Non Durable Consumption
Motivation
• Consumption is large share of GDP (about two thirds).
• Utility and welfare depends to a large extent on consumption.
• Consumption is linked to savings. Important macro-economic
variable.
– savings determines investment and growth.
– capital markets.
• Important to understand how consumption is linked to:
– income.
– income variability.
– labor supply.
– fertility.
– institutional features, retirement.
MACT1 2003-2004. I
Stylized Facts
Mean and standard deviations (annual rates of growth)
US
UK
Mean St. dev. Mean St. dev
Disposable income
0.032 0.025 0.026 0.026
Nondurable consumption 0.023 0.018 0.017 0.021
Durable expenditure
0.048 0.069 0.043 0.112
• Non durable consumption is smoother than income.
• Durable consumption is very volatile.
MACT1 2003-2004. I
Two Period Model
• Two periods: 0, 1. The consumer maximises:
t=1
X
β t u(ct ) = u(c0 ) + βu(c1 )
t=0
β ∈ [0, 1]
• At beginning of period 0, initial wealth A0 .
• Income in each periods: y0 , y1 .
• Budget constraints:
½
A1 = R0 (A0 + y0 − c0 )
A2 = R1 (A1 + y1 − c1 )
• Additional constraints:
– Consumption is non negative: c0 , c1 ≥ 0
– The consumer does not die with debts: A2 ≥ 0
• Combining both constraints:
A2 /(R1 R0 ) + c1 /R0 + c0 = (A0 + y0 ) + y1 /R0
MACT1 2003-2004. I
Optimal Consumption
• Maximisation with respect to c0 and c1 :
u0 (c0 ) = λ = βR0 u0 (c1 )
• λ is the multiplier on the budget constraint.
• Choice of A2 : λ = φ. So if λ > 0, φ > 0: it is not optimal to
leave money after period 2, so A2 = 0.
c1 /R0 + c0 = A0 + y0 + y1 /R0
= W0
– consumption decisions depends on life time wealth W0 .
– consumption decisions do not depend on timing of income:
consumption smoothing.
– savings decisions depend on timing of income.
MACT1 2003-2004. I
Stochastic Income
• In period 0, y1 is not known. Program of the agent:
max Ey1 |y0 [u(c0 ) + βu(R0 (A0 + y0 − c0 ) + y1 )]
c0
• We assume:
y1 = ρy0 + ε1
|ρ| ∈ [0, 1]
• Euler equation:
u0 (c0 ) = Ey1 |y0 βR0 u0 (R0 (A0 + y0 − c0 ) + y1 ).
• Example: u() is quadratic and βR0 = 1:
c0 = Ey1 |y0 c1 = R0 (A0 + y0 − c0 ) + Ey1 |y0 y1 .
• Solving for c0 and calculating Ey1 |y0 y1 yields:
c0 =
ρy0
R0 A 0
(R0 + ρ)
R0 (A0 + y0 )
+
=
+ y0
.
(1 + R0 )
(1 + R0 ) (1 + R0 )
(1 + R0 )
• Two sources of variation in consumption:
– variations in y0 affects life time wealth.
– variations in y0 bring information on future income.
(R0 + ρ)
∂c0
.
=
∂y0
(1 + R0 )
MACT1 2003-2004. I
Portfolio Choice
• Multiple assets:
– non stochastic asset with return Rs .
– stochastic asset with return R̃r .
• Consumer can hold both assets in quantity ar and as .
• Consumer’s choice problem:
max
u(y0 − ar − as ) + ER̃r βu(R̃r ar + Rs as + y1 ).
r s
a ,a
• First order conditions:
½ 0
u (y0 − ar − as ) = βRs ER̃r u0 (R̃r ar + Rs as + y1 )
u0 (y0 − ar − as ) = βER̃r R̃r u0 (R̃r ar + Rs as + y1 )
Hence
Rs ER̃r u0 (R̃r ar + Rs as + y1 ) = ER̃r R̃r u0 (R̃r ar + Rs as + y1 )
• Implication:
cov[R̃r , u0 (R̃r ar + Rs as + y1 )]
R = R̄ +
ER̃r u0 (R̃r ar + Rs as + y1 )
s
r
• if ar and as > 0 , cov[R̃r , u0 (R̃r ar + Rs as + y1 )] < 0
Hence
R̄r > Rs
MACT1 2003-2004. I
Borrowing Restrictions
• Consumption can not exceed income:
max[u(c0 ) + βu(R0 (A0 − y0 − c0 ) + y1 )]
c0 ≤y0
• Denote µ the multiplier on the borrowing constraint. First
order condition:
u0 (c0 ) = βR0 u0 (R0 (A0 + y0 − c0 ) + y1 ) + µ
– if µ = 0, constraint is not binding:
u0 (c0 ) = βR0 u0 (R0 (A0 + y0 − c0 ) + y1 )
– if µ > 0, constraint is binding: c0 = y0
u0 (y0 ) > βR0 u0 (y1 ).
