Economic growth and economic development 286

Introduction to Modern Economic Growth
number). By the converse argument,
g (x) ≤ f (x) + kg − f k

for any x ∈ X,

(T g) (x) ≤ T [f + kg − f k] (x)

for any x ∈ X,

(T g) (x) ≤ (T f ) (x) + β kg − f k

for any x ∈ X

Combining the last two inequalities implies
kT f T gk kf gk ,
Ô

proving that T is a contraction.

We will see that Blackwell’s sufficient conditions are straightforward to check in
many economic applications, including the models of optimal or equilibrium growth.

6.4. Proofs of the Main Dynamic Programming Theorems*
We now prove Theorems 6.1-6.6. We start with a straightforward lemma, which
will be useful in these proofs. For a feasible infinite sequence x = (x (0) , x (1) , ...)
∈ Φ(x (0)) starting at x (0), let
¯ (x) ≡
U

∞

X

β t U (x (t) , x (t + 1))

t=0

be the value of choosing this potentially non-optimal infinite feasible sequence. In
¯ (x) exists and is finite. The next lemma shows that U
¯ (x)
view of Assumption 6.1, U
can be separated into two parts, the current return and the continuation return.
Lemma 6.1. Suppose that Assumption 6.1 holds. Then for any x (0) ∈ X and

any x ∈ Φ(x (0)), we have that

¯ (x) = U(x (0) , x (1)) + β U(x
¯ 0)
U
where x0 = (x (1) , x (2) , ...).
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