Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Cross-price elasticity of demand for one health care good with respect to another good's price

eps_ij = (dQ_i / dP_j) * (P_j / Q_i)

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.

  • Arc cross-price elasticity between two observed price and quantity pairs

    eps_arc = ((Q_i1 - Q_i0) / (Q_i1 + Q_i0)) / ((P_j1 - P_j0) / (P_j1 + P_j0))

    Divides the change in the quantity of good i, as a proportion of the average of its two values, by the change in the price of good j, as a proportion of the average of its two prices. Using averages gives the same answer whichever observation is treated as the start; the factors of 2 in the averages cancel. With i equal to j the same formula gives the arc own-price elasticity, as in the RAND form HE-FM-MH-003.

  • Point cross-price elasticity on a straight-line demand relation

    eps_ij = b_ij * P_j / Q_i

    Multiplies the slope of good i's demand with respect to the price of good j by the ratio of that price to the quantity at the point of interest. On a straight line the slope is constant but the elasticity changes along it, and at the midpoint of two observations it equals the arc elasticity.

  • Quantity response to own and cross price changes under constant elasticities

    Q_i1 = Q_i0 * (P_i1 / P_i0)^eps_ii * (P_j1 / P_j0)^eps_ij

    Projects the new quantity of good i when its own price and the price of good j both change, using the log-log demand form in which each coefficient is an elasticity: the proportional change in log quantity is the elasticity-weighted sum of the log price changes. For small changes this equals the article's approximation, proportional change in Q_i about equal to the sum of eps_ij times the proportional change in P_j.

  • Compensated cross-price elasticity from the gross elasticity by the Slutsky equation

    eps_c_ij = eps_ij + w_j * eta_i

    Separates a gross (uncompensated) cross-price elasticity, which mixes substitution with an income effect, into its compensated part. In elasticity form the Slutsky equation is eps_ij = eps_c_ij minus w_j times eta_i, so the compensated elasticity adds back the budget share of good j times the income elasticity of good i. With a small budget share, as for most co-payments, the two nearly coincide.

  • Reverse compensated cross-price elasticity from Slutsky symmetry and budget shares

    eps_c_ji = w_i * eps_c_ij / w_j

    Uses the symmetry of the Slutsky matrix of compensated price derivatives, which in elasticity form reads w_i eps_c_ij = w_j eps_c_ji, to obtain the compensated response of good j to the price of good i from the response of i to the price of j. The two compensated elasticities have the same sign but are equal only when the budget shares are equal.