economics 2010c: lecture 3 the classical consumption modelsep 09, 2014 · 2 linearizing euler...
TRANSCRIPT
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Economics 2010c: Lecture 3The Classical Consumption Model
David Laibson
9/9/2014
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Outline:
1. Consumption: Basic model and early theories
2. Linearization of the Euler Equation
3. Empirical tests without “precautionary savings effects”
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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.
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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
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• 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.
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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
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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 =
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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:
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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
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It follows that
0() = ̃+10(+1) if 0() ≥ ̃+10(+1) if =
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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.
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• 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.
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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.
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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.
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• 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
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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+⎞⎠ ∀
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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.)
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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]
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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
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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).
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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: )
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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.
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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?
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∆ 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.
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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)
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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).
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• 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)