Macroeconomics 2 Lecture 6 - New Keynesian Business Cycles 2. Zsófia L. Bárány Sciences Po 2014 March
Main idea: introduce nominal rigidities Why? in classical monetary models the price level ensures money market clearing: M P = L(Y, i) = C L 0(i) = C L 0 (r + π e ) for example if money demand is specified as (Cagan): m t p t = αe t (p t+1 p t ) we can solve this forward and get p t = 1 1 + α i=0 the price level is like an asset price α i Tt (m t+1 ) 1 + α
the price level in reality is not like an asset price it comes from the aggregation of many prices, set by individual price setters, at different points in time P adjusts more slowly as a reaction to an increase in M the nominal interest rate will have to fall, and so will probably the real interest rate the Euler equation then implies an increased current demand for consumption output dynamics depend on how the price setters respond to the shift in demand older fixed price (wage) models: output given by the minimum of supply and demand at the given price in imperfect competition firms have an incentive to accommodate these shifts as long as P > MC demand affects output
A static model of price-setters simplified Blanchard-Kiyotaki (1987) model continuum of households all produce one differentiated good Y i = ZN i consume all differentiated goods U ( C i, M ) i P, N i = ( ) ( α Mi Ci P α 1 α where consumers have Dixit-Stiglitz preferences ) 1 α N1+γ i 1 + γ ( 1 C i 0 ) θ C θ 1 θ 1 θ ij dj θ > 1
the price index is: ( 1 P 0 ) 1 P 1 θ 1 θ j dj the budget constraint is 1 0 P j C ij dj + M i = P i Y i + M i where M i is initial money holdings household needs money for transactions, now MIU rather than CIA budget constraint short cut to a dynamic budget constraint
Demand for household j s product for now assume that each household spends a nominal amount X i on consumption goods the household s problem is then ( 1 max {C ij } 1 j=0 0 then we get that ) θ C θ 1 θ 1 θ ij dj C ij = X i P subject to 1 ( ) θ Pj and C i P = X i P 0 P j C ij dj = X i since P is the price index we wrote earlier, the demand for good j by household i can be written as C ij = C i ( Pj P ) θ given overall consumption, C i, the demand for each good depends on its relative price P j /P
Choice of consumption and money using the demand for goods, the household s choice between money and consumption can be written as: max C i,m i ( ) ( ) α Mi 1 α Ci P s.t. PC i + M i = P i Y i + M i α 1 α given Y i and M i the optimal consumption and money balances are C i = α P iy i + M i P and M i P = (1 α)p iy i + M i P hhs allocate their initial wealth in α and 1 α proportion to consumption and money balances (not surprising given Cobb-Douglas utility)
from the two FOCs we can get: C i = α M i 1 α P the demand for the good j by hh i is: ( ) θ Pj C ij = C i = α M i P 1 α P ( ) θ Pj P using the two FOCs in the utility function, the indirect utility can be written as: P i P Y i N1+γ i 1 + γ + M i P
Pricing and output decision the hh finally chooses output (equivalently hours: N i = Y i /Z) and its price: P i max P i,y i P Y i Y 1+γ i Z (1+γ) 1 + γ the price it charges and the demand for its product is connected through the overall demand for good i: Y i = 1 0 C ji dj = α M 1 α P ( ) θ Pi P where M = 1 0 M jdj is the total desired money holding in the economy & money demand must equal money supply M = M: Y i = α M 1 α P ( ) θ Pi P
the maximization yields: P i P = θ θ 1 Y γ i Z (1+γ) price equals marginal cost times a markup plugging in Y i once more gives: ( ) 1 P i θ P = θ 1 X γ Z (1+γ) 1+θγ α M where X 1 α P an increase in real money supply leads to an increase in the optimal relative price magnitudes depend on θ and γ, as γ 0 the effect gets smaller
Equilibrium all hhs are identical besides that they produce different goods i.e. their preferences for all goods are identical and the hhs preferences are identical in equilibrium the relative price must be 1 output by each hh: 1 = θ ( ) 1 θ 1 θ 1 Y γ Z (1+γ) Y = Z 1+γ θ ( γ θ 1 N = θ ) 1 γ Z lower equilibrium output than under perfect competition due to the markup (homework: check!) output lower than first best technological shock Y, N, the effect is bigger if γ is closer to 0
Price level determination closing the model: the price level must be such that the level of demand is right Y = α M 1 α P P = α M 1 α Y output determined by marginal cost and markup nominal money neutral - moves price one-for-one this does not seem like much progress
