part 3 macroeconomics of financial markets · motivation by now we have several important...

Macroeconomics Part 3

Macroeconomics of Financial Markets

Lecture 7

Permanent Income Hypothesis and Buffer-Stock Saving

Motivation

By now we have several important explanations of saving

Saving is needed to accumulate productive capital

Saving and borrowing (or consuming out of wealth) mean the reallocation

of uneven flow of income over time

Voluntary or mandatory saving for retirement is the feature of human life-

cycle (and social security system)

But this list is still incomplete

We have to study saving as a reallocation of wealth over time in more depth

We did not discuss any implication of uncertainty that people face

To reveal other motives for saving let’s abstract from previous

frameworks of capital and pension wealth accumulation

Macroeconomics of Financial Markets 2

Permanent Income Hypothesis

Friedman M. (1957) A Theory of the Consumption Function. Princeton

University Press: Princeton.

Hall R. E. (1978) “Stochastic Implications of the Life Cycle -

Permanent Income Hypothesis: Theory and Evidence”. Journal of

Political Economy, 86(6), pp. 971-87.


PIH

Consider a representative agent with infinite life-time horizon

Take Euler Eq. and intertemporal budget constraint from Lecture 2:

1

1

)(

)()1.7(

1

r

Cu

Cu

t

t


To simplify algebra assume that r = ρ

Then, regardless the dynamics of disposable income, the optimal

choice is constant consumption, Ct+1 = Ct = C

00 11)2.7(

r

YA

r

C d

tt

t

PIH

00

111

1

r

YA

rCCCC

d

tttt

tY

r

YA

r

rC P

defd

tt

0 11)3.7(


Permanent income is the annuity value of total wealth, which is the

sum of

assets

and human wealth (the discounted sum of future disposable income)

This intertemporal story differs from Keynesian assumptions:

consumption is determined by permanent income, not current income

Reminder


Assume that you buy an asset (annuity), which will provide you with the constant coupon (annuity) payment each period

The relation between the price of annuity X and annuity payment x is

Xx

r

r

r

x

r

xx

1...

112

Xr

rxor

1

You can write the same relation for the case of finite time horizon

It will be just a little more complicated

It is useful for everyone: think about a bank credit and credit payment each month/year

Permanent income definition reads like that!

Hint


If calculations at infinite horizon seem weird, take a simple example

Assume that a consumer lives for T periods, has some initial assets A0. And to simplify further, assume r = ρ = 0

C

T

Cu max)(1

0

1

0

0

1

0

Td

T

YAC

CCCCtCuCu Ttt 1101 ...)()(

1

0

0

1

0

0

1

0

1 Td

Td

T

YAT

CYACTC

Hint


Compare this result with the Eq. (7.3) for t =0:

They say the same thing

for the T-period horizon and zero interest rate, the annuity coefficient is

1/T (or the reciprocal of future periods)

for the infinite horizon, the annuity coefficient is r/(1+r)

But be careful in interpreting permanent income as average income

This is just the special case

1

0

0

1 TdYA

TC

0

011

r

YA

r

rC

d

tYAtT

Td

tt

1

0

1

t

r

YA

r

r d

tt

0 11

PIH

Define the difference between permanent income and current income

as transitory income:

0 11

)4.7(

t

tt

d

t

Pd

t

defT

t

r

YA

r

rY

YYY


Apparently, as far as Y P = C, transitory income is the same concept as

saving

It allows to transfer resources over time: save, when current income exceed

permanent income, and borrow (or consume out of wealth) when current

income is lower than permanent income

Test yourself

Assume that you won

$1 000 000

Let’s estimate your future life span as 50 years

Following PIH you will increase your yearly spending by $20 000

NO!!! I would like to spent a million NOW!!!

??? Do you think you are smart rational guy?


Test others

Do these “behavioral patterns” match PIH?

Poor people save a lesser fraction of their income than reach men do,

as they have to meet the minimal standard of living, buy necessities,

etc.

Some people save too little to “keep up with the Jones”


PIH and estimated consumption function

Keynesian cross looks like:

Keynes assumed that consumption function is stable

and that higher income leads to higher saving rate

Econometric estimations on panels of households confirm this hypothesis Macroeconomics of Financial Markets 12


But aggregate time-series show different relation (Kuznets puzzle)



Also, differences in estimates of consumption functions for different

groups of households can be hardly explained in a Keynesian tradition


Estimation of consumption function

The slope of estimated consumption function depends on the relative variation in YP и YT

An increase in current income leads to an increase in consumption if only it is associated with an increase in permanent income

iii ebYaC

TP

P

TP

PTP

YVarYVar

YVar

YYVar

YYYCov

YVar

CYCovb

,,ˆ

PTPP YbYYbYYbCa ˆ1ˆˆˆ



If Var(YT) >> Var(YP), then changes in current income mostly reflect changes in transitory income. Thus, they have a relatively weak impact on consumption (i.e., b < 1, a > 0)

If Var(YT) << Var(YP), then changes in current income mostly reflect changes in permanent income. Thus, consumption changes one-for-one with the current income

iii ebYaC

TP

P

TP

PTP

YVarYVar

YVar

YYVar

YYYCov

YVar

CYCovb

,,ˆ




For a household, most of the variation in income comes from personal issues (age, employment, career, etc.), i.e. Var(YT) >> Var(YP) Thus, panel-data estimates give b < 1, a > 0

