Consumption Notes Brown University

7/30/2019 Consumption Notes Brown University

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David N. Weil January 24, 2013

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 certainamount of income, and you can save it or consume it. So cant think about one without thinkingabout the other.

2. This topic is really part of both the long run and the short run analysis. In the long run, the savingrate determines the level of output (or the growth rate or output). But in the short run, thedetermination 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 theapproach 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 asbeing 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 tothink 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 ofthe thing it is determining.]

often this is written in a linear form:

C = c0 + c1(Y-T)

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

This is often called the Keynesianconsumption function. Keynes wrote that c0>0 and 0


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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 getsricher?

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. Thesaving 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 toanother. Thus saving should be viewed explicitly as an intertemporal problem. So for example, theMPC should depend on why your income has gone up. Put another way, the consumption functionshould have a lot more than just today's income in it -- for example, it should have tomorrow'sincome in it.

So we want a model of saving behavior that is more based on fundamentals. To build a such amodel, we start with the question: Why do people consume? Answer: Because it makes them happy.

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

1Historical note: This wasn't actually known for sure when Keynes wrote: Simon Kuznets, who

invented national income accounting i.e. how to measure GDP and stuff -- discovered theapproximate constancy of the US saving rate over a period of 100 or so years. His discovery set offa flurry of work on consumption in the 1950s that culminated in Friedman and Modiglianiscontributions. Interestingly, in most other developed countries (summarized in Angus Maddisonswork) the saving rate has risen over time -- although probably not in the way that Keynes' modelpredicted.

2The Kuznets finding can be put another way. If we go to the data (say annual data on incomeand consumption for a country over time) and run the regression C = c0 + c1 Y, we will get the resultthat in short samples the estimated value of c1 will be smaller than it will be in large samples. Whenwe talk about the Permanent Income Hypothesis we will see why this is true.

01

Y -T -C C cs = = 1- = 1- -c

Y -T Y -T Y -T


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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 twoissues: how we add up utility over many different periods of time, and how we deal with theexpected utility when there is uncertainty.

Adding up Utility Over Time

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

V = U(C1) + U(C2)

(In a little while we will introduce the notion ofdiscounting, by which utility in the futuremay 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 relationbetween consumption at different periods of time. Say, for example, that we have $300 to consumeover two periods (and we temporarily ignore things like the interest rate): How shall we divide itup?

The answer is that we would want to smooth it -- that is, consume the same amount in eachperiod. The way to see this is to look at the marginal utility of consumption. Suppose that weconsumed different amounts in different periods. Then the marginal utility of consumption wouldbe 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 behigher.

Utility under Uncertainty

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

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


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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 VonNeumann - Mortgenstern (VNM) utility. This is the approach that we always use. The secondmethod is called wrong.

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

The fact that uncertainty lowers your utility is called risk aversion. Notice that risk aversionis a direct implication of the utility function being curved. (The mathematical rule that shows this iscalled Jensens inequality: if U is concave, then U(E(C)) > E(U(C)), where E is the expectationoperator.) If the utility function were a straight line then the utility of $150 with certainty would bethe same as the utility of a lottery with equal chances of getting $100 and $200. A person whoindeed 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 likeinsurance, 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 twosides of the same coin: they are both results of the curvature of the utility function. If the utilityfunction were linear (and so the marginal utility of consumption constant) then people would notcare about smoothing consumption, and their expected utility would not be lowered by risk.

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

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


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the utility function.

The CRRA Utility Function

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

where > 0. Note that if > 1, then the CRRA formulation implies that utility is always negative,

although it becomes less negative as consumption rises. This does not matter (as long as whetheryou are alive or not is not subject to choice) although it often gets students confused.

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

is called the coefficient of relative risk aversionand it measures, roughly, the curvature

of the utility function. If is big, then a person is said to be risk averse. If is zero, the person is

said to be risk-neutral.5

4Proof: First re-write the utility function by adding a constant: u(c) = (c1--1)/(1-). Think of

this as a function of: g() = (c1--1)/(1-) re-write as

ln-( (c) )c -1eg( ) =

1-

since g(1) = 0/0, we apply LHopitals rule

lnlim lnln

- (c)-c (c)eg( ) = = (c)

1 -1

.

5

The Coefficient of Relative Risk Aversion is defined asU (C)C

-U (C)

It is easy to verify that this is constant in C for the CRRA utility function.

1-t

tC

U( ) =C1-


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To see how measures the curvature of the utility function, we can calculate the elasticity of

marginal utility with respect to consumption, that is

So the bigger is , 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 forconsumption smoothing, insurance, etc.

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

We solve:

We can use a calculator to find the value of x for different values of. 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 = 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.

x

1 8.6 (log utility)2 16.73 23.54 28.85 32.96 35.8

dUUdc = c = -

U U

c

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

11- 1-

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


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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 probably average

around 2-3, 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.

