Question

2. A society has two individuals, 1 and 2, and is considering three policy proposals, A, B, and C. It is known that the two individuals rank the three policies as follows: Individual 1: A the best, B the 2n best, and C the worst Individual 2: C the best, A the 2n best, and B the worst. e. From the ranking information given above, for each individual, assign a set of utility numbers to represent the given preference relation. f. Consider the utility numbers that you have just assigned to individual 1. Suppose individual 1's utilities are ordinal. Assign a different set of utility numbers that will represent individual 1's preferences. What care do you need to take when choosing the new set of utility numbers for this purpose? g. Consider again the utility numbers that you assigned to individual 1 in e. Suppose now that individual 1's utilities are cardinal. Assign a different set of utility numbers that will represent individual 1's preferences. What care do you need to take when choosing this new set of utility numbers? h. Consider the utility numbers that you assigned to both individuals in e. Suppose individual utilities are level comparable. Assign a different set of utility numbers to each individual that will reflect the fact that their utilities are level comparable. Again, what care do you need to take when choosing the new utility numbers for the two individuals? i. Consider the utility numbers that you assigned to both individuals in e. Suppose now that we can compare utility differences of one individual as well as of two individuals. Assign a different set of utility numbers to each individual that will reflect this fact. Again, what care do you need to take when choosing the new utility numbers for the two individuals? j. Consider the following statement: "Since every individual prefers A to B, they should not adopt policy B." For this statement to be meaningful, do we need to assume that each individual's utility is cardinal? Explain, k. Consider the following statement: "Person 1 gains 80 units of utility in the move from Policy C to Policy A, while person 2 loses 10 units of utility in the move from Policy C to Policy A. Because person 1's gain is greater than person 2's loss in the move from Policy Cto Policy A, nobody can dispute that A is better than C." What implicit assumption that this statement makes regarding k.1 each person's utility being ordinal or cardinal? k.2 interpersonal comparisons of individual utilities? L. Consider the following statement: "Clearly, comparing Policy C and Policy B, the worst off individual under Policy C is better off than the worst off individual under Policy B. What implicit assumption that this statement makes regarding interpersonal comparisons of individual utilities?

Answer 1

e)

We have to assign the utilities in such a way that we give higher magnitude to those alternatives whose rankings are higher in preference relation, so that preference relation is preserved.

Preference relation of Individual 1 : $A\succ B\succ C$

so we well assign highest utility to A, say 6 then to B and C respectively 4 & 2.

Entities	Utility
A	6
B	4
C	2

Similarly,

Preference relation of Individual 2 : $C\succ A\succ B$

hence, we will assign the utility values accordingly,

Entities	Utility
A	5
B	3
C	7

f)

In ordinal utility concept one can only order the preference relation of the consumer/individual but can not assign the cardinal numbers to them.

eg, for individual-1 the ordering of preference relation is,

$A\succ B\succ C$

g)

The care that we should keep in mind is that it should be positive monotone transformation.That is,

$v\left ( x_{i} \right ) = a\times u(x_{i}) + b$

where $u(x_{i})$ is the old assignment of utilities and $v(x_{i})$ is the new assignment of utilities to the preference relation.

Also a and b are constants here.

for a = 2 and b = 1 our new assignment of utilities will be,

$v\left ( A \right ) = (2\times 6 ) +1 = 13$

$v\left ( B \right ) = (2\times 4 ) +1 = 9$

$v\left ( C \right ) = (2\times 2 ) +1 = 5$

j)

No, we don't need to assume the utilities to be cardinal in order to make the given statement because it can be easily observed with the ordinal utility approach.

We have for Individual 1 : $A\succ B\succ C$

For, Individual 2 : $C\succ A\succ B$

so, For both of them $A\succ B$.