Signature
eps_c_ij = eps_ij + w_j * eta_i
| Inputs | Definition | Unit |
|---|---|---|
eps_ij | Uncompensated elasticity of good i's demand with respect to the price of good j | none |
w_j | Spending on good j as a proportion of the budget M, P_j Q_j / M | proportion |
eta_i | Proportional change in demand for good i per proportional change in income | none |
eps_c_ij | Response of the Hicksian demand for good i to the price of good j, in elasticity form | 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
Compensated cross elasticity from named gross elasticity, share and income elasticity
With GrossCrossElasticity, BudgetShareJ and IncomeElasticityI named, the formula returns the compensated elasticity, held in CompensatedCrossElasticity.
=GrossCrossElasticity+BudgetShareJ*IncomeElasticityI
Assumptions
Smooth demand at a utility-maximising choice for the Slutsky cross-price decomposition
The Slutsky equation holds for continuous, locally non-satiated preferences with single-valued, differentiable demands, evaluated where income equals the expenditure needed for the utility level.
Elasticities and budget share measured at the same point
eps_ij, w_j and eta_i refer to the same prices, income and population; the elasticity form follows from multiplying the Slutsky equation by P_j / Q_i.
Worked examples
Generic demand when the branded co-payment is a tiny budget share
If the branded co-payment takes 0.2 per cent of the budget and the income elasticity of generic use is 0.5, the compensated elasticity is 0.3846 plus 0.001, about 0.3856, so gross and compensated values nearly coincide (computed here for illustration).
eps_ij = 0.3846; w_j = 0.002; eta_i = 0.5; eps_c_ij = 0.3856
Gross cross-price complements that are net substitutes
With a gross elasticity of minus 0.1, a budget share of 0.3 for good j and an income elasticity of 1 for good i, the compensated elasticity is plus 0.2: the sign changes once the income effect is removed (computed here for illustration).
eps_ij = -0.1; w_j = 0.3; eta_i = 1; eps_c_ij = 0.2
Common errors
Subtracting the income term instead of adding it to a gross cross-price elasticity
Writing eps_c_ij = eps_ij minus w_j eta_i doubles the income effect in the wrong direction; in the second example it gives minus 0.4 instead of plus 0.2.
Classing goods as substitutes from a gross elasticity alone
Gross substitution need not be reciprocal and can differ in sign from the compensated effect, as the second example shows; substitutes in the Hicksian sense are defined on compensated demand.
Sources
Slutsky equation decomposing a price effect into substitution and income effects
Wolitzky A. Lectures 3-4: consumer theory. 14.121 Microeconomic Theory I, Fall 2015. Cambridge, MA: MIT OpenCourseWare; 2015. Slide 39, Slutsky equation theorem: the derivative of Marshallian demand for good i with respect to p_j equals the derivative of Hicksian demand minus the derivative of x_i with respect to income times x_j (total effect, substitution effect, income effect); slide 40: good i is a substitute for good j if Hicksian demand h_i is increasing in p_j. The elasticity form follows by multiplying both sides by p_j / x_i.
Canonical Identity
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