Assignment 4 - Public Goods
Winter 2014
ECON 3328b
Solutions:
1. Suppose that there are 100 identical individuals whose identical preferences can be
described by utility function ui   ( X i , G)  10 X i0.4 G 0.4 , where Xi is the quantity of
the private good and G is the quantity of a pure public good consumed by person i.
What is the efficient provision level of the public good if the PPF is described by
100,000 = X + G, where X and G are the total quantity of the private good and public
good produced by society.
The social marginal cost of the public good is given by the MRT (marginal rate of
transformation). This can be found by finding the absolute value of the slope of the PPF
(production possibilities frontier.
The social marginal benefit of the public good is given by the ∑MRSi. Each person’s
MRSi can be found by taking first derivatives of the utility function.
 u i


 4 X 0.4 G 0.6 X

G
u i

u
 
i
i
i
 4 X i0.6 Gi0.4 and
 4 X i0.4 Gi0.6 so that MRS i  
 i
 0.6 0.4
X i
G
G
 u i
 4 X i Gi

X i 

Since everyone has identical preferences, the social marginal benefit of the public good is
100
X
(1)
MRS i  100 i

G
i 1
100
Identical preferences also imply that:
X
i 1
i
 100 X i  X  100,000  G , so that
X i  1,000  0.01G (2)
Combining (1) and (2) yields the social marginal benefit of the public good
100
100 X i 100(1,000  0.01G ) 100,000  G
, which is clearly
SMB   MRS i 


G
G
G
i 1
diminishing in G. Setting SMB equal to SMC (social marginal cost) of the public good,
identifies the Pareto Efficient provision level of the public good. Setting the SMB=SMC
100,000  G
 1 so that 100,000 – G = G → 100,000 = 2G so that G = 50,000.
yields
G
2. Problem #1 page 198 of the textbook.
The net benefits of the club are: (B – C) = (36 − m2) ∙ s2/3 – 27∙(s/m)
The optimal membership and size are characterized by the conditions:
∂(B − C)
∂m
= −2s2/3 m+27s/m2 = 0
∂(B − C)
∂s
= (2/3)∙(36 −m2)∙s−1/3 −27/m= 0
These two conditions can be expressed as:
s1/3 =(2/27)∙m3
s1/3 =(2/3)∙(m/27)∙(36 − m2)
(1)
Combining these equations to eliminate s yields: 3m2 = 36 − m2
Implying that m∗ = 3. Substitution into (1) yields: s∗ = 8
3. Suppose that each person a natural disaster occurs in Victoria B.C. and families in
London Ontario would like to help. Assume that each family has identical preferences
given by u  f ( x, D)  x  D1 / 2 , where x = family’s consumption and D = total
amount of aid received by Victoria. Each family faces a budget constraint of
M  x  d , where d = the family’s contribution to the aid package.
a. Why would an economist believe that not enough aid will be forthcoming?
From the perspective of the donor, aid is a pure public good. It is both non-excludable
and non-rival. It is non-excludable because my utility depends on total donations (D)
made by the community. The community has no way of preventing me from enjoying the
knowledge that a deserving recipient has been helped. It is non-rival because if the
community collectively helps another, the pleasure that each member of the donor
community enjoys is independent of the number of members of the donating community.
In a situation like this, economists would expect that individuals will free-ride by offering
a small or zero donation.
b. Determine the efficient level of aid.
Suppose that n = population of London. The efficient level of aid can be found by
solving:
max
u  x  (nd )1 / 2
st
M  xd,
 d, x 
L  x  (nd )1 / 2   ( M  x  d )
L
 1    0,
x
1
(nd )  2 n  1
1
2
L

d

1
(nd )  2 n    0,
1
2
1
2
n
1
2
d
1

2
L
 M xd 0

n
d* 
4
Since D = nd, the efficient level of aid is
n2
*
*
D  nd 
4
This could also be solved by setting the sum of the MRS equal to the MRT
u
1
n
MU D D 12 ( D)  2
1
n
MRS i 




MRS i 
and
1
1

2

u
MU x
1
2D
2D 2
i 1
x
n
n
n2
*
Since
MRT  1,
MRS

MRT


1

D


i
4
2D
i 1
1
2
MRT  1
c. What would be the equilibrium level of aid given voluntary contributions?
Let D-i = (n-1)d = equals the contribution of everyone other than person i. Given this
contribution, a utility maximizing agent would:
max
u  x  ( d  D i ) 1 / 2
st
M  xd
 d, x 
L  x  ( d  D i ) 1 / 2   ( M  x  d )
L
 1    0,
x
1
(nd )  2  1
1
2
L

d

1
(d  Di )  2    0,
1
2
1
2
 (nd )
1
2

L
 M xd 0

dE 
Since D = nd, the efficient level of aid is DE = n∙dE = 1/4
1
n4
4. Suppose that the utilities possibilities frontier (UPF) for an economy with two
individuals (A and B) is given by UA = 100 – 0.01∙ (UB)2.
a. Sketch the UPF.
UA
100
75
N
W
R
50
25
25
50
75
100
UB
b. What is the welfare maximizing distribution of utility for a Utilitarian? N
A precise point on the UPF is found by finding the point where the slope of the UPF
equals 1.
dU A
 0.02U B  1
dU B

U B  50
and
U A  100  0.01  (50) 2  100  25  75
c. What is the welfare maximizing distribution of utility for a Rawlsian? R
A precise point can be found by finding the point on the UPF where UA=UB
UA=100 - 0.01(UA)2 which is a quadratic which yields to UA = UB = 61.8
d. What is the welfare maximizing distribution of utility if the Welfare function
is given by W = 2∙(UA)0.5 + 2∙(UB)0.5. W
This one is truly difficult to solve without a program that can solve using numerical
methods. Full marks if you can sketch the solution for b, c, and d.