Advanced Macroeconomics II · A little bit of history (1) Keynesian consumption function (1936) I Consumption is a constant fraction of disposable income. C t = Y t; 2(0;1) I Assumption

$Page 1: Advanced Macroeconomics II · A little bit of history (1) Keynesian consumption function (1936) I Consumption is a constant fraction of disposable income. C t = Y t; 2(0;1) I Assumption$
Advanced Macroeconomics II

Lecture 5

Consumption: Permanent Income Hypothesis

Isaac Baley

UPF & Barcelona GSE

January 25, 2016

1 / 60

Motivation

• Consumption is a large fraction (70%) of aggregate output.

• Households derive direct utility from consumption

⇒ key determinant of welfare, both at short and long run.

• Savings vary a lot over the life cycle.

⇒ key to understand investment in the long run (e.g. ageing populations)

• Relatively stable in the business cycle.

⇒ still key for business cycle fluctuations

• Consumption depends on real interest rates.

⇒ key to understand impact of monetary policy

• Consumption is affected by fiscal instruments (consumption tax, income tax

through its effects on labor supply, etc)

⇒ key to understand impact of fiscal policy

2 / 60

A little bit of history (1)

• Keynesian consumption function (1936)

I Consumption is a constant fraction of disposable income.

Ct = αYt , α ∈ (0, 1)

I Assumption of Solow’s growth model.

I Implication: Big role for government stimulus.

• Friedman’s Permanent Income Hypothesis [PIH] (1957)

I Individual consumption tracks permanent income, which is the

“normal” level of income.

cPi,t = αtYPi,t

Yi,t = Y Pi,t + εYi,t

ci,t = cPi,t + εCi,t

I Implication: Smaller role for government stimulus.

3 / 60

A little bit of history (2)

• Ando and Modigliani’s Life Cycle Theory (1963)

I Savings smooth consumption over the life cycle while income varies.

ci,t = αt,aYi,t , a = age i = individual

• Afterwards, microfoundation of consumption optimization problem.

I PIH is a special case of a general framework.

I Adding uncertainty, heterogenous agents, borrowing constraints, GE ...4 / 60

Roadmap

1 Facts about aggregate and household level consumption

2 Consumption-savings problem with certainty

I Euler Equation and Consumption Smoothing

I Full Solution

I Application: Ricardian Equivalence

3 Consumption-savings problem with uncertainty

I Special case: Quadratic utility

I Propensity to consume and income persistence

4 Empirical evidence

5 / 60

Facts about consumption

• Fact 1: Consumption is a large fraction of aggregate output.

I Focus on consumption of non-durables + services (∼ 60% of output).

I Later we will study durables.

0%

10%

20%

30%

40%

50%

60%

70%

80%

1947

1950

1954

1958

1962

1966

1970

1973

1977

1981

1985

1989

1993

1996

2000

2004

2008

2012

Durables/Y Services/Y NonDurables/Y C/Y

6 / 60

Facts about consumption (cont...)

• Fact 2: Consumption is less volatile than output.

• Fact 3: Relatively low correlation of consumption with output.

I HP-detrended, quarterly time series from US during 1954 –1991.

Variable St Dev (%) Correlation with GNP

GNP 1.72 1

Consumption (non-durables) 0.86 0.77

Investment (gross private domestic) 8.24 0.91

Hours Worked 1.59 0.86

Cooley and Prescott (1996)

• This facts point towards aggregate consumption smoothing.

7 / 60

Aside: Hodrick-Prescott Filter (1997)

• Two sided filter that allows for isolating cyclical component of a time series.

• Let yt = logYt , which has a trend τt and a cyclical component y ct :

yt = τt + y ct + εt

• Trend solves following problem for a given λ (1600 for quarterly data):

min{τt}Tt=1

(T∑t=1

(yt − τt)2 + λ

T−1∑t=2

[(τt+1 − τt)− (τt − τt−1)]2)

• Other filters: Baxter and King, BandPass filters.

