Recitation 6
Akshat Gautam and Susannah Scanlon *
October 28, 2021
We’ll cover: (i) the Permanent Income Hypothesis, (ii) the introduction of uncertainty, (iii)
Rational Expectations, (iv) The Hall Consumption Model. The presence of uncertainty has very
profound effects on what our models predict and how they operate.
1 Permanent Income Hypothesis
Throughout I’m going to assume that r is fixed, and we will focus attention only on the house-
hold. Remember the classic problem we’ve been trying to solve all this time:
max
{ct,bt+1}∞t=0
∞∑
t=0
βtu(ct)
such that ct + bt+1 ≤ yt + (1 + r)bt
How do we solve this? Well we can express it as a series of Lagrangians, as we’ve done up to this
point:
L =
∞∑
t=0
βt{u(ct)− λt(ct + bt+1 − yt − (1 + r)bt)}
First order conditions with respect to out two choice variables, ct and bt+1 gives us:
{ct} :βt(u′(ct)− λt) = 0
∴ u′(ct) = λt
{bt+1} :− βtλt + βt+1λt+1(1 + r) = 0
λt = λt+1β(1 + r)
By substitution, we get our Euler Equation again:
u′(ct) = β(1 + r)u
′(ct+1) (1)
Suppose that β(1 + r) 6= 1; what would happen? Well if β(1 + r) < 1, then u′(ct) < u′(ct+1), which
implies ct > ct+1. What is the limit as we go further and further into the future? 0! By identical
reasoning, if β(1+r) > 1, then consumption must diverge to infinity! Therefore, we usually assume
*Updated October 28, 2021. Please let me know if you find any errors—no matter how small; just shoot me an
email to . Thanks to Miguel, Joe, Eddie and Susie for writing these last years, and letting us
copy them with minimal edits.
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that β(1 + r) = 1; this is often called the complete markets assumption. We’ll come back
to why it’s called this later (probably in next recitation), but for now, just accept that we’re
assuming that β(1 + r) = 1.
So if β(1 + r) = 1, then we know that ct = ct+1 = c , i.e. some fixed cfor every t; consumption
is perfectly smoothed across all time periods! Remember how we’ve gone about solving these
problems? Find the EE, then plug it into the budget constraint! These are included in the notes
from R4, where we find that-
∞∑
t=1
1
(1 + r)t−1
ct = a0 +
∞∑
t=1
1
(1 + r)t−1
yt (2)
Substituting in our next condition that ct = ct+1 = c, we find:
∞∑
t=1
1
(1 + r)t−1
ct = c
∞∑
t=1
1
(1 + r)t−1
= c
∞∑
t=0
1
(1 + r)t
= c
(
1 + r
r
)
Equation 12 then becomes:
c
(
1 + r
r
)
= a0 +
∞∑
t=1
1
(1 + r)t−1
yt
Therefore:
c =
r
1 + r
(
a0 +
∞∑
t=1
1
(1 + r)t−1
yt
)
(3)
So we can pin down exactly what consumption will be in every period, assuming we know the
entire path of {yt}∞t=0. Suppose, then, that we increase the amount of income in time period 1
by $1; how much would consumption increase by? Only r
1+r
, which is considerably less than one.
Note that consumption is a function only of the present value of future income streams, plus some
initial wealth (and of course the interest rate, r). What this is telling us is that agents take into
account their entire future path of income when thinking about consuming something
today. In doing so, they’re able to perfectly smooth, even if their income path is very uneven, by
borrowing when their income is lower and saving when it is higher.
The PIH is often illustrated graphically as shown in Figure 1. The basic idea is this: when
we’re young, we borrow heavily, either to invest in our education or in housing, knowing that
when we are older we will make more money. Once we are wealthier, we recognise the need
to both (i) pay back our debts from when we were young, and (ii) save for when we retire, and
hence we save heavily. Once we stop working, we live on our savings. Throughout, we consume the
same amount. Now of course this isn’t exactly what happens, but it captures something which is
absolutely true that the naive Keynesian consumption function cannot; that we take into account
the future when we think about consumption and savings decisions today1.
