intermediate macroeconomics - lecture 9 -...

Intermediate MacroeconomicsLecture 9 - Consumption

Zsofia L. Barany

Sciences Po

2014 March

Keynesian consumption function

Keynes (1936)

”the amount of aggregate consumption mainly depends on theamount of aggregate income” and this relationship ”is a fairlystable function”

the consumption function:

C = a + bY where a > 0, 0 < b < 1

Hypothesis

1. a and b are constant

2. average propensity to consume (APC) ↓ as income ↑I APC = C

Y = aY + b

I if a > 0, then as Y ↑ → APC ↓I APS = 1− APC , average propensity to save

Consumption puzzle

What is the empirical relationship between consumption andcurrent income?

Studies did not find a consistent and stable relationship

Y

CI within a country over time,

APC was constanta = 0

I across households at a pointin time, APC was fallinga > 0

→ Kuznets consumption puzzle

Two theories from the 1950s to solve the puzzle:

I Milton Friedman: permanent income theory of consumption

I Franco Modigliani: life-cycle theory of consumption

both theories highlight the role of permanent or lifetime incomein determining consumption

The permanent income theory of consumption

I current income is composed of:I permanent income ∼ average income, which people expect to

get in every period in the futureI transitory income ∼ temporary deviations from the average

income

I consumption depends on permanent income

I if a change in the current income is mainly transitory, is hasvery little impact on consumptiona positive transitory shock mainly goes into savinga negative transitory shock implies dissaving or borrowing

Life-cycle theory of consumption

I income varies systematically over the phases of theconsumer’s ’life cycle’⇒ consumers plan over their entire lifetime to achieve smoothconsumption

I ⇒ consumption depends on life-time income, saving is used toachieve smooth consumption

I if a change in the current income has a very small impact onlife-time income, is has very little impact on consumption

I Life-cycle pattern

1. borrow when young ⇔ income is low2. save during middle age ⇔ income is high3. dissave during old age ⇔ retirement

Solving the puzzle

I both theories highlight the role of permanent or lifetimeincome in determining current and future consumption– saving is for future consumption

I they imply that the time pattern of income is not importantfor consumption as a consumer uses saving and borrowing tosmooth the path of consumption

I across households, much of the variation in income reflectsfactors such as unemployment and the fact that householdsare at different points in their life cycles

I over time for the aggregate economy, almost all variation inaggregate income reflects long run growthwhich are permanent increases in the economy’s resourcesthe transitory components across households cancel out

Solving the puzzle - PIH

The permanent income hypothesis implies:

C = αY P

APC =C

Y= α

Y P

Y

where Y P is permanent income and Y is current income

I if high-income households have higher transitory income thanlow-income households⇒ APC is lower in high-income households

I over the long run, income variation is due mainly to variationin permanent income⇒ stable APC

Solving the puzzle - Life-cycle theory

The life-cycle theory of consumption implies:

C = αW

APC =C

Y= α

W

Y

where W is lifetime wealth (total discounted value of lifetimeincome) and Y is current income

I across households, income varies more than wealth⇒ high-income households should have a lower APC thanlow-income households

I over time, aggregate wealth and income grow together⇒ stable APC

A two-period model

Two periods:

I can capture the dynamic nature of consumption and saving

I difference between current income and lifetime/permanentincome

I simple enough to analyze

Questions we look at:

I responses of consumer to changes in income and interest rates

I government budget deficits and the Ricardian Equivalence

Lifetime budget constraintThe consumer lives for two periods.

The current period’s budget constraint:

c + s = y − t

c consumption, s saving, y income, t taxes

The future period’s BC:

c ′ = y ′ − t ′ + (1 + r)s

r is the interest rate

Convert the future period’s value into current values (divide by1 + r) and sum the two:

c+s+c ′

1 + r= y−t+

y ′

1 + r− t ′

1 + r+

(1 + r)s

1 + r

c+c ′

1 + r= y +

y ′

1 + r− t − t ′

1 + r︸︷︷︸≡w wealth

Lifetime budget constraintThe consumer lives for two periods.

The current period’s budget constraint:

c + s = y − t

c consumption, s saving, y income, t taxes

The future period’s BC:

c ′ = y ′ − t ′ + (1 + r)s

r is the interest rate

Convert the future period’s value into current values (divide by1 + r) and sum the two:

c +c ′

1 + r= y +

y ′

1 + r− t − t ′


Lifetime budget constraint

I E - endowment

I consumer is aborrowerif optimalconsumption ison AE

I consumer is asaverif optimalconsumption ison EB

I we present valueof wealth

Consumer’s optimum

highest possible indifference curve→ tangent to the budget constraint⇒ MRSc,c ′ = 1 + r

A consumer who is a lender

I E - endowment

I A optimal point

I B vs Dc∗ < y − tconsumes lessthan his incomein period 1

I c ′∗ > y ′ − t ′

consumes morethan his incomein period 2

I LENDER

A consumer who is a borrower

I E - endowment

I A optimal point

I B vs Dc∗ > y − tconsumes morethan his incomein period 1

I c ′∗ < y ′ − t ′

consumes lessthan his incomein period 2

I BORROWER

An increase in income

Current income increases

I income effect: current and future consumption increases

I the consumer acts to smooth consumption over time

I → saving increases

Future income increases

I income effect: current and future consumption increases

I the consumer acts to smooth consumption over time

I → saving decreases

Non-durable and services consumption

I smooth relativeto aggregateincome

I this reflectsconsumptionsmoothing

Durable consumption

I more volatilethan income

I durablesconsumption iseconomicallymore likeinvestmentthanconsumption

Temporary vs permanent increase in income

temporary increase in income

I has a smaller impact onlifetime wealth

I AB to DE

I smaller effect on currentconsumption

I saves most of a purelytemporal increase

permanent increase in income

I has a larger impact onlifetime wealth

I AB to FG

I larger effect on currentconsumption

Application: Consumption and the stock market

How should a change in stock market prices affect consumption?

