Mario and Luigi both have the utility function U(x,y) = (1/3)^4/3 x^2/3 y^4/3. They each spend their income I on goods x and y. (a) Mario and Luigi only differ in their incomes, which are $324 for Mario and $450 for Luigi. What is the aggregate demand function for good x in terms of its price px? (b) Now forget about Luigi. Initially, prices are px = 4/3 and py = 3. Then the price of good x increases to px = 9/2. What is the total effect of the price change on Mario’s consumption (remember that Mario’s income is $324)? (c) What is the utility Mario obtained under the old prices? (d) What is the income Mario would need to be as well off after the price increase as before? (e) What are the income and substitution effects of that price change for Mario’s consumption of good x?

Mario and Luigi both have the utility function U(x,y) = (1/3)^4/3 x^2/3 y^4/3. They each spend their income I on goods x and y.

(a) Mario and Luigi only differ in their incomes, which are $324 for Mario and $450 for Luigi. What is the aggregate demand function for good x in terms of its price px?

(b) Now forget about Luigi. Initially, prices are px = 4/3 and py = 3. Then the price of good x increases to px = 9/2. What is the total effect of the price change on Mario’s consumption
(remember that Mario’s income is $324)?

(c) What is the utility Mario obtained under the old prices?

(d) What is the income Mario would need to be as well off after the price increase as before?

(e) What are the income and substitution effects of that price change for Mario’s consumption of good x?

 

 

 

 

Answer:

U(x,y) = (1/3)^4/3 x^2/3 y^4/3

Mm=324 Ml=450

a) MRS = px/py

mux/muy = px/py

mux/muy = 2/3x-1/3 y 4/3 / (4/3y1/3 x2/3)

y/2x=px/py

putting in b.c

y*= 2m/3py

x*=m/3px

b) initially px=4/3 py=3

x* = m/3(4/3) = m/4 = 324/4=81

y* = 2m/9=2*324/9=72

if px = 9/2

x0 = m/3px = 24

total change = x0-x1 = 24 – 81 = -57

c) u = 1/34/3 * 812/3 *724/3 = 0.2*13.9*259.7 = 721

d) income needed to be as well off after the price change be m’

m’ = 81*9/2 + 72*3 = 364.5 +216 = 580

e) now

x1=m’/3px = 580 / 3(9/2) = 50

x2 = 2m’/3py = 129

substutution effect = x1 – x*= 50 – 81= -31

income effct = sub.effect – total = –31 (-57)= -26

Asked on February 13, 2018 in economics.
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