Non Durable Consumption, Slides of Japanese

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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.

Two Period Model

• Two periods: 0, 1. The consumer maximises:

∑^ t=

t=

βtu(ct) = u(c 0 ) + βu(c 1 ) β ∈ [0, 1]

• At beginning of period 0, initial wealth A 0.
• Income in each periods: y 0 , y 1.
• Budget constraints: { A 1 = R 0 (A 0 + y 0 − c 0 ) A 2 = R 1 (A 1 + y 1 − c 1 )
• Additional constraints:
  • Consumption is non negative: c 0 , c 1 ≥ 0
  • The consumer does not die with debts: A 2 ≥ 0
• Combining both constraints:

A 2 /(R 1 R 0 ) + c 1 /R 0 + c 0 = (A 0 + y 0 ) + y 1 /R 0

Optimal Consumption

• Maximisation with respect to c 0 and c 1 :

u′(c 0 ) = λ = βR 0 u′(c 1 )

• λ is the multiplier on the budget constraint.
• Choice of A 2 : λ = φ. So if λ > 0, φ > 0: it is not optimal to leave money after period 2, so A 2 = 0.

c 1 /R 0 + c 0 = A 0 + y 0 + y 1 /R 0 = W 0

• consumption decisions depends on life time wealth W 0.
• consumption decisions do not depend on timing of income: consumption smoothing.
• savings decisions depend on timing of income.

Portfolio Choice

• Multiple assets:
  • non stochastic asset with return Rs.
  • stochastic asset with return R˜r.
• Consumer can hold both assets in quantity ar^ and as.
• Consumer’s choice problem:

max ar^ ,as^ u(y 0 − ar^ − as) + E (^) R˜r βu( R˜rar^ + Rsas^ + y 1 ).

• First order conditions: { u′(y 0 − ar^ − as) = βRsE (^) R˜r u′( R˜rar^ + Rsas^ + y 1 ) u′(y 0 − ar^ − as) = βE (^) R˜r R˜ru′( R˜rar^ + Rsas^ + y 1 )

Hence

RsE (^) R˜r u′( R˜rar^ + Rsas^ + y 1 ) = E (^) R˜r R˜ru′( R˜rar^ + Rsas^ + y 1 )

• Implication:

Rs^ = R¯r^ +

cov[ R˜r, u′( R˜rar^ + Rsas^ + y 1 )] E (^) R˜r u′( R˜rar^ + Rsas^ + y 1 )

• if ar^ and as^ > 0 , cov[ R˜r, u′( R˜rar^ + Rsas^ + y 1 )] < 0 Hence R¯r^ > Rs

Borrowing Restrictions

• Consumption can not exceed income:

max c 0 ≤y 0 [u(c 0 ) + βu(R 0 (A 0 − y 0 − c 0 ) + y 1 )]

• Denote μ the multiplier on the borrowing constraint. First order condition:

u′(c 0 ) = βR 0 u′(R 0 (A 0 + y 0 − c 0 ) + y 1 ) + μ

• if μ = 0, constraint is not binding:

u′(c 0 ) = βR 0 u′(R 0 (A 0 + y 0 − c 0 ) + y 1 )

• if μ > 0, constraint is binding: c 0 = y 0

u′(y 0 ) > βR 0 u′(y 1 ).

• Implication:
  • Consumption depends on timing of income.

More on Quadratic Utility

• Combining the budget constraint and the Euler equation gives:

ct =

R − 1

R

[

Rat− 1 + Et

∑^ ∞

i=

yt+i Ri

]

R − 1

R

Wt

where Wt is the expected total future wealth.

• Consumption is a constant fraction of future wealth: consump- tion smoothing.
• Consumption does not depend on variance of income. (cer- tainty equivalence).
• Consumption and income:

∂ct ∂yt

R − 1

R

Et

∑^ ∞

i=

R−i^ ∂yt+i ∂yt

• 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 = −

∑^ ∞

j=

R−j^ Et∆yt+j

This is the saving-for-the-rainy-day formula.

