(Calculus required) Prove Theorem 10.7.3 as follows: (a) Prove the only if part of the

Chapter 10, Problem 9

(choose chapter or problem)

(Calculus required) Prove Theorem 10.7.3 as follows: (a) Prove the only if part of the theorem; that is, show that if C is a productive consumption matrix, then there is a vector x 0 such that x > Cx. (b) Prove the if part of the theorem as follows: Step 1. Show that if there is a vector x 0 such that Cx < x, then x > 0. Step 2. Show that there is a number such that 0 << 1 and Cx < x. Step 3. Show that Cnx < nx for n = 1, 2,.... Step 4. Show that Cn 0 as n . Step 5. By multiplying out, show that (I C)(I + C + C2 ++ Cn1 ) = I Cn for n = 1, 2,.... Step 6. By letting n in Step 5, show that the matrix infinite sum S = I + C + C2 + exists and that (I C)S = I . Step 7. Show that S 0 and that S = (I C)1. Step 8. Show that C is a productive consumption matrix.