Wethen discussed some of the key features of this model

Spring Semester ’12-’13 Akila Weerapana

LECTURE 18: Consumption Under Uncertainty

I. INTRODUCTION

• In the last lecture we analyzed a very general model of consumption built on a dynamic optimization framework, allowing for interest rates, discount factors, inheritances, bequests, time-varying income streams and the inter-temporal elasticity of subsititution to all play a role.

• Wethen discussed some of the key features of this model. These include the ideas that con- sumers smooth out their consumption and, in a world with perfect information and diminish- ing marginal utility, would never choose a path that involves wild swings in consumption. We also showed that the relationship between the discount rate β and the return on assets (1 + r) determined whether the time-path of consumption was rising,falling or flat over time. We also showed that permanent increases in income have a much bigger impact on consumption than temporary income changes did, which suggests that temporary tax cuts are avery ineffective form of fiscal stimulus.

• Even though this model gave us several useful insights, the basic framework of complete certainty over the path of future income seems obviously unrealistic. So today’s lecture takes a look at solving the dynamic optimization problem faced by the consumer in an uncertain world. What we want to see is if the core intuition from the certainty model can flow through to the uncertainty case. It turns out that the results do not flow through except under some unrealistic assumptions.

II. ADDING UNCERTAINTY TO THE MODEL

• The model with certainty that we solved the last time, yielded a consumption equation of the form:

Ct= (1 + r − θ^σ)(θ^σ)^t

 A(1 + r)^T − A

(1 + r)^T − (θ^σ)^T + Y_P⁰ r

  (1 + r)^T − 1 (1 + r)^T − (θ^σ)^T



• We want to explore the implications of uncertainty. Given the complexity of the algebra in even the case without uncertainty, we can simplify things greatly by setting θ = 1 and assuming that A = A = 0. Recall that θ 6= 1 simply influenced the shape of the consumption path. Since the focus here is not on the shape of the path but on uncertainty, we can greatly simplify the analysis by setting θ = 1. The same applies for bequests and inheritances since they just raise or lower the consumption path.

• Imposing these conditions gives us a nice intuitive result that consumers will just consume their permanent income in every period.

C_t= Y_P⁰

• Let’s revisit the model without uncertainty that we derived in the previous lecture and con- sider the case where the consumer is unsure about the path of future income (and hence future consumption), the best she can do at time 0 is to make a plan to maximize her path of expected consumption

C0,C1,Cmax2,···CT −1

T −1

X

t=0

β^tE0U (Ct) subject to A_t+1= (1 + r)A_t+ Y_t− C_t for t = 0, 1, 2, · · · T − 1.

• Note that E_t[X_t+n] is our notation for the expectation of what X will be at some point t + n, based on the information we have at time t.

• The Bellman equation is Vt(At) = max

Ct

[U (Ct) + βEtVt+1(At+1)] where At+1= (1 + r)At+ Yt− C_t

• The FOC will be

U⁰(C_t) + βE_tV⁰(A_t+1)∂A_t+1

∂C_t = 0

⇒ U⁰(C_t) − βE_tV⁰(A_t+1) = 0

• The envelope theorem gives us the following equation

V⁰(A_t) = βE_tV⁰(A_t+1)∂A_t+1

∂At

⇒ V⁰(A_t) = β(1 + r)E_tV⁰(A_t+1)

• These are not as easy to combine as before because of the expectations term. Combining the two we get

V⁰(At) = (1 + r)U⁰(Ct)

• We can take expectations based on time t − 1 information to get E_t−1V⁰(A_t) = (1 + r)E_t−1U⁰(C_t))

• The first equation can be re-written lagged by one period as U⁰(C_t−1) = βE_t−1V⁰(A_t)

• Finally, then we can eliminate the V⁰(.) terms and get ^U0(C_β^t−1) = (1 + r)E_t−1U⁰(C_t) or U⁰(Ct−1) = β(1 + r)Et−1U⁰(Ct)

• This equation is more commonly written as U⁰(C_t) = β(1 + r)E_tU⁰(C_t+1) or using our θ = β(1 + r) notation as

U⁰(Ct) = θEtU⁰(Ct+1)

• So the Euler equation is essentially the same as before except the t + 1 terms enter in expec- tation rather than in actuality. Impose the θ = 1 assumption and we have

U⁰(Ct) = EtU⁰(Ct+1)

• What about the budget constraint? We can’t use the general method anymore because of the expectations term but we can use iteration.

A₁ = (1 + r)A₀+ Y₀− C₀ = A(1 + r) + Y₀− C₀

A₂ = (1 + r)A₁+ Y₁− C₁ = A(1 + r)²+ (Y₀− C₀)(1 + r) + (Y₁− C₁)

A₃ = (1 + r)A₂+ Y₂− C₂ = A(1 + r)³+ (Y₀− C₀)(1 + r)²+ (Y₁− C₁)(1 + r) + (Y₂− C₂) ... ...

