exercise 2

Exercise 1. (12 points) As we mentioned in class, a utility function U :X C 1R" > R represents the preference relation 2 when it agrees with the relation in the sense that: a > b (Eb u(a) Z u(b) a > b (=b u(a) > u(b) a ~ b 1:} u(a) = u(b) a) Suggest a utility function that represents preferences that are monotonic yet not strong monotonic. Justify your suggestion. (3 points) Now, consider the following three utility functions defined over a consumption set X c R2++. i) U(x, y) = max[2x,' 2y] (mind, this is the max, not the min). ii) U(X,y) = Xzy2 iv) U(x,y) = X2 + y2 b) For each utility functions above, assess whether it represents monotonic, strong monotonic or non-monotonic preferences. (3 points) c) For each of the utility functions above, draw an indifference curve map and by looking at the shape of the indifferenc curves assess whether it represents convex, strict/y con vex, or non-convex preferences. (6 points) Exercise 2. (6 points) Consider a consumer with monotonic preferences over the feasible set X = PM that are represented with the utility func U(x). Assume also that the utility levels are positive for all bundles in R M a) Show that the transformation f(U(x)) = aU(X) + b(U(x))2 where a, b >0 also represents the same preferences. b) Show that the transformation g(U(X)) = U(X) + 2?; x, does not represent the same preferences