LMU The Slutsky Relationship Between the Two Demands Question

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Intermediate microeconomics homework. 6 questions with parts. Must show work. questions are about: utility, hicksian and marshillian preference, Cobb Douglas, slutsky.

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K Re¤ett
ECN 312
Homework 5/Review Term Exam 2
Due to submit March 22, 2022
(Work this out though for second term Exam)
Do all the problems as this is also a study guide for the second term exam.
The second term exam will basically be the Marshallian and Hicksian theory of
consumer demand. One thing to focus on is comparative statics (normality of
demand, gross substitutes, and gross complements). I also want to you to also
thinking about the approach to comparative statics presented in class using the
increasing and decreasing di¤erence property of the consumers problem relative
to parameters of interest (i.e., income and prices) in the case of Marshallian
demand. Also, understand the “duality” between the two consumer problems
and how to construct Marshallian demand from Hicksian, and Hicksian Demand
from Marshallian) as well as envelope theorems for both problems. I …nally want
to you know how to explain the Slutsky decomposition with a picture (detailed),
and also using calculus.
1. Consider a consumer with utility u(x1 ; x2 ) = :5 ln x1 + :5 ln x2 :
(a). Does this consumer have preferences that have the “increasing difference” property? How about the “strict increasing di¤erence” property? In
answering this question, de…ne both the “increasing di¤erence property” and
the “strict increasing di¤erence” property for this utility function.
(b) Now, consider an equivalent representation of the above utility function
:5
x:5
x
1 2 : Does this utility function have the “increasing di¤erence property”? How
about the “strict increasing di¤erence” property?
(c). Show using the arguments on monotone comparative statics in class,
argue that the demand for both good 1 and good 2 are normal. In doing so,
de…ne a normal good.
2. Consider a consumer with CES utility u(x1 ; x2 ) = x1 + x2 .
(a) Does this consumer have preferences the “increasing di¤erence property”? How about the strict increasing di¤erence property? Make a detailed
argument.
(b) for the case that = :5; show using the arguments on monotone comparative statics in class that both goods are normal.
(b) for the case that = 2; show that the su¢ cient conditions based upon
the increasing di¤erence property for normal demand for both goods would fail.
3. Do the following:
(a) Consider a Hicksian consumers problem where the preferences are strictly
convex and strictly monotonic and represented by a continuous di¤erentiable
strictly concave strictly increasing utility function u(x). Assume preferences
1
are such that for any p >> 0 and feasible utility level u 2 U = fuju =
u(x); x 2 Cg: State the Hicksian demand problem for this consumer. What is
the interpretation of the “value function” for this problem?
(b) Now, using a Lagrangian approach, apply the envelope theorem to compute all the envelopes in all parameters for the value function again assuming
preferences u(x) are as stated in part (a).
(c) Now, for this consumer in part (a) and (b), state the Marshallian demand
problem. What is the interpretation of the value function in this problem?
(d) Now, compute all the envelopes of the value function using a Lagrangian
approach.
(e) Now, state the Marshallian problem for this consumer. Using the Lagrangian approach, compute the envelope theorems of the value function in each
price and construct Roy’s identity, and the envelope income.
(f) rewrite the problem in (e) using variable substitution. Apply the envelope
theorem to compute all the envelopes of the value function, and show that the
envelopes are the same is using the Lagrangian approach.
(g) Now, show how we get the Hicksian demands from the Marshallian problem, and Marshallian demand from the Hicksian problem.
4. Explain in what sense the expenditure minimization problem (the Hicksian demand problem) is a “dual” representation of the consumer choice problem
to the utility maximization problem (the Marshallian demand problem). That
is,
(a) write now both problems for the case of utility function that represents
strictly convex preferences (for example, when u(x1 ; x2 ) is strictly concave)
(b) De…ne the value function in each problem, and interpret it (with words).
(c) Then, using the value function de…nition in the two problems, show how:
(i) a Hicksian demand can be computed as a Marshallian demand.
(ii) a Marshallian demand can be computed as a Hicksian demand
(d) Finally, give a detailed annotated picture explain how this duality works.
5. (a) State the Slutsky relationship between the two demands (Hicksian
and Marshallian). In the Slutsky equation, using the mathematical formulation
the income e¤ect vs the substitution e¤ect, decompose the e¤ects of an “own
price change” for good 1 in a consumer choice problem with 2 goods.
(b) Can you develop a notion of a income and substitution e¤ect for good
2 when the price of good 1 changes? If so, provide a detailed explanation. If
not, explain why you cannot identify with the income or substitution e¤ect of
a “cross price” change.
(c) Now, provide a picture of the Slutsky decomposition carefully labeling
both substitution e¤ects and income e¤ects.
(d) In (c), how does this picture look for a normal good? for an inferior
good? Finally, Gi¤en good?
6. (a) Say consumers have Cobb Douglas preferences over types of goods.
Will this economy have have each consumer having normal demand for both
goods? Will each consumer’s demand satisfy the law of demand in own price?
2
(b) now, say preferences are perfect substitutes. How about this case per
the market demand curve for all goods necessarily satisfying the law of demand
(own price). (di¢ cult question).
3

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microeconomics

cobb douglas

marshillian preference

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