Question

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Suppose there are two people, Jack and Marie, who must split a fixed income of 2000. The marginal utilities of income for Jack and Marie are as follows:

MUj = 510- 3 IC

MUm = 510- 6 IN where Ij and Im are incomes of Jack and Marie respectively.

a) Draw the MU of Jack and Marie in the same Edgeworth box diagram.

b) What is the optimal distribution of income if social welfare function is additive?

c) What is the optimal distribution if society values only the utility of Jack?

Answer 1

a). The MU of Jack and Marie are drawn below :

b). If social welfare function is additive, utilities of both Jack and Marie matters equally. Therefore, the optimal distribution is at the point where both individual's MU are same.

SWF = U_j + U_m

Max SWF = U_j + U_m s.t. I_j +I_m = 2000

Solving the above maximization constraint using the langrangian function we get the following condition for optimum distribution:

MU_j = MU_m

510 - 3 I_j = 510 - 6 I_m , where I_j +I_m = 2000

I_m = 666.67

I_j = 1333.33

c). When society values only the utility of Jack,

SWF = min {U_j, U_m} = U_j s.t. I_j +I_m = 2000

Solving the above maximization constraint using the langrangian function we get the following condition for optimum distribution:

MU_j = 0

510 - 3I_j = 0

I_j = 170

I_m = 2000 - 170 = 1830