Monetary Economics Notes Nicola Viegi 2 University of Pretoria - School of Economics
Contents New Keynesian Models. Readings...............................2 Basic New Keynesian Model................... 2.2. Consumer Problem.................... 2.2.2 The Firm......................... 5 2 Optimal Monetary Policy 9 2..3 Optimal Policy Problem................. 9 2..4 Discretionary Solution.................. 2..5 Commitment - Timeless Perspective........... 2..6 Policy Inertia....................... v
Chapter 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.. Readings Two books provide a complete overview of a very large literature. They are: Woodford, Michael (23): Interest and Prices, Princeton University Press. Gali, Jordi (28) 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.
2 CHAPTER NEW KEYNESIAN MODELS.2 Basic New Keynesian Model.2. Consumer Problem E t X i= " C i t+i Z C t = Z P t = N # + t+i + c jt dj p jt dj (.) (.2) (.3) P t C t + B t = W t N t + ( + r t ) B t + t (.4) ) Optimal Allocation of Consumption Expenditure subject to Lagrangian FOC Noting that L t = Z min c ij Z Z C t = p jt c jt dj + @L t @c jt = p jt + " " Z p jt c jt dj (.5) c jt dj C t Z # c jt dj (.6) (.7) # c jt dj c jt = (.8) Rewrite (.8) as Z c jt dj = C t (.9)
.2 BASIC NEW KEYNESIAN MODEL 3 p jt C t c jt = (.) p jt p jt c jt Or, as usually expressed in the literature = C t c jt (.) = C t (.2) c jt = C t (.3) c jt = C t (.4) p jt Substituting (.5) in (.2) we get: c jt = C t (.5) C t = " Z # C t dj Z = p jt dj C t (.6) and solving for the Lagrange Multiplier / we get = Z = = Z Z p jt p jt p jt dj dj dj Substituting this back in FOC (.8), we get: c jt = P t (.7) (.8) P t (.9) C t (.2) 2) Optimal Dynamic Consumption/Leisure Decision
4 CHAPTER NEW KEYNESIAN MODELS In real terms, the consumer problem can be written as: " X C E t i t+i N # + t+i + i= (.2) Lagrangian C t + B t = W t N t + ( + r t ) B t + t (.22) P t P t P t P t L = E t FOC E t X i= X i= " C i t+i # N + t+i + + (.23) i Wt+i t+i N t+i + ( + r t+i ) B t+i + t+i B t+i C t+i (.24) @L = Ct + t+i = (.25) @C t @L = N W t t + t = (.26) @N t P t @L = t + E t t+ ( + r t+ ) = (.27) @B t P t Rearranging and using condition (.25) to eliminate the Lagrange multiplier, we get: N t = Ct W t C t = E t ( + r t+ ) (.28) P t Pt Ct+ (.29) As shown before, this implies the following log linear relationships (which we will use later) on w t p t = c t + n t (.3) c t = E t fc t+ g fi t E t t= ln g (.3)
.2 BASIC NEW KEYNESIAN MODEL 5.2.2 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) a downward sloping demand curve c jt = Z t N jt (.32) c jt = C t (.33) ) are ran- nominal inertia like in Calvo (983).- in each period ( domly chosen to set their prices Real Total Cost, Average Cost, Marginal Cost T C = W t P t N t = W t Z t P t c jt (.34) AC = MC = W t Z t P t (.35) Productivity shocks a ect marginal cost of the rm Firm pricing problem max p jt+i = E t X i= = E t X i= i= i i;t+i c jt+i " i p jt i;t+i P t+j W t+j c jt Z t+i C t+i MC t+i (.36) C t+i# (.37) " X # = E t i i;t+i MC t+i C t+i (.38) FOC for the optimal price p t
6 CHAPTER NEW KEYNESIAN MODELS E t X i= i i;t+i "( ) p jt X E t i i;t+i ( ) i= + MC t+i p jt + MC t+i p jt # C t+i (.39) = C t+i (.4) = Flexible Price Equlibrium p t P t = MC t (.4) = W t (.42) Z t P t Log linearizing W t = Z t P t = N t C t (.43) ln az t = ln N t C t taking total derivatives and evaluating at the steady state (.44) de ne thus Z dz t = N dn t + C dc t (.45) c x f t = dx t X (.46) Doing the same for the production function bz t = bn t + bc t (.47) by f t = bn t + bz t (.48) Knowing that (without government) in equilibrium consumption equal income