8 / 60

Facts about consumption (cont...)

• Fact 4: Within household’s life cycle, the correlation of consumption

and income is large and positive during early and late years, and

negative in the middle.

Souce: Gourinchas & Parker (2002), “Consumption over the Life Cycle”. CES data for US.

9 / 60

Roadmap




I Full Solution






10 / 60

The building blocks

• Time: Discrete and infinite t = 0, 1, 2, ....,∞.

• Many identical optimizing households who live forever

I Choosing consumption sequence to maximize utility U ({ct}∞t=0)

I Dynasty where current generations care about future ones (bequests).

I Discrete time version of the consumption problem in Ramsey model.

• Endowments:

I Labor income {yt}∞t=0 is a non-stochastic sequence.

I Initial wealth a0.

• Financial markets: borrowing and saving through a one period risk-free

bond with return R = 1 + r (with certainty, one asset is complete markets).

• Partial equilibrium approach:

I No production sector and no population growth.

I Interest rate R fixed, and labor income y is exogenous.

11 / 60

The building blocks

• Preferences:

U = max∞∑t=0

βtu (ct)

1 Stationarity: us(·) = u(·) and βs = β for all periods s.

2 Homogeneity: ui = u and βi = β for all households i .

3 No habit formation: u at t only depends on consumption at t.

4 Monotonicity and Risk Aversion: u′ > 0, u′′ < 0

• Later on, we change assumptions:

I u′′′ > 0⇐⇒ Prudence (precautionary saving)

I Habits

I Hyperbolic discounting

12 / 60

Net assets and budget constraint

• Define net assets at+1 at the beginning of period t + 1:

at+1 = Rat − ct + yt with initial condition a0 given

• Implicit timing assumption:

1 Begin period with net assets at

2 Receive interest payments rat and income yt

3 Decide consumption ct (or equivalently net assets for next period at+1).

• We can explore different timings, but no big changes in results.

• The maximization problem is then:

V0 = max{ct}∞t=0

∞∑t=0

βtu (ct)

subject to: at+1 = Rat − ct + yt , a0 given

13 / 60

What about transversality conditions?

• Substituting the budget constraint recursively forward, starting from t = 0:

a0 =1

R(a1 + c0 − y0) =

1

R

(1

R(a2 + c1 − y1) + c0 − y0

)= . . .

• Moving incomes to LHS and consumptions to RHS:

a0 +1

R

(y0 +

y1R

+ ...+ytR t

)− at+1

R t+1=

1

R

(c0 +

c1R

+ ...+ctR t

)• Keep substituting and take limits:

Ra0 +∞∑j=0

(1

R

)j

yj − limt→∞

at+1

R t+1=∞∑j=0

(1

R

)j

cj

• If allowed to increase its borrowing over time, the household will choose at+1

equal to negative infinity, and consumption equal to infinity.

14 / 60

No Ponzi schemes

• To eliminate this possibility a “No Ponzi game” condition is imposed:

limt→∞

at+1

R t+1≥ 0

I This condition is an exogenous constraint, not an optimality condition

necessary for the maximization.

• Since it is then not optimal for at to grow too large, the condition above

must hold with exact equality for the household to maximise utility.

limt→∞

at+1

R t+1= 0

• For now we abstract from borrowing constraints and only use “No Ponzi”.

15 / 60

Different borrowing constraint

• Ad-hoc borrowing constraints (usually binding):

I Borrowing only up to some value B ≥ 0:

at+1 ≥ −B

I Extreme case (no borrowing, only saving)

at+1 ≥ 0

• Natural borrowing constraint (NBC):

at+1 ≥ at where at ≡ −∞∑j=1

yt+j

R j

Why is it “natural” not to borrow more than the stream future income?

More on this later.