1It’s worth noting that some economists have taken this result very seriously and actually put in place the PIH
in their own lives; not uncommon for PhD Econ students to amass $10,000s in debt
2
Figure 1: Graphical Illustration of the Permanent Income Hypothesis
2 Introducing Uncertainty
Throughout the above, and indeed the entire course thus far, we haven’t once mentioned that fact
that some things are uncertain. Given that models of intertemporal decision making are motivated
by a desire to understand how choices today influence options tomorrow, we should be concerned
about the fact that the future is uncertain. Going back to our previous example, suppose we
didn’t know what y was going to be in the future, how would we choose consumption? We need to
choose consumption thats equal to the present value of future income streams… but now we don’t
know those future income streams!!
How do we solve this problem? Well, by having people form beliefs! In order to do that, we
have to implement a bit of structure on how we think yt might evolve in the future. In order to see
this, let’s think about a few real world examples:
� When you think about how hot or cold it will be tomorrow, how do you do it?
� When thinking about which library to study in based on how busy you think it will be, how
do you choose?
� How do you pick your lottery numbers?
Each one of these systems of thinking can be formalised. I’m going to argue that each of these
questions have answers that can be roughly modelled by the following three equations, where �t+1
is a random shock, i.e. something that we can’t predict, that affects our variable of interest at
time t+ 1:
yt+1 = ρyt + �t+1 (4)
yt+1 = y + �t+1 (5)
yt+1 = �t+1 (6)
Let’s consider each of these in turn, working with the assumption that �t is an iid random
variable, i.e. it’s distribution is fixed over time, and we draw from out randomly, without reference
3
to previous draws.
Equation 4
Equation 4 answers the first question; what the temperature is tomorrow is probably going to be
pretty similar to what it was today. When we think some variable Yt has this relationship, where
Yt+1 is a function of ρYt and some random shock, we call it an AR(1) process, which stands for
autoregressive(1); i.e. we can regress Yt+1 on itself one period back. As you might expect, an
AR(2) takes the form:
Yt+1 = ρYt + φYt−1 + �t+1
For simplicity, we’ll stick to AR(1)s for now. Remember that at time t, we don’t know what Yt+1
is going to be; why is that? Because �t+1 is not something we can predict. It is an iid random
variable. What is the expected value of Yt+1? Well if we take the expectation conditional on all
the information available at time t, which I will denote using the notation Et, then:
Et[Yt+1] = Et[ρYt + �t+1]
= Et[ρYt] + Et[�t+1]
= ρEt[Yt] + Et[�t+1]
But if we’re conditioning on all the information available at time t, then it must be the case that
we know what Yt is, right? The value of Yt has been realised! It is a non-random constant now.
Therefore:
Et[Yt+1] = ρYt + Et[�t+1]
What value do you think Et[�t+1] ought to take? Well if it is a random shock, that we cannot
predict, then we must think that in expectation it is 0. If instead we expected it to be some
positive constant, then that wouldn’t be a random shock; note that if we thought for example
that there was some trend in our process, i.e. Et[�t+1] = a, then we could equally describe the
process described in Equation 4 as:
yt+1 = a+ ρyt + �̃t+1
And here it should be obvious that the expected value of �̃t+1 = 0. So when we talk about random
shocks, we’re always talking about something that is mean equal to 0. So, our best prediction
of tomorrow’s Y ?
