Economic growth and economic development 271

Introduction to Modern Economic Growth
problems this added level of generality is not necessary. Yet another more general formulation would be to relax the discounted objective function, and write the
objective function as
sup U (x (0) , x (1) , ...).
{x(t)}∞
t=0

Again the added generality in this case is not particularly useful for most of the
problems we are interested in, and the discounted objective function ensures timeconsistency as discussed in the previous chapter.
Of particular importance for us in this chapter is the function V ∗ (x (0)), which
can be thought of as the value function, meaning the value of pursuing the optimal
strategy starting with initial state x (0).
Problem A1 is somewhat abstract. However, it has the advantage of being
tractable and general enough to nest many interesting economic applications. The
next example shows how our canonical optimal growth problem can be put into this
language.
Example 6.1. Recall the optimal growth problem from the previous chapter:
max

{c(t),k(t)}∞
t=0

subject to

∞
X

β t u (c (t))

t=0

k (t + 1) = f (k (t)) − c (t) + (1 − δ) k (t) ,
k (t) ≥ 0 and given k (0). This problem maps into the general formulation here with

a simple one-dimensional state and control variables. In particular, let x (t) = k (t)
and x (t + 1) = k (t + 1). Then use the constraint to write:
c (t) = f (k (t)) − k (t + 1) + (1 − δ) k (t) ,
and substitute this into the objective function to obtain:
max

{k(t+1)}∞
t=0

∞
X
t=0

β t u (f (k (t)) − k (t + 1) + (1 − δ) k (t))

subject to k (t) ≥ 0. Now it can be verified that this problem is a special case

of Problem A1 with U (k (t) , k (t + 1)) = u (f (k (t)) − k (t + 1) + (1 − δ) k (t)) and

the constraint correspondence G (k (t)) given by k (t + 1) ≥ 0 (which is the simplest
257