At the aggregate level (in dynamics), most changes in income reflect economic growth Thus, time-series estimates give b ≈ 1, a ≈ 0

“Blacks” and “Whites” have similar mpc’s, but different YP’s

iii ebYaC

TP

P

TP

PTP

YVarYVar

YVar

YYVar

YYYCov

YVar

CYCovb

,,ˆ



PIH when future incomes are uncertain In reality, households do not know their future income for sure

They have to maximize expected life-time utility subject to budget constraint in expected present value terms


00 11)6.7(

r

YEA

r

CE

d

ttt

tt

tCt

t

tCuE max

)1(

)()5.7(

0

0

In this case, Euler Eq. takes the form

1

1

)(

)()7.7(

1

r

CuE

Cu

tt

t

Assume again that r = ρ. Then

)()()8.7( 1

ttt CuECu

Random walk hypothesis Assume that

This corresponds to quadratic utility (linear marginal utility)

This assumption is known as certainty equivalence: the agent choose consumption as if he knows true values of future incomes

It simplifies Euler Eq. (7.8):


)()( 11

tttt CEuCuE

)()()()()8.7( 11

tttttt CEuCuCuECu

2

1111 ,0...~,)9.7( diiCCorCCE ttttttt

This is so called Random Walk Hypothesis: the change in consumption in each period is a random variable with zero mean

It implies that the best forecast for any future consumption is current consumption

ttt CCE

Random walk hypothesis

We arrive at the similar definition of permanent income


But now permanent income (and thus consumption) is not constant, it follows some stochastic process

What determines the random change in consumption?

ttt CCE

00 11)6.7(

r

YEA

r

CE

d

ttt

tt

P

t

defd

tttt Y

r

YEA

r

rC

0 11)9.7(


You can easily derive (7.10) from (7.9) by using dynamic budget constraint

The change in consumption between t and t+1 is determined by the change in expectations about future incomes

This change is random as far as the change in expectations is random

0

1111

11

r

YEA

r

rC

d

tttt

0

1111

11)10.7(

r

YEYE

r

rCCC

d

tt

d

ttttt


0 11)9.7(

r

YEA

r

rC

d

tttt


Only new information about future incomes leads to the change in

permanent income (consumption)

If household does not receive news, consumption remains the same

It implies, e.g., that only unexpected changes in taxes can affect

consumption

Consumption changes when new tax policy is announced

Temporal change in tax policy should have smaller impact on

consumption than permanent change

If incomes in the past can forecast future incomes, then change in

consumption is not purely random (this is so-called excess sensitivity)


0

1111

11)10.7(

r

YEYE

r

rCCC

d

tt

d

ttttt

Prudent consumer

While certainty equivalence is a concept of choice under uncertainty,

it ignores the impact of changing risks

It seems to be unrealistic

Assume, that you have to choose consumption today under

uncertainty of income in the next period. Consider two setups:

#1 with probability 0.5 your income will be $1200 and only $800 otherwise

#2 with probability 0.5 your income will be $1500 and only $500 otherwise

Under certainty equivalence this setups are identical, as they assume

the same expected income $1000

But would you choose the same level of consumption in both setups

or you would be prudent?

How to model your prudence?


Prudent consumer

To introduce the prudence motive we have to assume

In this case Jensen inequalities give us:


0)( u

11

tttt CEuCuE

2

1

2

2

2

11

2

21 ,)()( tttt CuECuE

That is, expected marginal utility is higher than marginal utility of

expected consumption

and the higher the risk, the higher expected marginal utility

As far as marginal utility is a decreasing function, it means that

higher risk implies lower Ct+1

That is, facing higher risk for tomorrow, consumer saves more today

Prudent consumer


)(Cu

2

21)(

tt CuE

2

11)(

tt CuE

1

ttCEu

1ttCE 1tC

Precautionary saving

Prudence motive introduces another reason for saving, so called precautionary saving

It differs from what we have discussed previously (saving to accumulate capital and increase future production, saving to reallocate resources over time, and saving for retirement period)

But it does not mean that RWH/PIH, which was based on certainty equivalence, is totally senseless

Prudence generates relatively small additional asset holding, so called buffer-stock saving

This motive become weaker as assets are accumulated

So it implies some upward trend in consumption: young household without enough wealth should consume less, but by accumulating wealth they will consume more in the future

It works in the opposite direction with respect to impatience (a high ρ)


Liquidity constraints

Throughout our analysis we assumed that households can save and

borrow at the same interest rate

The only limit to borrow (or over-save) was NPG condition

In reality, people often face liquidity constraints

Interest on saving is lower than the credit rate they face

Some people do not even have an access to credit, which means that At ≥ 0

Facing liquidity constraints (now or in the future) implies additional

motive for buffer-stock saving

If there is a difficulty to borrow in times of low income, it is better to

accumulate additional wealth to be able to smooth consumption in the future

Buffer-stock saving is a root to excess saving in time of crisis

This calls for Keynesian policy (remember Krugman)


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