Question

Consider a division problem in which 10 units of food are to be divided between persons A and B. The agents’ preferences are given by uA (x) = −(x − 2)2 and uB (x) = x , respectively.

Assume the initial allocation is (3,7). Is the allocation (2,8) individually rational? Is it envy- free?

Answer 1

Since we have 10 units of food bith of the persons A andB shiuld get the units that maximise their utility keeping in mind the condition of pareto optimality ie the good is allocated in the most efficient manner and is achieved when the utility of a person can be increased without decreasing the utility of another person.

uA (x) = −(x − 2)2   uB (x) = x

(3,7) =-(3-2)2=-2 =7

so utility of A is -2 whereas utility of B is 7.

next when we allocate (2,8) as the units of good between A and B we get

uA (x) = -(2-2)2=0 uB (x)= 8

given below is the table of all possible allocations

uA (x) = −(x − 2)2   uB (x) = x

0,10 4 10

1,9 2 9

2,8 0 8

3,7 -2 7

4,6 -4 6

5,5 -6 5

6,4 -8 4

7,3 -10 3

8,2 -12 2

9,1 -14 1

10,0 -16 0

so we see that when A consumes 2 units and B 8 units the utility of both the consumers is more than the utility when consumer A has 3 units and B has 7 units . also it is observed that the maximum utility of both consumers is when A gets 0 unit and B gets 10 units which is the pareto optimal condition.

the allocation (2,8) is envy free from given endowment of (3,7) but not from the pareto optimal (0,10) because the utility is maximum of both at (0,10) but they are better off at (2,8) from the initial point of (3,7)