16 / 60

Back to the problem

• Reformulate the sequential problem in terms of at+1 by substituting the

budget constraint:

V0(a0) = max{at+1}∞t=0

∞∑t=0

βtu (Rat + yt − at+1)

I FOC: −u′(ct) + βRu′(ct+1) = 0.

• Alternatively, use the recursive problem

V (a, y) = maxa′

u (Ra + y − a′) + βV (a′, y ′)

I FOC: −u′(c) + β ∂V (a′,y ′)∂a′ = 0

I Envelope: ∂V (a,y)∂a = Ru′(c)

I FOC + Forward Envelope: −u′(c) + βRu′(c ′) = 0.

17 / 60

Roadmap




I Full Solution






18 / 60

Euler Equation and Consumption Smoothing (1)

• The Euler equation for this problem is given by:

u′ (ct) = βRu′ (ct+1)

• Interpretation:

I Let ρ = subjective intertemporal discount rate, such that β = 11+ρ

.

I ρ = 0.02 means consuming tomorrow rather than today reduces u by 2%.

• Euler Eq can be written comparing subjective and objective discounts:

u′ (ct)

u′ (ct+1)=

1 + r

1 + ρ

Since u′′(c) < 0, then

I If r = ρ, then perfect smoothing c0 = c1 = .... = ct = ct+1.

I If r > ρ, intertemporal saving motive: ct < ct+1.

I If r < ρ, intertemporal borrowing motive: ct > ct+1.

19 / 60


• Example: Constant Relative Risk Aversion (CRRA) Utility

u (ct) =c1−γt − 1

1− γu′ (ct) = c−γt

• Euler equation:

c−γt

c−γt+1

=1 + r

1 + ρ=⇒ ct+1

ct=

(1 + r

1 + ρ

) 1γ

• Denote consumption growth with g c = log(

ct+1

ct

).

• Then Euler implies a constant trend:

g c =1

γlog

(1 + r

1 + ρ

)≈ r − ρ

γ

20 / 60


• More in general, we can linearize the FOC.

u′ (ct) =1 + r

1 + ρu′ (ct+1)

• A first order linear approximation of the RHS around ct :

u′ (ct) ≈1 + r

1 + ρ[u′ (ct) + u′′ (ct) (ct+1 − ct)]

• Dividing both sides by u′(ct) and multiply and divide by ct rearranging:

1 ≈ 1 + r

1 + ρ

[1 +

u′′ (ct)

u′ (ct)(ct+1 − ct)

]≈ 1 + r

1 + ρ

1 +u′′ (ct) ctu′ (ct)︸︷︷︸−γ(ct)

ct+1 − ctct

where the coefficient of relative risk aversion is defined as:

γ (ct) ≡ −u′′ (ct) ctu′ (ct)

21 / 60


• Under the linear approximation, consumption growth is given by:

−γ (ct)ct+1 − ct

ct≈ 1 + ρ

1 + r− 1

ct+1 − ctct

≈ r − ρRγ (ct)

I Consumption trend depends on ρ, r and γ (ct).

• Under the example of CRRA utility: γ (ct) = γ. Proof.

u (ct) =c1−γt

1− γ; u′ (ct) = c−γt ; u′′ (ct) = −γc−(1+γ)t

−u′′ (ct) ctu′ (ct)

=γc−(1+γ)t ct

c−γt

= γ

I Trend becomes constant: r−ρRγ

I Consumption is smoothed over time, even without uncertainty.

22 / 60

Roadmap




I Full Solution






23 / 60

Full solution

• Solution of the problem: consumption as a function of a0, {yt}∞t=0 . Close

form solution in the deterministic case.

• Solution: Euler equation + budget constraint

• So we have two equations:

∞∑t=0

(1

R

)t

ct = Ra0 +∞∑t=0

(1

R

)t

yt

u′(ct)

u′(ct+1)= βR

24 / 60

Solution with constant consumption (perfect smoothing)

• Let us impose r = ρ or equivalently βR = 1.

