Name:____Solutions_______
FE431: PUBLIC FINANCE
Assignment 5: Due Friday 2/19/16
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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
Part I - Answer the following questions which cover material from Chapters 1 and 2
of your textbook:
1. Jane has preferences over two goods, P and F, given by
UJ  P*F
Tarzan’s utility function over P and F is
UT  P  F
a. Calculate Jane’s marginal rate of substitution if she has 6 F and 24 P.
Here, Jane’s MRSPF 
U / P F 6
 
 1/ 4 .
U / F P 24
b. Assuming the price of a unit of P is 3, and the price of a unit of F is 1,
how much of each good would you expect Jane to purchase if she has
income of 42?
Jane seeks to maximize U = P*F subject to her budget constraint
42=3P+F.
Solving the budget constraint for F yields F = 42-3P
Substituting this into her utility function yields
U=P*(42-3P) = 42P-3P2
Take the derivative and set it = 0, which yields
42-6P=0
P* = 7
F*=42-3(7) = 21
She should purchase 7 P and 21 F to maximize her utility.
c. Calculate Tarzan’s marginal rate of substitution if he has 6 F and 24 P.
U / P 1
  1 (it is constant – he is always willing
Tarzan’s MRS PF 
U / F 1
to trade one F for one P and keep utility constant).
d. Assuming the price of a unit of P is 3, and the price of a unit of F is 1,
how much of each good would you expect Tarzan to purchase if he has
income of 42?
Since F and P are perfect one-for-one substitutes, but P is more expensive,
Tarzan would choose to purchase all F. His optimal consumption bundle is
F* = 42, P* = 0. This is a “corner solution” (note that it has no calculus
solution).
Part II - Answer the following questions which cover material from Chapter 2 of
your textbook (Externalities) and imperfect competition / monopoly power:
2. Assume that a fuel-cell powered vehicle is developed that has zero hazardous
emissions. Let the marginal private benefit (i.e. the market demand) for these
vehicles be
MPB = 50,000 – Q
where Q is the total quantity of fuel-cell vehicles purchased. Given their ability to
reduce pollution, these vehicles are considered to have a marginal external
benefit of
MEB = 0.2Q
Furthermore, assume there is a constant marginal (private) cost of producing
such vehicles of 30,000. There are no marginal external costs.
a. What is the economically efficient number of fuel-cell vehicles that should
be produced and purchased each year?
The efficient number of vehicles is where MSB = MSC.
MSB = MPB + MEB = 50000-Q+0.2Q=50000-0.8Q
MSC=30000
50000-0.8Q=30000
Q*=25000 vehicles per year
b. Calculate the equilibrium market quantity and price if such vehicles are
produced in perfectly competitive markets. Show (by calculation) that the
DWL at the perfectly competitive outcome is $10 million (per year). How
large of a Pigouvian subsidy would be necessary to lead to the efficient
number of vehicles being purchased?
The equilibrium market quantity and price is where Demand = Supply.
Demand = MPB = 50000-Q
Supply = MPC = 30000
50000-Q=30000
QE = 20000 cars per year
PE = $30000 per car
To calculate DWL, it helps to draw a picture and calculate the area of the
triangle given by the area below the MSB curve, above the MSC curve (which
is just MPC and is flat) between the efficient quantity and the market quantity.
DWL = ½ * base * height where “base” = (25000 – 20000) = 5000, “height” =
(34000-30000) = 4000
DWL = $10 million per year
The Pigouvian subsidy that is needed is $5000 per vehicle (which is the MEB
at the efficient quantity of 25000 vehicles.)
c. Suppose such vehicles were produced by a single-price monopoly. Calculate
the monopoly’s price and quantity of vehicles sold, and demonstrate that the
deadweight loss (per year) in this case would be $90 million per year.
The monopoly seeks to maximize profits and produces where MR = MC.
MC = 30000
Total revenue = PQ = (50000-Q)Q = 50000Q-Q2
MR = dTR/dQ = 50000-2Q
50000-2Q = 30000
QM = 10000 cars
PM = $40000 car
DWL = $90 million year (calculated the same way as in (b) above)