changes in stock prices affect the value of lifetime wealth

1. publicly traded stock is not a large fraction of national wealthhousing stock, privately held companies constitute a largefraction⇒ dampen the effect of stock prices on consumption

2. we expect changes in stock prices to be permanentfinancial theory: stock prices are martingalesbest predictor for tomorrow’s price is today’s price⇔ any change in the price is a surprise and is expected to bepermanent⇒ amplify the effect of stock prices on consumption

Application: Consumption and the stock market

An increase in the real interest rate

I increases the relativeprice of currentconsumption⇒ steeper slope

I substitution effect:decrease c , increase c ′

I income effectsaving is worth more (+)borrowing is moreexpensive (-)

An increase in the real interest rate - for a lender

I substitution effectA to Ddecrease c , increase c ′

I income effectsaving is worth more→ positiveincrease c , c ′

D to B

I c ambiguousc ′ increases

An increase in the real interest rate - for a borrower

I substitution effectA to Ddecrease c , increase c ′

I income effectborrowing is more expensive→ negativedecrease c , c ′

D to B

I c decreasesc ′ ambiguous

An increase in the real interest rate

Mathematical representation of the two period model

The consumer’s lifetime utility is:

U(c , c ′) = u(c) + βu(c ′)

where

I β is the discount factor: β ∈ (0, 1)the consumer cares about future consumption, but isimpatient

I u(·) is the instantaneous utility function

The consumer chooses c and c ′ to maximize the above subject tohis lifetime budget constraint

c +c ′

1 + r= y +

y ′

1 + r− t − t ′


We can simplify the problem:

c ′ = (1 + r)(w − c)

The consumer has to choose c to maximize:

u(c) + βu((1 + r)(w − c))

The first order condition is then:

u′(c)−β(1 + r)u′((1 + r)(w − c)) = 0 ⇒ u′(c) = β(1 + r)u′(c ′)

I at the optimal choice the marginal benefit of increasing c ,u′(c), equals the marginal cost of increasing it, β(1 + r)u′(c ′)

I at this (c , c ′) the indifference curve is tangent to the budgetline

MRSc,c ′ ≡MUc

MUc ′=

u′(c)

βu′(c ′)= 1 + r =

Pc

Pc ′

An example - log utility

Assume that u(c) = ln c ⇒ the marginal utility is u′(c) = 1c

The FOC becomes

1

c= β(1 + r)

1

c ′⇒ c ′ = β(1 + r)c

Plugging this into the BC

c +c ′

1 + r= c +

β(1 + r)c

1 + r= c + βc ′ = w ⇒ c =

w

1 + β

I current consumption depends on lifetime wealth

I the effect of changes in current income, y , on currentconsumption, c , depends on how y affects wealth

I easily extendable to T periods: the larger is T , the smaller isthe effect of current income, y , on wealth, and hence on c

Current income and consumption

I in this model, current income can affect current consumptiononly through affecting lifetime wealth

I In the example, c = w1+β ,

I ∂c∂y = ∂c

∂w ·∂w∂y = 1

1+β · 1I an increase in current income by 1, increase current

consumption by 11+β

I ⇒ savings increase by 1− 11+β = β

1+β

I ⇒future consumption increases by β1+β (1 + r)

Consumption Smoothing!

Current income and consumption

current income can affect current consumption only throughaffecting wealth

empirical studies suggest that current income may have a directeffect on current consumption

There are at least two explanations

1. borrowing constraints or liquidity constraintshome purchases require down payment, availability ofconsumer credit, etc

2. the psychology of instant gratificationimpatient at short horizons, but patient at long horizon

Borrowing constraints

c

c ′

y ′ − t ′

y − t

c c

I borrowing constraint⇒ c ≤ y − t has to hold

I budget line is the blue one

constraint not binding

I optimal consumption bundlewithout constraint:c< y − t

I optimal consumption bundlewith constraint is the same

constraint binding

I optimal consumption bundlewithout constraint:c> y − t

I optimal consumption bundlewith constraint: c = y − t


c

c ′

y ′ − t ′

y − tc

c






constraint binding




c

c ′

y ′ − t ′

y − tc c






constraint binding



The psychology of instant gratification

Extending the model used until now for T periods we get thefollowing lifetime utility

U(c1, c2, ..., cT ) = u(c1) + βu(c2) + β2u(c3) + ...+ βT−1u(cT )

the discount factor is β between any two consecutive periods (tand t + 1)

I in one survey 76% said they were not saving enough forretirement

I David Laibson claims this is due to the pull of instantgratificationthe discount factor between period 1 and 2 is smaller thanbetween any future period t and t + 1

A quick survey

1. Would you preferA) 100 euros now orB) 105 euros in a month?

2. Would you preferA) 100 euros in June orB) 105 euros in July?

I in studies, most people answer A to question 1 and B toquestion 2

I someone choosing B in question 2 nowmight choose A in June 2014 for question 1

I the pull of instant gratification

I dynamic inconsistency

I one representation: hyperbolic discounting

Summary

A more general consumption function

c = c(w , y)

current consumption depends both on lifetime wealth and currentincome

the dependence of consumption on total wealth has two mainimplications for the relation between consumption and income

I consumption is likely to respond less than one for one tofluctuations in current income

I current consumption may move even if current income doesnot change

intermediate macroeconomics - lecture 9 -...

Documents