Question

18. Suppose that Maggie cares only about chai and bagels. Her utility function is U(C,B) C 0.8B 0.2, where C is the number of cups of chai she drinks in a week, and B is the number of bagels she eats in a week. The price of chai is $2, and the price of bagels is S1. Maggie has S300 to spend per week on chai and bagels. a. What is Maggie's utility-maximizing bundle of chai and bagels? Use the back side of this sheet if you need more space. Suppose that there is another consumer (let's call him Anthony). You don't know anything about his utility function or his income. All you know is that he consumes both goods, chai and bagels. What do you know about Anthony's marginal rate of substitution at his optimal consumption bundle? Why? b.

Answer 1

U = C^0.8 B^0.2 price of chai (p_c) = $2 price of begel (P_B) = $1 Since (M) = $300. Budget Censtroitic C P_c + B P_B = M At utility maximizing point MRS_CB = P_c/P_B U = c^0.3 B^0.2 Mu_c = partial differential U/ partial differential C MU_B = partial differential U/ partial differential B MU_c = 0.3 c^-0.2 B^0.2 MU_c = 0.8 (B/C)^0.2 MU_B = 0.2 c^0.8 B^-0.8 MU_B = 0.2(c/B)^0.8 NRS_CB = MU_C/MU_B NRS_CB = 0.8(B/C)^0.2/0.2(c/B)^0.8 MRS_CB = 4(B/C)^0.2 (B/C)^0.8 MRS_CB = 4 B/C NRS_CB = P_C/P_B 4 B/C = 2/1 B = 2 c/4 \r\n B = 0.5 C Put (2) in budget cenbeint CP_c + B p_B = M c(2) + 0.5 C(1) = 300 2c + 0.5c = 300 c = 300/2.5 c = 120 Put c = 120 in (2) B = 0.5 C B = 0.5(120) B = 60 Maggie otiting maximizing bundle if chai and bagel is ch_a1 = 120, B_agel = 60.