Signature
eps_ij = b_ij * P_j / Q_i
| Inputs | Definition | Unit |
|---|---|---|
b_ij | Change in the quantity of i per unit change in the price of j, positive for substitutes | units of i per currency unit |
P_j | Price of good j at which the elasticity is computed | currency per unit |
Q_i | Quantity of good i demanded at that price | units per period |
eps_ij | Elasticity of demand for good i with respect to the price of good j at the point (P_j, Q_i) | 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
Point cross-price elasticity from a named slope, price and quantity
With the slope in CrossSlope and the price and quantity in PriceJ and QtyI, the formula returns the point elasticity, held in PointCrossElasticity. The slope from two observations is =(QtyAfter-QtyBefore)/(PriceAfter-PriceBefore).
=CrossSlope*PriceJ/QtyI
Assumptions
Linear cross-price demand relation over the observed price range
The slope b_ij is constant between the observed points; with curved demand the formula holds only at the point where the slope is measured.
Other prices and income held fixed for the point cross-price elasticity
The slope is a partial effect, so it applies only while the other prices and income stay where they were.
Worked examples
Generic demand at the starting co-payment of 8 pounds
The straight line through the article's two observations has a slope of 1,000 packs per 4 pounds, 250 packs per pound. At 8 pounds and 6,000 packs the point elasticity is 250 x 8 / 6,000, about 0.33 (computed here for illustration).
b_ij = 250; P_j = 8; Q_i = 6000; eps_ij = 0.3333
Generic demand at the midpoint co-payment of 10 pounds
At the midpoint, 10 pounds and 6,500 packs, the point elasticity is 250 x 10 / 6,500, about 0.3846, equal to the arc elasticity (computed here for illustration).
b_ij = 250; P_j = 10; Q_i = 6500; eps_ij = 0.3846
Generic demand at the new co-payment of 12 pounds
At 12 pounds and 7,000 packs the point elasticity is about 0.43 (computed here for illustration), so a single figure depends on where it is measured.
b_ij = 250; P_j = 12; Q_i = 7000; eps_ij = 0.4286
Common errors
Using the slope alone as the elasticity
A slope of 250 packs per pound is not unit-free; the elasticity rescales it by P_j / Q_i, here about 0.33 to 0.43.
Applying one point cross-price elasticity far from where it was measured
On a straight line the elasticity rises from about 0.33 to 0.43 between 8 and 12 pounds; carrying 0.33 to a much larger price change misstates the response.
Sources
Cross-price elasticity as the percentage change in demand for a 1 per cent change in another price
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. Introduction: cross-price elasticities quantify the percentage change in demand for one type of alcohol due to a 1 per cent change in the price of another type, and identify whether two types are substitutes or complements.
Canonical Identity
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