.2 BASIC NEW KEYNESIAN MODEL 7 Impulse response function with exible prices + by f t = bz t (.49) + 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: The expression for the optimal price p t P t = Pt = ( ) (p t ) + P Log linearising the price index, we get: Log linearising the second we get E t P i= i i;t+i MC t+i E t P i= i i;t+i t (.5) P t (.5) P t bp t = ( ) bp t + b P t (.52) bp t = E t X i= This can be quasi-di erentiated to get: i i dmct+i + b (.53) Combining the two equations gives: bp t = bp t+ + d MC t + b P t (.54) bp t b P t = bp t+! P b t + MC d t + P b t (.55) Solving for in ation, yields:
8 CHAPTER NEW KEYNESIAN MODELS t = E t t+ + e d MC t (.56) where ( ) ( ) e = (.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: Log linearizing, we get: MC = W t Z t P t (.58) Usind the labour supply condition, by t = bn t + bz t dmc t = c W t b Pt b Zt (.59) dmc t = W c t Pt b (by t bn t ) (.6) + = ( + ) by t bz t (.6) + h i = ( + ) by t (.62) Which makes possible to write the NKPC in term of deviation from the exible price equilibrium t = E t t+ + by t (.63) where = ( + ) e by f t by f t
Chapter 2 Optimal Monetary Policy 2..3 Optimal Policy Problem subject to min L t = E t y X = n 2 t+ T 2 + y 2 t+i o (2.) t = E t t+ + ky t + " t (2.2) y t = E t y t+ (i t E t t+ ) + t (2.3) Lagrangian min L t = E t i X = 8 >< >: FOC respect to i t+ is equal to 2 2 t+ + yt+i 2 + t+ [ t+ E t t++ ky t+ " t+ ] + t+ yt+ E t y t++ + (i t+ E t t++ ) t+ t+ = Problem can be stated just in term of and y 9 9 >= >; (2.4)
CHAPTER 2 OPTIMAL MONETARY POLICY 2..4 Discretionary Solution Period by period problem subject to min L = y 2 E 2 t + yt 2 (2.5) t = E t t+ + ky t + " t (2.6) " t+ = " t + v t ; < < ; v t is iid FOC dl dy t = k t + y t = (2.7) y t = k t (2.8) Solution Solving forward t = + k E t 2 t+ + + k " t. (2.9) 2 t = k 2 + ( ) " t (2.) 2..5 Commitment - Timeless Perspective subject to min L t = E t i X = 2 2 t+ + yt+i 2 (2.) t = E t t+ + ky t + " t where, as before, " t+ = " t + v t is the stochastic policy process FOC
min L t = E t i X = 2 2 t+ + yt+i 2 + t+ [ t+ E t t++ ky t+ " t+ ] FOC with respect to t and y t for = (2.2) and for > $ t = t + t = (2.3) $ y t = y t k t = (2.4) FOC (2) $ t+ = ( t+ + t+ t+ ) = (2.5) $ y t+ = y t+ k t = (2.6) for:! $ y t+ = y t+ k t = (2.7) for: =! $ = t + t = t (2.8) for: >! $ = t+ + t+ t+ t+ = (2.9) 2..6 Policy Inertia Combining (2.9) and (2.7) we get which can be rewritten as t + k (y t y t ) = y t = y t k t (2.2)
2 CHAPTER 2 OPTIMAL MONETARY POLICY Solution substituting in the supply equation k (y t y t ) = k (E ty t+ y t ) + ky t + " t k y t k y t ky t = k E ty t+ k y t + " t + + k2 y t = E t y t+ + y t k " t (2.2) Second order di erence equation to solve with undetermined coe cients method Undetermined Coe cients Method First posit a solution for y t that is a function of the state (y t ; " t ) : y t = ay t + b" t (2.22) This, togheter with the assumption of shocks following a AR() process, gives substituting in (2.2) we get E t y t+ = ay t + b" t = a 2 y t + b (a + ) " t (2.23) + + k2 (ay t + b" t ) = a 2 k y t + b (a + ) " t + yt " t (2.24) that is: + + k2 (ay t + b" t ) = a 2 y t + b (a + ) k " t (2.25) Find the value of a and b that match the coe cients: + + k2 + + k2 a = a 2 t (2.26) b = b (a + ) k (2.27)
Equation for a quadratic - choose the root jaj < : The solution for b is instead unique, i.e k b = (2.28) ( + ( a )) + k 2 Decision rule for t is: 3 t = k (y t y t ) (2.29) t = k ay t " k ( + ( a )) + k 2 t y t (2.3) t = k ( a) y t + " ( + ( a )) + k 2 t (2.3) Under both precommitment and discretion, monetary policy completely o sets the impacts of the demand shock, t, so t does not a ect either y t 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.