We can also do an exercise to show how the willingness to pay to avoid risk depends on thesize of the risk. Suppose that we take the above example and now multiply the size of consumptionin the two states of the world by some amount z. We can again solve for the amount that a person iswilling to pay to avoid that risk:

( [ ( ]( ) ) ( ) )1 1

1- 1- 1- 1-1- 1-x = z150-(.5 + .5 z z150-(.5 + .5z100 200 ) 100 200 )

Where the term in square brackets is just what you were willing to pay to avoid the originaluncertainty. In other words, the amount that you are willing to pay to avoid uncertainty relative to

certainty depends only on the risk relative to the certain outcome. If you are willing to pay $10 toavoid uncertainty of $50 relative to a base of $150, then you are willing to pay $10,000 to avoiduncertainty of $50,000 relative to a base of $150,000. That is why this formulation of the utilityfunction is called Constant RelativeRisk Aversion (CARA).

The other utility function that we use a lot is Constant Absolute Risk Aversion (CARA).

1 We can do the same exercise as above, considering a lottery where the outcomes are either, $100 or

$200, each with probability 50% or else $150-x with certainty. Setting the utilities equal to solve forx we get

1 0.5 0.5 0.5

0.5 This equation gives an implicit value forx.

Now, suppose that we change the value of consumption prior to the introduction of risk (i.e. the$150), without changing the size of the risk. In other words, instead of consumption being 150+50or 150-50, we make it so that consumption is 1500+50 or 1500-50. Looking at the derivationabove, it should be clear that the value of x will not change. That is why this is called constant


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absoluterisk aversion.

So, suppose that we start in a case where a person is willing to pay 10 to avoid a risk of 50, startingwith a base of 150. That is, she thinks that 140 with certainty give the same expected utility as 100and 200, each with probability 50%.

Under CRRA, we know that she would also view 1400 with certainty as having the same utility as1000 or 2000, each with probability 50%.

Under CARA, we know that she would view 1490 as having the same utility as 1550 and 1450,each with probability 50%. But how much would she be willing to pay to avoid the risk of 1000and 2000? The answer is that it must be more than 100. The way to see this is that for any utilityfunction with negative second derivative, the utility loss from uncertainty rises with the size of theuncertainty. So if she is willing to pay 10 to get rid of uncertainty of 50 (plus or minus), she mustbe willing to pay more than 100 to get rid of uncertainty of 500

Fisher Model

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

People live for two periods. They come into the world with no assets. And when they die, theyleave 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 areborrowing 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 ontheir 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


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budget constraint is a line with a slope of negative one. Note that the budget constraint runs throughthe point W1, W2 -- you always have the option of just consuming your income in each period. TheY 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 aregoing 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 sucha point, the marginal utility of first period consumption is high, and that of second periodconsumption is low. So it would take only a small gain in C1 to make up for a big loss of C2 in orderto keep the person having the same utility. So the indifference curve is steep. Similarly, when C1 islarge and C2 is small, the indifference curve is flat.

6 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:

and taking the first order conditions:

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

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

so C1 = C2

6More formally, one can use the implicit function theorem. Let F(C1,C2) = U(C1) + U(C2). Then fo

F(C1,C2) = k (where k is some constant):

1

2

2 1C

1 2C

d U ( )FC C=- =-

d U ( )C CF

1 2 1 2 1 2L = U( ) + U( ) + ( + - - )C C W W C C


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

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 periodswill 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 sameamount. How should consumption change? How about saving?

Already, we can see some problems with Keynes' way of looking at consumption. Consumptiondepends 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 bypeople 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

Y intercept is (1+r)W1 + W2,


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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 matterwhat 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 toconsumption in the first period?)

Answer is: it depends.

First, lets look at what happens in the case where the person was initially saving. Rememberfrom 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 bothperiods.

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

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

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 consumptionwill fall, and saving will rise.

Discounting

We might want to introduce some discounting of utility experienced in the future. For example,

suppose that is some discount factor that we use for discounting future utilities.

V = U(C1) + U(C2)/(1+)

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


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and get the first order conditions

dL/dC1 = U'(C1) -

dL/dC2 = U'(C2)/(1+) - /(1+r)

which can be solved for:

U'(C1)/U'(C2) = (1+r)/(1+)

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. Todo this, however, one needs to know the exact form of the utility function. This is done in one of thehomework exercises.

Liquidity Constraints

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

For person who would have wanted to save anyway, no big deal. But for person who would havewanted 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 morelike 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 ratefor 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 kinkpoint. Interesting result is that if the optimum is at the kink point, then small changes in one or bothinterest rates will not affect the optimal level of consumption.

