Coordinating Business Cycles · 2015-06-17 · Coordinating Business Cycles Edouard Schaal New York...
Transcript of Coordinating Business Cycles · 2015-06-17 · Coordinating Business Cycles Edouard Schaal New York...
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Coordinating Business Cycles
Edouard SchaalNew York University
Mathieu Taschereau-DumouchelUniversity of Pennsylvania
Wharton School
Multiple Equilibria and Financial Crises ConferenceMay 2015
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Motivation
90
95
100
105
1985 1990 1995 2000 2005 2010
Figure : US real GDP (log) and linear trend (2007Q4 = 100)
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Motivation
• Postwar US business cycles:
◮ Strong tendency to revert back to trend◮ 2007-09 recession: the economy seems to have fallen to a lower
steady state
• We propose an explanation based on coordination failures
◮ When complementarities are strong, can model the economy as acoordination game with multiple equilibria
• Diamond (1982); Kiyotaki (1988); Benhabib and Farmer (1994);...
◮ Hypothesis: the economy is trapped in a low output equilibrium asagents fail to coordinate on higher production/demand
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Our Contribution
• We develop a model of coordination failures and business cycles
• We respond to two key challenges in this literature:
◮ Quantitative
• Typical models are too stylized/unrealistic⇒ Our model is a small deviation from standard neoclassical model with
monopolistic competition
◮ Methodological
• Equilibrium indeterminacy limits welfare/quantitative analysis⇒ We adopt a global game approach to discipline equilibrium selection
• The model can be used as a benchmark for quantitative and policyanalysis
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Our Contribution
• We develop a model of coordination failures and business cycles
• We respond to two key challenges in this literature:
◮ Quantitative
• Typical models are too stylized/unrealistic⇒ Our model is a small deviation from standard neoclassical model with
monopolistic competition
◮ Methodological
• Equilibrium indeterminacy limits welfare/quantitative analysis⇒ We adopt a global game approach to discipline equilibrium selection
• The model can be used as a benchmark for quantitative and policyanalysis
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Our Contribution
• We develop a model of coordination failures and business cycles
• We respond to two key challenges in this literature:
◮ Quantitative
• Typical models are too stylized/unrealistic⇒ Our model is a small deviation from standard neoclassical model with
monopolistic competition
◮ Methodological
• Equilibrium indeterminacy limits welfare/quantitative analysis⇒ We adopt a global game approach to discipline equilibrium selection
• The model can be used as a benchmark for quantitative and policyanalysis
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Model Structure
• Standard neoclassical model with:
◮ Monopolistic competition
• Aggregate demand externality provides a motive to coordinate
◮ Non-convex capacity choice Evidence
• Breaks concavity of firm’s problem, locally increasing returns• Large evidence for investment, labor but also shifts/production lines• We capture these non-convexities in the simplest way
ut ∈ {uh > ul}
• Multiplicity?
◮ Multiplicity for relevant parameters under complete information,◮ Uniqueness everywhere under incomplete information (global game)
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Main Results
• Dynamics
◮ Multiple steady states in the multiplicity region◮ Deep recessions: the economy can fall in a coordination trap where
coordination on high steady state is difficult◮ Quantitatively consistent with various features of the recovery from
2007-2009 recession
• Policy
◮ Fiscal policy in general welfare reducing as coordination problemmagnifies crowding out
◮ But sometimes increases welfare by helping coordination close to atransition
◮ Optimal policy is a mix of input and profit subsidies
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I. Model: Complete Information Case
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Model
