Solutions for Public Finance problem sets
By Anita Gantner and Sofia Franco
Chapter 2: Public goods and Private goods
Exercise 2.1
(a) The equation implied by the Samuelson condition requires that:
1000
 MRS
i 1
i
 m arg inal cos t of public good
The marginal rate of substitution between skating rink and Labatt’s ale for a typical citizen:
U i  X i , Y 
100
Y
MRS i 
 2
U i  X i , Y  Y
X i
So, the Samuelson condition is
100000
 10 .
Y2
(b) Since the MRSi depends only on the public good, the left side of Samuelson condition will
also depend only on the public good. So, this equation uniquely determines the efficient rink
size for Muskrat because:
U i2  X i , Y 
200
  3  0;
2
 Y
Y
(ii) left side of Samuelson condition is a decreasing function of Y.
(i)
Given the marginal cost of the public good, there can be at most one value of y that satisfies that
equation .
The Pareto optimal amount of skating rink size is the Y=100.
1
Exercise 2.2
Since we have quasilinear utility functions, the sum of the MRS i will depend only on the fireworks.
So, we do not need to use the feasibility condition to determine Pareto optimal amount of fireworks
for Cowflop.
U i  X i , Y 
1100
1100
P
Y
The Samuelson condition is:  MRS i  
 Y
PX
i 1
i 1 U i  X i , Y 
X i
MRSi =
1000
ai
2000 Y
1100
So, we have
and note that
i 1
i
1
 (10 2000
i 1
 i  1  2  3  ....  1000 
Y
1000 *1001
.
2
)  1 and Y=626.251.
Exercise 2.3
(a) In order to find the optimal amount of public good as a function of the parameters of the
problem we have to solve the following:
Max
X i ,X j ,Y
U
L
1
 x1 Y 1

 xi Y 1  U iL
i  2,....,500

0

s.t  x j Y 1   U j
j  1,.....,500
 500
500
500
500
 xi   x j  Y   wi   w j  w
 i 1
j 1
i 1
j 1
From the resolution of this problem we get:
2
1   500
1 

Y

xi 


 i 1


 500
500
 x  x Y  w

i
j

i 1
j 1
500
x
j 1
j
( Samuelson
Condition )
( feasibilit y condition )
500
1  1

)
From this system we get: Y  (1   )W    x j (

j 1

As we see, the optimal amount of public good depends on:
(i)
(ii)
(iii)
aggregate income of the community;
on how the private goods are divided between the Larsons and the Olsens;
on the shares α and β.
(b) If α = β , then all individuals have identical Cobb-Douglas utility functions.
The optimal amount of public good, in this case, is reduced to:
Y  (1   )W
and it depends only on the aggregate income but not on how that income is distributed.
Thus, any redistribution of income that leaves the aggregate income in the community unchanged
will have no effect on the efficient allocation of Y.
An example would be:
Suppose the two following income distribution:


where,
w
L
1
L
O
, w2L , w3L ,......, w500
; w1O , w2O ,......., w500
w , w , w ,......, w
L
1
L
2
L
3
L
500

O
; w1O , w2O , w3O ,......., w500
w1L  w1L  
and
w3L  w3L   ,
w2O  w2O  
and
w3O  w3O  

.
3
It is easy to see that both will conduct to the same optimal amount of Y since the sum of all
individuals’ income is the same in both cases, that is, the aggregate income is unchanged.
Exercise 2.4
(a)
Max XC+ 2 Y
C ,T
X  Y  U
D
s.t.  D
X C  X D  Y  W
The Samuelson condition is:
MRS
C
X ,Y
 MRS
D
X ,Y
1 Y

1
2
1
1 
 Y 2 1
2
So, the Pareto optimal amount of Y if both persons are to have positive consumption of private
goods is:
Y=
9
4
9
9
units of private goods
 any allocation (XC,XD) that divides W4
4
between person C and person D in some proportions can be achieved.
And XC+XD = W-
(b)
We want the set of Pareto optimal allocations in which one or the other person consumes no private
goods:
(XC, XD, Y)
4

9
9
At allocation (W- , 0, ), person C gets all the private goods.
4
4
9
3
9
9
and UD = . The allocation (W- , 0, ) is Pareto optimal and the Samuelson
4
2
4
4
9
conditions apply (since Y =
is the solution of equation MRS XC ,Y  MRS XD,Y  1 ).
4
UC = W +

9 9
At allocation (0,W- , ), person D gets all the private goods.
4 4
UC = 3 and UD = W -
3
9
9
. Allocation (W- , 0, ) also is Pareto optimal and satisfies Samuelson
4
4
4
condition.
But we can find other Pareto optimal points in which one or the other person consumes no private
9
9
goods by making people act in a purely selfish way. These points, by contrast with (W- , 0, ) and
4
4
9
9
(W- , 0, ), do not maximize the sum of utilities.
4
4

