$©$ February 7, 2021, Christopher D. Carroll CRRA-RateRisk

Consumption out of Risky Assets

Consider a consumer with CRRA utility whose only available ﬁnancial asset has a risky return factor $ℜℜℜ$ which is lognormally distributed, $log ℜℜℜ ∼ 𝒩 (𝔯𝔯𝔯 − σ2 ∕2,σ2 ) t+1 r r$ .

With market assets $m$ , the dynamic budget constraint is:

m = (m − c )ℜℜℜ . t+1 t t t+1

(1)

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

[ ( ) ] ct+1 − ρ 1 = β 𝔼t ℜℜℜt+1 ----- ct

(2)

and postulate a solution of the form $ct = κmt$ :

[ ( ) ] κmt+1 − ρ 1 = β 𝔼t ℜℜℜt+1 -------- κmt [ ( ) ] (mt − ct)ℜℜℜt+1 − ρ = β 𝔼t ℜℜℜt+1 ----------------- mt [ ( ) − ρ] (1 − κ)mt ℜℜℜt+1 = β 𝔼t ℜℜℜt+1 ----------------- mt [ − ρ] = β 𝔼t ℜℜℜt+1 ((1 − κ )ℜℜℜt+1 ) − ρ [ 1− ρ] = β (1 − κ ) 𝔼t ℜℜℜ t+1 (1 − κ)ρ = β 𝔼t [ℜℜℜ1 t+−1 ρ] ( ) 1− ρ 1∕ρ (1 − κ) = β 𝔼t[ℜℜℜ t+1 ] ( )1 ∕ρ κ = 1 − β 𝔼 [ℜℜℜ1 − ρ] t t+1

which (ﬁnally) yields an exact formula for $κ$ :

( )1∕ρ κ = 1 − β 𝔼t[ℜℜℜ1t−+1ρ] .

(3)

Since $1− ρ log ℜℜℜ t+1 = (1 − ρ) log ℜℜℜt+1$ , fact $[ELogNormTimes ]$ implies that (using the deﬁnition $exp (∙) ≡ e∙)$ ,

1− ρ 2 2 2 𝔼t [ℜℜℜ t+1 ] = exp [(1 − ρ )(𝔯𝔯𝔯 − σ r∕2 ) + (1 − ρ) σ r∕2 ] = exp [(1 − ρ )𝔯𝔯𝔯 − (1 − ρ) (σ2∕2 ) + (1 − ρ)(σ2 ∕2 ) − ρ (1 − ρ )σ2∕2 ] r r r = exp [(1 − ρ )𝔯𝔯𝔯 − ρ (1 − ρ )σ2r ∕2].

Substituting in (3):

1∕ρ [ ( 2 ) ]1∕ρ κ = 1 − β exp ρ (1 ∕ρ − 1)𝔯𝔯𝔯 − (1 − ρ )σr ∕2 1∕ρ [ 2 ] = 1 − β exp (1 ∕ρ − 1)𝔯𝔯𝔯 − (1 − ρ )σr∕2 .

Now use $[OverPlus ]$ and $[TaylorOne ]$ ,

( )1 ∕ρ β1∕ρ = ---1--- 1 + 𝜗 − 1 ≈ 1 − ρ 𝜗 − 1 ≈ exp (− ρ 𝜗 )

which hold if $− 1 ρ 𝜗$ is close to zero. Substituting into (4) and using $[ExpPlus ]$ and $[LogEps ]$ gives

κ ≈ 1 − (1 + ρ − 1(𝔯𝔯𝔯 − 𝜗) − 𝔯𝔯𝔯 + (ρ − 1)σ2 ∕2 ) r = 𝔯𝔯𝔯 − ρ − 1(𝔯𝔯𝔯 − 𝜗) − (ρ − 1 )(σ2r∕2 )

which, when $σ2 = 0 r$ , reduces to the usual perfect foresight formula $− 1 κ = 𝔯𝔯𝔯 − ρ (𝔯𝔯𝔯 − 𝜗 )$ .

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 $σ2 r$ is negative). The reduction in consumption as risk increases reﬂects the precautionary saving motive.¹

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 $𝜗 − 𝔯𝔯𝔯 ≈ 0$ 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.

Relation Between MPC $κ$ and Parameters

Figure 1:Marginal Propensity to Consume Falls as Relative Risk Aversion $ρ$ Rises

Figure 2:Marginal Propensity to Consume Falls as Risk $σ$ Rises

References