Optimal monetary policy with incomplete markets
Optimal monetary policy
when asset markets are incomplete
R. Anton Braun1
1 University
2 Kyoto
Tomoyuki Nakajima2
of Tokyo, and CREI
University, and RIETI
December 19, 2008
Optimal monetary policy with incomplete markets
Outline
1
Introduction
2
Model
Individuals
Aggregation
Firms
Aggregate shocks
Government
3
Results
Permanent productivity shock
Temporary productivity shock
4
Conclusion
Optimal monetary policy with incomplete markets
Introduction
Outline
1
Introduction
2
Model
Individuals
Aggregation
Firms
Aggregate shocks
Government
3
Results
Permanent productivity shock
Temporary productivity shock
4
Conclusion
Optimal monetary policy with incomplete markets
Introduction
Inflation-output tradeoff in the representative-agent
framework
In the standard sticky price model, the optimal monetary
policy is approximately given by complete inflation
stabilization.
Schmitt-Grohé and Uribe (2007), etc.
Concerning the output-inflation tradeoff, the monetary
authority should place exclusive weight on the inflation
stabilization.
The welfare cost of business cycles is nil in the
representative-agent framework used in the standard New
Keynesian model.
Optimal monetary policy with incomplete markets
Introduction
Uninsured idiosyncratic shocks
Idiosyncratic income shocks are very persistent and their
variance fluctuate countercyclically.
Storesletten, Telmer and Yaron (2004), Meghir and
Pistaferri (2004), etc.
The existence of such idiosyncratic shocks may generate a
large welfare-cost of business cycles.
Krebs (2003), De Santis (2007), etc.
How does it affect optimal monetary policy? In particular,
how does it change the weight the monetary authority
should place on the inflation stabilization?
Optimal monetary policy with incomplete markets
Introduction
This paper
Individuals face uninsured idiosyncratic income shocks
with countercyclical variance.
The model is otherwise standard new Keynesian model
with:
monopolistic competition;
Calvo price setting;
capital accumulation.
Consider optimal monetary policy (Ramsey policy).
Optimal monetary policy with incomplete markets
Introduction
Main findings
Countercyclical idiosyncratic risk can generate a very large
welfare-cost of business cycles.
But it does not affect the inflation-output tradeoff much.
The optimal monetary policy is essentially characterized as
complete price-level stabilization.
Thus, the monetary authority should place almost exclusive
weight on the stabilization of inflation.
Optimal monetary policy with incomplete markets
Model
Outline
1
Introduction
2
Model
Individuals
Aggregation
Firms
Aggregate shocks
Government
3
Results
Permanent productivity shock
Temporary productivity shock
4
Conclusion
Optimal monetary policy with incomplete markets
Model
Composite good
Yt = aggregate output of a composite good:
1
Z
Yt =
0
1− 1
Yj,t ζ
!
dj
1
1− 1
ζ
which can be consumed or invested:
Yt = Ct + It
Pt = price index:
Z
Pt =
0
1
!
1−ζ
Pj,t
dj
1
1−ζ
Optimal monetary policy with incomplete markets
Model
Individuals
Preferences of individuals
A continuum of ex-ante identical individuals.
Preferences:
ui,0 = E0i
∞
X
t=0
βt
i1−γ
1 h θ
ci,t (1 − li,t )1−θ
1−γ
Let 1/γc = elasticity of intertemporal substitution of
consumption for a fixed level of leisure:
γc ≡ 1 − θ(1 − γ)
Optimal monetary policy with incomplete markets
Model
Individuals
Idiosyncratic shocks
Two assumptions for tractability
In general, with uninsured idiosyncratic shocks, the wealth
distribution, an infinite-dimensional object, must be
included in the state variable.
We circumvent this problem by assuming that
idiosyncratic shocks follow random walk processes;
idiosyncratic shocks affect both labor and capital income.
Optimal monetary policy with incomplete markets
Model
Individuals
Idiosyncratic shocks
Random walk with countercyclical variance
ηi,t = the idiosyncratic shock for individual i:
ln ηi,t = ln ηi,t−1 + ση,t η,i,t −
2
ση,t
2
where
η,i,t is i.i.d., and N(0, 1).
ση,t = variance of innovations to idiosyncratic shocks, which
is assumed to fluctuate countercyclically.
Optimal monetary policy with incomplete markets
Model
Individuals
Idiosyncratic shocks
Flow budget constraint
Assume that ηi,t affects i’s income in two ways.
