The Optimal Inflation Rate under Downward Nominal Wage Rigidity

Transcription

1 The Optimal Inflation Rate under Downward Nominal Wage Rigidity Mikael Carlsson and Andreas Westermark 1 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

2 Introduction/Motivation Puzzle introduced by Schmitt-Grohe & Uribe (2010): Cannot generate a significant positive Ramsey optimal inflation rate in standard monetary models, whereas central banks typically target 2% inflation. Tobin (1972): Inflation grease the wheels when wages cannot be adjusted downward. 2 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

3 Introduction/Motivation Puzzle introduced by Schmitt-Grohe & Uribe (2010): Cannot generate a significant positive Ramsey optimal inflation rate in standard monetary models, whereas central banks typically target 2% inflation. Tobin (1972): Inflation grease the wheels when wages cannot be adjusted downward. 2 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

4 Robust empirical finding: Money wages do not fall during an economic downturn. Data from personnel files: Altonji & Devereux (2000), Baker, Gibbs & Holmstrom (1994), Fehr & Goette (2005), and Wilson (1999). Survey/register data in Altonji & Devereux (2000), Akerlof, Dickens & Perry (1996), Dickens et. al. (2007), Fehr & Goette (2005), Holden & Wulfsberg (2008) and others. Interviews or surveys with wage setters like Agell & Lundborg (2003), and Bewley (1999). 3 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

5 Robust empirical finding: Money wages do not fall during an economic downturn. Data from personnel files: Altonji & Devereux (2000), Baker et al. (1994), Fehr & Goette (2005), and Wilson (1999). Survey/register data in Altonji & Devereux (2000), Akerlof et al. (1996), Dickens et. al. (2007), Fehr & Goette (2005), Holden & Wulfsberg (2008) and others. Interviews or surveys with wage setters like Agell & Lundborg (2003), and Bewley (1999). 3 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

6 Robust empirical finding: Money wages do not fall during an economic downturn. Data from personnel files: Altonji & Devereux (2000), Baker et al. (1994), Fehr & Goette (2005), and Wilson (1999). Survey/register data in Altonji & Devereux (2000), Akerlof et al. (1996), Dickens et. al. (2007), Fehr & Goette (2005), Holden & Wulfsberg (2008) and others. Interviews or surveys with wage setters like Agell & Lundborg (2003), and Bewley (1999). 3 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

7 Robust empirical finding: Money wages do not fall during an economic downturn. Data from personnel files: Altonji & Devereux (2000), Baker et al. (1994), Fehr & Goette (2005), and Wilson (1999). Survey/register data in Altonji & Devereux (2000), Akerlof et al. (1996), Dickens et. al. (2007), Fehr & Goette (2005), Holden & Wulfsberg (2008) and others. Interviews or surveys with wage setters like Agell & Lundborg (2003), and Bewley (1999). 3 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

8 Figure: Empirical distribution of yearly nominal wage changes for stayers in the US during the period (PSID, cleaned from measurement error) 4 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

9 Economic consequences: Real wages might be too high for some firms, leading to too much unemployment. Asymmetric dynamics. Inflation leads to decrease in real wages for firms that don t renegotiate wages. 5 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

10 Economic consequences: Real wages might be too high for some firms, leading to too much unemployment. Asymmetric dynamics. Inflation leads to decrease in real wages for firms that don t renegotiate wages. 5 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

11 Economic consequences: Real wages might be too high for some firms, leading to too much unemployment. Asymmetric dynamics. Inflation leads to decrease in real wages for firms that don t renegotiate wages. 5 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

12 Economic consequences: Real wages might be too high for some firms, leading to too much unemployment. Asymmetric dynamics. Inflation leads to decrease in real wages for firms that don t renegotiate wages. 5 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

13 Related literature Kim & Ruge-Murcia (2010) Fagan & Messina (2009) 6 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

14 Related literature Kim & Ruge-Murcia (2010) Fagan & Messina (2009) 6 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

15 Purpose The purpose of this paper is to study the implications for monetary policy in terms of optimal average inflation when: There is a role for money as a medium of exchange. Declining nominal wages might not be a viable margin for adjustment. State dependent price/wage setting - Lucas critique. Deterministic aggregate productivity growth - important, since it pushes optimal inflation down substantially. Stochastic idiosyncratic productivity to match the empirical wage change distribution. 7 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

16 Purpose The purpose of this paper is to study the implications for monetary policy in terms of optimal average inflation when: There is a role for money as a medium of exchange. Declining nominal wages might not be a viable margin for adjustment. State dependent price/wage setting - Lucas critique. Deterministic aggregate productivity growth - important, since it pushes optimal inflation down substantially. Stochastic idiosyncratic productivity to match the empirical wage change distribution. 7 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

17 Purpose The purpose of this paper is to study the implications for monetary policy in terms of optimal average inflation when: There is a role for money as a medium of exchange. Declining nominal wages might not be a viable margin for adjustment. State dependent price/wage setting - Lucas critique. Deterministic aggregate productivity growth - important, since it pushes optimal inflation down substantially. Stochastic idiosyncratic productivity to match the empirical wage change distribution. 7 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

