An Economical Business-Cycle Model Pascal Michaillat (LSE) & - - PowerPoint PPT Presentation

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An Economical Business-Cycle Model Pascal Michaillat (LSE) & - - PowerPoint PPT Presentation

Sep 16, 2022 165 likes •777 views

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

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SLIDE 1

An Economical Business-Cycle Model

Pascal Michaillat (LSE) & Emmanuel Saez (Berkeley) April 2015

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SLIDE 2

Slack and inflation in the US since 1994

1994 1999 2004 2009 2014 0% 10% 20% 30% 40%

idle labor (ISM) idle capacity (Census)

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SLIDE 3

Slack and inflation in the US since 1994

1994 1999 2004 2009 2014 0% 10% 20% 30% 40% 0% 2.5% 5% 7.5% 10%

unemployment (right scale) idle labor idle capacity

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SLIDE 4

Slack and inflation in the US since 1994

1994 1999 2004 2009 2014 0% 10% 20% 30% 40% 0% 2.5% 5% 7.5% 10%

core inflation (right scale) slack

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SLIDE 5

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

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SLIDE 6

The model

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SLIDE 7

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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SLIDE 8

Money and bonds

households hold B bonds at nominal interest rate i government circulates money M

  • pen 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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SLIDE 9

Behavior of representative household

supply k labor services choose consumption c, real money m, real wealth a to maximize utility

+∞

e−δ·t ·

  • ε

ε −1 ·c

ε−1 ε +φ(m)+ω(a

+)

  • dt

subject to law of motion of real wealth da dt = f(x

+)·k −

  • 1+τ(x

+)

  • ·c−i·m+r ·a+seigniorage

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SLIDE 10

Utility for real money

real money m utility money bliss point

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SLIDE 11

Utility for real wealth

real wealth a utility no aggregate wealth a=m+b=0

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SLIDE 12

Matching function and market tightness

v ¡help-­‑wanted ¡ads k ¡units ¡of ¡labor ¡services

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SLIDE 13

Matching function and market tightness

sales ¡= ¡ ¡ ¡ ¡ ¡ ¡ ¡ = purchases ¡= ¡ ¡ ¡ ¡ ¡ ¡ ¡=

  • utput: ¡ ¡y = h(k,v)

capacity ¡k ¡ tightness: ¡x = v / k ¡ help-­‑wanted ¡ads ¡v ¡

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SLIDE 14

Matching cost: ρ services per ad

  • utput =
  • 1+τ(x

+)

  • · consumption

proof: y

  • utput

= c

  • consumption

+ ρ ·v

  • matching cost

= c+ρ · y q(x) ⇒ y·

  • 1− ρ

q(x)

  • = c

⇒ y =  1+ ρ q(x

−)−ρ

 ·c ≡

  • 1+τ(x

+)

  • ·c

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SLIDE 15

Consumer’s first-order conditions

costate variable: λ = c−1/ε 1+τ(x) demand for real money balances: φ′(m) = i· c−1/ε 1+τ(x) consumption Euler equation: dλ/dt λ = 1+τ(x) c−1/ε ·ω′(a)+i−π −δ

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SLIDE 16

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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SLIDE 17

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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IS curve (from consumption Euler equation)

IS consumption c nominal interest rate i

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SLIDE 20

IS curve without utility of wealth

IS iIS(x, π) = π + δ consumption c nominal interest rate i

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SLIDE 21

LM curve (from demand for real money balances)

LM

consumption c nominal interest rate i

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SLIDE 22

LM curve with money > bliss point (liquidity trap)

LM

consumption c nominal interest rate i

iLM(x, m) = 0

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SLIDE 23

IS & LM determine interest rate and AD

consumption nominal interest rate IS LM cAD(x, π, m)

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SLIDE 24

IS & LM determine interest rate and AD

IS LM i consumption nominal interest rate cAD(x0 < x, π, m)

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AD curve

consumption c AD

cAD(x, π, m) =

  • δ + π

(1 + τ(x)) · (φ0(m) + !0(0))

