Optimal Fiscal and Monetary Policy: Equivalence Results

Transcription

1 Optimal Fiscal and Monetary Policy: Equivalence Results by Isabel Correia, Juan Pablo Nicolini, and Pedro Teles, JPE 2008 presented by Taisuke Nakata March 1st, 2011 () Optimal Fiscal and Monetary Policy March 1st, / 15

2 Introduction Question: How does sticky price affect the conduct of fiscal/monetary policy? A model in which a fraction α of firms set prices one period in advance. One-period non-state-contingent debt. Labor income and consumption tax. Answer: Not at all. Ramsey allocations are the same under sticky prices and under flexible prices. Policies that implement the Ramsey allocations are also the same. () Optimal Fiscal and Monetary Policy March 1st, / 15

3 Environment Discrete Time, Infinite Horizon. s t (s 0,s 1,s 2,...,s t ) where s t S t 3 types of agents: A representative household. Government. A continuum of firms, indexed by i [0,1]: Firms i [0, α): sticky-price firms that set their prices one period in advance. Firms i [α, 1]: flexible-price firms that set their price contemporaneously. y i,t (s t ) = A(s t )n i (s t ) 3 goods: [ C k (s t 1 θ/(θ 1),(k ) = 0 c k,i(s t ) di] (θ 1)/θ = 1,2) [ ] G(s t 1 θ/(θ 1) ) = 0 g i(s t ) (θ 1)/θ di () Optimal Fiscal and Monetary Policy March 1st, / 15

4 Household subject to max {c 1,i (s t ),c 2,i (s t ),N(s t ),M(s t ), B(s t ),B(s t+1 )} t=0,i [0,1] P c (s t )C 1 (s t ) M(s t ) M(s t ) + B(s t ) + t=0 s t+1 s t Q(s t+1 s t )B(s t+1 ) W(s t ) s t β t π(s t )u(c 1 (s t ), C 2 (s t ), N(s t )) where 1 W(s t+1 ) M(s t ) + R(s t ) B(s t ) + B(s t+1 ) pi c (st )c 1,i (s t )di 1 0 pi c (st )c 2,i (s t )di + [1 τ n (s t )]w(s t )N(s t ) 0 pi c (st ) = [1 + τ c (s t )]p i (s t ) [ 1 ] 1/(1 θ) P c (s t ) = [pi c (st )] (1 θ) di 0 W(s 0 ) = 0 () Optimal Fiscal and Monetary Policy March 1st, / 15

5 Household FONCs c 1,i (s t ) C 1 (s t ) u C,1 (s t ) u C,2 (s t ) u C,2(s t ) u N (s t ) u C,1 (s t ) P c (s t ) = [ pi c(st ) ] θ P c (s t & c 2,i(s t ) ) C 2 (s t ) = [ pi c(st ) ] θ P c (s t ) = R(s t )( 1) = P c (s t ) [1 τ n (s t )]w(s t ) = βr(s t )E t [ uc,1 (s t+1 ) P c (s t+1 ) Q(s t+1 s t ) = βπ(s t+1 s t ) u C,1(s t+1 ) u C,1 (s t ) ] P c (s t ) P c (s t+1 ) () Optimal Fiscal and Monetary Policy March 1st, / 15

6 Government 1 min gi (s t ),i [0,1] 0 pc i (st )g i (s t )di [ 1 θ/(θ 1) subject to 0 g i(s t ) di] (θ 1)/θ = G(s t ) g i (s t ) = G(s t ) [ p c i (st ) P c (s t ) ] θ A government policy consists of {g i (s t ),τ c (s t ),τ n (s t ),R(s t ),M g (s t ), B g (s t )} t=0 () Optimal Fiscal and Monetary Policy March 1st, / 15

7 Firms Flexible Price Firms (i [α,1]): max pi (s t ) p i (s t )y i (s t ) w(s t )n i (s t ) subject to y i,t (s t ) = [ p c i (st ) P c (s t )] θy (s t ) & y i,t (s t ) = A(s t )n i (s t ) p i (s t ) = θ w(s t ) θ 1 A(s t ) p f (s t ) Sticky Price Firms (i [0,α)): max pi (s t 1 ) s t+1 s t 1 Q(st+1 s t 1 )[p i (s t 1 )y i (s t ) w(s t )n i (s t )] subject to the same constraints as above. p i (s t 1 ) = E t 1 [v(s t )p f (s t )] p s (s t 1 ) At t=0, they charge p 1 which is exogenously given. () Optimal Fiscal and Monetary Policy March 1st, / 15