18 Purpose The purpose of this paper is to study the implications for monetary policy in terms of optimal average inflation when: There is a role for money as a medium of exchange. Declining nominal wages might not be a viable margin for adjustment. State dependent price/wage setting - Lucas critique. Deterministic aggregate productivity growth - important, since it pushes optimal inflation down substantially. Stochastic idiosyncratic productivity to match the empirical wage change distribution. 7 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

19 Purpose The purpose of this paper is to study the implications for monetary policy in terms of optimal average inflation when: There is a role for money as a medium of exchange. Declining nominal wages might not be a viable margin for adjustment. State dependent price/wage setting - Lucas critique. Deterministic aggregate productivity growth - important, since it pushes optimal inflation down substantially. Stochastic idiosyncratic productivity to match the empirical wage change distribution. 7 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

20 Purpose The purpose of this paper is to study the implications for monetary policy in terms of optimal average inflation when: There is a role for money as a medium of exchange. Declining nominal wages might not be a viable margin for adjustment. State dependent price/wage setting - Lucas critique. Deterministic aggregate productivity growth - important, since it pushes optimal inflation down substantially. Stochastic idiosyncratic productivity to match the empirical wage change distribution. 7 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

21 Overview 1 The model. 2 (Ramsey policy) 3 Calibration. 4 Numerical results. 8 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

22 Overview 1 The model. 2 (Ramsey policy) 3 Calibration. 4 Numerical results. 8 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

23 Overview 1 The model. 2 (Ramsey policy) 3 Calibration. 4 Numerical results. 8 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

24 Overview 1 The model. 2 (Ramsey policy) 3 Calibration. 4 Numerical results. 8 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

25 The Economic Environment Intermediate-goods Firms, Prices and Wages State dependent price setting as in Dotsey, King & Wolman (1999) and Lie (2010). Menu cost c p of changing prices. Follows cdf G P. Let α j t denote the endogenous probability of adjusting prices in period t, given that the firm last adjusted it s price j periods ago. There is J > 1 such that α J = 1. 9 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

26 The Economic Environment Intermediate-goods Firms, Prices and Wages State dependent price setting as in Dotsey et al. (1999) and Lie (2010). Menu cost c p of changing prices. Follows cdf G P. Let α j t denote the endogenous probability of adjusting prices in period t, given that the firm last adjusted it s price j periods ago. There is J > 1 such that α J = 1. 9 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

27 The Economic Environment Intermediate-goods Firms, Prices and Wages State dependent price setting as in Dotsey et al. (1999) and Lie (2010). Menu cost c p of changing prices. Follows cdf G P. Let α j t denote the endogenous probability of adjusting prices in period t, given that the firm last adjusted it s price j periods ago. There is J > 1 such that α J = 1. 9 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

28 The Economic Environment Intermediate-goods Firms, Prices and Wages State dependent price setting as in Dotsey et al. (1999) and Lie (2010). Menu cost c p of changing prices. Follows cdf G P. Let α j t denote the endogenous probability of adjusting prices in period t, given that the firm last adjusted it s price j periods ago. There is J > 1 such that α J = 1. 9 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

29 Firms faces demand Y j t = ( P j t P t ) σ Y t, (1) and uses a wholesale good as input. The wholesale sector is perfectly competitive where the market price is denoted by p w t. The price is chosen such that [ ] v 0 P 0 t t = max p w P 0 t Yt 0 P t t ( +E t Λ t,t+1 β α 1 t+1 v0 t+1 + (1 α 1 t+1 E t Λ t,t+1 βp w t+1 α1 t+1 Ξ 1,t+1, ( )) ) v 1 P 0 t t+1 P t+1 where α 1 t+1 Ξ 1,t+1 is expected price adjustment costs Λ t,t+1 is the ratio of Lagrange multipliers for consumers. (2) 10 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

30 Firms faces demand Y j t = ( P j t P t ) σ Y t, (1) and uses a wholesale good as input. The wholesale sector is perfectly competitive where the market price is denoted by p w t. The price is chosen such that [ ] v 0 P 0 t t = max p w P 0 t Yt 0 P t t ( +E t Λ t,t+1 β α 1 t+1 v0 t+1 + (1 α 1 t+1 E t Λ t,t+1 βp w t+1 α1 t+1 Ξ 1,t+1, ( )) ) v 1 P 0 t t+1 P t+1 where α 1 t+1 Ξ 1,t+1 is expected price adjustment costs Λ t,t+1 is the ratio of Lagrange multipliers for consumers. (2) 10 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

31 and Ξ j,t = 1 α j t ( G 1 P α j ) t 0 xdg P (x). The values v j t evolve according to ( v j P j ) [ t P j ] t t = p w t Y j P t P t t ( ( ( ) +E t Λ t,t+1 β α j+1 t+1 v0 t α j+1 t+1 v j+1 t+1 E t Λ t,t+1 βp w t+1 αj+1 t+1 Ξ j+1,t+1. P j t P t+1 )) 11 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

32 Price adjustment probabilities are α j t = G P ( v 0 t vj t p w t ). 12 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

