Introduction to Modern Economic Growth Therefore, the initial value of the costate variable µ (0) must be chosen so as to satisfy this equation Notice also that in this problem both the objective function, u (y (t)), and the constraint function, −y (t), are weakly monotone in the state and the control vari- ables, so the stronger form of the transversality condition, (7.55), holds You are asked to verify that this condition is satisfied in Exercise 7.20 7.6 A First Look at Optimal Growth in Continuous Time In this section, we briefly show that the main theorems developed so far apply to the problem of optimal growth, which was introduced in Chapter and then analyzed in discrete time in the previous chapter We will not provide a full treatment of this model here, since this is the topic of the next chapter Consider the neoclassical economy without any population growth and without any technological progress In this case, the optimal growth problem in continuous time can be written as: max∞ [k(t),c(t)]t=0 Z ∞ exp (−ρt) u (c (t)) dt, subject to k˙ (t) = f (k (t)) − δk (t) − c (t) and k (0) > Recall that u : R+ → R is strictly increasing, continuously differen- tiable and strictly concave, while f (·) satisfies our basic assumptions, Assumptions and Clearly, the objective function u (c) is weakly monotone The constraint function, f (k) − δk − c, is decreasing in c, but may be nonmonotone in k HowÊ Ô ever, without loss of any generality we can restrict attention to k (t) ∈ 0, k¯ , where ¡ ¢ k¯ is defined such that f k¯ = δ Increasing the capital stock above this level would reduce output and thus consumption both today and in the future When Ê Ô k (t) ∈ 0, k¯ , the constraint function is also weakly monotone in k and we can apply Theorem 7.14 Let us first set up the current-value Hamiltonian, which, in this case, takes the form (7.58) ˆ (k, c, µ) = u (c (t)) + µ (t) [f (k (t)) − δk (t) − c (t)] , H 351