February 7, 2021, Christopher D. Carroll PerfForesightCRRA
This handout solves the problem of a perfect foresight consumer with
intertemporally separable CRRA utility who discounts
future utility geometrically by a factor
per period. The finite horizon
solution, whose last period is
, extends to the infinite horizon case if intuitive
‘impatience’ and ‘finite human wealth’ conditions hold.
The consumer’s problem in period is to
| (1) |
subject to the constraints
| (2) |
where is ‘permanent labor income,’ which always grows by a factor
:
| (3) |
It will be convenient to think of both market resources and permanent
noncapital (labor) income
as state variables in this problem. Bellman’s
equation is
| (4) |
The first order condition for this maximization is
| (5) |
and the Envelope theorem tells us that
| (6) |
But the right hand sides of (5) and (6) are identical, so
| (7) |
and similar logic tells us that which (substituting
for
in (6)) gives us the Euler equation for consumption:
| (8) |
Thus, consumption grows in every period by a factor , where we
use the Old English letter
to measure what we will call the “absolute
patience” factor. Specifically, if
| (9) |
we will say that the consumer exhibits ‘absolute impatience’ because this is the
condition that guarantees that the level of consumption will be falling
(and what better definition of absolute impatience could there be than
deliberately spending so much that you will have to cut your spending in the
future?). If the consumer exhibits “absolute patience” (the consumer
wants to defer resources into the future in order to achieve consumption
growth).
The Intertemporal Budget Constraint tells us that the present discounted value of consumption must match the PDV of total resources:
| (10) |
Fact from MathFacts can be used to show that the PDV of labor
income (also called ‘human wealth’
) is
| (11) |
while the PDV of consumption is
| (12) |
We can solve the model by combining (12) and (11) using (10) to obtain:
| (13) |
where is the marginal propensity to consume (MPC) out of overall (human
plus nonhuman) wealth
.
In order to apply to move to the infinite-horizon case (
),
we need to impose the condition
| (14) |
Why? Because if income were expected to grow at a rate greater than the interest rate forever, then the PDV of future income would be infinite; with infinite human wealth, the problem has no well-defined solution. We henceforth call (14) the Finite Human Wealth Condition (FHWC).
Similarly, if consumption starts at a positive level and grows by the factor
, in order for the PDV of consumption to be finite we must
impose:
| (15) |
and we will henceforth call the ‘return patience factor’ whose log is the
‘return patience rate’
(
is the lower-case version of
) and
what (15) says is that the desired growth rate of consumption must be less than
the interest rate in order for the model to have a well-defined solution. This
condition therefore imposes a requirement that ‘impatience’ be greater than
some minimum amount. (For (much) more on the various definitions of
impatience used in this handout, their implications, and parallel conditions for
models with uncertainty, see Carroll (Forthcoming)).
If both the RIC and the FHWC hold, then the model has a well-defined infinite horizon solution,1 as can be seen by realizing that
| (16) |
Substituting these zeros into (13) yields
| (17) |
where is the consumer’s ‘overall’ or ‘total wealth,’ the sum of human and
nonhuman wealth, and
is the infinite-horizon marginal propensity to
consume.
Now consider the question ‘What is the level of that will leave total wealth
intact, allowing the same value of consumption in period
and forever after
(that is, allowing
)?’
The intuitive answer is that the wealth-preserving level of spending is exactly equal to the (properly conceived) interest earnings on one’s total wealth. We call this the ‘sustainable’ level of consumption.
