An Economical Business-Cycle Model
Pascal Michaillat (LSE) & Emmanuel Saez (Berkeley)
April 2015
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Slack and inflation in the US since 1994
40%
idle capacity (Census)
30%
20%
10%
idle labor (ISM)
0%
1994
1999
2004
2009
2014
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Slack and inflation in the US since 1994
40%
idle capacity
10%
30%
7.5%
20%
5%
10%
idle labor
0%
1994
1999
unemployment
(right scale)
2004
2009
2.5%
2014
0%
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Slack and inflation in the US since 1994
40%
10%
slack
30%
7.5%
20%
5%
10%
2.5%
core inflation (right scale)
0%
1994
1999
2004
2009
2014
0%
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Objective of the paper
develop a tractable business-cycle model in which
fluctuations in supply and demand lead to
I
some fluctuations in slack—unemployment, idle labor,
and idle capacity
I
no fluctuations in inflation
use the model to analyze monetary and fiscal
policies
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The model
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Overview
start from money-in-the-utility-function model of
Sidrauski [AER 1967]
add matching frictions on market for labor services
as in Michaillat & Saez [QJE 2015]
⇒ generate slack
⇒ accomodate fixed inflation in general equilibrium
add utility for wealth as in Kurz [IER 1968]
⇒ enrich aggregate demand structure
⇒ allow for permanent liquidity traps
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Money and bonds
households hold B bonds at nominal interest rate i
government circulates money M
open market operations impose M(t) = −B(t)
nominal financial wealth: A = M + B
price of labor services is p
inflation rate is π = p˙ /p
real variables: m = M/p, a = A/p, r = i − π
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Behavior of representative household
supply k labor services
choose consumption c, real money m, real wealth a
to maximize utility
Z +∞
ε−1
ε
e−δ ·t ·
· c ε + φ (m) + ω(a ) dt
+
ε −1
0
subject to law of motion of real wealth
da
= f (x ) · k − 1 + τ(x ) · c − i · m + r · a + seigniorage
+
+
dt
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Utility for real money
utility
money bliss point
real money m
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utility
Utility for real wealth
no aggregate wealth
a=m+b=0
real wealth a
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Matching function and market tightness
k units of labor services
v help-‐wanted ads
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Matching function and market tightness
tightness: x = v / k capacity k sales = =
output: y = h(k,v)
purchases = =
help-‐wanted ads v 10 / 45
Matching cost: ρ services per ad
output = 1 + τ(x ) · consumption
+
proof:
y
y = |{z}
c
+
ρ ·v
= c+ρ ·
|{z}
|{z}
q(x)
consumption matching cost
output
ρ
⇒ y· 1−
=c
q(x)


ρ
 · c ≡ 1 + τ(x ) · c
⇒ y = 1 +
+
q(x ) − ρ
−
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Consumer’s first-order conditions
costate variable:
c−1/ε
λ=
1 + τ(x)
demand for real money balances:
c−1/ε
φ (m) = i ·
1 + τ(x)
0
consumption Euler equation:
dλ /dt 1 + τ(x) 0
= −1/ε · ω (a) + i − π − δ
λ
c
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Equilibrium: 6 variables, 5 equations
[c(t), m(t), a(t), i(t), p(t), x(t)]+∞
t=0 satisfy
consumption Euler equation
demand for real money balances
no wealth in aggregate: a(t) = 0
matching process: (1 + τ(x(t))) · c(t) = f (x(t)) · k
m(t) = M(t)/p(t) and monetary policy sets M(t)
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Equilibrium selection: fixed inflation
price p(t) is a state variable with law of motion:
p˙ (t) = π · p(t)
p(0) and π are fixed parameters
given p(t), tightness x(t) equalizes supply to demand
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Steady-state equilibrium:
IS, LM, AD, and AS curves
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nominal interest rate i
IS curve (from consumption Euler equation)
IS
consumption c
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IS curve without utility of wealth
nominal interest rate i
IS
iIS (x, ⇡) = ⇡ +
consumption c
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LM curve (from demand for real money balances)
nominal interest rate i
LM
consumption c
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nominal interest rate i
LM curve with money > bliss point (liquidity trap)
iLM (x, m) = 0
LM
consumption c
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nominal interest rate
IS & LM determine interest rate and AD
LM
IS
cAD (x, ⇡, m)
consumption
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nominal interest rate
IS & LM determine interest rate and AD
LM
i
IS
cAD (x0 < x, ⇡, m) consumption
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AD curve
(x, ⇡, m) =

+⇡
(1 + ⌧ (x)) · ( 0 (m) + ! 0 (0))
✏
market tightness x
c
AD
AD
consumption c
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market tightness x
AS curve
capacity: k
quantity of labor services
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AS curve
market tightness x
capacity k
output: y = f(x) k
quantity of labor services
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AS curve
market tightness x
output y
capacity k
consumption:
quantity of labor services
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AS curve
market tightness x
output
capacity
consumption recruiting slack
quantity of labor services
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AS curve
market tightness x
cAS (x) = (f (x)
⇢ · x) · k
AS
consumption c
