# consumption

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```					David N. Weil                                                 January 25, 2012

Lecture Notes in Macroeconomics

Section 1: Consumption

Several ways to approach this subject.

1. Note that “saving” and “consumption” are really the same question: that is, you get a certain
amount of income, and you can save it or consume it. So can’t think about one without thinking

2. This topic is really part of both the long run and the short run analysis. In the long run, the
saving rate determines the level of output (or the growth rate or output). But in the short run, the
determination of consumption is also important for studying the business cycle.

3. Consumption theory is one of the most elegant branches of economic theory. Much of the
approach taken here to consumption is taken elsewhere in economics to e.g. fertility, schooling,
health, etc. Thus these tools (and the problems with them) are far more general than it might appear.

4. In all of this section of the course, we will be treating labor income as exogenous (Note:
“exogenous” does not mean “constant” or “certain.”) We will also mostly treat interest rates as
being exogenous, but also look at some cases of endogenous interest rates.

You may recall the approach taken to consumption in many undergraduate macro textbooks is to
think about a “consumption function” that relates consumption to disposable income:

C = C(Y-T) [where note that we are using c as both the name of the function and the name of
the thing it is determining.]

often this is written in a linear form:

C = c0 + c1(Y-T)

where the little c’s are coefficients. c1 is, of course, the marginal propensity to consume.
[picture]

This is often called the Keynesian consumption function. Keynes wrote that c0>0 and 0<c1<1

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due to a “psychological law” -- essentially that when you do not have a lot of income, you focus on
immediate needs; but when you have satisfied these, you look more to the future and save.

Ways to test: look cross sectionally; look at short time series. (flesh his out)
Both of these looked good for the Keynesian consumption function.

Two problems with the Keynesian view:

1. Empirical: What does this model predict will happen to the rate of saving as a country gets
richer?
Y -T - C       C         c
s=            = 1-      = 1 - 0 - c1
Y -T         Y -T      Y -T

where C/(Y-T) is often called the average propensity to consume. So the Keynesian consumption
function says that as a country gets richer the saving rate should rise. This just doesn't work. The
saving rate is pretty constant over long periods of time.12

2. Theory: Think about the act of saving: you are moving consumption from one period to
another. Thus saving should be viewed explicitly as an intertemporal problem. So for example, the
MPC should depend on why your income has gone up. Put another way, the consumption function
should have a lot more than just today's income in it -- for example, it should have tomorrow's
income in it.

So we want a model of saving behavior that is more based on fundamentals. To build a such a
happy.

1
Historical note: This wasn't actually known for sure when Keynes wrote: Simon Kuznets, who
invented national income accounting -- ie how to measure GDP and stuff -- discovered the
approximate constancy of the US saving rate over a period of 100 or so years. His discovery set off
a flurry of work on consumption in the 1950s that culminated in Friedman and Modigliani’s
contributions. Interestingly, in most other developed countries (summarized in Angus Maddison’s
work) the saving rate has risen over time -- although probably not in the way that Keynes' model
predicted.
2
The Kuznets finding can be put another way. If we go to the data (say annual data on income
and consumption for a country over time) and run the regression C = c0 + c1 Y, we will get the result
that in short samples the estimated value of c1 will be smaller than it will be in large samples. When
we talk about the Permanent Income Hypothesis we will see why this is true.

2
We represent the idea that consuming makes people happy with a utility function.

By utility function, we just mean some function that converts a level of consumption into a
level of utility. U = U(C) [picture]

Why should the utility function be curved? Try to motivate intuitively: think about the marginal
utility of additional consumption. Seems like this goes down.

Most of the interesting things that we can say about utility come from thinking about two
issues: how we add up utility over many different periods of time, and how we deal with the
expected utility when there is uncertainty.

How do we add up utility across time? Well essentially, we can just take the sum of
individual utilities. Say that we are considering just two periods. Let U( ) be the “instantaneous”
utility function. Then total utility, V, is just

V = U(C1) + U(C2)

(In a little while we will introduce the notion of discounting, by which utility in the future
may mean less to us than utility today. But for now, we will ignore this idea.)

What does our understanding about the utility function say about the optimal relation
between consumption at different periods of time. Say, for example, that we have \$300 to consume
over two periods (and we temporarily ignore things like the interest rate): How shall we divide it
up?

The answer is that we would want to smooth it -- that is, consume the same amount in each
period. The way to see this is to look at the marginal utility of consumption. Suppose that we
consumed different amounts in different periods. Then the marginal utility of consumption would
be lower in the period where we consumed more. So we could consume one unit less in that period,
and one unit more in the period where the marginal utility was higher, and our total utility would be
higher.

Utility under Uncertainty

Now let’s consider a case where there is only one time period, but in which there is
uncertainty about what consumption will be in that period. Suppose, for example, that I know that
there is a 50% chance that my consumption will be \$100 and a 50% chance that my consumption
will be \$200. How do we calculate my expected utility?

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There are two ways that you might consider doing it: could take the expected value of my
utilities, or the utility of my expected consumption.

V = .5*U(100) + .5*U(200)

or V = U (.5*100 + .5*200)

The first of these methods of calculating utility from a probabilistic situation is called Von
Neumann - Mortgenstern (VNM) utility. This is the approach that we always use. The second
method is called wrong.

How do we know the VNM utility is the right way to think about utility when there are
different possible states of the world? Here is a simple demonstration: Suppose that you can have
either \$150 with certainty, or a lottery where you have a chance of getting either \$100 or \$200, each
with a probability of .5. Which would you prefer? Almost everyone would say they prefer the
certain allocation. This is a simple example of risk aversion. But notice that if we chose the second
technique for adding up utility across states of the world, we would say that you should be
indifferent.

The fact that uncertainty lowers your utility is called risk aversion. Notice that risk aversion
is a direct implication of the utility function being curved. (The mathematical rule that shows this is
called Jensen’s inequality: if U is concave, then U(E(C)) > E(U(C)), where E is the expectation
operator.) If the utility function were a straight line then the utility of \$150 with certainty would be
the same as the utility of a lottery with equal chances of getting \$100 and \$200. A person who
indeed gets equal utility from these two situations is called risk neutral.3

What are the consequences of risk aversion? Clearly this is the motivation for things like
insurance, etc. Similarly, this is why in financial theory we say that people trade off risk and return:
to accept more risk, an investor has to be promised a higher expted return.

The Relation between Risk Aversion and Consumption Smoothing

Now we get to the really big idea: risk aversion and consumption smoothing are really two
sides of the same coin: they are both results of the curvature of the utility function. If the utility
function were linear (and so the marginal utility of consumption constant) then people would not
care about smoothing consumption, and their expected utility would not be lowered by risk.

3
One can come up with many instances of risk neutrality or even risk-loving (i.e. more
uncertainty raises utility) behavior, such as participating in lotteries, flipping a coin with your friend
for who will buy coffee, etc. However, it is unlikely that these exceptions tell us much about the
vast majority of consumption decisions.

4
This will be important for many reasons: among them is that even when we are talking about
a world with no uncertainty, we will often use the idea of risk aversion to measure the curvature of
the utility function.

The CRRA Utility Function

We will often use a particular form of the utility function, called the Constant
Relative Risk Aversion utility function.

1-
Ct
U( C t ) =
1- 

where sigma > 0. Note that if sigma > 1, then the CRRA formulation implies that utility is always
negative, although it becomes less negative as consumption rises. This does not matter, although it
often gets students confused.

Note that in the special case where σ=1, the CRRA utility function collapses to U(C) = ln(C).4

σ is called the coefficient of relative risk aversion and it measures, roughly, the curvature
of the utility function. If σ is big, then a person is said to be risk averse. If σ is zero, the person is
said to be risk-neutral.

To see how sigma measures the curvature of the utility function, we can calculate the
elasticity of marginal utility with respect to consumption, that is

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Proof: First re-write the utility function by adding a constant: u(c) = (c1-σ-1)/(1-σ). Think of
this as a function of σ: g(σ) = (c1-σ-1)/(1-σ) re-write as

ce -( ln (c) ) - 1
g(  ) =
1- 

since g(1) = 0/0, we apply L’Hopital’s rule

lim              -c ln(c)e - ln (c)
g(  ) =                       = ln(c) .
 1                     -1

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dU 
dc = c U  = - 
U      U
c

So the bigger is σ (in absolute value), the more rapidly the marginal utility of consumption declines
as consumption rises [picture]. And the larger is this change in marginal utility, the greater is the
motivation for consumption smoothing, insurance, etc.

As an exercise, we can show this by calculating the amount that a person is willing to pay to
avoid uncertainty. For example, calculate the value x such that the utility of \$150-x with certainty is
equal to the utility of a 50% chance of \$100 and a 50% chance at \$200. How does x change with σ?

We solve:

(150 - x )1- = .5* 100 1- + .5* 200 1-

1
x = 150 - (.5* 100 1- + .5* 200 1- )1-

We can use a calculator to find the value of x for different values of sigma. By thinking about what
value of x seems reasonable, we can then decide what is a reasonable value for sigma (see table).
For example, if sigma = 6, then x=35.8, so a person would be indifferent between a 50% chance of
consuming \$100 and a 50% chance of consuming \$200, on the one hand, and certain consumption of
\$114.20, on the other.

σ     x

1    8.6 (log utility)
2   16.7
3   23.5
4   28.8
5   32.9
6   35.8

Note that even though log utility is probably not reasonable a priori (based on the above) or
empirically, we use it a lot because it is so convenient. Empirical estimates of σ probably average

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around 3, but there is no agreement. Some anomalies in finance (such as the "equity premium
puzzle") can only be explained with what seem like unreasonably high values of σ.

[see homework question on CRRA and CARA.]

[ The difference between relative risk aversion and absolute risk aversion can be thought of this
way. Suppose that I am willing to pay 10 to avoid the uncertainty of a lottery that gives me either
150 or 50 each with probability 50% -- that is, I consider certain consumption of 90 to have utility
equal to the lottery. If utility is CRRA, then I will also be willing to pay 100 to avoid a lottery of
1500 or 500. If utility is CARA, then I will be willing to pay 10 to avoid a lottery of 1050 or 950.
Note that with CARA I will be willing to more than 100 to avoid a lottery of 1500 or 500 -- this
should be obvious for the following reason: the larger is uncertainty, the more (at the margin) you
are willing to pay to avoid it. So if a CARA consumer with expected income of 1000 will pay 10 to
avoid 50 worth of uncertainty, he will pay more than 100 to avoid 500 worth of uncertainty.]

Fisher Model

So now we look more formally at an intertemporal model of saving. The simplest model is
the two-period model of Irving Fisher.

People live for two periods. They come into the world with no assets. And when they die, they
leave nothing behind.

In each period they have some wage income that they earn: W1 and W2.

Similarly, in each period, they consume some amount C1 and C2.

The amount that they save in period 1 is S1 = W1 - C1. S1 can be negative, in which case they are
borrowing in the first period and repaying their loans in the second period.

For the time being assume that they do not earn any interest on their savings or pay any interest on
their borrowing. So the amount that they consume in the second period is

C2 = S1 + W2

that is, in the second period you consume your wage plus your savings.

We can combine these two equations to get the consumer's intertemporal budget constraint:

C1 + C2 = W1 + W2

We can draw a picture with C1 on the horizontal axis and C2 on the vertical axis. The
budget constraint is a line with a slope of negative one. Note that the budget constraint runs through

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the point W1, W2 -- you always have the option of just consuming your income in each period. The
Y and X intercepts of the budget constraints are both W1 + W2.

Consumer can consume any point along this line (or any beneath it, but that would be waste).

What is saving in this picture? Show which points involve saving or borrowing.

So where does the person choose to consume? Well, clearly along with a budget constraint we are
going to need some indifference curves.

Say that his total utility (V) is just the sum of consumption in each period:

V = U(C1) + U(C2)

Where U() is just a standard utility function.

To trace out an indifference curve, consider a point where C1 is low and C2 is high. At such
a point, the marginal utility of first period consumption is high, and that of second period
consumption is low. So it would take only a small gain in C1 to make up for a big loss of C2 in order
to keep the person having the same utility. So the indifference curve is steep. Similarly, when C1 is
large and C2 is small, the indifference curve is flat.5 So it has the usual bowed-in shape.

So optimal consumption is where the budget constraint is tangent to an indifference curve.

We can also solve the problem more formally, setting up the lagrangian:
L = U( C 1 ) + U( C 2 ) +  ( W 1 + W 2 - C 1 - C 2 )

and taking the first order conditions:

dL/dC1 = 0 = U'(C1) - λ ===> λ = U'(C1)

dL/dC2 = 0 = U'(C2) - λ ===> λ = U'(C2)

so C1 = C2

5
More formally, one can use the implicit function theorem. Let F(C1,C2) = U(C1) + U(C2). Then for
F(C1,C2) = k (where k is some constant):
d C2    F       U ( C 1 )
= - C1 = -
d C1    F C2    U ( C 2 )

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Combining this with the budget constraint gives: C1=C2= (W1 + W2 )/2 , which is not so shocking,

We can use this simple model to think about consumption in the face of different
circumstances.

What happens if income rises? This will shift out budget constraint. Consumption in both periods
will rise. What happens to saving? Answer: it depends on which periods income went up.

-- Suppose that your current income falls but that your future income rises by exactly the same
amount. How should consumption change? How about saving?

Already, we can see some problems with Keynes' way of looking at consumption. Consumption
depends not just on today's income but on future (or past) income.

Interest rates

Now we make the model slightly more complicated by considering interest rates:

let r be the real interest rate earned on money saved in period 1 -- or the interest rate paid by
people who borrow in period one.

Now the definition of saving is still:

S1 = W1 - C1

but consumption in the second period is now:

C2 = (1+r) S1 + W2

or combining these:

W1 + W2/(1+r) = C1 + C2/(1+r)

Can draw diagram as before

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Y intercept is (1+r)W1 + W2,
X intercept is W1 + W2/(1+r).

The budget constraint still goes through the point (W1, W2), which we call your “endowment point”
-- that is, if you consume your income in each period, that is a feasible consumption plan no matter
what the interest rate is.

What happens to the budget constraint when the interest rate changes??

Answer: it rotates around the endowment point. What does this do to saving in the first period (ie to
consumption in the first period?)

First, let’s look at what happens in the case where the person was initially saving. Remember
from micro that there are two effects: the income and the substitution effect.

Income effect is that we can get onto a higher indifference curve. This tends to raise C for both
periods.

Substitution effect: consumption in the second period has gotten cheaper. This tends to lower first
period consumption and raise second period consumption.

Upshot is that in this case, can't tell what happens to first period consumption, or first period
saving.

What if person had had negative saving in the first period, and then interest rate goes up?

Now income and substitution effects work in the same direction, so that first period consumption
will fall, and saving will rise.

Discounting

We might want to introduce some discounting of utility experienced in the future. For example,
suppose that Θ is some discount factor that we use for discounting future utilities.

V = U(C1) + U(C2)/(1+Θ)

Now we can once again solve for the optimal path of consumption with both interest and
discounting. We set up the Lagrangian:

10
U( C 2 )           W           C2 
L = U( C 1 ) +            +   W 1 + 2 - C1 -         
1+               1+ r      (1 + r) 

and get the first order conditions

dL/dC1 = U'(C1) - λ

dL/dC2 = U'(C2)/(1+Θ) - λ/(1+r)

which can be solved for:

U'(C1)/U'(C2) = (1+r)/(1+Θ)

this is one equation in the two unknowns of C1 and C2. It can be combined with the budget
constraint to give two equations in two unknowns, and so can be solved for the two values of C. To
do this, however, one needs to know the exact form of the utility function. This is done in one of the
homework exercises.

Liquidity Constraints

What happens if there are constraints on borrowing? What does the budget constraint look like
now?

For person who would have wanted to save anyway, no big deal. But for person who would have
wanted to borrow, they will be at corner. We say that such a person is "liquidity constrained."
Example of a college student.

What will such a person's consumption be? Just their current income. So they will look a lot more
like the Keynesian model, except that the MPC will be one.

Differential interest rates

It may also be the case that the interest rate for borrowing is different than the interest rate
for saving -- presumably the rate for borrowing will be higher.

What will the budget constraint look like in this case? It will be kinked at the endowment
point. In this case, there are three possible optima: either tangent to one of the arms, or at the kink
point. Interesting result is that if the optimum is at the kink point, then small changes in one or both
interest rates will not affect the optimal level of consumption.

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Extension to More than Two Periods

Now we can easily extend the model to an arbitrary number of periods:

Consider a person planning consumption over periods 0...T-1. (labeling the periods this way is
just slightly more convenient) She faces a path of wages {W0, ....WT-1}

she gets utility according to an instantaneous utility function U(C), which is discounted at rate Θ.
That is
T -1
U( C t )
V =
t=0 (1+  )
t

She faces interest rate r on any assets (negative or positive) that she has. In particular, call At the
assets that she has at the beginning of a period. This is equal to

At = (1+r)*(At-1 + Wt-1 - Ct-1 )

She starts life with zero assets: A0 = 0.

We also impose the rule that she must have zero assets at the end of her life -- that is AT = 0 (where
AT = (1+r)(AT-1 + WT-1 - CT-1) . Put another way, in the last period of life she spends her earnings
plus any accumulated assets [or less any accumulated debts). Dying in debt is not allowed.

How will we derive her inter-temporal budget constraint?

Start by writing down the expression for assets in each period

A1 = (1+r)(W0 - C0)       [since A0 = 0 ]

A2 = (1+r)(A1 + W1 - C1) = (1+r)(W1-C1) + (1+r)2(W0 - C0)

etc...

AT = (1+r)(WT-1 - CT-1) + (1+r)2(WT-2 - CT-2) + ... (1+r)T (W0 - C0)

We divide all the terms in this last expression by (1+r)T, and note that it is equal to zero, to get
T -1
-
0 =  W t C tt
t=0 (1+ r )

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Notice that we have gotten rid of all of the A's. This expression can be re-arranged to say that the
present discounted value of consumption is equal to the present discounted value of wages.

T -1           T -1

 (1+ rt )t =  (1+ r )t
t=0
W
t=0
Ct

This is the intertemporal budget constraint... which looks a lot like the two period version derived
above.

An Aside: The Budget Constraint in Continuous Time

We can also derive a similar intertemporal budget constraint in continuous time. The evolution of
assets is governed by the differential equation:
d A(t) 
= A = rA(t) + w(t) - c(t)
dt
This can be solved, along with the initial condition A(0)=0, to give
t
A(t) = (w(s) - c(s))e r(t-s) ds
0

This just says that assets at time t are the present values of the past differences between wages and
consumption.

Assets at the end of life are zero, that is, A(T)=0. So setting t=T in the above equation,
T                            T

e
0
r(T -s)
w(s) ds =  e r(T -s) c(s) ds
0

Multiplying by e-rT
T                        T

e-rs w(s) ds = e-rs c(s) ds
0                        0

[End of Aside]

Now with our budget constraint and our utility function, we can do a big Lagrangian....

T -1
U( C t )         T -1
-        
L=               +      W rC)t        t

t=0 (1+  )
t                         t
      (1+
t=0                 

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to solve this we would just find the T first order conditions which, combined with the budget
constraint, would allow us to solve for the T+1 unknowns: λ and the T values of consumption. In
many cases this is a big mess to solve, but we can get far by just looking at the FOCs for
consumption in two adjacent periods, t and t+1:

dL U ( C t )      1
=        -          =0
d C t (1+ )t
(1+ r )t

dL           U ( C t + 1 )       1
=                  -              =0
d C t +1       (1+ )    t +1
(1+ r )t + 1

these two can be combined to give

U ( C t )     1+ r
=
U ( C t +1 ) 1+ 

this is a key condition that relates consumption in adjacent periods. Notice that even if we don't
know the full solution to the consumer's problem (that is, what the level of consumption in each
period should be), we know that this condition should hold.

There is a huge amount of intuition built into this expression, so it is worth thinking about for a
while.

Let's start on the intuition by showing how we could have gotten a similar result without calculus:

Suppose that I have a discounted utility function, and that the interest rate is zero. I have some set
amount of total consumption that I want to do. How will I divide it between the periods?

To see the answer: consider a path of consumption (C0, C1,...) Suppose that I want to know
whether this path of consumption is optimal. Well, suppose that I consider consuming slightly less
(call it one unit, for convenience) in period zero, and then consuming the same amount more in
period one.

How much would I lose? answer: U'(C0)

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How much would I gain? answer: U'(C1)/(1+Θ)

Note that the (1+Θ) comes from the fact that utility that I get in the second period is not worth as
much to me as utility in the first period. Now if one of these was bigger than the other, then clearly
the path of consumption that I was considering was not the optimal one. So everywhere along the
optimal consumption path, it will be the case that

U'(Ct) = U'(Ct+1)/(1+Θ)

So what does this say about the optimal path of consumption in the presence of discounting? I
says that the marginal utility of consumption must be rising. So therefore consumption must be
falling along the optimal path.

