Monetary Economics Notes

Monetary Economics Notes Nicola Viegi1 2010 1 University of Pretoria - School of Economics Contents 1 New Keynesian Models 1.1 Readings . . . . . . . . . . . 1.2 Basic New Keynesian Model 1.2.1 Consumer Problem . 1.2.2 The Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Optimal Monetary Policy 2.0.3 Optimal Policy Problem . . . . . . . 2.0.4 Discretionary Solution . . . . . . . . 2.0.5 Commitment - Timeless Perspective . 2.0.6 Policy Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . 9 . 10 . 10 . 11 v . . . . Chapter 1 New Keynesian Models Modern New Keynesian models are the standard tool of modern macroeconomic modelling for policy analysis. These models bridge the gap between the methodology of the Real Business Cycle tradition and practical policy evaluation. Introducing price stickiness and imperfect competition in the basic RBC model they provide an useful tool to analyse response of economies to nominal and real shock. 1.1 Readings Two books provide a complete overview of a very large literature. They are: Woodford, Michael (2003): Interest and Prices, Princeton University Press. Gali, Jordi (2008) Monetary Policy, In‡ation and the Business Cycle, Princeton University Press The Gali’s book provide a very useful discussion of the literature at the end of each chapter. The objective of the following notes is just to clarify some of the maths passages necessary to follow the argument and are not substitute to a careful reading of the Gali book. 1 2 1.2 1.2.1 CHAPTER 1 NEW KEYNESIAN MODELS Basic New Keynesian Model Consumer Problem Et 1 X i i=0 " Ct = Z 1 # 1+ Nt+i 1+ 1 Ct+i 1 (1.1) 1 1 (1.2) cjt dj 0 Pt = Z 1 1 1 1 pjt dj (1.3) 0 Pt Ct + Bt = Wt Nt + (1 + rt ) Bt 1 + (1.4) t 1) Optimal Allocation of Consumption Expenditure Z 1 pjt cjt dj min cij (1.5) 0 subject to Ct = Z 1 1 1 (1.6) cjt dj 0 Lagrangian Lt = Z " 1 pjt cjt dj + 0 Z Ct 1 1 1 cjt dj 0 # (1.7) FOC " Z @Lt = pjt + @cjt 1 1 1 1 cjt dj 1 cjt 0 # =0 (1.8) Noting that Z 0 Rewrite (1.8) as 1 1 1 cjt dj 1 1 = Ct (1.9) 1.2 BASIC NEW KEYNESIAN MODEL 1 3 1 Ct cjt = 0 pjt (1.10) 1 1 pjt = (1.11) Ct cjt 1 1 pjt cjt = (1.12) Ct 1 1 cjt = pjt cjt = (1.13) Ct (1.14) Ct pjt Or, as usually expressed in the literature pjt cjt = (1.15) Ct Substituting (1.15) in (1.2) we get: Ct = "Z 1 pjt 1 Ct dj 0 # 1 Z 1 = 1 1 pjt 1 dj Ct (1.16) 0 and solving for the Lagrange Multiplier / we get Z 1 = 1 1 pjt 1 dj (1.17) 0 = Z 1 1 pjt 1 dj (1.18) 0 = Z 1 1 1 pjt 1 dj Pt (1.19) 0 Substituting this back in FOC (1.8), we get: cjt = pjt Pt Ct 2) Optimal Dynamic Consumption/Leisure Decision (1.20) 4 CHAPTER 1 NEW KEYNESIAN MODELS In real terms, the consumer problem can be written as: " # 1 1+ 1 X C N t+i t+i i Et 1 1+ i=0 Ct + (1.21) Wt Bt 1 Bt t = Nt + (1 + rt ) + Pt Pt Pt Pt (1.22) Lagrangian L = Et Et 1 X i=0 1 X i i " # + (1.23) Bt+i 1 Wt+i t+i Nt+i + (1 + rt+i ) + Pt+i Pt+i Pt+i t+i i=0 FOC 1+ Nt+i 1+ 1 Ct+i 1 Ct+i @L = Ct + t+i = 0 @Ct Wt @L = Nt + t =0 @Nt Pt 1 1 @L =0 = + Et t+1 (1 + rt+1 ) t @Bt Pt Pt+i Bt+i (1.24) Pt+i (1.25) (1.26) (1.27) Rearranging and using condition (1.25) to eliminate the Lagrange multiplier, we get: Nt = Ct Ct = Et Wt Pt (1.28) (1 + rt+1 ) Pt Pt+i Ct+1 (1.29) As shown before, this implies the following log linear relationships (which we will use later) on wt pt = c t + nt ct = Et fct+1 g (1.30) 1 fit Et t=1 ln g (1.31) 1.2 BASIC NEW KEYNESIAN MODEL 1.2.2 5 The Firm Firms are pro…t maximisers but they fact three constraints: Production Function linear in labour input (the simplest possible - there is no capital - just looking at the short run properties of the model) cjt = Zt Njt (1.32) a downward sloping demand curve cjt = pjt Pt+i (1.33) Ct nominal inertia like in Calvo (1983).- in each period (1 domly chosen to set their prices ) are ran- Real Total Cost, Average Cost, Marginal Cost Wt Wt Nt = cjt Pt Z t Pt Wt AC = M C = Zt Pt (1.34) TC = (1.35) Productivity shocks a¤ect marginal cost of the …rm Firm pricing problem max pjt+i = Et 1 X i i;t+i i=0 = Et 1 X i i;t+i i=0 = Et 1 X i=0 i i;t+i " " pjt cjt+i Pt+i pjt Pt+j FOC for the optimal price pt pjt Pt+i pjt Pt+i Wt+j cjt Zt+i Pt+i Ct+i 1 M Ct+i (1.36) M Ct+i pjt Pt+i pjt Pt+i # Ct+i # (1.37) Ct+i (1.38) 6 Et CHAPTER 1 NEW KEYNESIAN MODELS 1 X i i;t+i i=0 Et " (1 1 X ) i i;t+i 1 pjt (1 pjt Pt+i 1 + M Ct+i pjt Pt+i ) i=0 + M Ct+i 1 pjt 1 pjt pjt Pt+i pjt Pt+i # Ct+i(1.39) = 0 Ct+i(1.40) = 0 Flexible Price Equlibrium pt = Pt 1= Wt = Zt Pt 1 M Ct Wt 1 Z t Pt 1 = Nt Ct (1.41) (1.42) (1.43) Log linearizing ln aZt = ln Nt Ct (1.44) taking total derivatives and evaluating at the steady state 1 1 1 dZt = dNt + dCt Z N C de…ne thus c dXt xft = X (1.45) (1.46) zbt = n bt + b ct (1.47) bt + zbt ybf t = n (1.48) Doing the same for the production function Knowing that (without government) in equilibrium consumption equal income 1.2 BASIC NEW KEYNESIAN MODEL 1+ + ybf t = 7 zbt (1.49) Impulse response function with ‡exible prices Sticky Prices Equlibrium The price index and in‡ation is determined by the joint solution of the following dynamic equations: Price index evolves according to: Pt1 ) (pt )1 = (1 + Pt1 (1.50) 1 The expression for the optimal price Et pt = Pt 1 P1 i i=0 Et P1 i=0 Pt Pt+i i;t+i M Ct+i i i;t+i Pt Pt+i 1 (1.51) Log linearising the price index, we get: Pbt = (1 ) pbt + Pbt Log linearising the second we get pbt = Et 1 X i i i=0 This can be quasi-di¤erentiated to get: pbt = (1.52) 1 d M C t+i + Pbt+i d pbt+1 + M C t + Pbt (1.53) (1.54) Combining the two equations gives: 1 Pbt 1 Pbt 1 = Solving for in‡ation, yields: Pbt+1 1 ! 1 d Pbt + M C t + Pbt (1.55) 8 CHAPTER 1 NEW KEYNESIAN MODELS t = Et where t+1 (1 e= d + eM Ct ) (1 (1.56) ) (1.57) Which is the New Keynesian Phillips. Express the NKPC in term of deviation from the ‡exible price equilibrium Recall the marginal cost is given by: MC = Wt Z t Pt d ct M Ct = W Pbt (1.58) Log linearizing, we get: Zbt (1.59) n bt ) 1+ + i (1.60) Usind the labour supply condition, ybt = n bt + zbt d ct M Ct = W Pbt (b yt = ( + ) ybt h = ( + ) ybt ybtf zbt (1.61) (1.62) Which makes possible to write the