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AS curve and state of the economy
market tightness x
overheating economy
efficient economy
AS
slack economy
consumption c
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General equilibrium
output
market tightness
AS
capacity
general
equilibrium
AD
slack
x
c
y
k
quantity of labor services
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Dynamical system is a source
˙
˙ = ( + ⇡) ·
! 0 (0)
0
(m)
0
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Immediate adjustment to shock
˙
0
b
a
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Macroeconomic shocks
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nominal interest rate
Increase in AD: fall in MU of wealth
AD increases
LM
IS
consumption
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labor market tightness
Increase in AD: fall in MU of wealth
output
capacity
AS
AD
quantity of labor services
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labor market tightness
Increase in AS: rise in capacity
AS
output
capacity
AD
quantity of labor services
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Monetary and fiscal policies
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Increase in money supply
market tightness x
AS
output capacity
low tightness
and output
depressed AD
consumption c
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nominal interest rate i
Increase in money supply
AD increases
LM
IS
consumption c
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Increase in money supply
market tightness x
output capacity
efficient tightness
AS
AD
consumption c
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Money supply in a liquidity trap
market tightness x
AS
output capacity
very low tightness
and output
very depressed AD
consumption c
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nominal interest rate i
Money supply in a liquidity trap
LM in liquidity trap
IS
LM
consumption c
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Money supply in a liquidity trap
market tightness x
output capacity
inefficiently low
tightness
AS
AD in
liquidity trap
consumption c
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Alternative policy: helicopter money
government prints and distributes M h > 0
aggregate wealth is positive: a = mh > 0
IS curve depends on helicopter money:
ε
δ +π −i
IS
c =
(1 + τ(x)) · ω 0 (mh )
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nominal interest rate
Helicopter money always stimulates AD
AD increases
LM
IS
consumption
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nominal interest rate
Helicopter money always stimulates AD
IS
AD increases
LM in liquidity trap
consumption
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Alternative policy: tax on wealth
government taxes wealth at rate τ a > 0
IS curve depends on wealth tax:
δ + τa + π − i
c =
(1 + τ(x)) · ω 0 (0)
IS
ε
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nominal interest rate
Tax on wealth always stimulates AD
AD increases
LM
IS
consumption
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nominal interest rate
Tax on wealth always stimulates AD
IS
AD increases
LM in liquidity trap
consumption
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Alternative policy: government purchases
government purchases g(t) units of labor services
g(t) enters separately in utility function
g(t) financed by lump-sum taxes
AD curve depends on government purchases:
δ +π
cAD =
(1 + τ(x)) · (φ 0 (m) + ω 0 (0))
ε
+
g
1 + τ(x)
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labor market tightness
Government purchases stimulate AD
output
capacity
AS
AD
quantity of labor services
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Summary of policies
conventional monetary policy sets money supply M
M stabilizes economy out of liquidity trap
I
M → LM curve → AD curve
M is ineffective in liquidity trap
I
LM curve is stuck
alternative policies work in liquidity trap
I
I
helicopter money / wealth tax → IS curve → AD curve
government purchases → AD curve
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Inflation and slack dynamics
in the medium run
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Simplifying assumptions
1. no money growth
2. no liquidity trap
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Directed search [Moen, JPE 1997]
buyers search for best price/tightness compromise
in equilibrium, buyers are indifferent across markets:
(1 + τ(x)) · p = e in any market (x, p)
seller chooses p to maximize p · f (x) subject to
(1 + τ(x)) · p = e
⇔ seller chooses x to maximize f (x)/(1 + τ(x))
⇔ seller chooses efficient tightness x∗
if x < x∗ , seller wants to lower p and conversely
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Price-adjustment cost [Rotemberg, REStud 1982]
seller chooses p, π, and x
to maximize the discounted sum of nominal profits
Z +∞
κ 2
−I(t)
e
· p · f (x) · k− · π dt
2
0
subject to
p˙ = π · p
e
1 + τ(x) =
p
solution yields Phillips curve
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Dynamical system describing equilibrium
system of 3 ODEs: law of motion of price (˙p),
˙ consumption Euler equation (˙x)
Phillips curve (π),
state variable: p
jump variables: π, x
the unique steady state has x = x∗ and π = 0
system is a saddle around steady state
stable manifold is a line: dynamic determinacy
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Short-run/long-run effects of shocks
increase in:
aggregate demand
money supply
aggregate supply
x
π
p
y
+
+
0
+
0
0
+
0
+
+
0
+
0
0
+
0
−
−
0
0
0
−
−
+
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