Now imagine that we have an interest rate to contend with (and forget about discounting for a
second): Whatever we don't spend will grow in value at an interest rate r.

Again consider some allegedly optimal path of consumption. Suppose that I were to move one unit
of consumption from period 0 to period 1

would lose: U'(C0)

would gain U'(C1) (1+r)

Suppose that these two were not equal -- then clearly you were not on the optimal path. So the
condition for being on the optimal path is
U'(C0) = U'(C1) (1+r)

So what has to be happening to consumption in this case? The marginal utility must be falling --
so consumption must be rising.

So now say that I want to characterize the optimal path of consumption in the case where I have
both an interest rate r and a rate of time discount Θ. Clearly the first order conditions relating every
two adjacent periods' consumption will be:

U ( C t ) 1+ r
=
U ( C t+1 ) 1+

So what does this tell us? Suppose that Θ is greater than r? Then the marginal utility of
consumption in period t+1 is higher than the marginal utility of consumption in period t, and so
consumption must be falling. What if r>Θ? What if they are equal? So interest and discounting

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work against each other.

If we did know the exact form of the utility function, we could go further. For example, if
we know that the utility function is of the CRRA form

1-
Ct
U( C t ) =
1- 

then U'(C) = C-σ

and so the first order condition can be re-written
1
C t +1  1+ r  
=      
C t  1+ 

Before we discuss the interpretation of this first order condition, we can derive a similar one
in continuous time.

To re-write the first order condition with CRRA utility in continuous time:

First note that for small values of x, the approximation ln(1+x) x (or alternatively, 1+x ex )
is fairly accurate.

So for (1+r) we write er, and same for Θ. [being completely accurate, the r that we use in
continuous time, the “instantaneously compounded” interest rate, is not exactly equal to the r used in
discrete time.]

so we can rewrite the first order condition as

1/ 
c t +1  e 
r
=            = ( e r- )1/ 
ct  e 

re-write this allowing the unit of time used to be a parameter:

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c t + t
= ( e(r - ) t )1/ 
ct

where if Δt=1 then we have the previous equation.

                                        
define c as the time derivative of consumption: c = dc/dt.

c    -c 
c = lim  t + t t 

t 0
 t 
Thus the growth rate of consumption is given by

 c t +t - c t                      c t +t 
        -1
                                                                  ( e(r - )t )1/  - 1 
= lim  t
c
lim
 = t 0    ct        = lim                             
c t 0  c t                                t  t 0                      t           
                                             
          

The numerator and denominator of the last expression are both zero when Δt is zero, so we apply
L'Hopitals rule, taking derivatives of top and bottom with respect to Δt:

1                  1
( e(r- )t ) -1(r -  )e(r - )t

1

Evaluating at Δt=0, we end up with


c 1
= ( r - )
c 

Interpretation of the FOC

In both discrete and continuous time, the FOC says the same thing: the rate at which consumption
should fall or grow depends two things: first, the difference between r and theta; and second on the

17
curvature of the utility function.

--------------------------------------
Completing the solution

Often all we need to look at is the first order condition. But if we want to complete the
solution to the lifetime optimization problem, we can. The FOC tells us how consumption in
adjacent periods compares. So given one value of consumption (say, consumption in the first
period), we can figure out consumption in all periods -- that is, the entire path of consumption.

[note, by the way, what will happen to the FOC if r changes over time. This condition would then
have to be re-written with r(t) in it, but would be otherwise the same.]

From here, it is simply a matter of finding the value of consumption in the first period that
satisfies the budget constraint.

Completing the solution is easiest in the case of continuous time where we let the time
horizon (i.e. T) be infinite. Note that there are some technical problems that can crop up in
considering infinite time as opposed to just letting T be very large. For example, we can’t impose
the “no dying in debt” condition (A(T) = 0 ), and instead have to impose a different condition (often
called the “no Ponzi game condition” that I will not discuss here. For our purposes, it is sufficient to
state that the infinite PDV of consumption has to equal the infinite PDV of wages.

Consider a simple case where w(t) = 1 for all t. Utility is CRRA,  and r are given.

The first order condition for consumption growth can be integrated to give

c(t )  c(0)e(1/  )( r  )t

The budget constraint is thus

           

 e dt   e c(0)e
 rt      rt   (1/  )( r  ) t
dt
0           0

Notice that for this budget constraint to make sense, we have to have that the right hand side is
finite. If r>theta, then consumption is growing, but it must be growing slowly enough so that its
PDV is finite. Thus we assume:

18
1
r             (r   )


Integrating the budget constraint….

1       c(0)

r r  (1/  )(  r )

1   r 
c(0)  1 
 r 
      

From this we see
 If θ>r, then initial consumption is above 1, in which case consumption asymptotes to zero
 The bigger is σ, the closer is initial consumption to 1.

Dynamic Programming

(leaning very heavily on Blanchard and Fisher p. 280-282)

Dynamic programming is a method of writing and solving dynamic optimization problems that
differs from the Lagrangian. It uses recursive equations, also called Bellman equations, to break a T
period problem up into a bunch of much smaller one period problems. The most important thing to
note is that the problem itself is not any different, only the solution method has changed. So we still
have individuals trying to maximize their lifetime utility subject to some constraints on their lifetime
wages and assets. We should get out exactly the same Euler equation.

If you read a book like Stokey and Lucas (1989), or Sargent (1987), much of the text is spent
proving to you that it is theoretically possible to solve the following problem this using techniques
of dynamic programming:

     U  ct 
max V                                 s.t.   at 1  1  r  at  wt  ct 
1   
c1 ...c                            t
t 0

And subject to some Non-Ponzi Game (NPG) condition; or alternatively go through time T and
subject to the condition that AT=0

We’ll take that proof as a given, and just proceed to show you how the method actually works. The
proof depends a lot on the presence of time-separable preferences and then requires U(c) to be

19
concave, continuous, etc..

The first step is to write down the value function, which is an indirect utility function.

       U  Cs 
Vt  at   max                                      s.t.   at 1  1  r  at  wt  ct 
1   
ct ...c                     s t
s t

This value function tells us that V is the maximized value of utility I have, given an initial asset
level of at, from time period t until infinity, along my optimal path. Bellman then used the insight
that if you performed your optimization at time t+1, the path of consumption that you would choose
must follow the exact same path that you would have chosen for periods t+1 to infinity if you had
done your optimization at time t. (The crucial assumption for this is that preferences are time
separable).

This means I can write the above value function recursively, or as follows:

          V a     
(1)         Vt  at   max U  Ct   t 1 t 1  s.t.     at 1  1  r  at  wt  ct 
ct

           1    

This recursive, or Bellman, equation tells us that the value of my lifetime utility from time t forward
is equal to the utility of consumption at time t plus the value of my lifetime utility from time t+1
forward.

Suppose for the moment that I actually know what the V function looks like (and notice that the V
function can possibly change over time). Then my problem is no longer a many period problem but
only a one period problem. The question is trading off current utility at time t for more financial
wealth at time t+1 (which I already know I will spread optimally among the remaining periods of my
life).

So let’s do the maximization in the Bellman equation. We can substitute in the difference equation
for assets into the maximand in equation (1). We get

          V ((1  r )(at  wt  ct )) 
(2)                          Vt (at )  max  U (Ct )  t 1                       
ct
                   1                

Then take the derivative of (2) w.r.t. consumption to get the first order condition that

1
U '  Ct  
1
1  r V  a  '
t 1   t 1

which is already starting to look a lot like the Euler equation we found before. This says that I
should trade of the marginal utility of consumption today against the (suitably discounted) marginal

20
value of an extra unit of the asset tomorrow. But we don’t know what this V function looks like, so
this equation doesn’t help us a lot.

However, the next big insight in the dynamic programming method is that there is a simple envelope
relationship between V’ and U’ along the optimal path.6 To see this, take the derivative of (2) w.r.t.
assets in period t, applying the envelope theorem:
1
Vt '1  at 1       1  r Vt ' 2  at  2  .
1

The right hand side of the above equation is just equation (2), moved forward one period. So we can
just substitute to get:

Vt '1  at 1   U '  Ct 1  .

Plugging this into the original period t FOC gives us

1
U '  Ct  
1
1  r U '  Ct 1 
and this is obviously just the usual first order condition (also called the Euler equation).

Solving the model completely requires that you then solve the Euler equation for some consumption
path and utilize the budget constraint. This is just the same as before. The dynamic programming
method doesn’t necessarily offer any extra help during these last steps. It’s main value is that
certain problems are easier to set up as Bellman equations in the first place. The recursive equations
are also useful because they are easier to translate to computer code that can iterate through periods
quickly to find the optimal path (which allows you to calibrate your model).

An additional mathematical result of this technique that can be useful involves the nature of V.
Under a certain set of conditions (continuity, concavity, etc..) it can be shown that the Bellman
equation is an example of a contraction mapping, and that this means the V functions (which were
previously allowed to vary over time) will converge to a single functional form V(a). In addition,
this means that the control function, or the rule for setting consumption in time t as a function of
assets at time t, will be time invariant as well.

6
More formally, the envelope theorem says that if you have y  max f ( x, c ) , then the derivative
x

dy/dc can be evaluated as follows. First, define x*=g(c) as the optimal value of x given a value of c.
Write y=f(g(c),c). Now dy/dc = f1(g(c),c)g’(c)+f2(g(c),c). But we know that f1(g(c),c)=0 by the
first order conditions that made x*=g(c) in the first place. So the first term drops out and dy/dc =
f2(g(c),c). In other words, the derivative of y with respect to c is just the derivative of the original
f(x,c) function with respect to c.

21
[Skip the below or fix up?]

To see what this means, consider a problem with log utility, so that
1 r
Ct 1       Ct
1
and therefore consumption in any period s>t can be written as
s t
 1 r 
Cs          Ct .
 1 
The budget constraint at time t is the following
                    
Cs                   Ws
 1  r s t
s t 
 At  
         s t 1  r 
s t

And you can solve these together to get that
           
Ws 
Ct          At                     .
1         s  t 1  r 
s t
                          

This rule holds for any period t, so the consumption rule (or control rule) is identical for all periods.
This doesn’t mean that consumption itself is necessarily identical every period, but the rule for
setting it is. You may still have consumption rising or falling depending on the relationship of r and
the discount rate.
The real benefits of DP come when we extend it to uncertainty … (come back and do an example
with stochastic wages?).

Some Open Economy Applications

We can use the two period model of consumption to draw a helpful picture. Suppose that we
graph the interest rate on the vertical axis, and the level of (first period) saving on the horizontal,
with zero somewhere in the middle of the horizontal axis. What is the relation?

Obviously, the position of the curve will depend on the values of w1 and w2. (as well as the
parameters of the utility function). The bigger is w1 and the smaller is w2, the higher will be saving
at any given interest rate.

But what about the shape of the curve overall?

We know that if saving is negative, then an increase in the interest rate will raise the amount
of saving -- we know this because in this case the income and substitution effects are aligned. For
zero saving, we also know that the curve is upward sloping. But for positive saving, we don’t know.
The curve may well bend backward.

Question: what determines the degree to which the curve can bend backward? Answer: the

22
degree of risk aversion!

Why? The degree of risk aversion tells us how the person trades off smoothing of
consumption for taking advantage of the interest rate to get more consumption in a later period. If a
person is very risk averse, then he wants very smooth consumption. In this case, the curve will end
up bending backward

Now suppose that we have a two-period world, and we are thinking about a country, rather
than an individual.

Quick review of open economy national income accounting:

From this we derive the standard national income accounting equation

Y = C + I + G + NX

one problem: is Y GDP or GNP?

The answer is that we can make it either one; as long as we define imports and exports
appropriately.

In fact, for (almost) all of this course, the distinction will not matter. When we think about
capital flows, we will be thinking not about portfolio investment or foreign direct investment (FDI)
but rather about debt (denoted B). In this case, there will be no foreign ownership of factors of
production, and so GDP and GNP will be the same.

Y = C + I + G + NX

Y - C - G = national saving = I + NX

(Y - T - C)+ (T-G) = national saving

private saving + gov't saving = national saving = I + NX

Define Bt as net foreign assets at time t.

The Current Account is the change in net foreign assets. It is equal to NX plus interest on
the assets we hold abroad, minus interest on the debt that we owe foreigners.

In discrete time:      CA = Bt+1 - Bt = rBt + NXt

In continuous time:        
CA  B  rB  NX

23
So for our thinking about capital flows between countries, there are going to be a variety of
assumption that we can make about the different pieces.

Nature of openness (for this course, the only type of openness we will think about is capital flows.):

closed economy:     NX is zero; r is endogenous.

small open economy: r is exogenous and fixed at r*, the world level, which is exogenous; NX is
endogenous.

large open economy: economy is large enough to affect the world interest rate, so r=r*, but r* is
endogenous. Also, if this is a two-country world, then NX = -NX*

Well: a person saving in the first period and consuming more than his income in the second
period is exactly equivalent to running a CA surplus in the first period and a CA deficit in the
second period. (Even though the world only lasts for two periods, we can think of the requirement
that people do not die in debt as meaning that B3 = 0.)

Suppose that there is no trade between countries. Then, since there is no government, W=C
in both periods.

Note that this is not just a case of liquidity constraints in the standard sense. Rather, since
everyone is identical, there will be no borrowing or lending. But (key observation): there can still be
an interest rate! We think of the interest rate as being the level that clears the market for loans --
which will clear at the level where there neither borrowing or lending. This is called the “Autarky
interest rate”

To figure out the Autarky interest rate, we can just go back to the first order condition, but
now we know that consumption has be equal to Y, and so we can just substitute it:

U'(Y1)/U'(Y2) = (1+r)/(1+Θ)

Now, here is the big result:

===> If the autarky interest rate is lower than the world interest rate, then the open economy will
run a current account surplus in the first period. And if the autarky interest rate is higher than the

24
world interest rate, then the economy will run a current account deficit in the first period.

Intuitively, this is pretty obvious. We can also show it graphically

[The autarky interest rate is what arises in the closed economy version of our model. It is the place
where the curve derived above crosses zero. So we can also see here the result about the interest
rate!]
----

Large Open Economy model

Now we can do a large open economy model, making r (= r*) endogenous. We just draw
two versions of the saving vs interest rate diagram that we derived above, and look for the interest
rate where saving in one country is the negative of saving in the other. etc.

Intuition building problem:

Let’s look at the large open economy model with an infinite number of periods, instead of
just two.

Let’s think about two equally sized open economies. Equally sized in the sense that they
have the same endowment income.

Y1,t = Y2,t =Y for all t

we forget about G and I

θ1 < θ2

What will the equilibrium look like? The key to figuring this out is to realize that the
interest rate cannot remain constant! (At least if we assume that consumption can’t be negative).

===> In the long run, we know that the interest rate will be equal to the θ1. We can trace out the
path of interest rates and net assets pretty easily (at least graphically!).

The PIH and the LCH

25
the model just presented in very standard. The PIH and LCH are two ways of making the same
point.

Permanent Income Hypothesis

Developed by Milton Friedman

Rather than focusing on the whole life cycle, the PIH thinks about shorter period changes in income.

The PIH starts by separating income into two parts

Y = YP + YT       (note, could have used Y-T here...)

permanent income is the part of income that you expect to persist into the future, sort of like your
average future income. Transitory income is the other part of income - the part that is different from
the average (note that it can be positive or negative in a given year).

Take a person with a job. Their permanent income is their salary. If in some year they get a
bonus, or if in some year they have a smaller salary for some reason, that is positive or negative
transitory income.

Think about the following two changes in my income. One month I get a letter saying that I have
won \$1000 in the lottery. Is that a change in my permanent or transitory income? What about if I
get a letter from the dean saying that my salary is higher by \$1000 a month?

How will my consumption change in each of these scenarios?

This was Friedman's insight. Your consumption should just depend on your permanent income.
To the extent that transitory income is different from permanent income, you will just use your
saving to make up the difference.

Let's take an example: suppose you looked at two people, both of whom earned the same amount -
- say \$100,000 in a given year. One is a businessman, for whom this is the regular salary. The other
is a farmer, who has very unstable income, and for whom this was a good year. Which should have
higher saving?

So in the PIH, the consumption function is roughly,

C = α*Yp

Now we can go back and see how the PIH explains the facts about the consumption function that
Keynes failed on.

26
First think about the long run: over the long run, when income increases, this is clearly a change
in permanent income. So consumption and saving will just be constant fractions of income in the
long run.

What about in the short run (or looking across households)?

What would you see if all of the variation in income that you looked at was transitory? Then there
would be no relation between C and Y -- the short run consumption fn would be flat. What if some
of the variation were transitory and some permanent? Then you would see what is present in the
data. (see homework problem).

We can also give this result an econometric interpretation. Suppose that a researcher has
collected income and consumption data from a large population. Consumption in the population is
determined by the permanent income hypothesis: C = Yp, where Yp is permanent income and
0<α<1. Permanent income in the population is given by Yi = Y p + ρi, where ρ is distributed
normally, with variance σ2p The researcher does not observe permanent income, however. She only
observes current income, which is related to permanent income by Yci = Ypi + εi. εi is transitory
income. It is distributed normally, with variance σ2t. There is zero covariance between permanent
and transitory income.

The researcher estimates the “consumption function”

ˆ
Ci     Yi

The relationship between estimated beta and the true value is given by

ˆ         p2

 
 p   t2
2

(this is the formula for attenuation bias or measurement error bias. You can think of observed
income as permanent income measured with error).

One current issue in macro is what Friedman meant -- or what is the truth -- about how far into
the future one should look in thinking about "permanent" income. Is it for the rest of your life? For
your life and your children's lives? Or is it for some shorter period, like the next 5 years? This will
turn out to be important in some of the questions we look at below.

Life Cycle Hypothesis

27
Due to Franco Modigliani7

One direction to go with the analysis of consumption presented above is to look more realistically
at what determines saving of people in the economy.

T            T

 (1+ r )t =  (1+ rt )t
t=0
Ct
t=0
W

Now think about your income over the course of life (where we start life at the beginning of
adulthood). The biggest thing that you will notice is that there is a big change at retirement -- your
income goes to zero.

[picture]

Now think about your preferences. We know that in you are going to want to have smooth
consumption -- for example in the case where the interst rate is equal to the discount rate, you will
want constant consumption.

[picture]

What is the relation between the income and the consumption lines? Well, if the interest rate is
zero, then the areas under them have to be the same (that is, the sum of lifetime income has to be the
same as the sum of lifetime consumption). If the interest rate is not zero, it is a little more
complicated -- what matters is the present discounted value of income is equal to the PDV of
consumption.

What does this model say about a person's assets over the course of life?

[picture]

The LCH is also concerned with the total wealth of all of the people in the economy. Why is
this so important? Because, for a closed economy, the capital stock of the economy is made up of
the wealth of the people in the economy. [and, as you will see when we look at growth, the capital
stock is really important].

7
Once, when asked exactly what the difference was between the LCH and the PIH,
Modigliani replied that when the model fit the data well it was the LCH, and when it didn’t it was
the PIH.

28
We can see the aggregate amount saved in the economy by just adding up each age groups saving
or dissaving, multiplied by the number of people who are that age. What does this say should be
happening to the saving rate of the US as the population ages?

We can also see the effect of social security on saving or total assets in the economy. Social
Security lowers income during the working part of life, but raises it during the retirement part of life.
So it lowers the saving rate (and level of wealth) at any given age.

[note -- we will talk about the empirical implications of this model and how well they stand up
later.]

Income growth and Savings in the Life Cycle model
[To be added: do this with flat wage profile (or cross section) and do explicit examples]

How does the growth rate of income affect the saving rate in the life cycle model?
Specifically, if we compare two countries that have the same θ and r, and the same age structure, but
different growth rates of wage income, which will have higher saving rate.

Answer: it depends on the form of income growth. Two cases to look at.

1) Suppose that the shape of the life cycle wage profile is the same in the two countries (it could
be flat, or hump shaped, or whatever). Then in the high growth country, the growth rate of wages
between successive generations must be larger. This means that if we look at a cross section of the
population by age, the growth rate of aggregate wages will be reflected in it, i.e. the youngest people
will have relatively higher wages in the high growth country.

2) Suppose that the cross sectional profile of wages in the two countries is the same. Then any
individual’s lifetime wage profile will reflect this aggregate growth; in this case, people in the high
wage growth country will have rapidly growing lifetime wage profiles.

(Of course there could be a mixed case in between 1 and 2 as well)

Cases 1 and 2 yield very different results.

Case 1: here, the lifetime profile of the saving rate is unaffected by growth. The aggregate saving
rate is just a weighted average of this, where the weights depend on the number of people and their
income. Since young do saving and are richer when growth is higher, higher growth will raise the
aggregate saving rate!

Case 2: Now, higher growth affects the saving rate. Specifically, it lowers the saving rate of the

29
young. It also means that working age people (who are saving) earn more than did old people (who
are dis-saving) -- the effect which we saw in case 1 tends to raise the saving rate. For reasonable
parameters, the lowering effect dominates, so higher income growth lowers the saving rate.

