Introduction to Modern Economic Growth 7.5 Discounted Infinite-Horizon Optimal Control Part of the difficulty, especially regarding the absence of a transversality condition, comes from the fact that we did not impose enough structure on the functions f and g As discussed above, our interest is with the growth models where the utility is discounted exponentially Consequently, economically interesting problems often take the following more specific form: Z ∞ exp (−ρt) f (x (t) , y (t)) dt with ρ > 0, (7.46) max W (x (t) , y (t)) ≡ x(t),y(t) subject to (7.47) x˙ (t) = g (x (t) , y (t)) , and (7.48) y (t) ∈ R for all t, x (0) = x0 and lim x (t) ≥ x1 t→∞ Notice that throughout we assume ρ > 0, so that there is indeed discounting The special feature of this problem is that the objective function, f , depends on time only through exponential discounting, while the constraint equation, g, is not a function of time directly The Hamiltonian in this case would be: H (t, x (t) , y (t) , λ (t)) = exp (−ρt) f (x (t) , y (t)) + λ (t) g (x (t) , y (t)) = exp (−ρt) [f (x (t) , y (t)) + µ (t) g (x (t) , y (t))] , where the second line defines µ (t) ≡ exp (ρt) λ (t) (7.49) This equation makes it clear that the Hamiltonian depends on time explicitly only through the exp (−ρt) term In fact, in this case, rather than working with the standard Hamiltonian, we can work with the current-value Hamiltonian, defined as (7.50) ˆ (x (t) , y (t) , µ (t)) ≡ f (x (t) , y (t)) + µ (t) g (x (t) , y (t)) H which is “autonomous” in the sense that it does not directly depend on time The following result establishes the necessity of a stronger transversality condition under some additional assumptions, which are typically met in economic applications In preparation for this result, let us refer to the functions f (x, y) and 345