Evidence from Data

• Hall (1978) uses a quadratic utility:

ct = βREtct+

If βR = 1, consumption is a random walk:

ct+1 = ct + εt+

• Consumption growth should be unrelated to any variable dated t.
  • Quarterly data for US non durable consumption.
  • Lagged stock market prices significantly predicts consump- tion 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 de- partures 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 We- ber (1993), Blundell et al. (1994), Attanasio and Browning (1995)

Portfolio Choice

• N assets available. Let si denote the share of asset i = 1, 2 , ...N
• Define consumption as: c = A −

i si. With this in mind, the Bellman equation is given by:

v(A, y, R− 1 ) = max si u(A −

i

si) + βER,y′|R− 1 ,yv(

i

Risi, y′, R)

• The first order condition for the optimization problem holds for i = 1, 2 , ..., N and is:

u′(c) = βER,y′|R− 1 ,yRiu′(c′) for i = 1, 2 , ..N

• Estimation (Hansen- Singleton 1982): Define εit+1(θ) as

εit+1(θ) ≡ βRit+1u′(ct+1) u′(ct)

− 1 , for i = 1, 2 , ..N

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

mT =

T

∑^ T

t=

(εit+1(θ)zjt )

• The GMM estimator is defined as the value of θ that minimizes

JT (θ) = mT (θ)′WT mT (θ).

Here WT is an N qxN q matrix that is used to weight the various moment restrictions.

Endogenous Labor Supply

• Agent chooses both savings and how much labor to supply.
• Value function:

v(A, w) = max A′,n

U (A + wn − (A′/R), n) + βEw′|wv(A′, w′)

• 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 − (A′/R), n is a function of (A,w,A’).

n = φ(A, w, A′)

• Substituting:

v(A, w) = max A′^ Z(A, A′, w) + βEw′|wv(A′, w′)

where

Z(A, A′, w) ≡ U (A + wϕ(A, w, A′) − (A′/R), ϕ(A, w, A′))

• MaCurdy (1981)
  • Uses the PSID.
  • Estimation in several steps: ∗ estimates the first order condition for labor supply. ∗ concentrate on intertemporal choice.

Borrowing Constraints

• Numerical solution:
  • by value function iterations:

v(x) = max 0 ≤c≤x u(c) + βEy′^ v(R(x − c) + y′)

∗ defining a grid over x. ∗ interpolating the value v(x′) ∗ iterating until convergence.

• by iteration on Euler equation. ∗ defining a grid over x. ∗ defining a function c(x).

u′(c(x)) = max{u′(x), βREu′(c(x′))}.

∗ finding the function c(x) which satisfies the Euler equa- tion.

Borrowing Constraints

Consumption and Liquidity Constraints: Optimal Consumption Rule

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 re- moved?
• Attanasio, Banks, Meghir and Weber (1999), Gourinchas and Parker (2002)

Consumption over the Life Cycle

• Denote Yt the income of the individual:

Yt = PtUt

Pt = GtPt− 1 Nt

• Ut is iid. With probability p, Ut = 0, with probability 1 − p, log Ut ∼ N (0, σ u^2 ).
• U (c, z) = v(Z)c^1 −ρ/(1 − ρ).
• Budget Constraint:

Wt+1 = (1 + r)(Wt + Yt − Ct)

• As u′(0, z) = −∞ and P (Yt = 0) 6 = 0, =⇒ no borrowing will ever happen.
• Cash on hand:

Xt = Wt + Yt Xt+1 = R(Xt − Ct) + Yt+

• Value Function:

Vt(Xt, Pt) = max Ct [u(Ct) + βEtVt+1(Xt+1, Pt+1)]

• Optimal Consumption (Euler Equation):

u′(Ct) = βREtu′(Ct+1)

• Denote xt = Xt/Pt and ct = Ct/Pt. The normalized cash-on- hand evolves as:

xt+1 = (xt − ct)

R

Gt+1Nt+

• Ut+