2 2 21 1 1

U( )C W CL = U( ) + + + - -C W C

1 + 1 + r (1 + r)


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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 .

That is

She faces interest rate r on any assets (negative or positive) that she has. In particular, call At theassets 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 A T = 0 (whereAT = (1+r)(AT-1 + WT-1 - CT-1) . Put another way, in the last period of life she spends her earningsplus 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

Notice that we have gotten rid of all of the A's. This expression can be re-arranged to say that the

T-1t

tt=0

U( )CV =

(1+ )

T-1t t

tt=0

-W C0=(1+r )


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present discounted value of consumption is equal to the present discounted value of wages.

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

An Aside: The Budget Constraint in Continuous Time

We can also derive a similar intertemporal budget constraint in continuous time. The evolution ofassets 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 givetr(t-s)

0

A(t) = (w(s)- c(s)) dse

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

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

r(T-s) r(T-s)

0 0

w(s) ds c(s) dse e=

Multiplying by e-rTT T

-rs -rs

0 0

w(s) ds c(s) dse e=

[End of Aside]

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

to solve this we would just find the T first order conditions which, combined with the budget

T-1 T-1t t

t tt=0 t=0

W C=

(1+r (1+r) )

T-1 T-1

t t tt t

t=0 t=0

U( ) -C W CL= +(1+ (1+r) )


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constraint, would allow us to solve for the T+1 unknowns: 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 forconsumption in two adjacent periods, t and t+1:

these two can be combined to give

this is a key condition that relates consumption in adjacent periods. Notice that even if we don'tknow the full solution to the consumer's problem (that is, what the level of consumption in eachperiod 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 awhile.

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 setamount 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 knowwhether 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)

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

t

t tt

dL U ( ) 1C= - =0d (1+ (1+r) )C

t +1

t +1 t +1t +1

dL U ( ) 1C= - =0

d (1+ (1+r) )C

t

t +1

U ( ) 1+rC=

U ( ) 1+C


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

then U'(C) = C-

and so the first order condition can be re-written

Before we discuss the interpretation of this first order condition, we can derive a similar onein 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. [being completely accurate, the r that we use in

continuous time, the instantaneously compounded interest rate, is not exactly equal to the r used indiscrete time.]

so we can rewrite the first order condition as

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

1-t

tC

U( ) =C1-

1

t +1

t

1+rC=

1+C

1/r

1/t +1 r-

t

c e= = ( )e

c e

1/t + t (r - ) t

t

c= ( )e

c


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

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

Thus the growth rateof consumption is given by

The numerator and denominator of the last expression are both zero when t is zero, so we applyL'Hopitals rule, taking derivatives of top and bottom with respect to t:

Evaluating at t=0, we end up with

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 thecurvature of the utility function.

lim t + t tt 0

-c cc =

t

0 0 0lim lim lim

t + tt + t t1/(r- ) t

t

t t tt

c-c c -1c ( -1)ect= = =c t tc

1-1(r- ) t (r- ) t1( (r - ))e e

1

c 1= ( r - )

c


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

Often all we need to look at is the first order condition. But if we want to complete thesolution 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 firstperiod), 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 thenhave 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 thatsatisfies 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 inconsidering infinite time as opposed to just letting T be very large. For example, we cant imposethe no dying in debt condition (A(T) = 0 ), and instead have to impose a different condition (oftencalled the no Ponzi game condition that I will not discuss here. For our purposes, it is sufficient tostate 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

(1/ )( )( ) (0) r tc t c e

The budget constraint is thus

(1/ )( )

0 0

(0)rt rt r te dt e c e dt

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

1( )r r

Integrating the budget constraint.


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1 (0)

(1/ )( )

c

r r r

1

(0) 1

r

c r

From this we see

If>r, then initial consumption is above 1, in which case consumption asymptotes to zero The bigger is , 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 thatdiffers from the Lagrangian. It uses recursive equations, also called Bellman equations, to break a Tperiod problem up into a bunch of much smaller one period problems. The most important thing tonote is that the problem itself is not any different, only the solution method has changed. So we stillhave individuals trying to maximize their lifetime utility subject to some constraints on their lifetimewages 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 spentproving to you that it is theoretically possible to solve the following problem this using techniques ofdynamic programming:

11

...0

max . . 11

t

t t t ttc ct

U cV st a r a w c

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

Well take that proof as a given, and just proceed to show you how the method actually works. Theproof depends a lot on the presence of time-separable preferences and then requires U(c) to beconcave, continuous, etc..