33 Households Consumption purchases are subject to a proportional transaction cost as in Schmitt-Grohe & Uribe (2004). Payoff function ] E t β [u r t (c r ) κ L (h ir ) 1+ξ r=t i 1 + ξ di. (3) Given consumption c t and price level P t total consumption cost is ( )) ct P t c t (1 + s where P t is the price level, m t is real balances and m t s = A c t m t + B m t c t 2 AB. 13 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

34 Households Consumption purchases are subject to a proportional transaction cost as in Schmitt-Grohe & Uribe (2004). Payoff function ] E t β [u r t (c r ) κ L (h ir ) 1+ξ r=t i 1 + ξ di. (3) Given consumption c t and price level P t total consumption cost is ( )) ct P t c t (1 + s where P t is the price level, m t is real balances and m t s = A c t m t + B m t c t 2 AB. 13 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

35 Households Consumption purchases are subject to a proportional transaction cost as in Schmitt-Grohe & Uribe (2004). Payoff function ] E t β [u r t (c r ) κ L (h ir ) 1+ξ r=t i 1 + ξ di. (3) Given consumption c t and price level P t total consumption cost is ( )) ct P t c t (1 + s where P t is the price level, m t is real balances and m t s = A c t m t + B m t c t 2 AB. 13 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

36 The budget constraint is given by ( ct P t c t (1 + s m t P t 1 m t 1 + ϖ t + W t, )) + P t m t + b t R t + θ t+1 P t (F t Z t ) where b t is bonds R t the interest rate, θ t+1 is the share of intermediate product firms F t, the value of firms and Z t dividends. ϖ t is wealth at the start of time and W t = 1 0 E t W it di + (1 n t ) b r, where b r representing the value of home production. 14 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

37 The budget constraint is given by ( ct P t c t (1 + s m t P t 1 m t 1 + ϖ t + W t, )) + P t m t + b t R t + θ t+1 P t (F t Z t ) where b t is bonds R t the interest rate, θ t+1 is the share of intermediate product firms F t, the value of firms and Z t dividends. ϖ t is wealth at the start of time and W t = 1 0 E t W it di + (1 n t ) b r, where b r representing the value of home production. 14 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

38 The budget constraint is given by ( ct P t c t (1 + s m t P t 1 m t 1 + ϖ t + W t, )) + P t m t + b t R t + θ t+1 P t (F t Z t ) where b t is bonds R t the interest rate, θ t+1 is the share of intermediate product firms F t, the value of firms and Z t dividends. ϖ t is wealth at the start of time and W t = 1 0 E t W it di + (1 n t ) b r, where b r representing the value of home production. 14 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

39 Labor market I - production and matching Several wholesale firms with one employee that sell a good y it produced using labor (hours) h it given productivity a it to intermediate goods firms with technology where y it = (a it h it ) 1 γ. a it = e γ r t ε a it with γ r the growth rate of aggregate productivity and ε a it an idiosyncratic shock. 15 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

40 Matches separate with probability 1 ρ. Constant-returns matching function where and v t is aggregate vacancies. m a t = σ m u σ a t v 1 σ a t, u t = 1 n t. 16 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

41 Matches separate with probability 1 ρ. Constant-returns matching function where and v t is aggregate vacancies. m a t = σ m u σ a t v 1 σ a t, u t = 1 n t. 16 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

42 Labor market II - Wage determination Wholesale firms and workers adjust wages with some positive endogenously determined probability α j w t in the j w th period following the last renegotiation (state dependent). Note that α J w t = 1 for some J w > Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

43 Labor market II - Wage determination Wholesale firms and workers adjust wages with some positive endogenously determined probability α j w t in the j w th period following the last renegotiation (state dependent). Note that α J w t = 1 for some J w > Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

44 Bargaining When a firm/household pair renegotiates the wage, bargaining takes place in a setup similar to the model by Holden (1994). In the model, the parties bargain every period. Each bargaining round starts with one of the parties making a bid, then the other party responds yes or no. There are two key features of the Holden (1994) model. 18 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

45 Bargaining When a firm/household pair renegotiates the wage, bargaining takes place in a setup similar to the model by Holden (1994). In the model, the parties bargain every period. Each bargaining round starts with one of the parties making a bid, then the other party responds yes or no. There are two key features of the Holden (1994) model. 18 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

46 Bargaining When a firm/household pair renegotiates the wage, bargaining takes place in a setup similar to the model by Holden (1994). In the model, the parties bargain every period. Each bargaining round starts with one of the parties making a bid, then the other party responds yes or no. There are two key features of the Holden (1994) model. 18 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

47 Bargaining When a firm/household pair renegotiates the wage, bargaining takes place in a setup similar to the model by Holden (1994). In the model, the parties bargain every period. Each bargaining round starts with one of the parties making a bid, then the other party responds yes or no. There are two key features of the Holden (1994) model. 18 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

48 First: If the response is no, there is a choice whether to continue bargaining in good faith and post a counter offer or enter into disagreement. If the latter choice is made, there is a probability that the match breaks down and the wage is determined in a standard Rubinstein-Ståhl fashion. Moreover, in case a party initiate bargaining under disagreement, both parties face their own known fixed disagreement cost (randomly drawn at the beginning of each period). This cost may be due to deteriorating firm/worker relationships. Similar to Holden (1994), but with probability of breakdown instead of strikes. 19 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