But this is progress! think about an increase in nominal money supply the equilibrium requires a proportional change in the price index, but no change in the relative prices how will this happen? who is in charge of the price level? nobody each price setter has to increase the prices individually, i.e. modify his relative price, if γ not far above 0, then each firm only adjusts its price by a little the price adjustment can be very slow
Introducing nominal rigidities hh have to set nominal prices, they might want to do these at discrete time intervals (rather than continuously). why? menu costs: a small deviation from the optimum price has only second order effects on profits, but it has a first order effect on output and welfare - diagram depending on the MC curve, the desired change in price might be really small (if γ is very small) modify the timing of the model as follows: 1. each hh chooses the price of its product before the nominal money supply and the productivity is realized 2. consumption decisions are taken after observing the realization of M and Z
The choice of production and relative price of the hh is now: ( ) P i max E P i,y i P Y i Y 1+γ i Z (1+γ) 1 + γ subject to Y i = α M 1 α P ( ) θ Pi = X P ( ) θ Pi P Now M (or X ) and Z are random variables, the FOC is: ( ( ) θ ( ) ) (1+γ)θ 1 Pi E X (1 θ) + θx 1+γ Z (1+γ) Pi = 0 P P rearranging: P i P = ( θ θ 1 E ( X 1+γ Z (1+γ)) E(X ) ) 1 1+θγ higher E(X ) higher relative price
In equilibrium the relative prices must all be 1. The price level is implicitly defined by: 1 = θ E ( X 1+γ Z (1+γ)) θ 1 E(X ) α remember that X = 1 α if MC < P, then demand and output will be: M P and employment will be: Y = α M 1 α P N = Y Z
Implications: unanticipated movements in the nominal money supply affect the real money supply one-for-one, as prices are already set the real money supply affects output and consumption one-for-one demand affects output as long as MC < P holds (diagram) the relative prices will not change welfare moves together with Y and M increase money unexpectedly in order to increase welfare? unanticipated technological shocks do not affect demand nor output and they initially decrease employment introduce a more realistic price setting & dynamics New Keynesian model
A basic New Keynesian DSGE model take the imperfect competition model from last week assume Φ = 0, i.e. constant returns to scale there are profits in the steady state assume that the aggregation function f takes the Dixit-Stiglitz form: the price index is then ( 1 Y t = 0 0 ) ( ) θ θ 1 θ 1 Y i θ t di ( 1 ) 1 P t = (Pt) i 1 θ 1 θ di
Let the utility function be: U (C, H) = The consumer s problem is: subject to max {C t,h t,k t+1 } t=0 ( ) C 1 σ H1+γ 1 σ 1 + γ E 0 β t U(C t, H t ) t=0 P t C t + P t K t+1 = W t H t + (U t + P t (1 δ)) K }{{} t + Π t R t and let Λ t β t be the multiplier associated with the time t BC
the FOCs: C t : C σ t = Λ t P t H t : H γ t = Λ t W t K t+1 : Λ t P t = βe t (Λ t+1 R t+1 )
Calvo pricing in each period a fraction (1 η) of firms have the possibility to reset their prices the other η fraction keep their previous price η [0, 1] is the index of price stickiness the implied average price duration is 1 1 η completely unrealistic, but easy to handle don t have to keep track of the history of the firms, when they last changed prices does not matter not connected to the micro foundations of price rigidity, which is very history dependent
What price do the price-setters choose? firm sets its price to maximize expected profits however, the price it sets in period t might remain for several periods has to take into account the probability that the same price stays for 1, 2,.., n,.. periods the problem can be written as: max E t η j β j Λ t+j Π t+j (P Pt t ) Λ t j=0 Starting from the Dixit-Stiglitz aggregator, as before, it can be shown that the demand for good i is: Y i t = ( ) P i θ t Y t P t
Using the demand, the per period profit function is: ( P t ) θ Y t+j TC ( ( P t ) θ Y t+j) Π t+j (P t ) = P t P t+j P t+j assuming the Cobb-Doulas production function Y i t = (H i t) α (K i t ) 1 α the producers of the differentiated goods take wages and rental rates as given the total cost can be expressed as TC(Y ) = YW α t U 1 α t (1 α) 1 α α α
The optimum Pt satisfies: E t j=0 η j β j Λ ( ) t+j (1 θ)(pt ) θ Y t+j Pt+j θ + θy t+j S t+j (Pt ) θ 1 Pt+j θ = 0 Λ t where the nominal marginal cost is: S t+j = U1 α t+j W 1 α t+j (1 α) 1 α α α the optimal P t can be expressed as: what is a t? Pt = θ 1 θ E t j=0 ηj β j Λ t+j Λ t Y t+j Pt+j θ S t+j E t j=0 ηj β j Λ t+j Λ t Y t+j Pt+j θ =a t θ 1 θ Y tp θ t S t + (1 a t )E t P t+1