Et[Yt+1] = ρYt
What about Yt+j? Given our information at time t, what should our expectation of Yt+j be? Recall
that our process tells us that Yt+j = ρYt+j−1 + �t+j ; now consider the following:
Et[Yt+j ] = Et[ρYt+j−1 + �t+j ]
= ρEt[Yt+j−1]
By identical logic we can keep going, substituting in our governing process as outlined in Equation
14, until we get:
Et[Yt+j ] = ρjYt (7)
4
Typically in economics, we assume that ρ ≤ 1; why is that? Suppose ρ > 1; what would happen to
Yt? It would explode; put mathematically, we can see that if ρ > 1, then:
lim
j→∞
Et[Yt+j ] = lim
j→∞
ρjYt =∞
In economics, nothing goes to infinity, ever! Therefore any process that we’re using to model
economic behaviour should not lead to the conclusion that such behaviour will diverge to infinity!
Equation 5
The way you might think about how busy a given library is going to be is by assuming that it will
be as busy as it has been on average, plus some random shock. If we let Y t =
1
t
∑t
i=0 Yi, then
we get the expression in Equation 5. What should we expect in terms of how busy the library is
tomorrow? Using the same procedure outlined above:
Et[Yt+1] = Et[Y + �t+1]
= Y
Clearly,
Et[Yt+2] = Et[Y + �t+1 + �t+2]
= Y
And so on, for as far forward as you care to look.
Equation 6
This is just a simplified version of Equation 5; here, we have no idea what tomorrow’s Yt+1 is going
to be. So, as with the lottery, we just have to guess.
A Side Note; what do we mean by conditional on all information at time t?
I mentioned above that Et denotes the expectation conditional on all information at time t; I could
just have easily said, let Ωt denote the set of all information available at time t, and then written
E[.|Ωt]. But what constitutes the set Ωt?
This is a good and important question. In the world at large, identifying the set of all available
information is an impossible task; for example, when taking the expectation of tomorrow’s prices,
we have information on the price of copper in futures markets in Bulgaria, but do we really make
use of this information? It is available! But processing all the available information in the world
pertinent to our expectations is too difficult.
In our restricted context, i.e. in our model, we don’t need to worry about all of this extra
baggage. We have written the expression for Yt as determined solely by past observations of Yt
and shocks �t. Claiming that we’re using all information available today is simply to say that we
observe and take into account Yt when forecasting Yt+1.
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3 Rational Expectations
What are Rational Expectations?
Whilst it may not be immediately obvious, what we just did above in forecasting the expected
future values of Yt+1 was an exercise in Rational Expectations. What do we mean by Rational
Expectations? We can summarise this concept using two distinct notions; expectations are rational
if and only if :
� The belief framework is consistent with the model framework.
� All information available is used to form beliefs.
Consider the first component; what this is telling us is that if in our model, Yt evolves according
to the process outlined in Equation 4, then agents in our model believe that it evolves
according to Equation 4. This means they are using the correct system to forecast what will
happen to Yt in the future.
The second component builds on our previous discussion by emphasising it. Rational expec-
tations requires that agents do not ignore information that is pertinent to them. In our
model, that means they make use of all available information to forecast the future.
It should be clear to you that in all the examples above, we obeyed these two principles, and
hence we solved the problem as though we had rational expectations.
What are not Rational Expectations?
A common mistake is to assume that rational expectations means that the agents are always
’correct’. Let’s go back to our previous examples, and suppose that our random shock is distributed
continuously (for example, suppose it is a normal random variable); remember that we found that
our best guess of Yt+1 according to Equation 4 was ρYt? Well the probability that Yt+1 = ρYt is
zero. Why? Given �t+1 is continuous, although in expectation it is zero, the probability that it is
zero is zero. In formal language, we say that Yt+1 6= ρYt almost surely. So rational expectations
doesn’t require that agents can predict the future!
Why are Rational Expectation reasonable:
They are enormously useful. Under the framework of Rational Expectations, we can produce
consistent and coherent results from our models. To see this, suppose that we do away with the
first component of rational expectations, i.e. we allow agents to have false systems of beliefs. How
would you model this? How would you select an alternative belief system? What justification could
you give for this belief system?
Crucially, how could you claim that agents would persistently believe in a false system?