NKPC in term of deviation from the ‡exible price equilibrium t where = ( + )e = Et t+1 + ybt ybtf (1.63) Chapter 2 Optimal Monetary Policy 2.0.3 Optimal Policy Problem min Lt = Et y n 1 X 1 =0 2 T 2 t+ 2 + yt+i o (2.1) subject to = Et t+1 + kyt + "t yt = Et yt+1 (it Et (2.2) (2.3) t t+1 ) + t Lagrangian min Lt = Et i 1 X =0 8 > > > < > > > :+ 1 2 t+ t+ yt+ [ t+ 2 t+ 2 yt+i + Et t+1+ Et yt+1+ + (it+ + kyt+ Et "t+ ] t+1+ ) t+ 9 > > > = > > > ; (2.4) FOC respect to it+ is equal to =0 t+ Problem can be stated just in term of 9 and y 10 2.0.4 CHAPTER 2 OPTIMAL MONETARY POLICY Discretionary Solution Period by period problem 1 min L = E y 2 2 t + yt2 (2.5) subject to t = Et t+1 "t+1 = "t + vt ; + kyt + "t 0< < 1; (2.6) vt is iid FOC dL = k dyt t + yt = 0 (2.7) k yt = (2.8) t Solution t = Et + k2 t+1 + + k2 "t . (2.9) Solving forward t 2.0.5 = k 2 + (1 ) "t (2.10) Commitment - Timeless Perspective min Lt = Et i 1 X 1 =0 subject to t = Et 2 t+1 2 t+ 2 + yt+i + kyt + "t where, as before, "t+1 = "t + vt is the stochastic policy process FOC (2.11) 11 min Lt = Et i 1 X 1 2 =0 2 t+ 2 + yt+i + [ t+ Et t+ t+1+ kyt+ "t+ ] (2.12) FOC with respect to t and yt for $ = =0 + t =0 t (2.13) t and for $ = yt yt k = ( + t+ = yt+ t =0 (2.14) >0 $ t+ t+ 1) =0 (2.15) t+ $ yt+ k t =0 (2.16) FOC (2) f or: f or: $ 0! yt+ $ = 0! = = yt+ t + t k t =0 (2.17) =0 (2.18) t f or: $ > 0! = t+ + t+ t+ 1 =0 (2.19) t+ 2.0.6 Policy Inertia Combining (2.19) and (2.17) we get t + k (yt yt 1 ) = 0 which can be rewritten as yt = yt k 1 t (2.20) 12 CHAPTER 2 OPTIMAL MONETARY POLICY Solution substituting in the supply equation k k yt (yt k 1+ yt 1 ) = yt + kyt = k2 (Et yt+1 k Et yt+1 k yt = yt ) + kyt + "t Et yt+1 + yt k yt 1 k 1 + "t (2.21) "t Second order di¤erence equation to solve with undetermined coe¢cients method Undetermined Coe¢cients Method First posit a solution for yt that is a function of the state (yt 1; "t ) : yt = ayt 1 + b"t (2.22) This, togheter with the assumption of shocks following a AR(1) process, gives Et yt+1 = ayt + b "t = a2 yt 1 + b (a + ) "t (2.23) substituting in (2.21) we get 1+ + k2 a2 y t (ayt 1 + b"t ) = (ayt 1 + b"t ) = 1 1 + b (a + ) "t + yt k 1 "t (2.24) that is: 1+ + k2 a2 y t 1 + b (a + ) k "t (2.25) Find the value of a and b that match the coe¢cients: 1+ 1+ + + k2 k2 a= 1 a2 b = b (a + ) (2.26) t k (2.27) 13 Equation for a quadratic - choose the root jaj < 1: The solution for b is instead unique, i.e k b= Decision rule for t = t = t = k (yt k k t (1 (1 + (1 (2.28) )) + k 2 a is: yt 1 ) ayt (2.29) k 1 a) yt (1 + (1 1 + (1 + (1 )) + k 2 a a "t )) + k 2 yt "t 1 (2.30) (2.31) Under both precommitment and discretion, monetary policy completely o¤sets the impacts of the demand shock, t , so t does not a¤ect either yt or . Cost shocks, "t , have di¤erent impacts under precommitment and discretion because under discretion there is a stabilization bias. There is history dependence of optimal policy under precommitment, but none under discretion. The history dependence is a mean by which commitment is implemented.