Your task is to determine the optimal voucher value as a function of household income. The voucher must implement the government's target of 5kg 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?

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.

i) Derive the marginal rate of substitution for the typical household.

ii) 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.

iii) What is the lowest level of income at which a household's consumption of fruit and vegetables will meet the Government's target?

iv)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.

v)Suppose that, instead of transfers, the government decides to use vouchers redeemable for fruit and vegetables. What value of the 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.

vi)explain the significance of your analysis section findings in the context of the scenario. In this section, you must address all the issues raised in the scenario. Be sure to base your discussion on the results in your analysis section and avoid assertions or speculation. Use economic terminology as appropriate