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) ≥ 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) ≥ 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) ≥ (which is the simplest 257