AT ≡ A = A(1 + r)^T + (Y0− C₀)(1 + r)^{T −1}+ (Y1− C₁)(1 + r)^{T −2}+ (Y₂− C₂)(1 + r)^{T −3}+ · · · + (Y_{T −1}− C_{T −1})

A

(1 + r)^{T −1} = A(1 + r) + (Y0− C₀) +(Y1− C₁)

(1 + r) +(Y2− C₂)

(1 + r)² + · · · +(YT −1− C_{T −1}) (1 + r)^{T −1} A

(1 + r)^{T −1} = A(1 + r) +

T −1

X

t=0

(Y_t− C_t) (1 + r)^t

T −1

X

t=0

Ct

(1 + r)^t = A(1 + r) − A (1 + r)^{T −1} +

T −1

X

t=0

Yt

(1 + r)^t

• This simply states that the PDV of consumption has to equal the PDV of income. Using the assumption that bequests and inheritances are zero, we get

T −1

X

t=0

Ct

(1 + r)^t =

T −1

X

t=0

Yt

(1 + r)^t

• Finally, recall that we defined permanent income asPT −1 t=0

Y₀^P

(1+r)^t =PT −1 t=0

Yt

(1+r)^t. This implies that

T −1

X

t=0

C_t (1 + r)^t =

T −1

X

t=0

Y₀^P (1 + r)^t

⇒

T −1

X

t=0

E0Ct

(1 + r)^t =

T −1

X

t=0

E0Y₀^P (1 + r)^t

• So the system of equation that characterizes the solution in the uncertainty case looks very

very similar to the case with certainty

Certainty Case | Uncertainty Case

T −1

X

t=0

Ct

(1 + r)^t =

T −1

X

t=0

Y₀^P (1 + r)^t |

T −1

X

t=0

E0Ct

(1 + r)^t =

T −1

X

t=0

E0Y₀^P (1 + r)^t U⁰(C_t) = U⁰(C_t+1) | U⁰(C_t) = E_tU⁰(C_t+1)

III. QUADRATIC UTILITY AND CERTAINTY EQUIVALENCE

• So in the case of uncertainty a lot hinges upon whether marginal utility is linear or not. For the rest of today’s class, we will focus on the case where marginal utility is linear. If marginal utility is linear, then utility has to be quadratic. An example of such a utility function is one of the form U (C_t) = C_t−^aC₂^t2. In this case, U⁰(C_t) = 1 − aC_tand the Euler equation becomes 1 − aCt= Et(1 − aCt+1) which implies that Ct= EtCt+1

• In that case, we can derive the consumption path as

T −1

X

t=0

E0Ct

(1 + r)^t =

T −1

X

t=0

E0Y₀^P (1 + r)^t

T −1

X

t=0

C₀ (1 + r)^t =

T −1

X

t=0

E₀Y₀^P (1 + r)^t C₀

T −1

X

t=0

1 (1 + r)^t

!

= E₀Y₀^P

T −1

X

t=0

1 (1 + r)^t

!

C₀ = E₀Y₀^P

• This type of result should be compared to the certainty case where C₀ = Y₀^P. The scenario where the outcome under certainty is exactly the same as applying an expected value to the outcome under uncertainty is known as certainty equivalence. In this case, the crucial assumption required for certainty equivalence was that utility is quadratic.

• Finally, it is useful to get some intuition about the result we derived here. In the certainty case, consumption is equal in every period. Here, expected consumption is the same but clearly as time changes our expectations change and the path of consumption will change.

We can illustrate this crucial insight as follows.

• In the certainty case, under the assumptions given here, C₀ = C₁≡ Y₀^P so C₁− C₀ = 0.

• In the uncertainty case, under certainty equivalence, C₀= E₀C₁≡ E₀Y₀^P so E₀C₁− C₀= 0.

• But what about C₁−C₀in the uncertainty case? This is a lot more complicated. Think about someone making the decision to consumer from period 1 onwards, given that their period 0

consumption decision was made earlier. The consumption path will satisfy

T −1

X

t=1

E₁C_t (1 + r)^t−1 =

T −1

X

t=1

E₁Y_t

(1 + r)^t−1 + (1 + r)(Y₀− C₀)

T −1

X

t=1

E₁C_t (1 + r)^t−1 =

T −1

X

t=1

E₁Y₁^P

(1 + r)^t−1 + (1 + r)(Y₀− C₀) Using the fact that C1 = E1Ct+1 for t = 1, ..., T − 2

T −1

X

t=1

C1

(1 + r)^t−1 =

T −1

X

t=1

E1Y₁^P

(1 + r)^t−1 + (1 + r)(Y0− C₀) C₁

T −1

X

t=1

1 (1 + r)^t−1

!