The price index The price index is: ( 1 ) 1 P t = (Pt) i 1 θ 1 θ di = (η(p t 1 ) 1 θ + (1 η)(pt ) 1 θ) 1 1 θ 0 where P t 1 is the price index in the previous period note the role of η if η = 0 prices are fully flexible if η = 1 prices are fixed
Closing the model Money demand M t P t = Y γm t ln M t ln P t = γ m ln Y t ad hoc money demand function, which an be derived from different assumptions direct link between the real and the nominal side of the economy: Y i t = ( Mt P t ) 1 γm ( P i t P t ) θ if prices were flexible, any change in M t would be fully absorbed by a proportional change in P t and all the P i ts Money supply ln M t ln M t 1 = ρ(ln M t 1 ln M t 2 ) + e t
Log-linearization in the non-stochastic BGP: π t = Pt P t 1 > log-linearization of the optimal relative price p t = ln P t P t ln P t P t = 1 and P t P t = 1 = (1 ηβ)ŝ t + ηβe t ( π t+1 + p t+1) > log-linearization of the inflation, using the price index: π t = ln π t ln π t = 1 η η combining the two we get the New Keynesian Phillips curve: p t π t = 1 η (1 ηβ)ŝ t + βe t π t+1 η inflation depends on real marginal cost and is forward looking due to price stickiness
Results - Chari, Kehoe and McGrattan response to a 1 percent money shock (with fairly inelastic labor supply) contemporary output increases, since P t is not responding one-for-one price level adjusts quickly: returns to equilibrium level those who adjust, adjust very aggressively lack of persistence hours respond too much factor prices respond too much as well prices adjust very rapidly nominal interest rate increases
Refine the model 1. material inputs besides choosing K and N, firms choose material inputs, which are units of the composite good 2. variable capacity utilization of K at a cost of faster depreciation, capital can be used more intensively during booms 3. more elastic labor supply add an extensive margin: people can decide whether to join the workforce at a fixed cost
Results of the more refined model hours per worker response in right range employment responds a lot aggregate hours response is bigger persistent, hump-shaped and larger output response factor prices move less slower price adjustment nominal interest rate still increases
Understanding the results Adjusting firms trade-off: loss in sales if adjusting the price too much increase in marginal cost if adjusting too little The marginal cost depends on factor prices: if factor prices move a lot, then the marginal cost moves a lot, firms will find it optimal to adjust aggressively lots of price adjustment leads to low persistence
In the CKM specification capital constant, no materials need a large labor supply response to the demand shock but labor is quite inelastic large increase in the wage as the capital/labor ratio drops, the rental rate increases the marginal cost increases a lot in booms the firms that can adjust do adjust a lot in booms, more than one-for-one
In the complete model variable capacity utilization, materials labor supply needs to respond less moreover, labor supply is more elastic, since there is an extensive margin wage rate does not have to go up that much the capital/labor ratio also moves much less the rental rate does not increase that much either the marginal cost is less pro-cyclical firms find it optimal to adjust their prices less
inflexible economy steep MC curve adjustment through aggressive price change & small output change get to the same equilibrium price level faster flexible economy flatter MC curve adjustment through smaller price change & larger output change get to the same equilibrium price level in many smaller steps
the nominal interest rate has to go up in all cases due to the expected inflation this is a bit embarrasing look at interest rate based monetary rules mark-ups are counter-cyclical prices respond less this had to be the case: employment and wage move the same way and there are no technology shocks
State-of-the-art NK DSGE models canonical model by Christiano, Eichenbaum, and Evans (JPE 2004) key additional elements: Calvo nominal price and wage rigidities habit formation in consumption adjustment costs in investment best quantitative match with the empirical IRFs so far instead of calibration of parameters one by one: joint calibration by Method of Moments formal estimation of parameters also with MoM or ML