A Comparison of 2 popular models of monetary policy Petros Varthalitis Athens University of Economics & Business June 2011 Varthalitis (AUEB) Calvo vs Rotemberg June 2011 1 / 45
Aim of this work Obviously, CB s need models with nominal rigidities which allow a real role for monetary policy. 1. Calvo model (Staggered Pricing, Calvo 1983 JME) 2. Rotemberg model (Rotemberg 1982 JPE) The aim of this study is to compare them. This is interesting both in: Policy Level: Lombardo & Vestin ECB wp & EL (2008). Theory Level: Ascari & Rossi (2010) wp, Dellas & Collard (2007) J.Macro Varthalitis (AUEB) Calvo vs Rotemberg June 2011 2 / 45
Key Features of the model (in words): 1. Households consume, work, save in money, private bonds & capital. 2. Government nances government expenditures with seignorage revenues and lump-sum taxes. 3. Firms operate under Monopolistic Competition, i will examine two models of price setting: a. Calvo b. Rotemberg Varthalitis (AUEB) Calvo vs Rotemberg June 2011 3 / 45
Household s Problem (both models) indexed by j max fc t (j),c t (i,j),x t (i,j),m t (j),n t (j),k t (j),b t (j)g t=0 subject to: β t U (c t (j), m t (j), n t (j), g t (j)) t=0 (1) c t (j) + x t (j) + b t (j) + m t (j) P = R t 1 t 1 P t b t 1 (j) + P t 1 P t m t 1 (j) + w t n t (j) + rt k k t 1 (j) τ l t(j) (2) k t (j) = (1 δ) k t 1 (j) + x t (j) (3) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 4 / 45
Household s Problem (both models) There are i di erentiated goods.using the DS aggregator we have: Z 1 c t (j) 0 Z 1 x t (j) 0 c t (i, j) ε ε 1 x t (i, j) ε ε 1 ε ε 1 di ε ε 1 di (4) (5) P t c t (j) = P t x t (j) = Z 1 0 Z 1 0 P t (i) c t (i, j) di (6) P t (i) x t (i, j) di (7) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 5 / 45
Households- FOC (both models) [Euler for Bonds] : [Euler for Capital] : 1 c t+1 (j) = σ P t βe t R t c t (j) σ (8) P t+1 c t (j) σ = βe t h(1 δ) + r k t+1 i c t+1 (j) σ (9) [Money Demand] : [Labour Supply] : ν [m t (j)] c t (j) σ = R t 1 (10) R t n t (j) φ c t (j) σ = w t (11) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 6 / 45
Households- FOC (both models) [Demand for i] : where P t = Pt (i) ε c t (i, j) + x t (i, j) = fc t (j) + x t (j)g (12) R 1 0 P t (i) 1 ε 1 1 ε di. P t Varthalitis (AUEB) Calvo vs Rotemberg June 2011 7 / 45
Government (both models) Government Budget Constraint: P t τ l t + M t = M t 1 + P t g t (13) where Z 1 g t 0 g t (i) ε ε 1 ε ε 1 di (14) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 8 / 45
Firms Step A: Cost Minimization (both models) Firms i 2 [0, 1]. Perfect Competition in factor markets. subject to: Ψ t (Y t (i)) = n min W t n t (i) + Rt k k t fn t (i), k t 1 (i)g o 1 (i) (15) Y t (i) = A t k t 1 (i) a n t (i) 1 a (16) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 9 / 45
Firms- Calvo Model 1 Monopolistic Competition. Y t (i), i 2 [0, 1]. 2 In Calvo model each period t there are two fraction of rms: Cannot Reoptimize 1 θ C Reoptimize θ C where θ C is an exogenous probability. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 10 / 45
Firms- Calvo Model A rm which cannot reoptimize (belongs to θ C ) just set its previous period price: P t (i) = P t 1 (i) (17) θ C is Calvo nominal rigidity parameter. In Calvo model there is heterogeneity across rms. Symmetry fails. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 11 / 45
Firms- Calvo Model Step B: Pro t Maximization max Pt (i) k=0 subject to: θ C k Et fq t,t+k (P t (i)y t+k (i) Ψ t+k (Y t+k (i)))g (18) P Y t+k (i) = t (i) ε Yt+k d (19) P t+k and given Yt d C t + X t + G t is aggregate demand, Q t,t+k is the stochastic discount factor and Ψ t (..) is the minimum cost function. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 12 / 45
Firms Calvo Price Setting Rule ( θ C k P Et Q t (i) ε t,t+k Y d P k=0 t+k t+k P t (i) ε ε 1 Ψ0 t+k ) = 0 (20) Firm i sets its price P t (i) at period t as the weighted sum of the expected nominal marginal costs of the next k periods. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 13 / 45
Firms- Rotemberg Model 1 Monopolistic Competition. Y t (i), i 2 [0, 1]. 2 Each rm i faces a convex price adjustment cost. All rms solve an identical problem each period: PAC = ϑr 2 Pt (i) P t 1 (i) 1 2 Y t (i) (21) where ϑ R measures nominal price rigidity. As ϑ R increases so does nominal price rigidity. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 14 / 45
