Economic growth and economic development 285

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 ≥ 0 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
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