Arizona ECN 312 - Intermediate Microeconomic

Intermediate Microeconomic Theory

Fall 2016

Homework 1 - Part A
The homework exercises are increasingly difficult. You should start as soon as
possible so as to not fall behind. Contact me or the tutoring center (ask for
Keegan Kells or Ryan Theisen) for help.

Math Review:
Exercise 1
Find the maxima of the following functions:
1.

f (x) = (A + 1) ln(x) ? x

2.

g(x) = ? x2 + 4 x + 2

3.

h(x) = 10(A + 1) e?x , for x ? 0

where A must be replaced with the last digit of your ASU ID#.
Exercise 2
1. For the following function:
y = h(x) = a ln(x + b), for x ? 0,
determine the possible values of paramaters a and b such that the function takes
a value of zero when x = 0 and is always strictly increasing in x.
2. For the following function:
y = f (x) = ?x? , for all x ? 0 and where ? > 0,
determine the possible values of parameter ? such that the function is strictly
increasing in x, but at an ever-decreasing rate (hint: Take the first and second
derivatives and force the first to be strictly positive. That will give you a
condition on the possible values of ?, knowing that x and ? are positive. Then
force the second derivative to be negative. That will give you a second condition
on the possible values of ?).
3. For ? = 1, graph the function f (x) for the following values of ?:

1

1 1
4, 2,

1, 2.

Exercise 3
1. Using first order conditions only, propose solutions to the following optimization problems. For problem (ii) use both the Substitution and Lagrangian
techniques:



(i) max x1 [Ax1 x2 + 32 (A + 1)] + (A + 2) x2 x12 ? x2 + x22 ,
x1 ,x2

where A must be replaced with the last digit of your ASU ID#.


(ii) max 4 x11/2 x21/2
x1 ,x2

s.to: B (4 ? 2x1 ) + 4 = (B + 1)x2 + 2x1
where B must be replaced with the second-to-last digit of your ASU ID#.
2. (Optional ) Spot for which one of the two problems your proposed solution is
wrong. Explain (you will have to use second order conditions).

Budget Sets:
Exercise 4
In the country of Bolivia, bread costs 3bs (bolivianos) per loaf and milk costs
4bs per gallon.The government has decided to attack poverty by guaranteeing 8
loafs of bread and 3 gallons of milk per person per week. Only individuals that
cannot afford this bundle qualify.
1. Obtain the range of income levels for individuals that qualify for the program.
2. Assume that the government gives out 8 loafs of bread and 3 gallons of milk
to all qualifying individuals. Also, assume that people cannot trade the goods
they receive from the government.Graph the new budget set for a qualifying
individual of your choice.
3. Is it possible that some people end up asking for a salary reduction?
4. Graph the budget set you chose in part 1 but now assume that there is a
black market for government provided bread and milk (same prices as before).
5. Determine a cheaper way for the government to implement the guarantee.
Graph the new budget set for the individual of part (2).
Exercise 5
You are willing to work up to 80 hours a week. You make $10(B + 1) an
hour, where B is the second-to-last digit of your Student ID#.
1. Define and graph your budget set in terms of leisure (equal to non–worked
hours) -Good 1- and ”money spent on every other good ” -Good 2-.
2. Assume that above 40 hours per week you get an overtime hourly rate of
$16(B + 1) per hour. Define and graph the new budget set.
3. Assume that in addition to the over-time rate you also have $200(B + 1) in
your checking account. Define and graph the new budget set.

2

Preferences and Utility:
Exercise 6
Belinda loves chocolate and always thinks that more of it is better than less.
She would love to take piano lessons as well. However, she knows from her
sister’s experience that for the first four hours of piano lessons her satisfaction
level will steadily decline as compared to taking no lessons at all. After that
every extra second of piano lessons will make her happier.
1. Graph at least three of Belinda’s indifference curves in a way that is consistent
with her preferences.
Now assume that her love for chocolate is not as strong. After the sixth bar
she gets satiated and her happiness would decline were she to consume more.
2. Graph at least three of Belinda’s indifference curves in a way that is consistent
with this new feature.
3. For each of the following three utility functions, decide whether it is possible
that it represents Belinda’s preferences in (1), in (2) or none. Explain why.
u(x1 , x2 ) = 40 + 3x21 ? 24x1 + x22 + 12x2
v(x1 , x2 ) = 10 + 3x21 + 24x1 ? x22 + 12x2
w(x1 , x2 ) = 20 + 3x21 ? 24x1 ? x22 + 12x2
4*. Consider the following bundles: x = (5, 2) and y = (3, 2). Find a new
bundle z that is a weighted average of x and y and use this bundle to show that
preferences represented by u(x1 , x2 ) and w(x1 , x2 ) are not convex.

3

Intermediate Microeconomic Theory

Fall 2016

Student Name:

Student ID

Homework 1 - Part A
Front Page
Selected Answers:
Exercise 1.1.
x? =

Exercise 3.1.(i).
x?1 =

Exercise 3.1.(ii).
x?2 =

Exercise 5.1.
The budget set is represented by all bundles (x1 , x2 ) that satisfy the following
equation:

Exercise 5.3.
The budget set is represented by all bundles (x1 , x2 ) that satisfy the following
equations: