Question

Samantha purchases food (F) and other goods (Y ) with the utility function U = FY. Her income is 12. The price of a food is 2 and the price of other goods 1. a) How many units of food does she consume when she maximizes utility? b) The government has recently completed a study suggesting that, for a healthy diet, every consumer should consume at least F = 8 units of food. The government is considering giving a consumer like Samantha a cash subsidy that would induce her to buy F = 8. How large would the cash subsidy need to be? Show her optimal basket with the cash subsidy on an optimal choice diagram with F on the horizontal axis and Y on the vertical axis. c) As an alternative to the cash subsidy in part (b), the government is also considering giving consumers like Samantha food stamps, that is, vouchers with a cash value that can only be redeemed to purchase food. Verify that if the government gives her vouchers worth $16, she will choose F = 8. Illustrate her optimal choice on an optimal choice diagram. (You may use the same graph you drew in part (b).)

Answer 1

Samantha purchases food (F) and other goods (Y ) with the utility function U = FY. Her income is 12. The price of a food is 2 and the price of other goods 1.

a) How many units of food does she consume when she maximizes utility?

She consumes a bundle that satisfies the condition MRS = -price ratio. Here MRS = -Y/F and price ratio is 2/1. This gives us -Y/F = -2/1 or Y = 2F. The budget equation is 12 = 2F + Y and this now becomes 12 = 2F + 2F. This gives F* = 3 and Y* = 6. The optimal bundle is (3, 6).

b) The cash subsidy need to be as large so that her income increases by subsidy S and he now consumes F = 8. This makes the budget equation

(12 + S) = 2(F = 8) + Y

(12 + S) = 2(F = 8) + 2(F = 8)

12 + S = 32

Subsidy should be 32 – 12 = $20.

c) As an alternative to the cash subsidy in part (b), the government is also considering giving consumers like Samantha food stamps, that is, vouchers with a cash value that can only be redeemed to purchase food. If the government gives her vouchers worth $16, her income will now be same as $12. But her consumption of food will now be 8 units because she gets a free voucher of $16 to be spent on good F. With the remaining income of $12, she buys 12 units of good Y. The new bundle is (8, 12).