Cross-Price Elasticity

Cross-Price Elasticity Between Health Care Markets

Cross-price elasticity links the price of one product or service to the use of another. In health economics it is the measure behind questions such as how many prescriptions move to a generic when a branded medicine becomes dearer to the patient, whether a tobacco tax pushes smokers towards cheaper products, and whether higher charges for primary care shift use into hospitals. Checked evidence on it comes from medicines, tobacco, alcohol and a randomised insurance experiment, and it feeds into budget impact calculations when a co-payment changes. Own-price responses are measured by the price elasticity of demand, and the underlying theory is that of consumer choice.

The formula and what its sign means

A cross-price elasticity always involves two goods and a direction, namely the good whose use is measured and the good whose price changes. Swapping them gives a different number, so the order of the subscripts matters. For a smooth demand function, the point elasticity is defined as follows.

$$\varepsilon_{ij} = \frac{\partial Q_i}{\partial P_j} \cdot \frac{P_j}{Q_i}$$

where $\varepsilon_{ij}$ is the cross-price elasticity of demand for good $i$ with respect to the price of good $j$, $Q_i$ is the quantity of good $i$ demanded, $P_j$ is the price of good $j$, and $\partial Q_i / \partial P_j$ is the change in demand for $i$ per unit change in the price of $j$, with income, preferences and other prices held constant.

A positive value means the goods are substitutes, so when $j$ becomes dearer, people use more of $i$, as when patients move from a branded medicine to its generic. A negative value means they are complements, so a dearer $j$ also reduces use of $i$, as when a charge for a diagnostic visit reduces use of the treatment that follows it. A value near zero means near independence at the margin studied, which says nothing about whether the two are clinically related. With only two observations, the midpoint (arc) version gives the same answer whichever observation is treated as the start. It is calculated as follows.

$$\varepsilon^{arc}_{ij} = \frac{(Q_{i,1} - Q_{i,0}) / \bar{Q}_i}{(P_{j,1} - P_{j,0}) / \bar{P}_j}$$

where $Q_{i,0}$ and $Q_{i,1}$ are the quantities of good $i$ before and after the change, $P_{j,0}$ and $P_{j,1}$ are the prices of good $j$ before and after, and $\bar{Q}_i$ and $\bar{P}_j$ are the averages of the two quantities and the two prices.

Gross and compensated effects under Slutsky symmetry

Elasticities estimated from market or claims data are usually gross (uncompensated) effects. They mix a pure substitution response with an income effect, because a dearer $j$ also leaves the household poorer. The Slutsky equation separates the two, and in elasticity form it reads as follows.

$$\varepsilon_{ij} = \varepsilon^{c}_{ij} - w_j \eta_i$$

where $\varepsilon_{ij}$ is the gross cross-price elasticity, $\varepsilon^{c}_{ij}$ is the compensated cross-price elasticity (utility held constant), $w_j = P_j Q_j / M$ is the share of the budget $M$ spent on good $j$, and $\eta_i$ is the income elasticity of demand for good $i$.

Slutsky symmetry applies to compensated effects only, since the matrix of price derivatives of compensated demand is symmetric. Converted into elasticities, the equality is weighted by budget shares, as follows.

$$w_i \varepsilon^{c}_{ij} = w_j \varepsilon^{c}_{ji}$$

where $w_i$ is the budget share of good $i$ and the other terms are as defined above. If $i$ is a net substitute for $j$, then $j$ is a net substitute for $i$, but the two compensated elasticities are equal only when the budget shares are equal, so a product with a small share can respond strongly to the price of a product with a large share while the reverse response is small. In principle the gross elasticities that most studies report can differ even in sign, because the income effects of the two goods differ. In practice, a single health service or medicine co-payment usually takes a very small share of the budget, so the income term $w_j \eta_i$ is close to zero and gross and compensated cross elasticities nearly coincide for most care prices. Demand-system estimators often impose symmetry as a restriction, and a reported matrix should say whether it did.

Substitution between branded and generic medicines through prescribers and pharmacists

Medicines are the clearest health care case of close substitutes, but the price response passes through several decision makers. The prescriber chooses a molecule and the pharmacist may choose which version to dispense, so a cross-price elasticity for a medicine summarises that whole chain, not the patient alone.

Ellison, Cockburn, Griliches and Hausman studied four cephalosporin antibiotics in the United States in the late 1980s, three of which lost patent protection and saw generic entry during the study period. Modelling demand in stages that match prescribing and dispensing, and using wholesale prices and quantities, not the prices patients paid, they found fairly high elasticities between branded and generic versions of the same drug. Between different drugs that treat many of the same conditions, their estimates section reports lower, often insignificant, cross-price elasticities, while their conclusion describes demand elasticity there as smaller but often significant. They also noted that by 1989 every US state had a law at least allowing pharmacists to dispense a cheaper generic unless the physician prohibited substitution. Their results suggest some price sensitivity at both the prescribing and dispensing stages, with stronger evidence at dispensing.

