Indifference Curve

Indifference Curve in Health Economics

An indifference curve is a line on a two-axis diagram passing through every combination of two quantities that a person ranks as equally good. In health economics the two quantities might be life years and quality of life, or the health of two population groups, and the slope of the line shows how much of one would be given up for a little more of the other. Time trade-off and standard gamble answers are points of indifference of this kind, and the shape of the curves decides whether adding up QALYs describes what patients or decision makers prefer.

Indifference sets built from a ranking of options

Osborne and Rubinstein model preferences as a complete and transitive ranking of alternatives. The indifference relation holds when each of two alternatives is at least as desirable as the other, and it divides the alternatives into disjoint classes called indifference sets, which for bundles of two goods are sometimes called indifference curves. When a utility function represents the ranking, each set is one contour of that function.

$$I(a) = \{ x \in X : x \sim a \} = \{ x \in X : u(x) = u(a) \}$$

where $I(a)$ is the indifference set through the bundle $a$, $X$ is the set of all bundles, $x \sim a$ means the person is indifferent between $x$ and $a$, and $u$ is any utility function representing the ranking.

Because the sets form a partition, one person's curves never cross. The OpenStax authors present the curves as a way of describing preferences without putting numbers on utility, with higher curves standing for preferred bundles.

$$\left. \frac{d x_2}{d x_1} \right|_{u(x) = u(a)} = -\frac{\partial u / \partial x_1}{\partial u / \partial x_2}$$

where $x_1$ and $x_2$ are the quantities of the two goods, the left side is the slope of the curve through $a$, and the right side is minus the ratio of the two marginal utilities. That ratio is the marginal rate of substitution, whose estimation from choice data has its own page.

Osborne and Rubinstein call preferences convex when every bundle between two bundles is at least as good as the worse one. The OpenStax authors describe the usual curve as downward sloping, steeper on the left and flatter on the right, and link the shape to diminishing marginal utility. A patient who already attends many physiotherapy sessions would therefore give up little other spending for one more.

Shapes of indifference curves in health valuation

Osborne and Rubinstein give several shapes besides the smooth convex case. Each has a health counterpart.

With a constant trade-off the curves are straight lines with slope $-v_1/v_2$, where $v_1$ and $v_2$ are fixed values per unit of each good. A decision maker who adds unweighted QALYs across two patients has such curves with slope minus 1. With complementary goods the preference is represented by the smaller of the two quantities and the curves are L-shaped, as for a medicine that is useless without its companion diagnostic test. When the person prefers bundles closer to an ideal bundle, the curves ring that point, as for a drug dose above which more causes harm.

Lexicographic preferences give priority to one good and use the second only to break ties. Osborne and Rubinstein show that each indifference set is then a single point and that no utility function represents the preference. Bleichrodt reports that 5 of 30 answers to time trade-off questions posed by Pliskin and colleagues gave up no life years at all. Refusing every trade of length for quality looks lexicographic, although the state may simply have been judged close to full health.

The budget line and money measures of welfare

The OpenStax authors show the best affordable choice as the single point where the budget constraint just touches the highest attainable indifference curve. They split a price change by drawing a line parallel to the new budget line and tangent to the original curve, so the move along the original curve is the substitution effect and the shift between curves is the income effect. The consumer choice and utility maximisation pages work through co-payment cases with numbers.

The income that would return the person to the original curve after a change is the compensating variation, and the income change at the old prices that reaches the new curve is the equivalent variation. These distances between curves are what willingness to pay and willingness to accept questions try to measure in cost-benefit analysis.