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


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1

...max . . 1

1t

s

t t t t t ts tc cs t

U CV a st a r a w c

This value function tells us that V is the maximized value of utility I have, given an initial asset levelof a

t, 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 mustfollow the exact same path that you would have chosen for periods t+1 to infinity if you had doneyour 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:

(1)

1 1 1max . . 11t

t t

t t t t t t tc

V aV a U C st a r a w c

This recursive, or Bellman, equation tells us that the value of my lifetime utility from time t forwardis 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 Vfunction can possibly change over time). Then my problem is no longer a many period problem butonly a one period problem. The question is trading off current utility at time t for more financialwealth at time t+1 (which I already know I will spread optimally among the remaining periods of mylife).

So lets do the maximization in the Bellman equation. We can substitute in the difference equationfor assets into the maximand in equation (1). We get

(2) 1((1 )( ))( ) max ( )1t

t t t tt t t

c

V r a w cV a U C

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

' 1 11

' 11

t t tU C r V a

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

value of an extra unit of the asset tomorrow. But we dont know what this V function looks like, sothis equation doesnt help us a lot.

However, the next big insight in the dynamic programming method is that there is a simple envelope


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relationship between V and U along the optimal path.7 To see this, take the derivative of (2) w.r.t.assets in period t, applying the envelope theorem:

' '1 1 2 21

11

t t t tV a r V a

.

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

' 1 1 1't t tV a U C .

Plugging this into the original period t FOC gives us

11

' 1 '1

t tU C r U C

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 consumptionpath and utilize the budget constraint. This is just the same as before. The dynamic programmingmethod doesnt necessarily offer any extra help during these last steps. Its main value is that certainproblems are easier to set up as Bellman equations in the first place. The recursive equations arealso useful because they are easier to translate to computer code that can iterate through periodsquickly 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 Bellmanequation is an example of a contraction mapping, and that this means the V functions (which werepreviously 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 ofassets at time t, will be time invariant as well.[Skip the below or fix up?]

To see what this means, consider a problem with log utility, so that

7 More formally, the envelope theorem says that if you have max ( , )x

y f x c , then the derivative

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 firstorder 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 originalf(x,c) function with respect to c.


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1

1

1t t

rC C

and therefore consumption in any period s>t can be written as

1

1

s t

s t

rC C

.

The budget constraint at time t is the following

1 1s s

ts t s ts t s t

C WA

r r

And you can solve these together to get that

1 1s

t t s ts t

WC A

r

.

This rule holds for any period t, so the consumption rule (or control rule) is identical for all periods.This doesnt mean that consumption itself is necessarily identical every period, but the rule forsetting it is. You may still have consumption rising or falling depending on the relationship of r andthe discount rate.

The real benefits of DP come when we extend it to uncertainty (come back and do an examplewith stochastic wages?).

Some Open Economy Applications

We can use the two period model of consumption to draw a helpful picture. Suppose that wegraph 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 theparameters of the utility function). The bigger is w1 and the smaller is w2, the higher will be savingat 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 amountof saving -- we know this because in this case the income and substitution effects are aligned. Forzero saving, we also know that the curve is upward sloping. But for positive saving, we dont know.The curve may well bend backward.

Question: what determines the degree to which the curve can bend backward? Answer: thedegree of risk aversion!

Why? The degree of risk aversion tells us how the person trades off smoothing of


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consumption for taking advantage of the interest rate to get more consumption in a later period. If aperson is very risk averse, then he wants very smooth consumption. In this case, the curve will endup 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 aboutcapital 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 ofproduction, 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 theassets 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

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


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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 isendogenous.

large open economy: economy is large enough to affect the world interest rate, so r=r*, but r* isendogenous. 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 secondperiod is exactly equivalent to running a CA surplus in the first period and a CA deficit in the secondperiod. (Even though the world only lasts for two periods, we can think of the requirement thatpeople do not die in debt as meaning that B3 = 0.)

There is another way that we can think about this same issue, which is more international.

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

Note that this is not just a case of liquidity constraints in the standard sense. Rather, sinceeveryone is identical, there will be no borrowing or lending. But (key observation): there can still bean 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 Autarkyinterest rate

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

U'(Y1)/U'(Y2) = (1+r)/(1+)

Now, here is the big result:

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


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where the curve derived above crosses zero. So we can also see here the result about the interestrate!]----

Large Open Economy model

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

Intuition building problem:

Lets look at the large open economy model with an infinite number of periods, instead ofjust two.

Lets think about two equally sized open economies. Equally sized in the sense that theyhave the same endowment income.

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

we forget about G and I

1 < 2

Two countries start with B=0.

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

===> In the long run, we know that the interest rate will be equal to the 1. We can trace out the

path of interest rates and net assets pretty easily (at least graphically!).

