1
Supplementary Note on the OG Model
Econ 204A  Prof. Bohn
Here are some comments on Romer’s exposition and on general OG dynamics. Read Romer’s Section
2.9 carefully as it lays out the individual problem. In Romer’s Section 2.10, the key equation for the
dynamics is (2.59), or equivalently (2.67). Logutility/CobbDouglas is a special case. The ‘Speed of
Convergence’ section examines convergence in this special case. Romer then discusses the general
case, but without examining convergence. This note examines convergence in general, as in class, but
provides more specialized cases as example.
Slight change to notation: Let me follow Romer and use w
t
for the wage
per efficiency unit
. Let’s denote perworker
wage income by W
t
= w
t
A
t
= wage per A
t
units of work.
The individual problem with general utility function
Preferences:
u
(
C
1
t
)
+
!
"
u
(
C
2
t
+
1
)
Budget equation for workers:
C
1
t
+
a
t
=
W
t
Budget equation for retirees:
C
2
t
+
1
=
(1
+
r
t
+
1
)
!
a
t
For a graphical analysis, combine the budget equations to obtain the Intertemporal Budget Constraint
W
t
=
C
1
t
+
1
1
+
r
t
+
1
!
C
2
t
+
1
.
For the algebraic solution, there are several approaches: set up a Lagrangian with preferences subject
to IBC; solve IBC for C
1
and insert into preferences, then maximize with respect to C
2
; or solve the
budget equations for consumption, insert into preferences, and maximize with respect to assets. The
latter yields
u
(
W
t
!
a
t
)
+
"
#
u
[(1
+
r
t
+
1
)
#
a
t
]
as the objective function. The first order condition is
!
u
'(
W
t
!
a
t
)
+
#
(1
+
r
t
+
1
)
#
u
'((1
+
r
t
+
1
)
#
a
t
)
=
0
The solution is a function
a
t
=
a
(
W
t
,
r
t
+
1
)
of the exogenous current wage and next period’s interest
rate. Its derivatives can be determined by taking total differentials (or equivalently, apply the Implicit
Function Theorem):
[
!
u
"(
C
1
t
)](
dW
t
!
da
t
)
+
u
'
#
dr
t
+
1
+
(1
+
r
t
+
1
)
#
u
"(
C
2
t
+
1
)
#
[
a
t
dr
t
+
1
+
(1
+
r
t
+
1
)
#
da
t
]
=
0
=>
da
t
=
[
!
u
"(
C
1
t
)]
dW
t
+
[
u
'
#
dr
t
+
1
!
a
t
(1
+
r
t
+
1
)
#
u
"(
C
2
t
+
1
)]
#
dr
t
+
1
[
!
u
"(
C
1
t
)
!
(1
+
r
t
+
1
)
2
#
u
"(
C
2
t
+
1
)]
Because u”<0, the denominator is positive; also
0
<
da
t
+
1
/
dW
t
<
1
. The sign of
da
t
+
1
/
dr
t
+
1
is
ambiguous because substitution (positive) and income effects (negative) conflict. The properties of the
savings rate s=a/W follow from the properties of a. Note that if utility is homothetic, optimal
consumption and asset are proportional to W, so the savings rate depends only on r and not on W.
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 Fall '08
 Staff
 Economics, Capital accumulation, savings rate, Dkt

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