4 3 Welfare theory (answers)

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Microeconomics II 
Undergraduate degree in Economics 
 
Review Exercises – 4.3. Welfare theory 
 
Exercise 33.5. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 403-404 
 
Norton and Ralph have a utility possibility frontier that is given by the following 
equation, 1002 =+ NR UU (where R and N signify Ralph and Norton respectively). 
 
a) If we set Norton’s utility to zero, what is the highest possible utility Ralph can 
achieve? If we set Ralph’s utility to zero, what is the best Norton can do? 
b) Sketch, in a graph, the utility possibility frontier. 
c) Derive an equation for the slope of the above utility possibility curve 
d) Both Ralph and Norton believe that the ideal allocation is given by maximizing 
an appropriate social welfare function. Ralph thinks that 5,75 == NR UU is the 
best distribution of welfare, and presents the maximization solution to a 
weighted-sum-of-the-utilities social welfare function that confirms this 
observation. What was Ralph’s social welfare function? (Hint: What is the slope 
of Ralph’s social welfare function?) 
e) Norton, on the other hand, believes that 9,19 == NR UU is the best distribution. 
What is the social welfare function Norton presents? 
 
Answers: 
 
a) 100
0
1002 =⇒
⎩⎨
⎧
=
=+
R
N
NR U
U
UU
 ; 10
0
1002 =⇒
⎩⎨
⎧
=
=+
N
R
NR U
U
UU
 
 
b) 
 
 
 
 
 
 
 
 
 0 
25 
50 
75 
100 
15 5 10 0 20
UR 
UN 
Utility possibility frontier 
Microeconomics II 
Undergraduate degree in Economics 
 
c) 
22 100100 NRNR UUUU −=⇔=+ 
N
N
R U
dU
dU 2−= 
 
d) 
The generic formula of the weighted-sum-of-the-utilities social welfare function is: 
( ) ( ) RNRN UUUUW αα −+= 1, 
At the optimal point, the isowelfare curves must have the same slope as the utility 
possibility frontier. 
, 2 2 2 21N R
N N
U U N N N N
R
R
W
dU UMRS U U U UWdU
U
α
α
∂
∂= − ⇔ = − ⇔ − = − ⇔ − = −∂ −
∂
 
With 5,75 == NR UU , the slope of the utility possibility frontier is 
10522 −=×−=− NU , therefore 
( ) ( ) RNRN UUUUW 11
1
11
10,
11
1011010
1
+=⇒=⇔−=⇔−=−− αααα
α 
Or, multiplying by 11 (if we multiply the welfare function by a positive constant it does 
not change the structure of the preferences), ( ) RNRN UUUUW += 10, 
Note that 10−=
∂
∂
∂
∂
−
R
N
U
W
U
W
 is still true. 
 
e) 
With 9,19 == NR UU , the slope of the utility possibility frontier is 
18922 −=×−=− NU , therefore 
( ) ( ) RNRN UUUUW 19
1
19
18,
19
1811818
1
+=⇒=⇔−=⇔−=−− αααα
α 
Or, multiplying by 19 (if we multiply the welfare function by a positive constant it does 
not change the structure of the preferences), ( ) RNRN UUUUW += 18, . 
Note that 18−=
∂
∂
∂
∂
−
R
N
U
W
U
W
 is still true.