Person C has no interest in person’s D consumption,
Max XC+ 2 Y
X C ,Y
s.t. X C  Y  W
The solution of this problem is Y=1. So, the allocation that we have is (w-1, 0,1) and the utilities
associated to it are:
U C  w 1  w 
3
3
and U D  1  .
4
2
5
We have a set of Pareto optimal allocations where the utility distributions result from allocations
 9
 9
(W-Y, 0, Y) with Y  1,  . But for Y  1,  , the allocations do not satisfy the Samuelson
 4
 4
conditions.
1
1

1 
We know that MRSC = Y 2 and MRSD = Y 2 . So, take for example:
2

Y = 1 MRSD + MRSC =

Y=
P
1
3
1   1  Y .
2
2
PX
3
MRSD + MRSC = 0.41 + 0.82 =1.23 > 1
2
 Person D has no interest in person’s C consumption and acts in a selfish way.
Max XD+
X D ,Y
Y
s.t. X D  Y  W
1
1 1
. The allocation we have now is ( 0, W- , ) and the utilities
4
4 4
1
3
 w  w .
4
4
The solution of this problem is Y=
are U C  1  3 and U D
By the same reason used in the case before, we have a set of Pareto optimal allocations in which
Person C gets no private goods and where the utility distribution result from allocations of the type
1 9
1 9 
(0, W-Y, Y), with Y   ,  . As before, the allocations (0, W-Y, Y) with Y   ,  do not meet
4 4
4 4
Samuelson condition.
But for all these points be pareto optimal points is necessary to impose a restriction to W.
W has to be superior to 2.25, as we will see in the next question.
In summary,
The set of Pareto optimal allocations in which one or the other person consumes no private goods is:

9 9  9
9

 9
S   X C , X D , Y  :  w  ,0, ;  0, , w  ; w  Y ,0, Y   Y  1,   W  2.25;
4 4  4
4

 4

0, w  Y , Y 

 9
 Y  1,   W  2.25
 4

6
All these points are Pareto optimal since we cannot improve one person’s utility without making the
other person worse off. On the other hand as it was shown, at these Pareto optimal points the
Samuelson condition does not necessarily apply.
( c ) Before look at the graph take a look in the following two tables:
 9
Consider the case where (W-Y, 0, Y) with Y  1,  .
 4
(i)
Y
1
1.5
2
2.25
UC
W+1
W+0.95
W+0.83
W+0.75
UD
1
1.2
1.4
1.5
1 9 
Consider the case where (0, W-Y, Y) with Y   ,  .
4 4
(ii)
Y
UC
UD
0.25
1
W+0.25
0.5
1.41
W+0.21
0.75
1.73
W+0.12
1
2
W
1.5
2.45
W-0.28
2
2.83
W-0.59
2.25
3
W-0.75
In order for all these points (W-Y, 0, Y) be better than points (0, W-Y, Y) for person C we need to
assure that w + 0.75 > 3  w >2.25.
7
By the same logic, for the points (0, W-Y, Y) be better than points (W-Y, 0, Y) for person D we
need to assure that w > 1.25.
If we consider both restrictions then we need to assure that the aggregate income is bigger than 2.25
in order for those allocations be Pareto Optimal.
The graph for an aggregate income W=3 is then:
The utility possibility frontier (UPF) is the set CBAD  . Any point between the straight line AB
9
9
can be achieved by supplying
units of public good and dividing (w- ) units of private goods
4
4
between C and D in some proportions. The curved line CB corresponds to the utility distributions
1 9
that result from allocations (0, W-Y, Y) with Y   ,  and w =3. The curved line AD
4 4
8
 9
corresponds to the utility distributions that result from allocations (W-Y, 0, Y) with Y  1,  and w
 4
=3.
The curved segment DE  does not belong to the UPF since it consists of the utility distributions for
the allocations where person C has all the private goods but the amount of public good is less than
the amount person C would prefer to supply for himself. Symmetrically, the curved segment CF 
also does not belong to the UPF since it consists of the utility distributions corresponding to
allocations where person D gets all the private goods and the amount of public good is inferior to the
amount person d would like to supply for himself.
(d) The straight-line portion of the UPF is the zone achievable with positive private consumption
for both people.
At
the
UPF:
UC U D  X C  2 Y *  X D  Y *
with
Y*=
9
.
4
So
we
have
that
9
. Since the amount of income left for C and D after the public good has
2
9
9
been paid is w - then we have U D  w   U C .
4
4
UC U D  X C  X D 
So, the straight-line portion of UPF is: U D ,U C ;U C  3;3.75 : U D  w 
9
UC
4
.
(e)
For the curved portion (CB) we have that XC = 0 and XD =W-Y. So UC =2 Y and UD=W-Y +
U C2
+ Y . From person C utility function we have that Y 
and plugging this expression into
4
U2 U
person D utility function we describe the top curved part of UPF as being U D  W  C  C .
4
2
For the bottom curved part we set XC = W-Y and XD = 0 and we get UC = W-Y + 2 Y and
UD= Y . So, the bottom part is described as U C  W  U D2  2U D .
9