The Optimal Use of Government Purchases for Stabilization

Transcription

1 The Optimal Use of Government Purchases for Stabilization Pascal Michaillat (Brown) Emmanuel Saez (Berkeley) December / 35

2 the policies for business-cycle stabilization extensive research on monetary policy but monetary policy is sometimes constrained zero lower bound (Japan, USA, EU) monetary union (EU, USA) leading to high unemployment 2 / 35

3 in this paper: government purchases framework: matching model of the business cycle outcome: a formula linking optimal budget-balanced government purchases to unemployment gap unemployment multiplier social value of government consumption 3 / 35

4 relationship to other frameworks competitive model [Samuelson 1954] matching with efficient unemployment but matching also has inefficient unemployment disequilibrium model [Mankiw & Weinzierl 2011] matching with no matching cost but matching has supply in addition to demand New Keynesian model [Woodford 2010] alternative representation of slack but matching is a bit more tractable 4 / 35

5 a matching model 5 / 35

6 structure dynamic matching model building on Michaillat & Saez [2015] identical self-employed households government 2 consumption goods traded on 1 matching market public services + private services 1 asset for saving 6 / 35

7 matching market capacity of each household: k services household purchases: C(t) private services government purchases: G(t) public services output: Y(t) = C(t) + G(t) < k unemployment rate: u(t) = 1 Y(t)/k price of services: p(t) 7 / 35

8 matching function and market tightness number of vacancies: v(t) matching function: m(t) = ω (k Y(t)) η v(t) 1 η market tightness: x(t) = v(t)/(k Y(t)) selling rate and buying rate: f (x(t)) = m(t) k Y(t) = ω x(t)1 η q(x(t)) = m(t) v(t) = ω x(t) η 8 / 35

9 market flows relationships separate at rate s given x, output and unemployment converge to Y(x,k) = f (x) s + f (x) k, u(x) = s s + f (x) convergence to steady state is extremely fast so we assume Y(t) = Y(x(t),k) and u(t) = u(x(t) ) + 9 / 35

10 matching cost: ρ services per vacancy output (Y) = consumption (y) + matching cost Y = y + ρ v = y + s Y ρ q(x) matching wedge: τ(x + ) = s ρ/[q(x) s ρ] total consumption: y = Y/[1 + τ(x)] private consumption: c = C/[1 + τ(x)] public consumption: g = G/[1 + τ(x)] 10 / 35

11 supply structure: summary tightness x capacity: k public + private services 11 / 35

12 supply structure: summary capacity k tightness x output: Y (x, k) =(1 u(x)) k public + private services 11 / 35

13 supply structure: summary output Y(x,k) capacity k tightness x idle capacity: u(x) k public + private services 11 / 35

14 supply structure: summary output Y(x,k) capacity k tightness x consumption: Y (x, k) y(x, k) = 1+ (x) public + private services 11 / 35

15 supply structure: summary consumption y(x,k) output Y(x,k) capacity k tightness x matching cost: y(x, k) (x) public + private services 11 / 35

16 demand structure: example asset: housing h(t) in fixed supply H traded on a competitive market Iacoviello [2005] and Liu, Wang, Zha [2013] households choose c(t) and h(t) to maximize utility + e δ t [U (c,g) + V (h)]dt 0 subject to flow budget constraint dh dt = p [1 u(x)] k p [1 + τ(x)] c T 12 / 35

17 aggregate demand in the example market clearing on housing market: h = H aggregate demand from Euler equation: U c (cd (x,g,p),g) = p (1 + τ(x)) V (H) δ price of services relative to housing: p = p(g) assumption required in matching model here: general mechanism 13 / 35

18 equilibrium tightness is x = x(g) x y(x,k) c d (x, g, p(g)) + g x(g) y Y k y, Y 14 / 35

19 efficient unemployment rate x demand x(g)=x* u* supply y* Y* y, Y 15 / 35

20 inefficiently high unemployment rate x demand x* x(g) supply u > u* y Y y, Y 16 / 35

21 inefficiently low unemployment rate x demand x(g) u < u* x* supply y Y y, Y 17 / 35

22 formulas for optimal government purchases 18 / 35

23 government s problem households instantaneous utility is U (c, g) government purchases are financed by a lump-sum tax to maintain a balanced budget given x(g), the government chooses g to maximize U y(x(g),k) g,g }{{} c 19 / 35

24 correcting the Samuelson formula first-order condition of government s problem is 0 = U g U c + U c y x dx dg optimal government purchases satisfy 1 = MRS }{{ gc + y } x dx dg Samuelson formula }{{} correction correction: effect of g on welfare through x 20 / 35