Tobacco and alcohol taxes and switching between products

The effect of a health tax on total consumption depends on where users go when one product becomes dearer. The International Agency for Research on Cancer (IARC) handbook on tax and price policies for tobacco control finds that the limited survey-based research suggests that, at least in high-income countries, tobacco products are generally substitutes. It reports US evidence that higher cigarette taxes encourage some substitution to smokeless tobacco, with positive cross-tax elasticities, while smokeless tobacco taxes were generally insignificant in models of cigarette smoking. One study from India, by John, gave mostly negative though rarely significant cross-price estimates, suggesting complementarity. For tax design, the handbook warns that tobacco taxation can be made ineffective if substitutable products remain taxed at a much lower rate, and concludes that a similar tax level across products would reduce the incentive to substitute.

Meng and colleagues estimated own- and cross-price elasticities for ten UK alcohol categories (five beverage types, each bought in shops or in pubs and bars) from the Living Costs and Food Survey for 2001/2 to 2009, for appraising minimum unit pricing and taxation. The cross-price elasticities were smaller than the own-price ones and mixed in sign, and only 6 of the 90 were statistically significant, 5 of them positive. Across the two markets, the IARC handbook reports a UK study that found no significantly complementary behaviour between cigarettes and alcoholic drinks.

Outpatient, inpatient and prescription drug use under cost sharing

Cost sharing usually applies to some services and not others, so it changes relative prices across the health system. Whether care discouraged in one setting reappears in another is a cross-price question that a single figure such as the price elasticity of healthcare cannot answer, and two well-known studies point in different directions.

The RAND Health Insurance Experiment included an individual deductible plan with free inpatient care and 95% coinsurance for outpatient services up to an annual cap. Compared with free care, outpatient expenses fell by about one-third, the fall in inpatient expenses was small and not significant, and the admission rate lay roughly midway between the free plan and the family coinsurance plans, which Manning and colleagues read as a nontrivial cross-price elasticity between inpatient and outpatient services. Chandra, Gruber and McKnight later summarised this result as hospital use falling rather than rising on that plan, suggesting that physician and hospital care were complements.

Chandra, Gruber and McKnight themselves studied retired California public employees, whose supplemental insurance wraps around Medicare, after co-payments for physician visits and prescription drugs rose. Unlike the RAND experiment, they found that higher hospital use offset part of the savings, mostly among people with chronic disease. They frame the risk as downstream hospitalisations caused by reduced access to beneficial care, which is a cross effect running through health, not a direct switch between services. The savings fell mainly to the supplemental insurer and the hospital costs mainly to Medicare, so an analysis confined to one payer would have missed the offset.

Worked example of a higher branded co-payment and a shift to the generic

The figures are illustrative. A payer covers a branded medicine with a list price of £28 per pack and its generic at £4 per pack. Patients pay £2 per pack for the generic, and the payer raises the branded co-payment from £8 to £12. Monthly dispensing is as follows.

1. Midpoint change in generic use.

$$\frac{7000 - 6000}{(6000 + 7000)/2} = \frac{1000}{6500} = 0.1538$$

where 7,000 and 6,000 are generic packs after and before the change and 6,500 is their midpoint.

2. Midpoint change in the branded co-payment.

$$\frac{12 - 8}{(8 + 12)/2} = \frac{4}{10} = 0.40$$

where 12 and 8 are the branded co-payments in pounds and 10 is their midpoint.

3. Arc cross-price elasticity.

$$\varepsilon^{arc}_{GB} = \frac{0.1538}{0.40} \approx 0.38$$

where $\varepsilon^{arc}_{GB}$ is the arc cross-price elasticity of generic demand with respect to the branded co-payment. The positive value shows the two versions behaving as substitutes. Over this range, a 1% rise in the branded co-payment goes with about a 0.38% rise in generic dispensing.

4. Branded own-price response. Branded dispensing changed by $(2800 - 4000)/3400 = -0.3529$, an arc own-price elasticity of $-0.3529 / 0.40 \approx -0.88$. Of the 1,200 branded packs lost, 1,000 were replaced by generic packs and 200 were not replaced, so total dispensing fell from 10,000 to 9,800 packs.

5. Budget impact for the payer. The payer's cost per pack is list price less co-payment, which is £20 for the branded version before, £16 after, and £2 for the generic throughout. Monthly payer spending falls from £92,000 (4,000 × £20 plus 6,000 × £2) to £58,800 (2,800 × £16 plus 7,000 × £2), a saving of £33,200. An analysis that modelled the own-price response but held generic use at 6,000 packs would estimate spending of £56,800 and a saving of £35,200, overstating the saving by £2,000, about 6% of the true saving. Of the £33,200, £3,600 is a rise in patients' co-payments (from £44,000 to £47,600 a month), and only £29,600 is a fall in spending at list prices (from £136,000 to £106,400). The 200 unreplaced packs matter too if they reflect lower adherence.

Budget impact analysis and policy simulation

A budget impact analysis turns on how the mix of treatments changes. The ISPOR good practice task force lists among its key elements the current mix of treatments, the expected mix after the new intervention, the cost of each mix and any change in condition-related costs. When relative prices drive the change in mix, as with generic entry, a new co-payment tier or a tax, cross-price elasticities are one way to estimate it.