It is possible to produce Rational Expectations using a learning model, where agents are initially
agnostic about the system governing a process, but through observation come to understand the
process. If you’re going to claim that they’re not doing that, then why aren’t they? Are people
stupid? Do people not respond to incentives?
Suppose we relax the second term, so that we allow agents to ignore some information; which
information do we allow them to ignore? Why that information and not some other? Is the
information that agents do observe something you’re imposing? If so, what justification can you
give for that? Why wouldn’t agents use all available information?
6
The fact remains that even at the research frontier, we (as in, humanity, not just
economists) do not know how people form expectations. It is very difficult to credibly
offer an alternative mechanism that is robust to the learning argument.
Why are Rational Expectations not reasonable:
Despite all of this, it should be clear to all of you that people don’t have rational expectations. It is
clearly the case that people don’t perfectly understand the systems that govern random processes
relevant to them, and it’s also clearly the case that we don’t think about the future price of futures
markets of copper in Bulgaria, even though it might be relevant and available information to us.
Don’t worry, economists know this. Frontier work in economics focuses extensively on what
happens when we relax both of these features of Rational Expectations.
A Small but ’Important’ rant:
Something that really bothers me when I hear it in the news is when people say, ’God, these
stupid economists believing that people form rational expectations! What idiots!’ Whilst there
are a few eccentrics out there who really do believe in rational expectations, the vast majority of
the economics profession are aware that rational expectations are not an actual representation of
reality; they are a model. It’s also worth noting that it’s a model that does extremely well in
most cases, in that it produces results that have been enormously useful, not just in understanding
the world but in producing more effective policy that has improved the lives of millions.
Challenging a model by offering a credible alternative framework is both a noble and valuable
pursuit. Shouting that rational expectations aren’t true is just a waste of air; there is nothing
impressive or valuable in just decrying a model framework because of its implausibility, nothing
creative in that, nothing worthy of praise. The reason why Tversky and Kahneman, and Thaler
won Nobel Prizes in this area is not because they said ’Rational Expectations is nonsense’. They
won Nobel Prizes because they were able to make statements of the following form; ’expectations
are not formed rationally, because of this consistent and modellable characteristic, that
governs broad human behaviour across individuals and over time. By identifying a system by
which behaviour differs from the rational paradigm, these economists were able to provide a
coherent and credible system that can challenge rational expectations, and the whole rational
framework. This is an extraordinary achievement, and is worthy of praise.
For those of you unconvinced of the progress economics has made since the rational expecta-
tions revolution (which occurred nearly 50 years ago, at a time when less than 1% of the US
population owned a computer), I would draw your attention to work by Mike Woodford, here
at Columbia. Or Hassan Afrouzi, or Mark Dean, or Alessandra Casella. So there are four Pro-
fessors just at Columbia, whose research deals almost exclusively with deviations from Rational
Expectations.
However I do want to reiterate that although Rational Expectations are clearly false, that does
not mean they are not useful. In most settings, rational expectations are a very good proxy
for real world behaviour. They are also extremely tractable; this feature may not appear hugely
important to you now, but this is an enormously non-trivial advantage of this modelling framework.
Without rational expectations, many models become literally impossible to solve. Okay, that’s
my rant over!
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4 The Hall Consumption Model
So now we’re going to bring in uncertainty and Rational Expectations to think about a more
complicated version of the problem we saw in Section 2. Now we suppose that agents are uncertain
about the future, because {yt}∞t=0 is random and unknown. For solving this problem, recall the
Lagrangian we found towards the start of Section 2; we will replicate it here, including the fact
that we must take expectations about future variables:
L =
∞∑
t=0
βtEt [u(ct)− λt(ct + bt+1 − yt − (1 + r)bt)] (8)
Take first order conditions, starting with consumption
{ct} : Et
[
βt(u′(ct)− λt)
]
= 0
: u′(ct) = λt
How did we move from the first to the second line? Because the assumption of Rational Expectations
tells us that the agent takes into account all information available today; ct and λt are made up
of information available today, which is known! Therefore, we can get rid of the expectation term.
