An Economical Business-Cycle Model

Transcription

1 An Economical Business-Cycle Model Pascal Michaillat (LSE) & Emmanuel Saez (Berkeley) April / 45

2 Slack and inflation in the US since % idle capacity (Census) 30% 20% 10% idle labor (ISM) 0% / 45

3 Slack and inflation in the US since % idle capacity 10% 30% 7.5% 20% 5% 10% idle labor unemployment (right scale) 2.5% 0% 0% / 45

4 Slack and inflation in the US since % 30% slack 10% 7.5% 20% 5% 10% 2.5% core inflation (right scale) 0% % 2 / 45

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 3 / 45

6 The model 4 / 45

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 5 / 45

8 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 = ṗ/p real variables: m = M/p, a = A/p, r = i π 6 / 45

9 Behavior of representative household supply k labor services choose consumption c, real money m, real wealth a to maximize utility + [ ] ε e δ t ε 1 c ε 1 ε + φ(m) + ω(a ) dt + 0 subject to law of motion of real wealth [ ] da dt = f (x) k 1 + τ(x ) c i m + r a + seigniorage / 45

10 Utility for real money money bliss point utility real money m 8 / 45

11 Utility for real wealth utility no aggregate wealth a=m+b=0 real wealth a 9 / 45

12 Matching function and market tightness k units of labor services v help- wanted ads 10 / 45

13 Matching function and market tightness capacity k tightness: x = v / k sales = = output: y = h(k,v) purchases = = help- wanted ads v 10 / 45

14 Matching cost: ρ services per ad [ ] output = 1 + τ(x ) consumption + proof: y }{{} output = c }{{} consumption [ y 1 ρ ] = c q(x) ρ y = 1 + c q(x) ρ y + ρ v = c + ρ }{{} q(x) matching cost [ ] 1 + τ(x ) c + 11 / 45

15 Consumer s first-order conditions costate variable: λ = c 1/ε 1 + τ(x) demand for real money balances: φ (m) = i consumption Euler equation: dλ/dt λ = 1 + τ(x) c 1/ε c 1/ε 1 + τ(x) ω (a) + i π δ 12 / 45

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) 13 / 45

17 Equilibrium selection: fixed inflation price p(t) is a state variable with law of motion: ṗ(t) = π p(t) p(0) and π are fixed parameters given p(t), tightness x(t) equalizes supply to demand 14 / 45

18 Steady-state equilibrium: IS, LM, AD, and AS curves 15 / 45

19 IS curve (from consumption Euler equation) nominal interest rate i IS consumption c 16 / 45

20 IS curve without utility of wealth IS nominal interest rate i i IS (x, ) = + consumption c 17 / 45

21 LM curve (from demand for real money balances) LM nominal interest rate i consumption c 18 / 45

22 LM curve with money > bliss point (liquidity trap) nominal interest rate i i LM (x, m) =0 LM consumption c 19 / 45

23 IS & LM determine interest rate and AD nominal interest rate LM c AD (x,, m) IS consumption 20 / 45

24 IS & LM determine interest rate and AD nominal interest rate i LM IS c AD (x 0 <x,,m) consumption 20 / 45

25 AD curve apple c AD + (x,, m) = (1 + (x)) ( 0 (m)+! 0 (0)) market tightness x AD consumption c 21 / 45

26 AS curve market tightness x capacity: k quantity of labor services 22 / 45

27 AS curve capacity k market tightness x output: y = f(x) k quantity of labor services 22 / 45

28 AS curve output y capacity k market tightness x consumption: quantity of labor services 22 / 45

29 AS curve output capacity market tightness x consumption recruiting slack quantity of labor services 22 / 45

30 AS curve c AS (x) =(f(x) x) k market tightness x AS consumption c 22 / 45

31 AS curve and state of the economy overheating economy market tightness x AS efficient economy slack economy consumption c 23 / 45

32 General equilibrium market tightness x AS AD output general equilibrium slack capacity c y k quantity of labor services 24 / 45

33 Dynamical system is a source =( + )! 0 (0) 0 (m) 0 25 / 45

34 Immediate adjustment to shock 0 b a 26 / 45

35 Macroeconomic shocks 27 / 45

36 Increase in AD: fall in MU of wealth nominal interest rate AD increases LM IS consumption 28 / 45

37 Increase in AD: fall in MU of wealth labor market tightness AS output capacity AD quantity of labor services 28 / 45

38 Increase in AS: rise in capacity labor market tightness AS output capacity AD quantity of labor services 29 / 45

39 Monetary and fiscal policies 30 / 45

40 Increase in money supply AS output capacity market tightness x low tightness and output depressed AD consumption c 31 / 45

41 Increase in money supply nominal interest rate i AD increases IS LM consumption c 31 / 45

42 Increase in money supply output capacity market tightness x AS efficient tightness AD consumption c 31 / 45

43 Money supply in a liquidity trap AS output capacity market tightness x very low tightness and output very depressed AD consumption c 32 / 45

44 Money supply in a liquidity trap nominal interest rate i LM in liquidity trap IS LM consumption c 32 / 45

45 Money supply in a liquidity trap output capacity market tightness x AS inefficiently low tightness AD in liquidity trap consumption c 32 / 45

46 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: [ c IS δ + π i = (1 + τ(x)) ω (m h ) ] ε 33 / 45

47 Helicopter money always stimulates AD nominal interest rate AD increases LM IS consumption 34 / 45

48 Helicopter money always stimulates AD nominal interest rate IS AD increases LM in liquidity trap consumption 34 / 45

49 Alternative policy: tax on wealth government taxes wealth at rate τ a > 0 IS curve depends on wealth tax: c IS = [ δ + τ a ] + π i ε (1 + τ(x)) ω (0) 35 / 45

50 Tax on wealth always stimulates AD nominal interest rate AD increases LM IS consumption 36 / 45

51 Tax on wealth always stimulates AD nominal interest rate IS AD increases LM in liquidity trap consumption 36 / 45

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: [ ] c AD δ + π ε g = (1 + τ(x)) (φ (m) + ω + (0)) 1 + τ(x) 37 / 45

53 Government purchases stimulate AD labor market tightness AS output capacity AD quantity of labor services 38 / 45

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

55 Inflation and slack dynamics in the medium run 40 / 45

56 Simplifying assumptions 1. no money growth 2. no liquidity trap 41 / 45

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 42 / 45

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 0 ṗ = π p 1 + τ(x) = e p solution yields Phillips curve 43 / 45

59 Dynamical system describing equilibrium system of 3 ODEs: law of motion of price (ṗ), Phillips curve ( π), consumption Euler equation (ẋ) 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 44 / 45

60 Short-run/long-run effects of shocks increase in: x π p y aggregate demand money supply aggregate supply / 45