[Exercise: think about the solution if preferences are CARA (with c 0 )instead of CRRA]

The PIH and the LCH

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

Permanent Income Hypothesis


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

We can also give this result an econometric interpretation. Suppose that a researcher hascollected 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


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Due to Franco Modigliani8

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

Lifetime budget constraint is:

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

[picture]

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

[picture]

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

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 isthis so important? Because, for a closed economy, the capital stock of the economy is made up ofthe wealth of the people in the economy. And, as you know from studying growth, the capital stockis really important. Indeed, the OLG model is the life cycle model, with a very simple life cycle.

8Once, 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 didnt it wasthe PIH.

T Tt t

t tt=0 t=0

C W=(1+r (1+r) )


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We can see the aggregate amount saved in the economy by just adding up each age groups savingor dissaving, multiplied by the number of people who are that age. What does this say should behappening 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 uplater.]

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 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 formof 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 couldbe flat, or hump shaped, or whatever). Then in the high growth country, the growth rate of wagesbetween successive generations must be larger. This means that if we look at a cross section of thepopulation by age, the growth rate of aggregate wages will be reflected in it, i.e. the youngest peoplewill 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 anyindividuals lifetime wage profile will reflect this aggregate growth; in this case, people in the highwage 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 savingrate is just a weighted average of this, where the weights depend on the number of people and theirincome. Since young do saving and are richer when growth is higher, higher growth will raise theaggregate saving rate!

Case 2: Now, higher growth affects the saving rate. Specifically, it lowers the saving rate of theyoung. It also means that working age people (who are saving) earn more than did old people (whoare dis-saving) -- the effect which we saw in case 1 tends to raise the saving rate. For reasonable


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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 notdepend on aggregate growth rate of income.

Ricardian Equivalence

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

The most controversial implication is the so-called Ricardian Equivalence proposition (which wasmentioned, 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, lets look at the simplest modelwith taxes, one with just two periods.

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

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

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

You can see that the Z's will just cancel out, and the budget constraint is left unaffected. What

2 221 11

( - )C Y T+ = - +C Y T

1+r 1+r

1 2= - Z = (1+r)ZT T

2 221 11

( - [ +(1+r)Z])C Y T+ = - [ -Z]+C Y T

1+r 1+r


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about savings, though? Since the budget constraint has not changed, first period constumption willnot 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 goup by Z? No: because the government is going to have to borrow to finance its tax cut. In fact, it isgoing to have to borrow exactly Z (or, if it was running a deficit already, it will have to borrow Zmore 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 isconcerned they are not "net wealth," since they represent the governments liabilities, which will inturn 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 the2000'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 byan 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 permanentincome, and so do not affect consumption.

-- Note that Ricardian equivalence is about the timingof taxes -- it does not say that if thegovernment spending increases this should have no effect on consumption. That is, Ric Equiv saysthat you care about the present value of the taxes you pay. Government spending, either today ortomorrow, will affect this present value, and so affect consumption. For example, if the govt fights a


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war today, your consumption will fall, because you will have to pay for the war. But whether thewar is tax financed or bond financed will not matter to your consumption today. [but note that theresponse 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 willraise 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 extrataxes and consume less. This objection has generated the most debate, and often discussions ofRicardian Equivalence lapse into discussions about intergenerational relations. Before going alongthis path, we should note that even if this objection were true, most of the present value of any taxcut today will be paid back by people who are alive today; in which case even if there were norelations between generations, Ricardian Equivalence would be mostly true.

The intergenerational argument in defense of Ricardian Equivalence goes: Since we see peopleleaving bequests to their children when they die, we know that they must care about their children'sutility. Now suppose that we take money away from their children and give it to them. Clearly, ifthey 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 ofthe bequest, rather than from their children's consumption (or utility), then a shift out in the parent'sbudget constraint will lead them to consume more of both bequests and consumption today. Slightvariation (Bernheim, Shlieffer, and Summers) is that bequests are payment for services (letters,phone calls) from kids. Same result in response to a tax cut.

-- 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. Whenyou see it (with all its discussion of bequests, annuities, etc.), remember why it is relevant to thedebate about Ricardian equivalence.

-- 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).


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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 usualconsumption smoothing models, they would still raise their consumption only very slightly inresponse 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 tolive, and the real interest rate were 5%, and they kept consumption constant, then a tax cut of $100that they never expected to pay back would increase consumption by approximately $8. This ispretty close to the zero dollar increase predicted by Ricardian Equivalence. By contrast, if onebelieved in a Keynesian consumption function (where empirically estimated MPC's are in the roughneighborhood 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 cutin 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 (orhousehold). What are the characteristics that I can expect to see in it, assuming that the household isbehaving according to lifetime optimization model described above.

I want to argue that one of the most important is that the level of consumption will neverjump, by which I mean that it will never change dramatically from period to period. Whenconsumption 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 inthe list that they are all violations of the simple model presented above.