• Implication:
– Consumption depends on timing of income.
MACT1 2003-2004. I
Infinite Horizon
• Bellman’s equation:
v(A, y, R) = max u(c) + βEy0 ,R0 |R,y v(A0 , y 0 , R0 )
c
A0 = R0 (A + y − c).
• If only income is stochastic:
v(A, y) = max u(c) + βEy0 |y v(A0 , y 0 )
c
• Euler equation:
u0 (c) = βREy0 |y u0 (c0 )
• Special cases:
– Quadratic utility:
c = βREy0 |y c0
1−γ
– CRRA utility function: u(c) = 1c− γ
c0
1 = βREy0 |y ( )−γ
c
MACT1 2003-2004. I
More on Quadratic Utility
• Combining the budget constraint and the Euler equation gives:
"
#
∞
X
R−1
yt+i
ct =
Rat−1 + Et
R
Ri
i=0
R−1
Wt
=
R
where Wt is the expected total future wealth.
• Consumption is a constant fraction of future wealth: consumption smoothing.
• Consumption does not depend on variance of income. (certainty equivalence).
• Consumption and income:
∞
∂ct
R − 1 X −i ∂yt+i
R
Et
=
∂yt
R
∂yt
i=0
– if income is i.i.d., then consumption does not depend on
current income.
– if income is persistent, consumption and current income
are linked.
• Further manipulation gives an expression for savings:
st = at − at−1 = −
∞
X
R−j Et ∆yt+j
j=1
This is the saving-for-the-rainy-day formula.
MACT1 2003-2004. I
Evidence from Data
• Hall (1978) uses a quadratic utility:
ct = βREt ct+1
If βR = 1, consumption is a random walk:
ct+1 = ct + εt+1
• Consumption growth should be unrelated to any variable dated t.
– Quarterly data for US non durable consumption.
– Lagged stock market prices significantly predicts consumption growth.
– Rejects the PIH model.
• Flavin (1981) allow for a general ARMA process for income.
Rejects the PIH model because current consumption appears
to be too related to current income.
• The sensitivity of consumption to income has led to many departures from the simple quadratic model:
– Liquidity constraints: Hall and Mishkin (1982),Zeldes (1989),
Campbell and Mankiw (1989)
– Introduction of durable goods: Hayashi (1985).
– Bounded rationality and heterogeneity: Caballero (1992).
– Effect of demographics on preferences: Attanasio and Weber (1993), Blundell et al. (1994), Attanasio and Browning
(1995)
MACT1 2003-2004. I
More on Non Quadratic Utility
• With quadratic utility: certainty equivalence.
• If marginal utility is non linear, variance of income will matter:
precautionary motives for saving.
• Taylor approximation of Euler equation:
·³
h
i
´¸
³
´
ct+1 − ct
ct+1 − ct 2
ρ
1
1
=
+ o(ct+1 )
1−
+ 2 Et
Et
ct
ct
ζ
βR
00
with ζ = −c uu0
000
ρ = −c uu00
– ζ is the coefficient of relative risk aversion.
– ρ is the coefficient of relative prudence.
• The growth in consumption depends on two terms:
– the interest rate.
– the variance of consumption. Higher variance, higher consumption growth, because higher savings.
• This linearized Euler equation can be estimated on data.
MACT1 2003-2004. I
Portfolio Choice
• N assets available. Let si denote the share of asset i = 1, 2, ...N
P
• Define consumption as: c = A − i si .
