The Optimal Use of Government Purchases for Macroeconomic Stabilization. Pascal Michaillat (LSE) & Emmanuel Saez (Berkeley) July 2015

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1 The Optimal Use of Government Purchases for Macroeconomic Stabilization Pascal Michaillat (LSE) & Emmanuel Saez (Berkeley) July / 29

2 You are Barack Obama. It is early The unemployment rate has reached 9% and the federal funds rate is 0%. By how much should you increase government purchases? we propose an answer based on 2 sufficient statistics: the government-purchases multiplier the elasticity of substitution between government and personal consumptions 2 / 29

By how much should you increase government purchases?

3 A model of unemployment with government purchases 3 / 29

4 Overview dynamic continuous-time model measure 1 of identical self-employed households benevolent government a matching market where labor services are traded not all services are sold in equilibrium unemployment modeling follows Michaillat and Saez [QJE, 2015] market represents both a labor and a product market 4 / 29

traded not all services are sold in equilibrium unemployment modeling follows

5 The matching market for labor services the productive capacity of each household is 1 households buy C(t) services government buys G(t) services households sell Y(t) = C(t) + G(t) < 1 services the unemployment rate is u(t) = 1 Y(t) services are traded through long-term relationships 5 / 29

services households sell Y(t) = C(t) + G(t) < 1 services the

6 Matching function and market tightness households and government advertise v(t) vacancies matching function: m = ω (1 Y(t)) η v(t) 1 η market tightness: x(t) v(t)/(1 Y(t)) rates at which new relationships are formed: f (x(t)) + m 1 Y(t) = ω x(t)1 η q(x(t) ) m v(t) = ω x(t) η 6 / 29

market tightness: x(t) v(t)/(1 Y(t)) rates at which new relationships

7 Market flows relationships separate at rate s output is a state variable: Ẏ = f (x) (1 Y) s Y assumption: flows are balanced, f (x) (1 Y) = s Y output, unemployment become jump variables Y(x + ) = f (x) s + f (x), u(x ) = s s + f (x) 7 / 29

balanced, f (x) (1 Y) = s Y output, unemployment become

8 Matching cost: ρ services per vacancy Y }{{} gross output = y }{{} + ρ v }{{} net output matching cost market flows are balanced so s Y = v q(x) and Y = y + ρ s Y q(x) [ Y 1 s ρ ] = y q(x) [ Y = 1 + s ρ q(x) s ρ Y = [1 + τ(x + )] y ] y 8 / 29

9 Gross and net consumptions net consumptions enter households utility function C: gross personal consumption c = C/(1 + τ(x)): net personal consumption G: gross government consumption g = G/(1 + τ(x)): net government consumption 9 / 29

c = C/(1 + τ(x)): net personal consumption G: gross

10 Gross output, unemployment, net output market tightness x capacity: 1 labor services 10 / 29

11 Gross output, unemployment, net output market tightness x gross output: Y (x) = f(x) s + f(x) unemployment rate: u(x) =1 Y (x) capacity 1 labor services 10 / 29

12 Gross output, unemployment, net output gross output Y(x) capacity 1 market tightness x net output: y(x) = Y (x) 1+ (x) matching cost: y(x) (x) labor services 10 / 29

13 Feasible allocation and equilibrium a feasible allocation [c,g,y,x] satisfies y = y(x) and y = c + g an equilibrium function is g [c,g,y,x] the equilibrium function reduces to g x(g) 11 / 29

14 Efficient unemployment: x(g) = x x* c g u* y* Y* 1 12 / 29

15 Inefficiently high unemployment: x(g) < x x* x c g u > u* y Y 1 13 / 29

16 Inefficiently low unemployment: x(g) > x x c g u < u* x* y Y 1 14 / 29

17 Sufficient-statistics formula for optimal government purchases 15 / 29

18 Value of government purchases we follow Samuelson [REStat, 1954] households instantaneous utility is U (c, g) U is homothetic so the marginal rate of substitution MRS gc U / g U / c is a decreasing function of g/c = G/C MRS gc is a decreasing function of G/Y 16 / 29

the marginal rate of substitution MRS gc U / g U / c is a

19 Government s problem given an equilibrium function x(g), the government s problem is to determine g to maximize welfare U y(x(g)) g,g } {{ } c 17 / 29