ηi,t equals the productivity of individual i’s labor.
ηi,t also affects the return to savings of individual i.
The flow budget constraint of i is given by
ci,t + ki,t + si,t
ηi,t
=
Rk ,t ki,t−1 + Rs,t si,t−1 + ηi,t wt li,t
ηi,t−1
where ki,t = physical capital and si,t = value of shares.
Optimal monetary policy with incomplete markets
Model
Individuals
Idiosyncratic shocks
Remarks
The assumption that ηi,t also operates as a shock to the
return to individual savings is artificial, but ...
Without this assumption, the wealth distribution would have
to be included as a state variable.
With this assumption, the effect of the presence of
idiosyncratic shocks would be overemphasized.
Our finding is that the tradeoff faced by the monetary
authority is little affected by the presence of idiosyncratic
shocks.
Hence, dropping this assumption would strengthen our
result.
Optimal monetary policy with incomplete markets
Model
Aggregation
Associated representative-agent problem
Consider a representative-agent’s utility maximization
problem:
max U0 = E0
∞
X
t=0
βt
h
i1−γ
1
νt Ctθ (1 − Lt )1−θ
1−γ
subject to
Ct + Kt + St = Rk ,t Kt−1 + Rs,t St−1 + wt Lt
Here, νt is a preference shock defined by
"
#
t
X
1
2
νt ≡ exp
γc (γc − 1)
ση,s
2
s=0
=
1−γc
Et [ηi,t
]
Optimal monetary policy with incomplete markets
Model
Aggregation
Aggregation result
Proposition
Suppose that {Ct∗ , L∗t , Kt∗ , St∗ }∞
t=0 is a solution to the
representative agent’s problem. For each i ∈ [0, 1], let
∗
ci,t
= ηi,t Ct∗
∗
= L∗t
li,t
∗
ki,t
= ηi,t Kt∗
∗
si,t
= ηi,t St∗
∗ , l ∗ , k ∗ , s ∗ }∞ is a solution to the problem of
Then {ci,t
i,t i,t i,t t=0
individual i.
Optimal monetary policy with incomplete markets
Model
Aggregation
Remark 1
The utility of the representative agent is indeed the
cross-sectional average of individual utility:
U0 = E0 [ui,0 ]
Optimal monetary policy with incomplete markets
Model
Aggregation
Remark 2
How idiosyncratic shocks affect the aggregate economy
can be understood by looking at the “effective discount
factor”:
β̃t,t+1 ≡ β
νt+1
νt
= β exp
1
2
γc (γc − 1)ση,t+1
2
Thus
(
2
↑ ση,t+1
=⇒
↑ β̃t,t+1
if γc > 1
↓ β̃t,t+1
if γc < 1
Optimal monetary policy with incomplete markets
Model
Aggregation
Remark 3
The SDF used by individual i is
λi,t+1
λt+1 ηi,t+1 −γc
β
=β
λi,t
λt
ηi,t
λt+1
γc 2
=β
exp −γc ση,t+1 η,i,t+1 + ση,t+1
λt
2
It follows that individuals agree on the present value of the
profit stream of each firm.
In particular, they agree with the representative agent,
whose SDF is given by β
λt+1 νt+1
λt ν t .
Optimal monetary policy with incomplete markets
Model
Firms
Firms
Standard model with monopolistic competition and Calvo
price setting.
Production technology of firm j:
α 1−α
Yj,t = zt1−α Kj,t
Lj,t − Φt
where zt is aggregate productivity shock, and Φt is a fixed
cost of production.
Demand for variety j:
Yj,t =
Pj,t
Pt
−ζ
Yt
1 − ξ = probability of arriving an opportunity to change the
price of each variety.
Optimal monetary policy with incomplete markets
Model
Aggregate shocks
Aggregate shocks
Productivity shock is either permanent or temporary.
1
The case of permanent productivity shock:
ln zt = ln zt−1 + µ + σz z,t −
2
ση,t
= σ̄η2 + bσz z,t
σz2
2
Optimal monetary policy with incomplete markets
Model
Aggregate shocks
Aggregate shocks
Productivity shock is either permanent or temporary.
