Signature
eps_arc = ((Q_i1 - Q_i0) / (Q_i1 + Q_i0)) / ((P_j1 - P_j0) / (P_j1 + P_j0))
| Inputs | Definition | Unit |
|---|---|---|
Q_i1 | Quantity of good i demanded after the price of j changes, such as generic packs dispensed per month | units per period |
Q_i0 | Quantity of good i demanded before the price of j changes | units per period |
P_j1 | Price of good j faced at the decision margin after the change, such as the branded co-payment | currency per unit |
P_j0 | Price of good j before the change, different from P_j1 | currency per unit |
eps_arc | Midpoint proportional change in the quantity of i divided by midpoint proportional change in the price of j; positive for substitutes, negative for complements | none |
|---|
Function
Cross-price elasticity of demand for one health care good with respect to another good's price
Maps a change in the price of good j to the proportional change in the quantity of good i demanded, holding income, preferences and other prices constant. A positive value marks substitutes, as with a branded medicine and its generic; a negative value marks complements; the order of the subscripts matters. Own-price elasticity is the case i = j. The notation follows the Cross-Price Elasticity article.
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Implementations
Excel
Arc cross-price elasticity from named quantities and prices
With the quantities of good i in QtyAfter and QtyBefore and the prices of good j in PriceAfter and PriceBefore, the formula returns the arc elasticity, held in ArcCrossElasticity.
=((QtyAfter-QtyBefore)/(QtyAfter+QtyBefore))/((PriceAfter-PriceBefore)/(PriceAfter+PriceBefore))
Assumptions
Only the price of good j changes between the two arc cross-price observations
Income, preferences, the price of good i and other prices are the same in both observations, so the change in Q_i is attributed to the price of j.
Cross-price elasticity price measured where the prescribing or buying decision is made
P_j is the price faced by whoever chooses, such as the patient's co-payment; for medicines the response passes through prescribers and pharmacists, so the elasticity summarises the whole chain.
Worked examples
Generic dispensing when the branded co-payment rises from 8 to 12 pounds
Generic packs rise from 6,000 to 7,000 a month when the branded co-payment rises from 8 to 12 pounds: 1,000 / 6,500, or 0.1538, divided by 4 / 10, or 0.40, gives about 0.38, as in the article.
Q_i1 = 7000; Q_i0 = 6000; P_j1 = 12; P_j0 = 8; eps_arc = 0.3846
Arc cross-price elasticity of generic dispensing read in the opposite direction
Treating the later month as the start (7,000 to 6,000 packs as the co-payment falls from 12 to 8 pounds) gives the same 0.3846, the reason for using midpoints.
Q_i1 = 6000; Q_i0 = 7000; P_j1 = 8; P_j0 = 12; eps_arc = 0.3846
Arc own-price elasticity of the branded medicine
With i equal to j, branded packs falling from 4,000 to 2,800 as the branded co-payment rises from 8 to 12 pounds give minus 0.3529 / 0.40, about minus 0.88, as in the article.
Q_i1 = 2800; Q_i0 = 4000; P_j1 = 12; P_j0 = 8; eps_arc = -0.8824
Common errors
Dividing by starting values instead of averages in an arc cross-price elasticity
Using 6,000 and 8 as bases gives 0.1667 / 0.50, or 0.33, and reversing the start gives 0.1429 / 0.3333, or 0.43; neither is the arc value of 0.38.
Assuming every branded pack lost becomes a generic pack gained
In the article's example 1,200 branded packs are lost but only 1,000 generic packs gained; holding total use fixed would put generic use at 7,200. A budget impact model that ignores the cross response altogether, keeping generic use at 6,000, overstates the payer's saving by 2,000 pounds a month, about 6 per cent of the true saving.
Sources
Arc elasticity defined relative to the average of the two values
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. Footnote 11 and Table 4: the arc elasticity of x with respect to y is the ratio of the percent change in x to the percent change in y, each computed relative to the average, (x2 minus x1)/((x2 + x1)/2); it converges to the standard elasticity as the two values get closer.
Positive cross-price elasticities between branded and generic versions
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. Table 6 (conditional elasticities between branded and generic versions of a drug, expenditure on that drug held constant): for drugs 1 and 3 own-price elasticities are negative and below minus 1 and cross-price elasticities are positive; Table 7: fairly large and significant unconditional elasticities between products and their generic substitutes; abstract and page 429: lower, often insignificant, cross-price elasticities between therapeutic substitutes; demand modelled in stages matching prescribing and dispensing.
Canonical Identity
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