Introduction to Modern Economic Growth The previous two theorems show that the contraction mapping property is both simple and powerful We will see how powerful it is as we apply to obtain several important results below Nevertheless, beyond some simple cases, such as Example 6.2, it is difficult to check whether an operator is indeed a contraction This may seem particularly difficult in the case of spaces whose elements correspond to functions, which are those that are relevant in the context of dynamic programming The next theorem provides us with sufficient conditions for an operator to be a contraction that are typically straightforward to check For this theorem, let us use the following notation: for a real valued function f (·) and some constant c ∈ R we define (f + c)(x) ≡ f (x) + c Then: Theorem 6.9 (Blackwell’s Sufficient Conditions For a Contraction) Let X ⊆ RK , and B(X) be the space of bounded functions f : X → R defined on X Suppose that T : B(X) → B(X) is an operator satisfying the following two conditions: (1) (monotonicity) For any f, g ∈ B(X) and f (x) ≤ g(x) for all x ∈ X implies (T f )(x) ≤ (T g)(x) for all x ∈ X (2) (discounting) There exists β ∈ (0, 1) such that [T (f + c)](x) ≤ (T f )(x) + βc, for all f ∈ B(X), c ≥ and x ∈ X, Then, T is a contraction with modulus β Proof Let k·k denote the sup norm, so that kf − gk = maxx∈X |f (x) − g (x)| Then, by definition for any f, g ∈ B(X), f (x) ≤ g (x) + kf − gk (T f ) (x) ≤ T [g + kf − gk] (x) (T f ) (x) ≤ (T g) (x) + β kf − gk for any x ∈ X, for any x ∈ X, for any x ∈ X, where the second line applies the operator T on both sides and uses monotonicity, and the third line uses discounting (together with the fact that kf − gk is simply a 271