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MICRO ECONOMICS WORKS
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Question 5 (Choice) Mrs. Gordon has 3 sons. She spends $1013 on each boy during the holiday season to buy them chocolate and gummy bears. Assume the price of chocolate (good 1) is $2 and the price of gummy bears [good 2) is $5 per pound. Being a clever economist, she has estimated that the utilitj,r functions of the three boys are \"Fred't-Tlax =$i$i \"Ted($l;$2) = it: + 23:2 and \"Ed($11$2) =$1 +522. a. For each boy nd his marginal utility of chocolate and his marginal utility of gummy bears {M U!) b. Find each boy's marginal rate of substitution (REES) between the two goods. For each boy, how does the MRS change along an indifference curve (that is as you go down an indifference curve increasing $1 and decreasing 32)? c. How much chocolate and candy will l'vlrs. Gordon give each boy [assume she wants to maximize each boys utility subject to spending exactly at = $100 for that boy's gift]? d. If she had decided to spend m = $120 on each boy, how would her choice change? e. Ifshe had decided to spend even more? can you predict how her choice would change? Show this graphically1 no need to derive the new choice. Question 2 (30 points): Suppose an individual's utility function for two goods X and Y is given by U193 1,3. : F3'4J1'4) Denote the price of good X by Pr, price of good Y by P; and the income of the consumer by I. a) (2 points) Write down the budget constraint for the individual. 1)) (4 points) Derive the marginal utilities of X and Y. c) (3 points) Derive the expression for the marginal rate of substitution of X for Y. Write down the tangency condition for the utility maximization problem of the individual. {Very Important: In the questions below you must show the steps. In particular, you must clearly write the equations from which you get your answers. Otherwise, you are not going to get any credit.} d) (6 points) Suppose Px = 3, P): = 1, and I = 100. Find out the utility maximizing amounts of X and Y consumed by the consumer. e) (5 points) Now suppose Px falls to 2 while i\":'( = 1, and I = 100. What are the new levels of demand for X and Y? t) (5 points) Diagrammatically show the income and substitution effects of a change in in}[ from 3 to 2 on the consumption of X. You must explicitly specify the optimal consumption bundles for the different sets of prices (appropriately graph and label the actual quanties) but you may qualitatively specify anything else that is deemed relevant (e. g. budget constraints or intermediary changes in X and Y). g) (5 points) Which effect (substitution or income) will move you along a given indifference curve? And for this question, do you move to the to}? or right along the given indifference curve and desc1ibe what this means for the MRS (Le. does the MRS increase or decrease?). Also, explicitly specify what the MRS is equal to before and aer you move along the indifference curve. Extra Credit: (5 points) Numerically derive the income and substitution effect of a change in the price of X from 3 to 2 on the consumption of X. (1) My utility function U x1.*2 depends on the two commodities x1, x2. When p, = 3 and p2 = 7, my demand functions are such that I spend all my income on commodity 2 but never buy any of commodity 1, no matter how large or small my income may be. Does it follow that x, is an economic "bad" for me? If so, explain why. If not, give an alternative explanation. (2) If x1(P1.P2,m) and x2(P1,P2, m) are my demand functions for commodities 1 and 2, and MRS(x1, *2) denotes my Marginal Rate of Substitution at the commodity bundle *1, X2 , what is the numerical value of the expression MRS(x, (1,3,12), x2 (1,3,12))? (Assume x1 and x2 are positive.) (3) Suppose x1=m/(3p1). (a) What is the numerical value of Ex, p, when the price of good 1 is $3? (b) What is the numerical value of Ex, p, when the price of good 2 is $3? (4) I need 1 bike frame and 2 tires to make a bicycle. My utility function for the two goods is U(XF,XT)=min(XF,XT/2). What is my demand function for tires? How many tires do I buy when PF=20, pr=5 and my income is $120 per week? (5) My utility function is U(x1,*2)=2x1 xz and my utility-maximizing bundle (at existing prices and my income) consists of 3 units of x, and 2 units of x2. If p1=4, what must pz equal? (6) Suppose MU,(2,3)=7. Explain in words what this means. (7) Explain in words what means to say: Ex, p, = -2. (8) A consumer consumes only 2 goods, x, and x2. Can both be inferior? If so, use 2 indifference curves and 2 budget lines to show the effects of an increase in m on consumption of good 1 and good 2. If not, draw the same graph, and show how the consumption bundle chosen after the increase in I would have to be in 2 different places. (Bonus: Prove your answer using calculus.) (9) U x,y = 2x+ y. (a) What is the demand for y if the budget constraint is given by x + y = 10? (b) What is the demand for y if the budget constraint is given by 2x + y = 10?Problem 1 (20 points) In this problem, you can assume that the optimal choice is interior. Let Chelsea's utility function be U(C, L) = log(C) + 9 x log(L), where C denotes con- sumption and L leisure. Let T denote time available to split between leisure and work, w denote the wage rate and V 2 0 denote non-labor income (in particular, V can be zero). (a) Derive her optimal choice of C and L as a function of w, T, and V. (5 points) (b) Let h = T- L denote hours worked. Calculate the first derivative of Chelsea's optimal choice of h with respect to w. (5 points) (c) Does your result have the same sign as the substitution or the income effect? What does this tell you? What if V = 0? (5 points) (d) What is the slope of Chelsea's indifference curve at her optimal choice? What is the slope of her budget line? Is it true that the budget line is tangent to the indifference curve at the optimum? (5 points)