Which case is right? Probably 2 is closer to the truth. For example, wage profiles do not
depend on aggregate growth rate of income.

Ricardian Equivalence

We will now talk about some of the implications of the optimal consumption/saving models that
we have discussed. Later we will look at more direct empirical evidence.

The most controversial implication is the so-called Ricardian Equivalence proposition (which was
mentioned, and dismissed, by David Ricardo, and was given its modern rebirth by Robert Barro).

Consider the effect of changes in the timing of taxes. To do so, let’s look at the simplest model
with taxes, one with just two periods.

Let T1 and T2 be taxes in the first and second periods. Lifetime budget constraint is now:

C2              ( - )
C1+        = Y 1 - T 1+ Y 2 T 2
1+ r               1+ r

Now consider a change in tax collections that leaves the present value of tax collections
unchanged:

T 1 = - Z   T 2 = (1+ r)Z

For example, if Z is positive (the usual case that we will think about), this would mean that we
were cutting taxes today, and raising them in the future. What does this do to the budget constraint?

30
C2                      ( - [ +(1+ r)Z])
C1+        = Y 1 - [ T 1 - Z] + Y 2 T 2
1+ r                           1+ r

You can see that the Z's will just cancel out, and the budget constraint is left unaffected. What
about savings, though? Since the budget constraint has not changed, first period constumption will
not change. But saving of the people in this economy is equal to

S = Y1 - T1 - C1

So if we reduce taxes by Z, we should raise saving by the same amount. So does the capital stock go
up by Z? No: because the government is going to have to borrow to finance its tax cut. In fact, it is
going to have to borrow exactly Z (or, if it was running a deficit already, it will have to borrow Z
more dollars).
The government will issue bonds, paying interest r, and people will hold them instead of capital --
so the amount of capital will not change. (just like giving people a piece of paper with "bond"
written on one side and "future taxes" written on the other.).
Notice that although people who hold the bonds think of them as wealth, as far as the economy is
concerned they are not "net wealth," since they represent the governments liabilities, which will in
turn be payed by the people.

This is essentially all there is to the Ricardian Equivalence idea.

-- idea has generated a huge amount of discussion among economists.

-- natural application is the explosion of the US government debt in the 1980's and again in the
2000's. One way to look at it is:

Y = C + I + G + NX

Y - C - G = national saving = I + NX

(Y - T - C)+ (T-G) = national saving

private saving + gov't saving = national saving = I + NX

Ricardian equivalence says that if we cut T, it will lower gov't saving, but raise private saving by
an equal amount.

-- Can also look at Ricardian equiv in the life cycle model....

-- Similarly, in PIH, tax cuts and increases are just transitory shocks; they do not affect permanent

31
income, and so do not affect consumption.

-- Note that Ricardian equivalence is about the timing of taxes -- it does not say that if the
government spending increases this should have no effect on consumption. That is, Ric Equiv says
that you care about the present value of the taxes you pay. Government spending, either today or
tomorrow, will affect this present value, and so affect consumption. For example, if the govt fights a
war today, your consumption will fall, because you will have to pay for the war. But whether the
war is tax financed or bond financed will not matter to your consumption today. [but note that the
response of consumption will depend on how long you expect the extra spending to last].

Potential problems with Ricardian Equivalence:

-- different interest rates. If the government can borrow for less than the rate at which people can,
then gov't debt may expand budget constraint (at least for borrowers).

-- If people are liquidity constrained in first period consumption, then government borrowing will
raise their consumption (show in fisher diagram).

-- If people are myopic whole thing doesn't wash. This is probably true, but hard to model.

--If people are life-cyclers, and will not be alive when the tax increase comes along, then their
budget constraints will be expanded and they will consume more. Later generations will get extra
taxes and consume less. This objection has generated the most debate, and often discussions of
Ricardian Equivalence lapse into discussions about intergenerational relations. Before going along
this path, we should note that even if this objection were true, most of the present value of any tax
cut today will be paid back by people who are alive today; in which case even if there were no
relations between generations, Ricardian Equivalence would be mostly true.

The intergenerational argument in defense of Ricardian Equivalence goes: Since we see people
leaving bequests to their children when they die, we know that they must care about their children's
utility. Now suppose that we take money away from their children and give it to them. Clearly, if
they were at an optimum level of transfer before, they will just go back to it by undoing the tax cut
(by raising the bequest that they give).

Much ink has been spilled attacking this proposition. For example:

-- Can specify the motive for bequests in a number of ways: if parents get utility from the giving of
the bequest, rather than from their children's consumption (or utility), then a shift out in the parent's
budget constraint will lead them to consume more of both bequests and consumption today. Slight
variation (Bernheim, Shlieffer, and Summers) is that bequests are payment for services (letters,
phone calls) from kids. Same result in response to a tax cut.

32
-- Alternatively, can argue that bequests are not for the most part intentional, but rather accidental.
Consider the life cycle model with uncertain date of death. This model will be covered later. When
you see it (with all its discussion of bequests, annuities, etc.), remember why it is relevant to the

-- Interaction of precautionary savings and Ricardian Equivalence (Barsky, Mankiw, and Zeldes,
AER.) -- Don't do in lecture -- just do in HW. (precautionary saving will be discussed below).

One more thing to think about with Ricardian Equivalence: What if people were completely
myopic, and never expected to pay back their tax cut. Note that if they were following our usual
consumption smoothing models, they would still raise their consumption only very slightly in
response to a tax cut (since they would spread their windfall out over the whole of their lives). So
Ricardian Equivalence is still almost true in such a case: for example, if people had 20 years left to
live, and the real interest rate were 5%, and they kept consumption constant, then a tax cut of \$100
that they never expected to pay back would increase consumption by approximately \$8. This is
pretty close to the zero dollar increase predicted by Ricardian Equivalence. By contrast, if one
believed in a Keynesian consumption function (where empirically estimated MPC's are in the rough
neighborhood of .75), then there would be a \$75 increase in consumption.

[but, of course, if RE were true and the tax cut were perceived to be permanent (due to a cut
in government spending), then C would rise by the full amount of the tax cut].

Deep thought:

Suppose that I look at data on the path of consumption followed by some person (or
household). What are the characteristics that I can expect to see in it, assuming that the household is
behaving according to lifetime optimization model described above.

I want to argue that one of the most important is that the level of consumption will never
“jump,” by which I mean that it will never change dramatically from period to period. When
consumption does change, it will be because of the difference between theta and r.

So if we do observe consumption jumping up or down, what are we to conclude from it?

I will list some possibilities, but it will take us a while to cover them. But you should see in
the list that they are all violations of the simple model presented above.

1)Liquidity constraints -- we had been assuming that these didn't exist

1)New information -- we had been assuming a world with certainty.

33
2) “non-convex budget sets,” specifically things like means tests -- we had been assuming these
away since we made income exogenous.

Liquidity constraints under certainty:

Let's return to the issue of liquidity constraints that came up when we looked at the two
period model.

Suppose that you have data on the income and consumption of a large number of individuals,
over a long period of time. Each individual is assumed to have known in advance (that is, from the
beginning of the sample period), what her income would be for the rest of her life. Individuals in
this data set chose their consumption to maximize a usual utility function, with u'(c)>0, u''(c)<0.
They were able to save money at a real interest rate which was exactly equal to their time discount
rate. However, they were not able to borrow money at all.
Individuals did not have smooth income. That is, for each individual there was a good deal
of year-to-year variation in income. Furthermore, some individuals had income that rose over the
course of the time period examined, while others had income that fell over the time period.
Remember, however, that each individual knew in advance what the time path of her income would
be.

What would you expect the data on consumption by individuals to look like? In particular,
discuss the following two points: First, what general statements can you make about what the time
paths of consumption of individuals can look like. What sorts of paths can you rule out? What sorts
of paths would you expect to see? Second, what will be the relationship between changes in income
and changes in consumption experienced by individuals? Are large decreases in consumption going
to occur in the same years as large decreases in income? Will large consumption increases come in
the same years as large income increases?

1) consumption can never jump down. When theta = r, consumption can never fall at all.

2) when consumption jumps up, it must be the case that assets are zero. If theta=r, then it must be
the case that assets are zero when consumption rises at all.

Uncertainty

So far, we have looked at consumption only in a certainty framework. We now look at the effects of
uncertainty.

34
Let's start by reviewing what we mean by expectation. Let x be a random variable, with probability
density function f(x). Then the expectation of x is


E(x)=  x f(x) dx
-

The relation between the actual realization of a random variable x and its expectation can be written
as

x = E(x) + ε

where ε is a random variable with mean zero.

First cut at uncertainty: lifespan uncertainty

In our presentation of the life cycle model, we assumed that the date of death was known. In
reality, of course, there is a good deal of uncertainty. To incorporate this into the LC model, we
apply the insight that, if you are not alive, you get no utility from consumption.
Let Pt be the probability of being alive in period t. Then an individual maximizes

T    PU (ct )
      t
t  0 (1   )
t

note that we still allow for Θ to measure pure time discount.

Consider the problem of a person who may die over period 0..T. Assume that there is no

The formal problem is
PU (ct )
T
Max  t
t 0
t
(1   )

s.t. At = (1+r)(At-1 + Wt-1 - Ct-1)

At ≥ 0        for all t

35
A0 given

Note that here, W, C, and A are the paths of wages, consumption, and assets that the person will
have if they are alive. That is, since the only uncertainty in this model is when you will die, and that
uncertainty is not resolved until it happens, you might as well plan out your whole conditional paths
of consumption and assets from the beginning (put another way: no new information arrives until it
is too late to do anything about it).

Note that the constraint on assets differs from before: as before we say that you cannot die in debt.
With certain lifespan, this implies that you have to have zero assets at period T. But now, it implies
that you have to have non-negative assets at all periods!

Maximization problems of this form are generally unpleasant (it will be discussed a little
below, when we look at the “Buffer Stock” model of saving). To get around it, assume that we are
looking at an elderly person with no labor income, who has only some initial stock of wealth. Such
a person would never let wealth become negative, because then she would have zero consumption
for the rest of her life.

We set up the lagrangian:
T
U( )           T -1

L =  P t C tt +   A0 -  C t t 
t=0 (1+  )          t=0 (1+ r ) 

The first order condition relating consumption in adjacent periods is
U ( C t +1 ) 1+ P t
=
U ( C t ) 1+ r P t +1

we can re-write the second part of the right hand side as

Pt     P +( P t - P t +1 )     P -P
= t +1               = 1+ t t +1  1+  t
P t +1        P t +1              P t +1

where small ρt is the probability of dying in a given period conditional on having lived to that age
[ρt=(Pt - Pt+1)/Pt]

The FOC is

36
U ( C t +1 ) (1+ )(1+  t ) (1+ +  t )
=               
U ( C t )       (1+ r)         (1+ r)

Note that the path of consumption that we are solving for here is the path that the person will
So the probability of dying functions just like a discount rate in this case.

------------------------------------
A note of realism: Obviously, the probability of dying rises with age.

Empirically, the probability of dying in old age turns out to conform very closely to a log-
linear specification:
ln(rho) = β_0 + β_1 age

This regularity is known as “Gomperetz's rule”

What will consumption paths of people look like given that this is true?

Suppose that initially, theta+rho<r. Them consumption should be rising. But over time, rho
will rise, and consumption will begin to fall. So there should be these hump-shaped paths of
consumption.

------------------------------------

Note, by the way, that even though I said that new information can be one of the reasons for
a jump in consumption, this model with uncertain lifespan does not get you any jumps in
consumption -- since when the new information arrives there is nothing that you can do about it!

Annuities

The person faced with the above problem will almost certainly die holding assets. Only if she lives
as long as was remotely possible ex-ante will she die with zero wealth. Assuming that she does not
value leaving a bequest, what could make her better off? Answer: an annuity.

Consider a cohort of people with a probability of dying  , and some market interest rate, r.
Suppose that a company makes a deal with each person, saying: "Give me your money, and I will
pay you some rate of interest z, but if you die before next year I will get to keep you money." What
would z have to be such that the insurance company earned zero profits?

Pays:       Earns
(1+z)(1-  ) = (1+r)

37
z = (1+r)/(1-  ) - 1  (1+r+  ) -1 = r+ 

An annuity is an example of such a contract. You give the company money, and they pay you a
yearly payment until you die. (Actual annuities are not like the ones described here, in that you pay
a lump sum up front, and then they pay you a constant level of income each year. )

What is the consumption path of an old person with access to an annuity? The FOC is just
U ( C t+1 ) (1+  +  )
=
U ( C t ) (1+ r +  )

so in the case where  =r, the person would have flat consumption even though her probability of
death was rising. Thus the payments from a real life annuity are consistent with the consumption
path that a individual with r=  theta would choose. )

Cost of Lifespan Uncertainty (Ryan Edwards) [should this and Bommier be moved to after
becker etc.?]

Why this interesting: We know that dying is bad (since you miss out on utility). Here we look at

One way this is bad is that you may have money left over at the end of life. However, annuities can
take care of that problem. But it turns out that there is still a cost.

Consider a person born at time zero with some known survival curve P(t). For simplicity, we will
give him some initial wealth A(0) and no labor income. Also for simplicity, we set the interest rate r
and the time discount rate  to be equal and greater than zero. There is a perfect market for
annuities. This means that the man will have flat consumption. The actual level of that
consumption will depend on the survival curve (it would be (   r ) A(0) if mortality were constant
at rate rho, for example). But we will not worry about this. Call c* the optimal level of
consumption. His instantaneous utility is thus U(c*). We will assume that this is positive (Recall
that for a CRRA utility function, instantaneous utility can be negative unless you add some constant
in front of it. We will get back to this issue in a little bit).

If he lives to age T, then his lifetime utility is
T
u (c*)
e
 t
u (c*)dt             (1  e T )
0


But from the perspective of the beginning of life, T is a random variable. So his expected lifetime

38
utility is the integral of the above thing over all the possible realizations of T. call f(T) the
probability that he will die at age T and call V the expected utility:

u (c*)
V                    (1  e  T ) f (T )dT
0



u (c*)
            (1   e  T f (T )dT )
             0

So from this expression we can see directly that uncertainly in the lifespan makes you worse off
[draw a picture of e^(-theta T); it is convex to the origin, so uncertainty raises the expected value by
Jensen’s inequality].

Now, suppose that we know something about the uncertainty of life span. It turns out that
the age of death (viewed from birth) has a hump shaped distribution that is not really normal, but
is not too far off. So let’s take it as normal. The standard deviation of age of death is around 15
(the mean, which is to say life expectancy at birth, is around 75). So let T be distributed
N(M,S2 )

We will use the fact that if x is normally distributed with variance  x , then
2

2
E (e x )  e E ( x ) x / 2

(This is a fact that is often very useful when paired with the CARA utility function in all sorts of
problems involving uncertainty). This in turn implies
2       2
E (e x )  e E ( x )  x /2

So then we can write expected lifetime utility as

u (c*)                    2 2
                (1  e  M    S /2
)


Now we can ask, what variation in M would be equivalent to eliminating uncertainty in lifespan (i.e.
setting S2 to zero.8 The answer is

8
I am doing a slight cheat here in holding c* constant. However, I could come up with a way of justifying that if I
really needed to.

39
S2
2

If we choose theta=.03, and for S=15, this gives 3.38 years. So this is how much you would be
willing to reduce mean life to eliminate uncertainty.

Notice that all the “benefit” here comes from discounting, which gives us the convexity.

An Alternative View of Lifetime Utility
(Bommier, “Mortality, Time Preference, and Life Cycle Models, working paper, 2006)

The model of lifetime utility with uncertainty that we have been using is the standard one, first
developed by Yaari (1965). The model is (in continuous time)


E (V )   P(t )e t u (c(t ))dt
0

[Note, I am using exponential time discounting, but Bommier uses a more general time discounting
where instead of the exponential term there is just some term  (t ) which represents the weight on
utility from a period, which we assume is non-declining]

This formulation comes, in turn, form applying the usual Von-Neumann Morgenstern model of
expected utility under uncertainty to a model of utility form a certain lifetime:

T
V   e t u (c(t ))dt
0

Bommier proposes the following alternative model for lifetime utility in the case of certainty:

T              
W     u (c(t ))dt 
0              

Where  () is a function that we will (in the usual case) assume has positive first and negative
second derivatives.

He argues that this function looks different than what we are used to, but that it meets the
same axioms that we want. For example, it says that the marginal rate of substitution between

40
consumption at any two points in time is independent of consumption at other points in time and of
the length of life, that more years of consumption make us happier, that more consumption makes us
happier, etc.

The big thing that this formulation does not have is a pure time discount rate. This is an old
argument. Back in 1928, Ramsey famously argued against a pure time discount rate in the absence
of mortality uncertainty (he said that it arose from “weakness of the imagination.”) Similarly Pigou
(1920) says that pure time discount is “wholly irrational.” It is not clear whether Ramsey and Pigou
meant these as statements about what was an appropriate model of human behavior (that is, a
positive view) or as normative statements. But anyway….

The justification for the  function is along the lines that a person gets “filled up” with
instantaneous utility, so that further increments do not do as much as initial increments. It is sort of
the lifetime equivalent of the explanation for the curvature of the instantaneous utility function. For
example, when you choose what books to read, you read the great ones first, and the less great, and
so on. Or similarly, you might spend \$100 on a restaurant meal every night, but some of those will
give you more pleasure because they are a new experience.

An interesting difference between the Bommier formulation and the standard one is
regarding “temporal risk aversion.” Consider some consumption levels
c1<C1 and c2<C2

Now consider two sets of lotteries. Lottery # 1 gives (c1, c2) and (C1, C2) each with equal
probability. Lottery #2 gives (c1, C2) and (C1, c2), each with equal probability. Loosely speaking,
Lottery # 1 gives you a chance of winning in both periods or losing in both periods, whereas in
lottery #2 you always win in one and lose in one.

According to the Yaari formulation, you are indifferent between lottery #1 and lottery #2.
According to the Bommier formulation (assuming  is concave) you prefer lottery #2. This seems
(I guess) more reasonable [at least until we introduce stuff like habit formation later on.]

Another advantage of the Bommier formulation is that it separates risk aversion from the
intertemporal elasticity of substitution. In the standard model, the curvature of the instantaneous
utility function determines both of these. In the Bommier formulation, the  function affects risk
aversion but has nothing whatsoever to do with intertemporal elasticity of substitution.

In the absence of lifetime uncertainty, the Bommier formulation implies that if there is a
positive interest rate, consumption will be rising. Indeed, without lifetime uncertainty, we don’t
ever need to know anything about the  function – we just maximize the thing inside it (which is
standard utility without time discounting) and we are done. [Note that in the case of certainty, there
is no reason to have the good meals or read the good books early in life, which is why you would

41
never get a declining path of consumption. You might get this if you allowed for utility from
memory, but then you would also have to allow for utility from anticipation… but this is all not the
point.]

The Bommier formulation becomes more useful when we allow for lifetime uncertainty.
Since  is a function, we can’t just pass the expectation sign through the integral. So expected

 T               
E (W )  E    u (c(t ))dt  
 0               

[We adopt the same setup as for the Yaari model, in which death is uncertain and unpredictable, so
one simply picks a feasible consumption path at time zero and sticks with it until death]

To see how this affects consumption, think about the example of reading books. If you know that
you will live exactly 80 years and can read one book per year, then with the Bommier formulation, it
doesn’t matter in which order you read the books. But if lifespan is uncertain, then you will start
with the best and read down the list from there. Similarly, with time discounting would have done
the same thing. So the point is that once we allow for mortality uncertainty, the Bommier
formulation gives us “discounting-like” behavior.

Unlike the Yaari model, however, the effect of mortality on consumption paths is not just
like incorporating a higher time discount rate. In fact, there is no simple closed form solution for
consumption paths (I think). The whole thing has to be solved numerically. This makes it less
attractive…. But Bommier says that now that we all have computers, this should not be a big
obstacle.

The Value of Being Alive vs. Dead and its implications for Convergence of Full Income

(very simplified discussion of Murphy and Topel, NBER 11405, and Becker et al, “The Quantity
and Quality of Life, AER March 2005).

The starting point for estimates of the utility of being alive vs. dead is people’s willingness to
trade off risks of death for money. This can be seen in e.g. the wage premium required to get an
individual to take a risky job, or the willingness of people to pay for safety features of a product.
We generally look at the effects of small changes in the probability of death, but to make such
measures useful, we blow them up to the “value of a statistical life.” For example, if a person is

42
indifferent between paying \$1000 and taking a 1 in 10,000 risk of death, then the value he is putting
on a statistical life is \$10,000,000. Estimates of the value of a statistical life in the US are around
\$6,000,000.

We consider a very simple setup. An individual has constant mortality probability  . He never
retires, and has constant wage w. The interest rate r and time discount rates  are equal and greater
than zero. Finally, there is an annuity market, so that the interest rate that the individual can earn on
his savings is r +  . These conditions deliver the result that the individual will want flat
consumption, and since he is born with zero assets, he will just consume his wage at every instant.