Now we take derivative with respect to bt+1:
{bt+1} : Et
[
−βtλt + βt+1λt+1(1 + r)
]
= 0
: λt = β(1 + r)Et[λt+1]
At time t, all components except λt+1 are known! But we don’t know λt+1… why? We know that
λt+1 = u
′(ct+1), but we don’t know what income will be tomorrow! So we can’t possibly now what
we’ll consume tomorrow. Substituting in our two FOCs, we can get the following expression:
u′(ct) = β(1 + r)Et[u′(ct+1)] (9)
This looks familiar! It is our Euler Equation, only now we have an expectation term in there.
A sidenote on expectations
Let’s assume, as we did before, that β(1 + r) = 1. Will consumption therefore be the same (in
expectation) in every period?
NO!!!!
To see this very clearly, suppose we have some random variable, xt that has mean 0 and variance
σ2. Then it follows that:
E[x2t ] = σ
2
It is not true, that:
E[xt] = σ
Why does this happen? Well because we can’t just plug inverses and rearrange equations when
we’re operating with expectations like we can when we work with algebra. Remember that the
strict interpretation of the expectation of a variable, x, whose support is from a to b, is given by:
E[x] =
∫ b
a
xf(x)dx
8
Similarly:
E[x2] =
∫ b
a
x2f(x)dx
Clearly, then, in general it will not be the case that:
(
E[x2]
) 1
2 =
√∫ b
a
x2f(x)dx =
∫ b
a
xf(x)dx = E[x]
So be careful!! In general:
u′(ct) = Et[u′(ct+1)] ; ct = Et[ct+1] (10)
Back to the model
It turns out that there is one and only one utility function where the above implicative statement
does hold: quadratic utility. To see this consider the following:
u(ct) = ct −
a
2
c2t
∴ u′(ct) = 1− act
∴ Et[u′(ct+1] = Et[1− act+1]
Going back to Equation 9, and assuming β(1 + r) = 1, then we find that:
1− act = Et[1− act+1]
= 1− aEt[ct+1]
So it must be the case that, with quadratic utility:
ct = Et[ct+1] (11)
For the Hall model, we’re going to assume quadratic utility. Note, that Hall’s model results are
robust to other forms of utility! It is just that quadratic is very easy to use and makes the argument
very clearly. So now we’ve found our Euler Equation, we need to stick it into our budget constraint
to solve for consumption. Recall that, today, we face the following budget constrain:
ct + bt+1 = yt + (1 + r)bt (12)
It follows that:
ct+1 + bt+2 = yt+1 + (1 + r)bt+1
∴ bt+1 =
1
1 + r
(ct+1 + bt+2 − yt+1)
But we know that yt+1 is unknown! So our value for bt+1 must be in expectation. Using this,
and implementing the result into Equation 12, we find that today’s budget constraint is given by:
ct + Et
[
1
1 + r
(ct+1 + bt+2 − yt+1)
]
= yt + (1 + r)bt
9
Re-arrangement gives us:
ct +
1
1 + r
Et[ct+1] +
1
1 + r
Et[bt+2] = yt +
1
1 + r
Et[yt+1] + (1 + r)bt (13)
We might recognize this pattern from before, no? We can keep substituting in expressions for bt+j ,
forever! In the end, we get an expression that looks like this:
∞∑
j=0
(
1
1 + r
)j
Et[ct+j ] = (1 + r)bt +
∞∑
j=0
(
1
1 + r
)j
Et[yt+j ] (14)
This is basically identical to the budget constraint we laid out before, only now we have expectations!