Firms- Rotemberg Model In Rotemberg model, all rms i solve an identical problem. So, there is symmetry. ϑ R is Rotemberg nominal rigidity parameter. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 15 / 45
Firms- Rotemberg Model Step B: Pro t Maximization max fp t (i)g t=0 Q t,t+1 eω t = 8 < Q t,t+1 : t=0 9 P t Y t (i) Ψ r t (Y t (i)) = ϑ R Pt (i) 2 2 P t 1 1 Yt (i) (i) ; (22) P t (i) subject to: Pt (i) ε Y t (i) = Y t (23) P t Varthalitis (AUEB) Calvo vs Rotemberg June 2011 16 / 45
Firms - Rotemberg Rotemberg price setting rule Q t,t+1 (1 ε) Y t + Q t εψ r t 0 (.) Y t + Q t,t+2 ϑ R Pt+1 1 P t = Q t ϑ R Pt P t 1 1 Pt+1 Pt P t 1 (24) P t Varthalitis (AUEB) Calvo vs Rotemberg June 2011 17 / 45
Aggregation (in words) In Calvo model there is an aggregation issue which arises from the presence of two fraction of rms in each period t, θ C which cannot reoptimize and 1 θ C which reoptimize. In Rotemberg model, rms are symmetric and aggregation is trivial. (Next slides appendix) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 18 / 45
Calvo Model: Aggregation Household aggregation is trivial: x t Z 1 0 x t (j) dj (25) where x t (j) = c t (j) x t (j) k t (j) n t (j) eω t (j) b t (j) m t (j) 0 and x t = 0 c t x t k t n t eω t b t m t. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 19 / 45
Calvo Model: Aggregation Aggregate Supply: Y s t Z 1 0 Y t (i) ε ε 1 ε ε 1 di (26) Each rm i faces an identical technology: Y t (i) = A t k t 1 (i) a n t (i) 1 a (27) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 20 / 45
Calvo Model: Aggregation Aggregation: Y s t Z 1 0 A t k t (i) a n t (i) 1 a ε 1 ε ε ε 1 di =? = At kt a 1nt 1 a (28) We denote two auxiliary indices Y 0 t R 1 0 A tk t (i) a n t (i) 1 a di and R P 0 1 t 0 P t (i) di ε 1 ε. We can proove that (an analytical appendix will be available): Yt s 1 a = h 0 i ε A t kt nt 1 a (29) P t P t Varthalitis (AUEB) Calvo vs Rotemberg June 2011 21 / 45
Calvo Model: Price Dispersion Yun (1996) denotes: t " # P 0 ε t (30) P t This is a measure of price dispersion and measures the loss of output in the Calvo economy due to price dispersion, t 1 where equality holds only when P t P t 1 = 1. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 22 / 45
Calvo Model: Evolution of Aggregate Price Level Z Pt 1 ε = θpt 1 1 ε + S θ (P t (i)) 1 ε di = θp 1 ε t 1 + (1 θ) (P t ) 1 ε (31) All rms i which reoptimize at period t solves an identical problem so P t (i) = P t. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 23 / 45
Aggregation Rotemberg No aggregation problems in Rotemberg model. Households (j) are symmetric, so: x t = Z 1 Firms i are symmetric, i.e. aggregate supply: 0 x t (j) dj (32) Y s t Z 1 0 Y t (i) ε ε 1 = A t k a t 1n 1 a t ε ε 1 Z 1 di = A t k t (i) a n t (i) 1 a ε ε 1 ε ε 1 di 0 (33) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 24 / 45
Transformations In Calvo model we de ne 3 new endogenous variables which subsitute the price levels P t P 0 t Pt 0 : Θ t P t P t (34) t Π t " # P 0 ε t (35) P t P t P t 1 (36) In Rotemberg model we only subsitute the aggregate price level with in ation: Π t P t P t 1 (37) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 25 / 45
Decentralized Equilibrium Calvo Model (given policy) 13 endogenous variables: Yt c t k t n t x t w t r k t m t s l t mc t Π t 0 and t Θ t 0 for 13 equilibrium equations. Given policy R t s g t 0, and an exogenous shock At. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 26 / 45
Decentralized Equilibrium Calvo Model h i c σ σ t = βe t c t+1 rt+1 k + (1 δ) (38) c t σ σ 1 = βe t c t+1 (39) R t Π t+1 ν m t c σ = R t 1 (40) t R t n t φ c t σ = w t (41) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 27 / 45
Decentralized Equilibrium Calvo Model k t = (1 δ) k t 1 + x t (42) w t = mc t (1 a)a t k t 1 a n t a (43) r k t = mc t aa t k a 1 t 1 n1 a t (44) τ l t + m t = m t 1 1 Π t + g t (45) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 28 / 45
Y t = c t + x t + g t (46) θ C k Et k=0 8 >< >: Q t,t+k 2 6 4 k i=1 Θ t Π t+i 3 7 5 ε Y t+k 0 B @ Θ t Π t ε ε 1 MC t+k k i=0 Π t+i 9 1 >= C A = 0 >; (47) Y t = 1 t A t k a t 1n 1 a t (48) [Π t ] 1 ε = θ C + (1 θ C ) [Θ t Π t ] 1 ε (49) t = θ t 1 Π ε t + (1 θ) Θ ε t (50) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 29 / 45