49 First: If the response is no, there is a choice whether to continue bargaining in good faith and post a counter offer or enter into disagreement. If the latter choice is made, there is a probability that the match breaks down and the wage is determined in a standard Rubinstein-Ståhl fashion. Moreover, in case a party initiate bargaining under disagreement, both parties face their own known fixed disagreement cost (randomly drawn at the beginning of each period). This cost may be due to deteriorating firm/worker relationships. Similar to Holden (1994), but with probability of breakdown instead of strikes. 19 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

50 First: If the response is no, there is a choice whether to continue bargaining in good faith and post a counter offer or enter into disagreement. If the latter choice is made, there is a probability that the match breaks down and the wage is determined in a standard Rubinstein-Ståhl fashion. Moreover, in case a party initiate bargaining under disagreement, both parties face their own known fixed disagreement cost (randomly drawn at the beginning of each period). This cost may be due to deteriorating firm/worker relationships. Similar to Holden (1994), but with probability of breakdown instead of strikes. 19 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

51 First: If the response is no, there is a choice whether to continue bargaining in good faith and post a counter offer or enter into disagreement. If the latter choice is made, there is a probability that the match breaks down and the wage is determined in a standard Rubinstein-Ståhl fashion. Moreover, in case a party initiate bargaining under disagreement, both parties face their own known fixed disagreement cost (randomly drawn at the beginning of each period). This cost may be due to deteriorating firm/worker relationships. Similar to Holden (1994), but with probability of breakdown instead of strikes. 19 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

52 First: If the response is no, there is a choice whether to continue bargaining in good faith and post a counter offer or enter into disagreement. If the latter choice is made, there is a probability that the match breaks down and the wage is determined in a standard Rubinstein-Ståhl fashion. Moreover, in case a party initiate bargaining under disagreement, both parties face their own known fixed disagreement cost (randomly drawn at the beginning of each period). This cost may be due to deteriorating firm/worker relationships. Similar to Holden (1994), but with probability of breakdown instead of strikes. 19 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

53 Second: There is an old contract in place at the firm and if there is no new wage agreement, workers work according to the old contract. As pointed out by Holden (1994), this is a common feature of many western European countries. As soon as there is bargaining under disagreement, payoffs are determined in a standard Rubinstein-Ståhl bargaining game and the disagreement costs is paid out of the parties respective pockets. A credible threat leads to immediate renegotiation and hence no disagreement in equilibrium. To derive only downward nominal rigidity, asymmetries in disagreement costs are required. Gives a standard formulation of the bargaining problem when there is disagreement, as in Christoffel, Kuester & Linzert (2009). 20 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

54 Second: There is an old contract in place at the firm and if there is no new wage agreement, workers work according to the old contract. As pointed out by Holden (1994), this is a common feature of many western European countries. As soon as there is bargaining under disagreement, payoffs are determined in a standard Rubinstein-Ståhl bargaining game and the disagreement costs is paid out of the parties respective pockets. A credible threat leads to immediate renegotiation and hence no disagreement in equilibrium. To derive only downward nominal rigidity, asymmetries in disagreement costs are required. Gives a standard formulation of the bargaining problem when there is disagreement, as in Christoffel et al. (2009). 20 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

55 Second: There is an old contract in place at the firm and if there is no new wage agreement, workers work according to the old contract. As pointed out by Holden (1994), this is a common feature of many western European countries. As soon as there is bargaining under disagreement, payoffs are determined in a standard Rubinstein-Ståhl bargaining game and the disagreement costs is paid out of the parties respective pockets. A credible threat leads to immediate renegotiation and hence no disagreement in equilibrium. To derive only downward nominal rigidity, asymmetries in disagreement costs are required. Gives a standard formulation of the bargaining problem when there is disagreement, as in Christoffel et al. (2009). 20 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

56 Second: There is an old contract in place at the firm and if there is no new wage agreement, workers work according to the old contract. As pointed out by Holden (1994), this is a common feature of many western European countries. As soon as there is bargaining under disagreement, payoffs are determined in a standard Rubinstein-Ståhl bargaining game and the disagreement costs is paid out of the parties respective pockets. A credible threat leads to immediate renegotiation and hence no disagreement in equilibrium. To derive only downward nominal rigidity, asymmetries in disagreement costs are required. Gives a standard formulation of the bargaining problem when there is disagreement, as in Christoffel et al. (2009). 20 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

57 Second: There is an old contract in place at the firm and if there is no new wage agreement, workers work according to the old contract. As pointed out by Holden (1994), this is a common feature of many western European countries. As soon as there is bargaining under disagreement, payoffs are determined in a standard Rubinstein-Ståhl bargaining game and the disagreement costs is paid out of the parties respective pockets. A credible threat leads to immediate renegotiation and hence no disagreement in equilibrium. To derive only downward nominal rigidity, asymmetries in disagreement costs are required. Gives a standard formulation of the bargaining problem when there is disagreement, as in Christoffel et al. (2009). 20 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

