m)}{\partial m}$$

where $h_1(p_1, p_2, u)$ is compensated demand for the health good at utility $u$, set equal to the utility reached at $(p_1, p_2, m)$, the first term on the right is the substitution effect and the second is the income effect, the response of demand to income multiplied by the quantity bought. Autor's lecture notes state that, with a diminishing marginal rate of substitution, the substitution effect is always negative when utility is held constant, while the income effect can have either sign.

For a normal good the two effects work in the same direction, so the Marshallian curve is flatter than the compensated curve through the same point, meaning that quantity responds more to price. For an inferior good the income effect works against the substitution effect. A Giffen good, whose Marshallian demand rises with its own price, needs the income effect to dominate. Marshall attributed the case to Giffen, citing poorer labouring families who bought more bread when its price rose, and judged such cases rare.

Multiplying the Slutsky equation through by $p_1 / x_1$ gives the elasticity form, which is often easier to use with published estimates.

$$\varepsilon_{11} = \varepsilon^{h}_{11} - s_1 \eta_1$$

where $\varepsilon_{11}$ is the Marshallian (uncompensated) own-price elasticity, $\varepsilon^{h}_{11}$ is the compensated own-price elasticity, $s_1 = p_1 x_1 / m$ is the budget share of the health good and $\eta_1$ is its income elasticity of demand. The gap grows with the budget share, so the two elasticities nearly coincide for a single small charge and can differ substantially for self-funded treatment that absorbs a large share of a low income.

What estimated demand curves for care and health taxes measure

Demand curves estimated from observed purchases and prices are Marshallian unless the study adjusts for income. People facing a new charge or a health tax are not compensated, so the observed change in quantity contains both effects. Aron-Dine, Einav and Finkelstein draw the question behind the RAND Health Insurance Experiment in this way, with health care spending chosen at the tangency between an indifference curve and the budget set that a level of cover creates.

Health tax studies show why the label matters. Di Cosmo and Tiezzi estimated an implicit Marshallian demand system for 16 food groups in Italy to simulate a tax on saturated fat. They used the uncompensated elasticities to predict how purchases, and through them weight, would change, used compensated elasticities to describe substitution between foods net of income effects, and measured the welfare cost to consumers by compensating variation, about EUR 13.40 a month for single adults at mean income and regressive relative to income. Swapping one elasticity for the other would misstate either the predicted change in purchases and weight or the substitution between foods.

Not every published elasticity is Marshallian in the full sense. In their model of a 20 per cent tax on sugar sweetened drinks in the UK, Briggs and colleagues estimated elasticities within separate demand systems that were conditional on spending on each group of goods staying constant, and then calculated unconditional elasticities that let spending on a group change when one of its prices changes. An elasticity taken into a model should therefore come with a note of what the original study held fixed.

The Marshallian curve also assumes that price is the only thing rationing use. Where care is free at the point of use, observed use at a zero money price reflects waiting times, travel and clinical gatekeeping as well as demand, so a single observation at a zero charge is not a point on the curve that a money price would trace.

Welfare readings and the welfarist debate

The area to the left of the Marshallian curve between two prices is Marshallian consumer surplus. The exact money measures of a price change, compensating variation and equivalent variation, are areas to the left of compensated demand curves at the old and new utility levels, so the Marshallian area lies between them when there is an income effect.

Roy's identity links the Marshallian curve to the indirect utility function, as follows.

$$x_1(p_1, p_2, m) = -\frac{\partial V(p_1, p_2, m) / \partial p_1}{\partial V(p_1, p_2, m) / \partial m}$$

where $V(p_1, p_2, m)$ is the indirect utility function, the highest utility attainable at prices $p_1$ and $p_2$ and income $m$, and the denominator is the marginal utility of income. A demand system derived from a utility or cost function therefore carries the information needed for the exact measures, which is why a study such as the Italian one can report compensating variation from an estimated demand system.

The use of these curves in valuation rests on welfarism, which judges outcomes by individual utility. Brouwer and colleagues contrast cost-benefit analysis, which uses effective demand and willingness to pay, with cost-utility analysis, where something like need is used instead. In the extra-welfarist approach, which they see archetypically in cost-utility analysis, outcomes are measured in health, often in QALYs, and the Marshallian curve plays no part in valuing them. It still enters such models as a behavioural input, predicting uptake after a charge or purchases after a health tax.

Worked example of a price rise for self-funded physiotherapy

The figures below are illustrative. A person on a low income pays for physiotherapy sessions out of pocket, with a monthly budget of GBP 1,200, and always spends a quarter of the budget on sessions, which corresponds to Cobb-Douglas utility with $\alpha = 0.25$. Osborne and Rubinstein use this fixed-share demand as an example. The price of a session rises to GBP 40 from GBP 30, with the prices of other goods unchanged.

1. Quantities on the Marshallian curve. Demand at each price is calculated as follows.

$$x_1 = \frac{\alpha m}{p_1}, \qquad x_1^0 = \frac{0.25 \times 1200}{30} = 10, \qquad x_1^1 = \frac{0.25 \times 1200}{40} = 7.5$$

where $\alpha$ is the budget share spent on sessions, and $x_1^0$ and $x_1^1$ are sessions a month at the old and new prices.

