An Economical Business-Cycle Model Pascal Michaillat (LSE) & Emmanuel Saez (Berkeley) April 2015 1 / 45
Slack and inflation in the US since 1994 idle capacity (Census) 40% 30% 20% 10% idle labor (ISM) 0% 1994 1999 2004 2009 2014 2 / 45
Slack and inflation in the US since 1994 idle capacity 40% 10% 30% 7.5% 20% 5% 10% 2.5% unemployment idle labor (right scale) 0% 0% 1994 1999 2004 2009 2014 2 / 45
Slack and inflation in the US since 1994 40% 10% slack 30% 7.5% 20% 5% 10% 2.5% core inflation (right scale) 0% 0% 1994 1999 2004 2009 2014 2 / 45
Objective of the paper develop a tractable business-cycle model in which fluctuations in supply and demand lead to ◮ some fluctuations in slack— unemployment, idle labor, and idle capacity ◮ no fluctuations in inflation use the model to analyze monetary and fiscal policies 3 / 45
The model 4 / 45
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 5 / 45
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 6 / 45
Behavior of representative household supply k labor services choose consumption c , real money m , real wealth a to maximize utility � + ∞ � � ε ε − 1 e − δ · t · ε + φ ( m )+ ω ( a + ) ε − 1 · c dt 0 subject to law of motion of real wealth � � da dt = f ( x + ) · k − 1 + τ ( x + ) · c − i · m + r · a + seigniorage 7 / 45
Utility for real money money bliss point utility real money m 8 / 45
Utility for real wealth utility no aggregate wealth a=m+b=0 real wealth a 9 / 45
Matching function and market tightness k ¡units ¡of ¡labor ¡services v ¡help-‑wanted ¡ads 10 / 45
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 = + ρ · v = c + ρ · y c q ( x ) ���� ���� ���� consumption output matching cost � � 1 − ρ = c ⇒ y · q ( x ) � � ρ ⇒ y = 1 + 1 + τ ( x + ) · c ≡ · c q ( x − ) − ρ 11 / 45
Consumer’s first-order conditions costate variable: c − 1 / ε λ = 1 + τ ( x ) demand for real money balances: φ ′ ( m ) = i · c − 1 / ε 1 + τ ( x ) consumption Euler equation: d λ / dt = 1 + τ ( x ) · ω ′ ( a )+ i − π − δ c − 1 / ε λ 12 / 45
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 ) 13 / 45
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 14 / 45
Steady-state equilibrium: IS, LM, AD, and AS curves 15 / 45
IS curve (from consumption Euler equation) nominal interest rate i IS consumption c 16 / 45
IS curve without utility of wealth IS nominal interest rate i i IS ( x, π ) = π + δ consumption c 17 / 45
LM curve (from demand for real money balances) LM nominal interest rate i consumption c 18 / 45
LM curve with money > bliss point (liquidity trap) nominal interest rate i i LM ( x, m ) = 0 LM consumption c 19 / 45
IS & LM determine interest rate and AD nominal interest rate LM IS c AD ( x, π, m ) consumption 20 / 45
IS & LM determine interest rate and AD nominal interest rate LM i IS c AD ( x 0 < x, π, m ) consumption 20 / 45
AD curve � � � δ + π c AD ( x, π, m ) = (1 + τ ( x )) · ( φ 0 ( m ) + ! 0 (0)) market tightness x AD consumption c 21 / 45
AS curve market tightness x capacity: k quantity of labor services 22 / 45
AS curve capacity k market tightness x output: y = f(x) k quantity of labor services 22 / 45
AS curve output y capacity k market tightness x consumption: quantity of labor services 22 / 45
AS curve output capacity market tightness x consumption recruiting slack quantity of labor services 22 / 45
AS curve c AS ( x ) = ( f ( x ) − ρ · x ) · k market tightness x AS consumption c 22 / 45
AS curve and state of the economy overheating economy market tightness x efficient economy slack economy AS consumption c 23 / 45
General equilibrium output capacity AS general market tightness equilibrium AD slack x c y k quantity of labor services 24 / 45
Dynamical system is a source ˙ ˙ λ = ( δ + π ) · λ − ! 0 (0) − φ 0 ( m ) λ 0 λ λ 25 / 45
Immediate adjustment to shock ˙ λ 0 λ b λ a λ 26 / 45
Macroeconomic shocks 27 / 45
Increase in AD: fall in MU of wealth nominal interest rate AD increases LM IS consumption 28 / 45
Increase in AD: fall in MU of wealth output capacity labor market tightness AS AD quantity of labor services 28 / 45
Increase in AS: rise in capacity output capacity labor market tightness AS AD quantity of labor services 29 / 45
Monetary and fiscal policies 30 / 45
Increase in money supply AS output capacity market tightness x low tightness and output depressed AD consumption c 31 / 45
Increase in money supply AD increases nominal interest rate i LM IS consumption c 31 / 45
Increase in money supply output capacity market tightness x efficient tightness AD AS consumption c 31 / 45
Money supply in a liquidity trap AS output capacity market tightness x very low tightness and output very depressed AD consumption c 32 / 45
Money supply in a liquidity trap LM in liquidity trap nominal interest rate i IS LM consumption c 32 / 45
Money supply in a liquidity trap output capacity market tightness x inefficiently low tightness AD in liquidity trap AS consumption c 32 / 45
Alternative policy: helicopter money government prints and distributes M h > 0 aggregate wealth is positive: a = m h > 0 IS curve depends on helicopter money: � � ε δ + π − i c IS = ( 1 + τ ( x )) · ω ′ ( m h ) 33 / 45
Helicopter money always stimulates AD nominal interest rate AD increases LM IS consumption 34 / 45
Helicopter money always stimulates AD nominal interest rate AD increases IS consumption LM in liquidity trap 34 / 45
Alternative policy: tax on wealth government taxes wealth at rate τ a > 0 IS curve depends on wealth tax: � δ + τ a + π − i � ε c IS = ( 1 + τ ( x )) · ω ′ ( 0 ) 35 / 45
Tax on wealth always stimulates AD nominal interest rate AD increases LM IS consumption 36 / 45
Tax on wealth always stimulates AD nominal interest rate AD increases IS consumption LM in liquidity trap 36 / 45
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: � � ε δ + π g c AD = + ( 1 + τ ( x )) · ( φ ′ ( m )+ ω ′ ( 0 )) 1 + τ ( x ) 37 / 45
Government purchases stimulate AD output capacity labor market tightness AS AD quantity of labor services 38 / 45
Summary of policies conventional monetary policy sets money supply M M stabilizes economy out of liquidity trap ◮ M → LM curve → AD curve M is ineffective in liquidity trap ◮ LM curve is stuck alternative policies work in liquidity trap ◮ helicopter money / wealth tax → IS curve → AD curve ◮ government purchases → AD curve 39 / 45
Inflation and slack dynamics in the medium run 40 / 45
Simplifying assumptions 1. no money growth 2. no liquidity trap 41 / 45
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 42 / 45
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