1
The case of permanent productivity shock:
ln zt = ln zt−1 + µ + σz z,t −
σz2
2
2
ση,t
= σ̄η2 + bσz z,t
2
The case of temporary productivity shock:
ln zt = ρz ln zt−1 + σz z,t −
2
ση,t
= σ̄η2 + b ln zt
σz2
2(1 + ρz )
Optimal monetary policy with incomplete markets
Model
Government
Government
Fiscal policy: no taxes, no debt, etc.
Monetary policy: Set the state-contingent path of the
inflation rate {πt }.
Two monetary policy regimes:
1
Ramsey regime: Set {πt } so as to maximize the ex ante
utility of individuals.
2
Inflation-targeting regime: Set πt = 1 at all times.
Optimal monetary policy with incomplete markets
Results
Outline
1
Introduction
2
Model
Individuals
Aggregation
Firms
Aggregate shocks
Government
3
Results
Permanent productivity shock
Temporary productivity shock
4
Conclusion
Optimal monetary policy with incomplete markets
Results
Experiments
Most parameters are calibrated following Boldrin,
Christiano and Fisher (2001) and Schmitt-Grohé and Uribe
(2007).
We compare the following cases:
γc = 0.7, 2;
b = 0, −0.8;
productivity shock is either permanent or temporary;
the monetary policy regime is either Ramsey or
inflation-targeting.
Optimal monetary policy with incomplete markets
Results
Welfare measures
∆bc = welfare cost of business cycles:
∞
X
β t ν̄t
t=0
i1−γ
1 h
((1 − ∆bc )C̄)θ (1 − L̄)1−θ
1−γ
= E−1
∞
X
β t νt
t=0
i1−γ
1 h rbc θ
1−θ
(Ct ) (1 − Lrbc
)
t
1−γ
∆inf = welfare cost of the inflation-targeting regime:
E−1
∞
X
β t νt
t=0
= E−1
i1−γ
1 h
1−θ
((1 − ∆inf )Ctram )θ (1 − Lram
)
t
1−γ
∞
X
t=0
β t νt
i1−γ
1 h inf θ
1−θ
(Ct ) (1 − Linf
)
t
1−γ
Optimal monetary policy with incomplete markets
Results
Permanent productivity shock
Permanent productivity shock
Welfare costs of business cycles and the inflation-targeting regime
γc
0.7
0.7
2
2
b
0
-0.8
0
-0.8
∆bc (%)
-0.8191
-1.2983
2.0938
7.3301
∆inf (%)
0.0000
0.0000
0.0002
0.0006
Optimal monetary policy with incomplete markets
Results
Permanent productivity shock
Permanent productivity shock
Impulse responses when γc = 0.7 and b = 0.
output growth
1.8
0.45
1.6
consumption growth
4
0.35
1.2
0.3
1
3
0.25
0.8
2
0.2
0.6
0.15
0.4
0.2
0.1
0
0.05
-0.2
investment growth
5
0.4
1.4
1
0
0
5
10
15
20
labor
1
0
5
0.9
0
x 10
5
10
15
20
15
20
-1
0
5
10
15
20
inflation
-3
4
0.8
3
0.7
0.6
2
0.5
1
0.4
0
0.3
0.2
0
5
10
15
20
-1
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Permanent productivity shock
Permanent productivity shock
Impulse responses when γc = 0.7 and b = −0.8.
output growth
1.8
0.45
1.6
consumption growth
4
0.35
1.2
0.3
1
3
0.25
0.8
2
0.2
0.6
0.15
0.4
0.2
0.1
0
0.05
-0.2
investment growth
5
0.4
1.4
1
0
0
5
10
15
20
labor
1
0
5
0.9
0
x 10
5
10
15
20
15
20
-1
0
5
10
15
20
inflation
-3
4
0.8
3
0.7
0.6
2
0.5
1
0.4
0
0.3
0.2
0
5
10
15
20
-1
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Permanent productivity shock
Permanent productivity shock
Impulse responses when γc = 2 and b = 0.
output growth
1.4
1
1.2
1
consumption growth
1.4
0.8
1.2
0.7
1
0.6
0.8
investment growth
1.6
0.9
0.8
0.5
0.6
0.6
0.4
0.4
0.3
0.4
0.2
0.2
0.2
0
0
0.1
0
5
10
15
20
labor
0.21
0
18
0.2
16
0.19
14
0
x 10
5
10
15
20
15
20
-0.2
0
5
10
15
20
inflation
-4
12
0.18
10
0.17
8
0.16
6
0.15
4
0.14
2
0.13
0.12
0
0
5
10
15
20
-2
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Permanent productivity shock
Permanent productivity shock
Impulse responses when γc = 2 and b = −0.8.