With this in mind, the Bellman equation is given by:
v(A, y, R−1 ) = max u(A −
si
X
si ) + βE
i
R,y 0 |R
−1 ,y
v(
X
Ri si , y 0 , R)
i
• The first order condition for the optimization problem holds
for i = 1, 2, ..., N and is:
u0 (c) = βER,y0 |R−1 ,y Ri u0 (c0 ) for i = 1, 2, ..N
• Estimation (Hansen- Singleton 1982): Define εit+1 (θ) as
εit+1 (θ)
βRit+1 u0 (ct+1 )
≡
− 1, for i = 1, 2, ..N
u0 (ct )
εit+1 (θ) is a measure of the deviation for an asset i.
• Orthogonality restrictions:
– Et (εit+1 (θ)) = 0 for i = 1, 2, ..N.
– E(εit+1 (θ) ⊗ zt ) = 0 for i = 1, 2, ..N.
• Let
T
1X i
mT =
(εt+1 (θ)ztj )
T t=1
• The GMM estimator is defined as the value of θ that minimizes
JT (θ) = mT (θ)0 WT mT (θ).
Here WT is an N qxN q matrix that is used to weight the various
moment restrictions.
MACT1 2003-2004. I
Endogenous Labor Supply
• Agent chooses both savings and how much labor to supply.
• Value function:
v(A, w) = max
U (A + wn − (A0 /R), n) + βEw0 |w v(A0 , w0 )
0
A ,n
• Note that:
– savings decision is dynamic.
– labor supply decision is static.
• First order condition for labor supply:
wUc (c, n) = −Un (c, n).
• using c = A + wn − (A0 /R), n is a function of (A,w,A’).
n = φ(A, w, A0 )
• Substituting:
0
0
0
0 |w v(A , w )
v(A, w) = max
Z(A,
A
,
w)
+
βE
w
0
A
where
Z(A, A0 , w) ≡ U (A + wϕ(A, w, A0 ) − (A0 /R), ϕ(A, w, A0 ))
• MaCurdy (1981)
– Uses the PSID.
– Estimation in several steps:
∗ estimates the first order condition for labor supply.
∗ concentrate on intertemporal choice.
MACT1 2003-2004. I
Borrowing Constraints
• Deaton (1991)
– Infinitely lived agent.
– income uncertainty.
– no borrowing.
– Agent chooses consumption/ savings to maximize total
flow of utility.
• Let x = A + y be the cash-on-hand.
A0 = R(x − c)
• Suppose y is iid, the value function can be written:
v(x) = max u(c) + βEy0 v(R(x − c) + y 0 )
0≤c≤x
• borrowing constraint: c ≤ x.
• Euler equation:
u0 (c) = max{u0 (x), βREu0 (c0 )}.
MACT1 2003-2004. I
Borrowing Constraints
• Numerical solution:
– by value function iterations:
v(x) = max u(c) + βEy0 v(R(x − c) + y 0 )
0≤c≤x
∗ defining a grid over x.
∗ interpolating the value v(x0 )
∗ iterating until convergence.
– by iteration on Euler equation.
∗ defining a grid over x.
∗ defining a function c(x).
u0 (c(x)) = max{u0 (x), βREu0 (c(x0 ))}.
∗ finding the function c(x) which satisfies the Euler equation.
MACT1 2003-2004. I
Borrowing Constraints
Consumption and Liquidity Constraints: Optimal Consumption
Rule
MACT1 2003-2004. I
Borrowing Constraints
Simulations of Consumption and Assets with Serially Correlated
Income
MACT1 2003-2004. I
Consumption over the Life Cycle
• Understand the dynamics of consumption and savings.
• At least two different motives:
– precautionary motive as income uncertainty.
– retirement motive.
• How important are these two motives?
• How would savings decrease if income uncertainty were removed?
• Attanasio, Banks, Meghir and Weber (1999), Gourinchas and
Parker (2002)
MACT1 2003-2004. I
Consumption over the Life Cycle
• Denote Yt the income of the individual:
Yt = P t U t
Pt = Gt Pt−1 Nt
• Ut is iid. With probability p, Ut = 0, with probability 1 − p,
log Ut ∼ N (0, σu2 ).
• U (c, z) = v(Z)c1−ρ /(1 − ρ).
• Budget Constraint:
Wt+1 = (1 + r)(Wt + Yt − Ct )
• As u0 (0, z) = −∞ and P (Yt = 0) 6= 0, =⇒ no borrowing will
ever happen.