58 Second: There is an old contract in place at the firm and if there is no new wage agreement, workers work according to the old contract. As pointed out by Holden (1994), this is a common feature of many western European countries. As soon as there is bargaining under disagreement, payoffs are determined in a standard Rubinstein-Ståhl bargaining game and the disagreement costs is paid out of the parties respective pockets. A credible threat leads to immediate renegotiation and hence no disagreement in equilibrium. To derive only downward nominal rigidity, asymmetries in disagreement costs are required. Gives a standard formulation of the bargaining problem when there is disagreement, as in Christoffel et al. (2009). 20 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

59 Value for the family of a worker at firm i is in period t is, V j w t ( w j w t, a t ) ( )) ( ) (h = w j w t h t w j t w j 1+ξ w w t, a t κ L t, a t 1 + ξ +β E t Λ t,t+1 ϑ (a t+1, a t ) [ ( +ρ a t+1 A ρα j w+1 t+1 ( w j w+1 1 α j w+1 t+1 + (1 ρ) U t+1 ], ) ( ) w 0 t+1, a t+1 )) V 0 t+1 t+1, a t+1 ( w j w+1 t+1, a t+1 V j w+1 t+1 ( w j w+1 t+1, a t+1 where ϑ (a t+1, a t ) denotes the transition probability from productivity state a t to a t+1. ( ) Since the firm has the right to manage, hours h t w j w t, a t are determined by the firm by maximizing the per-period payoff in p w t y it w j w t h it. (4) ) 21 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

60 Value for the family of a worker at firm i is in period t is, V j w t ( w j w t, a t ) ( )) ( ) (h = w j w t h t w j t w j 1+ξ w w t, a t κ L t, a t 1 + ξ +β E t Λ t,t+1 ϑ (a t+1, a t ) [ ( +ρ a t+1 A ρα j w+1 t+1 ( w j w+1 1 α j w+1 t+1 + (1 ρ) U t+1 ], ) ( ) w 0 t+1, a t+1 )) V 0 t+1 t+1, a t+1 ( w j w+1 t+1, a t+1 V j w+1 t+1 ( w j w+1 t+1, a t+1 where ϑ (a t+1, a t ) denotes the transition probability from productivity state a t to a t+1. ( ) Since the firm has the right to manage, hours h t w j w t, a t are determined by the firm by maximizing the per-period payoff in p w t y it w j w t h it. (4) ) 21 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

61 The value when being unemployed is U t = b r + βe t Λ t,t+1 (s t+1 V x,t+1 + (1 s t+1 ) U t+1 ), where s t is the hiring rate and where V x,t = J w 1 j w =0 a t A ( ) ( ) ω j w t w j w+1 t+1, a t V j w t w j w t, a t, (5) ( ) where ω j w t w j w+1 t+1, a t denotes the share of workers with wage w j w t and productivity a t or, if new hires have flexible wages ( ) ( ) V x,t = ω 0 t w 1 t+1, a t Vt 0 w 0 t, a t, (6) a t A The bargaining surplus is (similar to Christoffel et al. (2009)) ( ) ( ) H j w t w j w t, a t = V j w t w j w t, a t U t, (7) 22 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

62 The value when being unemployed is U t = b r + βe t Λ t,t+1 (s t+1 V x,t+1 + (1 s t+1 ) U t+1 ), where s t is the hiring rate and where V x,t = J w 1 j w =0 a t A ( ) ( ) ω j w t w j w+1 t+1, a t V j w t w j w t, a t, (5) ( ) where ω j w t w j w+1 t+1, a t denotes the share of workers with wage w j w t and productivity a t or, if new hires have flexible wages ( ) ( ) V x,t = ω 0 t w 1 t+1, a t Vt 0 w 0 t, a t, (6) a t A The bargaining surplus is (similar to Christoffel et al. (2009)) ( ) ( ) H j w t w j w t, a t = V j w t w j w t, a t U t, (7) 22 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

63 The value for the firm is ( ) ( )) J j w t w j w t, a t = p w t (a t h t w j 1 γ ( ) w j t, a t w w t h t w j w t, a t Φ +β Λ t,t+1 ϑ (a t+1, a t ) [ ( a t+1 A α j w+1 t+1 ( 1 α j w+1 t+1 w j w+1 ) ( t+1, a t+1 ( w j w+1 t+1, a t+1 ρj 0 t+1 )) ( )) w 0 t+1, a t+1 ( w j w+1 t+1, a t+1 ρj j w+1 t+1 where Φ are fixed consisting of a fixed labor cost Φ L and a fixed capital cost Φ K as in Christoffel et al. (2009). (8) )], 23 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

64 In case there is bargaining under disagreement, wages are determined according to ( ( )) ϕ ( ( 1 ϕ max H 0 Wit 0 t w 0 t, a t Jt 0 w 0 t t)), a, (9) where w 0 t = W0 t P t and ϕ denotes the bargaining power of workers. A firm that last renegotiated wages j periods ago can credibly call for bargaining under disagreement if the gain from adjusting the wage is larger that the disagreement cost. Similarly, the worker calls for bargaining under disagreement if the gain from adjusting is larger than the disagreement cost. 24 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

