Chapter 9. First Applications to
Optimal Growth
Our aim is now to look for optimal growth paths, given initial conditions
of the economy. The reader will see that for most problems the calculus of
variations is quite su¢ cient; but we feel that we should present applications
of the maximum principle as well because it is so widely used.
Most important however, is the following caveat: in this chapter we will
present the
traditional
results of optimal growth theory as they have been
expounded in the literature for more than four decades. This bulk of litera
ture is an outgrowth of the seminal paper by Frank Ramsey (1928) in which
the author was looking for optimal investment trajectories maximizing a sum
of
utility
°ows entailed by consumption.
We should stress that the great part of this literature remained very much
theoretical in the sense that it just posited the existence of a concave utility
function that would be accepted by society as a whole. Results were discussed
from a qualitative point of view, on the basis of the phase diagram that gave
the directions of the fundamental variables of the economy ±in general, the
capitallabour ratio, on the one hand, and consumption per person on the
other. Whenever the di/erential equations were solved (through numerical
methods) at the beginning of the sixties, the strangest of results appeared
whatever the utility functions used: for instance, exceedingly high savings
rates (in the order of 6070%). The consequence of this dire situation is that
optimal economic growth always remained in the realm of theory, and no
serious attempt to compare optimal investment policies to actual time paths
was ever carried out.
In the next chapter we will show that the culprit is the very utility func
tion itself. We will analyze in a systematic way the consequences of using
any member of the whole spectrum of utility functions, and we will show the
damage done by them. We will then suggest another, much more direct, way
of approaching optimal growth, leading to applicable results.
For the time being, however, we want to stick to the traditional approach
because the reader should be familiar with the hypotheses, methods and
results of this mainstream approach.
1
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The mainstream problem of optimal growth:
a simpli°ed presentation
We will now present a problem of optimal growth with a single aim in mind:
to illustrate the methods of economic dynamics that were presented in the
last chapter. For that purpose, we shall deal with the simplest model we can
imagine; it is stripped of any complication which might interfere with the
exposition of the way methods work
1
.
Thus we will suppose that capital is the only factor of production; net
income (net of depreciation) is given by
Y
t
=
F
(
K
t
)
, where
F
(
:
)
is strictly
concave
(
F
0
(
:
)
>
0
;
F
00
(
:
)
<
0)
. The problem is to determine
K
t
such that
Max
K
t
Z
1
0
U
(
C
t
)
e
°
it
dt
(1)
subject to the constraint
_
K
t
°
I
t
=
Y
t
±
C
t
(2)
where
i
is a discount rate which applies to all utility °ows
U
(
C
t
)
. The utility
function,
U
(
:
)
, is supposed to be concave (thus
U
0
(
:
)
>
0
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 Fall '08
 OLIVIERDELAGRANDVILLE
 Economics

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