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

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

3) non-convex budget sets, specifically things like means tests -- we had been assuming theseaway 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


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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 thebeginning 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)


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

x = E(x) +

where 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. Inreality, of course, there is a good deal of uncertainty. To incorporate this into the LC model, weapply 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

0

( )

(1 )

Tt t

tt

PU c

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

Consider the problem of a person who may die over period 0..T. Assume that there is noadvanced warning.

The formal problem is

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

At 0 for all t

A0 given

Note that here, W, C, and A are the paths of wages, consumption, and assets that the person willhave 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 pathsof consumption and assets from the beginning (put another way: no new information arrives until itis 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.

Max0

( )(1 )

T t tt

t

PU c


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With certain lifespan, this implies that you have to have zero assets at period T. But now, it impliesthat you have to have non-negative assets at all periods!

Maximization problems of this form are generally unpleasant (it will be discussed a littlebelow, 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. Sucha person would never let wealth become negative, because then she would have zero consumptionfor the rest of her life.

We set up the lagrangian:

The first order condition relating consumption in adjacent periods is

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

where small t is the probability of dying in a given period conditional on having lived to that age

[t=(Pt - Pt+1)/Pt]

The FOC is

Note that the path of consumption that we are solving for here is the path that the person will

follow ifshe is alive.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.

T T-1t t t

0t tt=0 t=0

U( )C CPL= + -A

(1+ (1+r) )

tt +1

t +1t

U ( ) 1+C P=

U ( ) 1+rC P

t t +1 t t +1 t t +1

tt +1 t +1 t +1

+( - ) -P P P P P P= = 1+ 1+

P P P

t +1 t t

t

(1+ )(1+ ) (1+ + )U ( )C=

U ( ) (1+r) (1+r)C


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Empirically, the probability of dying in old age turns out to conform very closely to a log-

linear specification:

ln(rho) = _0 + _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


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The other, more intuitive way, in this simple case, would be as follows. First, observe that c3 will beequal to c2, live

so we have c2, die = (W0 c1) and c3 = c2, live = (W0 c1)/2

Now we need to know the relationship between marginal utility in the first period and expectedmarginal utility in the second. Answer: they had better be equal, for optimality. So we can directlywrite:

1 .5 1 112

2) Suppose an individual is born at time zero with wealth W0. She gets no labor income. Timeis continuous. Let r=theta=0. Initially, her probability of death is zero. She knows starting at

time zero that at some later time, t, she will learn her probability of death (which will beconstant thereafter). Suppose that there are two possible probabilities she will learn: 0 , where each will happen with probability 50%. We can solve this problembackward: first, figure out consumption after the information is revealed at time t as afunction of Wt in each of the two possible states of the world. Then figure out W t as afunction of initial consumption c0 (which in this simple case is the same as ct -- that is,consumption just before the information is revealed.). Then impose the FOC, which is thatthe marginal utility of consumption just before the information is revealed has to be equal tothe expected marginal utility of consumption just after.

This gives us a very important rule that we can generalize: consumption can be expected to jump,but it has to be the case that the marginal utility of consumption just before the expected jump isequal to the expected marginal utility of consumption just after.

Annuities

The person faced with the above problem will almost certainly die holding assets. Only if she livesas long as was remotely possible ex-ante will she die with zero wealth. Assuming that she does notvalue 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 willpay you some rate of interest z, but if you die before next year I will get to keep you money." Whatwould z have to be such that the insurance company earned zero profits?


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Pays: Earns

(1+z)(1-) = (1+r)

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 ayearly payment until you die. (Actual annuities are not like the ones described here, in that you paya 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

so in the case where =r, the person would have flat consumption even though her probability ofdeath 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 afterbecker etc.?]

Why this interesting: We know that dying is bad (since you miss out on utility). Here we look atanother bad, which is uncertainty about when you die.

One way this is bad is that you may have money left over at the end of life. However, annuities cantake 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 willgive 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 forannuities. This means that the man will have flat consumption. The actual level of that

consumption will depend on the survival curve (it would be ( ) (0)r A if mortality were constantat rate rho, for example). But we will not worry about this. Call c* the optimal level ofconsumption. His instantaneous utility is thus U(c*). We will assume that this is positive (Recallthat for a CRRA utility function, instantaneous utility can be negative unless you add some constantin 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

0

( *)( *) (1 )

Tt Tu ce u c dt e

t+1

t

U ( ) (1+ + )C=

U ( ) (1+r+ )C


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setting S2 to zero.9 The answer is

2

2

S

If we choose theta=.03, and for S=15, this gives 3.38 years. So this is how much you would bewilling 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, firstdeveloped by Yaari (1965). The model is (in continuous time)

0

( ) ( ) ( ( ))tE V P t e u c t dt

[Note, I am using exponential time discounting, but Bommier uses a more general time discountingwhere 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 ofexpected utility under uncertainty to a model of utility form a certain lifetime:

Utility from a certain lifespan T is :

0

( ) ( ( ))T

tV T e u c t dt

To get expected utility, we just integrate this over all the possible life spans (Where F(T) is the PDFof lifespans)

0

( ) ( ) ( )E V V T F T dT

9I 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.