8 Firms Flexible Price Firms (i [α,1]): max pi (s t ) p i (s t )y i (s t ) w(s t )n i (s t ) subject to y i,t (s t ) = [ p c i (st ) P c (s t )] θy (s t ) & y i,t (s t ) = A(s t )n i (s t ) p i (s t ) = θ w(s t ) θ 1 A(s t ) p f (s t ) Sticky Price Firms (i [0,α)): max pi (s t 1 ) s t+1 s t 1 Q(st+1 s t 1 )[p i (s t 1 )y i (s t ) w(s t )n i (s t )] subject to the same constraints as above. p i (s t 1 ) = E t 1 [v(s t )p f (s t )] p s (s t 1 ) At t=0, they charge p 1 which is exogenously given. () Optimal Fiscal and Monetary Policy March 1st, / 15

9 Market-Clearing c 1,i (s t ) + c 2,i (s t ) + g i (s t ) = A(s t )n i (s t ) for each i. [ ] C 1(s t ) + C 2(s t ) + G(s t 1 [ p c ) i (s t ) θ P c (s )] di = A(s t )N(s t ) t N(s t ) = 1 0 n i(s t )di M(s t ) = M g (s t ) B(s t ) = B g (s t ) B(s t+1 ) = 0 0 () Optimal Fiscal and Monetary Policy March 1st, / 15

10 Competitive Equilibrium A competitive equilibrium consists of Allocations: {C 1 (s t ),C 2 (s t ),N(s t )} t=0, {c 1,i(s t ),c 2,i (s t ),n i (s t )} t=0 for all i, and {M(s t ), B(s t ),B(s t+1 )} t=0. Prices/Policies: {p i (s t ),P c (s t ),w(s t ),Q(s t+1 s t ), g i (s t ),G(s t ),τ c (s t ),τ n (s t ),M g (s t ), B g (s t ),R(s t )} t=0 such that the allocation solves the household s problem given prices/policies. p i (s t ) solves the firm s problem given prices/policies. g i (s t ) solves the government s problem given p i (s t ),P c (s t ), and G(s t ). markets clear. () Optimal Fiscal and Monetary Policy March 1st, / 15

11 Two objects of interest Ω f : The set of implementable allocations for goods and labor {C 1 (s t ),C 2 (s t ),N(s t )} t=0 when α = 0 (i.e., under flexible prices). Ω s (α): The set of implementable allocations for goods and labor {C 1 (s t ),C 2 (s t ),N(s t )} t=0 when 0 < α < 1 (i.e., under sticky prices.) () Optimal Fiscal and Monetary Policy March 1st, / 15

12 Logic: A Big Picture Show Ω f Ω s (α) for any 0 < α < 1 Define Ω R ( relaxed set ) such that Ω f Ω s (α) Ω R for any 0 < α < 1. Show optimal allocation in Ω R is in Ω f. Therefore, optimal allocation in Ω s (α) is in Ω f for any 0 < α < 1. () Optimal Fiscal and Monetary Policy March 1st, / 15

13 Ω f Proposition 1 (Part 1): Ω f is characterized by E 0 t=0 β t[ ] u C,1 (s t )C 1 (s t ) + u C,2 (s t )C 2 (s t ) + u N (s t )N(s t ) = 0 u C,1 (s t ) u C,2 (s t ) C 1 (s t ) + C 2 (s t ) + G(s t ) = A(s t )N(s t ) Proposition 1 (Part 2): Given P c (s 0 ), each allocation in Ω f is implemented with a unique path for {R(s t ),M g (s t ), B g (s t ), 1+τc (s t ) 1 τ n (s t ), w(st ) p f (s t ),[1 + τc (s t )]p f (s t )} t=0. Corollary 1: There are multiple combinations of {τ c (s t ),τ n (s t ),w(s t ),p f (s t )} consistent with each implementable allocation. One of them supports constant p f (s t ) = P for any P > 0. () Optimal Fiscal and Monetary Policy March 1st, / 15

