Question

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t) and soda (s), and these provide him a utility of: utility = U(t,s) = Suppose the price of twinkies is P each and the price of soda is Ps per cup. Also suppose Paul has I, given to him by his mother, to spend on lunch at school For this general case, which of the following are first order conditions for a utility maximizing bundle of twinkies and soda for Paul? Check all that apply. di ds dl, dL dt After solving these first order conditions, it is clear that t- and s* = Suppose twinkies cost $0.1 each and soda costs $0.25 per cup. Also suppose Paul has $1 , given to him by his mother, to spend on lunch at school At the given prices and income, Paul's utility maximizing combination of twinkies and soda is 1* and s*- Which of the following functions represents Paul's indirect utility function?

Answer 1

U (t, s) = squareroot ts (we have to maximize this) subject to p_t t + p_s s = I (This is budget constraint) We form Lagrangian function, denoted by L max_{t, s} L = squareroot ts - lambda [P_t t + P_s s - I] first order conditions are {t}: dL/dt = 1/2 s^1/2 t^-1/2 - lambda P_t = 0 dL/dt = 1/2 squareroot s/t - lambda P_t = 0 (i) {s}: dL/ds = 1/2 s^-1/2 t^1/2 - lambda P_s = 0 dL/ds = 1/2 squareroot t/s - lambda P_s = 0 (ii) Matching these two conditions with the options, we can see that second from the top & second from the bottom is the answer. From (i) we get 1/2 squareroot s/t = l P_t (iii) From (ii) we get 1/2 squareroot t/s = lambda P_s (iv) Dividing (iii) by (iv) we get s/t = P_t/P_s = > s = P_t/P_s t (v)