Models covering several products extend the single elasticity to a matrix. For small changes in several prices at once, the approximation is as follows.

$$\frac{\Delta Q_i}{Q_i} \approx \sum_{j} \varepsilon_{ij} \frac{\Delta P_j}{P_j}$$

where $\Delta Q_i / Q_i$ is the proportional change in demand for product $i$, $\Delta P_j / P_j$ is the proportional change in the price of product $j$, and the sum runs over all products including $i$, so that $\varepsilon_{ii}$ is the own-price elasticity. Meng and colleagues describe earlier UK work that estimated such a matrix for 16 beverage categories to model minimum unit pricing; a matrix can be applied to aggregate sales or to individual records in a microsimulation. Because most cross terms are imprecise, whether to use the point estimates or set insignificant terms to zero is a structural choice for scenario and probabilistic sensitivity analysis.

Estimation choices that change the answer

Cross-price elasticities need independent movement in relative prices, and estimates become unstable when prices move together; the IARC handbook gives a Dutch example in which negative cross-price elasticities between manufactured and hand-rolled cigarettes were attributed to collinearity between their prices. The price must be the one faced at the decision margin. A cross-tax elasticity is not a cross-price elasticity, since it equals $\varepsilon_{ij} \cdot \partial \ln P_j / \partial \ln T_j$, where $T_j$ is the tax on good $j$, so converting it needs the pass-through rate and the tax share of the price. With a specific tax passed through one for one, the factor is $T_j / P_j$, which is below one, and the two elasticities are equal only if a 1% rise in the tax raises the price by 1%. Restrictions such as symmetry or grouping products into budget stages shape the estimates, and Ellison and colleagues note that such structure is often imposed in order to estimate cross-price elasticities at all. The time horizon must also be long enough to capture effects, such as hospital offsets, that appear only after health has changed.

Cross-price elasticity beside own-price and income elasticity

Own-price elasticity describes the response to a good's own price and income elasticity the response to income; cross-price elasticity describes the response to the price of something else. An offset effect is a cross-price effect measured across settings, budgets and time. Therapeutic substitution between comparators in an economic evaluation is a clinical choice, and it becomes a cross-price question only when it responds to relative prices.

Common misreadings of cross-price estimates

Four errors recur when cross-price elasticities enter health economic work. Each is listed with the reason it fails.

• Reading a small value as clinical independence. Ellison and colleagues found smaller cross-price elasticities between drugs that treat many of the same conditions than between generic versions, with some price sensitivity at both the prescribing and dispensing stages.
• Assuming the two directions are equal. Only compensated effects are symmetric, and then weighted by budget shares; reported gross effects can differ in size and sign.
• Treating every unit lost by one product as gained by another. In the worked example, one-sixth of the lost branded packs were not replaced.
• Transferring estimates between settings or payers. Tobacco evidence differs between high-income countries and India, and the RAND and Chandra studies differ on outpatient and inpatient care.

Sources

• 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)
• Ellison SF, Cockburn I, Griliches Z, Hausman J. Characteristics of demand for pharmaceutical products: an examination of four cephalosporins. RAND Journal of Economics. 1997;28(3):426-446. https://economics.mit.edu/sites/default/files/publications/Characteristics%20of%20Demand.pdf
• International Agency for Research on Cancer. Effectiveness of Tax and Price Policies for Tobacco Control. IARC Handbooks of Cancer Prevention, Tobacco Control, Volume 14. Lyon: IARC; 2011. Chapters 2, 4, 5 and 11. https://publications.iarc.who.int/Book-And-Report-Series/Iarc-Handbooks-Of-Cancer-Prevention/Effectiveness-Of-Tax-And-Price-Policies-For-Tobacco-Control-2011
• Manning WG, Newhouse JP, Duan N, Keeler E, Benjamin B, Leibowitz A, Marquis MS, Zwanziger J. Health Insurance and the Demand for Medical Care: Evidence from a Randomized Experiment. R-3476-HHS. Santa Monica, CA: RAND Corporation; 1988. Summary and Section IV. https://www.rand.org/pubs/reports/R3476.html
• Meng Y, Brennan A, Purshouse R, Hill-McManus D, Angus C, Holmes J, Meier PS. Estimation of own and cross price elasticities of alcohol demand in the UK: a pseudo-panel approach using the Living Costs and Food Survey 2001-2009. Journal of Health Economics. 2014;34:96-103. doi:10.1016/j.jhealeco.2013.12.006
• Sullivan SD, Mauskopf JA, Augustovski F, et al. Budget impact analysis: principles of good practice. Report of the ISPOR 2012 Budget Impact Analysis Good Practice II Task Force. Value in Health. 2014;17(1):5-14. doi:10.1016/j.jval.2013.08.2291
• Wolitzky A. Lectures 3-4: consumer theory. 14.121 Microeconomic Theory I, Fall 2015. Cambridge, MA: MIT OpenCourseWare; 2015. https://ocw.mit.edu/courses/14-121-microeconomic-theory-i-fall-2015/