gr5215 fall 2021 lec 11 Hall consumption
1
The Hall (1978) consumption
model
Miller GR5215
An even simpler T-period
consumption problem
2
Individual lives T periods, zero interest and
no discounting
U = u(C
t
) ′u >0
t=1
T
∑ ′′u <0 C t ≤ A0 + Yt t=1 T ∑ t=1 T ∑ ℓ = u(C t ) t=1 T ∑ +λ A0 + Yt t=1 T ∑ − Ct t=1 T ∑ ⎛ ⎝⎜ ⎞ ⎠⎟ A simpler T-period consumption problem (II) 3 Take FOC: ′u (Ct )= λ ℓ = u(C t ) t=1 T ∑ +λ A0 + Yt t=1 T ∑ − Ct t=1 T ∑ ⎛ ⎝⎜ ⎞ ⎠⎟ So MU is constant over time. But we’re not discounting, so this means C is constant over time, so: C t = 1 T (A0 + Yt t=1 T ∑ ) ∀t Permanent and transitory income 4 The RHS of above is called permanent income, the present value of income, divided by T. Consumption is perfectly smoothed and the timing of income doesn’t matter. Y trans =Y t −Y perm C t = 1 T (A0 + Yt t=1 T ∑ ) ∀t Define transitory income: Permanent and transitory income (II) 5 Consider an increase in Y1, what is the response in C1? From solution above: ΔC1 = 1 T ΔY1 So almost all of transitory income is saved (if T is big). Sometimes people summarize as: “You consume out of permanent income and save transitory income.” Version of the Permanent Income Hypothesis (PIH), Friedman (1957) Implies consumption should be stable and savings volatile Add uncertainty 6 Assume quadratic utility, following Hall (1978) This incorporates expected utility theory and adds the (strong) assumption that the same utility function can be used over both time and states of the world U = E (C t − a 2 C t 2) t=1 T ∑ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a>0
Budget constraint is the same as the certainty case
Find the FOC
7
Using the standard perturbation argument:
1−aC1 = E1 1−aCt⎡⎣ ⎤⎦ ∀t
Where E1 denotes the mathematical expectations
operator conditional on information available at
time 1. Thus:
C1 = E1 Ct⎡⎣ ⎤⎦ ∀t
The budget constraint will be met with equality, so
we can substitute it into the above
“Random walk” consumption
8
Take expectations of the budget constraint:
LHS of above is TC1, by the FOC, so:
This result is called certainty equivalence – you
consume the same amount as if you knew what
the future values of Y would be – need quadratic
utility for this as depends on linear MU
E1[Ct ]= A0 + E1[Yt ]
t=1
T
∑
t=1
T
∑
C1 =
1
T
A0 + E1[Yt ]
t=1
T
∑
⎛
⎝⎜
⎞
⎠⎟
“Random walk” consumption
(II)
9
Write:
where et is a random variable with zero
expectation. This is true for any random variable.
Now use the optimizing result that:
to get:
Ct = Et−1 [Ct ]+et
Et−1 [Ct ]=Ct−1
Ct =Ct−1 +et
“Consumption is a random walk”
(technically a martingale)
Hall’s empirical work
• Hall then tested the RW implication, by
regressing the change in aggregate
consumption on information available at
time t
• He only found (marginal) statistical
significance for stock market prices as a
predictor
• So claimed that RW consumption was a
good approximation of the US data
10
Consumption as a function of
information
11
From above:
C2 =
1
T −1
A1 + E2[Yt ]
t=2
T
∑
⎛
⎝⎜
⎞
⎠⎟
C2 =
1
T −1
A0 +Y1 −C1 + E2[Yt ]
t=2
T
∑
⎛
⎝⎜
⎞
⎠⎟
C2 =
1
T −1
A0 +Y1 −C1 + E1[Yt ]
t=2
T
∑ + E2[Yt ]
t=2
T
∑ − E1[Yt ]
t=2
T
∑
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜
⎞
⎠
⎟
Consumption as a function of
information (II)
12
C2 =
1
T −1
A0 +Y1 −C1 + E1[Yt ]
t=2
T
∑ + E2[Yt ]
t=2
T
∑ − E1[Yt ]
t=2
T
∑
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜
⎞
⎠
⎟
C2 =
1
T −1
TC1 −C1 + E2[Yt ]
t=2
T
∑ − E1[Yt ]
t=2
T
∑
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜
⎞
⎠
⎟
C2 =C1 +
1
T −1
E2[Yt ]
t=2
T
∑ − E1[Yt ]
t=2
T
∑
⎛
⎝⎜
⎞
⎠⎟
Where the expression in parentheses has the
interpretation of the revision of the expected
permanent income
Comments on information and
consumption
• The change in consumption depends on
a random (uncorrelated) shock and new
information about future income
• In the old Keynesian consumption
function a tax cut now increases
consumption, but in the Hall framework
a future tax cut announcement can
increase consumption
13
Empirical work on C, following
Hall
• If a stochastic process is specified for Y,
then can use the information revision
equation directly to test
• Lots of empirical work has been based
on this approach
• Starting with Flavin (1981) authors
investigated this equation and found
“excess sensitivity,” with C responding
“too much” to movements in Y
14
Empirical work on C, following
Hall (II)
Other authors have found C responding to
predictable movements in Y – depends on
the representation of the process for Y
One representation that approximates US
output:
15
ΔYt =0.2ΔYt−1 + εt
With this representation, lagged changes in
output have predictive power, for changes in
consumption
Empirical work on C, following
Hall (III)
• Finding rejections of the Hall model, many
authors looked for ways to explain
observed behavior
• Deaton noted that individual income
processes look very different from
aggregate processes; much more
variable, more mean reversion
• Zeldes (1989) and others, examined
“liquidity constraints” – limits on the
ability to borrow 16
Empirical work on C, following
Hall (IV)
• Deaton (1991) looked at the full
optimization problem for households
facing liquidity constraints
• Turns out, if constraints are important,
optimizing households will rarely face
them, because they will self-insure (save
more)
• This makes levels and stocks matter
17
Empirical work on C, following
Hall (V)
• Campbell and Mankiw (1989) showed that
Hall’s predictability test had low power
against reasonable alternatives
• They set up a model with two types of
consumers: Hall-type and “rule of thumb”
consumers who consume their current
income (see Romer 8.3)
• They estimate that aggregate data is
consistent with 50% “rule of thumb”
18
Empirical work on C, following
Hall (VI)
Hall answered back (1988) that there could
also be predictable movements in interest
rates that would break the RW implication
19
Recall that with CRRA utility:
C
t+1
C
t
=
1+ r
t+1
1+ ρ
⎛
⎝⎜
⎞
⎠⎟
1/θ
But direct empirical estimates of interest
rate sensitivity are small