I This is a sensible assumption: in GE, with homogeneous agents, the

steady state implies constant consumption.

• From EE it yields immediately ct = c ∀t.

• It implies that the LHS of the budget reads:

∞∑t=0

(1

R

)t

ct = c∞∑t=0

(1

R

)t

=c

1− 1R

= cR

r

• Therefore:

c =r

1 + r︸︷︷︸annuity value

[Ra0 +

∞∑t=0

(1

R

)t

yt

]︸︷︷︸≡wp

0 permanent wealth

≡ yp

• Consumption is constant since the value of wp0 is perfectly known at t = 0.

25 / 60

Permanent Income Hypothesis

• This is the permanent income hypothesis (PIH).

ct = c = yp =r

1 + rwp0 for any t

• Consumption equals permanent income, or the annuity value of wealth.

• An increase in current income has a very minor effect in current

consumption, because the increase is spent evenly during all lifetime.

∂ct∂yt

=∂ct∂yp

t

∂ypt

∂yt= 1

∂ypt

∂yt=

r

1 + r

where the last equality can be obtained from:

ypt =

r

1 + rwpt =

r

1 + r

Rat + yt +∞∑j=1

(1

1 + r

)j

yt+j

• Hence: ∂ct

∂yt= r

1+r < 1

26 / 60

Roadmap




I Full Solution






27 / 60

The Ricardian Equivalence (1)

• Imagine a government has to finance a given amount of spending.

• Does the timing of taxation matter?

• Does it matter whether

a) it sets taxes today or

b) it issues debt and collects taxes in the future to pay the debt back?

• Ricardian Equivalence: No, it does not matter.

28 / 60


• In the Keynesian model, temporary tax cuts can have a large stimulating

effect on demand.

• PIH suggests that a consumer will spread out the gains from a temporary tax

cut over a long horizon, and so the stimulus effect will be much smaller.

• Furthermore, PIH implies that a increase in income today that is

accompanied by a decrease of income in the future with the same present

value (⇒ the permanent income does not change) will not change

consumption decisions.

• Ricardian Equivalence is consistent with PIH.

• Let’s add a government to the previous model.

29 / 60


• The government collects lump sum taxes τt and issues debt Bt to finance a

given stream of public spending {gt}∞t=0.

• The government has its own budget constraint:

RBt + gt = Bt+1 + τt

• Substituting forward recursively:

Bt =1

R

(1

R(Bt+2 + τt−1 − gt−1) + τt − gt

)• After imposing a transversality condition, obtain the intertemporal budget:

∞∑j=0

gt+j

R j=∞∑j=0

τt+j

R j− RBt

The present value of government spending has to be equal to its paying

capacity, which is equal to the present value of taxes minus debts.

30 / 60


• The household budget constraint becomes:

ct + at+1 = (yt − τt) + Rat

The corresponding intertemporal budget constraint will be:

∞∑j=0

ct+j

R j= Rat +

∞∑j=0

yt+j − τt+j

R j

• If we impose constant consumption and we solve, we get:

ct = r

at +1

R

∞∑j=0

yt+j

R j−∞∑j=0

τt+j

R j

Timing of the taxes does not matter, only the NPV of taxes matters.

31 / 60

Ricardian Equivalence: Failure

1 Capital Market imperfections

I If there are borrowing constraints.

2 Finite lifetimes

I What if a cut of taxes today comes with higher taxes in 30 years time,

once a big fraction of today’s population is already dead?

3 Distortionary taxation

I If taxes are distortionary, then there exist an optimal intertemporal

profile of taxes (the solution to the Ramsey problem).

4 Income uncertainty

I If changes in government taxation alter the degree of uncertainty that

households face in their income.

32 / 60

Ricardian Equivalence: Why bother?

• The Ricardian equivalence has value as a reference point.