• Infinitely-lived representative household that solves
maxCt ,Lt ,Kt+1
E
∞∑
t=0
βt
[
1
1− γ
(
Ct −L1+νt
1 + ν
)1−γ]
, γ > 0, ν > 0
under the budget constraints
Pt (Ct + Kt+1 − (1− δ)Kt) 6 WtLt + RtKt +Πt
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Production
• Two types of goods:
◮ Final good used for consumption and investment◮ Differentiated goods j ∈ [0, 1] used in production of final good
• Competitive final good industry with representative firm
Yt =
(ˆ 1
0
Yσ−1σ
jt dj
)
σ
σ−1
, σ > 1
yielding demand curve and price index
Yjt =
(
Pjt
Pt
)−σ
Yt and Pt =
(ˆ 1
0
P1−σjt dj
)
11−σ
and we normalize Pt = 1
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Intermediate Producers
• Unit continuum of intermediate goods producer under monopolisticcompetition
Yjt = AeθujtKαjt L
1−αjt
• Aggregate productivity θ follows an AR(1)
θt = ρθt−1 + εθt , εθt ∼ iid N(
0, γ−1θ
)
• Capacity utilization ujt
◮ Binary decision ujt ∈ {1, ω} with ω > 1◮ Operating at high capacity ω costs f◮ Acts as a TFP shifter:
Ah (θt) ≡ ωAeθt > Ae
θt ≡ Al (θt)
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Equilibrium Definition
DefinitionAn equilibrium is policies for the household {Ct (θ
t) ,Kt+1 (θt) , Lt (θ
t)},policies for firms {Yjt (θ
t) ,Kjt (θt) , Ljt (θ
t)} , j ∈ {h, l}, a measuremt (θ
t) of high capacity firms, prices {Rt (θt) ,Wt (θ
t)} such that
• Household and firms solve their problems, markets clear,
• Mass of firms with high capacity is consistent with firms’ decisions
mt
(
θt)
≡
1 if Πht − f > Πlt
∈ (0, 1) if Πht − f = Πlt
0 if Πht − f < Πlt
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Characterization
• The intermediate producer faces a simple static problem
• Producers face a positive aggregate demand externality
Πjt = PtY1σ
t Yσ−1σ
jt −WtLjt − RtKjt
where σ determines the strength of externality
• In partial equilibrium, the capacity choice collapses to
Π = max
[
1
σ
Yt
Pσ−1ht
− f ,1
σ
Yt
Pσ−1lt
]
with the cost of a marginal unit of output
Pjt =σ
σ − 1MCjt and MCjt ≡
1
Ajt (θ)
(
Rt
α
)α (
Wt
1− α
)1−α
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Characterization
• Incentives to use high capacity increase with aggregate demand Yt
Yt
−f
Πht(Yt)− f
Πlt(Yt)
Πt
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Characterization
• Under GHH preferences,
◮ Labor supply curve independent of C ,◮ Production side of the economy can be solved independently of
consumption-saving decision!
• We thus proceed in two steps:
◮ First, study static equilibrium (production and capacity choice)◮ Then, return to the dynamic economy (C and K ′ decisions)
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Static Equilibrium
• Simple aggregate production function:
Yt = A (θt ,mt)Kαt L
1−αt
Lt =
[
(1− α)σ − 1
σA (θt ,mt)K
αt
]1
ν+α
• Endogenous TFP:
A (θ,m) =(
mAh (θ)σ−1
+ (1−m)Al (θ)σ−1
)1
σ−1
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Static Equilibrium
• Simple aggregate production function:
Yt = A (θt ,mt)Kαt L
1−αt
Lt =
[
(1− α)σ − 1
σA (θt ,mt)K
αt
]1
ν+α
• Endogenous TFP:
A (θ,m) =(
mAh (θ)σ−1
+ (1−m)Al (θ)σ−1
)1
σ−1
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Static Equilibrium
• Simple aggregate production function:
Yt = A (θt ,mt)Kαt L
1−αt
Lt =
[
(1− α)σ − 1
σA (θt ,mt)K
αt
]1
ν+α
• Endogenous TFP:
A (θ,m) =(
mAh (θ)σ−1
+ (1−m)Al (θ)σ−1
)1
σ−1
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Static Equilibrium: Multiplicity
Proposition 1
Suppose that 1+να+ν > σ − 1, then there exists cutoffs BH < BL such that
there are multiple static equilibria for BH 6 eθKα 6 BL.Productivityθ
Capital K
Multiple equilibria
High equilibrium only
m = 1
Low equilibrium only
m = 0
BL
BH
Intuition
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Static Equilibrium: Multiplicity
OutputY
Capital K
High equilibrium Yh(Kt , θt)
Low equilibrium Yl(Kt , θt)
Mixed equilibrium Ym(Kt , θt)
BH BL
Multiplicity vs. Uniqueness
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Static Equilibrium: Efficiency
Is the static equilibrium efficient?
Proposition 2
For 1+να+ν > σ − 1, there exists a threshold BSP < BL such that
• For eθKα 6 BSP , the planner chooses m = 0,
• For eθKα > BSP , the planner chooses m = 1.
In addition, for σ low enough, BSP < BH .
