Definition. Let p = (P1, P2) E R + denote price vectors, and m E R++ represent income. Define v(p, m): R4+ XR++ +R be a function defined as v(p, m) = max u(x) subject to p.X= m. v is called the indirect utility function, and the value of x that solves the above maximization problem is called the consumer's demanded bundle. We assume that there is a unique demanded bundle at each budget for purposes of convenience. Definition. Let f be a function of n variables defined on a set S for which (tx1, ..., txn) ES whenever t > 0) and (x1, ..., xn) E S. Then, f is said to be homogeneous of degree k if f(t11, ..., tun) = + f(21,..., In) for all (21, ..., In) S and all t > 0. Problem 1. Answer the following questions: (a) Show that v(p, m) is non-increasing in p; that is, if p' p, v(p', m) s v(p, m.). Similarly, v(p, m) is non-decreasing in m. (b) v(p, m) is homogeneous of degree 0 in (p, m). (c) v(p, m) is continuous at all (p, m). Definition. Let p = (P1, P2) E R + denote price vectors, and m E R++ represent income. Define v(p, m): R4+ XR++ +R be a function defined as v(p, m) = max u(x) subject to p.X= m. v is called the indirect utility function, and the value of x that solves the above maximization problem is called the consumer's demanded bundle. We assume that there is a unique demanded bundle at each budget for purposes of convenience. Definition. Let f be a function of n variables defined on a set S for which (tx1, ..., txn) ES whenever t > 0) and (x1, ..., xn) E S. Then, f is said to be homogeneous of degree k if f(t11, ..., tun) = + f(21,..., In) for all (21, ..., In) S and all t > 0. Problem 1. Answer the following questions: (a) Show that v(p, m) is non-increasing in p; that is, if p' p, v(p', m) s v(p, m.). Similarly, v(p, m) is non-decreasing in m. (b) v(p, m) is homogeneous of degree 0 in (p, m). (c) v(p, m) is continuous at all (p, m)