C .D. Carroll Economics Letters 68 2000 67 –77
71
5
≠v ≠v ≠
c
t t
t h
] ]]
v 5
1
t
≠ h
≠ c ≠h
t t
t
16 ≠
h
t 11 h
h
]] 5
u 1 bE v
F G
t t
t 11
≠ h
t h
h
5 u 1 1 2
lbE [v ]
t t
t 11
2.2. Numerical solution As with the standard time-separable model, no analytical solutions to this model appear to exist for
general forms of uncertainty. Numerical solution proceeds as follows. The derivative of uc,h with respect to c can be substituted into Eq. 12 to yield:
2 g 2r
2 g
x h
˜ c h
h 5
bE R v
2 v l
17
s d
t t
t t
t 11 t 11
t 11 2
r g 2gr
x h
˜ c
5 h
bE R v
2 v l
18
s d
t t
t t 11
t 11 t 11
2 1
r g 121 r
2 1
r x
h
˜ c 5 h
b E R
v 2 v
l 19
s d
t t
t t 11
t 11 t 11
x h
Given the existence of the marginal value functions in the next period v and v
, Eqs. 19 and
t 11 t 11
13 can be jointly solved numerically for optimal c and w at some set of grid points in x, h space,
t t
and approximate policy functions can be constructed using any of several methods see Judd, 1998, for a catalog of options. The approximated marginal value functions can be constructed on the same
x, h grid by substituting the optimal values of c and w into the envelope relations 14 and 16.
t t
With these marginal value functions in hand, it is then possible to solve for optimal policy in period t 2 1 and so on to any earlier period by backward recursion.
Thus, to solve the finite-lifetime version of the model, simply note that in the final period of life T the future marginal utilities are equal to zero so that:
c c
v 5
u c , h
T t
t h
h
v 5
u c , h
T t
t
and backward recursion provides policy functions for all previous periods of life. An infinite-horizon solution to the model can be defined as the finite-horizon solution as the horizon approaches infinity.
3. The steady-state
The discussion of numerical solution methods was suited to the use of the model to describe a microeconomic problem like that examined by Dynan 1993 or van de Stadt et al. 1985. Models
with habits have also recently been applied in macroeconomic problems where the representative agent’s steady-state infinite-horizon solution is relevant. It turns out that it is possible to solve
analytically for the steady-state of the perfect-foresight version of the model, as follows.
72 C
.D. Carroll Economics Letters 68 2000 67 –77
Roll Eq. 16 forward one period to get:
h h
h
v 5
u 1
1 2 lb [v
] 20
t 11 t 11
t 12
which can be substituted into 15 to yield:
c x
h h
u 5 v 2 lb [u
1 1 2
lbv ]
21
t t
t 11 t 12
h
Now Eq. 15 can also be rolled forward one period and solved for b [v
]
t 12
1
h x
c
S
]
D
b [v ] 5
v 2
u
f g
t 12 t 11
t 11
l which can be substituted into Eq. 21:
1 2 l
c x
h x
c
F S
]]
D G
u 5 v 2 lb u
1 v
2 u
22
s d
t t
t 11 t 11
t 11
l
c x
x h
c
u 5 v 2 1 2 lb [v
] 2 b lu
2 1 2
lu 23
f g
t t
t 11 t 11
t 11
1 2 l
c x
x h
c
]]] u 5 v 2
v 2
b lu 2
1 2 lu
24
f g
t t
t t 11
t 11
R R 2 1 2
l
c x
h c
S
]]]]
D
u 5 v
2 b lu
2 1 2
lu 25
f g
t t
t 11 t 11
R
x
which can be rolled forward one period and solved for v :
t 11
R
x h
c c
]]]] v
5 b lu
2 1 2
lu 1
u 26
f f g
g
S D
t 11 t 12
t 12 t 11
R 2 1 2 l
Finally, from Eqs. 25 and 14 we have: R 2 1 2
l
c x
h c
S
]]]]
D
u 5 [R bv
] 2
b lu 2
1 2 lu
27
f g
t t 11
t 11 t 11
R
c h
c c
h c
u 5 R b b lu
2 1 2
lu 1
u 2
b lu 2
1 2 lu
28
f s f g
dg f
g
t t 12
t 12 t 11
t 11 t 11
which is the Euler equation for this problem. An alternative form is:
c c
h c
h c
u 2 b [Ru
] 5 b Rb lu
2 1 2
lu 2
lu 2
1 2 lu
29
f s d s
dg
t t 11
t 12 t 12