65 In case there is bargaining under disagreement, wages are determined according to ( ( )) ϕ ( ( 1 ϕ max H 0 Wit 0 t w 0 t, a t Jt 0 w 0 t t)), a, (9) where w 0 t = W0 t P t and ϕ denotes the bargaining power of workers. A firm that last renegotiated wages j periods ago can credibly call for bargaining under disagreement if the gain from adjusting the wage is larger that the disagreement cost. Similarly, the worker calls for bargaining under disagreement if the gain from adjusting is larger than the disagreement cost. 24 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

66 In case there is bargaining under disagreement, wages are determined according to ( ( )) ϕ ( ( 1 ϕ max H 0 Wit 0 t w 0 t, a t Jt 0 w 0 t t)), a, (9) where w 0 t = W0 t P t and ϕ denotes the bargaining power of workers. A firm that last renegotiated wages j periods ago can credibly call for bargaining under disagreement if the gain from adjusting the wage is larger that the disagreement cost. Similarly, the worker calls for bargaining under disagreement if the gain from adjusting is larger than the disagreement cost. 24 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

67 Hiring and other constraints Entrant firms chooses vacancies so that the vacancy cost is equal to the expected value of filling a vacancy. Thus, determined by κ t v t = m a J w 1 t β j w =0 a t+1 A Flow equation of prices and wages. ( ) ( ) E t ω j w t w j w+1 t+1, a t+1 Λ t,t+1 J j w t+1 w j w t+1, a t+1. p j t = pj 1 t π t, (10) 25 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

68 Hiring and other constraints Entrant firms chooses vacancies so that the vacancy cost is equal to the expected value of filling a vacancy. Thus, determined by κ t v t = m a J w 1 t β j w =0 a t+1 A Flow equation of prices and wages. ( ) ( ) E t ω j w t w j w+1 t+1, a t+1 Λ t,t+1 J j w t+1 w j w t+1, a t+1. p j t = pj 1 t π t, (10) 25 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

69 Optimal policy - the Ramsey Problem We solve for the Ramsey optimal policy. The policymaker maximizes ] E t β [u r t (c r ) κ L (h ir ) 1+ξ r=t i 1 + ξ di subject to the constraints from the competitive equilibrium described above. 26 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

70 Calibration u (c t ) = log c t and Deep Parameters 1 β σ 10 ξ 2 γ 1/3 prod gr κ Φ K 1/3 Φ L and Deep Parameters 2 b r 0.48 ρ 0.9 σ a 0.6 σ m 0.83 A B ϕ 0.5 κ L Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

71 Productivity growth is on a quarterly basis. To model the idiosyncratic productivity process, we use a four-state Markov chain with a quarterly persistence of 0.6 (bounded from above due to numerical reasons) and with a ratio between the max and the min state of Replacement rate (the ratio of home production value to the average wage) of around Hiring cost is around 0.14% of steady state output. 28 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

72 Productivity growth is on a quarterly basis. To model the idiosyncratic productivity process, we use a four-state Markov chain with a quarterly persistence of 0.6 (bounded from above due to numerical reasons) and with a ratio between the max and the min state of Replacement rate (the ratio of home production value to the average wage) of around Hiring cost is around 0.14% of steady state output. 28 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

73 Productivity growth is on a quarterly basis. To model the idiosyncratic productivity process, we use a four-state Markov chain with a quarterly persistence of 0.6 (bounded from above due to numerical reasons) and with a ratio between the max and the min state of Replacement rate (the ratio of home production value to the average wage) of around Hiring cost is around 0.14% of steady state output. 28 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

74 Productivity growth is on a quarterly basis. To model the idiosyncratic productivity process, we use a four-state Markov chain with a quarterly persistence of 0.6 (bounded from above due to numerical reasons) and with a ratio between the max and the min state of Replacement rate (the ratio of home production value to the average wage) of around Hiring cost is around 0.14% of steady state output. 28 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

75 Bargaining power set to ϕ = 0.5. Intermediate goods producing firms price adjustment costs follows a beta distribution with parameters l = 2.1, r = 1 and upper bound Disagreement costs in the wholesale sector also follows the beta distribution with parameters l H = 2.1, r H = 1 and l J = 2.1, r J = 1 and upper bounds B H for workers and B J for firms. To find the bounds B H and B J, we fit the dispersion of yearly wage changes in the model (given a yearly inflation rate of 2 %) to the empirical dispersion of yearly wage changes in the US during the period using a minimum distance estimator. 29 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

76 Bargaining power set to ϕ = 0.5. Intermediate goods producing firms price adjustment costs follows a beta distribution with parameters l = 2.1, r = 1 and upper bound Disagreement costs in the wholesale sector also follows the beta distribution with parameters l H = 2.1, r H = 1 and l J = 2.1, r J = 1 and upper bounds B H for workers and B J for firms. To find the bounds B H and B J, we fit the dispersion of yearly wage changes in the model (given a yearly inflation rate of 2 %) to the empirical dispersion of yearly wage changes in the US during the period using a minimum distance estimator. 29 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