The individual has CRRA utility with coefficient of relative risk aversion sigma. In addition, the
individual has utility  just from being alive. His instantaneous utility function is

c1
u        
1

Now, consider a case in which the individual has the opportunity to trade a very small risk to his life
for more money. For example, he can spend \$500 more to take a safe flight vs. a risky one. We
consider a trade between life and risk that leaves the individual indifferent. [Note that the constant
mortality assumption buys us that willingness to trade life risk for money is not a function of age. In
real life, old folks should be less willing to pay to take a safe flight.] Let  be the probability of
dying, and let x be the amount of extra consumption that he gets. Since x is small, it does not affect
the marginal utility of consumption (it doesn’t matter if we imagine him consuming it all at once or
spreading it out over the rest of his life.) The cost in terms of expected life utility lost is

 w1    
       

w 1
          1
  e  (   ) t          dt            
0                 1                  

The gain is the marginal utility of consumption, which is w  , multiplied by the extra consumption,
x. Putting these together,

 w1    
       
1
         
x
w
              

The term x /  is called “the value of a statistical life.” [Note, in this derivation, there is a missing
(1-epsilon) term that does not matter as long as epsilon is small. Really, you only get the extra
marginal utility if you are alive]

43
We can rearrange this to solve for alpha

x                   w1 
     w  (    )        
                   1 

From here, we can just plug in numbers.
I use the following (mostly from Becker et al.)

W = c = \$26,000 this is GDP per capita in the US

x /  = \$2,000,000 this is on the low end of estimates for the US. (This is roughly what is implied
by Becker et al.’s formulation).

 = .8 This is their reading of the literature. It seems too low to me, but no one has a good
estimate

r =  = .03
  .02 (this gives a 50 year life expectancy)

Putting these together gives a value of  = -8.81. Becker et al., using a fancier approach, get a
value of -16.2. [Note that the value of  depends on  . Since their   1 , utility from
consumption is positive, and so  can be negative. For   1 , utility from consumption is negative,
so  has to be positive.]

In what follows below, I will use their value.

Given a value of alpha, we can ask at what level of consumption an individual is indifferent between
being alive or dead. That is, setting utility to zero

c1
0      
1

c  ( (1   ))1/(1 )  \$357

[Note: the level of consumption that gives indifference between being alive and dead is incredibly
dependent on  . To demonstrate this, I did the following. Using the above setup, including  =.8, I
chose the value of life so that I replicated Becker et al.’s value of  (this involved setting the value
of life to around \$1.5 million). This then replicated their value of the indifference level of
consumption. Holding the other values of the parameters constant, I then changed  to 3. The

44
implied value of the indifference level of consumption is roughly \$9,900! In fact, if  is 10
(admittedly an unreasonable value), then the break even level is around \$18,000, implying that being

The intuition for this large effect of  is that when  is large, the marginal utility of consumption
falls rapidly as the level of consumption rises. If sigma is big, then it means that reductions in
consumption raise the marginal utility of consumption a lot, so sufficient reductions in consumption
very rapidly get you to the point where utility from being alive is zero. (That is, if you are willing to
take any life risk at all, the U’ must be not too small relative to the utility of being alive. So then if
U’ is not too small and lowering c raises U’ a lot, then at some point lowering c will also make it not
worth being alive.]

Implications for Economic Growth

We usually look at economic growth by looking at growth rates of GDP per capita. But if
people get utility from being alive as well as consumption, we should consider their “full income.”
(Note: we don’t look at growth by looking at growth of utility. Why not? Because utility is not
observable.)

Consider an indirect utility function of an individual with annual income y(t) and survival function
(probability of being alive in year t) of S(t)


V (Y , S )  max  e  t S (t )u (c(t ))dt
0

s.t.

                               

e            S (t ) y (t )dt   e  rt S (t )c(t )dt
 rt

0                               0

[Fix up notation. Look back at paper.]

Note that we are assuming a perfect annuity market, so that expected lifetime income is equal to
expected lifetime consumption. Define Y as the PDV of expected lifetime income. So the indirect
utility is a fn of Y and S.

Consider a country at two points in time, with lifetime incomes Y and Y’ and similar survival
functions S and S’.

45
We are interested in the extra income that we would have to give the person so that he would have
the same utility he had in the second period, but with the mortality rates observed in the first. Call
this extra income W(S,S’)

V(Y’ + W(S,S’),S) = V(Y’, S’)

The growth rate of full income is the change in actual income plus this imputed change in income (I
think that this is called “compensating variation,” which is discussed more below in the context of
Jones and Klenow)

G = [Y’ + W(S,S’)]/Y – 1

(a few adjustments, not discussed here, have to be made to turn this from PDVs into annual income
growth).

1960       1960      2000       2000         Value of      Growth   Growth
life       GDP       Life       GDP          life expect   Rate     Rate
expect     p.c.      Expect     p.c.         gains (in     GDP      full
terms of      p.c.     income
ann income
Poorest    41         896       64         3092         1456          3.1%     4.1%
50% of
countrie
s in
1960
Richest    65         7195      74         18162        2076          2.3%     2.6%
50%

So poorest countries get big income growth boost

[Do Jones and Klenow here – I have a full handout on this! ]

Implications for Health Care Expenditures
(based on Hall and Jones, QJE 2006)

A man is born at time zero. Time is continuous. He receives exogenous, constant income at a rate of
y per period that he is alive.

The man cannot borrow or save. He can use his income for two things: he can purchase a

46
consumption good, which gives him utility, or he can spend it on health, which raises his probability
of staying alive (note that spending on health is not considered part of consumption in this setup. In
the real world, health spending is considered part of consumption, although it really shouldn’t be.)
Spending on the consumption good is denoted c; spending on health is denoted h. Thus the budget
constraint is:

y = c(t) + h(t).

His pure rate of time discount is zero. However, he only gets utility when he is alive.

The man’s probability of dying at any point in time,  (t), is a function of his health
spending at that point in time. Specifically,

 (t) = 1 / h(t)

His instantaneous utility is

u (t )  ln( c (t ))

Note: you do not have to do any fancy dynamic optimization, Hamiltonians, etc. to solve this
problem.

A.    Characterize the optimal paths of spending on health and consumption. You should not do
any math at all to answer this part of the problem. Simply state in words what these paths
look like and how you reached this conclusion.

B.    Write out expected lifetime utility as a function of c(0). Solve this integral to get the PDV
of lifetime utility as a function of c(0).

C.    Use the expression you just derived to find an implicit expression for the optimal value of
c(0). You will not be able to solve this explicitly.

D.    Suppose that income is

y = 2e,

where e is the base of natural logarithms (2.71828….). Solve for the exact values of h and c at time
zero. [Hint: Try writing the answer to part C in terms of the ratio h/c or c/h.]

E. [You can solve this part without solving parts C or D.] Consider several individuals who have
different levels of income. Each individual’s income is constant. How does the share of income
devoted to health vary with the level of income? (You could answer this by applying the implicit

47
the fact that rich countries spend more of their income on health than do poor countries?

1                   1
                        
U   ln(c)e    y c
 ln(c)  e   y c
 ln(c)( y  c)
0                         0

Derivative w.r.t. c implies (y-c)/c = ln(c) = h/c

Other things to do with this:

What is implied risk behavior as income gets really low?
      Willing to risk life
      Willing to risk wealth if there is an option not to be alive.

Relate this to Stone Geary – very different implications.
       SG says that risk aversion toward wealth rises as wealth falls.

Precautionary saving

We now ask a different question about uncertainty: how does uncertainty affect behavior?

It is easy to show that uncertainty makes you worse off. In the single period case, we
know that, if c is some random variable:

E(u(c)) < u(E(c)) .

So, if you could pay to reduce uncertainty, you would do so (for example, buying insurance). But
now we want to ask a different question: does uncertainty raise saving?

Let's examine the question in a simple two period model. For simplicity, we will assume that
there is no discounting and no interest rate. In period 1 you get income Y and consume C1. In
period 2, you participate in some lottery: with probability .5 you get some amount L given to you.
With probability .5 you have the same amount taken away from you. So your consumption in period
2 is

C2 = Y - C1 + L with probability .5 and

Y - C1 - L with probability .5

48
Your only choice variable is C1. We will want to answer the question: how does saving (or first
period consumption) change when the size of the lottery changes (that is, as uncertainty is
increased).

Expected utility is just:

E(V) = U(C1) + E(U(C2))

(Notice that we don't need to put an expected value symbol in from of C1, since it is known).

= U(C1) + .5*U(Y - C1 + L) + + .5*U(Y - C1 - L)

We maximize this by taking the derivative with respect to C1 and setting it equal to zero:

0 = U'(C1) - .5*U'(Y - C1 + L) - .5*U'(Y - C1 - L)

The optimal value of C1 is the value that solves this equation. That is, the equation implicitly tells
us C1 as a function of L. What we care about is how C1 changes as L changes. To find this
derivative, we use the implicit function theorem (See Chaing). If we write this as 0 = F(C1,L), the
implicit function theorem tells us that :

dC1                -.5[U (Y - C 1 + L) -U (Y - C 1 - L)]
=- FL =-
dL     F C1  U ( C 1 ) - .5[-U (Y - C 1 + L) -U (Y - C 1 - L)]

Notice that since U''(C)<0, the term in the denominator that is in square brackets is positive, and so
the whole denominator is negative. So the key term is:

U''(Y - C1 + L) - U''(Y - C1 - L)

When will this term be positive? When U'''(C) > 0. In this case, the whole expression is negative,
and so

dC1
<0
dL

in which case there is precautionary saving.

Assuming U'''>0 is what we have been doing in drawing the marginal utility curve as [picture]

49
The intuition is that spreading out consumption in the second period raises the expected value of
the marginal utility of consumption in the second period. This means that in taking part of
consumption away from the first period and moving it to the second period (ie by saving more), you
can, in expectation, get higher marginal utility.

This result about U'''>0 being necessary for precautionary saving is not unique to the two period
model -- it holds generally.

Many of our favorite utility functions, such as log, CRRA, and CARA have positive third
derivatives, and thus imply precautionary savings.

A utility function that doesn't imply precautionary savings is quadratic utility:

U(C) = ß0 + ß1C - ß2 C2

Here the first derivative is positive (for low enough C), the second negative, and the third is
zero.

(We know that quadratic utility cannot be globally correct, since it implies that marginal
utility becomes negative at some point. But it can still serve as a useful approximation as long as we
restrict our attention to a limited range of values of C)

We often use quadratic utility for mathematical simplicity, and also sometimes precisely
because it gets rid of precautionary savings.

If there is no precautionary savings (and also no precautionary dis-savings), then the
economy or consumer is said to display “certainty equivalence.” That is, they act as if future
income at every period t were certain to be equal to E(Yt).

====> Another application of this idea is precautionary childbearing [expand this; point out that
positive third derivative is a natural assumption.]

====> The general way to deal with these sort of uncertainty problems is via dynamic
programming, which you will see later in your math course.

3rd approach to uncertainty: Stochastic Income (leading up to Hall’s Euler Equation approach)

Example of a stochastic process and permanent income.

50
Suppose that we are the consumer, and we have some form of expectations about future
income: what is the optimal level of consumption that we should choose. [This example is taken
from Abel's chapter in Handbook of Monetary Economics.]

We consider optimal consumption for a person with income that follows a stochastic process:

wt 1  w   (wt  w)   t 1

where ε is iid mean zero.

Describe what is meant by this stochastic process. If α is zero, then income is iid with mean
w bar. If α is one, then income is a random walk. More generally, the size of α tells us how
persistent the income process is -- that is, how long a shock lasts.

Consider a person with
2) r = Θ
3) infinite horizon

1) will give us "certainty equivalence." That is, consumption will be the same as if there was no
uncertainty (even though utility will be lower).

2) will give us flat desired consumption

3) This is for convenience -- things would look almost the same with a "long" horizon.

So for certainty equivalence, we simply have to set consumption today at the level that
would be sustainable in expectation. That is, if all future values of ε were zero.

so the budget constraint is


            
 (1+cr ) t
s-t
= At + E t   w s s-t 
s=t                          s=t (1+ r ) 

notice that the subscript on consumption is t, while on wages it was s.

Of the two terms on the right hand side, the first is just assets (ie wealth), and the second is
the pdv of future wages -- which we could call "human wealth."

51
We use the rule that


1
 x = 1- x
s=0
s

so the left hand side can be replaced with

ct((1+r)/r)

so ct = (r/(1+r))*( A + HW)

(or, in continuous time, c = r*(A + HW) -- this should be fairly intuitive looking. It just says
that if you want to have constant consumption, you should just consume the interest on your wealth,
where wealth is actual plus human).

Notice that if w were constant, then the pdv of future wages would be w /r -- but because
of out timing assumption, HW would include today's wages, so HW would be w ((1+r)/r), and so
consumption would be

c = (r/(1+r))At + w

Now we can apply our same rule to the right had side of the equation. First re-write wt as
w + (wt - w )

The sum of expected is just ((1+r)/r)* w .

As to the other term, from the stochastic process, we know that

Et (wt+1 - w ) = α(wt - w )
Et (wt+2 - w ) = α2(wt - w ) etc.

52
So we get

1+ r
= At +
1+ r
w+ 
 s-t ( wt - w )
ct
r              r      s=t   (1+ r )s-t

re-arrange and get:

r               r
ct =        At + w +          ( wt - w )
1+ r          1+ r - 
This says that the amount by which current income affects consumption depends on the
persistence of income. The term r /(1  r   ) is the MPC, which will be very low unless income
is very persistent.

The Euler Equation Approach to Consumption

In the above example, consumption ends up being a function of only current income and
current assets, with the MPC out of current income depending on the degree of persistence in the
time series (ie on α).

The above approach used a particularly simple stochastic process for income (ie an AR(1)).
However, income could follow a much more complex stochastic process ( for example
Autoregressive Moving Average process, called ARMA -- you can learn about these in a time series
course.) In such a case, to predict future income we need not only know today’s income, but also
income from several past periods. Since consumption depends on the PDV of income, this would
imply that consumption was also a function of not only today’s income, but also income from
several past periods.

Going even further, there might be things other than past income that were observable today
and which also helped predict future income. For example,

-- suppose that we are looking at data on an individual. Say that she is a professor. Then her
future income growth depends on the number of publications. So her optimal consumption depends
on her number of publications.

-- Suppose that we have data on people’s professions (plumber vs surgeon). Different
professions have different wage profiles (flat for plumbers, rising for surgeons). Since the
individual knows this, he should take it into account in figuring out the PDV of future wages, and
thus this should affect consumption today as well.

53
-- if we are looking at the aggregate economy, there may be pieces of data available at time t
that tell us about expectations of future income, for example the value of the stock market or the
consumer confidence index. These should affect consumption today as well.

The upshot is that consumption today should be a function of wages today and of lots of
things that predict how wages will change in the future. So one could imagine estimating a very
complicated “consumption function” that incorporated all of this kind of stuff on the right hand side.

And indeed, people do this. The problem is that it is very hard to interpret what you get in
terms of any theory.

Now, suppose that instead of looking at the level of consumption, you look at the change in
consumption, that is, c(t+1) - c(t). What do we expect to be the predictive power of all of this stuff
(current income, past income, other stuff) that went into the consumption function? After all, this
stuff went into the consumption function in the first place because it helped to predict how income
would change over time.

The answer is that under the PIH, this stuff should have no predictive power at all!

So now suppose that you tried to predict consumption in period t+1 using consumption in
period t as well as other information available at time t. The test that Hall proposed is that nothing
else available at time t should matter. This turns the usual sort of econometrics that we do on its
head: usually, we do empirical work hoping that what we look at will matter (as judged by a t
statistic, for example). Here, the theory succeeds if other things do not come in significantly.

More formal description of the Euler Equation approach:

Consider the consumption plan formulated at some date t. That is, consider a person at time period t
who is deciding on consumption today. We know that, along this consumption path, the relation
between Ct and the Ct+1 is
1+ 
E t [U ( C t+1 )] =        U ( C t )
1+ r
where now consumption in period t+1 has an expectation sign in front of it because it is not actual
consumption in that period, but just what is expected to be consumed.9

9
This first-order condition relating consumption in adjacent periods is also known as the
Euler equation, and so this approach to studying consumption is often called the “Euler Equation
Approach.”

54
Note that all of the things on the right hand side are known already at time t.

Recall that the relation between the actual realization of a random variable x and its
expectation can be written as

x = E(x) + ε

where ε is a random variable with mean zero.

Or in this case,

U ( C t+1 ) = E t(U ( C t+1 ))+  t+1

That is, the actual marginal utility of consumption at time t+1 will be equal to the expected marginal
utility plus a mean-zero error.

If we assume that r= theta, then we can combine these last two equations to get

U ( C t+1 ) = U ( C t )+  t+1

This says that the marginal utility of consumption follows a random walk. Of course we
can’t observe the marginal utility of consumption -- we can only observe consumption itself.

U(C) = ß0 + ß1C - ß2 C2

then U'(c) = ß1 -2ß2 C

So the above equation becomes
 1 - 2  2c t+1 =  1 - 2  2c t +  t+1

or
C t+1 = C t +  t+1

55
where the epsilon is slightly differently scaled, but is the same mean zero error term.

Now, consider what should happen when we regress the change in consumption on all of the
“stuff” observed in period t that went into the consumption function for the level of consumption (ie
current and past income, consumer confidence, the stock market, papers published, etc.). All these
things predicted changes in income, but they should not predict changes in consumption. The
reason is that changes in income that they predicted were already taken into account when optimal
consumption in period t was calculated.

This is the test of the PIH proposed by Robert Hall. Specifically, suppose that you found
some variable that plausibly predicted changes in income, and which also came in significantly
when you regressed the change in consumption on it (and was in the information set of people at
time t when they made their consumption decisions). This would violate the PIH.

Hall's test of the permanent income hypothesis is so clever because it can be performed by an
observer (i.e. the econometrician looking at data) who knows very little about how the consumption
decision is being made. For example, we don't have to know anything about what the person's
expectations of future income are.

(One of the clever aspects of the Hall approach is that it does not rely on being able to actually
estimate the consumption function. The essence of the “Lucas Critique” is that the consumption
function should not be stable when the economic environment changes, and so estimating
consumption functions went out of style. Since the Lucas Critique has not yet been presented in this
course, this comment will not be very meaningful the first time you read it. But later in the course it
should make sense.)

More generally, if r is not equal to  , or if utility is not quadratic, the implication of the PIH
is that consumption should just be predictable by last period's consumption and some constant
growth term. If there are any other ways in which consumption is predictable, then the LCH/PIH
does not hold.

Empirical evidence on Euler equation results: it turns out that the question gets complicated. Can
see Hall for a summary.

An Aside: Tax Smoothing

The discussion earlier of Ricardian Equivalence took the movements of taxes from one period to

56
another as being exogenous. But a related literature has asked what is the optimal pattern of taxes.
That is, given that the government has a certain amount of spending that it wants to do, how should
it arrange taxes to finance it?

To answer this question we have to think a little harder about the question of how the taxes are
collected.

We often assume that taxes were lump sum -- for example, \$1 per person. Why is this the
easiest assumption to make? In this case, taxes do not have any effect on behavior other than
through the budget constraint. (In fact, even if taxes fall on labor income, if the supply of labor is
inelastic, then there is still no distortion, so taxes might was well be lump sum.)
In the case of lump sum taxes, we have seen that the timing of taxes may not matter at all.
But in the real world, very few taxes are lump sum. Instead, taxes are levied on income or
consumption -- that is, you have to pay the tax when you engage in economic activity. We have
learned in Micro that such taxes are distortionary.

As an example, consider a tax on wages. We have some supply for labor and some demand for
labor, an equilibrium price and quantity, and an associated consumer and producer surplus (note that
in this case the "consumer" is the firm, while the "producer" is the worker.). Now the government
imposes a tax on wages -- it doesn't matter if this tax is imposed on the worker or the firm, the result
is the same. The tax imposes a wedge between what the producer gets and what the consumer pays.
In equilibrium, quantity goes down. We can show the areas of consumer surplus, producer surplus,
and government revenue.

Note that a little triangle got lost -- this is the loss of efficiency due to the tax. This is the "dead
weight loss."

Now suppose we wanted to graph a function that related the loss of efficiency per unit of
revenue collected. When the tax rate is zero, revenue is zero. But the marginal revenue from
imposition of a tax is high, while the marginal efficiency loss is zero. As the tax rate rises, each rise
in the tax rate costs more in efficiency, and earns less revenue. So the cost in efficiency per tax
dollar collected is an increasing and convex function (that is, its second derivative is positive).
Now you should see the relation between the government's problem in choosing tax collections
over time and the individual's problem of choosing consumption over time. Individual has a
concave utility function, and wants to maximize utility. Government has a convex "distortion"
function, and it wants to minimize total distortions. In both cases, the optimal thing is to smooth.
So in the presence of distortionary taxes, if Ricardian Equivalence holds, gov't wants constant taxes.
And all of the Permanent Income type thinking that we are used to doing for consumption should
also hold for taxes: for example, suppose that the government has to fight a war. Should it pay for it
by taxing, or by borrowing? Clearly, this is a temporary change in it's spending. We know that an
individual's consumption should not react much to a temporary change in income; similarly, taxes
should not react much to a temporary change in spending.

