The Labor Market in the New Keynesian Model: Incorporating a Simple DMP Version of the Labor Market and Rediscovering the Shimer Puzzle

Transcription

1 The Labor Market in the New Keynesian Model: Incorporating a Simple DMP Version of the Labor Market and Rediscovering the Shimer Puzzle Lawrence J. Christiano April 1, 2013

2 Outline We present baseline NK model with a Diamond-Mortensen-Pissarides type labor market. First paper to combine NK and DMP is Walsh (2003). Now this is a huge literature. The New Keynesian DMP model. Describe equilibrium conditions of household and goods producers These coincide with those in the simple, standard NK model ( Clarida-Gali-Gertler model ) Details: Describe the DMP labor market. Compare the DMP model with a version in which specification in which labor market is competitive. discover Shimer puzzle for monetary models : employment/unemployment respond very little to an expansionary monetary policy. replace hiring with search costs.

3 Households Many identical households, each having a unit measure of workers. All workers are sent to the labor market, where they are either employed or unemployed. Household problem: max E 0 Â t=0 b t log C t, subject to: P t C t + B t+1 W t + number of employed households z} { l t number of unemployed households, times unemployment benefits z } { (1 l t ) D +R t 1 B t + Profits net of taxes t First order condition: 1 1 = be t C t C t+1 R t p t+1

4 Obtaining the NK IS Curve Let X t C t exp ( a t ) and substitute into household Fonc: or, 1 X t exp (a t ) = be 1 t X t+1 exp (a t+1 ) 1 1 = E t X t X t+1 Rt+1 log-linearizing: R t p t+1 R t p t+1, R t+1 1 b exp (a t+1 a t ). ˆX t = E t ˆX t+1 ˆR t b p t+1 ˆR t+1. linearizing about a zero inflation steady state: x t = E t x t+1 E t [r t p t+1 r t ] r t = log (b) + E t [a t+1 a t ].

5 Goods Production Final good firms solve Z 1 Z 1 max P t Y t P i,t Y i,t dj, s.t. Y t = 0 Demand curve for i th monopolist: # Pt Y i,t = Y t. P i,t Production function: Y i,t = exp (a t ) h i,t, a t = r a a t 1 + # a t, P i,t 1 0 Y # 1 # # 1 # i,t dj. where h i,t is a homogeneous input (not labor!) with after-subsidy nominal price, (1 n) J t P t. Calvo Price-Setting Friction: P i,t = P t with probability 1 q. with probability q

6 Optimal Price Setting As in baseline NK model: p t = K t, p t P t, F t = 1 + bqe t p F t P t+1 # 1 F t+1 (a) t # K t = p t = # 1 s t + bqe t p t+1 # K t+1 (b) " # 1 q p t # 1 1 " 1 #! K t = 1 q F t # 1 q p t # # 1 q (c) s t = (1 n) J t exp (a t ) Log-linearizing (a)-(c) around zero inflation steady state (and, assuming 1 n = (# 1) /#) p t = (1 q)(1 bq) q ŝ t + bp t+1 p t p t 1.

7 Linearized Equilibrium Conditions for Households and Goods Producing Firms Equations p t = (1 q)(1 bq) q ŝ t + bp t+1 x t = E t x t+1 E t [r t p t+1 r t ] r t = log (b) + E t [a t+1 a t ]. ŝ t = Ĵ t a t. If we also have a monetary policy rule, would have 5 equations in 6 variables: p t, ŝ t, x t, r t, rt, Ĵ t Need on net, one more equation. In simple model with competitive labor markets, have J t = C t l j t. Here, j is a utility parameter and this equation is not available now. Equations provided by search/matching framework of DMP.

8 Labor Market Large number of identical and competitive firms; produce homogeneous output using only labor, l t,forrealprice,j t. At the start of period t, afirmmaypostasinglevacancyfor fixed cost k, inwhichcaseitmeetsaworkerwithprobability, Q t. Immediately after they meet, worker and firm agree on a wage and begin employment. Match survives into period t + 1 with fixed, exogenous probability r. Number of workers searching for work at the end of period t 1: 1 l t 1 unemployed workers plus the (1 r) l t 1 employed workers that separate. total searching workers 1 l t 1 + (1 r) l t 1 = 1 rl t 1.