77 Bargaining power set to ϕ = 0.5. Intermediate goods producing firms price adjustment costs follows a beta distribution with parameters l = 2.1, r = 1 and upper bound Disagreement costs in the wholesale sector also follows the beta distribution with parameters l H = 2.1, r H = 1 and l J = 2.1, r J = 1 and upper bounds B H for workers and B J for firms. To find the bounds B H and B J, we fit the dispersion of yearly wage changes in the model (given a yearly inflation rate of 2 %) to the empirical dispersion of yearly wage changes in the US during the period using a minimum distance estimator. 29 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

78 Bargaining power set to ϕ = 0.5. Intermediate goods producing firms price adjustment costs follows a beta distribution with parameters l = 2.1, r = 1 and upper bound Disagreement costs in the wholesale sector also follows the beta distribution with parameters l H = 2.1, r H = 1 and l J = 2.1, r J = 1 and upper bounds B H for workers and B J for firms. To find the bounds B H and B J, we fit the dispersion of yearly wage changes in the model (given a yearly inflation rate of 2 %) to the empirical dispersion of yearly wage changes in the US during the period using a minimum distance estimator. 29 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

79 Figure: Empirical distribution of yearly nominal wage changes for stayers in the US during the period (PSID, cleaned from measurement error) 30 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

80 Figure: The yearly nominal wage change distribution implied by the model. 31 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

81 This procedure yields parameters B H = for workers and for firms B J = When imposing a symmetry restriction, we find the upper bounds to equal B H = B J = Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

82 Results Baseline: Inflation around 1.2 %. Table: Yearly optimal inflation rate under the Ramsey policy Asymmetric Symmetric Flexible wage frictions wage frictions wages Baseline Flex wages for new hires Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

83 Conclusions Inflation around 1.2 percent a year. Decomposition : Flexible wages gives deflation of 0.96 percent. Symmetric adjustment costs gives inflation of 0.36 percent. Asymmetric adjustment costs gives inflation of 1.2 percent. 34 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

84 Conclusions Inflation around 1.2 percent a year. Decomposition : Flexible wages gives deflation of 0.96 percent. Symmetric adjustment costs gives inflation of 0.36 percent. Asymmetric adjustment costs gives inflation of 1.2 percent. 34 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

85 Conclusions Inflation around 1.2 percent a year. Decomposition : Flexible wages gives deflation of 0.96 percent. Symmetric adjustment costs gives inflation of 0.36 percent. Asymmetric adjustment costs gives inflation of 1.2 percent. 34 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

86 Conclusions Inflation around 1.2 percent a year. Decomposition : Flexible wages gives deflation of 0.96 percent. Symmetric adjustment costs gives inflation of 0.36 percent. Asymmetric adjustment costs gives inflation of 1.2 percent. 34 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

87 Conclusions Inflation around 1.2 percent a year. Decomposition : Flexible wages gives deflation of 0.96 percent. Symmetric adjustment costs gives inflation of 0.36 percent. Asymmetric adjustment costs gives inflation of 1.2 percent. 34 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

88 Extensions One distortion at a time. Hosios condition. Aggregate uncertainty/dynamics - first and second order. Then the ZLB/liquidity trap becomes interesting as well. 35 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

89 Extensions One distortion at a time. Hosios condition. Aggregate uncertainty/dynamics - first and second order. Then the ZLB/liquidity trap becomes interesting as well. 35 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

90 Extensions One distortion at a time. Hosios condition. Aggregate uncertainty/dynamics - first and second order. Then the ZLB/liquidity trap becomes interesting as well. 35 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

91 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market: Inflation redistributes between workers and firm: Consider some π and average real wage in economy 36 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

92 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market: Inflation redistributes between workers and firm: Consider some π and average real wage in economy 36 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

93 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market: Inflation redistributes between workers and firm: Consider some π and average real wage in economy 36 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

94 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market: Inflation redistributes between workers and firm: Consider some π and average real wage in economy 36 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

95 w j π 1 2 j 37 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

96 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market Inflation redistributes between workers and firm Consider some π and average equilibrium real wage in economy Decrease π to π and keep average real wage constant (NOT equilibrium) 38 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

97 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market Inflation redistributes between workers and firm Consider some π and average equilibrium real wage in economy Decrease π to π and keep average real wage constant (NOT equilibrium) 38 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

98 w j π 1 2 j 39 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

99 w j π π < π 1 2 j 40 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

100 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market Inflation redistributes between workers and firm Consider some π and average equilibrium real wage in economy Decrease π to π and keep average real wage constant (NOT equilibrium) Firm gains and worker looses because wage costs are a good for worker and a bad for firm Equilibrium average real wage goes up 41 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

101 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market Inflation redistributes between workers and firm Consider some π and average equilibrium real wage in economy Decrease π to π and keep average real wage constant (NOT equilibrium) Firm gains and worker looses because wage costs are a good for worker and a bad for firm Equilibrium average real wage goes up 41 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

102 Intuition Real wages too high due to DNWR. - higher inflation erodes real wages at firms that have too high wages No DNWR: Still inefficient labor market Inflation redistributes between workers and firm Consider some π and average equilibrium real wage in economy Decrease π to π and keep average real wage constant (NOT equilibrium) Firm gains and worker looses because wage costs are a good for worker and a bad for firm Equilibrium average real wage goes up 41 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

