ECON 3202/5402 — HW Assignment 05THE UNIVERSITY OF NEW SOUTH WALES School of Economics Term 1, 2020 ECON 3202/5402 — HW Assignment 05 Instructions 1. Solve all the following problems unless indicated as optional. 2. Write your answers as clearly and concisely as possible. 3. Show your work. Partial credit may be awarded only if a substantial part of the answer is provided. 4. The lecturer and tutor reserve the right to deduct marks from and/or refuse to grade illegible homework. 5. Submit your answer sheets as a pdf fifile in the Moodle website for the course. 6. Due date is Monday 30 March 23:00. 7. There is a total of 5 marks available for HW Assignment 05. 1Problem 1 (5 marks). Consider the following functions. (i) f : R → R defifined by f(x) = a x, for a ∈ R. (ii) g : Rn → R defifined by g(x) = a · x, for a ∈ Rn. (iii) h : Rn → Rm defifined by h(x) = A x, for the (m × n) matrix A. Show that all these functions are linear. Problem 2 (optional). The following is the standard formulation of the utility maxi- mization problem (UMP) in classic demand theory. There are n goods in the economy. A bundle of goods is x = (x1, . . . , xn), where xi ≥ 0 for all i = 1, . . . , n. Each good i is sold in a competitive market at price pi > 0. A consumer has preferences represented by a continuous utility function u: Rn+ → R. The consumer is endowed with wealth w > 0 and maximizes her utility subject to a budget constraint: max x∈ Rn+ u(x) subject to n ∑i=1 pi xi ≤ w. (i) Show that the UMP has an optimal solution. (ii) What can go wrong when pi = 0 for some i = 1, . . . , n? Problem 3 (optional). Show that the collection { (1, 1),( −1, 1) } forms a basis for R2; i.e., it is linearly independent and spans R2. Problem 4 (optional). The unit sphere in Rn is the set of points whose distance to 0 is exactly 1: S1 = {z ∈ Rn : 'z' = 1} . A nice property of linear functions is that we can understand their behavior by look- ing at how they transform elements in the unit sphere alone. The following lemma makes this claim precise. Lemma Let T : Rn → R be a linear function. For every x ∈ Rn, there exist z ∈ S1 and α ∈ R such that T(x) = α T(z). Prove the lemma. 2Hint: Note that we can scale (up or down) any vector x ∈ Rn, x = 0, to obtain a vector that points to the same direction and that lies in S1. Namely, for x = 0 consider the vector z = 1 #x# x. See the fifigure below. S1 x z Problem 5 (optional). Let T1 and T2 be two linear functions mapping Rn to R, and let α1, α2 ∈ R be given. Show that the function T defifined on Rn by T(x) = α1 T1(x) + α2 T2(x) is also linear. 3 |