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Microeconomics FHS Problem Set 3 Sketch Solutions
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Official sketch solutions for select questions in Problem Set 3: Applied Welfare Economics: Public Goods and Externalities; the Environment
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UNIVERSITY OF OXFORD Last updated: September 17, 2014Department of Economics
Undergraduate - Microeconomics
Tutorial/Class 3: Applied Welfare Economics:
Public Goods and Externalities; the Environment
Sketch solutions
FOR USE BY TUTORS ONLY
DO NOT CIRCULATE, ESPECIALLY TO STUDENTS
Note: For each tutorial/class, a collection of questions related to the topic is provided. In
general there are too many questions for students to tackle in a single week. Tutors will decide
which of these questions their students should do in preparation for the tutorial or class.
, Problems and Multipart Questions
1. Externalities I: Gains from Trade
Two students, a and b, live in college. Student a has only one hobby: listening to
“******” music.1 Student a’s music system can produce noise levels of up to 100
decibels. Her utility depends on the loudness of the sound, D, and the amount of
money she has, ma . Her preferences are represented by the utility function:
1
ua (ma , D) = 10D 2 + ma .
Student b lives next door to student a and is irritated by the loud music he has to put
up with. His preferences are represented by the utility function:
1
ub (mb , D) = 10(100 D) 2 + mb ,
where mb denotes the amount of money he has.
(a) Draw student a’s marginal benefit curve and student b’s marginal cost curve.
2.0
MB,MC
1.5
MC b
1.0
0.5
MB a
0.0
0 20 40 60 80 100
Decibels, D
Figure 1: Marginal benefit and cost curves
(b) Term is nearly over and student a has run out of money. What is her utility?
At a competitive equilibrium student a must be choosing D 0 to maximise:
1
10D 2 ,
subject to D 100. Given local non-satiation, the constraint binds and the
outcome is D⇤ = 100. Her utility is:
1
ua = 10(100) 2 = 100 .
1
Insert your least preferred genre of music.
1
, (c) What would be a Pareto efficient noise level?
At any Pareto optimal allocation in which Do is the level of D and mi is student
i’s wealth, it must be impossible to change D and re-allocate wealth so as to
make one consumer better off without making the other worse off. This implies
that Do must maximise the joint surplus of the two students.2 That is, Do
0 must be chosen to maximise:
1 1
10D 2 + 10(100 D) 2
subject to D 100. From the FOC:
5 5
p =p ) Do = 50.
D 100 D
(d) Student b has £100. What is the maximum “bribe” he is willing to pay to student
a to turn down her music to a Pareto-efficient level? Would £50 be sufficient?
Note the assumption that the property rights have been assigned to student a.
Let B denote the bribe student b pays to student a.
The biggest bribe student b will pay to reduce the noise to Do = 50 is the value
that leaves him just indifferent between h50, 100 Bi and h100, 100i. Using
student b’s preferences, this is:
1 1
10(100 50) 2 + 100 B = 10(100 100) 2 + 100 .
Simplifying: p
1
10(100 50) 2 = B ) B = 50 2 > 50 .
The smallest bribe student a will accept is the value that leaves her just
indifferent between h50, Bi and h100, 0i. Using student a’s preferences, this is:
1 1 p
10(50) 2 + B = 10(100) 2 ) B = 100 50 2 < 50 .
So, yes, £50 would be sufficient.
(e) Draw an Edgeworth box to illustrate this problem, carefully indicating the initial
endowment, the contract curve, and the outcome if the £50 bribe is accepted.
To draw the Edgeworth box, it is helpful to work out the students’ marginal
rates of substitution. Here B is on the x axis and D on the y axis.
Student a’s MRS between noise and a bribe from student b:
p
a M UB D
M RSDB = =
M UD 5
2
See, e.g., MWG p. 353 for a fuller explanation. Note the implication that we are looking for the
intersection of MB and MC in Figure 1.
2
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