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.
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Questions d) and e)
![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 gooď's budget share with respect to income (es, I= dsx/ƏI · I/ x) is equal to e, 1- 1. Interpret
this conclusion with a few numerical examples.
b. Show that the elasticity of a goods budget share with respect to its own price (e, p. = dsz/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, pe = 0(Px · x)/ðpx · 1/x] is also equal to ex, pe +1.
d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good
(esss py = dsx/Opy · Py/$x) is equal to ex, py-
e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given
by Sx = 1/(1+ pp,"), where k = d/(8 – 1) = 1 – 6. Use this share equation to prove Equation 5.56: ex, p. = -(1 – Sx)o.
Hint: This problem can be simplified by assuming px= Py, in which case s, = 0.5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bf67c0c-5683-45bb-853c-68207b37e65c%2Feffd6cdd-c776-4da6-a8f0-1aa1f659228c%2F7u5ph9h_processed.png&w=3840&q=75)
Transcribed Image Text:5.9 Share elasticities
In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good,
x, the income share is defined as sx = Pxx/I. In this problem we show that most demand elasticities can be derived from
corresponding share elasticities.
a. Show that the elasticity of a gooď's budget share with respect to income (es, I= dsx/ƏI · I/ x) is equal to e, 1- 1. Interpret
this conclusion with a few numerical examples.
b. Show that the elasticity of a goods budget share with respect to its own price (e, p. = dsz/dpx · Px/sx) is equal to
ex, p. + 1. Again, interpret this finding with a few numerical examples.
c. Use your results from part (b) to show that the "expenditure elasticity" of good x with respect to its own price
[ex p, pe = 0(Px · x)/ðpx · 1/x] is also equal to ex, pe +1.
d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good
(esss py = dsx/Opy · Py/$x) is equal to ex, py-
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+ pp,"), where k = d/(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.
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