• Cash on hand:
Xt = W t + Y t
Xt+1 = R(Xt − Ct ) + Yt+1
• Value Function:
Vt (Xt , Pt ) = max [u(Ct ) + βEt Vt+1 (Xt+1 , Pt+1 )]
Ct
• Optimal Consumption (Euler Equation):
u0 (Ct ) = βREt u0 (Ct+1 )
• Denote xt = Xt /Pt and ct = Ct /Pt . The normalized cash-onhand evolves as:
xt+1 = (xt − ct )
MACT1 2003-2004. I
R
+ Ut+1
Gt+1 Nt+1
Consumption over the Life Cycle: Numerical Solution
• evaluate ct (x) at each point of the grid using:
Z Z
0
0
µ
µ
ct+1 (x − ct )
u (ct (x)) = βR(1 − p)
u
µ
µ
Z
+βRp u0 ct+1 (x − ct )
R
Gt+1 N
R
¶
+ U Gt+1 N
Gt+1 N
¶
¶
Gt+1 N dF (N )
¶
dF (N )dF (U )
– first term: expected value of future marginal utility conditional on zero income.
– second term: expected value of future marginal utility conditional on positive income.
MACT1 2003-2004. I
Consumption over the Life Cycle
• Estimation: Simulated Method of Moments.
• Unconditional mean of consumption
Z
ln Ct (θ) = ln Ct (x, P, θ)dFt (x, P, θ)
• Objective function:
It
S
1X
1X
ln Ct (Xts , Pts , θ)
g(θ) =
ln Cit −
It i=1
S s=1
• Criteria to Minimize:
g(θ)0 W g(θ)
MACT1 2003-2004. I
Consumption over the Life Cycle
Optimal Consumption Rule
MACT1 2003-2004. I
Consumption over the Life Cycle
Observed and Predicted Consumption Profiles
MACT1 2003-2004. I
Consumption over the Life Cycle
Observed Consumption and Income
MACT1 2003-2004. I
Durable Consumption
PIH and Durable Expenditures
• Budget constraint:
A0 = R(A + y − c − pe).
• Accumulation of durables:
D0 = D(1 − δ) + e
δ parameterizes the depreciation of the durable good.
• Bellman Equation:
V (A, D, y, p) = max
u(c, D) + βEy0 ,p0 |y,p V (A0 , D0 , y 0 , p0 )
0
0
D ,A
with
c = A + y − (A0 /R) − p(D 0 − (1 − δ)D)
• First order conditions:

 uc (c, D) = βREy0 ,p0 |y,p VA (A0 , D0 , y 0 )

uc (c, D)p = βEy0 ,p0 |y,p VD (A0 , D0 , y 0 ).
In both cases, these conditions can be interpreted as equating
the marginal costs of reducing either nondurable or durable
consumption in the current period with the marginal benefits
of increasing the (respective) state variables in the next period.

 uc (c, D) = βREy0 |y uc (c0 , D0 )

puc (c, D) = βEy0 ,p0 |y,p [uD (c0 , D0 ) + p0 (1 − δ)uc (c0 , D0 )]
MACT1 2003-2004. I
Quadratic Utility
• Mankiw (1982) studied the pattern of durable expenditure when
U (c, D0 ) is separable and quadratic.
• durable expenditures follows an ARMA(1,1) process given by:
et+1 = a0 + a1 et + εt+1 − (1 − δ)εt
where a1 = βR.
• Estimation:
Table 1: ARMA(1,1) Estimates on US and French Data
Specification
No trend
Linear trend
α1
δ
α1
δ
US durable expenditures
1.00(.03)
1.5 (.15)
0.76 (0.12) 1.42 (0.17)
US car registration
0.36(.29)
1.34 (.30) 0.33 (0.30) 1.35(0.31)
France durable expenditures 0.98 (0.04) 1.20 (0.2) 0.56 (0.24) 1.2 (0.36)
France car expenditures
0.97(0.06) 1.3 (0.2)
0.49 (0.28) 1.20 (0.32)
France car registrations
0.85 (0.13) 1.00 (0.26) 0.41 (0.4) 1.20 (0.41)
Notes: Annual data. For the US, source FRED database, 1959:11997:3. French data: source INSEE, 1970:1-1997:2. US registration: 1968-1995.
