Question

An individual has the utility function: U(x,y)=xy

x = $ spent on education

y = $ spent on other goods

The individual receives a voucher that pays $S for education or a lump-sum subsidy of $S.

1. What is the optimal bundle with the voucher?

2. What is the optimal bundle with the lump-sum subsidy?

3. With which option is she strictly better off?

Answer 1

1. An optimal bundle will be where marginal utility from education=marginal utility from other goods.

A voucher pays $S of subsidy on education. Since this voucher mandates that the person must spend that $S on education, it means that the utility function of the person becomes

U(x+s,y)=(x+s)y=xy+sy

Hence, optimal bundle will be where

dU/dx=dU/dy

dU/dx=y

dU/dy=x+s.

At optimal bundle,

y=x+s

Which means that the person must spend equally on other goods as he/she spends on education (including subsidy). Which means the person will reduce their spending on education by the same amount as subsidy and increase their spending on other goods by the same amount.

Utiity at this level= y(x+s)=xy+sy

2. A lump sum subsidy means that the person can use it as he/she sees fit. Its a lumpsum increase in their income. As the person must spend equally on both to have the highest utility, at optimal bundle now

x+s/2=y+s/2.

Utility at this level

(x+s/2)(y+s/2)

=xy+(x+y)s/2+s²/4

3. The person is better off in the lumpsum subdidy because the total utility is higher in that case.