© February 20, 2012, Christopher D. Carroll CRRA-RateRisk

Consumption out of Risky Assets

Merton (1969) and Samuelson (1969) solve the optimization problem for a consumer who receives no labor income1 and whose only available financial asset has a risky return factor R which is lognormally distributed, 2 2 log Rt+1 ~ N (r - σr∕2, σ r) .

With market assets m , the dynamic budget constraint is:

mt+1 = (mt - ct)Rt+1. (1 )

Start with the standard Euler equation for consumption under CRRA utility:

[ ] ( ) - ρ 1 = β Et Rt+1 ct+1- (2 ) ct
and postulate a solution of the form ct = κmt :
[ ( ) ] κmt+1 - ρ 1 = β Et Rt+1 -------- κmt [ ( ) ] (mt - ct)Rt+1 - ρ = β Et Rt+1 ---------------- mt [ ( ) - ρ] (1 - κ)mtRt+1 = β Et Rt+1 ----------------- mt [ - ρ] = β Et Rt+1 ((1 - κ )Rt+1 ) - ρ [ 1- ρ] = β (1 - κ) Et R t+1 (1 - κ)ρ = β Et[R1t-+ ρ1] ( )1∕ρ 1- ρ (1 - κ ) = β Et [R t+1 ] ( )1∕ρ κ = 1 - β E [R1 - ρ] . (3 ) t t+1

Since logR1 - ρ = (1 - ρ) log R t+1 t+1 , fact ELogNormTimes implies that (using the definition ∙ exp (∙) ≡ e ) ,

E[R1-ρ]= exp [(1 - ρ )(r - σ2∕2 ) + (1 - ρ)2σ2 ∕2 ] (4) tt+1 r r = exp [(1 - ρ )r - (1 - ρ )(σ2r∕2 ) + (1 - ρ )(σ2r∕2 ) - ρ (1 - ρ)σ2r ∕2] 2 = exp [(1 - ρ )r - ρ(1 - ρ)σ r∕2]. (5)

Substituting in (3):

κ= 1 - β1∕ρ exp [ρ ((1 ∕ρ - 1)r - (1 - ρ)σ2 ∕2 )]1∕ρ [ r] = 1 - β1∕ρ exp (1 ∕ρ - 1)r - (1 - ρ)σ2 ∕2 . (6 ) r

Now use OverPlus and TaylorOne,

( ) 1∕ρ 1 1∕ρ β = ------- 1 + ϑ ≈ 1 - ρ- 1ϑ ≈ exp (- ρ - 1ϑ )
which hold if - 1 ρ ϑ is close to zero. Substituting into (6) and using ExpPlus and LogEps gives
- 1 2 κ ≈ 1 - (1 + ρ (r - ϑ ) - r + (ρ - 1)σr ∕2) - 1 2 = r - ρ (r - ϑ ) - (ρ - 1 )(σ r∕2) (7 )
which, when σ2r = 0 , reduces to the usual perfect foresight formula -1 κ=r-ρ(r - ϑ ) .

This equation implies the plausible result that as unavoidable uncertainty in the financial return goes up ( 2 σ r rises) the level of consumption falls (because ρ > 1 , so - (ρ - 1) which multiplies 2 σr is negative), reflecting the precautionary saving motive.2

The top figure plots the marginal propensity to consume as a function of the coefficient of relative risk aversion (for both the true MPC and the approximation derived above), under parameter values such that ϑ - r ≈ 0 so that a change in ρ does not affect the MPC through the intertemporal elasticity of substitution channel. As intuition would suggest, as consumers become more risk averse, they save more (the MPC is lower; that is, the plotted loci are downward-sloping).

The other way to see the precautionary effect is to examine the effect on the MPC of a change in risk. For a consumer with relative risk aversion of 3, the bottom figure shows that as the size of the risk increases, the MPC κ falls.


Figure 1: Relation Between MPC κ and Parameters
PIC
(a) Marginal Propensity to Consume Falls as Relative Risk Aversion ρ Rises
PIC
(b) Marginal Propensity to Consume Falls as Risk σ Rises

References

   CARROLL, CHRISTOPHER D., AND MILES S. KIMBALL (1996): “On the Concavity of the Consumption Function,” Econometrica, 64(4), 981–992, http://econ.jhu.edu/people/ccarroll/concavity.pdf.

   MERTON, ROBERT C. (1969): “Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case,” Review of Economics and Statistics, 50, 247–257.

   SAMUELSON, PAUL A. (1969): “Lifetime Portfolio Selection By Dynamic Stochastic Programming,” Review of Economics and Statistics, 51, 239–46.