output growth
1.4
1
1.2
1
consumption growth
1.4
0.8
1.2
0.7
1
0.6
0.8
investment growth
1.6
0.9
0.8
0.5
0.6
0.6
0.4
0.4
0.3
0.4
0.2
0.2
0.2
0
0
0.1
0
5
10
15
20
labor
0.21
0
18
0.2
16
0.19
14
0
x 10
5
10
15
20
15
20
-0.2
0
5
10
15
20
inflation
-4
12
0.18
10
0.17
8
0.16
6
0.15
4
0.14
2
0.13
0.12
0
0
5
10
15
20
-2
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Temporary productivity shock
Temporary productivity shock
Welfare costs of business cycles and the inflation-targeting regime
γc
0.7
0.7
2
2
b
0
-0.8
0
-0.8
∆bc (%)
-0.0171
-0.6191
-0.0073
12.2258
∆inf (%)
0.0000
0.0001
0.0000
0.0024
Optimal monetary policy with incomplete markets
Results
Temporary productivity shock
Temporary productivity shock
Impulse responses when γc = 0.7 and b = 0.
output
1.5
consumption
0.35
5
0.25
1
investment
6
0.3
4
0.2
3
0.15
2
0.1
0.5
1
0.05
0
0
0
0
5
10
15
20
labor
0.8
-0.05
0.5
0
x 10
5
10
15
20
15
20
-1
0
5
10
15
20
inflation
-3
0.7
0
0.6
0.5
-0.5
0.4
0.3
-1
0.2
-1.5
0.1
0
-2
-0.1
-0.2
0
5
10
15
20
-2.5
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Temporary productivity shock
Temporary productivity shock
Impulse responses when γc = 0.7 and b = −0.8.
output
2.5
consumption
0.6
investment
12
0.4
10
0.2
8
0
6
-0.2
4
-0.4
2
2
1.5
1
0.5
-0.6
0
0
5
10
15
20
labor
2
-0.8
0.5
0
0
x 10
5
10
15
20
15
20
-2
0
5
10
15
20
inflation
-3
0
1.5
-0.5
1
-1
-1.5
0.5
-2
0
-2.5
-0.5
0
5
10
15
20
-3
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Temporary productivity shock
Temporary productivity shock
Impulse responses when γc = 2 and b = 0.
output
1.1
consumption
0.34
1
investment
3.5
0.32
0.9
3
0.3
2.5
0.8
0.28
0.7
0.26
2
0.24
1.5
0.6
0.5
0.22
0.4
1
0.3
0.2
0.2
0.18
0.1
0
5
10
15
20
labor
0.4
0.16
1
0.5
0
x 10
5
10
15
20
15
20
0
0
5
10
15
20
inflation
-3
0.35
0
0.3
0.25
-1
0.2
-2
0.15
0.1
-3
0.05
-4
0
-0.05
0
5
10
15
20
-5
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Results
Temporary productivity shock
Temporary productivity shock
Impulse responses when γc = 2 and b = −0.8.
output
-0.04
consumption
1.6
investment
0
1.4
-0.06
-1
1.2
-0.08
1
-0.1
-2
0.8
-0.12
-3
0.6
-0.14
0.4
-0.16
-0.2
-4
0.2
-5
-0.18
0
0
5
10
15
20
labor
0.2
-0.2
5
0
0
x 10
5
10
15
20
15
20
-6
0
5
10
15
20
inflation
-3
4
-0.2
3
-0.4
2
-0.6
1
-0.8
0
-1
-1.2
0
5
10
15
20
-1
0
5
10
Solid lines: Ramsey policy; dashed lines: inflation targeting.
Optimal monetary policy with incomplete markets
Conclusion
Outline
1
Introduction
2
Model
Individuals
Aggregation
Firms
Aggregate shocks
Government
3
Results
Permanent productivity shock
Temporary productivity shock
4
Conclusion
Optimal monetary policy with incomplete markets
Conclusion
Conclusion
We have developed a New Keynesian model with
uninsurable idiosyncratic income shocks.
The welfare cost of business cycles can be very large
when the variance of idiosyncratic shocks fluctuates
countercyclically.
Nevertheless, the optimal monetary policy is roughly the
same as the zero-inflation policy. The presence of
countercyclical idiosyncratic shocks does not affect the
inflation-output tradeoff.