Question

The government is considering variou

s subsidy and incentive programs to induce low

-

income families to live in better quality housing than they would otherwise live in. Three plans

are

i.

Income subsidy: provide additional income I to a family that can be spent in any way.

ii.

Price subsidy: pay a fixed percentage

p

of a family’s rent.

iii.

Voucher: pay an amount

s

toward a family’s rent, provided the normal rent is at least R.

Suppose commodities can be separated into housing (in amount x) and a general commodity

representing all ot

her goods (in amount z). Suppose a family has uti

l

ity function

U(x,z) = z

4/5

x

1/5

And a monthly income of $500.

The government wishes this family to live in a house that rents for at

least $150. Suppose the prices are each $1.00 per unit.

a.

Without any subsid

y, how much will the family spend on rent?

b.

Under each of the plans above, how much subsidy would be required to induce the family to live

in a $150 house?

Answer 1

A) utility maximizing condition:

MUx/Px= MUz/Pz

0.2*z^0.8/(x^0.8)=0.8*x^0.2/(z^0.2)

z/x=4

z=4x

Budget constraint:,

500=4x+x=5x

X=500/5=100

Z=4x=100*4=400

B) optimal condition:

Z=4x

M=4x+x

M=5x

Required x=150

M=5*150=750

Plan a required, government to give additional 250 money to him.

Plan b) Utility maximizing condition:

MUx/px= MUz/Pz

0.2*z^0.8/(px*x^0.8)=0.8*x^0.2/(z^0.2)

z/x=4px

Z=4*x*px

Budget constraint;,

500=4x*px+px*x

Required x=150

500=4*150*px+px*150

500=600px+150px

Px=500/750=0.67

Required percentage Decrease in price for Consumer=(1-0.67)/1*100=33%

So government needs to pay 33% of the rent price.

Plan 3) government needs to pay 50$ voucher.