t 11 t 11
Note that if l 5 0 so that the reference level for ‘habits’ never changes, or if g 5 0 so that habits
c c
should not matter, the problem simplifies, as it should, to u 5 [R bu
] which is the Euler equation
t t 11
for the standard time-separable problem without habits. Now let us assume that there is a perfect-foresight solution to the model in which the growth rate of
consumption and the habit stock are both equal to a constant s, so that the ratio of consumption to
habits is constant at c h 5 x which implies that h 5 c x. Substituting this formula for h into the
t t
t t
equations for the derivatives of c and h gives:
C .D. Carroll Economics Letters 68 2000 67 –77
73
c 2
r g r 21
u 5 c c
x
t t
t c
rg 2g 2r g 12r
u 5 c x
t t
h c
u 5 2 gu x
t t
Note that this implies that we can rewrite:
h c
c
lu 2 1 2 lu 5 u 2glx 2 1 2 l 30
t t
t
Defining k 5 b2glx 2 1 2 l, rolling Eq. 30 forward one and two periods and substituting
into Eq. 28 gives:
c c
c c
u 5 R b u
k 1 u 2
u k
31
s d
t t 12
t 11 t 11
c c
c
u 5 R b u
k 1 u R
b 2 k 32
s d
t t 12
t 11 rg 2g 2r
rg 2g 2r rg 2g 2r
c 5
R b c
k 1 c R
b 2 k 33
s d
t t 12
t 11 2
Now if consumption is growing at rate s each period, then c
5 sc and c
5 s c . Substituting
t 11 t
t 12 t
rg 2g 2r
these expressions into Eq. 33 and dividing both sides by c gives:
t 2
rg 2g 2r rg 2g 2r
1 5 R b
s
s k
d
2 s
R b 2 k
34
rg 2g 2r
or defining h 5 s
this becomes a quadratic equation in h:
2
0 5 1 2 hRb 2 k 2 Rbh k
35 which has the two solutions:
1 ]
R b
h 5 36
1
5
] 2
k yielding the two possible solutions for steady-state growth:
1 r 1g 12r
bR s 5
37
1 r 1g 12r
5
b [gxl 1 1 2 l]
1 r
The first of these potential solutions reduces to s 5 bR
if g 5 0, which matches the usual
formula for consumption growth in the time-separable case. By contrast, the second solution does not
8
reduce to the optimal time-separable solution when g 5 0 and so cannot be an optimum.
8
Another way to see that the second solution cannot be optimal is to note that the implied growth rate is independent of interest rates.
74 C
.D. Carroll Economics Letters 68 2000 67 –77
We can also solve for the steady-state value of x, the ratio of consumption to habits. Expand the
accumulation equation for h: h
5 lc 1 1 2 lh
t 11 t
t
5 lc 1 1 2 l lc
1 1 2
lh
t t 21
t 21 2
5 lc 1 1 1 2 lc
c 1 1 2 l c
c . . .
t t 21
t t 22
t
1 ]]]]]
5 lc
t 2
1
1 2 s
1 2 l
2 1
c sh 5 1 l
f
1 2 s
1 2 l
g
t t
x 5 1 l s 2 1 2 l f
g It is also possible to solve for the level of consumption in a version of the model where labor
income is growing by a constant factor G from period to period and the gross interest factor R is constant both of these conditions will hold in the steady-state of a standard neoclassical growth
model. If consumption grows at rate s every period, then the present discounted value of
9
consumption is:
2
PDV c 5 c 1 1 s R 1 s R 1 . . .
t t
1 ]]]
5 c
t
1 2 s R
Assuming G , R, the present discounted value of labor income is: PDV y 5 y 1 1 G R 1 . . .
t t
y
t
]]] 5
1 2 G R Equating the present discounted value of consumption with the PDV of resources, we have:
y 1
t
]]] ]]]
c 5
1 x
t t
1 2 s R
1 2 G R y
t
]]]
F G
c 5 1 2 s R
1 x
t t
1 2 G R or, substituting the solution for
s from above: y
t 2
1 1
r 1g 12r
]]]
F G
c 5 1 2 R R
b 1
x
t t
1 2 G R
9
In order for this derivation to be valid, it is necessary to have s , R.
C .D. Carroll Economics Letters 68 2000 67 –77
75
4. Dynamics of the perfect foresight model
