Question

6. Demonstrate that an unequal distribution of goods at a welfare maximum may occur when the weights attached to individual utilities are not equal, and/or when individuals have different utility functions.

Answer 1

Unequal distriburion of goods leads to decline in social welfare maximum when weight attached to individual's utility functions are not equal or thay are having different utility functions.

For example there are two individuals in the economy , with their respective income levels I1 and I2. which is being spent on single good X.

U(X) = ln(X) is the same utiity functions for both the individuals.

Utilitarian social welfare function : This utility fiunction sumps up the utility functions of all the individuals so that the total welfare of society can be obtained. All people are treated equally, reagrdless initial level of utility function. Suppose U1 and U2 the utility levels of two individuals. The utlity function can be expressed as below

W ( U1, U2 ) = U1 + U2

The unequal distribution or loss in utility leads to fall in social welfare. However, any rise in utility level of both the individuals leads to increase in total social welfare in economics.

Rawlsian welfare function : This welfare function measure the level of social welfare by measuring the utility level of worst off members of society and and maximising welfare would mean maximising the income of the poorest person without regard for the incomes of the others. The utility function can be expressed as below

W(U1, U2) = min (U1, U2)

For example U1 = 4.38 and U2 = 2.3

This functions states to redistribute income to individual who is having less utility ie., W = 2.3 \to bring equality and increase welfare of poorest individual.

Bergson-Samuelson Social Welfare Function: This welfare function is an ordinal index of society's welfare and sum up utilities of each and every individuals. The utility function can be seen as below

W= W(U₁, U₂,U₃…………. , U_n)

W = Social welfare and U1, U2 ,..., Un is the utility levels.It represent the ordinal utility indices of different individuals of the society.