Question

Problem 1. [30 points) Jane's favorite flowers are tulips I and roses 22. Suppose P1 = 10, P2 = 5 and Y = 100. a) (5) Write down Jane's budget constraint (an inequality) and plot all Jane's affordable bundles on the graph (her budget set). Find the slope of a budget line (number). b) [5] Jane's utility function is given by U(11, 12) = V(Intı + ln 12)2 + 7 Propose a simpler utility function that represents the same preferences (give a formula). Explain why your utility represents the same preferences (one sentence). If you do not know the answer to b), assume further U(11, 12) = 1122. c) [5] Plot Jane's indifference curve map (graph), find MRS analytically (give a formula) and find its value at bundle (1,12)=(4,8). Give the economic interpretation of this number (one sentence). d) [5] Write down two equalities that determine the optimal bundle. e) [5] Find the optimal bundle (1,1%). S) [5] Is your solution interior? (a yes-no answer). Argue why corner points can not be optimal here. (Hint: What is the value of Jane's utility function at corner points?)

Please show your work

Answer 1

X . Q) a) BC 10 X,+ 5X23100, Pixi+ P2 X2=Y. Slope = - 12 = -10 = -2 Ans. Xah Budget b.) 4X1, X2) = (en xi ten x2]2+7 7 u2 = (lnxit ln x2)2 + 7. (u²7) 12 = ln xi + ln x2 Now simple utility funcn, u= ln CX,.&2) , MRS=X2/X, Now both shows same preferences, buz two utility funeb have same MRS

C) 4 = X1X2MRS= 240X = Xa UləX2 X, at (4,8), MRS = 8/4=2. Eco Intespoetation, at point (4.8), in order to consume additional Xl, give up a units of Xa, Keeping utility level un changed. d) two inequality, loxi+ 5x2 < 100. at Eqm MRS = PilP2