Question

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Samantha purchases housing (h) and other goods (?) with the utility function ? = h?. Her income is 120. Housing is measured in units of square feet. The price of a housing is 2 (per square foot) and the price of other goods 1.

a. How much housing does she consume when she maximizes utility?
b. The government has recently completed a study suggesting that everyone should have at least 80 square feet of housing (i.e., h = 80). The government is considering giving a consumer like Samantha a cash subsidy that would enable to get that amount. How large would the cash subsidy need to be? Show her optimal basket with the cash subsidy on an

optimal choice diagram with h on the horizontal axis and ? on the vertical axis.
c. As an alternative to the cash subsidy in part (b), the government is also considering giving consumers like Samantha housing vouchers, that is, vouchers with a cash value that can only be redeemed to buy housing. Verify that if the government gives her vouchers worth $160, she will choose h = 80. Illustrate her optimal choice on an optimal choice diagram.

You may use the same graph you drew in part (b).

Answer 1

Samantha's budget constraint: ph*h+py*y = 2h + y = 120

a) Maximising Samantha's utility constriant

U = hy

Put y = 120 - 2h in the utility function

U = h(120 - 2h) = 120h - 2h²

Maximise wrt h

dU/dh = 120 - 4h = 0

h = 30

y = 120 - 2h = 120 - 60 = 60

At (h,y) = (30,60), U =hy = 30*60 = 1800

b) h = 80

Samantha's new budget constraint : 2h + y = 120 +S

Maximise U = hy wrt the above budget constraint

L = hy + (120 + S - 2h -y)

dL/dh = y - 2 = 0 ....> = y/2

dL/dy = h - = 0 ........> = h

Hence, y = 2h = 2*80 = 160

120 + S = 2h + y = 2*80 + 160 = 320

S = 320 - 120 = 200

c)