• Durable goods depreciate at a rate of 100%!
MACT1 2003-2004. I
Extensions
• Simple PIH model fails to match data. Number of extensions:
– adjustment costs.
– other sources of shocks.
– discrete choice model at individual level.
MACT1 2003-2004. I
Quadratic Adjustment Costs
• Bernanke 1985 goes beyond PIH formulation by adding in price
variations and costs of adjustment.
V (A, D, y, p) = max
u(c, D, D0 ) + βEy0 |y V (A0 , D0 , y 0 , p0 )
0
0
D ,A
with
a
d
1
u(ct , Dt , Dt+1 ) = − (c̄ − ct )2 − (D̄ − Dt )2 − (Dt+1 − Dt )2
2
2
2
• Model is estimated out of first order conditions.
• Model is rejected by the data.
MACT1 2003-2004. I
Non Convex Adjustment Costs
• Infrequent purchases at household level.
• [s, S] model of purchases. Bar-Ilan and Blinder (1992) and
Bar-Ilan and Blinder (1988).
Figure 3: [s,S]
MACT1 2003-2004. I
General Setting
• Non convexity arises for instance if resell price ps is less than
purchase price (normalized to 1).
• Three possible actions: buy, sell, do nothing.
V (A, D, y) = max(V b (A, D, y), V s (A, D, y), V i (A, D, y))
with:
V b (A, D, y) = maxe,A0 u(A + y − (A0 /R) − e, D) + βEy0 |y V (A0 , D(1 − δ) + e, y 0 )
V s (A, D, y) = maxs,A0 u(A + y − (A0 /R) + ps s, D) + βEy0 |y V (A0 , D(1 − δ) − s, y 0 )
V i (A, D, y) = maxA0 u(A + y − (A0 /R), D) + βEy0 |y V (A0 , D(1 − δ), y 0 )
• This is a complex problem:
– extensive margin: buy,sell / inaction.
– intensive margin: choice of e or s.
• Simpler problems have been analyzed by:
– Deaton and Laroque (1990): durable consumption and
portfolio choice.
– Eberly (1994) and Attanasio (2000) estimate an ad-hoc
version of the model.
– Caballero (1993) studies the aggregation of an [s, S] model.
– Adda and Cooper (2000) study a simplified version.
MACT1 2003-2004. I
Dynamic Discrete Choice Model
• Let (z, Z) denote the state variables, an idiosyncratic component and aggregate variables. z = y, Z = Y, p, ε.
• Consumer can either keep of scrap the durables.
Vi (z, Z) = max[Vik (z, Z), Vir (z, Z)]
with
Vik (z, Z) = u(si , y + Y, ε) + β(1 − δ)EVi+1 (z 0 , Z 0 ) +
βδ{EV1 (z 0 , Z 0 ) − u(s1 , y 0 + Y 0 , ε0 ) + u(s1 , y 0 + Y 0 − p0 + π, ε0 )}
and
Vir (z, Z) = u(s1 , y + Y − p + π, ε) + β(1 − δ)EV2 (z 0 , Z 0 ) +
βδ{EV1 (z 0 , Z 0 ) − u(s1 , y 0 + Y 0 , ε0 ) + u(s1 , y 0 + Y 0 − p0 + π, ε0 )}.
• utility function:
·
u(si , c) = i
−γ
ε(c/λ)1−ξ
+
1−ξ
¸
• Process for aggregate variables: VAR(1)
Yt = µY + ρY Y Yt−1 + ρY p pt−1 + uY t
pt = µp + ρpY Yt−1 + ρpp pt−1 + upt
εt = µε + ρεY Yt−1 + ρεp pt−1 + uεt
MACT1 2003-2004. I
Solving the model
• Model is solved by value function iterations.
• Let J(zt , Zt ; θ) represent the optimal scrapping age.
• Let hk (zt ;θ) represent the probability that a car of age k is
scrapped:
hk (zt , Zt ; θ) = δ
if k < J(zt , Zt ; θ)
hk (zt , Zt ; θ) = 1
otherwise
• At aggregate level:
H(Zt , θ) =
Z
h(zt , Zt , θ)φ(zt )dzt
• Aggregate expenditures are determined as:
X
St (Zt , θ) =
Hk (Zt , θ)ft (k)
k
where ff (k) is the amount of cars of age k in period t.