9 Notation and Identities Probability a searching worker finds a period t job denoted f t. Law of motion of employment is: l t = workers that remain attached to their firm z} { rl t 1 + flow from unempl to empl plus flow from empl to empl z } { f t (1 rl t 1 ) Alternative law of motion of employment: l t = (r + x t ) l t 1, where x t denotes the hiring rate, as a proportion of l t 1. For the two laws of motion to be compatible, require: f t = x tl t 1 1 rl t 1 = net new jobs number of workers searching.

10 Labor Market Tightness Probability a firm meets a worker is denoted Q t : Q t = net new jobs number of firms posting vacancies = x tl t 1 v t l t 1 = x t v t, where v t denotes the vacancy rate, as a proportion of l t 1. Matching function 0 1 # new hires # people searching s 0 1 # vacancies 1 s z } { z } { z } { x t l t 1 = s 1 rl t 1 l t 1 v t A. divide by 1 rl t 1 : f t = x tl t 1 1 rl t 1 = s m lt 1 v t 1 rl t 1 where Q t denotes labor market tightness. note, f t = Q t Q t, so Q t = s m Q s t 1 s = s m Q 1 s t,

11 Value Functions J t is the value to a firm of an employed worker: J t = J t w t + re t m t+1 J t+1, w t W t P t. J t and m t+1 are given to firm and worker. Value of employment to a worker: V t = w t + E t m t+1 [rv t+1 + (1 r)(f t+1 V t+1 + (1 f t+1 ) U t+1 )] Value of unemployment to a worker: U t = D + E t m t+1 [f t+1 V t+1 + (1 f t+1 ) U t+1 ] Free entry and zero profits dictate: k = Q t J t.

12 Nash Bargaining Workers and firms choose a wage that solves a joint optimization problem, taking as given future wages. Surplus of worker: S t w t + Ṽ t U t, Ṽ t E t m t+1 [rv t+1 + (1 r)(f t+1 V t+1 + (1 f t+1 ) U t+1 )] Surplus of firm: Agreed-upon wage rate solves J t w t, J t J t + re t m t+1 J t+1 max w t wt + Ṽ t U t h J t w t 1 h, taking Ṽ t and J t as given. FONC (Nash sharing rule): J t = 1 h h (V t U t ).

13 Labor and Goods Market Clearing Total purchases of homogeneous inputs by intermediate good producer: h t = Z 1 0 h i,t di. Labor market clearing h t = l t. Goods market clearing C t + v t l t 1 k = p t exp (a t ) l t, where 2 pt = 4(1 q) 1 q p (# 1)! # 1 # t + q p# t 1 q p t

14 Equilibrium Conditions from Labor Market 13 equilibrium conditions associated with 12 new variables: (6) J t = 1 h h l t, x t, Q t, J t, V t, U t, m t, f t, Q t, Y t, v t, w t (1) l t = (r + x t ) l t 1, (2) k = Q t J t (3) J t = J t w t + re t m t+1 J t+1 (4) V t = w t +E t m t+1 [rv t+1 + (1 r)(f t+1 V t+1 + (1 f t+1 ) U t+1 )] (5) U t = D + E t m t+1 [f t+1 V t+1 + (1 f t+1 ) U t+1 ] (V t U t ), (7) Q t = s m Q s t, (8) Q t = l t 1v t 1 rl t 1 (9) m t+1 = b u0 (C t+1 ) u 0 (C t ), (10) C t + kv t l t 1 = Y t, (11) f t = x tl t 1 (12) Y t = pt exp (a t ) l t, (13) Q t = x t 1 rl t 1 v t with pt = 1 bec we linearly approximate around zero inflation ss.