20 Formula in sufficient statistics the first-order condition is 0 = U g U c + U c dy dx dx dg U / g 1 = U / c + dy dx dx dg optimal government purchases satisfy 1 = MRS gc + dy } {{ } dx dx dg Samuelson formula } {{ } correction term 18 / 29

dg optimal government purchases satisfy 1 = MRS gc + dy }

21 Optimal G/Y versus Samuelson s (G/Y) x y(x) G/Y = (G/Y)* x* u* y 19 / 29

22 Optimal G/Y versus Samuelson s (G/Y) x y(x) G/Y > (G/Y)* x* u* y 19 / 29

23 Optimal G/Y versus Samuelson s (G/Y) x y(x) G/Y < (G/Y)* x* u* y 19 / 29

24 Assessment of actual US government purchases 20 / 29

25 Introducing estimable statistics linearize abstract formula around efficient tightness x* and Samuelson s ratio (G/C)*: 1 MRS gc = dy dg 1 G/C (G/C) (G/C) x x x dy dg 21 / 29

26 Introducing estimable statistics linearize abstract formula around efficient tightness x* and Samuelson s ratio (G/C)*: 1 MRS gc = dy dg 1 G/C (G/C) (G/C) x x x dy dg 21 / 29

27 Introducing estimable statistics linearize abstract formula around efficient tightness x* and Samuelson s ratio (G/C)*: 1 MRS gc = dy dg 1 G/C (G/C) (G/C) x x x dy dg 21 / 29

28 An implicit formula G/C (G/C) (G/C) ε dy dg x x x dy/dg: government-purchases multiplier ε: elasticity of substitution between g and c ε = 0: Leontief preferences, bridges to nowhere ε = 1: Cobb-Douglas preferences, benchmark case ε + : linear preferences, perfect substitute 22 / 29

29 Public employment/private employment in the US G/C (G/C)* / 29

30 Labor market tightness in the US x x* / 29

31 Assessment of US government purchases G/C (G/C) (G/C) + ϵ dy dg x x x G/C (G/C) (G/C) / 29

32 Assessment of US government purchases G/C (G/C) (G/C) + ϵ dy dg x x x x x x 25 / 29

33 Assessment of US government purchases G/C (G/C) (G/C) + ϵ dy dg x x x excessive government purchases dy dg =0.5 insufficient government purchases / 29

34 Assessment of US government purchases G/C (G/C) (G/C) + ϵ dy dg x x x excessive government purchases dy dg =0.03 insufficient government purchases / 29

35 Optimal response to an increase in unemployment rate from 5.9% to 9% 26 / 29

36 An explicit formula consider a shock to x(g) bringing the economy from [x,(g/c) ] to [x 0,(G/C) ] the optimal response of government purchases is G/C (G/C) (G/C) ε dy x(g/c) x dg x 27 / 29

37 An explicit formula consider a shock to x(g) bringing the economy from [x,(g/c) ] to [x 0,(G/C) ] the optimal response of government purchases is G/C (G/C) (G/C) ε dg dy 1 + ε ( x 0 x ) dy 2 (G/Y) dg (1 (G/Y) ) x (1 η) u 27 / 29

38 (u u )/u (1 η) (x x )/x ln(u/u ) ln(x/x ) 28 / 29

39 Optimal increase in G/Y when unemployment rises from 5.9% to 9% 5 Percentage points Multiplier 29 / 29

40 Optimal increase in G/Y when unemployment rises from 5.9% to 9% 5 Percentage points Multiplier 29 / 29

41 Optimal increase in G/Y when unemployment rises from 5.9% to 9% 5 Percentage points Multiplier 29 / 29

42 Optimal increase in G/Y when unemployment rises from 5.9% to 9% 5 Percentage points Multiplier 29 / 29

43 Optimal increase in G/Y when unemployment rises from 5.9% to 9% Percentage points 5 4 =2 3 =1 2 1 = Multiplier 29 / 29

44 Optimal increase in G/Y when unemployment rises from 5.9% to 9% Billions of USD =2 450 = = Multiplier 29 / 29

45 Optimal increase in G/Y when unemployment rises from 5.9% to 9% Percentage points 5 increase in G/Y reduction in u Multiplier 29 / 29

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