57
(Note: We have been assuming that the interest rate is invariant to all of this moving around of
taxes. Maybe that is right for a small open economy fighting a war. But for a closed economy, there
will be a reaction of the interest rate to the fact that the government wants to borrow a huge amount
during a war. So all of the conclusions of this section have to be revised).

Notice that given what we said about tax smoothing by an optimizing government, the
random walk property should also apply to taxes: nothing other than today's taxes should predict
tommorrows taxes. So if r=Θ, there should never be any predicatble changes in taxes. (See Barro
article on this topic).

Precautionary Savings and Stochastic Income

Now we look at a similar problem of solving for optimal consumption when income is
stochastic.10 But this time we are interested in precautionary savings, so we no longer set up the
problem with certainty equivalence.

Recall the difference between risk aversion and precautionary savings. Risk aversion says
that you are made worse off by uncertainty. This is just due to u''<0. Precautionary saving says that
you will change your behavior (i.e. save more) in response to uncertainty about future income. This
requires that u'''>0.

For this problem we will use a utility function where u'''>0. However, we are not going to be
able to use our favorite CRRA utility function, because in this case it is not analytically tractable.
Instead we use the Constant Absolute Risk Aversion utility function:

-1       - c
U(c) =        e


u = e- c u = -  e- c u =  2e- c

[(move this earlier?) What is the difference between CRRA and CARA? Consider the question of
how much someone is willing to pay to get rid of a certain risk. For example, how much would you
pay to get rid of a risk that consumption will go up or down by \$100, each with probability 50%.
Say that the base level of consumption is \$500 (so that consumption might be either \$400 or \$600),
and you find that a person is willing to pay \$30 to eliminate the risk (that is, they are indifferent
10
Blanchard and Fischer, p. 288-291 and notes from Cecchetti.

58
between \$470 with certainty and the lottery of \$400 or \$600). Now suppose that we raise their base
consumption to \$1,000 but keep the size of uncertainty constant: their consumption will be either
\$900 or \$1,100. How much will they pay to insure this risk. Under CARA, the answer would still
be \$30. Under CRRA, it would be less than \$30, since the risk relative to consumption has declined.
Under CRRA, it would be the case that they would pay \$60 to insure a risk of \$200 on a base of
\$1,000..... extend this example? [Put in Kimball measure of prudence.]

Notice that a property of CARA is that when consumption is zero, the marginal utility of
consumption is finite. By contrast, for CRRA, when consumption is zero the marginal utility of
consumption is infinite. Indeed, the CARA utility function is defined for negative consumption,
while CRRA is not. This makes CARA suitable (mathematically) for problems like the one we will
do now, where consumption may sometime end up being negative in order to satisfy a lifetime
budget constraint. Whether this makes sense economically is a different question. Consumption
can't really be negative, but on the other hand, the property of the CRRA utility function that
marginal utility is infinite at zero consumption may overstate the importance of that state of the
world... even a low probability of infinite marginal utility will dominate optimization. In fact, if
consumption gets near zero in advanced countries, other mechanisms kick in... we will address these
below.]

We will assume that theta=r=0 for convenience. We also assume a finite horizon.

We assume that income follows a random walk:

Yt = Yt-1 + εt

Where ε is distributed N(0,σ2)

Notice that income can become negative.

So the optimization problem at time t is:

 T -1 -1 - c s 
Max E t   e 
 s=t           

where assets and income at time t are known, subject to the constraint that AT=0.

I will state the solution, and then prove that it is correct.

The solution is:

59
1               (T - t - 1) 2
ct =        At + Y t -              
T -t                 4

We will first check that it satisfies the first order condition. Given zero interest and discount rates,
the FOC is just:

U ( c t ) = E t [U ( c t+1 )]

First we will solve for ct+1 in terms of things that we know at time t.

The evolution of assets if given by

At+1 = At + Y t - c t

 1                (T - t - 1) 2 
= At + Y t -       At + Y t -               
T - t                 4          

 T - t - 1   (T - t - 1) 2
= At            +             
 T -t            4

Now, we write ct+1 in terms of quantities at time t+1, being very careful to write in the proper time
index!

1                    (T - t - 2) 2
c t+1 =           At+1 + Y t+1 -              
T -t -1                     4

We can substitute in for assets and income at time t+1:

60
 1         2                 (T - t - 2) 2
c t+1 =       At +  + Y t +  t+1 -              
T -t      4                       4

 1                        (T - t - 3) 2
c t+1 =       At + Y t +  t+1 -              
T -t                          4

So


c t+1 = c t +  t+1 +         2
and                                                       2


E t( c t+1 ) = c t +       2
2

Notice that this is similar to the Hall random walk result.

From the utility function:

U ( c t+1 ) = e - c t+1

But we care about the expectation of this. Luckily, there is a rule that we can use in this
case: Let x be some random variable distributed normally with mean E(x) and variance σ2x. Then

E( e x ) = e E(x) +  x /2
2

Notice that this rule corresponds to Jensen's inequality: the exponential function is convex, so the
expectation of exponential of a random variable is greater than the exponential of the expectation of

61
the random variable (say that five time fast).

For the current problem, ct+1 is a random variable when viewed from the perspective of time
t. We already solved for its expectation, and its variance is just σ2.

E t(U ( c t+1 )) = E t( e
- c t+1

) = e E t (- c t+1 ) +
Var(- c t+1 )
2           
we know that

2
E t(- c t+1 ) = -  E t( c t+1 ) = -  c t -                             2
2

Var(- c t+1 ) =  2Var( c t+1 ) =  2 2

So

E t(U ( c t+1 )) = e                  = U ( c t )
- c t

So the first order condition checks!

Checking the budget constraint is much easier. We only have to check that if we follow the
suggested policy, end-of-life assets will be zero. As usual, T-1 is the last period of life, and so we
just check that assets at the beginning of period T would be zero: we substitute into the formula for
assets in period t+1 above, letting period t be period T-1:

 1                (T-(T-1)-1) 2
AT = AT-1+YT-1 - CT-1= AT-1+YT-1 -         AT-1+YT-1+             =0
T-(T-1)                4        

So what do we learn from this?

62
Most important result is that expected consumption rises over time. Further, the speed with
which it rises depends on the degree of uncertainty: σ2. This confirms the result of our two-period
model of precautionary saving.

We can derive a testable prediction from the model of precautionary saving: Think if you
followed a cohort of people over their lives, where each person had uncertain income, but on
average there was no uncertainty. What would you expect to see for average consumption of the
people in the cohort? -- it would be rising over time.

Now suppose that we grouped people into occupations. Within each occupation, there is an
expected wage profile (say: flat for mechanics, rising for brain surgeons, etc.) Should the slope of
the wage profile affect the slope of people's consumption profiles? No, of course. But suppose that
occupations differed in their degrees of uncertainty? We could look at people in risky vs non-risky
occupations. The model predicts that people in less risky occupations should have lower average
rates of consumption growth.

This is a generally applicable point: there are many situations where on average there is not
much uncertainty, but where at the individual level there is a good deal of uncertainty. Some
examples: health, date of death, success in career, etc. Macro-economists used to ignore this sort
of uncertainty, precisely because at the societal level the law of large numbers them go away. But if
they affect behavior, then the uncertainty is relevant.

Liquidity constraints under uncertainty:

if Θ>r, this leads to buffer stock saving (do more formally?).

Here is the simplest buffer stock model:

E(w) is constant, but there is variance (iid): wt  w   t   with iid shock.
time horizon is infinite (or long)
theta > r
liquidity constraint

== if there was no uncertainty and no liquidity constraint, would get a declining path of
consumption

63
== if there was uncertainty but no liquidity constraint, would also get a declining path, adjusted for
shocks (or could have a rising path due to precautionary savings, I guess).
== if liquidity constraint but no uncertainty, would get consumption = wages as constrained
optimum.

== if both, it is complicated. We have to solve by dynamic programming. (flesh this out!)

Define cash on hand as w + A -- that is, all that you have available to spend this period.
Since income is iid, this is all you can base your decision on. So we are looking for c as a function
of cash on hand.

===> handout picture from Deaton book. (Deaton’s parameters:   10% , r  5% )

To be added (maybe): non-linear budget sets.

In real life, the assumption that income is completely exogenous to consumption choices
does not hold. One of the most important feedbacks from consumption to income is in the form of
means tested government programs, which provide income to you only if you are poor enough to
qualify. Examples are Medicaid, Food Stamps, Welfare, financial aid for college, etc.11

[flesh out simplest example : see former homework problem below ]

33. Consider a world in which people live for two periods. In the first period of life, person i has
income of xi. In the second period, all people have income of zero.

Individuals can save money at a real interest rate of zero. They cannot borrow. They are
born and die with zero assets. Individuals have log utility, and their time discount rate is zero.

There is a government program which has the following setup: if the sum of your income
and your savings (from the previous period, if any) in a given period is less than a cutoff level, c,
then the government will give you enough money so that you can consume c. If they sum of your
income and savings in a period is higher than c, then the government will not give you anything.

Describe the relationship between the first-period saving rate (that is, saving divided by first
period income) and first period income. If you can't get exact solutions, try drawing a picture and
discussing how you think it might be done.

11
There is also an important feedback at the aggregate level: the more that people in a country
save, the higher will be the capital stock, and thus the higher will be wages. We will deal with this
feedback extensively when we study growth.

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Alternative forms of the utility function

We have been assuming time-separable preferences -- so utility at a given time depends on
just consumption at that time.

Is this reasonable? Quite possibly not: two people with the same consumption might have
different happiness depending on what they were used to.

There is nothing radical about non-time-separability (unlike some other alternatives to the
usual utility function) -- when we use time separable preferences it is really only for convenience.
One form of time non-separability is durability [think about food or vacations]. If utility from
consumption is durable, then the more I consume in period t, the lower my marginal utility of
consumption is in period t+1. [Note: if c in one period makes me happy in another period without
affecting the marginal utility of consumption, then we can just add this happiness back into the
initial period’s utility and we are (almost) back to time separability.]

We will look at another form of non-separability called habit formation, in which consuming
more in a period raises the marginal utility of consumption in the future.12

Consider a utility function like

1-
 c 
 
U(c,z) =  z 
1- 

where z is the quantity of habit.

Notice that habit makes you worse off. If gamma is zero, then habit is irrelevant. If
gamma=1, then only the ratio matters. So γ indexes the importance of habits. The closer it is to 1,
the more people care about consumption relative to habit. The closer it is to zero, the more people
care about the absolute level of consumption. For example if γ=.5, then a person with consumption
of 4 and habit stock of 4 will have the same utility as a person with consumption of 2 and a habit
stock of 1.

Habit in turn evolves according to the level of consumption:
12
It is perfectly possible for durability and habit-formation to coexist at different time
horizons. For example, if I consume a fine French meal at noon, it lowers the marginal utility of
French food at dinner time, because I am still full. But it also raises the marginal utility of French
food in future days because I will not longer be full and now have acquired a taste for good food.

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zt+1 = ρct + (1-ρ) zt

The bigger is ρ, the faster habits adjust.

We can rewrite this as an infinite sum:


zt 1    ct i (1   )i
i 0

So this says that the smaller is ρ, the more consumption in the distant past matters for today’s habit
stock.

(As an aside: we have been taking “habit” or “what you are used to” to depend on your own
past consumption. Another possibility is that it depends on the consumption that you observe
around you. This is the idea of “keeping up with the Jonses.”)

This can also be incorporated into a utility function. The classic case was Dusenberry, who
formulated the “relative income hypothesis” in an attempt to explain the same facts that Friedman
successfully explained by the PIH.)

Anyway, these things are a mess to solve, but they may provide better fit to reality that the
function that we use.

Interesting thoughts: do people understand their own future habits? See Keynes on Economic
Prospects! Discuss COW. Example of tenure, loss of limbs, etc. Also example of lunch.

More formal discussion of optimization in habit formation model. Can do something based on
Deaton formulation (see deaton book or Paxson eer): quadratic sub utility (muellbaur form) gives
simple first order condition even in stochastic model. Or can go over COW

 Also discuss Stevenson and Wolfers re: Easterlin Paradox

---------------------------------------------------
Inter-temporal consistency.

Go back to time-separable preferences.

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You will see when we you study monetary policy later on, that one of the key problems that
faces a policy maker is time inconsistency (example: negotiating with terrorists; nuclear war;
monetary policy).

Question: In a world of certainty (which we have been doing so far), would you ever want to re-
optimize. That is, if you can make all of your decisions at time one, are you then happy to stick with
them?

The classic paper that deals with this problem is Stroz (1956), which I confess I haven't read.

Stroz shows that the only form of discount factors which does not lead to inconsistency of the
lifetime utility function is our usual exponential discounting (i.e. dividing utility in period t by
(1+ θ )t in discrete time, or multiplying by e^{- θ t} in continuous.

Any other form of discounting leads to the paradox that the relative importance of two periods=
consumption depends on what point in time the problem is being viewed from

[picture]

Some people (such as Angus Deaton in his book) think that Stroz’s result shows that any
form of discounting other than exponential is irrational, and thus should not be considered.

Recent work by David Laibson takes on this challenge: argues that preferences are not, in
fact, exponential, and further that there is loads of evidence of inconsistency.

examples of inconsistency: revisions of plans, diets, Christmas clubs and other forms of hand-
tying, etc.

Laibson claims that the natural discount function is hyperbolic: so discount the immediate
future a lot, but not much between immediate and far future. [note: “hyperbolic” is not the
mathematically correct term for what Laibson posits, but the term has stuck.]

The actual utility function that he uses in his paper is

                                           

T -t
 1 
Ut = E t u( c t )+              u( c t +    )
 =1  1   

where ß<1

hyperbolic discounting means that my (today) self will want to precommit: forced saving,

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purchase of durables, etc.

Laibson then models a game between different “selves” at different periods.

Notice that the selves will want to pre-commit later selves. For example, I might want to
save my money in a way that it can only be taken out with one period's advance request... (Thaler’s
proposal: save more tomorrow)

Laibson also argues that the recent decline in the saving rate comes from the ability to get around
these sort of precommitment devices.

[for somewhere: talk about Phillipe Weil & non-expected utility].

Empirical evidence on LCH/PIH

In this section I will present a grab-bag of pieces of evidence relating the LCH/PIH. Since the
theory is a little amorphous and the data is imperfect, there is never going to be any one test that is
completely convincing. Three points to looking at all of these tests: First, get an overall impression
of how good the whole approach is. A second goal is to give some examples of how one applies
data to the testing of a theory. Third, to see how theory can be twisted around to make it fit the data.

1.1 Tests of the LCH looking at the wealth of old people. If one takes the life cycle model
seriously, and doesn't think that inter-generational relations are important, then one would expect to
see people's assets declining in old age. Since there is uncertainty, we do not expect to see assets hit
zero on the day before death, but we still expect to see them going down.

Many people have tried to test the LCH by seeing if old people have negative saving rates. The
answer tends to be that they either keep constant assets or they dissave slightly. To justify this with
just uncertainty, one would have to posit extreme risk aversion.

1.2 Variation on this test is to ask whether a bequest motive can explain this failing of the pure
LCH. Test is to look seperately at the behavior of people with and without children, under the
assumption that people with childen have stronger bequest motives. Result is that there is no
difference between the behavior of the two: this is bad news for the intergenerational version of the
LCH.

2. Tests of the life cycle model as an explanation for the aggregate capital stock. One point of the
LCH is to explain the size of the aggregate K stock. Can test this by simulation. Start with realistic
parameters for discount rate, interest rate, wage path over course of life cycle, population growth,

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etc. Then can figure out what each person's wealth should be. Sum this up across people, and
compare it to aggregate income [also derived by summing up individual income across age groups].
This gives you a wealth/income or K/Y ratio. Now compare this to what we observe in the
economy [roughly 3]. Result: ratio delivered by model is too small. [cite: White?] In other words,
the LCH does not explain most of the capital stock.

3. Relation between lifetime income and consumption profiles.

Note that we don't expect consumption profile to be flat, since tastes may change over lifetime.
For example, when you have kids, you may want to spend more. Similarly, when you are old, you
may have less utility from consumption (ie can't go on vacations).

The key prediction of the LCH/PIH is that your consumption profile should be invariant to
the shape of your income profile. That is, two people with the same discounted lifetime value of
income but different patterns in which it is received should have the same consumption profile. For
example, an athlete who makes all his money early in life, vs a brain surgeon who doesn't start
earning until late in life, but then makes a lot.

This is one of the things that Carroll and Summers look at in their paper. They find that, breaking
people down by occupation, the averages income and consumption profiles are, in fact, very similar.
They interpret this as meaning that the LCH/PIH is just wrong.

One counter argument to their results: suppose that utility is a function of some combination of
consumption and leisure. What is the price of leisure? It is just the wage. So when wages are high,
then leisure is expensive, and so people consume less of it. This may raise the marginal utility of
consumption, and so people will consume more (since, ignoring interest and discounting, they are
setting the marginal utility of consumption equal in every period.). Key the sign of d2u / dc dL
(homework problem).

This model is particularly appropriate if much of what we measure as “L” is really home production.
High wage people, and people in the high wage part of their careers, substitute out of L and into
more c. For example, we often see a big drop in consumption at the (predictable) time of
retirement, when there is predictable fall in wage income. Is this a violation of the LCH? Maybe
not, since people are using more time in order to shop around and get more units of goods for their
consumption dollars.

4. Evidence on the size of assets: one can just look directly at the size of assets that people hold.
Ask: are they holding as much wealth as the LCH says that they should be? Find that, in fact,
people hold little wealth. For example median financial wealth (that is, not counting house or
pension or NPV of future Social Security) of families with head aged 45-54 was \$4,131 in 1983. So
it doesn't look like life cycle saving.

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But there are some mitigating points here. First, most people have their retirement saving done
for them in the form of SS and pensions. So maybe they are even saving too much in these forms,
and don't want to save more -- maybe they would even save less, if they could. Also, people hold
houses, which are very valuable. But in any case, assets seem so close to zero that it is hard to
believe that they represent some optimum choice [note: it is possible that people set their pensions
so that non-pension assets are zero...].

5. Campbell and Mankiw -- time series tests of the response of consumption to income. As we have
seen above, there is no problem with consumption responding to changes in income. It may simply
be that the change in income contains information about future changes in income. But suppose that
we could look only at predictable (in advance) changes in income. PIH/LCH says that consumption
should not change in response to these. This is what Cambell and Mankiw look at

Yt = total disposable income. Suppose divided into two streams: to PIH people (group 2)
and to people who consume current disposable income (group 1). λ goes to group one.

C1,t = Y1,t = λ Yt
Y2,t = (1-λ) Yt

in differences:
C 1,t = Y 1,t = Y t

While for the PIH groups

C2,t = Y2,tp = (1-λ)Ytp

C 2,t =  +(1-  ) t

where mu is the trend growth rate (due to OLG and growth) and epsilon is the innovation in
_permanent_ income. (and is orthogonal to past...)

C t =  + Y t +(1 -  ) t

Of course this can't be estimated directly, since change in Y is correlated with error in permanent

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income. (If there were never any shocks to permanent income, then ε would be zero all the time, and
we could estimate the equation directly).

The solution is to find instruments for Delta Y -- lagged things that predict changes in
income.

The solution is to use twice-lagged income -- see my notes for 284 for further discussion (or
see the article itself or Deaton's book).

They estimate λ to be around .5 -- so half of all consumption is done by rule-of-thumb (or
liquidity constrained) consumers. (This does not mean that half of people are this way -- since it is
probably the poorer people who are more likely to be liquidity constrained, more than half of actual
people will be liquidity constrained if half of consumption is done by such people.).

Shea

(not the paper discussed in the Romer book)

Same framework, but examine the possibility that liquidity constraints are the problem.

Divide changes in income into pos and neg. It should be for Pos where consumption follows
income -- so it should have a larger lambda.

Unfortunately, there are few drops in income, and fewer predictable (at least at the agg level)

More realistically: divide into above and below trend.

Table: perverse finding: consumption responds more to predictable changes in income when
income is going down than when it is going up.

Kubler Ross? (see quote).

Wilcox

changes in SS benefits announced at least 6 (and usually 8) weeks in advance. Under PIH, there
should be no change in consumption in the months in which change actually occurs.

Coefficients say that there is a 1.4% change in consumption for a 10% increase in SS benefits.
Note this does not say that consumption of the SS recipients goes up by 14% of their income change
-- this is total consumption.

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1986 SS payments 200 billion, PCE 2.8 trillion. So by point estimate, more than dollar for dollar
spending.

