b. Find the MRS for every consumption bundle. Is the MRS decreasing in x1? c. In a new figure graph her budget set if p1= 1, p2= 1 and m = 5 d. Find her optimal consumption bundle(x1*, x2*) and draw it in the figure. e. Find her optimal consumption bundle (x1**, x2**) if p1= 3, p2= 1 and m = 5 and draw it in the figure. f. How high must m be for the consumer to be able to reach the old utility level U = 10 at the new prices? Draw this new budget line.
Central University of Finance and Economics School of Economics Intermediate Microeconomics, Spring 2011 Homework 2 （Due Date: Wednesday, March 23, 2011） 1. (16 points) In this problem we review your knowledge of calculus. a) Fill out the followingU=x11/3x22/3 U=x1x23 U=lnx1+3lnx2 b) Consider utility function U=x1x23. Interpret the value of the MRS at (2,3) (see last column of the table in a). Is good 1 more or less “valuable” than good 2? How much (approximately) of good 2 would you have to give to a consumer, after taking away 0.0001 of good 1, to keep him indifferent? c) Knowing that ln(x) is a strictly increase function and that ln(xαyβ)=αlnx+βlny, explain why the MRS of functions in the last two rows in the table coincide. 2. (9 points) George is a stamp (x1) collector, but he also likes fancy clothes (x2). His utility function is given by U(x1,x2)=x1+10x2 - 0.5x22 Each stamp costs p1=1 and a piece of his favorite clothing costs p2=3. a) Assuming that his total income is given by m=$15, find his optimal choice of x1 and x2. Is it interior? b) Suppose next year George’s salary doubles, resulting in his higher income m=$30. Find his new demanded quantities of stamps and clothes. Is it interior? c) In point a) and b) what is the marginal utility from one dollar invested in stamps, and in clothing (at the optimal demand). Are they equal? 3. (14 points) Tim consumes two goods, x and y, whose prices are respectively px and py. His income is I. Tim’s utility function can be represented as: U=x0.6y0.4 a) If Tim gives up one unit of good y , how much more of good x must he consume to hold his utility constant? Explain. b) Derive expressions for Tim’s demand for each of the goods in terms of prices and income. c) If Tim’s income equals $200, px=4 and py=2, how much x and y will he consume? d) What is Tim’s marginal utility of income? Explain what it means intuitively, and solve for the number. e) Show that the utility function given by: U=0.6lnx+0.4lny results in the same demand function as in part b).
f) Is good y normal or inferior? Explain, using an Engel curve in your explanation. 4. (15 points) As in the previous problem, Benjamin’s preferences are now described by U(x1,x2)=4lnx1+lnx2 His total income is m=$100, the price of MP3 is one dollar (per each song) (p2=$1). Suppose that the price of a movie drops from p1=$10 to p1=$5. a) By how much the “consumption” of movies changes due to the price drop? b) Are movies ordinary or Giffen goods? Explain why. c) By how much x1 changes because movies are cheaper relative to MP3? (find substitution effect) d) How about the effect of increased purchasing power of Benjamin’s income? (find income effect) e) Is the income effect in d) positive or negative? Why? (Hint: is a movie a normal or inferior good?) f) Show the total change, and the substitution and income effects on the graph. 5. (8 points) Suppose demand for TV is estimated to be Q=3000-10p+2px-4pz+0.2m, if p=60, px=50, pz=150, and m=30,000 Answer the following questions: a) What is the price elasticity of demand? b) What is the cross price elasticity with respect to commodity x? Give an example of what commodity x might be. c) What is the cross elasticity with respect to commodity z? give an example of what commodity z might be. d) What is the income elasticity? 6. (18 points) Consider the following utility function: U=x11/2x21/2 a. Draw the indifference curve that yields U = 4 (find x2 as a function of x1 with U = 4). b. Find the MRS for every consumption bundle. Is the MRS decreasing in x1? c. In a new figure graph her budget set if p1 = 1, p2= 1 and m = 8. d. Find her optimal consumption bundle (x1*, x2*) and draw it in the figure. e. Find her optimal consumption bundle (x1**, x2**) if p1= 2, p2= 1 and m = 8 and draw it in the figure. f. How high must m be for the consumer to be able to reach the old utility level U = 4 at the new prices? Draw this new budget line. g. Find ther optimal consumption given the new prices and the income from point f and draw it in the figure. h. Find the income and substitution effects based on points e. and g. 7. (20 points) Consider the following utility function: U = 2 x1 + x2. a. Draw the indifference curve that yields U = 10.