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

0

( ) ( ( ))T

W T u c t dt

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 thesame axioms that we want. For example, it says that the marginal rate of substitution betweenconsumption at any two points in time is independent of consumption at other points in time and ofthe length of life, that more years of consumption make us happier, that more consumption makes ushappier, etc.

The big thing that this formulation does nothave is a pure time discount rate. This is an oldargument. Back in 1928, Ramsey famously argued against a pure time discount rate in the absenceof 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 Pigoumeant these as statements about what was an appropriate model of human behavior (that is, apositive 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 ofthe lifetime equivalent of the explanation for the curvature of the instantaneous utility function. Forexample, when you choose what books to read, you read the great ones first, and the less great, andso on. Or similarly, you might spend $100 on a restaurant meal every night, but some of those willgive you more pleasure because they are a new experience.

An interesting difference between the Bommier formulation and the standard one isregarding temporal risk aversion. Consider some consumption levelsc1


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intertemporal elasticity of substitution. In the standard model, the curvature of the instantaneousutility 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.

Now lets think about lifetime uncertainty.

In the absence of lifetime uncertainty, the Bommier formulation implies that if there is apositive interest rate, consumption will be rising. Indeed, without lifetime uncertainty, we dont everneed 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, thereis no reason to have the good meals or read the good books early in life, which is why you wouldnever get a declining path of consumption. You might get this if you allowed for utility frommemory, but then you would also have to allow for utility from anticipation but this is all not thepoint.]

The Bommier formulation becomes more useful when we allow for lifetime uncertainty.

Since is a function, we cant just pass the expectation sign through the integral. So expectedlifetime utility is

0 0

( ) ( ( )) ( )T

E W u c t dt F T dT

[We adopt the same setup as for the Yaari model, in which death is uncertain and unpredictable, soone 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, itdoesnt matter in which order you read the books. But if lifespan is uncertain, then you will startwith the best and read down the list from there. Similarly, with time discounting would have donethe same thing. So the point is that once we allow for mortality uncertainty, the Bommierformulation gives us discounting-like behavior.

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

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


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

The starting point for estimates of the utility of being alive vs. dead is peoples willingness totrade off risks of death for money. This can be seen in e.g. the wage premium required to get anindividual to take a risky job, or the willingness of people to pay for safety features of a product.We generally look at the effects ofsmall changes in the probability of death, but to make suchmeasures useful, we blow them up to the value of a statistical life. For example, if a person isindifferent between paying $1000 and taking a 1 in 10,000 risk of death, then the value he is puttingon 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 neverretires, 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 onhis 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, theindividual has utility just from being alive. His instantaneous utility function is

1

1

cu

Now, consider a case in which the individual has the opportunity to trade a very small risk to his lifefor more money. For example, he can spend $500 more to take a safe flight vs. a risky one. Weconsider a trade between life and risk that leaves the individual indifferent. [Note that the constantmortality assumption buys us that willingness to trade life risk for money is not a function of age. Inreal 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 affectthe marginal utility of consumption (it doesnt matter if we imagine him consuming it all at once orspreading it out over the rest of his life.) The cost in terms of expected life utility lost is

1

1( )

0

1

1

t

w

we dt


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The gain is the marginal utility of consumption, which is w , multiplied by the extra consumption,

x. Putting these together,

1

1

w

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 extramarginal utility if you are alive]

We can rearrange this to solve for alpha

1

( )1

x ww

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 impliedby 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 fromconsumption 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 betweenbeing alive or dead. That is, setting utility to zero


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1

01

c

1/(1 )( (1 )) $357c

[Note: the level of consumption that gives indifference between being alive and dead is incrediblydependent 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 ofconsumption. Holding the other values of the parameters constant, I then changed to 3. The

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 beinga grad student is no better than being dead!

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 inconsumption raise the marginal utility of consumption a lot, so sufficient reductions in consumptionvery rapidly get you to the point where utility from being alive is zero. (That is, if you are willing totake any life risk at all, the U must be not too small relative to the utility of being alive. So then ifU is not too small and lowering c raises U a lot, then at some point lowering c will also make it notworth being alive.]

Implications for Economic Growth

We usually look at economic growth by looking at growth rates of GDP per capita. But ifpeople get utility from being alive as well as consumption, we should consider their full income.(Note: we dont look at growth by looking at growth of utility. Why not? Because utility is notobservable.)

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)

0

( , ) max ( ) ( ( ))tV Y S e S t u c t dt

s.t.