Wilcox also finds that the increase is biggest in durable goods (and of these, biggest in autos)
-- this seems like people are buying a car with their extra cash flow!

Many extensions of this work possible: look at consumption of old-people sensitive goods
(big, slow cars, eg) or by location...

New Material on Consumption

More evidence for and against the PIH:

Souleles (1999) Consumption rises when income tax refunds are received, even though this is
predictable.

Parker ( 1999) Consumption rises when take home pay rises as a result of cessation of Social
Security payments. This consumption is concentrated in durables and goods that can more easily be
postponed.

(also discuss first of the month papers) (this would be a good place to discuss M-Pesa material if not

-- Shapiro and Slemrod (AER 2003, and an earlier paper): looked at survey evidence from two
episodes. In 1992, there was a reduction in withholding but no change in eventual tax liabilities. In
2001 there was a change in law that cut taxes semi permanently (for 10 years), and a rebate of
excess withholding. In both cases, they asked households how they planned to deal with the extra
money: consume, save, reduce debt (which is the same as saving). Very odd results. In 1992, 40%
of households said they would mostly consume the extra take-home pay; in 2001, 22% of
households said that they would mostly raise spending. But this is the opposite of what theory says:
there should be little or now consumption out of the first change, and (ignoring long-term Ricardian
considerations) the second really was a semi-permanent increase in disposable income.

       Parker, Souleles, Johnson, and McClelland (2011, NBER WP 16684) Study fiscal
stimulus payments of 2008. Payments were large: 300-600 per for individuals, \$600-
1,200 for couples, with an addition \$300 per child. Find that consumers spent between
50 and 90 percent of payments, with the largest part of spending going to durables. How
can they measure? Answer: there was a random component to the timing of the
payments that households received, based on the last two digits of the social security

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number. The authors used the Consumer Expenditure Survey, where they could look at
consumption spending and knew SSN (and date) so they could look at whether

       monthly income is relatively constant throughout the year, but consumption
expenditure rises significantly in December (in US, 21 percent.) This is actually good
news for the PIH, even though consumption is non-smooth. The reason consumption
rises is obviously because of preferences for holiday spending. The fact that
consumption responds to this, rather than matching income, is thus good news for PIH.

       Browning and Collado (2001): in Spain, the majority of workers receive a double
paycheck in June and December -- but there are some workers who do not receive such
payments. (This depends on the worker’s job -- it is _not_ like a bonus that is uncertain.)
Finding is that the seasonal pattern of expenditure is the same in two groups. This is a
big win for PIH.

[Possible reconciliation: in the Spanish case, the cost of not smoothing would be large. In the US
cases, the non-smoothness is small, so no big cost from just having consumption move with income.

 Adams et al. AER Mar 2009, car purchases by low income people. Even though interest rate
is 20% or more, most pay mininimum down payment. Also peak in purchases in Feb, at time
of tax refunds. (nice pictures); sort into how big their EITC was; show that groups that get
bigger refunds have bigger purchase spikes.

One-time Payments:

B 1950, unanticipated payments to subset of US veterans holding National Service Life Insurance
policies. MPC = .3 - .5. Similarly, reparations payments from Germany to certain Israelis 1957-58,
MPC = .2 (these were very large payments, equal to about one year’s income). Carroll points out
that when these cases were originally analyzed, they were seen as victories for the Friedman PIH
since the MPC was much less than one. But in modern view, they are failures, since the MPC is
much higher than .05. Carroll’s point is that the modern PIH is not the same as the original
Friedman PIH.

Browning and Crossley, JEP, Summer 2001. Also Carroll JEP Summer 2001.

Yet more to be added on this: Stephens AER 2003 “3rd of tha Month…”,

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Toward reconciling all (some?) of this stuff:

1. Liquidity constraints prevent borrowing

2. Many people would like to borrow (either because income is growing rapidly and/or because
they have a high discount rate).

3. People don't hold zero assets because of a precautionary motive: income is stochastic and they

4. Optimal strategy (ala Deaton and Carroll) is to hold a buffer stock of assets -- say a few months
worth of income.

[could do a little exercise to show this... or go over Deaton’s version.]

5. How explain the aggregate capital stock? ==> rich people are not subject to the above model.
They hold most of the wealth. They are not subject to any of the models that we know.

6. Maybe some room for LCH/PIH people in the middle

7. Maybe more folks would optimize if they had to

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Problems

1) The Coefficient of Absolute Risk Aversion is defined as
U (C)
-
U (C)

and the Coefficient of Relative Risk Aversion is defined as
U (C)C
-
U (C)

where U'(C) is the first derivative and U''(C) is the second derivative of the utility function.

Consider the following two utility functions:

(1)   U(C) = -(1/α)e(-α C)

(2)   U(C) = C1-σ/1-σ

In which case is the coefficient of absolute risk aversion invariant to the level of C? (this is called
the “constant absolute risk aversion utility function”). In which case is the coefficient of relative
risk aversion invariant to the level of C? (this is called the “constant relative risk aversion utility
function”).
1.5) [core exam, 2006] A researcher is trying to determine the parameters of a woman’s utility
function. He knows that she has CRRA utility, but he does not know the value of her coefficient of
relative risk aversion,  , or her time discount rate,  . The woman is infinitely lived. Time is
continuous.

The researcher has presented the woman with different possible paths of consumption, asking which
was preferred to which. The woman answered that she was indifferent among the following three
paths of consumption:

c(0) = 1, g=.01
c(0) = 2, g = 0
c(0) = 4, g = -.0025

where c(0) is initial consumption and g is the annual growth rate of consumption.

Based on this information, solve for the woman’s values of  and  . Note: solving this in general
would require a computer. However, I will make things easier by telling you that  is equal to
either 2 or 3.

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2. Consider a two period Fisher model in which utility in each period is given by the function

U(C) = C1-σ/1-σ

Solve for the derivative of saving with respect to the interest rate: dS/dr (this is just the
negative of the derivative of first period consumption w.r.t. the interest rate). What condition on the
values of w1 and c1 guarantees that dS/dr is positive? Assuming that this condition does not hold
true, how (informally) does the value of σ affect whether dS/dr is positive or negative.

3.     An individual lives for two periods. In the first period she earns a wage of 100. In the
second period she earns a wage of zero. She earns interest on her savings at some interest rate r>0.
Her within-period utility function is
1-
c
U( c t ) = t
1- 

She has a discount rate of zero.

For what values of σ will her first-period consumption be equal to 50? For what values of σ
will it be less than 50? For what values will it be greater than 50?

3.5. [midterm 2005] An individual is born at time zero with assets of 100. She will live forever.
Time is continuous. She will receive no labor income. There is no uncertainty. She has CRRA
utility with coefficient of relative risk aversion (  ) equal to 2. The interest rate is 5%. Her initial
consumption is 3.

Solve for the value of  .

4. [midterm exam, 2005] A woman is born at time zero and will live forever. Time is continuous.
She has initial assets of zero. She can borrow or lend at interest rate r=0.04. Her time
discount rate is θ = .02. Here wage at time zero, w(0) = 1. Here wage grows at a rate of 2%
per year. She has CRRA utility, with a coefficient of relative risk aversion of 2. Solve for
her consumption at time zero.

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4.25) [Midterm exam, 2010] Mr. X is born at time zero with initial assets of 200. He receives no
labor income. He has a pure time discount rate of 4%. Mr. Y is born at time zero with initial
assets of 50. He also receives no labor income. His pure time discount rate is zero. Time is
continuous. The interest rate is 2%. There is no mortality. Both men have CRRA utility,
with coefficient of relative risk aversion equal to two.
A) [16 points] After how many years will the assets of the two men be equal? You should be
able to answer with an actual number (using a small approximation), but if that doesn’t work,
you can just write down the correct expression.

B) [16 points] After how many years will the consumption of the two men be equal? In this
case, you should supply the correct expression but you will not be able to (easily)
approximate the number of years.

4.5) [midterm exam, 2008] A woman has Stone-Geary utility of the logarithmic form:

u (c )  ln(c  c )           c 0

She is born at time zero and will live forever. Assume that r    0 . She has labor income
that is constant and equal to w per period. Assume that w  c .

A)     [5 points] Write out the discrete-time first order condition relating consumption in two
adjacent time periods t and t+1.

B)      [20 points] Solve for consumption at time zero. This can be done by either solving the
whole problem in discrete time, or switching to continuous time. If you switch to continuous time,
you will have to “guess” or derive the FOC for consumption in continuous time. However, I find it
easier to do this than to solve in discrete time. [Hint that was not on the midterm exam: things are
much easier if midway through the problem you define a new variable like c  c  c .]


5. Consider the optimal consumption path of a person who will live exactly T periods. In the first
period, she has labor earnings of one unit. Subsequently, her wage grows at rate g, so that in the
second period she earns 1+g, in the third period (1+g)2, etc. She can borrow or lend at an interest
rate of zero, and she discounts future utility at rate zero. Her instantaneous utility function is of the
CRRA form. She starts and ends life with zero assets.

Calculate saving in the first period of life (that is, first period income minus first period
consumption). What is the effect of increasing g on first period saving? Explain.

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6.A man lives for 40 periods. He has constant absolute risk aversion utility:

-1       - c t
U( c t ) =        e


He earns a wage of 100 in each period. The interest rate is r and the time discount rate is q. He is
born with zero assets and will die with zero assets, but he is able to borrow during his life (i.e. he is
not liquidity constrained). There is no uncertainty.

A) Derive (or state) the first-order condition relating consumption in adjacent periods of his life.

B)Assume that α=1, q=.05, and r=0. Derive his optimal first period consumption. Note: you will
have to use the approximation that ln(1+x) = x, which holds true for values of x near zero.

6.5 [midterm 2005] An individual is born at time zero. Time is continuous. The individual has
instantaneous utility function

u  ln(c ) if alive

(Note: this is the same as the utility function being u = ln (c) +      where  =0).

The individual has initial assets A(0) of 20. She has no labor income. There is no
uncertainty. The interest and time discount rates are both zero.

The individual has a maximum life to 20 years. She can choose to stop living earlier,
however.

Solve for her optimal path of consumption. You should calculate the exact level of c(0).

7.      A researcher has collected income and consumption data from a large population.
Consumption in the population is determined by the permanent income hypothesis: C = α Yp, where
Yp is permanent income and 0<α<1. Permanent income in the population is given by Ypi = Y p + ρi,
where ρ is distributed normally, with variance σ2p The researcher does not observe permanent
income, however. She only observes current income, which is related to permanent income by Yci =

78
Ypi + εi. εi is transitory income. It is distributed normally, with variance σ2t. There is zero
covariance between permanent and transitory income.

The researcher estimates the “consumption function”

Ci = ß0 + ß1 Yci

What value will she get for ß1, the marginal propensity to consume? How will ß1 compare to
α? Under what circumstances will it be a good estimate, and under what circumstances a bad
estimate?

7.5 [midterm exam, 2005] There are two kinds of people in a country: blue and red. Each person
has annual labor income given by the following equation:

wi ,c ,t  wc   i  i ,t

Where wc is the mean permanent income of people of color c (c=red or blue),  i is a random
variable that determines the permanent income of person i relative to the mean for his color group,
and  i ,t is transitory income for individual i in year t.  i and  i ,t are distributed iid normally with
means of zero and standard deviations   and   . Assume that wred  wblue .

A researcher has collected data on income, color, and consumption for a large number of
individuals in a single year. The researcher looks at how saving rates of red and blue people
compare, holding income constant (that is, the researcher looks at the saving rates of red and blue
people who have the same income). What will the researcher find? Why? Discuss how the relative
sizes of   and   will affect the magnitude of the researcher’s findings.

8.       [final exam, 2001] In response to the current economic slowdown, some economists have
proposed a temporary reduction in sales taxes in order to stimulate consumption. The following
question is inspired (loosely) by that proposal.

Consider the problem of optimal consumption in the presence of sales taxes. Let et be
expenditure on consumption in period t (that is, what the household spends). Let τt be the tax rate in
period t. The relation between consumption, expenditure, and taxes in a period is

ct = (1 - τt) et

We assume that all tax collections are thrown away.

Consider a person who lives for two periods. He has wealth A1 at the beginning of the first
period. He earns no wage income. There is an interest rate r and time discount rate θ, where

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r = θ >0. The utility function is CRRA.

A) Suppose that tax rates in the two periods are equal, that is τ1 = τ2 = τ > 0. Solve for optimal
consumption and expenditure in each period.

B) For what values of σ will a decrease in the first period tax rate lead to an increase in first period
expenditure?

C) For what values of σ will a decrease in the first period tax rate lead to an increase in first-period
tax revenue?

9. [midterm exam, 2001] So far, we have been ignoring changes in family composition when we
studied consumption. In fact, however, such changes -- for example the addition of children to a
household -- have an important effect on consumption.

Consider the case of a household that lives for two periods. In the first period there are M1
members of the household, and in the second period there are M2 members. The interest rate and
time discount rates are both zero. The present discounted value of lifetime wages (of all members of
the household) is w. Let ct be total consumption of the household in period t. We assume that
consumption is split evenly among the members of the household. We will consider two different
possible ways of modeling the way in which consumption affects utility. In each case, solve for the
optimal level of total household consumption, c, in both periods.

A) Suppose that consumption is split evenly among members of the household, and that total
household utility in a period is equal to per-person utility multiplied by the number of people in the
household. Per-person utility, in turn, is just given by the CRRA utility function. So total utility is:

 ( c / M t )1- 
Ut =Mt  t              
     1-        

B) Suppose that consumption is split evenly among members of the household, and that total
household utility in a period is equal to average per-person utility. So the total utility function is:

( c t / M t )1-
Ut=
1- 

Problems with Liquidity Constraints

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10. Consider the following version of the Fisher model. Individuals live for two periods. They can
borrow at a real interest rate of 100% (that is, if they borrow one dollar in period 1, they must repay
two dollars in period 2). They can lend at a real interest rate of zero. Their preferences are given
by:

U = ln(C1) + ln(C2)

Find optimal first period consumption for the following three individuals:

Ms X: W1 = 32 W2 = 32
Ms Y: W1 = 0 W2 = 64
Ms Z: W1 = 24 W2 = 40

10.5 [final exam, 2006] A woman is finishing graduate school and deciding on her career. She
has zero assets and is infinitely lived. The two careers she can choose are labeled A and B. Wages
in career A begin at w0,A and grow at rate 2% per year. Wages in career B begin at w0,B and grow at
rate 4% per year.

She can save at interest rate zero, but she cannot borrow. She discounts the future at rate 4%
per year. She has CRRA utility with coefficient of relative risk aversion   2 .

Solve for the ratio of initial wages in the two careers such that she is indifferent about which
to choose.

11.      (midterm, 2001) In our previous analysis of differential interest rates we assumed
(realistically) that the interest rate on borrowing was higher than the interest rate for saving. Now
we make the opposite assumption, which is not all that realistic.

Consider a person who lives for two periods. The interest rate for saving is 100%, so that
one dollar saved in period one turns into two dollars in period 2. The interest rate on borrowing is
zero.

A woman has labor income w1 = 2 and w2 = 3. She has log utility and a time discount rate
of zero.

Draw the set of feasible consumption possibilities. Show how to solve optimal consumption
in each period. It turns out that in figuring out optimal consumption, the very last step requires the
use of a calculator.

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12.     Mr. A and Mr. B have the same preferences (that is, the same instantaneous utility function
and the same discount rate). Both are born at time zero and die (with certainty) at time T. Both face
the same interest rate. Each is born with zero assets and dies with zero assets. Both are also
liquidity constrained: they are never allowed to have negative assets. They have different lifetime
wage profiles. There is no uncertainty: both individuals know in advance their entire lifetime wage
profiles.

It is observed that Mr. A's consumption grows at a constant (positive) rate over the course of
his life, while Mr. B's consumption declines at a constant rate over the course of his life. The rate at
which Mr. B's consumption declines is smaller than the rate at which Mr. A's consumption grows.
However, the present discounted values of lifetime consumption for the two men are the same.

Which man has higher lifetime utility? Explain.

13. [core exam, 2004] A woman is born at time zero and will live forever. Time is continuous.
She is born with zero assets. Her instantaneous utility function is of the CARA form:

1       - c
U(c) = -        e


where α =1. She discounts the future at rate θ =0.10. She can save at interest rate zero. She cannot
borrow.

Her labor income is exogenous. From time zero to time t=10, her labor income is two per
period. From time t=10 to time t=20, her labor income is one per period. After time t=20, her labor
income is two per period.

Solve for her path of consumption. What is her consumption immediately before and after
t=10? What is her consumption immediately before and after t=20? If there are any jumps or
inflections in her time path of consumption, calculate the exact point in time at which they take
place.

Note: the First Order Condition for consumption with CARA utility is

1

c=       (r -  ) .


This is analogous to the condition that we look at in the more common case of CRRA utility, which

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is


c 1
= (r -  )
c 

14.   (Final exam, 2001) This question was going to be on the Core exam, but I realized that it
was way too hard. So I have broken it down into steps and given you instructions as you go along.

A woman has labor income that is constant at one. Time is continuous. She is liquidity
constrained, so that she can never have negative assets. She is born with zero assets. From time
zero to time s, she faces an interest rate of zero. Starting at time s, she will be able lend at interest
rate r>0. Her time discount rate is θ, where r> θ >0. Her instantaneous utility function is
logarithmic.

Sketch her optimal time path of consumption. Under what conditions will optimal
consumption at time zero be equal to one?

A. As a first step, solve for the path of optimal consumption starting at time s. Solve for
consumption at time s, c(s), as a function of her assets at time s, A(s). You should be able to find a
closed-form expression for c(s). Make a graph with A(s) on the horizontal axis and c(s) on the
vertical axis.

B. Now consider what happens before time s. It turns out that there are several possible paths for
consumption in the period before time s, depending on the values of s, r, and θ. Sketch out these
possible paths.

C. Based on your answer to part B, draw a graph with A(s) on the horizontal axis and c(s) on the
vertical axis, showing how assets at time s are related to consumption at time s. Show (with words
or arrows) how the different points on this curve are related to the different consumption paths in
part B.

D. Putting together your answers to parts A and C, along with the usual condition that
consumption cannot jump, will give the optimal values of consumption and assets at time s.

E. Holding r and θ constant, how will changing s affect the value of consumption at time zero?
Specifically, under what conditions will consumption at time zero be equal to one?

Problems with Endogenous Interest Rates

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15. Consider a world with two countries, which have equal sized populations. Income is
exogenous and is identical in the two countries. There is no means of storing output, so income in a
given period has to be consumed in that period. There is no population growth.

Both countries have CRRA utility of the form

1-
c .
u(c) =
1-

But the two countries have different values of σ. Specifically, σ1 > σ2. In both countries, the time
discount rate is zero.

Income per capita at time zero is equal to one. Following this it grows at constant rate g.

A. Suppose that there were no trade between the countries. What would the interest rate be in each
country?

B. Suppose that starting at time zero, the two countries are able to trade with each other. Sketch
out what the path of consumption per capita will look like in each country. Also sketch out the path
of the world interest rate. Toward what level will the world interest rate asymptote? What will the
asymptotic growth rates of consumption in the two countries be, and what will be the asymptotic
ratio of consumption in the two countries? How will consumption at time zero in the two countries
compare?

Note: doing this problem with fancy math is neither required nor recommended.

15.5) Consider an economy in which there are two kinds of people: red and green. There are equal
numbers of each kind of person. The economy lasts for a two periods. In the first period, red and
green people each receive income of one apple per person. In the second period, red people receive
income of one apple per person, and green people receive income of zero.

Both red and green people have log utility, and the time discount rate is zero. There is no
way that apples can be stored between periods.

What is the market-clearing interest rate that will be changed on loans between period 1 and
period 2? Also, what will the per-capita borrowing and lending of red and green people be?

16.    [final exam, 2001] An economy is composed of equal numbers of two types of people,
purple and green. People live for two periods. Their lifetime utility functions are

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Green: U = ln(c1) + ln (c2)
Purple: U = ln(c1) + β ln (c2)

The world lasts for two periods. In each period, each person receives labor income of one. There is
no capital or other way to store output. Green and purple people can, however, borrow from or lend
to each other.

Is the equilibrium interest rate positive or negative? How does the answer depend on the
value of β? Why? You can answer the question with intuition or by grinding through the math.

17.     [midterm exam, 2003] Consider a world with two countries, which have equal-sized
populations. Time is discrete. The world lasts for 101 periods, starting in period zero and ending
after period 100. Labor income is equal to 1 unit per capita in both countries in all periods. Output
cannot be stored, but can be traded between the two countries. The interest rate is endogenous.

Within each country, instantaneous utility is of the Constant Absolute Risk Aversion
(CARA) form:

1
U = -   e - c
 

The two countries have identical values of α, specifically α = 1. The two countries have
different time discount rates, however. Specifically, θ1 = 0.01 and θ2 = -0.01 (this means that
people in country 2 care more about the future than the present).