Now we make use of the result in Equation 11; note that if ct = Et[ct+1], then it must also be true
that:
ct+k = Et+j [ct+j+1]
∴ Et[ct+k] = Et [Et+j [ct+j+1]]
By the Law of Iterated Expectations:
Et [Et+j [ct+j+1]] = Et [ct+j+1]
So, if we have ct = Et[ct+1], then we know that:
Et[ct+1] = Et[ct+2] = ct
So, for all j,
Et[ct+j ] = ct (15)
We can therefore simplify the left hand side of Equation 14:
∞∑
j=0
(
1
1 + r
)j
Et[ct+j ] =
∞∑
j=0
(
1
1 + r
)j
ct
= ct
∞∑
j=0
(
1
1 + r
)j
= ct
1 + r
r
So we can update Equation 14, to show that:
ct
1 + r
r
= (1 + r)bt +
∞∑
j=0
(
1
1 + r
)j
Et[yt+j ]
With simple rearrangement:
ct = rbt +
1 + r
r
∞∑
j=0
(
1
1 + r
)j
Et[yt+j ] (16)
10
Assuming a process for yt
Without further restrictions on how yt changes over time, we can go no further. So, in order for
us to see a bit more about what the model is telling us, let’s assume the process described in
Equation 14; namely that:
yt+1 = ρyt + �t+1 (17)
I’m going to assume the following about �t+1; namely that it is:
� IID
� Distributed normally with mean 0 and variance σ2
Back to the model again!
We now simplify the right hand side of Equation 14:
∞∑
j=0
(
1
1 + r
)j
Et[yt+j ] =
∞∑
j=0
(
ρ
1 + r
)j
yt
= yt
∞∑
j=0
(
ρ
1 + r
)j
= yt
1 + r
1 + r − ρ
Plugging this into Equation 16, we find that:
ct =
r
1 + r − ρ
yt + rbt (18)
We can also think about how savings evolves over time; recall that:
bt+1 = yt − ct + (1 + r)bt
∴ bt+1 − bt = yt − ct + rbt
Substituting in the value for ct we just found:
bt+1 − bt = yt −
(
r
1 + r − ρ
yt + rbt
)
+ rbt
Which, with re-arrangement, gives us:
bt+1 − bt =
1− ρ
1 + r − ρ
yt (19)
Summary up to now
We’ve derived two main results:
� ct = Et[ct+1]
� ct =
r
1+r−ρyt + rbt
11
But so far, we’ve not really said anything that differs too much from what we’ve seen so far; it seems
as though agents consume a fixed proportion of their income at time t, they expect to consume
tomorrow what they consume today, boring right? Well, we haven’t quite finished yet! We’ll now
start to explore further some of the implications of these expressions, implications that lie at the
root of why Hall’s model was so influential.
Diving deeper into Hall’s model
Suppose we were interested in how consumption between periods actually changes, not just how
it changes in expectation. Then consider the following derivation; Equation 18 gives us that:
ct+1 =
r
1 + r − ρ
yt+1 + rbt+1
By Equation 17, we know that we can re-write the above as:
ct+1 =
r
1 + r − ρ
(ρyt + �t+1) + rbt+1 (20)
Equation 18 also gives us that:
yt =
1 + r − ρ
r
ct − (1 + r − ρ)bt
It follows that:
ρyt = ρ
(
1 + r − ρ
r
ct − (1 + r − ρ)bt
)
Substituting this into Equation 20, we get:
ct+1 =
r
1 + r − ρ
(
ρ
(
1 + r − ρ
r
ct − (1 + r − ρ)bt
)
+ �t+1
)
+ rbt+1
With re-arrangement, we find:
ct+1 = ρct + r(bt+1 − ρbt) +
ρr
1 + r − ρ
�t+1 (21)
Note that we can write:
r(bt+1 − ρbt) = r(bt+1 − bt) + r(1− ρ)bt
Equation 19 gave us an expression for bt+1 − bt:
bt+1 − bt =
1− ρ
1 + r − ρ
yt
=
1− ρ
1 + r − ρ
(
1 + r − ρ
r
ct − (1 + r − ρ)bt
)
=
1− ρ
r
ct − (1− ρ)bt
12
It therefore follows that:
r(bt+1 − ρbt) = r
(
1− ρ
r
ct − (1− ρ)bt
)
+ r(1− ρ)bt
= (1− ρ)ct
Putting that back into Equation 21, we find:
ct+1 = ct +
ρr
1 + r − ρ
�t+1 (22)
This is the famous Hall result; consumption follows a random walk; that is to say, the only thing
that tells us any information about what consumption is likely to be tomorrow is what consumption
is today.
