advanced macroeconomics ii · a little bit of history (1) keynesian consumption function (1936) i...
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Advanced Macroeconomics II
Lecture 5
Consumption: Permanent Income Hypothesis
Isaac Baley
UPF & Barcelona GSE
January 25, 2016
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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
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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.
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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
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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.
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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.
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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.
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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
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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.
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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
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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
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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.
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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”.
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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?
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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.
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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
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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.
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Euler Equation and Consumption Smoothing (2)
• 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
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Euler Equation and Consumption Smoothing (3)
• 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)
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Euler Equation and Consumption Smoothing (4)
• 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.
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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
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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
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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.
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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
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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
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.
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The Ricardian Equivalence (2)
• 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.
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The Ricardian Equivalence (3)
• 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.
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The Ricardian Equivalence (3)
• 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.
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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.
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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.
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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
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 .
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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
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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.
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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)]
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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.
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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 , ...}
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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
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Roadmap
1 Facts about aggregate and household level consumption
2 Consumption-savings problem with certainty
I Application: The Ricardian equivalence
3 Consumption-savings problem with uncertainty
I Special case: Quadratic utility
I Propensity to consume and income persistence
4 Empirical evidence
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.
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Special case: Quadratic utility and βR = 1
• 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).
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Special case: Quadratic utility and βR = 1
• 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.
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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.
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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
R j(Et+1 − Et) yt+1+j
• Result: Changes in consumption are proportional to the revision in expected
earnings due to the new information (news) arriving in that same period.
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Random Walks
• Change in consumption is given by:
∆ct+1 =r
1 + r
∞∑j=0
1
R j(Et+1 − Et) yt+1+j
• 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.
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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.
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Roadmap
1 Facts about aggregate and household level consumption
2 Consumption-savings problem with certainty
I Application: The Ricardian equivalence
3 Consumption-savings problem with uncertainty
I Special case: Quadratic utility
I Propensity to consume and income persistence
4 Empirical evidence
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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
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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.
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Roadmap
1 Facts about aggregate and household level consumption
2 Consumption-savings problem with certainty
I Application: The Ricardian equivalence
3 Consumption-savings problem with uncertainty
I Special case: Quadratic utility
I Propensity to consume and income persistence
4 Empirical evidence
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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.
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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.
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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?
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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.
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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!!
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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.
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