I It defines the conditions we need for the equivalence to hold and we can

easily work out cases when it does not.

I Having in mind the Ricardian equivalence (or PIH) when we look at the

real world is a good framework to understand reality.

• From the policy point of view, it is a quantitative issue.

• Are the departures quantitatively important?

• Two papers to read for next week in Box.

33 / 60

Roadmap




I Full Solution






34 / 60

Adding uncertainty

• Basic problem same as before.

• Two potential sources of exogenous uncertainty:

I Labor income: {yt}∞t=0 is stochastic (we focus on this now).

I Capital income {Rt}∞t=0 is stochastic (later with asset pricing).

• First we focus on stochastic labor income and assume a stationary process.

• Only asset that can be traded is a one period bond with return R.

I Incomplete financial markets: With a continuum of states, a one-period bond

is not enough to move wealth across future states.

• Interest rate Rt = R is still exogenous and constant over time.

I GE with aggregate uncertainty: Rt is stochastic and correlated to yt .

35 / 60

Stationary Income Process

• A process {yt}∞t=0 is strictly stationary if the joint probability distribution

does not change when shifted in time:

F (yt1 , yt2 , . . . , ytk ) = F (yt+τ , yt1+τ , . . . , ytk+τ ), ∀k, τ, t1, t2, ...

• It implies 2nd order stationarity: unconditional mean E[yt ] = µy , variance

V[yt ] = σ2y and autocovariance Cov [yt , yt+τ ] = γ(τ) are finite and invariant.

• For example, a stationary process for income could be an AR(1) structure:

yt = (1− ρ)y + ρyt−1 + εt , εt ∼ iid(0, σ2ε)

Homework: µy = y γ(τ) = ργ(τ − 1), γ(0) = σ2y =

σ2ε

1− ρ2

• Cases:

I If ρ = 1, yt is a random walk

I if ρ = 0, yt is iid

I if ρ ∈ (0, 1), yt is persistent

36 / 60

Consumption-saving problem with uncertainty

• The problem:

max{ct}∞t=0

E0

[ ∞∑t=0

βtu (ct)

]subject to

at+1 = Rat + yt − ct , a0 given

yt = (1− ρ)y + ρyt−1 + εt , εt ∼ iid(0, σ2ε)

limt→∞

E0

[ at+1

R t+1

]≥ 0

• Timing:

1 Start period t with net assets at .

2 Receive interest payments rat .

3 εt is realized, and household learns realization of yt

4 Consumption ct(at , yt) is decided.

5 Residual wealth at+1 is invested (if positive) or borrowed (if negative).

• For now we assume there are no borrowing constraints or they don’t bind.

37 / 60

Recursive problem with uncertainty

• Problem in recursive form (substituting budget constraint):

V (at , yt) = maxat+1

u (Rat + yt − at+1) + βEt [V (at+1, yt+1)|yt ]

• Note: yt is a state variable for two reasons:

I it determines disposable income Rat + ytI it provides information on future income when ρ 6= 0

• FOC & Envelope & Forward Envelope:

u′ (ct) = βEt

[∂V (at+1, yt+1)

∂at+1

]∂V (at , yt)

∂at= Ru′ (ct) =⇒ ∂V (at+1, yt+1)

∂at+1= Ru′ (ct+1)

• Euler equation:

u′ (ct) = βREt [u′ (ct+1)]

38 / 60

Policy

• The solution is a policy function ct = c (at , yt) that satisfies:

I Bellman equation:

c (at , yt) = arg max u [c (at , yt)] + βEt [V (at+1, yt+1)]

I Feasibility:

at+1 = Rat + yt − c (at , yt)

I Euler equation: ∀(at , yt)

u′ [c (at , yt)] = βREt

u′c

at+1︷︸︸︷Rat + yt − c (at , yt), yt+1

• We discuss the Euler equation first, then full solution.

