Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 6: Compensated Demand and Convex Preferences2/17/10Please put your name, section time and TA on your problem set.This problem s et is due on Wednesday, 2/24/10.1. A function f : RN! R is called concave if for all x; y 2 RNand all 2 [0; 1], we havef (x + (1  ) y)  f (x) + (1  ) f (y) .A function f : RN! R is called quasi-concave if for all x; y 2 RNand all 2 [0; 1]:f (x + (1  ) y)  min ff (x) ; f (y)g .(Hint: It su¢ ces to make all the observations for RN= R, i.e. N = 1 andthen notice that your argument does not depend on N = 1.(a) Give a graphical example of a concave and quasi-concave functionfor RN= R, i.e. N = 1. Carefully illustrate the role of the convexcombination formed by two points x; y, namely x + (1  ) y for all 2 [0; 1].(b) In particular, draw a function which is quasiconcave, but fails to beconcave. What is the di¤erence between concave and quasiconcavefunction?(c) Argue formally that every concave function is also a quasi-concavefunction.(d) Argue that all extrema (critical points) of a concave f un ction areglobal maxima. Argue that therefore the …rst-order conditions arenecessary and su¢ c ient for …nding the global maxima with concavefunctions in unconstrained optimization problems on RN.(e) Can you argue that all critical points of a quasiconcave function arelocal maxima. Why or why not?2. A function f : RN! R is called strictly quasi-concave if for all x 6= y 2 RNand all  2 (0; 1):f (x + (1  ) y) > min ff (x) ; f (y)g . (1)An alternative de…nition of a strictly quasi-concave function is that theupper contour set of f is convex, .i.e. for all z 2 RUC (z) =x 2 RNjf (x) > zis c onvex. (2)1(a) Give a graphical example of a quasi-concave function for RN= R,i.e. N = 1 which is not strictly quasiconcave. Carefully illustrate therole of the convex combination formed by two points x; y, namelyx + (1  ) y for all  2 [0; 1].(b) From the example, argue that a strictly quasiconcave function has(at most) one global maximum, whereas a quasiconcave function mayhave many global maxima.(c) The upper contour set is a collection of points (a set). A set S isconvex if x; y 2 S, then any convex combination of x and y is also inthe set, i.e.x+(1  ) y 2 S for all  2 [0; 1]. Illustrate the elementsin the de…nition (2) graphically with the function f being a utilityfunction u (x1; x2) in the two dimensional commodity space (x1; x2).(d) Argue that strict quasiconcavity (1) implies the convexity of the up-per contour sets: (2). A good graphical representation and explana-tion in RN= R, i.e. N = 1 is su¢ cient.(e) Now consider a consumer who has a strictly quasi-concave utilityfunction. Argue that the same consumer has strictly convex prefer-ences. You may recall that we said a consumer has strictly convexpreferences if for all x; y with x  y, we have for all  2 (0; 1) thatx + (1  ) y  x and of course x + (1  ) y  y as well.(f) Now use the geometry of strictly convex preferences (or equivalentlythe strictly quasi-concave u tility function) to show that the own-pricee¤ect of the compensated deman d i is always negative, i.e.@xcip;U@pi< 0. (3)(Hint: It su¢ ces to illustrate this with the in di¤erence curve in thetwo dimensional commodity space (x1; x2).) Is the negative own-pricee¤ect also true for the uncompensated demand?3. A good i is called a normal good if@xi(p; M)@M 0and a good i is called an inferior good if@xi(p; M)@M< 0:A good i is called a Gi¤en good if@xi(p; M)@pi> 02where xiis th e uncompe nsated de mand. Using the Slutsky equation:@xi(p; M)@pj=@xcip; U@pj xj@xi(p; M)@M: (4)explain the relationship between Gi¤en goods and inferior goods.Reading Assignment: NS Chapter