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Fall 2023: Problem Set 3: Micro 1 Due Date: September 29, 2023

The document outlines Problem Set 3 for a Microeconomics course, due on September 29, 2023, and consists of various questions related to utility functions, preference relations, optimal choices, and economic principles. It includes tasks such as checking equivalence of utility functions, demonstrating conditions for preference and utility, and analyzing Cobb-Douglas utility functions. Additionally, it presents a data set for analysis of WARP and SARP conditions, along with statements for evaluation regarding their truthfulness.

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Sneha Gupta
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8 views2 pages

Fall 2023: Problem Set 3: Micro 1 Due Date: September 29, 2023

The document outlines Problem Set 3 for a Microeconomics course, due on September 29, 2023, and consists of various questions related to utility functions, preference relations, optimal choices, and economic principles. It includes tasks such as checking equivalence of utility functions, demonstrating conditions for preference and utility, and analyzing Cobb-Douglas utility functions. Additionally, it presents a data set for analysis of WARP and SARP conditions, along with statements for evaluation regarding their truthfulness.

Uploaded by

Sneha Gupta
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Fall 2023: Problem set 3

Micro 1
Due date: September 29, 2023

Total: 30 points

1. (5 points) Consider the following utility functions:

u(x1 , x2 ) = x21 + x22


v(x1 , x2 ) = x21 x22
w(x1 , x2 ) = ln x1 + ln x2

Check which one of the above utility functions is equivalent to the other. Clearly
show your work.

2. (3 points) In this problem, we will show that the equivalence between preference
and utility requires an ”if and only if condition”. We will show this using a coun-
terexample.
Write down an example of a three-element choice set with a complete and transitive
preference relation defined over it. Write down the utility values for each element
such that the following condition holds, for any two elements a and b in the choice
if

u(a) ≥ u(b) ⇒ a ≿ b

Show that in your example the set of maximal elements derived using the preference
relation and the utility function are not identical.

3. Consider the following utility function.

u(x1 , x2 ) = x21 x32

(a) (4 points) Solve for the optimal choice. You can use any method to solve the
problem. Make sure all steps are clearly articulated.
(b) (2 points) Find the Engel curve equation. Assume p1 = p2 = 1

4. Consider the following utility function.


1/2
u(x1 , x2 ) = x1 + x2

1
(a) (3 points) Find the optimal choice for m = 5, p1 = 0.25, p2 = 1.
(b) (1 point) What will be the optimal level of x2 if m drops to m′ = 0.75
You can use any method to solve the problem. Make sure all steps are clearly
articulated.

5. Consider the following Cobb-Douglas utility function:

u(x1 , x2 ) = ln x1 + ln x2

Suppose the price for good 1 decreases from p1 to q1 .


(a) (1 point) Find the price effect.
(b) (1 point) Find the hypothetical income m′ .
(c) (2 points) Find SE and IE.
(d) (1 point) Verify Slutsky decomposition.
Answers for Q 1-3 should be written in terms of the parameters of the model. You
can use the expression for optimal choice from lecture notes.

6. Consider the following data set

p1 = (1, 1, 2), x1 = (3, 3, 1), m1 = 8


p2 = (2, 1, 1), x2 = (1, 3, 3), m2 = 8
p3 = (1, 2, 1), x3 = (3, 1, 3), m3 = 8

(a) (2 points) Does this choice data satisfy WARP?


(b) (2 points) Does this choice data satisfy SARP?

7. (3 points) Consider the following statements

A. If a choice data satisfies SARP it also satisfies WARP.


B. If a choice data violates SARP it also violates WARP.

Argue if either one of them is true. Are both of them true?

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