Solve for the path of the real interest rate and for the initial level of consumption in each
country.

Note: you will have to use the approximation that ln(1+x) x, which holds for values of x
near zero.

18.     [final exam, 2003] The world is composed of two countries with equal populations. In both
countries, output is exogenous. There is no capital or other means to store consumption from period
to period, but people can make loans between the countries. In both countries, people have log
utility with time discount rate θ>0.

In country 1, output is equal to yh in even periods and yl in odd periods, where yh > yl. In

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country 2, output is equal to yl in even periods and yh in odd periods. The world begins in period
zero (which is, of course, an even period).

Solve for the path of interest rates, and also find the path of consumption in each country.

19) [Core Exam, 2005]. Consider a world with two countries, which have equal sized
populations. Time is continuous. Labor income per capita in each country is constant, exogenous,
and equal to 1. There is no means of storing output. There is no population growth. People in both
countries have the following instantaneous utility functions

u(c) = - e -c if     c0

=-      if      c <0

Individuals in country i discount the future at rate θi (i =1,2), where

0< θ1 < θ2

Starting at time zero, individuals in each country are allowed to borrow or lend at the market-
clearing world interest rate.

Note: you do not have to write out any equations at all in order to get full credit for this
problem, but you should carefully describe the principles that determine the answer you give.
Because this problem is very difficult, I have broken it down into smaller steps.

A. Suppose that at some point in time, c>0 in both countries. What must the interest rate be?

B. Suppose that the interest rate is the value that you derived in part A. What does the path of
consumption (at that point in time) look like in each country?

C. Explain why based on the above, the interest rate cannot permanently have the value you
derived in part A.

D. Suppose that at a point in time, consumption is zero in country 1 and positive in country 2. What
must the interest rate be? What do the paths of consumption look like in each country?

E. Is it possible that the interest rate will have the value you derived in part D from time zero until
infinity? Explain.

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F. Suppose that at a point in time, consumption is zero in country 2 and positive in country 1. What
must the interest rate be? What do the paths of consumption look like in each country?

G. Is it possible that the interest rate will have the value you derived in part F from time zero until
time infinity? Explain.

H. Putting all of the above together, sketch out the possible time paths of consumption in each
country as well as the world interest rate. If there are different possible cases, you should
briefly describe these.

19.5) [core exam, 2008] [note: this question has three parts.] An economy is populated by a
continuum of individuals. The economy last for two periods. Each individual is endowed with one
unit of the consumption good in the first period and one unit of the consumption good in the second
period. The good cannot be stored. Individuals can loan the consumption good to each other at
market clearing interest rate r.

Individuals have instantaneous utility functions of the CARA form

1
U (c )         e  c


A) Individuals all have the same value of  . However, they differ in terms of their time
discount rates. Specifically,  , the time discount rate, is distributed uniformly on the
interval [0,0.1]. Solve for the equilibrium value of r.

Note: for all of the parts of this question, you should use the approximation that ln(1+x)  x, which
is generally OK for x near zero.

B) Now suppose that  varies among individuals as well. Specifically let  and  be jointly
uniformly distributed on the interval   [1, 2] ,   [0, 0.1] . Once again, solve for the
equilibrium interest rate.

C) Now suppose that  and  are distributed over the same range as in part B, but they are not
jointly uniform. Specifically, the marginal distributions of  and  are uniform, but two variables
are perfectly correlated, so that the individual who has the lowest value of  also has the lowest
value of  , and so on. Explain how (i.e. higher or lower) and why the interest rate in this case will
differ from part B. You do not have to solve for the interest rate.

20. A man is born at time zero and will live forever. Time is discrete. The only thing he consumes

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is wheat. His within-period utility function is U = ln(c), where c is consumption of wheat. His time
discount rate is θ>0. At the beginning of period zero, he has some quantity A0 of wheat.

There are two things that he can do with any wheat that is left over at the end of a period. He
can store it in his warehouse or he can plant it. If he stores it in his warehouse, then a fraction δ will
decay before the beginning of the next period (where 0< δ <1). Alternatively, he can plant the
wheat in the ground. Wheat planted at the end of period t is harvested at the beginning of period
t+2. One unit of wheat planted at the end of period t yields (1+φ) units of wheat at the beginning of
period t+2, where φ>0. Wheat planted at the end of period t is not available at all in period t+1.

A. Solve for his path of consumption. Draw a picture showing what the path looks like. Carefully
describe any interesting features of the consumption path, and explain why these interesting
features are present. Also solve for his first period consumption (this expression is a bit
messy. You don=t have to simplify it).

B. Now suppose that there are a large number of identical farmers. All start at time zero with
identical quantities of wheat and face the same technology for production and storage
described above. Solve for the path of interest rates on one-period loans of wheat. That is,
what will be the interest rate on loans from period 0 to period 1; from period 1 to period 2;
and so on.

Problems with Uncertainty

20.5) [midterm exam 2008] A woman has assets A0 at the beginning of time zero. She has no
labor income. Time is discrete. She has a constant mortality hazard  . In other words, if she is
alive in period t, the probability that she will be alive in period t+1 is (1   ) . The interest rate is
r>0. Her pure time discount rate is zero. Her utility function is of the CRRA form:

c1
U (c ) 
1

Calculate the ratio of consumption to assets in period zero. (Note: you can solve this problem by
doing some infinite sums, but there is a much easier and more elegant way if you apply economic
reasoning.)

21. [midterm exam, 2003]

A) A woman has assets A(0) at time zero. Time is continuous. She has no labor income. The

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interest rate and time discount rate are both zero. Her instantaneous utility function is u(c) =
ln(c). She has a constant probability of death ρ>0. Solve for her initial level of
consumption, c(0).

B) A woman has assets A(0) at time zero. Time is continuous. She has no labor income. The
interest rate and time discount rate are both zero. Her instantaneous utility function is u(c) =
ln(c). From time zero to time 1, she has zero probability of death. Starting at time 1, she has
a constant probability of death ρ>0. Solve for her initial level of consumption c(0).

C) A woman has assets A(0) at time zero. Time is continuous. She has no labor income. The
interest rate and time discount rate are both zero. Her instantaneous utility function is u(c) =
ln(c). From time zero to time 1, she has zero probability of death. At time 1, she will have a
constant probability of death. However, she will not find out that probability until time one.
Specifically, what she knows at time zero is that the probability of death starting at time 1
will be ρ1 with probability 50% and ρ2 with probability 50%. Solve for her initial level of
consumption, c(0) [Note: to solve this problem you have to deal with a nasty quadratic
equation. You shouldn=t try to solve this. Just derive it, then say what you would do with
the solution if you could solve it.]

21.5   [core exam, 2011] A woman is born at time zero, with initial assets of A0 >0. Time is
continuous. From time zero until time R (retirement), she has zero probability of death.
From time R onward, her probability of death is constant at  per unit time.

From time zero to time R, she earns labor income at a rate of w per unit time. From time R
onward, she has no labor income.

The interest rate is zero. Her pure time discount rate is zero. She has log utility.

Solve for her initial consumption in terms of A0, R, w, and .

22. A man has a potential lifespan of 100 years. For the first 50 years, he will live with certainty.
After 50 years, he will enter a battle, and there is a 50% chance that he will survive. If he survives
the battle, then he will live for another 50 years and die at age 100.

At birth he has assets of \$100. He does not earn any additional wage income. He cannot die
in debt.

He has log utility: u(c) = ln(c). The interest rate and time discount rate are both zero.

Solve for his initial consumption, c0.

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23.     (Midterm exam, 2001) A person may live for one, two, or three periods. At the end of
period 1, after he has done his consumption for that period, there is a 50% chance that he will die,
and a 50% chance that he will live into period 2. He does not find out which happens until the event
actually takes place. If he does live into period 2, then at the end of period 2, there is once again a
50% chance that he will die and a 50% chance that he will live into period 3. Once again, he does
not find out which happens until the event actually takes place. His wealth at the beginning of
period 1 is A1. He earns no wage income. The interest rate and time discount rate are both zero. He
has log utility. Solve for his optimal consumption (if he is alive) in each period.

24.      Consider the problem of an old person trying to decide how much to consume as she
approaches the end of her life. The interest rate is r, there is no time discounting, and she has no
labor income. She has some initial amount of wealth, A0. She faces a constant probability, p, of
dying each year. Thus her probability of being alive in year t is (1-p)t. Of course she only gets
utility if she is alive. Her utility function is

          1-
E(U)= (1- p )t
ct
t=0       1- 

A. What is the relationship between her coefficient of relative risk aversion (σ) and the expected
size of the bequest that she will leave? Be sure to distinguish between the case where p>r and the
case where p<r. Explain.

B. Suppose that σ = 1 , so that the woman has log utility. Solve for her assets if she is still alive at
the beginning of period t, At , as a function of p, r, t, and A0 .

24.15) [final exam, 2008] A large cohort of individuals is born at time zero (you can think of it as a
continuum). Time is continuous. Individuals have a constant probability of death         .
Individuals do not know in advance when they are going to die. Their pure rate of time discount is
.

Individuals are infinitely risk averse. That is, they behave as if they have CRRA utility in the limit
as the coefficient of relative risk aversion approaches infinity.

At birth, each individual is endowed with wealth W0. They receive no labor income. Individuals
can save at an exogenous interest rate     .

There are no annuities. When an individual dies, his remaining wealth is distributed evenly among

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all surviving individuals.

Solve for the path of consumption.

24.25 [midterm exam, 2006] This question has five parts, of varying difficulty. You do not have to
get the early parts of this question right in order to correctly answer the later ones.

A) A woman does all of her consumption in one period. Her utility function is

c1
U            , where   2
1

Call w her certainty level of income. If she pays x dollars, she gets to earn w with certainty, so that
w
her consumption is equal to w-x. If she does not pay x, then she will earn either 2w or , each
2
with probability 50%.

Solve for the level of x such that she is indifferent between paying for certainty or not.
Express your answer in terms of the ratio of x to w.

B)    Suppose that a woman has the following utility function

c1
U  1           if alive
1

where   2. Solve for the level of consumption at which she is indifferent between being alive or

C) Consider a woman who has a utility function as in part (B) and faces the uncertainty described
in part (A). Suppose also that after she observes her income, she has the option of no longer being
alive, and thus getting zero utility. For what range of values of w will the answer that you derived in

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part (A) still be correct? Explain very briefly.

D. Consider the setup from part (C). Solve for the level of w at which the woman’s value of x, that
is, her willingness to pay for certainty, is exactly zero. Note: you should ignore values of w that are
less than or equal to zero. Note #2: I am not asking you to solve in general for x as a function of w,
even though this is one possible way to solve the problem.

E. Consider yet another utility function, the Stone-Geary utility function which is

(c   )1
U               ,
1

where   2 and   1 . Consider the form of uncertainty described in part (A), but assume that
there is no possibility of voluntarily dying. Draw a graph showing willingness to pay for insurance
as a fraction of income (x/w) on the vertical axis and w on the horizontal axis. Consider only values
of w>2. Just sketch the rough form of this curve, showing what it asymptotes to as w gets large and
what it looks like in the neighborhood of w=2. You do not have to derive the actual formula for
willingness to pay. Indeed, you can actually answer this question without doing any math at all.

24.5 [final exam, 2005] A man is born at time zero. Time is continuous. He has a probability of
dying,  , that is a function of his age. Specifically,

 (t )  .01t

To be clear, this means that the man has a 1% per year chance of dying when he is one year old; a
2% per year chance of dying when he is two years old (if he didn’t die already), and so on.

The man’s instantaneous utility is logarithmic. His pure time discount rate,  , is negative. In other
words, ignoring the possibility of death, he cares more about the future than about the present.
Specifically,  = -.05.

The man has wage that is constant and equal to one per period. He can save at an interest
rate of zero. He cannot have negative assets. His initial assets are zero.

Sketch out the man’s paths of consumption and assets. You do not have to solve for the
exact path or the exact value of c(0). However, you should label as much of the path as you can.
Specifically, you should point out any inflections, jumps, maxima or minima, etc. You should also
give the dates of any of these points if you can, and if you can’t give the exact date, then you should
say whatever you can (for example, “this jump takes place after date x and before date y.”)

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24.75) An individual is born at time zero with initial wealth w0 . Time is continuous. He has no
labor income. The interest rate is zero and his time discount rate is also zero. He has instantaneous
utility

u (t )  ln(c (t ))  

where  is the utility of being alive.

The individual has constant hazard of death  . For some strange reason, the individual can
choose  before he is born. However, he cannot change the value of  once he is living.

A) Assume that there are no annuities. Solve for the optimal value of  .

Note: to solve this problem, you will have to know the following helpful fact (which I wouldn’t


1
te
-xt
dt =       2
0                  x

B) Now assume that there are actuarially fair annuities. The annuity rate of interest that an
individual receives will be based on his true value of  (that is, the value that he chose). Solve for
the optimal value of  . [Note: I have given you all the information about the annuity market that I
think you will need. However, if you feel that you need to make additional assumptions about the
annuity market, you should explicitly state what these are.]

24.85) (final exam, 2008) There is a group of individuals who all have wealth A0 at time zero. The
individuals have no labor income. Time is continuous. Every individual i has a constant mortality
hazard i that is known to him, but not to outside observers. Everyone knows the distribution of  .

Individuals all have log utility and pure time discount rate   0 . There are two investment
options. First, individuals may put their money in a standard asset that pays interest r=  . Any
wealth that is left over when the individual dies is thrown into the sea.

Alternatively, individuals may participate in an annuity market at time zero, but not at any other
time. That is, they can turn over their full endowment to the annuity company at time zero in return
for a constant stream of payments equal to r a A0 , where r a is the annuity interest rate. To make
things easier, we will assume that a person with an annuity cannot save any of his annuity payments.
He must consume them as they arrive.

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A) [8 points] Consider the problem of the individual. What is the condition on i and r a so
that an individual will choose the annuity vs. not?

Note: to solve this problem, you will have to know the following helpful fact (which I wouldn’t


1
te
-xt
dt =       2
0                  x

B) [7 points] Now consider equilibrium in the annuity market. Annuity firms invest the initial
payments they received at rate r. Solve for the expected profit (measured at time zero) on
an annuity that pays rate r a that is sold to an individual with mortality probability i .

C) [5 points] Annuities are supplied by zero-profit firms that are risk neutral. The CDF for
individual mortality probabilities is f (  ) . Define  * as the cutoff level of individual
mortality, above which the individual will find it optimal to purchase an annuity. Write
down the condition relating  * and r a that produces zero expected profits in the annuity
market.

24.9) [Midterm exam, 2008) A woman has instantaneous utility of the form

u (c )  ln(c )  

Time is continuous. The interest rate and her pure time discount rate are both zero. The
woman is indifferent between two career options. She can be a coal miner earning a wage of w1 and
experiencing a constant probability of dying of 1 , or she can be a factory worker earning a wage of
w2 and experiencing a constant probability of dying of  2 , where w1>w2 and 1   2  0 .

There is an annuity market in which she can save money at interest rate that reflects the
death probability associated with her job.

A)          [20 points] Solve for  .

B)          [15 points] Assume 1 =.02,  2 =.01, w1= e, w2 = 1.

Calculate the value of a statistical life if the woman decides to be a coal miner and if she
decides to be a factor worker. Recall that the value of a statistical life is the ratio of the payment she
would receive to the instantaneous (one time) risk of death she would accept if she was indifferent to

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making such a trade. You should assume that the risk of death in question is very small.

24.925 [Final Exam 2010] An individual is born at time zero with zero assets. Time is continuous
She has log utility and a pure time discount rate of negative five percent per year. In other words,
except for the effect of mortality she cares more about the future than the present. The interest rate
is zero per period. She has mortality hazard given by the equation

a

1000
where a is her age and  is the annual probability of death. In other words, when she is 10 years old,
her mortality hazard is 1% per year; when she is 20 years old, the mortality hazard is 2% per year,
and so on.
She has receives a constant income flow of one unit per period. She cannot go into debt.
I don’t want you to solve for her complete path of consumption, because that turns out to be really
hard. Instead, I want you to characterize the paths of consumption and assets as well as you can. In
particular, you should describe the equations that you would solve or the method that you would use
to find c0. What are the conditions that hold at different critical points? At what ages do
consumption and assets peak, etc.

24.95 [Midterm exam, 2010] A researcher was assigned by the government to calculate the value
of statistical life by offering individuals sums of money in return for taking very small risks to life.
However, the researcher did not understand economic theory (or his instructions), so instead he
offered individuals sums of money in return for taking large risks to their lives. Your job in this
question is to calculate the properly defined value of a statistical life.

The individual in question had a constant mortality probability ρ=.02, pure time discount rate θ=.03,
and faced an interest rate r=.03. He is assumed to have instantaneous utility of the form

U(c) = ln(c) + α

The individual has no labor income. He has \$1,000,000 in assets. There is a perfect annuity market.
He has no utility from bequests.

Asked what amount of money would be required to get him to take a 50% chance of instant death,
the individual replied that he would require \$999,000,000. The researcher thus concluded that the
value of a statistical life was almost two billion dollars.

A) [16 points] Calculate the individual’s value of α.

B) [16 points] Calculate the (properly measured) value of a statistical life for this individual.

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Note: when I gave this problem on the midterm, I included a table of logarithms.

25. An individual lives for two periods. In each period she earns labor income of 1. Her utility
function is

U = ln(c1) + ln(c2)

She can borrow or lend some financial asset that pays a real interest rate of zero. However, there is
a 50% chance that between periods 1 and 2 all debts and financial wealth we be wiped out (there is
no other way that she can save between periods other than the financial asset). That is, if she
borrows, there is a 50% chance that she will not have to pay back the loan, and if she saves, there is
50% chance that all her saving will disappear. Solve for her optimal first period consumption.

26.     An individual lives for two periods with certainty and for a third period with probability
50%. He finds out at the end of period one (after he has done his consumption) whether he will die
at the end of period two or at the end of period three. He has earnings of one unit in the first period,
and no labor income thereafter. His instantaneous utility function is

U(Ct) = Ct1-σ/(1-σ)

He can borrow or lend at an interest rate of zero, and has a time discount rate of zero. He cannot die
in debt.

A.      Derive the equation that you would solve to get his optimal first period consumption. You
don't need to solve it.

B.       A government program is introduced that takes away τ from every worker in the first period
of life, and pays 2τ to all people who survive until the third period of life. What is the value of τ
that maximizes expected utility? (You don't have to do algebra to get this answer -- you can just
reason it out and explain your reasoning).

26.5 [Midterm exam, 2010]

A. A closed economy is populated by a large number of identical individuals. People live for two
periods. They have log utility with a time discount rate of zero. In the first period, everyone has
labor income of two units. In the second period, everyone will either have labor income of one unit,
or everyone will have labor income of three units, each with probability 50%. People do not know
until the beginning of period two what that labor income level will be. Individuals can lend or
borrow from each other at interest rate r. There is no way of moving output from one period to
another other than lending or borrowing from someone else. Solve for the equilibrium value of r.

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B) Consider the same setup as in part A, but now instead of all individuals having the same labor
income in period two, suppose that half of the people have labor income of one and half have labor
income of three – however, people still do not learn until the beginning of the second period what
their labor income will be. Solve for the equilibrium value of r. (Note: we assume that there is no
risk sharing, insurance, or state-contingent contracts. That is, there is no arrangement by which
people who are lucky enough to have good wage outcomes transfer income to people with bad
outcomes.)

C) Now, consider a similar setup with three periods. In periods one and two, everyone will have
labor income of two units per period. In period three, half the people will have labor income of one
and half will have labor income of three. However, people find out at the beginning of period two
what their period three income will be. As above, there are no state contingent contracts and no way
to store output between periods other than by lending it to someone else. Solve for the equilibrium r
that will hold between periods one and two, and also for the equilibrium r that will hold between
periods two and three.

27.     [midterm exam, 2002] A man has a maximum lifespan of three periods. In the first two
periods, he is alive with probability one. At the beginning of the second period (before he has made
his consumption decision), he finds out what his health status is. There are two possible values of
health status, “healthy” and “unhealthy,” which each occur with a probability of 0.5. If he is
healthy, then he will be alive in the third period with probability one. If he is unhealthy, he will be
alive in the third period with probability 0.5. The information about whether he will be alive in the
third period is not revealed until after second period consumption has taken place.

The interest rate and time discount rates are zero. He has labor income W in the first period
and zero in all subsequent periods. His instantaneous utility function is logarithmic.

Solve for his optimal first period consumption.

28. A person lives for two periods. In the first period she has income of 8 dollars. In the second
period, she has income of either 0 or 8 dollars, each with probability .5. The interest rate is zero.

Her instantaneous utility function is:

U(c) = c - .05*c2

Her discount rate is zero.

There is a test that can tell the person what her second period income will be before she
makes her first period consumption decision. If she does not take the test, then she will not know

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her second period income until after the first period is over. The test costs 2 dollars. Calculate
whether she should take the test or not.