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Static Equilibrium: Efficiency
Productivityθ
Capital K
CE: Multiple equilibria
CE: High equilibrium only
CE: Low equilibrium only
BL
BH
BSP
SP: High capacity
SP: Low capacity
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Static Equilibrium: Coordination Failure
Productivityθ
Capital K
CE: Multiple equilibria
CE: High equilibrium only
CE: Low equilibrium only
SP: High capacity
SP: Low capacity
Coordination failure
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II. Model: Incomplete Information Case
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Model: Incomplete Information
• Model remains the same, except:
◮ Capacity choice is made under uncertainty about current θt
• New timing:
1 Beginning of period: θt = ρθt−1 + εθt is drawn2 Firm j observes private signal vjt = θt + εvjt with εvjt ∼ iid N
(
0, γ−1v
)
3 Firms choose their capacity uj ∈ {ul , uh}4 θt is observed, production takes place, Ct and Kt+1 are chosen
Capacity choice
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Uniqueness of Static Game
Proposition 3
For γv large and if √γv
γθ>
1√2π
ωσ−1 − 1
σ − 1,
then the equilibrium of the static global game is unique and takes the
form of a cutoff rule v̂ (K , θ−1) ∈ R ∪ {−∞,∞} such that firm j choose
high capacity if and only if vj > v̂ (K , θ−1). In addition, v̂ is decreasing
in its arguments.
• Remark: the number of firms choosing high capacity is
m ≡ 1− Φ (√γv (v̂ (K , θ−1)− θ))
where Φ is the CDF of a standard normal
Details
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Uniqueness of Static Game
OutputY
Capital K
Y ∗(Kt , θt−1, θt)
BH BL
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Dynamics: Multiple Steady States
Kt+
1
Kt
Medium
θ
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Dynamics: Multiple Steady States
Kt+
1
Kt
High θ
Lowθ
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Dynamics: Phase Diagram
Productivityθ
Capital K
0
High regime∆K = 0
Low regime∆K = 0
⋆
•
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III. Quantitative Evaluation
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Quantitative Exercise
• The model is calibrated in a standard way Calibration
• We then evaluate the model on the following dimensions:
◮ Business cycle moments: similar performance to standard RBCmodel RBC moments
◮ Skewness: outperforms standard models due to existence of largerecessions (fat left tail) Skewness
◮ Impulse responses: secular stagnation, 2007-2009 recession?
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Impulse Responses
• The model dynamics display strong non-linearities
• We hit the economy with negative θ shocks:
1 Small2 Medium and lasts 4 quarters3 Large and lasts 4 quarters
• Results:
◮ The response to small shock is similar to standard RBC model◮ Strong amplification and propagation for larger shocks◮ Large, long-lasting shocks can push the economy towards low steady
state: coordination trap
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Impulse Responses
(a) θ
0 50 100 150 200 250−0.025
−0.020
−0.015
−0.010
−0.005
0.000
(b) Endogenous TFP
0 50 100 150 200 250−0.04
−0.03
−0.02
−0.01
0.00
(c) Output
0 50 100 150 200 250
−0.08−0.06−0.04−0.020.00
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Impulse Responses
(d) Labor
0 50 100 150 200 250
−0.06
−0.04
−0.02
0.00
(e) Investment
0 50 100 150 200 250−0.20
−0.15
−0.10
−0.05
0.00
(f) Capacity m
0 50 100 150 200 250
0.00
0.25
0.50
0.75
1.00
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2007-2009 RecessionLogdeviation
YILC
TFP
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
2006 2008 2010 2012 2014
Logdeviation
YILC
TFP
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
2006 2008 2010 2012 2014
Figure : US series centered on 2007Q4 (left) vs model (right)
TFP
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IV. Policy Implications
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Policy Implications
• The competitive economy suffers from two (related) inefficiencies:
1 Monopoly distortions on the product market,
• Correct this margin immediately with input subsidy skl that offsetsmarkup 1− skl =
σ−1σ
,
2 Inefficient capacity choice due to aggregate demand externality.
• We analyze:
◮ Impact of fiscal policy◮ Optimal policy and implementation
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Policy: Summary of Results
• Fiscal policy:
◮ Government spending is in general detrimental to coordination
• Crowding out effect magnified by coordination problem Crowding
• This effect dominates in most of the state space
◮ But negative wealth effect can overturn this result
• When preferences allow for wealth effect on labor supply, fiscal policy
may be welfare improving by helping coordination Welfare
• Possibly large multipliers without nominal rigidities Multiplier
• Optimal policy:
◮ A mix of constant input and profit subsidy implements theconstrained efficient allocation Optimal Policy
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V. Conclusion
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Conclusion
• We construct a dynamic stochastic general equilibrium model withcoordination failures
◮ Provides a foundation for Keynesian-type effects without nominalrigidities
• The model generates:
◮ Deep recessions: secular stagnation?◮ Fiscal policy can be welfare improving
• Future agenda:
◮ Quantitative side:
• Understand the role of firm-level heterogeneity• Use micro-data to discipline the non-convexities
◮ Learning, optimal fiscal policy, etc.