Indifference in time trade-off and standard gamble valuation

Health state valuation rests on stated indifference. Lipman, Brouwer and Attema describe the time trade-off weight as coming from the duration in full health that leaves the respondent indifferent to a longer duration in the impaired state, and the standard gamble weight from the probability at which a certain outcome and a risky prospect become equally attractive. Under expected utility and the linear QALY model, these weights are the ratio of the durations and the probability. Each answer therefore places two health profiles on the same indifference curve drawn with quality of life on one axis and life years on the other, and the linear QALY model makes every such curve a rectangular hyperbola.

$$q \times T = k, \qquad -\frac{dT}{dq} = \frac{T}{q}$$

where $q$ is the quality weight of a constant health state, $T$ is the number of life years spent in it, $k$ is the number of QALYs, fixed along one curve, and $T/q$ is the number of life years the model says would be given up per unit gain in quality.

As an illustration, the 8 QALY curve passes through 8 years at full health, 10 years at 0.8 and 16 years at 0.5, since 10 × 0.8 = 8 and 16 × 0.5 = 8. Its slope is 10 / 0.8 = 12.5 at the second point and 16 / 0.5 = 32 at the third, while a discrete move from 0.8 to 0.9 gives up 1.11 years, since 8 / 0.9 = 8.89, and a move from 0.5 to 0.6 gives up 2.67 years, since 8 / 0.6 = 13.33. Bleichrodt lists among the model's conditions a constant proportional trade-off, under which the proportion of remaining life years given up for a given improvement in health does not depend on how many years remain. The evidence against this condition is reported on the marginal rate of substitution page. When the true curves are not hyperbolas, a cost-utility analysis that sums QALYs can rank two treatments differently from the people whose answers produced the weights.

Social indifference curves over the health of two groups

The same geometry describes social preferences when the axes show the health of two groups. Dolan and Tsuchiya show that the contours of a social welfare function for two equally weighted groups are straight lines with a gradient of minus 1 under the utilitarian form and approach the maximin form as inequality aversion rises without limit.

In a tutorial on distributional cost-effectiveness analysis, Asaria, Griffin and Cookson explain that inequality aversion parameters describe how much total health a decision maker would sacrifice for a more equal distribution. They express the parameter through the equally distributed equivalent (EDE) of the Atkinson index and report an estimated Atkinson parameter of about 10.95 from work with the general public in England. For two groups of equal size the EDE and the slope of the social indifference curve are as follows.

$$h_{EDE} = \left[ \frac{1}{2} \left( h_A^{1-\varepsilon} + h_B^{1-\varepsilon} \right) \right]^{\frac{1}{1-\varepsilon}}, \qquad -\frac{d h_B}{d h_A} = \left( \frac{h_B}{h_A} \right)^{\varepsilon}$$

where $h_A$ and $h_B$ are average lifetime health in groups A and B, $\varepsilon$ is the inequality aversion parameter, $h_{EDE}$ is the equally distributed equivalent, which is the geometric mean $\sqrt{h_A h_B}$ when $\varepsilon = 1$, and $-dh_B/dh_A$ is the health of group B that could be given up for one more unit in group A with welfare unchanged. The EDE is where the social curve through a distribution meets the line of equal health. Against a health production possibility frontier, the highest attainable social curve picks one efficient allocation, as a budget line does for a consumer.

Worked example with two programmes and rising inequality aversion

All numbers in this example are illustrative. A health board can fund one of two equal-cost programmes for two groups of equal size. Group A, the more deprived, averages 60 lifetime QALYs and group B averages 80. Programme 1 adds 2 QALYs per person in group A, giving 62 and 80, and programme 2 adds 3 per person in group B, giving 60 and 83.

1. No inequality aversion. With $\varepsilon = 0$ the EDE is the mean, (62 + 80) / 2 = 71.0 for programme 1 and (60 + 83) / 2 = 71.5 for programme 2, so programme 2 is preferred, as under unweighted QALY maximisation.

2. Moderate aversion. With $\varepsilon = 1$ the EDE is the geometric mean, $\sqrt{4{,}960} = 70.43$ for programme 1 and $\sqrt{4{,}980} = 70.57$ for programme 2, so programme 2 still wins.

3. Stronger aversion. With $\varepsilon = 2$ the EDE is the harmonic mean, 9,920 / 142 = 69.86 for programme 1 and 9,960 / 143 = 69.65 for programme 2, so programme 1 is preferred.