  • market tightness x

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SLIDE 26

AS curve

capacity: k market tightness x quantity of labor services

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SLIDE 27

AS curve

  • utput: y = f(x) k

quantity of labor services capacity k market tightness x

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SLIDE 28

AS curve

  • utput y

consumption: capacity k quantity of labor services market tightness x

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SLIDE 29

AS curve

slack recruiting capacity

  • utput

consumption

quantity of labor services

market tightness x

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AS curve

AS cAS(x) = (f(x) − ρ · x) · k consumption c market tightness x

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SLIDE 31

AS curve and state of the economy

consumption c AS market tightness x

  • verheating economy

efficient economy slack economy

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General equilibrium

AS AD capacity

  • utput

general equilibrium quantity of labor services x c y k market tightness slack

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SLIDE 33

Dynamical system is a source

λ ˙ λ λ ˙ λ = (δ + π) · λ − !0(0) − φ0(m)

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SLIDE 34

Immediate adjustment to shock

λb λa λ ˙ λ

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SLIDE 35

Macroeconomic shocks

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SLIDE 36

Increase in AD: fall in MU of wealth

IS LM consumption nominal interest rate AD increases

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SLIDE 37

Increase in AD: fall in MU of wealth

AS AD capacity

  • utput

labor market tightness quantity of labor services

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SLIDE 38

Increase in AS: rise in capacity

labor market tightness quantity of labor services capacity

  • utput

AS AD

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SLIDE 39

Monetary and fiscal policies

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SLIDE 40

Increase in money supply

consumption c AS market tightness x capacity

  • utput

low tightness and output depressed AD

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SLIDE 41

Increase in money supply

LM consumption c AD increases IS nominal interest rate i

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SLIDE 42

Increase in money supply

consumption c AS market tightness x AD capacity

  • utput

efficient tightness

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SLIDE 43

Money supply in a liquidity trap

consumption c AS market tightness x capacity

  • utput

very low tightness and output very depressed AD

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Money supply in a liquidity trap

LM consumption c LM in liquidity trap IS nominal interest rate i

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Money supply in a liquidity trap

consumption c AS market tightness x capacity

  • utput

inefficiently low tightness AD in liquidity trap

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SLIDE 46

Alternative policy: helicopter money

government prints and distributes Mh > 0 aggregate wealth is positive: a = mh > 0 IS curve depends on helicopter money: cIS =

  • δ +π −i

(1+τ(x))·ω′(mh) ε

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SLIDE 47

Helicopter money always stimulates AD

IS LM consumption nominal interest rate AD increases

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SLIDE 48

Helicopter money always stimulates AD

IS LM in liquidity trap consumption nominal interest rate AD increases

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SLIDE 49

Alternative policy: tax on wealth

government taxes wealth at rate τa > 0 IS curve depends on wealth tax: cIS = δ +τa +π −i (1+τ(x))·ω′(0) ε

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SLIDE 50

Tax on wealth always stimulates AD

IS LM consumption nominal interest rate AD increases

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SLIDE 51

Tax on wealth always stimulates AD

IS LM in liquidity trap consumption nominal interest rate AD increases

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SLIDE 52

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: cAD =

  • δ +π

(1+τ(x))·(φ′(m)+ω′(0)) ε + g 1+τ(x)

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SLIDE 53

Government purchases stimulate AD

AS AD capacity

  • utput

labor market tightness quantity of labor services

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SLIDE 54

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

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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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SLIDE 57

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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SLIDE 58

Price-adjustment cost [Rotemberg, REStud 1982]

seller chooses p, π, and x to maximize the discounted sum of nominal profits

+∞

e−I(t) ·

  • p·f(x)·k−κ

2 ·π2 dt subject to ˙ p = π ·p 1+τ(x) = e p solution yields Phillips curve

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SLIDE 59

Dynamical system describing equilibrium

system of 3 ODEs: law of motion of price (˙ p), Phillips curve ( ˙ π), consumption Euler equation (˙ x) 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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SLIDE 60

Short-run/long-run effects of shocks

increase in: x π p y aggregate demand

+ + + +

money supply

+ + + +

aggregate supply

− − − − +

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