14 Ω f Proposition 1 (Part 1): Ω f is characterized by E 0 t=0 β t[ ] u C,1 (s t )C 1 (s t ) + u C,2 (s t )C 2 (s t ) + u N (s t )N(s t ) = 0 u C,1 (s t ) u C,2 (s t ) C 1 (s t ) + C 2 (s t ) + G(s t ) = A(s t )N(s t ) Proposition 1 (Part 2): Given P c (s 0 ), each allocation in Ω f is implemented with a unique path for {R(s t ),M g (s t ), B g (s t ), 1+τc (s t ) 1 τ n (s t ), w(st ) p f (s t ),[1 + τc (s t )]p f (s t )} t=0. Corollary 1: There are multiple combinations of {τ c (s t ),τ n (s t ),w(s t ),p f (s t )} consistent with each implementable allocation. One of them supports constant p f (s t ) = P for any P > 0. () Optimal Fiscal and Monetary Policy March 1st, / 15

15 Ω s (α) Proposition 2 (Part 1): Ω f Ω s (α) for any 0 < α < 1. Proof: Let p f (s t ) = p 1. Then, p s (s t 1 ) = p f (s t ) = p 1. The equilibrium conditions of the sticky price model collapse to the ones under flexible prices, plus the restriction that p f (s t ) = p 1. Proposition 2 (Part 2): Each allocation in Ω f can be implemented with policies that are independent of α. () Optimal Fiscal and Monetary Policy March 1st, / 15

16 Ω s (α) Proposition 2 (Part 1): Ω f Ω s (α) for any 0 < α < 1. Proof: Let p f (s t ) = p 1. Then, p s (s t 1 ) = p f (s t ) = p 1. The equilibrium conditions of the sticky price model collapse to the ones under flexible prices, plus the restriction that p f (s t ) = p 1. Proposition 2 (Part 2): Each allocation in Ω f can be implemented with policies that are independent of α. () Optimal Fiscal and Monetary Policy March 1st, / 15

17 Ω R The relaxed set Ω R is the set of aggregate allocations {C 1 (s t ),C 2 (s t ),N(s t )} t=0 such that there exist consumer prices {p c i (st ),P c (s t )} satisifying: E 0 t=0 β t[ ] u C,1 (s t )C 1 (s t ) + u C,2 (s t )C 2 (s t ) + u N (s t )N(s t ) = 0 [ C 1 (s t ) + C 2 (s t ) + G(s t ) ] 1 0 u C,1 (s t ) u C,2 (s t ) [ pi c(st ) ] θdi = A(s t P c (s t )N(s t ) ) Proposition 3 (Part 1): Ω s (α) Ω R for any 0 < α < 1 (thus Ω f Ω R ) Why? Any allocation in Ω s (α) satisfies the conditions above. () Optimal Fiscal and Monetary Policy March 1st, / 15

18 Ramsey allocations and policies Proposition 3 (Part 2): Optimal allocation in Ω R is in Ω f. Proof: Optimal allocation is attained when 1 0 [ p c i (st ) P c (s t )] θdi is minimized. This distortion term is minimized when prices are the same across firms. When that happens, the conditions collapse to the ones under flexible prices. Since Ω f Ω s (α) Ω R, optimal allocation in Ω s (α) is in Ω f for any 0 < α < 1. Proposition 4: Policies/prices that implement the Ramsey allocation do not depend on α. () Optimal Fiscal and Monetary Policy March 1st, / 15

19 Ramsey allocations and policies Proposition 3 (Part 2): Optimal allocation in Ω R is in Ω f. Proof: Optimal allocation is attained when 1 0 [ p c i (st ) P c (s t )] θdi is minimized. This distortion term is minimized when prices are the same across firms. When that happens, the conditions collapse to the ones under flexible prices. Since Ω f Ω s (α) Ω R, optimal allocation in Ω s (α) is in Ω f for any 0 < α < 1. Proposition 4: Policies/prices that implement the Ramsey allocation do not depend on α. () Optimal Fiscal and Monetary Policy March 1st, / 15

20 Ramsey allocations and policies Proposition 3 (Part 2): Optimal allocation in Ω R is in Ω f. Proof: Optimal allocation is attained when 1 0 [ p c i (st ) P c (s t )] θdi is minimized. This distortion term is minimized when prices are the same across firms. When that happens, the conditions collapse to the ones under flexible prices. Since Ω f Ω s (α) Ω R, optimal allocation in Ω s (α) is in Ω f for any 0 < α < 1. Proposition 4: Policies/prices that implement the Ramsey allocation do not depend on α. () Optimal Fiscal and Monetary Policy March 1st, / 15