39 / 60

Stochastic Euler Equation

• Euler Equation:

u′ (ct) = βREt [u′ (ct+1)]

• Implies that u′ (ct) is a sufficient statistic to predict u′ (ct+1) :

Et [u′ (ct+1)] =u′ (ct)

Rβ

• Let the expectation error be given by εut+1 ≡ u′ (ct+1)− Et [u′ (ct+1)], then

u′ (ct+1) =u′ (ct)

Rβ+ εut+1, Et

[εut+1

]= 0

• From rational expectations, εut+1 is independent of all the information

available at time t, and of ct in particular.

I Orthogonality conditions for GMM estimation:

Et

[εut+1xt

]= 0, where xt ∈ {ct , yt , ...}

40 / 60

Special case Rβ = 1

• Assume Rβ = 1u′ (ct+1) = u′ (ct) + εut+1

• Marginal utility is a random walk (a thus a martingale).

I Martingale: Et [xt+1] = xtI Submartingale: Et [xt+1] ≥ xtI Supermartingale: Et [xt+1] ≤ xt

• Recall that with certainty, Rβ = 1 meant perfect smoothing:

u′ (ct+1) = u′ (ct) ⇐⇒ ct+1 = ct = c

• Which preferences would imply that consumption itself is a martingale?

Et [ct+1] = ct

41 / 60

Roadmap



I Application: The Ricardian equivalence





42 / 60

Special case: Quadratic utility

• Consider a linear quadratic utility function

u (ct) = α0ct −α1

2c2t , αi > 0

• Provided that ct <α0

α1, the utility satisfies required conditions:

u′ (ct) = α0 − α1ct > 0 u′′(ct) = −α1 < 0

• Substituting in the Euler equation:

Et [u′ (ct+1)] =u′ (ct)

βR=⇒ Et [α0 − α1ct+1] =

α0 − α1ctRβ

• Solving for expected consumption at t + 1:

Et [ct+1] =1

βRct +

α0

α1

(1− 1

βR

)43 / 60

Special case: Quadratic utility and βR = 1

• By further assuming that βR = 1:

Et (ct+1) = ct

or alternatively

ct+1 = ct + εt+1, Et [εt+1] = 0

• Consumption is a random walk (martingale).

• By the Law of Iterated Expectations and the martingale property:

Et [ct+2] = Et [Et+1[ct+2]] = Et [ct+1] = ct

therefore in general for any future period:

Et [ct+j ] = ct , ∀j ≥ 0

i) The best predictor of future consumption is current consumption.

ii) No other information available at the current period can help predict

next periods’ consumption.

44 / 60


• To solve the model, we use both the information in the Euler equation

(Et [ct+j ] = ct ,∀j ≥ 0) and the budget constraint.

• Let us iterate forward the budget constraint from time t, use the No Ponzi

condition, and obtain:

∞∑j=0

1

R jEt [ct+j ] = Rat︸︷︷︸

Financial Wealth

+∞∑j=0

1

R jEt [yt+j ]︸︷︷︸

Ht Human Wealth

• Note: We used conditional expectations to deal with uncertain future

realizations of income and consumption as seen from time t.

• Note: Wealth is now stochastic (vs. certainty case where was a constant).

45 / 60


• Use the martingale property on the LHS:

∞∑j=0

1

R jEt [ct+j ] = ct

∞∑j=0

1

R j=

(R

R − 1

)ct =

(1 + r

r

)ct

and substitute back to get the optimal consumption:

ct =r

1 + r︸︷︷︸annuity value

Rat +∞∑j=0

1

R jEt [yt+j ]

︸︷︷︸

Wt = Wealth

≡ ypt

where we define permanent income yPt as the annuity value of total wealth.

• Note that permanent income is a random variable (vs. deterministic case).

• Result: If preferences are quadratic and βR = 1, then consumption

follows a martingale process and equals permanent income.

