Samantha purchases food (F) and other goods (Y ) with the utility function U = FY, with MUF= Y and MUY= F . 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 : U = FY , MUF= Y,MUY= F, Income (M) = 12 PF = 2 , PY = 1 a) Utility Maximization condition : MRS = MUF/MUY = PF/PY So,MUF/MUY = PF/PY Y/F = 2 or , Y = 2F (i) Budget constraint : 2F + Y = 12 2F + 2F = 12 Using (i) F = 3 and Y = 6 b) Budget Constraint :2F + Y = M Equilibrium condition :Y = 2F So , 4F = M For , F = 8 , Put F = 8 in above equation . We get M = 32 Hence ,cash subsidy need to be $20 (32 – 12). So , we have , F = 8 , Y = 16 and M = 32 As shown , Our budget line is OADand optimal point is Awhere indifference curve intersects budget line. c) Government give food stamps of$16 that can be only redeemed on food. Hence , As PF = $2 , Samantha can purchase 8 (16/2) units of food….
ce , His new utility function : (f+8)Y (As total Food (F) =f+ 8 i.e funits purchased by her income and 8 units by food stamps) MRS = MUF/MUY = Y/f+8 Equilibrium condition : MRS = PF/PY Y/f+8 = 2 or, Y = 2f+ 16 Budget constraint : 2f+ Y = 12 2f+2f+ 16 = 12 or , f= -1 , Which is not possible , Hence f= 0 Therefore , F = 0 + 8 = 8 So, We have , F = 8 and Y =12 As shown , Our new budget line is CBE and optimal point is B where indifference curve intersects budget line