Because human wealth is exactly like any other kind of wealth in this perfect
foresight framework, it is possible to work directly with the level of total wealth
to find the sustainable level of spending. Suppose we assume the consumer
will spend fraction
of total wealth in each period; the
that leaves wealth
intact will be given by
in
|
Thus, the consumer can spend only the interest earnings on wealth, divided
by the return factor
. (The division occurs because the requirement
is to be able to spend the same amount next period, so you need to
account for the time cost of today’s spending by dividing by
which
connects today’s spending to tomorrow’s wealth.) Note that the coefficient
multiplying total wealth in (17) is also divided by
. Thus, whether
the consumer is spending more than the sustainable amount, exactly
the sustainable amount, or less than the sustainable amount depends
upon whether the numerator in (17) is greater than, equal to, or less
than
. As noted before, the consumer will be ‘absolutely impatient’
if
|
Finally, if (which is to say, the interest rate exactly offsets
the time preference rate), then
regardless of the value
of
so that the consumer is ‘poised’ on the knife-edge between
patience and impatience. We refer to such a consumer as ‘absolutely
poised.’ Similarly, we say that a consumer for whom
is ‘return
poised.’2
Equation (17) can be simplified into something a bit easier to handle by
making some approximations. If , then we can use facts from
MathFacts to discover that
|
Substituting this into (17) gives
| (18) |
From this we can see again that whether the consumer is return patient, return
poised, or return impatient depends on the relationship between and
.
Note also that if
then the consumer is infinitely averse to changing the
level of consumption, and so once again the consumer spends exactly the
sustainable amount. (This consumer is ‘absolutely poised’ but ‘return
impatient’).
Now a brief digression on what ‘income’ means in this model. Suppose
for simplicity that the consumer had no capital assets (‘bank balances’
), and suppose that income was expected to stay constant at level
forever. In this case human wealth would be:
|
We found in equation (2) that the level of consumption that leaves ‘wealth’
intact was
| (19) |
So in this case, spending the ‘interest income on human wealth’ corresponds to
spending exactly your labor income. This seems less mysterious if you think of
income as the ‘return’ on your human capital, which is an asset whose value
is
. If you ‘capitalize’ your stream of income using the interest factor
and then spend the interest income on the capitalized stream, it
stands to reason that you are spending the flow of income from that
source.
With constant we can rewrite (18) as
| (20) |
appears three times in this equation, which correspond (in order) to
the income effect, the substitution effect, and the human wealth effect.
To see this, note that an increase in the first
reflects an increase in
the payout rate on total wealth (set
and refer to our formula
above for
, realizing that for small
,
.) That is, it simply
reflects the consequence for consumption of an increase in interest income
– so it captures the ‘income effect’ of interest rates. The second term
corresponds to the subsitution effect, as can be seen from its dependence on the
intertemporal elasticity of substitution
. Finally, the
term
clearly corresponds to human wealth, and therefore the sensitivity of
consumption to
coming through this term corresponds to the human wealth
effect.
The whole problem can be restated more simply by ‘dividing through’ by the
level of permanent income before solving. Hereafter, nonbold variables will be
the normalized bold-letter equivalent, e.g. , and note that if
then from the standpoint of date
,
| (21) |
which means that
| (22) |
Furthermore, the accumulation equations can be rewritten by dividing both
sides by :
| (23) |
| (24) |
Now if we define and
, the original problem can be
rewritten as:
| (25) |
subject to the constraints
| (26) |
and we can go through the same steps as above to find that the solution is
| (27) |
subject to the ‘finite human wealth’ condition
| (28) |
which is the same condition (14) as above, and also subject to the ‘return impatience condition’
| (29) |
which is also the same as above in (15).
Now note that (27) can be rewritten
| (30) |
where is the consumer’s total wealth-to-permanent-labor-income ratio, and
is the ‘marginal propensity to consume’ out of wealth.
As before, whether is rising or falling depends upon the relationship
between
and
. A consumer will be drawing down his
wealth-to-income ratio if
| (31) |
Now substituting the definitions of and
we see that whether
is
rising or falling depends on whether
| (32) |
where is the ‘growth patience factor.’ We call (32) the ‘growth impatience condition’
(GIC),3
and we say that the consumer is ‘growth impatient’ if (32) holds.
Thus, whether the consumer is patient or impatient in the sense of
building up or drawing down a wealth-to-income ratio depends on whether
the growth rate of labor income is less than, equal to, or greater than
the growth rate of consumption. Analogously to our earlier usages, a
consumer for whom (equivalently,
) would be ‘growth
poised.’
