$©$ February 7, 2021, Christopher D. Carroll CARAModelWithYRisk

Consumption with Constant Absolute Risk Aversion (CARA) Utility

Consider the optimization problem of a consumer with a constant absolute risk aversion instantaneous utility function $u (C ) = − (1∕α )e− αC$ implying $u ′(C ) = e− αC$ facing an interest rate that is constant at $r = R − 1$ .¹ The consumer’s optimization problem is

{ } ∑T max 𝔼 βs − tu (C ) {C}Tt t s s=t

(1)

subject to the constraints

B = (M − C )R t+1 t t Mt+1 = Bt+1 + Yt+1

(2)

where $Yt+1$ is the consumer’s idiosyncratic income, which exhibits a random-walk deviation from an exogenously-growing trend:

¯P = Γ P¯ t+1 t Yt+1 = P¯t+1 + Pt+1 Pt+1 = Pt + Ψt+1.

Bellman’s equation for this problem is

¯ ¯ Vt(Mt, Pt, Pt) = max T u (Ct) + 𝔼t[βVt+1 (Mt+1, Pt+1, Pt+1 )]. {C }t

(3)

The ﬁrst order condition (FOC) for the CARA utility problem is

u′(C ) = R β 𝔼 [VM ] t t t+1

(4)

and the Envelope theorem tells us that

VM = Rβ 𝔼 [VM ]. t t t+1

(5)

In the perfect foresight version of the model in which $Ψt = 0 ∀ t$ , the Euler equation will be

u′(C ) = R βu ′(C ) t t+1 exp [− αCt ] = R β exp [− αCt+1 ] 1 = R β exp [− α (Ct+1 − Ct)] exp [α (Ct+1 − Ct )] = R β α (C − C ) = log R β t+1 t C = C + log (R β)1∕α. t+1 t

(6)

The $log (R β)1∕α$ term reﬂects the intertemporal substitution factor in consumption. Notice that intertemporal substitution takes the form of additive changes in the level of consumption in the CARA utility model, rather than multiplicative changes that aﬀect the growth rate of consumption, as in the CRRA model.

Now suppose we are interested in the case where permanent income shocks are distributed normally, $Ψ ∼ 𝒩 (0,σ2 ) t Ψ$ . Then it turns out that the process

C = C + log (R β)1∕α + ασ2 ∕2 + Ψ t+1 t Ψ t+1

(7)

satisﬁes the FOC under uncertainty:

1 = R β 𝔼 [exp [− α(c − c )]] t t+1 t 1 = R β 𝔼t [exp [− α(α σ2 ∕2 + Ψt+1 + (1∕ α) log(R β ) + ct − ct)]] 2 2 Ψ 1 = R β exp [− α σ Ψ∕2 ]𝔼t {exp [− α Ψt+1 ]} exp [− α(1 ∕α )log R β ] 2 2 2 2 − 1 1 = R β exp [− α (σ Ψ∕2 )]exp [α (σΨ ∕2 )]exp [log (R β ) ] 1 = R β (R β)− 1 1 = 1.

(8)

Deﬁne $κ = log (R β )1∕α + α σ2 ∕2 Ψ$ , so that (7) becomes:

Ct+1 = Ct + Ψt+1 + κ.

(9)

The expected present discounted value of consumption is

ℙ (C ) = C + (C + Ψ + κ )∕R + (C + Ψ + κ + Ψ + κ )∕R2 + ... t t t t+1 t t+1 t+2 𝔼t [ℙt (C )] = Ct + Ct ∕R + Ct ∕R2 + ...+ κ ∕R + 2κ ∕R2 + 3 κ∕R3 + ... ∞ − 1 − 2 ∑ i = Ct (1 + R + R + ...) + κ i∕R . i=1

Now we need the following fact:

Fact 1. If $R > 1$ , then $∞ ( ) ∑ i R i∕R = --------2- i=0 (R − 1)$

Thus, the expectation of the inﬁnite horizon PDV of consumption is:

( ) ( ) 1 κR 𝔼t[ℙt (C )] = Ct ---------- + ---------- . 1 − 1 ∕R (1 − R )2

(10)

Given the process for income described above, we have

ℙt (Y ) = Yt + Yt+1 ∕R + Yt+2 ∕R2 + ... = ¯Pt + Pt + (Γ ¯Pt + Pt+1 )∕R + (Γ 2 ¯Pt + Pt+2 )∕R2 + ... ¯ ( 2 ) = Pt 1 + Γ ∕R + (Γ ∕R ) + ... + 2 Pt + (Pt + Ψt+1 )∕R + (Pt + Ψt+1 + Ψt+2 )∕R + ... ( P¯ ) ∑∞ 𝔼t [ℙt (Y )] = -----t---- + Pt R − s 1 − Γ ∕R ( ) ( s=0 ) P¯t Pt = ---------- + ---------- 1 − Γ ∕R 1 − 1∕R

(11)

The IBC says

ℙ (C ) = B + ℙ (Y ), t t t

(12)

Because the intertemporal budget constraint must hold in every state of the world, the expectation of the PDV of consumption must equal current wealth plus the expectation of the PDV of income. Thus,

𝔼 [ℙ (C )] = B + 𝔼 [ℙ (Y )] ( t t ) t ( t t ) ( ) ( ) 1 P¯t Pt κR Ct ---------- = Bt + ---------- + ---------- − ---------- 1 − 1∕R 1 − Γ ∕R 1 − 1∕R (1 − R)2 ( r ) [ ( P¯ ) ( κR ) ] Ct = Pt + -- Bt + -----t---- − ---------- R 1 − Γ ∕R (1 − R )2 ( ) [ ( ) ] ( 1∕α 2 ) r- ---P¯t----- log-(Rβ-)----+--ασ-Ψ∕2-- = Pt + R Bt + 1 − Γ ∕R − r (1 − R)2

The $Pt$ term reﬂects the consumer’s idiosyncratic level of permanent income, which has no systematic growth (or decline). The next term reﬂects the MPC out of total ‘certain’ wealth, human and nonhuman. The ﬁnal term reﬂects the combination of the intertemporal substitution motive (in the $log (R β )1∕α$ term) and the precautionary motive in the $2 α σ Ψ$ term, as is evident from the fact that it equals zero if either there is no precautionary motive ( $α = 0$ ) or there is no uncertainty $2 σ Ψ = 0$ .

Note some peculiar aspects of this solution. First, observe that, marginally, the consumer spends exactly the interest income on capital, $dCt ∕dBt = r∕R$ . The reason this is peculiar is that the MPC out of capital does not depend on how impatient the consumer is. Impatience is reﬂected in the change in consumption over time, but not in the level of consumption except as that is aﬀected by the budget constraint.

Second, notice that the eﬀect of income uncertainty on saving is the same in absolute dollars regardless of the level of resources or permanent income.

References

Caballero, Ricardo J. (1990): “Consumption Puzzles and Precautionary Savings,” Journal of Monetary Economics, 25, 113–136, http://ideas.repec.org/p/clu/wpaper/1988_05.html.