5.9 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is defined as sx = Pxx/I. In this problem we show that most demand elasticities can be derived from corresponding share elasticities. a. Show that the elasticity of a good's budget share with respect to income (e,, 1 = dsx/ÔI · I/ Sx) is equal to ex, 1 – 1. Interpret this conclusion with a few numerical examples. b. Show that the elasticity of a gooď's budget share with respect to its own price (es, Pe = dsx/dpx · Px/Sx) is equal to ex, p. + 1. Again, interpret this finding with a few numerical examples. c. Use your results from part (b) to show that the "expenditure elasticity" of good x with respect to its own price [ex p., p. = (Px · x)/@px · 1/x] is also equal to ex, p. + 1. d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good (es, Py = Osx/Opy · Py/$x) is equal to ex, p.- e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given by sx = 1/(1+p,P,*), where k = 8/(8 – 1) = 1 – 6. Use this share equation to prove Equation 5.56: ex, p. = -(1 – sx)o. Hint: This problem can be simplified by assuming px= Py» in which case s, = 0.5.
5.9 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is defined as sx = Pxx/I. In this problem we show that most demand elasticities can be derived from corresponding share elasticities. a. Show that the elasticity of a good's budget share with respect to income (e,, 1 = dsx/ÔI · I/ Sx) is equal to ex, 1 – 1. Interpret this conclusion with a few numerical examples. b. Show that the elasticity of a gooď's budget share with respect to its own price (es, Pe = dsx/dpx · Px/Sx) is equal to ex, p. + 1. Again, interpret this finding with a few numerical examples. c. Use your results from part (b) to show that the "expenditure elasticity" of good x with respect to its own price [ex p., p. = (Px · x)/@px · 1/x] is also equal to ex, p. + 1. d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good (es, Py = Osx/Opy · Py/$x) is equal to ex, p.- e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given by sx = 1/(1+p,P,*), where k = 8/(8 – 1) = 1 – 6. Use this share equation to prove Equation 5.56: ex, p. = -(1 – sx)o. Hint: This problem can be simplified by assuming px= Py» in which case s, = 0.5.
Chapter5: Income And Substitution Effects
Section: Chapter Questions
Problem 5.9P
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