Signature
eps_c_ji = w_i * eps_c_ij / w_j
| Inputs | Definition | Unit |
|---|---|---|
w_i | Spending on good i as a proportion of the budget | proportion |
eps_c_ij | Known compensated cross-price elasticity | none |
w_j | Spending on good j as a proportion of the budget, above zero | proportion |
eps_c_ji | Reverse compensated cross-price elasticity | 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
Reverse compensated elasticity from named shares
With CompensatedCrossElasticity, BudgetShareI and BudgetShareJ named, the formula returns the reverse compensated elasticity, held in ReverseCompensated.
=BudgetShareI*CompensatedCrossElasticity/BudgetShareJ
Assumptions
Cross-price elasticity symmetry applies to compensated effects only
The Hicksian price derivatives are symmetric; gross (Marshallian) cross effects are not, because income effects differ between goods.
Same population and prices for both compensated cross-price elasticities
Both elasticities and both budget shares refer to one consumer's or one population's demand system at the same prices.
Worked examples
Compensated cross-price elasticity of a small-share product and its large-share counterpart
If good i takes 1 per cent of the budget and good j 10 per cent, a compensated elasticity of 0.5 for i with respect to the price of j implies only 0.05 for j with respect to the price of i (computed here for illustration).
w_i = 0.01; eps_c_ij = 0.5; w_j = 0.1; eps_c_ji = 0.05
Equal budget shares give equal compensated elasticities
With both goods at 5 per cent of the budget the two compensated elasticities are equal (computed here for illustration).
w_i = 0.05; eps_c_ij = 0.5; w_j = 0.05; eps_c_ji = 0.5
Common errors
Assuming cross-price elasticities are equal in both directions
Only share-weighted compensated elasticities are equal; in the first example the response in one direction is ten times that in the other.
Applying symmetry to gross elasticities from claims data
Reported gross cross elasticities can differ in size and even sign between the two directions; symmetry restrictions in a demand system should be stated.
Sources
Symmetry of the Slutsky matrix and reciprocity of net substitutes
Wolitzky A. Lectures 3-4: consumer theory. 14.121 Microeconomic Theory I, Fall 2015. Cambridge, MA: MIT OpenCourseWare; 2015. Slide 36: if h(p, u) is continuously differentiable the matrix of its price derivatives is symmetric and negative semi-definite; slide 41: by symmetry of the Slutsky matrix i is a substitute for j if and only if j is a substitute for i, which is not true of gross substitutes, because income effects are not symmetric. Multiplying the symmetric derivative by p_i p_j / M gives w_i eps_c_ij = w_j eps_c_ji.
Canonical Identity
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