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Introduction to Modern Economic Growth 7.5.. Discounted Infinite-Horizon Optimal Control Part of the difficulty, especially regarding the absence of a transversality condi-tion, comes from

Introduction to Modern Economic Growth 7.5 Discounted Infinite-Horizon Optimal Control

Part of the difficulty, especially regarding the absence of a transversality condi-tion, 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:

x(t),y(t)W (x (t) , y (t))≡

Z ∞

0

exp (−ρt) f (x (t) , y (t)) dt with ρ > 0, 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

t→∞x (t)≥ x1 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

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) H (x (t) , y (t) , µ (t))ˆ ≡ f (x (t) , y (t)) + µ (t) g (x (t) , y (t))

which is “autonomous” in the sense that it does not directly depend on time The following result establishes the necessity of a stronger transversality con-dition under some adcon-ditional assumptions, which are typically met in economic applications In preparation for this result, let us refer to the functions f (x, y) and

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