Economics 202A
Lecture #2 Outline (version 1.4)
Maurice Obstfeld
I have commented on the ad hoc nature of the saving behavior postulated
by Solow. The next model assumes instead that people plan ahead in making
saving decisions. One advantage of this assumption is that we can do welfare
analysis of economic changes.
The model delivers °normative± answers to
questions such as, °How much
should
a country save?±In its various forms,
the following model has many applications in macroeconomics and public
²nance beyond the analysis of growth.
The RamseyCassKoopmans Model in Discrete Time
I will initially develop this model in discrete time. Then I will go to the
continuoustime limit to derive a mathematical framework comparable to
the Solow model³s. This will also serve to illustrate the principles of
optimal
control theory
, a very useful tool. There are many other approaches to the
derivation, such as the one based on dynamic programming in my notes at
http://www.econ.berkeley.edu/~obstfeld/ftp/perplexed/cts4a.pdf.
Another
possible source is Martin Weitzman³s book
Income, Wealth, and the Maxi
mum Principle
(Harvard University Press, 2003).
Assumptions:
°
There is a single composite good produced with the constantreturns
production function for total output,
Y
=
F
(
K; N
)
:
Here,
N
is popu
lation, which I assume equal to the (fully employed) labor force. (Feel
free to add laboraugmenting technical change as an exercise.)
°
Population growth is
N
t
= (1 +
n
)
N
t
°
1
.
°
A °generation±lives for a period
t
and maximizes
U
t
=
u
(
c
t
) +
°
(1 +
n
)
U
t
+1
,
where
°
(1 +
n
)
<
1
and
c
t
is the consumption of a representative
family member on date
t
. The idea is that you care about your own
consumption and the welfare of your
1 +
n
children.
1
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°
Capital depreciates at the rate
±
2
[0
;
1]
.
Because
U
t
=
u
(
c
t
)+
°
(1+
n
)
u
(
c
t
+1
)+
°
2
(1+
n
)
2
u
(
c
t
+2
)+
°
3
(1+
n
)
3
U
t
+3
,
etc., we may assert that the generation born on date
t
= 0
maximizes
U
0
=
1
X
t
=0
°
t
(1 +
n
)
t
u
(
c
t
)
subject to
K
t
+1
±
K
t
=
F
(
K
t
; N
t
)
±
N
t
c
t
±
±K
t
; K
t
²
0
; K
0
given.
Alternatively, we can express the constraints in the intensive form
k
t
+1
=
1
1 +
n
[
f
(
k
t
) + (1
±
±
)
k
t
±
c
t
]
; k
t
²
0
; k
0
given,
where
k
³
K=N; f
(
k
) =
F
(
k;
1)
:
Ramsey looked at the case
n
= 0
and
°
= 1
. The latter assumption may
seem paradoxical from a mathematical point of view (isn³t the in²nite sum
de²ning
U
0
likely to be divergent then?), but a problem set will show how
Ramsey handled it.
One simpli²cation is to assume the Inada condition on consumption that
lim
c
!
0
u
0
(
c
) =
1
. In that case, we can forget about the interim nonnegativity
constraints on the capital stock. We will never optimally get close to zero
capital, because the marginal utility of consumption would be very high.
It will be useful ²rst to solve the ²nitehorizon problem
max
f
c
t
g
T
X
t
=0
°
t
(1 +
n
)
t
u
(
c
t
)
subject to
k
t
+1
=
1
1 +
n
[
f
(
k
t
) + (1
±
±
)
k
t
±
c
t
]
;
(1 +
n
)
k
T
+1
²
0
; k
0
given.
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 Fall '08
 Robert
 Economics, Boundary value problem, Exogenous growth model

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