Name:____Solutions____
FE431: PUBLIC FINANCE
Assignment 2: Due Friday 1/29
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Make sure you show your work and that ALL answers are complete and neatly
written.
Only work that follows these instructions and represents a “Good Faith Effort” to
answer the questions posed will receive a check. (Copying answers from a classmate
without working through the problems is unacceptable conduct).
Stop by during my office hours (MWF period 4, Thurs period 9) if you have
questions, or set up another time to meet with me. A “Review of Numerical
Problems” is posted at www.usna.edu/Users/econ/swope/FE431/homeworks.html
Read Chapter 1 of your textbook (including the appendix). Answer the following questions
that deal with consumer utility maximization.
Indifference Curves
1. Draw the general shape of Hannah’s indifference curves under each of the following
situations. Show the general direction of increasing utility. Calculate her MRSXY. Pick two
goods for which the description applies to you.
a. Hannah likes both good X and good Y (e.g. UH = X*Y).
Y
0
X
MRSXY = Y/X (varies)
b. Hannah likes good X, but is indifferent (i.e. doesn’t care either way) about good Y (e.g.
UH = X).
Y
0
X
MRSXY = 1/0 = ∞ (constant)
c. For Hannah X is a “good”, but Y is a “bad” that lowers her utility. (e.g. UH = X - Y2).
(No need to calculate her MRS here, to avoid confusion).
Y
0
X
d. Hannah will always be willing to trade 2 Y for 1 X. That is, they are “perfect 2 to 1
substitutes” (e.g. UH = X +0.5 Y).
Y
0
X
MRSXY = 1/0.5 = 2 (constant)
Consumer Utility Maximization
2. Tom’s utility over apples (A) and bananas (B) is
UT (A, B) = A*B
where A is the number of lbs of apples he consumes in a week, and B is the number of lbs of
bananas. Tom has $10 to spend on apples and bananas each week. The price of bananas is PB
= $1/lb.
a. Solve the appropriate constrained maximization problem to find how many lbs of
apples and bananas Tom purchases each week if the price of apples is PA = $2/lb.
Max UT = A*B
Subject to 10=2A+1B or B = 10 – 2A
Substitution yields
UT = A*(10-2A)=10A-2A2
Taking the derivative and setting equal to zero yields
10 – 4A = 0
Solving yields
A*=2.5 lbs; using the budget constraint above gives B*=5 lbs.
b. How many lbs of apples would Tom purchase if PA = $1/lb? What if PA = $0.50/lb?
Solving similar constrained maximization problems as in (a) yields A*=5 lbs, B* = 5
lbs when PA = $1/lb, and A*= 10 lbs, B*=5 lbs when PA = $0.50/lb.
c. Plot Tom’s demand curve for apples based on your answers to parts (a) and (b)
above. Plot the price of apples on the y-axis and Tom’s corresponding quantity
demanded on the x-axis.
The demand curve should be a downward-sloping (though non-linear) line
connecting the price and quantity pairs for apples from parts (a) and (b) above.
3. Keith gets utility from the consumption of two goods, burritos (B) and nachos (N). His
utility is represented by the function U(B,N) = 4B + 2N. This implies a constant
marginal utility of burritos of 4 and a constant marginal utility of nachos of 2. Keith has
$M per week to spend on burritos and nachos. Burritos cost $3 each, while nachos cost
$1.
Keith is planning to spend half of his income on burritos and half on nachos. Would
Keith be maximizing his utility? If yes, explain why. If not, how would you recommend
he spend his income. Support your answer with an indifference curve / budget constraint
diagram with nachos on the y-axis and burritos on the x-axis. Label all relevant parts of
your diagram. You should also support your answer with any calculations you feel are
relevant.
Keith would not be maximizing his utility by spending half his income on each good
– he should only buy nachos. Even though he gets half as much utility form each
nachos, they cost only one third as much. If nachos are on the y-axis, his MRSBN =
4/2 = 2. That is, his indifference curves are straight, downward-sloping lines with
slope = -2 (similar to 1d above). His budget constraint is also a straight, downwardsloping line, but slope = Px/Py = 3/1 = 3. That is, his budget constraint is steeper than
his indifference curves, so you should be able to show that the highest indifference
curve he can get to is a B=0, N = M/PN (spending all income on nachos). This is a
“corner solution” (see #4).
4. Of the two questions (2a) and (3) above, one results in what we call an “interior
solution” and the other results in a “corner solution”. Which one is an interior solution
and which is a corner solution? Briefly explain why we call this a “corner solution in
economics”? (You are free to search the internet for the answer, if needed).
Question 3 results in a “corner solution” (i.e. an outcome where it is optimal to
consume zero of a good). This occurs because of the “linear” utility function.
Burritos and nachos here are “perfect 2-for-1 substitutes”. But Keith can get three
nachos for each burrito, so he would prefer to purchase all nachos at those prices, a
corner solution.