Prove Theorem 11.9.3 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 λ such that 0 < λ < 1 and Cx* < λx*.
Step 3. Show that CMx * < AMx * .
Step 4. Show that Cn -> 0 as ,
Step 5. By multiplying out, show that
(I - C)(I + C + C2 + ... + Cn-1) = 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 = 1.
Step 7. Show that S ≥ 0 and that S= (1 - C)-1.
Step 8. Show that C is a productive consumption matrix.