= E₁Y₁^P

T −1

X

t=1

1 (1 + r)^t−1

!

+ (1 + r)(Y₀− C₀)

C₁

T −2

X

t=0

1 (1 + r)^t

!

= E₁Y₁^P

T −2

X

t=0

1 (1 + r)^t

!

+ (1 + r)(Y₀− C₀)

• Define R_{0,T −1}=

PT −1

t=0 1

(1+r)^t



and R_{0,T −2}=

PT −2

t=0 1

(1+r)^t

 . Then

C1R0,T −2 = E1Y₁^PR0,T −2+ (1 + r)(Y0− C₀) C₁ = E₁Y₁^P +

 1 + r R0,T −2



(Y₀− C₀)

• Therefore, we can calculate, using the fact that C₀ = E₀Y₀^P C₁− C₀ = E₁Y₁^P − E₀Y₀^P +

 1 + r R_{0,T −2}



(Y₀− C₀) C₁− C₀ = [E₁Y₁^P − E₀Y₁^P] + [E₀Y₁^P − E₀Y₀^P] +

 1 + r R_{0,T −2}



(Y₀− C₀) C1− C₀ = [E1Y₁^P − E₀Y₁^P] + E0(Y₁^P − Y₀^P) +

 1 + r R_{0,T −2}



(Y0− C₀)

• Now consider the relationship between Y₀^P and Y₁^P.

T −1

X

t=0

Y₀^P (1 + r)^t =

T −1

X

t=0

Y_t

(1 + r)^t ⇒ Y₀^PR_{0,T −1}=

T −1

X

t=0

Y_t (1 + r)^t

T −1

X

t=1

Y₁^P

(1 + r)^t−1 =

T −1

X

t=1

Y_t

(1 + r)^t−1 ⇒ Y₁^PR_{0,T −2}=

T −1

X

t=1

Y_t (1 + r)^t−1 Y₁^PR_{0,T −2} = (1 + r) Y₀^PR_{0,T −1}− Y₀

Y₁^P = (1 + r) R_{0,T −1} R_{0,T −2}



Y₀^P − Y₀

 1 + r R_{0,T −2}



Y₁^P − Y₀^P =  (1 + r)R_{0,T −1}− R_{0,T −2} R0,T −2



Y₀^P − Y₀

 1 + r R0,T −2



• Plugging into above, we get

C1− C₀ = [E1Y₁^P − E₀Y₁^P] + E0(Y₁^P − Y₀^P) +

 1 + r R_{0,T −2}



(Y0− C₀) C₁− C₀ = [E₁Y₁^P − E₀Y₁^P] + E₀ (1 + r)R_{0,T −1}− R_{0,T −2}

R0,T −2



Y₀^P − Y₀

 1 + r R0,T −2



+

 1 + r R0,T −2



(Y₀− C₀) C₁− C₀ = [E₁Y₁^P − E₀Y₁^P] + (1 + r)R_{0,T −1}− R_{0,T −2}

R0,T −2

 E₀Y₀^P



−

 1 + r R0,T −2

 (C₀) Since C₀= E₀Y₀^P, we have that

C₁− C₀ = [E₁Y₁^P − E₀Y₁^P] + (1 + r)R_{0,T −1}− R_{0,T −2}− (1 + r) R0,T −2

 E₀Y₀^P

• Finally, given that R_{0,T −1} = ¹⁻(_1+r¹ )^T

1−_1+r¹ ≡ _(1+r)^(1+r)_T−(1+r)^T−1_{T −1} and R0,T −2 = ¹⁻(_1+r¹ )^{T −1}

1−_1+r¹ ≡

(1+r)^{T −1}−1

(1+r)^{T −1}−(1+r)^{T −2}, we can show that (1 + r)R0,T −1− R_{0,T −2} = (1 + r)

 (1 + r)^T − 1 (1 + r)^T − (1 + r)^{T −1}



− (1 + r)^{T −1}− 1 (1 + r)^{T −1}− (1 + r)^{T −2}

=

 (1 + r)^T − 1 (1 + r)^{T −1}− (1 + r)^{T −2}



−

 (1 + r)^{T −1}− 1 (1 + r)^{T −1}− (1 + r)^{T −2}



=

 (1 + r)^T − (1 + r)^{T −1} (1 + r)^{T −1}− (1 + r)^{T −2}



≡ (1 + r)

Therefore, we have that

C₁− C₀ = E₁Y₁^P − E₀Y₁^P

the actual change in consumption is the revision of information about permanent income received between period 0 and period 1.