In the first problem set, you solved the consumer problem for the utility function (₁,₂)= ₁+2√₂. Throughout this question, assume that the consumer has the same utility function, and that wealth is high enough to ensure an interior solution to the utility maximization problem. (a) Verify that the Slutsky equation holds for cross-price changes (i.e. for Good 1 when p2 changes, and for Good 2 when på changes). (b) Suppose that the price of Good 1 increases from pi to p while the price of Good 2 remains unchanged. Let u and u² respectively denote the utilities before and after this change. Using the Hicksian demand function from the first problem set, write expressions for the equivalent and compensating variations in terms of prices and utilities and ¹. How do they compare? (e) Instead of a change in p₁, suppose that the price of Good 2 increases from p; to p while the price of Good 1 remains unchanged. Write expressions for the equivalent and compensating variations. How do they compare? Explain why the comparison does or does not differ from that in part (b). (a) The Slutsky equation. Әр For good 1 and P2 8x1 8p₂ əhi 8p₂ بية || Əh₁ apj ៧គីឡូ = - Xi -(1) -----(2) from (1) and (2) equation 201 Əx - Xi Jw 1 2
In the first problem set, you solved the consumer problem for the utility function (₁,₂)= ₁+2√₂. Throughout this question, assume that the consumer has the same utility function, and that wealth is high enough to ensure an interior solution to the utility maximization problem. (a) Verify that the Slutsky equation holds for cross-price changes (i.e. for Good 1 when p2 changes, and for Good 2 when på changes). (b) Suppose that the price of Good 1 increases from pi to p while the price of Good 2 remains unchanged. Let u and u² respectively denote the utilities before and after this change. Using the Hicksian demand function from the first problem set, write expressions for the equivalent and compensating variations in terms of prices and utilities and ¹. How do they compare? (e) Instead of a change in p₁, suppose that the price of Good 2 increases from p; to p while the price of Good 1 remains unchanged. Write expressions for the equivalent and compensating variations. How do they compare? Explain why the comparison does or does not differ from that in part (b). (a) The Slutsky equation. Әр For good 1 and P2 8x1 8p₂ əhi 8p₂ بية || Əh₁ apj ៧គីឡូ = - Xi -(1) -----(2) from (1) and (2) equation 201 Əx - Xi Jw 1 2
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.11P
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Transcribed Image Text:. In the first problem set, you solved the consumer problem for the utility function (₁,₂)=
2₁+2√72. Throughout this question, assume that the consumer has the same utility function,
and that wealth is high enough to ensure an interior solution to the utility maximization
problem.
(a) Verify that the Slutsky equation holds for cross-price changes (i.e. for Good 1 when p2
changes, and for Good 2 when p₁ changes).
(b) Suppose that the price of Good 1 increases from pi to p/ while the price of Good 2
remains unchanged. Let u and u¹ respectively denote the utilities before and after this
change. Using the Hicksian demand function from the first problem set, write expressions
for the equivalent and compensating variations in terms of prices and utilities and
¹. How do they compare?
(c) Instead of a change in p₁, suppose that the price of Good 2 increases from pa to p
while the price of Good 1 remains unchanged. Write expressions for the equivalent and
compensating variations. How do they compare? Explain why the comparison does or
does not differ from that in part (b).
(a) The Slutsky equation.
əx
api
For good 1 and p2
8x1
8p2
əhi
8p2
8x1
dw
=
Əh₁
Әр
P₁
=
·Xi aw
Əh
opi
➖➖➖➖➖➖-(1)
-----(2)
from (1) and (2) equation
8x1 2p1
Əx₁
-Xi dw
1
P1
11
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