MACT1 2003-2004. I
Estimation
• Estimation by simulated method of moments.
Figure 4: Estimated Hazard Functions
•
MACT1 2003-2004. I
Labor
Introduction
• Some main topics:
• Distribution of wages.
• What are the determinants of wages?
– return to schooling.
– return to on-the-job training.
– heterogeneity in ability.
– unionization.
– discrimination.
• Labor market transitions:
– transition out of the labor force.
– transition into the labor force.
– job to job mobility.
MACT1 2003-2004. I
Topics
• Change in the return to schooling in the 70-90s.
• Increasing importance of unobserved components in explaining
wages.
• Policy evaluation:
– number of labor market policy directed towards low end of
labor market.
– effect of minimum wage?
– effect of unemployment policy measures?
– effect of in-work benefits.
MACT1 2003-2004. I
Labor and Dynamic Models
• Particular fruitful area.
• Some of early contributions in structural models are in labor:
– Dynamic problems. Should I accept a job today or wait to
find a better one?
– Important institutional features that have to be modelled.
e.g. timing of unemployment benefits are important. Need
to have a coherent model of behavior and institutional features.
– Importance of unobserved heterogeneity and selection. Example: best students go to university and earn a high wage.
How do we separate ability from the effect of education?
∗ either with some instruments in an IV approach.
∗ by writing explicitly the full model with heterogeneity
and endogenous education choice.
•
MACT1 2003-2004. I
A Simple Labor Search Model
• Agent starts as unemployed.
• The agent gets a job offer ω every period. ω is i.i.d.
• Agent decides whether to take the job or wait an additional
period.
– take the job: stays on the job forever.
– wait: unemployment benefits b.
• Bellman equation:
V (ω) = max
½
¾
u(ω)
, u(b) + βEω0 V (ω 0 ) .
1−β
• V (.) can be computed numerically using value function iterations.
MACT1 2003-2004. I
Reservation Wage
• Reservation wage: ω ∗ such that the agent is indifferent between
accepting and rejecting the offer:
u(ω ∗ )
= u(b) + βEω0 V (ω 0 )
1−β
• Denote F (.) the cumulative distribution of ω.
• Because ω is i.i.d., Eω0 V (ω 0 ) is a constant:
Z +∞
V (w)dF (w)
κ = EV (w) =
−∞
Z +∞
Z ω∗
V (ω)dF (ω) +
V (ω)dF (ω)
=
−∞
ω∗
Z ∞
u(w)
= F (w∗ ) (u(b) + βκ) +
dF (w)
w∗ 1 − β
• Reservation wage depends on:
– benefits b.
– discount factor β.
– Distribution of wage offers F (.)
– utility function.
ω ∗ = ω ∗ (θ)
MACT1 2003-2004. I
Estimation
• Suppose data on unemployment duration and accepted wages.
• Likelihood of accepting (any) wage:
Prob(accepting job, ω) = Prob(ω > ω ∗ (θ))
= 1 − F (ω ∗ (θ))
• Likelihood of accepting a job after t periods:
Li (θ) = F (ω ∗ (θ))t−1 (1 − F (ω ∗ (θ)))
• Likelihood of accepting a given wage:
Li (θ) =
f (ω)
I(ω > ω ∗ )
∗
1 − F (ω (θ)
• Likelihood of entire sample:
L(θ) =
Y
i
MACT1 2003-2004. I
Li (θ)
Extensions
• More realistic model of working: the agent can loose his job.
V (ω) = max{V W (ω), V U (b)}
with
V W (ω) = u(ω) + βδV U (b) + β(1 − δ)V W (ω)
V U (b) = u(b) + βEV (ω 0 )
• Unemployment benefits only last for 2 periods:
V W (ω) = u(ω) + βδV U (b, 1) + β(1 − δ)V W (ω)
 U
V (b, 1) = u(b) + βE max{V W (ω 0 ), V U (b, 2)}





V U (b, 2) = u(b) + βE max{V W (ω 0 ), V U (b, 3)}




 U
V (b, 3) = u(0) + βE max{V W (ω 0 ), V U (b, 3)}
MACT1 2003-2004. I
References
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