Economics 2010c: Lecture 3The Classical Consumption Model

David Laibson

9/9/2014

Outline:

1. Consumption: Basic model and early theories

2. Linearization of the Euler Equation

3. Empirical tests without “precautionary savings effects”

1 Application: Consumption.

Sequence Problem (SP): Find () such that

(0) = sup{}∞=0

0

∞X=0

()

subject to a static budget constraint for consumption,

∈ Γ ()

and a dynamic budget constraint for assets,

+1 ∈ Γ³ ̃+1 ̃+1

´

Here is the vector of assets, is consumption, is the vector of financialasset returns, and is the vector of labor income.

For instance, consider the case where the only asset is cash-on-hand (so iscash-on-hand) and consumption is constrained to lie between 0 and Then,

∈ Γ () ≡ [0 ]

+1 ∈ Γ³ ̃+1 ̃+1

´≡ ̃+1( − ) + ̃+1

0 = 0

• We will always assume that ̃ is exogenous and iid.

— However, in the welfare state, ̃ is not independent of See Hubbard,Skinner, and Zeldes (1995).

• We will always assume that is concave (00 0 for all 0).

• We will usually assume lim↓0 0() =∞ so 0 as long as 0

— I’ll highlight exceptions to this rule.

1.1 Bellman Equation representation

• The state variable, is stochastic, so it is not directly chosen (rather adistribution for +1 is chosen at time ).

• It is more convenient to think about as the choice variable.

Bellman Equation:

() = sup∈[0]

{() + (+1) } ∀

+1 = ̃+1(− ) + ̃+1

0 = 0

1.2 Necessary Conditions

• First Order Condition:0() = ̃+10(+1) if 0 0() ≥ ̃+10(+1) if =

• Envelope Theorem: 0() = 0(). Prove this. What if = ?

• Euler Equation:

0() = ̃+10(+1) if 0 0() ≥ ̃+10(+1) if =

1.3 Perturbation intuition behind the Euler Equation:

• What is the cost of consuming dollars less today?

Utility loss today = · 0()

• What is the (discounted, expected) gain of consuming ̃+1 dollars moretomorrow?

Utility gain tomorrow = h³̃+1

´· 0(+1)

i

Let’s now rederive the Euler Equation:

1. Suppose 0() ̃+10(+1) Then cut by and raise +1 bỹ+1 to generate a net utility gain:

·h−0() + ̃+10(+1)

i 0

This perturbation is always possible on the equilibrium path, so:

0() ≥ ̃+10(+1)

2. Suppose 0() ̃+10(+1) then raise by and cut +1 bỹ+1 to generate a net utility gain:

·h0()− ̃+10(+1)

i 0

This perturbation is possible on the equilibrium path as long as so:

0() ≤ ̃+10(+1) as long as

It follows that

0() = ̃+10(+1) if 0() ≥ ̃+10(+1) if =

1.4 Important consumption models:

1.4.1 Life Cycle Hypothesis: Modigliani & Brumberg (1954)

• ̃ = = 1

• Perfect capital markets (and no moral hazard), so that future labor incomecan be exchanged for current capital.

• Bellman Equation:

() = sup≤

{() + (+1) } ∀

+1 = (− )

0 = ∞X=0

− e• Sometimes referred to as “eating a pie/cake problem.”

• Euler Equation implies,

0() = 0(+1) = 0(+1)

• Hence, consumption is constant.

Budget constraint:∞X=0

− = 0∞X=0

− eSubstitute Euler Equation, 0 = to find

∞X=0

− 0 = 0∞X=0

− eSo Euler Equation + budget constraint implies

0 =µ1− 1

¶⎛⎝0 ∞X=0

− e⎞⎠ ∀

Consumption is an annuity. What’s the value of your annuity?

Remark 1.1 Friedman’s Permanent Income Hypothesis (Friedman, 1957) ismuch like Modigliani’s Life-Cycle Hypothesis.

1.4.2 Certainty Equivalence Model: Hall (1978)

• ̃ = = 1

• Quadratic utility: () = − 22

— This admits negative consumption.

— And this does not imply that lim↓0 0() =∞

• Can’t sell claims to labor income.