Decentralized Equilibrium Rotemberg Model (given policy) 11 endogenous variables: Yt c t k t n t x t w t r k t m t s l t mc t Π t 0 for 11 equilibrium equations. Given policy R t s g t 0, and an exogenous shock At. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 30 / 45
Decentralized Equilibrium Rotemberg Model h i c σ σ t = βe t c t+1 rt+1 k + (1 δ) (51) c t σ σ 1 = βe t c t+1 (52) R t Π t+1 ν m t c σ = R t 1 (53) t R t n t φ c t σ = w t (54) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 31 / 45
Decentralized Equilibrium Rotemberg Model k t = (1 δ) k t 1 + x t (55) w t = mc t (1 a)a t k t 1 a n t a (56) r k t = mc t aa t k a 1 t 1 n1 a t (57) τ l t + m t = m t 1 1 Π t + g t (58) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 32 / 45
Decentralized Equilibrium Rotemberg Model Y t = c t + x t + g t + ϑr 2 (Π t 1) 2 Y t (59) Y t = A t k a t 1n 1 a t (60) Q t,t+1 (1 ε) Y t + Q t,t+1 εmc t Y t + Q t,t+2 ϑ R (Π t+1 1) Π t+1 Y t = Q t,t+1 ϑ R (Π t 1) Π t Y t (61) Varthalitis (AUEB) Calvo vs Rotemberg June 2011 33 / 45
Comparison of Calvo versus Rotemberg (Nominal Rigidity) The di erent price setting mechanism causes: Price dispersion in Calvo model. A (price adjustment) cost in Rotemberg model. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 34 / 45
Comparison of Calvo versus Rotemberg DEs 1. Steady-State 2. Transition (Dynamics up to a rst-order approximation). Varthalitis (AUEB) Calvo vs Rotemberg June 2011 35 / 45
Steady-state with zero long-run in ation Calvo If Π = 1, we can proove that: = Θ = 1 (62) So price dispersion vanishes. Rotemberg So the price adjustment cost is zero. PAC = 0 (63) The two steady-state solutions are identical. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 36 / 45
Dynamics around zero in ation steady-state Production function in Calvo: by t = ba t + abk t 1 + (1 a) bn t b t (64) However b t = θb t 1 is a deterministic univariate autoregressive process which does not a ect the dynamics of the other endogenous variables of the model. In Rotemberg model the price adjustment cost is zero both o and in steady state so the rst-order approximation of the resource constraint is identical in two models. The equilibrium relations of the two models (except from the price setting rules) are equivalent up to a rst-order approximation. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 37 / 45
In ation Dynamics around zero in ation steady-state The linear New Keynesian Phillips Curves: Calvo: bπ t = βe t bπ t+1 + λ C cmc t (65) Rotemberg: bπ t = βe t bπ t+1 + λ R cmc t (66) where λ C and λ R depend on structural parameters of each model. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 38 / 45
Condition for equivalent Dynamics around zero in ation steady-state The two models deliver equivalent dynamics up to a rst-order when: λ C = λ R (67) 1 θ C 1 βθ C θ C = ε 1 ϑ R (68) Calvo θ C (Nominal Rigidity) β (Time preference) Rotemberg ϑ R (Nominal Rigidity) ε (Price elasticity of demand) The slope of Linear NKPC depends on the nominal rigidity parameters and on DIFFERENT structural parameters of the models Varthalitis (AUEB) Calvo vs Rotemberg June 2011 39 / 45
Slope of NKPC for equivalent dynamics θ C implies a ϑ R such that λ C = λ R, given β and ε : Varthalitis (AUEB) Calvo vs Rotemberg June 2011 40 / 45
A Sensitivity analysis of a non-zero in ation steady-state Output with respect to In ation Varthalitis (AUEB) Calvo vs Rotemberg June 2011 41 / 45
A Sensitivity analysis of a non-zero in ation steady-state Consumption with respect to In ation Varthalitis (AUEB) Calvo vs Rotemberg June 2011 42 / 45
A Sensitivity analysis of a non-zero in ation steady-state Output with respect to Nominal Rigidity Parameter Varthalitis (AUEB) Calvo vs Rotemberg June 2011 43 / 45
Sum up: 1 Rotemberg is a simpler model (at least in algebra). 2 Zero long-run in ation generates: i. identical steady-state solution ii. Equivalent dynamics given that the nominal rigidity parameters satisfy a speci c condition. 3 Non- zero long-run in ation generates important di erences between the two models. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 44 / 45
How we will proceed 1. Which model does better vis-à-vis the data (Calibration). 2. Macroeconomic & Welfare Implications of di erent policy rules. Varthalitis (AUEB) Calvo vs Rotemberg June 2011 45 / 45