103 Additional - resource constraint The resource constraint is 1 0 y it di + (1 n t ) b r = J 1 ) ( ( ))) ω j (p j ε ct t (c t 1 + s m j=0 t + κ t x 2 J 1 it n it 1di + ω j Ξ j,t j=0 42 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

104 Additional - consumer first-order conditions ( ) ( ct u c (c t ) = λ t (1 + s + c t s ct m t m t ( E t βλ t+1 = λ t ( c2 t m 2 s ct m t t λ t R t = βe t λ t π t+1 ) ) + 1 ) ) 1 m t 43 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

105 Additional - employment transition equations Employment transition equations are and, for j > 0, n 0 t = n j t = (ρ + x j t J w ( ) ρ + x 0 t α j wn j 1 t 1 j=1 ) ( ) 1 α j w n j 1 t 1 44 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

106 Additional - firm values The value of the firm is F (w it ) = p w t y it w it n it κ t 2 (x it) 2 n it 1 r k t k it + βe t Λ t,t+1 F (w it+1 ) Let J t (w it ) be the value of a worker at the firm, given that the worker is at the firm, ( ) J t (w it ) = p w t y it w j t n it r k t k it + βe t Λ t,t+1 F (w it+1 ) n it 45 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

107 The value of an additional employee is F t (w it ) n it 1 = κ t + 2 x2 it ( ( p w t y it w it n it r k t k it) = κ t 2 x2 it + (ρ + x it) J t (w it ) n it + βe t Λ t,t+1 F (w it+1 ) n it ) (ρ + x it ) The effect on firm profits of an additional employee is J t (w it ) = p w t (1 γ) y ( it w n it + βe t Λ t,t+1 κ ) t it 2 x2 it+1 + (ρ + x it+1) J t (w it+1 ) 46 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

108 Additional - adjustment probabilities Let df j t = J t (w 0 t du j t = H t (w 0 t ) J t ( w j t ) ) ( ) H t w j t The fraction of firms that calls for bargaining under disagreement is ( 1 ) if B F <F j t G F df j t 0 df j t BF 0 df j t < 0 47 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

109 Similarly, the fraction of workers that has an incentive to call for bargaining under disagreement to force a renegotiation of the wage contract is ( 1 ) if B U <du j t G U du j t 0 du j t BU 0 du j t < 0 48 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

110 The adjustment probabilities are then ( ) 1 ( ) if B F <df j t or if BU <du j t G F df j t + G U du j ( t ) ( ) 0 df j α j t = G U du j t G F df j t BF t and 0 du j ( t ) BU G F df j t 0 df j t BF and du j ( t ) < 0 G U du j t df j t < 0 and 0 duj t BU 0 df j t < 0 and duj t < 0 49 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

111 Agell, J. & Lundborg, P. (2003), Survey evidence on wage rigidity and unemployment: Sweden in the 1990s, Scandinavian Journal of Economics 105, Akerlof, G., Dickens, W. T. & Perry, G. (1996), The macroeconomics of low inflation, Brookings Papers on Economic Activity (1). Altonji, J. & Devereux, P. (2000), The extent and consequences of downward nominal wage rigidity, in S. Polachek, ed., Research in Labor Economics, Vol. 19, Elsevier, pp Baker, G., Gibbs, M. & Holmstrom, B. (1994), The wage policy of a firm, Quarterly Journal of Economics 109, Bewley, T. (1999), Why Wages Don t Fall During a Recession., Harvard University Press, Cambridge, MA. Christoffel, K., Kuester, K. & Linzert, T. (2009), The role of labor markets for euro area monetary policy. FRBP Working Paper No Dotsey, M., King, R. & Wolman, A. (1999), State-dependent pricing and the general equilibrium dynamics of money and output, Quarterly Journal of Economics 114, Fagan, G. & Messina, J. (2009), Downward wage rigidity and optimal steady state inflation. ECB Working Paper Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

112 Fehr, E. & Goette, L. (2005), Robustness and real consequences of nominal wage rigidity, Journal of Monetary Economics 52, Holden, S. (1994), Wage bargaining and nominal rigidities, European Economic Review 38, Holden, S. & Wulfsberg, F. (2008), Downward nominal wage rigidity in the oecd, The B.E. Journal of Macroeconomics 8 : Iss. 1 (Advances), Article 15. Kim, J. & Ruge-Murcia, F. (2010), Monetary policy when wages are downwardly rigid: Friedman meets tobin. forthcoming Journal of Economic Dynamics and Control. Lie, D. (2010), State-dependent pricing and optimal monetary policy. Federal Reserve Bank of Boston Working Paper Schmitt-Grohe, S. & Uribe, M. (2004), Optimal fiscal and monetary policy under sticky prices, Journal of Economic Theory 114, Schmitt-Grohe, S. & Uribe, M. (2010), The optimal rate of inflation. Forthcoming in Handbook in Macroeconomics. Tobin, J. (1972), Inflation and unemployment, American Economic Review 62, Mikael Carlsson and Andreas Westermark Optimal Inflation Rate

113 Wilson, B. A. (1999), Wage rigidity: A look inside the firm. Mimeo, Federal Reserve Board of Governors. 49 Mikael Carlsson and Andreas Westermark Optimal Inflation Rate