To get the intuition for this, consider the case of a consumer with no
nonhuman wealth, . This consumer’s absolute level of consumption will
grow at
and absolute level of income grows at
, but the PDV of
future consumption and future income must be equal. If income is growing faster
than consumption but has the same PDV, consumption must be starting out at a
level higher than income - which is the sense in which this consumer
is impatient (spending more than his income). ‘Growth impatience’ is
therefore the condition that causes consumers with no assets to want to
borrow.
We can now apply the model to answer our first useful question: How large does the model imply the ‘human wealth effect’ is?
For simplicity, assume that . Then the original version of the
approximate formula (18) tells us that the level of consumption will be given
by:
| (33) |
We are interested only in calibrations of the model in which the consumer is
‘growth impatient’ so that so if we define the rate of growth
impatience as
| (34) |
we can write this as
| (35) |
Remembering that imposition of the growth impatience condition is equivalent
to assuming , while the FHWC requires
, it is clear that the
expression
will be positive: The consumer will spend more than
his permanent labor income.
Now suppose we choose plausible values for .
Then (33) becomes:
| (36) |
Now suppose the interest rate changes to , while all other parameters
remain the same. Then (33) becomes:
| (37) |
The point of this example is that for plausible parameter values, the human wealth effect is enormously stronger than the income and substitution effects, so that we should see large drops in consumption when interest rates rise and conversely strong gains when interest rates fall. This is a summary of the main point of the famous paper by Summers (1981); Summers derives formulas for an economy with overlapping generations of finite-lifetime consumers, but those complications do not change the basic message.
The level of saving can be defined as total income minus total consumption:
| (38) |
but since
| (39) |
this can be rewritten as
| (40) |
(where the last approximations come from the assumptions that ) and
that
is ‘small.’ The saving rate (for which we use the letter
to distinguish it from
above) is the ratio of saving to total income (not
just labor income):
| (41) |
The first thing to notice about this expression is that as approaches
infinity, the saving rate asymptotes to
| (42) |
and whether the saving rate is positive or negative depends on whether the consumer is absolutely impatient, absolutely poised, or absolutely patient.4
Finally, if we rewrite this as
| (43) |
then it is apparent that the response of the saving rate to the interest rate is
| (44) |
If we consider almost any plausible configuration of parameter values, say
and
, this translates to a very large response of the saving
rate with respect to
(in the case of the parameter values mentioned above,
).
Consider first a circumstance in which the RIC holds (). In this case, the
perfect foresight unconstrained model does not have a sensible solution because
human wealth is infinite while the model implies that the optimal policy is to
consume a positive proportion of human wealth.
is not a useful
(or plausible!) solution.
The alternative case is when the RIC fails (). Here, the only way to
make sense of the model is to think about the limit of the finite horizon model as
the horizon extends to infinity. This is because behavior reflects a competition
between two pathologies that characterize the infinite horizon solution: It
exhibits a limiting MPC of zero out of total wealth, which includes human
wealth – which approaches infinity. A limiting solution of
is
even less useful than
!
It turns out that the limiting solution is not ambiguous, however. The finite
horizon solution implies that consumption out of human wealth when the end of
life is periods in the future is
| (45) |
whose limit is given by
| (46) |
since the if the FHWC condition fails () then if the RIC
holds, the GIC
must hold, which guarantees
so that
approaches zero as
Given the result from (8) that
|
we can rewrite the value function as
|
but since ,5
this reduces to
| (48) |
where is the discounted value of future consumption growth (that
is, the discounted value of the ratio of future consumption to today’s
consumption).
Carroll (Forthcoming) shows (in an appendix) that , which means
that we can write value as
| (49) |
Carroll, Christopher D. (Forthcoming): “Theoretical Foundations of Buffer Stock Saving,” Quantitative Economics.
Summers, Lawrence H. (1981): “Capital Taxation and Accumulation in a Life Cycle Growth Model,” American Economic Review, 71(4), 533–544, http://www.jstor.org/stable/1806179.