25 introducing estimable statistics elasticity of substitution between g and c: 1 MRS gc 1 ε unemployment gap: unemployment multiplier: y x u u u g/c (g/c) (g/c) dx dg m y 1 u du dg 21 / 35

26 an implicit formula g/c (g/c) (g/c) ε 1 η m u u u (g/c) : Samuelson level ε: elasticity of substitution between g and c η: elasticity of matching wrt unemployment u u : unemployment gap m: unemployment multiplier 22 / 35

27 departure from the Samuelson level unemployment multiplier m < 0 m = 0 m > 0 u > u g/c < (g/c) g/c = (g/c) g/c > (g/c) u = u g/c = (g/c) g/c = (g/c) g/c = (g/c) u < u g/c > (g/c) g/c = (g/c) g/c < (g/c) 23 / 35

28 the marginal value of public services ε = 0: digging holes or building pyramids g/c = (g/c) : no stabilization ε + : perfect substitution u = u : perfect stabilization entirely fill unemployment gap ε (0, + ): medium substitution medium stabilization: g/c (g/c) but u u partially fill unemployment gap 24 / 35

29 making the formula explicit implicit formula: not useful for quantitative results because u in RHS responds to g/c in RHS we start from (g/c) and u 0 u : g/c (g/c) (g/c) ε m u(g/c) u 1 η u first-order Taylor expansion of u at u((g/c) ) = u 0 u u u u 0 u u z m g/c (g/c) (g/c) 25 / 35

30 an explicit formula optimal g/c depends on fixed quantities: g/c (g/c) (g/c) ε 1 η m ε 1 + z 1 η u0 u m2 u optimal u depends on fixed quantities: u u z ε 1 η m2 (u 0 u ) z = (g/y) [1 (g/y) ] (1 u 0 )/u 26 / 35

31 results with distortionary taxation endogenous capacity: U (c,g,k) with U / k < 0 linear income tax: T = τ L (1 u(x)) k everything remains valid but (g/c) lower because of tax distortions however: link between multipliers changes no tax distortions: m = dy/dg tax distortions: m > dy/dg with taxes, we may have dy/dg < 0 but m > 0 27 / 35

32 numerical application: Great Recession in the US 28 / 35

33 calibration starting point: winter u = 6%, u 0 = 9%, and G/Y = 16.5% assumption: u = 6% and (G/Y) = 16.5% ε: input into policy problem η [0.5, 0.7]: Pissarides & Petrongolo [2001] m [0.2,1] on average in the US [Ramey 2013] in addition, m may be higher in recessions see Auerbach & Gorodnichenko [2012] 29 / 35

34 optimal stimulus spending Increase in G (% GDP) 6% 4% 2% =1 0% Unemployment multiplier 30 / 35

35 optimal stimulus spending Increase in G (% GDP) 6% 4% 2% 0% Unemployment multiplier 30 / 35

36 optimal stimulus spending Increase in G (% GDP) 6% 4% 3.6% $520 billion 2% 0% Unemployment multiplier 30 / 35

37 optimal stimulus spending Increase in G (% GDP) 6% 4% 2% 0% Unemployment multiplier 30 / 35

38 optimal stimulus spending Increase in G (% GDP) 6% 4% 2% =2 =1 =0.5 0% Unemployment multiplier 30 / 35

39 unemployment with optimal stimulus Unemployment rate 9% 8% 7% =0.5 =1 =2 6% Unemployment multiplier 31 / 35

40 quality of approximations in formula Unemployment rate 11% 9% 7% 6.1% 5% 3% Constant G=Y Aggregate demand 32 / 35

41 quality of approximations in formula Unemployment multiplier Constant G=Y Aggregate demand 32 / 35

42 quality of approximations in formula Govt. purchases/output 21% 19% 17% 16.5% 15% 13% Constant G=Y Optimal G=Y Explicit formula Aggregate demand 32 / 35

43 summary: implications for policy 33 / 35

44 1. dy/dg > 1 is not necessary for stimulus spending sufficient: du/dg > 0 available evidence suggests that du/dg > 0 2. tax distortions smaller stimulus tax distortions only reduce the average G 3. tax distortions du/dg < 0 tax distortions do reduce dy/dg but they raise du/dg 34 / 35

45 4. low marginal value of g no stimulus optimal: use g to reduce unemployment gap except if g = bridges to nowhere 5. positive value of g filling the unemployment gap optimal: partially filling unemployment gap except if g and c are perfect substitutes 6. bang-for-the-buck logic does not hold strongest stimulus for m = 0.4 same stimulus for m = 0.1 and m = / 35