46 / 60

Certainty equivalence

• Recall that without uncertainty and βR = 1, the FOC was given by:

u′ (ct+1) = u′ (ct) ⇐⇒ ct+1 = ct

• And the solution was given by:

c = ct =r

1 + r

Rat +∞∑j=0

1

R jyt+j

• Certainty equivalence: to solve the stochastic problem one can

1 solve the deterministic problem

2 substitute conditional expectations of the forcing variables (yt+j) in

place of the variables themselves.

• Implication: Variance and higher moments of the income process DO NOT

matter for the determination of consumption, only its expectation.

• However, ex-post consumption is not constant because ypt is not constant.

47 / 60

Consumption and news

• Let us compute how consumption changes between periods:

∆ct+1 = ct+1 − ct = ct+1 − Et [ct+1] =r

1 + r(Wt+1 − Et [Wt+1])

• Now lets compute the innovation (unexpected change) in wealth:

Wt+1 − Et [Wt+1] = R(at+1 − Et [at+1]︸︷︷︸at+1 known at t

) +∞∑j=0

1

R j

Et+1[yt+1+j ]− Et [Et+1[yt+1+j ]]︸︷︷︸Et [yt+1+j ]

=

∞∑j=0

1

R j(Et+1 − Et) yt+1+j

• Therefore, change in consumption is given by:

∆ct+1 =r

1 + r

∞∑j=0

1


• Result: Changes in consumption are proportional to the revision in expected

earnings due to the new information (news) arriving in that same period.

48 / 60

Random Walks

• Change in consumption is given by:

∆ct+1 =r

1 + r

∞∑j=0

1


• From budget constraint, we find the change in assets:

∆at+1 = rat + yt − ct

= rat + yt −r

R

Rat +∞∑j=0

1

R jEt [yt+j ]

= yt −

r

R

∞∑j=0

1

R jEt [yt+j ]

• Since ∆ct+1 and ∆at+1 are stationary, it means that the original series have

unit roots (integrated of order 1).

• Implication: ct and at do not converge.

49 / 60

Cointegration

• Cointegration: Two non-stationary series of order 1 are said to be

cointegrated if there exists a linear combination that is stationary.

• Claim: at and ct are cointegrated series.

I From the solution of consumption, we obtain a linear combination of at

and ct that is stationary:

[1 −r

] [ctat

]= ct − rat =

r

R

∞∑j=0

1

R jEt [yt+j ]

I According to Granger and Engle, [1 − r ]′ is a cointegrating vector

that, when applied to the non-stationary vector process [ct at ]′, yields

a process that is asymptotically stationary.

I Note: The cointegration vector is not unique. [ 1r − 1] (and many

others) are also cointegrating vectors.

50 / 60

Roadmap








51 / 60

Propensity to consume and income persistence (1)

• Since consumption depends only on permanent income:

ct = ypt

• The marginal propensity to consume out of current income is given by:

∂ ct∂ yt

=∂ ct∂ yp

t︸︷︷︸=1

∂ ypt

∂ yt=∂ yp

t

∂ yt

• In the stochastic case, ∂ ypt

∂ ytdepends on the process governing income.

1 Permanent shocks: getting a better job

2 Transitory shocks: win lottery, become temporarily unemployed

3 Persistent shocks

52 / 60

Propensity to consume and income persistence (2)

• Let us assume that income follows an AR(1) process:

yt+1 = (1− ρ)y + ρyt + εt+1, εt+1 ∼ iid(0, σ2ε), ρ ∈ [0, 1]

• In the problem set, you will compute the policy using two methods.

c (at , yt) = rat +R − 1

R − ρyt

• Propensity to consume out of current income: ∂ ct∂ yt

= R−1R−ρ

(i) If shocks are permanent (ρ = 1), then ∂ ct∂ yt

= 1.

• Consumption responds one to one to changes in income.

(ii) If shocks are purely transitory (ρ = 0), then ∂ ct∂ yt

= r1+r

.