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Evidence of Non-Convexities
• Typical neoclassical model assumes convex cost functions
◮ Well-defined maximization problem with unique equilibrium
• However, large evidence of non-convexities in cost functions:
◮ Firms adjust output along various margins which differ inlumpiness/adjustment/variable costs
• Cooper and Haltiwanger (2006): lumpy adjustments in labor andinvestment,
• Bresnahan and Ramey (1994): lumpy changes in production atplant-level with plant shutdowns/restart,
• Hall (1999): non-convexities in shift adjustments across Chryslerassembly plants.
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Evidence of Non-Convexities
• Ramey (JPE 1991) estimates cost functions◮ Example food industry:
Ct (Y ) = 23.3wtY − 7.78∗∗Y 2 + 0.000307∗Y 3 + . . .
Figure : Non-convex cost curve (Ramey, 1991)
Model33 / 33
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Static Equilibrium: Multiplicity
• Condition for multiplicity is
1 + ν
α+ ν> σ − 1
• This condition is more likely to be satisfied if
◮ σ is small: high complementarity through demand,◮ ν is small: low input competition (sufficiently flexible labor),◮ α is small: production is intensive in the flexible factor (labor).
Return
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Static Equilibrium: Multiplicity vs. Uniqueness
OutputY
Capital K
multiplicity1+να+ν = σ − 1uniqueness
Multiplicity
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Model: Incomplete Information
• Firms now solve the following problem:
u∗j = argmaxuj∈{uh,ul}
{
E [Uc (C , L) (Πh (K , θ,m)− f ) | θ−1, vj ] ,
E [Uc (C , L) Πl (K , θ,m) | θ−1, vj ]}
where
◮ Expectation term over θ and m◮ m is now uncertain and firms must guess what others will choose!
Return
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Uniqueness of Static Game
• Condition for uniqueness
√γv
γθ>
1√2π
ωσ−1 − 1
σ − 1
• This condition requires:
1 Uncertainty in fundamental θ (γθ low),2 High precision in private signals (γv high)
• Ensure that beliefs about fundamental (in γv ) dominates feedbackfrom others (in
√γv )
Return
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Parametrization
Standard parameters:
Parameter Value Source/Target
Time period one quarterCapital share α = 0.3 Labor share 0.7Discount factor β = 0.951/4 0.95 annual
Depreciation rate δ = 1− 0.91/4 10% annualElasticity of substitution σ = 3 Hsieh and Klenow (2014)
Risk aversion γ = 1 log utilityElasticity of labor supply ν = 0.4 Jaimovich and Rebelo (2009)Persistence θ process ρθ = 0.95 Cooley and Prescott (1985)
Stdev of θ σθ = 0.006 Stdev output
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Parametrization
Three parameters remain: γv , ω and f
• Precision of private information γv :
◮ Target dispersion in forecasts about GDP growth from SPF◮ One quarter ahead: γv = 124, 232 ≃ 0.2% stdev
• Capacity utilization ratio ω = uhul:
◮ Match pre-2008/post-2010 averages ≃ 1.017
• Fixed cost f :
◮ Chosen to match the tail probability of large crises in SPF(growth6-4%),
◮ Set f = 0.019 of GDP
Return
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Business Cycle Moments
Correlation with output
Correlation with output Output Investment Hours Consumption
Data 1.00 0.87 0.86 0.94Full model 1.00 0.89 1.00 0.99
RBC (f = 0, σ → ∞) 1.00 0.96 1.00 0.99
Table : Correlation with output
• Again, similar performance to a standard RBC model
Return
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Skewness
• The model does well for skewness and asymmetry of business cycles:
Skewness Output Investment Hours Consumption
Data -0.59 -0.31 -0.35 -0.44Full model -0.16 -0.14 -0.16 -0.14
RBC (f = 0, σ → ∞) 0.00 -0.01 0.00 0.01
Table : Skewness
Return
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Skewness and Fat Tail
• The negative skewness is due to ability to generate deep recessions:
−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015
−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06
Figure : Ergodic distribution of θ (top) vs. output (bottom)
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Skewness and Fat Tail
• Histogram of output in the data:
0
5
10
15
20
25
30
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
Figure : Distribution of log real GDP (1967-2014, linear trend)
Return
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Business Cycle Moments
Standard deviations
Stddev Rel. to Output Output Investment Hours Consumption
Data 1.00 3.27 1.46 0.94Full model 1.00 2.06 0.72 0.88
RBC (f = 0, σ → ∞) 1.00 1.72 0.71 0.84
Table : Standard deviation relative to that of Output
• The full model behaves similarly to a standard RBC model
Return
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Solution of the Model
2002 2004 2006 2008 2010
4.5
4.55
4.6
4.65
4.7
Log TFP
Notes: Linear trend from 2001Q1!2008Q2 (dashed!dotted). Forecast 2008Q3 and beyond based on linear trend (dotted).
