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. (4 marks)

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.) (16 marks)

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.) (10 marks)

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.) (10 marks)

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.) (10 marks)