• Extra income is spread evenly among all periods.

(iii) Propensity to consume increases with persistence (ρ)

• Current income carries information about future realizations and the

update of permanent income is bigger.

53 / 60

Roadmap








54 / 60

Prediction I

Prediction I: ct summarizes all time t information to forecast ct+1.

• If βR = 1 and certainty equivalence, then consumption is a random walk:

Et (ct+1) = ct =⇒ ct+1 − ct = εt+1

• Time t variables, besides ct , don’t have additional predictive power on ∆ct+1.

• Hall (1978) tests Prediction I.

I Confirms random walk hypothesis when adding lagged consumption...

I ... but rejects it when lagged stock market prices appear significant.

• This evidence stimulated research in the direction of borrowing constraints

and precautionary saving.

55 / 60

Prediction II

Prediction II: Predictable changes in yt have no effect on ct .

• The optimal consumption rule is ct = ypt and ct+1 = yp

t+1, which implies:

ypt+1 − yp

t = ypt+1 − Et

(ypt+1

)= εt+1

• εt+1 is the shock (unpredictable component) to permanent income, it is the

only thing that should alter consumption.

• Campbell-Mankiw (NBER Macro Annuals, 1989) test Prediction II.

I Full rationality is too strong: many agents are not rational or constrained.

I Fraction λ of agents consume all income and 1− λ follow PIH:

ct − ct−1 = λ (yt − yt−1) + (1− λ) εt

I Recall that: εt = ypt − Et−1 (yp

t )

I Clearly if the PIH is correct λ should be zero.

56 / 60

Prediction II

• Implementation: treat εt as an error, and estimate:

∆ct = λ∆yt + υt

υt = (1− λ) εt

I What is the problem you face if you estimate λ with OLS?

I Instrument using ∆ct−j with j ≥ 1.

• They find a consistent estimate of λ around 0.4 - 0.5:

I Rejects PIH: consumption responds to lagged (or predictable )

information.

I A finding of λ > 0 is called excess sensitivity of consumption.

I Around 40-50% are rule of thumb consumers?

57 / 60

Prediction III

Prediction III: V[∆ct ] should be equal to V[∆ypt ].

• Consumption changes 1 to 1 with respect to unexpected changes in

permanent income, thus variances should be the same.

• If ∆ct = ∆ypt , then it must be that V(∆ct) = V(∆yp

t ).

• Empirical evidence shows that V[∆ct ] < V[∆ypt ].

I Consumption seems to be excessively smooth with respect to

unexpected changes in future income.

I This is called excess smoothness of consumption.

58 / 60

Prediction III

• Important: ypt is not observed, so V[∆yp

t ] is computed by estimating a

stochastic process for yt .

• Idea behind? If ρ = 1 (income is a random walk), then

∆yt = ηt =⇒ V[∆yp] = V[∆y ]

• In fact, the process that fits the data best is

∆yt = ρ∆yt−1 + ηt with ρ = 0.44

• In this case, ηt has more than a one-for-one permanent effect on income:

one shock ηt and one lagged effect ρηt .

• Hence the Deaton paradox: V[∆c] = V[∆yp] > V[∆y ]

I But data: V[∆c] < V[∆y ]

I Consumption is excessively smooth compared to what we would expect

from the model!!

59 / 60

Problems with PIH

• Empirical tests of PIH basically fail.

• Because of certainty equivalence, risk does not matter!

• We will introduce two instances where risk (volatility of income shocks) will

matter for consumption and saving decisions:

1 Prudence

2 Borrowing constraints (similar to investment irreversibility)

• Both elements will generate precautionary savings that increase with the

volatility of income process.

60 / 60

Documents

Advanced Macroeconomics II · A little bit of history (1) Keynesian consumption function (1936) I Consumption is a constant fraction of disposable income. C t = Y t; 2(0;1) I Assumption