• Bellman Equation:

() = sup{() + (+1) } ∀

+1 = (− ) + ̃+1

0 = 0

∞X=0

− ≤∞X=0

− e (BC)• Euler Equation implies, = +1 = + (consumption is a randomwalk w/o drift):

+1 = + +1

• So ∆+1 can not be predicted by any information available at time

Budget constraint at date :

∞X=0

− + = +∞X=1

− e+

∞X=0

− + = +∞X=1

− e+Substitute = + to find

∞X=0

− = +∞X=1

− e+So Euler Equation + budget constraint implies

=µ1− 1

¶⎛⎝ + ∞X=1

− e+⎞⎠ ∀

2 Linearizing Euler Equation

Recall Euler Equation:

0() = +10(+1)

Want to transform this equation so it is more amenable to empirical analysis.

Assume that +1 is known at time .

Assume is an isoelastic (i.e., constant relative risk aversion) utility function,

() =1− − 11−

(Aside: lim→11−−11− = ln . Important special case.)

Note that

0() = −

We can rewrite the Euler Equation as

− = +1

−+1

1 = exphln³+1

−+1

´i1 = exp [−+ +1 − ln(+1)]

where − ln = and ln+1 = +1

Since, ln(+1) = ln(+1)− ln() we write,

1 = exp [+1 − − ∆ ln +1]

Assume that ∆ ln +1 is conditionally normally distributed. So,

1 = exp∙+1 − − ∆ ln +1 +

1

22∆ ln +1

¸

Taking the natural log of both sides yields,

∆ ln +1 =1

(+1 − ) +

2∆ ln +1

3 Empirical tests without precautionary savings

effects

Recall Euler Equation:

0() = +10(+1)

We write the linearized Euler Equation in regression form:

∆ ln +1 =1

(+1 − ) +

2∆ ln +1 + +1

where +1 is orthogonal to any information known at date

The conditional variance term is often referred to as the “precautionary savingsterm,” (more on this later).

We sometimes (counterfactually) assume that ∆ ln +1 is constant (i.e.,independent of time). So the Euler Equation reduces to:

∆ ln +1 = constant+1

(+1 − ) + +1

When we replace the precautionary term with a constant, we are effectivelyignorning its effect (since it is no longer separately identified from the otherconstant term: )

Hundreds of papers have estimated a linearized Euler Equation:

∆ ln +1 = constant+1

+1 + + +1

The principal goals of these regressions are twofold:

1. Estimate 1 , the elasticity of intertemporal substitution (EIS) =∆ ln +1+1

For example, see Hall (1988).

• For this model, the EIS is the inverse of the CRRA.

2. Test the orthogonality restriction: {Ω ≡ information set at date } ⊥+1

• In other words, test the restriction that information available at time does not predict consumption growth in the following regression

∆ ln +1 = constant+1

+1 + + +1

• For example, does the date expectation of income growth, ∆ ln+1predict date + 1 consumption growth?

∆ ln +1 = constant+1

+1 + ∆ ln+1 + +1

∈ [01 08] so ∆ ln+1 covaries with ∆ ln +1 (e.g., Campbell andMankiw 1989, Shea 1995, Shapiro 2005, Parker and Broda 2014).

• Orthogonality restriction is violated: information at date predicts con-sumption growth from to + 1

• In other words, the assumptions (1) the Euler Equation is true, (2) theutility function is in the CRRA class, (3) the linearization is accurate, and(4) ∆ ln +1 is constant, are jointly rejected.

A note on Shea’s methodology (for estimating ∆ ln+1)

1. Assign respondents to unions with national or regional bargaining

• national: trucking, postal service, railroads

• regional: lumber in Pacific Northwest, shipping on East Coast

2. Assign respondents to dominant local employer

• automobile worker living in Genesee County, MI (GM’s Flint plant)

Why does expected income growth predict consumption growth?

• Leisure and consumption expenditure are substitutes (Heckman 1974, Ghezand Becker 1975, Aguiar and Hurst 2005, 2007)

• Work-based expenses

• Households support lots of dependents in mid-life when income is highest(Browning 1992, Attanasio 1995, Seshadri et al 2006)

• Households are liquidity constrained and impatient (Deaton 1991, Carroll1992, Laibson 1997).

• Some consumers use rules of thumb: = (Campbell and Mankiw1989, Thaler and Shefrin 1981)

• Welfare costs of smoothing are second-order (Cochrane 1989, Pischke1995, Browning and Crossley 2001)