14.451 Lecture Notes Economic Growth

George-Marios Angeletos
path. But with infinite horizon it is better to consume less and invest more in
period 0, so as to never be forced to consume zero at finite time.
— On the other hand, any c0 that puts the economy below thesaddlepathleads
to so much capital accumulation in the limit that the transversality condition
is violated. Actually, in finite time the economy has cross the golden-rule and
will henceforth become dynamically inefficient. One the economy reaches kgold,
where f 0 (kgold) − δ =0, continuing on the path is dominated by an alternative
feasible path, namely that of investing nothing in new capital and consuming
c = f(kgold) − δkgold thereafter. In other words, the economy is wasting too much
resources in investment and it would better increase consumption.
• Let the function c(k) represent the saddle path. In terms of dynamic programming,
c(k) is simply the optimal policy rule for consumption given capital k. Equivalently,
the optimal policy rule for capital accumulation is given by
or in discrete time
˙k = f(k) − δk − c(k),
kt+1 ≈ G(kt) ≡ f(kt)+(1− δ)kt − c(kt).
• Finally, note that, no matter what is the form of U(c), you could also write the dynamics
in terms of k and λ:
˙λt
λt
= f 0 (kt) − δ − ρ
˙kt = f(kt) − δkt − c(λt),
where c(λ) solves Uc(c) =λ, that is, c(λ) ≡ U −1
c (λ). Note that Ucc < 0 implies c 0 (λ) < 0.
As an exercise, you can draw the phase diagram and analyze the dynamics in terms of
k and λ.
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