Question

3. Suppose Natalie's income is $200 per month, a concert ticket costs $30, a film ticket costs $5, and her preferences are represented by the utility function U(C, F) where C stands for the number of concerts and F stands for the number of films. (a) What is Natalie's marginal utility of film tickets? (b) What is Natalie's marginal utility of concert tickets? (c) What is Natalie's MRS for film tickets with concert tickets? (d) What is the optimal affordable choice of Natalie? e) How much utility does she derive from the optimal affordable choice? (f) Draw to scale Natalie's budget line, the optimal affordable choice, the indifference curve passing through the optimal affordable choice

Answer 1

Income = 200 let quantities be c & f for concert & film tickets respectively. Then, 200 = P_1 c + P-2 f given P_1 = 30 P_2 = 5 are prices of their tickets. 200 = 30 c + 5 f (1) Also, Given utility f^n: U = C^0.75 F^0.25 Marginal utility of concert ticket: a) du/dc = (0.75) (F^0.25/C^0.25) b) du/dF = (0.25) (C^0.75/F^0.75) c) MRS = MU_c/MU_f = (0.75) (f/c)^0.25/(0.25) (c/f)^0.75 \r\n MRS = (3) (f/c) d) optimal Affordable choice: Using (1) & (2) in lagranges Multiplier: dv/d lambda = 200 - 30 c - 5 f (3) dv/dc = (0.75) (f/c)^0.25 - 30 lambda (4) dv/df = (0.25) (c/f)^0.75 - 5 lambda (5) solving (3), (4), (5), (0.75) (f/c)^0.25/30 = (0.25) (c/f)^0.75/5 \r\n 2 = f/c 2 c = f using this in (3), 200 - 30 c - 5 f = 0 200 - 30 c - 10 c = 0 200 = 40 c c = 5 f = 10 Thus, required optimal/affordable choic is 3 concert ticket & 10 film ticket