BLS (Private Business)
BLS (Manufacturing)
BLS (Total)
2002 2004 2006 2008 2010 2012
4.52
4.54
4.56
4.58
4.6
4.62
4.64
4.66
4.68
Log TFP
Fernald (Raw)
Fernald (Utilization Adjusted)
Penn World Tables
Figure 5: Measures of Total Factor Productivity (TFP): 2001 to 2013
TFP (% Deviation from Trend)Figure : Various measures of TFP (source: Christiano, Eichenbaum and
Trabandt, 2014)
Return
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Fiscal Policy: Crowding Out
• Crowding out:Kt+
1
Kt
Basin of attraction
for low regime
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Fiscal Policy: Crowding Out
• Crowding out: decline in investmentKt+
1
Kt
Basin of attraction
for low regime
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Fiscal Policy: Crowding Out
• Coordination is worsened by crowding out:
◮ Capital K plays a crucial role for coordination,◮ By crowding out private investment, government spending makes
coordination on high regime less likely in the future!◮ Large dynamic welfare losses
• Result: Under GHH preferences,
◮ For γv large, firms’ choice of m unaffected by G ,◮ Government spending is always welfare reducing
Return
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Fiscal Policy: Wealth Effect
• How can a negative wealth effect be welfare improving?
Welfare
t
pure equilibrium m = 1
pure equilibrium m = 0
global game 0 < m < 1
G ր
Return
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Fiscal Policy
(a) Impact of G on capacity choice m
6.5 6.6 6.7 6.8 6.9 7.0 7.1
K
0.0
0.2
0.4
0.6
0.8
1.0
m
G=0
G>0
(b) Fiscal multiplier
6.5 6.6 6.7 6.8 6.9 7.0 7.1
K
0.0
0.5
1.0
1.5
2.0
∆Y/∆G
(c) Welfare gains in consumption equivalent
6.5 6.6 6.7 6.8 6.9 7.0 7.1
K
−0.010
−0.005
0.000
0.005
0.010
∆c c e
quiv
. gain
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Optimal Policy
• We study a constrained planner with same information as outsideobserver:
◮ At the beginning of period, only knows θ−1
◮ Does not observe firms’ private signals
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Constrained Planner Problem
• The planner chooses a probability to choose high capacity z (vj) forall signals vj
V (K , θ−1) = maxz,C ,L,K ′
Eθ
[
1
1− γ
(
C − L1+ν
1 + ν
)1−γ
+ βV (K ′, θ)
]
subject to
C + K ′ = A (θ,m)KαL1−α + (1− δ)K −mf
m (θ) =
ˆ √γvφ (
√γv (v − θ)) z (v) dv
A (θ,m) =(
mAh (θ)σ−1
+ (1−m)Al (θ)σ−1
)1
σ−1
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Constrained Planner Problem
Proposition 4
The competitive equilibrium with imperfect information is inefficient, but
the efficient allocation can be implemented with:
1 An input subsidy 1− skl =σ−1σ to correct for monopoly distortions,
2 A profit subsidy 1 + sπ = σσ−1 to induce the right capacity choice.
• Remark:
◮ The profit subsidy is just enough to make firms internalize theimpact of their capacity decision on others
Return
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Calibration Government Spending
• Utility function: U(C , L) = logC − (1 + ν)−1L1+ν
Parameter Value Source/Target
Time period one quarterCapital share α = 0.3 Labor share 0.7Discount factor β = 0.951/4 0.95 annual
Depreciation rate δ = 1− 0.91/4 10% annualElasticity of substitution σ = 3 Hsieh and Klenow (2014)
Risk aversion γ = 1 log utilityElasticity of labor supply ν = 0.4 Jaimovich and Rebelo (2009)Persistence θ process ρθ = 0.95 Cooley and Prescott (1985)
Stdev of θ σθ = 0.006 Stdev outputFixed cost f = 0.01485
High capacity ω = 1.017Government spending G = 0.00665 0.5% of steady-state output
Return
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