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0 0

( ) ( ) ( ) ( )rt rte S t y t dt e S t c t dt

[Fix up notation. Look back at paper.]

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

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

We are interested in the extra income that we would have to give the person so that he would havethe same utility he had in the second period, but with the mortality rates observed in the first. Callthis 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 (Ithink that this is called compensating variation, which is discussed more below in the context ofJones 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 incomegrowth).

1960lifeexpect

1960GDPp.c.

2000LifeExpect

2000GDPp.c.

Value oflife expectgains (interms ofann income

GrowthRateGDPp.c.

GrowthRatefullincome

Poorest50% ofcountriesin 1960

41 896 64 3092 1456 3.1% 4.1%

Richest

50%

65 7195 74 18162 2076 2.3% 2.6%

So poorest countries get big income growth boost


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J ones and Klenow Beyond GDPHere is a whole writeup of a paper on this (and many other) topic. As of 2013, this is still a workingpaper. You can find it on Charles Joness web site:

Utility:

v(l) is utility from leisure; log utility from consumption is used for convenience this can be doneaway with but then dont get such clean expressions; ubar is the utility from being alive.

Mortality stuff

S(a) is the probability of living to age a (from birth). e is life expectancy at birth. They calculate

this thing p which is the probability that Rawls is alive and gets to consume in this year

They normalize the utility of not being alive to zero, so they write:

So they will basically multiply utility of alive people by life expectancy (they drop the divided by100 part for convenience).

Critique: what is going on here? First, notice that another way to describe what they arecalculating is expected utility from birth, with zero time discount factor (where we are

ignoring life cycle type stuff and just giving everyone the same per-period consumption; and

we are also assuming that people will have their whole lives at this constant level of

consumption. So it is sort of utility for a synthetic person.) What they are calculating is not

the happiness of people that you will meet on the street, because it includes the people of a

cohort who are not actually alive. This raises all sort of philosophical issues. For example,

holding consumption per person constant, this formulation says that utility is the same in a

country where everyone dies as 40 as in a country where half the people die at birth and half

at 80. Seems like the right model could have some sort of investment in utility by societyin people or by people in themselves.

Inequality:


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Assume that consumption is distributed lognormally, with mean consumption c and standarddeviation of log consumption sigma.

ln ln 2 Variance of log GDP ranges from about 0.4 to 1.2 in the data (and is lower in rich countries onaverage). So going from a high to a low variance of GDP raises (E(ln (c ) ) by about .4 that is thesame as raising average income by exp(.4), which is about raising it by 50%.

Income-metric measure of utility differences

A) Equivalent Variation

By what factor do we have to adjust Rawls consumption in the US to make him indifferent betweenbeing born in the US and being born in country i?

Plugging these numbers into the utility function and re-arranging, we can get:

Rewrite this as

Where in practice the ln(y) will move to the other side.

Compensating VariationBy what factor would be have to change income of Rawls in a poor country to make him indifferent


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between being born there and being born in the US?

Difference between CV and EV is only in the stuff in parenthesis in the first term.

Equivalent variation weighs differences in life expectancy by a countrys own flow utility,while compensating variation weighs differences in life expectancy by US flow utility.

Data and calibration

Leisure

Assume 16 hours/day time endowment (sleep doesnt count)

So looking at just adults (issue?)There is some utility from leisure thing that I dont understand, but it doesnt matter much.

Ubar

This is key parameter, utility of being alive. They derive it from willingness to pay to avoid risks tolife (see Econ 207 notes). hey look at the literature and based on it set the value of statistical life fora 40 year old in the US at $4 million. This give ubar of 5.54.

They find that ubar is big enough so that even in the poorest countries, Rawls is ex-antehappier being alive then dead. J&K say that this is a good thing in terms of their theorymaking sense. However, given inequality, there will be some fraction of the population in

poor countries that is ex-post less happy being alive than if they had never been born. This

seems like a problem.


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Prinz and Weil (as yet unwritten) paper on Value of a Statistical L ife

Present the initial puzzle here and the solution later on.

The model of JK or BSL implies that for a certain level of consumption, it is better to bedead than alive.

For levels of consumption that are above the critical level, VSL is low for two reasons: first,VSL is always low for poor people because they cant pay a lot; but second, VSL is lowbecause the utility of being alive is not very high.

We can calculate an interesting object: VSL / consumption. [derive this] For our standard (JK or BSL) formulation, this rises with income When we look at the data, we see that VSL/consumption does not vary much at all with the

level of consumption!

Implications of Curved Utility for Health Care Expenditures

(based on Hall and Jones, QJE 2006)

Here is a core exam question from 2007 about this issue:

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

The man cannot borrow or save. He can use his income for two things: he can purchase aconsumption good, which gives him utility, or he can spend it on health, which

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