Econ 301Intermediate MicroeconomicsProf. Marek WeretkaProblem set 3(due Thursday, February 11th, before c lass)Problem 1 (Cobb Douglas Utility function)Tony likes nuts x1and berries x2, and his preferences are d esc ribe d by the following u tility funct ionU (x1; x2) = xa1xb2: Find the following variables:1) the optimal fraction (percentage) of income spent on berries;2) the optimal amount of total cash (d ollars) spent on berries;3) the optimal quantity of nuts consumed;4) the slope of the indi¤erence curve at the o ptimal bundle fo r the following values of parameters:a) a = 4, b = 8, p1= 5; p2= 10, m = 60b) a =13, b =13, p1= 4; p2= 1, m = 12c) a =12, b =32, p1= 5; p2= 1, m = 20Hi nt: Inst ead of calculati ng optimal choices using "two secrets of happiness," take advantage of the demandformula s f or Cobb Douglas utility that we deri ved in the class.Problem 2Benjami n spends his time either watching movies (x1) (he use s "on demand" option, cable TV) or listeningto songs - MP3 downloaded from the Internet (x2) . His preferences are described byU (x1; x2) = ln x1+ ln x2Answer the following questio ns:a) Derive Benj amin’s demand for movies and MP3 …les as a function of pr ices p1, p2and h is income m:(do not use Cobb Douglas formula but rathe r deri ve demand u sing "two secrets of happi ness").b) Fix the price of MP3 at p2= 1 a nd income on m = 10: Fi nd the price o¤er curve (give an exactformula x2= f (x1)) and plot it in the commodity space. Find the demand curve x1= f (p1) and plo t it inthe graph (with p1on vertical axis and x1on horizontal axis).c) Is x1an ordinary good or a Gi¤en good? Explain.d) Now …x p1= 1 and p2= 1: In the commodity space, plot the income o¤er curve. In addition, intwo separate gr aphs, plot Engel curves for both movies and M P3 …les. Argue that the two commodities arenormal (not inferior).e) For the demand functio ns from po int a), determine whet her the two goods are gross comp lements,subst itutes or neither.Problem 3 (Perfect Complements)Consider Trevor from our previous problem set who begins his day with strawberry mil kshake. To prepareit, he mixes milk, x1; strawberries x2and does so always in his favorite: proportion 1 glass of milk to 2strawberries. What is his utility function? Answer all the questions from Problem 2, from a) to e) using the sepreferenes.Problem 4 (Perfect Substitutes)Kate’s preferences are de…ned over consumption of two types of app les: Red Delicious (x1) and Jonagold(x2)U (x1; x2) =2x1+x2:Answer al l the quest ions from Problem 2, starting from a) to e) using these preferenes.Problem 5 (Quasilinear preferences)George is a stamp (x1) collector, but he also likes fancy clothes (x2). His ut ility function is given byU (x1; x2) = x1+ 10x212x22:Each stamp costs p1= 1 and a pi ece of his favorite clothing c osts p2= 2110a) Assuming that his total income is given by m = $10; …nd his optimal choice of x1and x2: (Is itinter ior?)b) Suppose next year George’s salary doubles, resulting in his higher income m = $20. Fin d his newdemanded quantities of stamps and clothes. (Is it interior?).c) Hard er: In point a) and b) what is the marg inal utility from one dollar investe d in stamps, and inclothing (at the optimal demand). Are the y equal?Hi nt: Unlike in a Cobb- Douglas utility function, with quasilinear preferen ces we might have