2. Slutsky compensation. Income is raised just enough to buy the original 10 sessions at the new price, and demand at that income is found, as follows.

$$m' = 1200 + (40 - 30) \times 10 = 1300, \qquad x_1' = \frac{0.25 \times 1300}{40} = 8.125$$

where $m'$ is the compensated income and $x_1'$ is sessions a month at the new price with that income. Osborne and Rubinstein show that a consumer compensated in this way buys no more of a good whose price has risen.

3. Splitting the change. The Slutsky substitution effect is $x_1' - x_1^0 = 8.125 - 10 = -1.875$ sessions and the income effect is $x_1^1 - x_1' = 7.5 - 8.125 = -0.625$ sessions. The two add to the total fall of 2.5 sessions, and the income effect is a quarter of it, equal to the budget share. Holding utility fixed instead, which needs GBP 89.48 rather than GBP 100 of compensation, gives 8.06 sessions, a substitution effect of $-1.94$ and an income effect of $-0.56$.

4. Elasticities. With fixed-share demand the Marshallian own-price elasticity is $-1$ and the income elasticity is 1, so the compensated elasticity is found from the elasticity form of the Slutsky equation.

$$\varepsilon^{h}_{11} = \varepsilon_{11} + s_1 \eta_1 = -1 + 0.25 \times 1 = -0.75$$

where the terms are as defined earlier. An analyst who estimated $-1$ from data and treated it as a pure substitution response would overstate substitution by a third.

5. Money measures of the loss. The Marshallian area integrates demand over the price change. The Cobb-Douglas expenditure function scales with $p_1^{\alpha}$, which gives the Hicksian measures, here written as positive amounts for a loss.

$$\Delta CS = \alpha m \ln\frac{40}{30} = 300 \times 0.287682 = 86.30$$

where $\Delta CS$ is the loss of Marshallian consumer surplus in pounds a month and $\alpha m$ is the GBP 300 spent on sessions. The Hicksian measures follow below.

$$CV = 1200 \times ((40/30)^{0.25} - 1) = 89.48, \qquad EV = 1200 \times (1 - (30/40)^{0.25}) = 83.27$$

where $CV$ is the compensation that would restore the original utility after the rise and $EV$ is the income loss at the old price that would leave the person as badly off as the rise does. The Marshallian loss lies between the two, about GBP 3 from each.

6. A small budget share. If the same person spent 2 per cent of the budget on sessions, use would fall to 0.6 sessions from 0.8, and the income effect would be only 0.004 sessions of that fall. The three measures become $\Delta CS = 24 \times 0.287682 = 6.90$, $CV = 1200 \times ((40/30)^{0.02} - 1) = 6.92$ and $EV = 1200 \times (1 - (30/40)^{0.02}) = 6.88$, each about 0.3 per cent away from the Marshallian figure.

7. Interpretation. For predicting use after the price rise, the Marshallian quantities in step 1 are the relevant ones, because real users are not compensated. For valuing the loss, the Marshallian area is close to the exact measures when the budget share is small and less close when out-of-pocket care takes a large share of a low income.

Marshallian, compensated and Engel curves compared

Nearby curves answer different questions. The Marshallian curve holds money income fixed and the compensated curve holds utility fixed, while the Engel function, in Osborne and Rubinstein's terms, traces demand against income at fixed prices. The price elasticity of demand summarises one of these curves at a point, so it should be reported with the curve it comes from.

Common misreadings of the Marshallian curve in health economics

Several errors recur in health applications. Treating the area as an exact willingness to pay is reasonable for small charges, less so for treatments that take a large share of income. Taking an uncompensated elasticity as the substitution response overstates, for a normal good, how far people switch between goods, which matters when a health tax is designed to shift purchases towards healthier foods.

A further error is to read a downward-sloping curve as a theorem. Osborne and Rubinstein note that demand under the standard assumptions on preferences need not fall as price rises, and that the model assumes prices convey no information about quality, which may fail where patients judge a medicine by its price.

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. Author manuscript: https://pmc.ncbi.nlm.nih.gov/articles/PMC3943162/
• Autor D. Lecture 5, Consumer theory and the carte blanche principle, and Lecture 9, Giffen goods and subsistence consumption. 14.03/003 Microeconomic Theory and Public Policy, Fall 2025. MIT OpenCourseWare; 2025. https://ocw.mit.edu/courses/14-03-microeconomic-theory-and-public-policy-fall-2025/
• Briggs ADM, Mytton OT, Kehlbacher A, Tiffin R, Rayner M, Scarborough P. Overall and income specific effect on prevalence of overweight and obesity of 20% sugar sweetened drink tax in UK: econometric and comparative risk assessment modelling study. BMJ. 2013;347:f6189. doi:10.1136/bmj.f6189
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. doi:10.1016/j.jhealeco.2007.07.003
• di Cosmo V, Tiezzi S. Fat vs. sugar: the case for a saturated fat tax in Italy. Health Economics. 2025;34(4):727-740. doi:10.1002/hec.4933
• Marshall A. Principles of Economics. 8th ed. London: Macmillan; 1920 (first published 1890). Book III, Chapters III and VI.
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded 2nd ed. Cambridge: Open Book Publishers; 2023. Chapter 5. doi:10.11647/OBP.0362