4. Reading the slopes. Programme 2 swaps 2 QALYs in group A for 3 in group B, a ratio of 1.5. At the starting point the curve's slope is $(80/60)^{\varepsilon}$, which is 1.00, 1.33 and 1.78 at the three values above. Programme 2 wins while the slope is below 1.5, which holds until $\varepsilon = \ln 1.5 / \ln(4/3) = 0.405 / 0.288 = 1.41$. Solving for equal EDEs gives an exact switch point of about 1.40.

5. The estimate for England. At $\varepsilon = 10.95$ the slope is about 23, and the EDEs are 65.97 for programme 1 and 64.08 for programme 2.

6. Interpretation. The programmes buy similar amounts of health, so the choice turns on the curvature of the social curves. Ranking by total QALYs chooses $\varepsilon = 0$, and the equity-efficiency trade-off page covers the health a stronger weighting gives up.

Whose curves count in welfarist and extra-welfarist analysis

Whose curves count divides the two traditions of welfare economics in health. Brouwer and colleagues quote Hurley's summary that the welfarist tenets require any policy to be judged solely by the resulting utilities of individuals, as assessed by those individuals. The yardstick is then each person's own indifference map, and money distances between curves are the benefits in a cost-benefit analysis.

Brouwer and colleagues describe extra-welfarism as supplementing utility with other characteristics of individuals and note that many health economists treat health itself as the maximand of a health care system. The curves that matter are then social contours over health, elicited from citizens or set by the decision maker.

Misreadings and limits of indifference curves

Indifference curves describe a ranking and nothing more, and most errors arise when they are asked to carry information they do not hold. The common ones in health applications are these.

1. Reading the gap between two curves as an amount of utility. Any utility function with the same contours represents the same preferences, so the curves show order and not intensity.
2. Assuming trades run the same way in both directions. Knetsch found that 89% of students given a coffee mug kept it rather than take a chocolate bar, while only 10% of those given the chocolate traded it for a mug, against 56% choosing the mug when neither was owned. He links this asymmetry to the gap between what people demand to give up a good and what they will pay for it, the same gap as between willingness to accept and willingness to pay for health.
3. Taking smooth convex curves for granted. Non-trading time trade-off answers, fixed-proportion treatments and doses with an ideal level all break the textbook shape.
4. Treating unweighted QALY sums as free of value judgements. Adding QALYs imposes straight-line social curves with a gradient of minus 1, the inequality-neutral special case in Dolan and Tsuchiya's family.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. https://doi.org/10.1177/0272989X15583266
• Bleichrodt H. QALYs and HYEs: under what conditions are they equivalent? Journal of Health Economics. 1995;14(1):17-37. https://doi.org/10.1016/0167-6296(94)00033-Z
• 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
• Dolan P, Tsuchiya A. The social welfare function and individual responsibility: some theoretical issues and empirical evidence from health. HEDS Discussion Paper 03/01. University of Sheffield; 2003. https://eprints.whiterose.ac.uk/id/eprint/11035/
• Greenlaw SA, Shapiro D, MacDonald D. Principles of Economics 3e. OpenStax; 2022. Appendix B, Indifference Curves. https://openstax.org/books/principles-economics-3e/pages/b-indifference-curves
• Knetsch JL. The endowment effect and evidence of nonreversible indifference curves. American Economic Review. 1989;79(5):1277-1284. https://cyber.harvard.edu/bridge/LawEconomics/knetch1.txt.htm
• Lipman SA, Brouwer WBF, Attema AE. QALYs without bias? Nonparametric correction of time trade-off and standard gamble weights based on prospect theory. Health Economics. 2019;28(7):843-854. https://doi.org/10.1002/hec.3895
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded 2nd ed. Open Book Publishers; 2023. Chapters 1 and 4. https://doi.org/10.11647/OBP.0362