Introduction to Modern Economic Growth Exercise 6.4 * (1) Prove the claims made in Example 6.3 and that the differential equation in (6.7) has a unique continuous solution (2) Recall equation (6.8) from Example 6.3 Now apply the same argument to T g and T g˜ and prove that ° ° °T g − T g˜° ≤ M × s × kg − g˜k (3) Applying this argument recursively, prove that for any n ∈ Z+ , we have sn × kg − g˜k kT n g − T n g˜k ≤ M n × n! (4) Using the previous inequality, the fact that for any B < ∞, B n /n! → as n → and the result in Exercise 6.2, prove that the differential equation has a unique continuous solution on [0, s] for any s ∈ R+ Exercise 6.5 * Recall the Implicit Function Theorem from the Mathematical Appendix Here is a slightly simplified version of it: consider the function f (y, x) such that that f : R× [a, b] → R is continuously differentiable with bounded first derivatives In particular, there exists < m < M < ∞ such that ∂f (y, x) ≤M m≤ ∂y for all x and y Then the Implicit Function Theorem states that there exists a continuously differentiable function y : [a, b] → R such that f (y (x) , x) = for all x ∈ [a, b] Provide a proof for this theorem using the Contraction Mapping Theorem, Theorem 6.7 along the following lines: (1) Let C1 ([a, b]) be the space of continuously differentiable functions defined on[a, b] Then for every y ∈ C1 ([a, b]), construct the operator f (y (x) , x) for x ∈ [a, b] M Show that T : C1 ([a, b]) → C1 ([a, b]) and is a contraction T y = y (x) − (2) Applying Theorem 6.7 derive the Implicit Function Theorem Exercise 6.6 * Prove that T defined in (6.15) is a contraction Exercise 6.7 Let us return to Example 6.4 308