This is really a massive big deal
Why? Well there are two reasons:
� It means consumption’s path over time ought to be completely random and unpredictable.
� It implies that the long run variance of consumption ought to be infinite.
Why does it imply these things? Let’s start with the first part: we have that consumption tomorrow
will be consumption today plus some totally and completely random shock, that could be positive
or negative. So too the case for consumption the following day. It is just a random walk; i.e. at
every point in time i just randomly move in any direction and map out a completely chaotic path.
The second takeaway is that long run variance of consumption ought to be infinite. This is
probably not immediately obvious if you’re not familiar with random walks, so we will show why
this is the case. Note that we found:
ct+1 = ct +
ρr
1 + r − ρ
�t+1
To simplify the algebra, we will define the variable ut as follows:
ut =
ρr
1 + r − ρ
�t+1 (23)
Thus, the variance of ut, which I’ll denote by γ, is given by the following expression:
var(ut) = γ =
(
ρr
1 + r − ρ
)2
× var(�t+1)
=
(
ρr
1 + r − ρ
)2
× σ2
Note that given that �t+1 is an iid random variable, so too is ut. It therefore follows that:
var(ct+1) = var(ct + ut+1)
= var(ct) + var(ut+1)
= γ
13
Note that var(ct) = 0, because it has already been realised. So we’ve found the value for the
expected variance of ct+1; how about for ct+j? Well, we know that:
ct+j = ct+j−1 + ut+j
But we also know that:
ct+j−1 = ct+j−2 + ut+j−1
That means we can write:
ct+j = ct+j−2 + ut+j−1 + ut+j
We can continue with this process of substitution, until we get something that looks like:
ct+j = ct +
j∑
i=0
ut+j−i
Therefore, by iid of {ut+j}∞j=0, it follows that:
var(ct+j) = var
(
ct +
j∑
i=0
ut+j−i
)
= var(ct) +
j∑
i=0
var(ut+j−i)
=
j∑
i=0
γ = jγ
Thus, it is very clear that:
lim
j→∞
var(ct+j) = lim
j→∞
jγ =∞ (24)
Comparison with example in Section 1
Let’s take a moment just to think about this; recall in the previous section without uncertainty,
we had that consumption had zero variance. We fixed a value of consumption determined by the
present value of future income streams. Now, by introducing uncertainty, we have an extraordi-
narily opposite prediction: consumption will ultimately have an infinitely high variance. This
is an extraordinary difference. Even with β(1 + r) = 1, we get this result.
How do we interpret these two results?
What was Hall trying to say? Was he trying to say, ’A-ha! I’ve shown that consumption is
completely unpredictable, and totally random, and will explode to infinity!’. Maybe. I think he
actually did believe that consumption followed a random walk, but that’s not what you should take
away from this (or at least, it isn’t what I think you should take away from this!)
Perhaps what is notable about Hall’s results is that they seem to be so clearly false. And yet,
they sprang out of what seemed to be a reasonably justifiable framework. Hall’s results shook
the economics world, and caused a great deal of soul searching, mostly because they appeared to
undermine the framework that existed at that time. Much research following Hall sought to resolve
these two seemingly impossible predictions.
14
Permanent Income Hypothesis
Introducing Uncertainty
Rational Expectations
The Hall Consumption Model