The government of the island nation of Autarka has become concerned by recent reports that many low income families are residing in substandard housing. (Housing that is too small for the family, or that has significant defects.) The government has committed to implementing new policies to ensure that all families receiving welfare payments can afford housing that meets a "minimum acceptable standard".

Two alternative schemes are under consideration:

1. Increase the weekly welfare payment made to low income families, to a level at which the families can afford to rent acceptable housing.

2. Subsidise the rents of families on welfare to bring the price of acceptable housing down to an affordable level.

The Institute for Cold-Hearted Economics has criticised both proposals for being too expensive. They advocate converting $210 of the existing weekly welfare payment to a rental voucher, thereby forcing low income families to prioritise housing over other consumption. They claim that using a voucher in this way, would meet the government's goals without incurring additional expenditure.

Each low income family receives a weekly payment of $600 from the government (this payment is their only source of income). Surveys of low income families suggest that their preferences are represented by the utility function, U = xy 2 , where x represents the number of acceptable homes rented by the family, and y represents the quantity of the composite good consumed. The associate marginal utilities are, MUx = y 2 and MUy = 2xy. Finally, the weekly rent on an acceptable family home is Px = $210 per week, and the price of the composite good is normalised to Py = 1. (You should interpret x < 1 as the family renting a home that does not meet the minimum acceptable standard.)

Step 1: Derive the marginal rate of substitution for a low income family.

Step 2: Find the family's demands for acceptable housing, and for the composite good, as functions of the family's weekly income I and the price of housing Px . (Note: Do not substitute for family income or the price of housing at this step.)

Step 3: What is the lowest level of income at which a low income family's consumption of acceptable housing will meet the Government's target? What level of utility does the family experience at this level of income? (Hint: Use the price Px = $210 at this step.)

Step 4: What is the highest housing price at which a low income family's consumption of acceptable housing will meet the Government's target? What level of utility does the family experience at this price? (Hint: Use the income I = $600 at this step.)

Step 5: Find a low income family's optimal consumption basket when its income is a $390 payment, plus a $210 voucher redeemable for housing. What level of utility does the family experience in this case? (Hint: Use the price Px = $210 at this step.)