15 Solving the Model and Assigning Values to Parameters Shortage of one equation in goods and household sector. The labor market added, on net, one equation (i.e., 12 extra variables and 13 extra equations). The model has the following 10 parameters household and goods production labor market z } { z } { b, r a, s a, q, #, D, k, h, s m, s Interestingly, # plays no role in the solution to the model, to a first order approximation This reflects, in part, how we set the subsidy to monopolists. Also reflects constant marginal costs of intermediate good producers (not obvious). Parameter values of goods and household sector easy to choose, since there is a lot of experience with them b = 0.99, r a = 0.95, s a = 0.01, q = 0.75.

16 Monetary Policy Taylor rule: R t R = a Rt 1 R exp {(1 a)[f p ( p t 1) + f x log (C t /C)] u t }, where u t is an iid (expansionary) policy shock. Parameters were set to the following (fairly conventional) values: a = 0.80, f p = 1.5, f x = 0.05.

17 Calibration of Labor Market Parameters Generally, follow Ravenna-Walsh (2008) and Walsh (2003), but also Shimer (2012) and Hall-Milgrom (2008). Variable name symbol value end/ex unemployment rate 1 l 0.05 endog replacement ratio D w 0.4 endog search cost/gross output of homog good kvl/y 0.01 endog vacancy filling rate Q 0.7 endog elasticity of matching w.r.t. unemp s 0.5 exog discount factor b 0.99 exog match survival rate r 0.9 exog Since we exogenously set the values of four endogenous variables, we must allow four exogenous parameters to be endogenous: D, k, h, s m.

18 Steady state computations first, go down first column, then go down second. In each case, which equilibrium condition that is used is indicated. R = 1/b (intertemporal Euler) (9) m = b k = (kvl/y) Y/(lv) (1) x = 1 r (2) J = k Q (8) Q = xl Q(1 rl) (3) w = J + rbj J (11) f = xl 1 rl D = D w w (7) s m = QQ s (10) C = l kvl J = 1 (pricing equations) (4)-(5) V U = w D (12) Y = l (5) U = D+bf (V U) 1 b (13) v = x/q (6) h = V U V U+J. 1 br(1 f )

19 Comparing the DMP Labor Market with Competitive Labor Markets The DMP version of the model was described in detail above. The competitive labor market version of the model is obtained by deleting equations 1-13 above and: Interpret J t as the real wage. Allow households to set employment, l t, on the intensive margin and equate J t = C t l j t, where 1/j denotes the Frisch labor supply elasticity (see the first set of lectures on the labor market in the New Keynesian model for details).

20 Comparing...results Run the Dynare file, DMP.mod, to see responses of the system to an expansionary monetary policy shock. "Shimer puzzle" for monetary economics: note that the response of output is weaker in the DMP model than it is in the (already very weak) version of the model in which the labor market is competitive. The intuition is that in an expansion, the labor market tightens (i.e., Q t rises) and the probability that a vacancy draws a worker, Q t, decreases, with costs fixed in terms of vacancies, the implied cost of meeting a worker rises, this negative feedback loop limits the expansion itself.

21 Hiring versus Search Costs One response to this Shimer puzzle is to replace the search costs of meeting a worker in the DMP model with hiring costs. Approach taken in New Keynesian model versions of DMP by GT and GST. Hiring cost specification: firm pays k to meet a worker. vacancies must be posted too, but they are costless. the only actual cost of meeting a worker is k. Replace equations (2) and (10) above by k = J t (2) 0 C t + kx t l t 1 = Y t (10) 0 do same replacement in the steady state equations, also replace the calibration equation for k : k = (kxl/y) Y/(lx)

22 The E ect of Switching to Hiring Costs from Search The switch cancels the negative feedback loop mentioned earlier. With a monetary expansion, the rise in employment is greater and inflation, smaller. This can be confirmed by running the Dynare software, DMP.mod. Unfortunately, this change is not su cient to allow the model to display the response to shocks observed in the data. in practice, exogenous frictions in the adjustment of wages are still required (see GT, GST and others). Christiano-Eichenbaum-Trabandt (2013) stress this point too, and suggest an alternative to exogenous stickiness in sticky wages. this is the subject of the next handout.