29. Consider a two-period model in which people work during both periods. The number of hours
is fixed at 1 in each period. People start off with no assets, and die with no assets. They can borrow
or lend at an interest rate of zero, and there is no time discounting. Their instantaneous utility
function is U(C)=ln(C).

In the first period, everyone's wage is \$10. In the second period, half of the people will make
\$15, and half will make \$5. But in the first period, people do not know to which group they will
belong.

A) The government is considering cutting taxes by \$1 per person in the first period, and
increasing taxes in the second period to pay back the debt. Taxes in the second period can
either be lump sum (\$1 per person) or proportional (10% of each person's wage).

How would each of these programs affect people's utility and the national saving rate?

B) A test is invented that will tell people at the beginning of period 1 what their income will be in
period 2. How would the availability of this test affect average utility of people in the population?
How would it affect total saving in the first period?

29.5 [midterm 2005] An individual lives two periods. His first period income, w1 is certain. His
second period income, w2 is distributed N( w ,  2 ). The interest rate and time discount rate are both
zero. He has CARA utility:

u (c )  e  c

A. [10 points] Derive the first order condition linking consumption in the first and second
periods of life. You will have to use the fact that if x is normally distributed with variance
 x2 , then

2
E (e )  e E ( x ) x / 2
x

Hint: write lifetime utility in terms of first period consumption.

B. [5 points] Solve for first period consumption

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C. [5 points] Consider two individuals, one named Homer and one named Mr. Burns. Homer
has w1 = 20,000. Mr. Burns has w1 = 20,000,000. Second period incomes (and
uncertainty) are the same for Homer and Mr. Burns. Suppose that initially, second period
income is 20,000 with no uncertainty. Solve for first period consumption for each man.
Now suppose that in the second period income is normally distributed with a mean of 20,000
and a variance of 10,000. Calculate the effect of this rise in uncertainty on each man’s first
period consumption.

D. [5 points] How would your answer to C have differed if the men had CRRA utility?
You should answer qualitatively. Do not try to calculate the exact answer.

29.7) [midterm exam, 2008] A person lives for two periods. In the first period she has income of
A. In the second period, she has income that is a normally distributed random variable  with mean
of zero and variance of  2 . She does not learn her second period income until after she has done
her first period consumption. The interest rate is zero. Her time discount rate is zero.

Her utility function is of the CARA form

1
u (c )         e  c


A) [15 points] solve for her first period consumption

You will have to use the fact that if x is normally distributed with variance  x , then
2

2
E (e )  e E ( x ) x / 2
x

B) [5 points] Economic conditions change in such a fashion that A falls and  2 goes to zero.
Given these new conditions, the individual finds it optimal to choose exactly the same level
of first period consumption that she chose in part A. What is the new value of A?

[10 points] Is her expected utility higher under the conditions of part A or part B? Show how you

29.75 [final exam, 2006] Consumption in period t+1 is a normally distributed random variable with
mean c and variance  c2 . Consumption at time t is c . Individuals have CARA utility of the form

99
U (c )  e  c

They have a time discount rate of zero.

A. [10 points] What is the risk free interest rate that holds between periods t and t+1?

Note: you will have to use the rule that if x is a normally distributed random variable then
E( e -x ) = e -E (x) +  x / 2
2

B. [10 points] What is the effect of increasing  c2 on the interest rate? Explain in economic terms
why this is true.

30. An individual faces the following problem. He lives for three periods. In the both the first and
the second period, his income is 1 per period. In the third period, his income is either 1 or 2, each
with a probability of .5. Further, he will find out at the end of the first period (that is, after he has
done his first period consumption, but before he has done his second period consumption) what his
income in period 3 will be. The interest rate and the person's time discount rate are both zero. His
instantaneous utility function is U(Ct)=ln(Ct). How would you solve for his consumption in the first
period. (Don't worry about solving any nasty quadratic equations that arise: just show the equation
that you would solve.)

30.5) [midterm exam, 2008] A man is born at time zero and will live forever. Time is discrete.
r=  =1 (note this means that the interest rate between periods is 100%.)

In the first period of life (period zero), his labor income is 1. In the next period (period 1)
there is a coin flip: with probability 50%, his labor income will be 1 again. With probability 50% he
will lose his job and his labor income will be it will be zero. If he loses his job, he never has labor
income again. If his income in period one is 1, then again in period two there will be a coin flip, and
he has a 50% chance of losing his job (and never finding a new one) and a 50% chance of keeping
his job. This goes on forever.

The man has quadratic utility. You should assume that the utility function is well behaved
(i.e. u’>0, u’’<0) for the entire range relevant to this problem. (It turns out that technically this
assumption cannot be correct, but you should ignore this technicality).

A) [10 points] Solve for consumption in period zero.

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B) [10 points] What are the two possible levels of consumption in period 1?

C) [10 points] How many different possible levels of consumption are there in period n?

D) [10 points. Hard!] List all the possible values of consumption in period n.

31) [midterm exam 2004] A man will live for three periods. He has time discount rate of zero.
His initial assets are A0. He has no labor income. He has CRRA utility with a coefficient of relative
risk aversion of 2.

The interest rate between periods 0 and 1 is zero. In period 1 (before he makes his
consumption decision for that period) he will find out what the interest rate between periods 1 and 2
will be. Specifically, it will be zero with probability 50% and 3 with probability 50%.

Solve for his first period consumption. [Note: this final expression for c0 is sort of ugly. You
should at least derive the equation that implicitly gives c0 as a function of A0.]

32.[Core exam, 2003] A married couple is composed of a wife and a husband. Time is continuous.
The wife lives forever. The husband is alive at time zero, and has a constant probability of
death ρ>0.

All consumption decisions in the house are made by the wife. Her instantaneous utility
function is

U = ln (cw) + ln(ch)          If the husband is alive
ln(cw) If the husband is not alive

where cw is the consumption of the wife and ch is the consumption of the husband.

The wife discounts future utility at rate θ. The interest rate is r. Assume r= θ >0.

The household has assets at time zero A(0)>0. In addition, while the husband is alive, the
household receives labor income at a rate of one unit per period. Note that when the husband dies,
all of the household’s assets remain in possession of the wife.

Your job is to sketch the possible time paths of the wife’s consumption and of household
assets, that is cw(t) and A(t), depending on the value of A(0) and on how long the husband lives.
Specifically,

A)For what value of A(0) will the path of assets be flat? Call this value A*.

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B)Consider the case where A(0)>A*. Sketch the possible paths of assets and wife’s consumption.
Specifically, show what the paths of consumption and assets look life before and after the
husband’s death. Show what the paths look like for different possible dates of the husband’s
death. Note: your answers to part B may be mostly or entirely in terms of words and pictures,
rather than equations.

33.2) [final exam, 2008] Consider the Bommier utility function. The pure time discount rate is
zero. An individual who lives for T periods (from zero to T-1) has lifetime utility:

  T 1       
V     u (ct )   .
  0          

Assume that
u (ct )  ln(ct )
and

 ( x)  e  x

Suppose that individuals live for two periods with certainty. They are endowed in period
zero with assets A0. They earn no labor income. They earn interest on their savings are rate r.

An economist is trying to understand consumption. She believes (correctly) that people
have a pure time discount rate of zero. She believes (incorrectly) that the people have CRRA utility
of the standard form (that is, not the Bommier form), with coefficient of relative risk aversion  .
She tries two experiments in order to estimate  .

A)   [10 points] Suppose that the economist varies r among individuals and looks at how the
path on consumption responds. What will she conclude is the value of the coefficient of
relative risk aversion?

B)   [10 points] Suppose that the economist attempts to estimate  by looking at choices with
respect to lotteries. Specifically, before individuals receive their time zero endowment, A0,
she is able to offer them choices of lotteries. For example, she offers them proposals like
“you can have A0=150 with certainty, or you can have A0=200 or A0=100+x, each with
probability 50%.” She then varies x until people are indifferent between the two lotteries.
What value of  will she estimate? [Hint: write lifetime utility in terms of initial assets.]

33.4) [final exam, 2008] Consider the Bommier utility function. The pure time discount rate is

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zero. An individual who lives for T periods (from zero to T-1) has lifetime utility:

  T 1       
V     u (ct )   .
  0          

Assume that
u (ct )  ln(ct )
and

 ( x)  e  x

Consider the problem of a person who will live either one or two periods. She is endowed at time
zero with assets A0=1. The interest rate is zero and there are no annuities. She cannot die in debt.
At the end of period zero, she dies with probability one half. If she does not die after period zero,
then she dies at the end of period 1 with certainty.

Solve for her first period consumption.

33.45) [core exam, 2009] A group of individuals is born at time zero. Time is continuous. The
death rate is constant at ρ>0.

At birth, each individual is endowed with wealth W(0), which is equal for all individuals.
There is no labor income. Individuals save in capital that pays return r. They cannot borrow or die
in debt. Individuals have pure time discount rate θ, and log utility. Assume r=ρ= θ >0.

Consider two different scenarios.

1) Individuals have no information about their dates of death, but there is an actuarially fair
annuity market. Individuals who deposit money in an annuity receive interest equal to r+ρ if
they live.

2) Individuals are informed at birth of the date of their death.13 There is no annuity market.

13
Note that there is no inconsistency between the death rate being ρ and people being informed of
their own dates of death. In our normal model we assumed that there was no advanced information
about who would die. But you could just as well imagine that everyone was told in advance when
they would die and that those known dates of death were such that a fraction ρ died every period.

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A) Describe what the paths of consumption look like in the two different scenarios. Discuss
both average consumption and individual consumption paths. You do not have to derive
precise expressions, but you can if you like.

B) In which scenario is expected utility (not conditioning on an individual’s date of death)
higher? Explain why. [Note: you do not need to grind through a whole lot of math to
answer this question. Economic reasoning should be sufficient. On the other hand, Boris
said that he found the math easier than intuition.]

C) Suppose that we consider a Bommier style utility function of the form

T                    
V     e t ln(c(t ))dt 
0                    

Where  '  0 ,  ''  0 and V is the lifetime utility of someone who lives T years. How
might the results of part B be altered in this case? Explain economically why this would be
so.

Problems with Nonlinear Budget Sets

33.5 [midterm, 2006] Consider the following two-period model. A man has labor income of W in
the first period and zero in the second period. He has log utility. His time discount rate is zero. He
can save at an interest rate determined by how much he saves. Specifically, for saving less than \$10,
he earns an interest rate of zero. If he saves more than \$10, all of his savings earns an interest rate
of 100%. (Note that the high interest rate does not only apply to his savings above \$10. So for
example, if he saves \$11 in period 1, he will be able to consume \$22 in period 2.)

A. Solve for the value of W for which the man is indifferent between saving enough to earn the
high interest rate and not doing so.

Note: you should get as close to the exact solution as you can, even if this involves some algebra. In
particular, if you end up with an expression that yields multiple solutions, you should show how all
but one can be ruled out.

B. Solve for the value of W at which the consumption of a person facing the interest rates described
above would be the same as the consumption of a person who could earn 100% interest no matter
how much he saved.

C. Based on your answers to (A) and (B), draw a graph with the first period saving rate on the
vertical axis and W on the horizontal axis.

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34. A woman lives for two periods. In the first period she has income of 3. In the second period,
she will have income of zero. The interest rate and the time discount rate are both zero. She
cannot borrow. Her utility function is U(ct) = ln(ct).

There is a government welfare program that provides a consumption floor of cmin in the
second period. In other words: if she does not have enough money left over to afford to consume
cmin , then the government will give her enough money so that she can afford to consume it.

A)Obviously, for high enough values of cmin , the consumer will decide to consume all of her wages
in the first period and go on welfare in the second period. For low enough values of cmin she
will choose consumption as if the welfare program did not exist. Calculate the critical value
of cmin for which she is indifferent between these two strategies.

B)Now suppose that there is only a 50% chance that she will be alive in the second period.
Calculate the critical value of cmin at which she will be indifferent between the two strategies
discussed in part A.

35.    People live for two periods. In the first period they have income of 1, in the second period
they have income of zero. The interest rate and the time discount rate are both zero. The utility
function is U(ct) = ln(ct).

There is a 50% probability that there will be a government welfare program in the second
period of people’s lives. If there is such a program, it will provide a consumption floor of cmin. In
other words: if they do not have enough money left over to afford to consume cmin , then the
government will give them enough money so that she can afford to consume it.

Solve for the critical level of cmin such that, if cmin is below this level, people’s choice of first
period consumption will not be affected by the existence of the program.

The algebra on this question may get a little tedious. If you can’t solve it, then show clearly

36. An individual has wealth A0 at time zero. She will live infinitely, and will not receive any
income. The interest rate is zero. She is unable to borrow. Her instantaneous utility function is
U(c) = ln(c). She has a discount rate of Θ>0. Time is continuous.

There exists a government welfare program that works in the following manner. If a person
has any wealth at all, she will receive nothing. If her wealth is equal to zero, then she will receive
cmin.

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Figure out her optimal consumption strategy. Solve for her initial consumption, c0.

Hint #1: Try thinking about the case where Θ=0.
Hint #2: Do not try thinking about this problem in discrete time. Doing so will make you insane.

37. [Core exam, 2001] A person is born at time zero and will live forever. He is born with assets
of 10. His labor income is one per period. His time discount rate is 5%. His instantaneous utility
function is of the CARA form:

U(c) = -e-c

There are two interest rates in the economy. For borrowing, the interest rate is 5%. For saving, the
interest rate is zero.

Solve for his optimal path of consumption. What is the consumption at time zero? Draw a picture
of the time path of consumption, showing any inflection points, etc.

Note: Solving this problem in continuous time requires optimization techniques that we did not
cover in my class. It can be more easily solved in discrete time. However, to do so, you have to
use the approximations that ln(1+x) x and ln(1/(1+x)) -x, both of which hold for x near zero.

38) [Core Exam, 2005] An individual lives for one period with certainty and may live into a
second period with probability ρ, where 0 <ρ <1. He knows the value of ρ. He does not find out
whether he lives in the second period until after he has done his first period consumption. He has
labor income w=2 in the first period of life, and no labor income in the second period of life.

His first period utility is ln(c1)

His second period utility is      ln(c2)       if     c2  c1
ln(c2) - 1   if      c2 < c1

The time discount rate and interest rate are both zero.

Make a graph showing his optimal first period consumption as a function of ρ. If there are
any notable jumps or kinks in this function, you should write out the implicit equation for the value
of ρ at which they take place. You do not have to solve explicitly for the value(s) of ρ.

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1) 38.5 [Core exam, 2011] A man does all of his consumption in one period. He has initial
wealth of \$100. Before he gets to do his consumption, there is some random bad event that
occurs with probability 20%. If that event occurs, he loses \$80. If the event doesn’t occur,
he doesn’t have to pay anything. Whatever is left, he gets to consume. He has CRRA utility
with coefficient of relative risk aversion   2 .

Before it is observed whether the event occurs, the man can buy insurance.

A) Suppose that the insurance plan works as follows. For each \$1 that the man spends on
insurance, he receives a payment of \$2 in the state of the world where the bad event occurs.
Notice that this is not actuarially fair insurance. Solve for his optimal insurance purchase.

B) Suppose instead that there is a different insurance policy offered. The man must pay a fixed
cost of \$z in order to have the right to buy insurance. Once he spends \$z, he is entitled to
buy actuarially fair insurance in whatever quantity he wants. Solve for the value of z for
which he is indifferent between buying insurance and not buying insurance. (Note: I want an
actual number of dollars for z, not an expression involving coefficients. However, the math
does not work out as neatly as I would have liked in this case, so it is OK to end up with an
expression involving just numbers that you would have to plug into a calculator to solve.
Even in this case, you should write down approximately how big z is.)

Problems with nonstandard utility functions

39.    [midterm exam 2002] A woman lives for two periods. Her wage income is W in the first
period and zero in the second period. The interest rate and time discount rates are zero.

She has a habit-formation utility function of the following form:

ut = ln( ct - γ zt)          γ < 1,

where zt is the habit stock period t. We assume that habit stock in a period is equal to last period’s
consumption

zt+1 = ct .

We further assume that habit stock in the first period of life is zero.

Solve for optimal consumption in the first and second periods of life.

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39.5) [midterm exam, 2008] A country is populated by a large number of identical individuals.
Individuals live for two periods. All individuals live for the same two periods (this is not an
overlapping generations model). Their utility functions are

 c2 
V  ln(c1 )  ln  1/ 2 
z 

Where z is the habit stock in the second period. The interest rate is zero. Individuals have wage w
in period 1 and zero in period 2.

A) [10 points] Consider the case where the individual’s habit stock in the second period is his
own consumption in the first period. Solve for consumption in each period.

B) [10 points] Consider the case where the individual’s habit stock in the second period is the
average level of consumption in the first period. Solve for consumption in each period.

40.[final exam, 2003] A person will be alive for three periods, labeled 1, 2, and 3. The interest rate
is zero. He has initial assets A1. His utility function in period t (for t = 1, 2) is of the
“hyperbolic” form,

 3            
Ut = ln ( c t ) +    ln ( c s ) 
 s=t+1        

where β<1.

Solve for his path of consumption in periods 1, 2, and 3 under the following two scenarios:

A) [10 points] At time 1, he can decide his consumption in the current period as well as all future
periods -- that is, he can pre-commit himself to a lifetime path of consumption.

B) [10 points] At time 1, he can decide his consumption in the current period, but cannot pre-
commit his future consumption path.

41) [midterm exam 2004] A woman is born at time zero and will live for exactly 9 periods, labeled
t=0, 1, ...8. She has initial assets A0. The interest rate and time discount rate are both zero. She has
hyperbolic preferences of the form

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T -1

V t = ln ( c t ) +     ln ( c
s=t+1
s   )

where β= 2.

A) Suppose that at time 0 she chooses her entire consumption path in periods 0-8. Further, there
will be no subsequent revisions in her plans -- that is, her hands will be tied in future periods. Solve
for consumption in each period.

B) Suppose that in each period the person makes a consumption plan as if she were able to bind
the hands of her future selves, and then does her consumption in that period according to the plan.
So, for example, in period 0 she will do the consumption that found in part A. But further suppose
that in each future period, she is able to re-make her plans (and she again assumes, incorrectly, that
her future selves will stick to that plan.).

Solve for her consumption in each period 0-8. You don=t have to literally solve for each
period, but rather show the general rule that will generate these values.

41.5) [midterm exam, 2008] An individual is born at time 0 and will live for exactly T periods (in
other words, the last period of life will be T-1). He has initial assets A0, and receives no wage
income. He is not allowed to hold negative assets. The interest rate is zero.

The individual has time t preferences given by the equation

1 T 1
Vt  ln(ct )              ln(cs )
2 s t 1

At time t, an individual can only choose his consumption for that period. He cannot “bind the
hands” of his future selves.

A) [8 points] Solve for the consumption of an individual who has assets AT-2 at the beginning
of period T-2 (the second to last period of life).

B) [7 points] Do the same for an individual who has assets AT-3 at the beginning of period T-3,
and for an individual who has assets AT-4 at the beginning of period T-4.

C) [8 points] Based on the above answers, what is the consumption of an individual who has
assets of A0 at the beginning of period 0?

D) [7 points] What is the consumption in period T-1 of the individual who had assets of A0 at
time zero? (The answer is actually a very simple expression).

109
42.    [midterm exam, 2003] A woman is born at time zero and will live until time T. Time is
continuous. There is no uncertainty. She is born with zero assets and will die with zero assets. Her
wage per unit of time is constant and equal to w. The interest rate is zero.

The woman has an unusual instantaneous utility function. It is

u(c(t))         =     α c(t)                  if c(t) c

=     αc + β (c(t) -c ) if c(t) >c

where α > β.

Her lifetime utility is given by

T
V =  e - t u(c(t)) dt
0

where θ > 0.

In the four parts of the question below, I ask you to solve for the optimal path of
consumption under different assumptions about the values of the parameters. In each case, you
don=t have to give me exact values, although you may if you want to. Mostly, I want you to show
me a picture of what the path looks like and briefly explain its key features.

A) Suppose that w =c and α e -θT > β . Solve for the optimal path of consumption. Also, explain
why the assumption that α e -θT > β is important.

B) Suppose that w >c and that α e -θT > β . Solve for the optimal path of consumption.

C) Suppose that w <c and that α e -θT > β . Solve for the optimal path of consumption.

D) Suppose that w =c and that α e -θT < β. Solve for the optimal path of consumption.

Problems not elsewhere classified

110
43)    [midterm exam 2004] Time is continuous. A man is born at time zero and will live forever.
He has initial assets A(0) = 100. He has no labor income. Assume that r=0 and θ=0.5.

There are two things on which the man can spend his money. He can buy consumption
goods (denoted c) or he can give his money to charity (denoted g). His instantaneous utility is
given by

U(c,g) = ln(c) + g

Note that utility from gifts to charity has non-decreasing marginal utility, because it is assumed that
one individual’s contributions have an infinitesimal effect on the world’s suffering.

Solve for his optimal paths of consumption and charitable contributions.

111

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