The government of the island nation of Autarka has become increasingly concerned about the health and wellbeing of low income households. Medical experts stress the importance of a healthy diet in this regard. They have determined that, ideally, a household would consume at least 10kg of fresh fruit and vegetables each week.

The government has proposed implementing a new voucher scheme to ensure that the 10kg per week consumption target is met. Under the government's proposal, the vouchers would be redeemable for fresh fruit and vegetables, but could not be used for other types of food, or for non-food consumption. Moreover, to limit the cost of the voucher scheme, the value of the vouchers sent to a household would be linked to the household's income.

Community groups in Autarka have expressed concern that a voucher scheme is unnec- essarily paternalistic; depriving low income households of the choice of how to spend their income. These groups argue that the government's consumption targets could be met just as well using transfers (monetary payments from the government to the low income households). Transfers, they argue, would empower low income households to make their own consumption decisions.

2.1 Your task

Your task is to determine the optimal voucher value as a function of household income. The voucher must implement the government's target of 10kg of fresh fruit and vegetables consumed per week, while minimising the cost of the vouchers to the government. You are also to assess the community groups' claim that a transfer scheme would implement the government's targets just as well. Do you think that the community groups are motivated by the health outcomes associated with a good diet? Or might they have another objective?

2.2 Household preferences

Using data from household surveys, you have determined that the 'typical' household's preferences are represented by the utility function,

U = (x + 36)y ,

where x represents the quantity of fruit and vegetables consumed, and y represents the

quantity of the composite good consumed. The associate marginal utilities are, MUx=y and MUy=x+36.

The price of fruit and vegetables is Px = $12 per kg, and the price of the composite good is normalised to Py = 1.

question:

Step 1: Derive the marginal rate of substitution for the typical household. (4 marks)

Step 2: Find the household's demands for 'fruit and vegetables', and for the composite good, as functions of the household's weekly income I. Hint: There is a critical income level below which the household's optimal consumption basket is a corner solution. (23 marks)

Step 3: What is the lowest level of income at which a household's consumption of fruit and vegetables will meet the Government's target? (5 marks)

Step 4: Suppose that the government decides to implement its target consumption of fruit and vegetables using transfers. How much would the government need to transfer to a household each week to achieve the 10kg consumption target? Hint: Your answer will be a function of the household's income I. You should assume that transfers cannot be negative. (4 marks)

Step 5: Suppose that, instead of transfers, the government decides to use vouchers re- deemable for fruit and vegetables. What value of voucher would the government need in order to achieve the 10kg consumption target? Hint: Your answer will be a function of the household's income I. You should assume that vouchers cannot have a negative value. (14 marks)