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Unemployment and Business Cycles
Lawrence J. ChristianoMartin EichenbaumMathias Trabandt
Background
• Key challenge for business cycle models.— How to account for observed volatility of labor marketvariables?
• Standard diagnosis— For plausibly parameterized models, in a boom, wages rise toorapidly, limiting expansion of employment.
— Classic RBC models (Chetty), standard effi ciency wage models(Alexopoulos), standard DMP models (Shimer).
Sticky Wages...
• New Keynesian DSGE models successful in matching time seriesdata, including hours worked, employment and real wages.
— but, they assume the result by positing that wages areexogenously sticky.
• model provides no rationale for wage stickiness.
— approach criticized on micro data grounds• good macro fit requires wage indexation, so that all wageschange all the time.
• but, in micro data individual wages constant for lengthy spells.
— underlying ‘monopoly power’theory of unemployment• on questionable empirical grounds (Christiano (2010))
— does not contribute to contemporary policy discussions (e.g.,effects of extending unemployment benefits).
Outline• Describe variants of DMP labor market environments, definethe problem, and several proposed solutions.
— Standard DMP model: Shimer critique (AER)— Early solutions: Hagedorn and Manovskii (AER),Ljungqvist-Sargent (working paper).
• Current solutions:— Hiring versus search costs (see Pissarides (P),Christiano-Trabandt-Walentin (CTW),Christiano-Eichenbaum-Trabandt (CET)).
— Hall-Milgrom (2008, HM).
• Steady State Properties.
• Integrate framework into New Keynesian model.
Simple model• Households
— have large numbers of workers, provide consumption insuranceto each one.• Consumption Euler equation:
1 = Etmt+1R̃t+1
— workers• Value function of unemployed worker (D−unemploymentbenefits, ft+1 job finding rate):
Ut = D+ Etmt+1[ft+1Vt+1 +
(1− ft+1
)Ut+1
]• Employed worker (wt wage rate)
Vt = wt + Etmt+1[ρVt+1 + (1− ρ)(
ft+1Vt+1 +(
1− ft+1)
Ut+1)
• Actually, only the difference, Sw,t, matters in the model:
Sw,t ≡ Vt −Ut = wt −D+ ρEtmt+1(
1− ft+1)
Sw,t+1.
Simple Model, cnt’d
• A firm that wants to meet worker must first post a vacancy, atcost c.
— Probably that a vacancy is filled is denoted qt.
• After meeting a worker, the firm must pay a fixed cost, κ,before bargaining.
• Free entry condition:
c = qt (Jt − κ) .
• Value of a worker to the firm, Jt :
Jt = ϑt −wt + ρEtmt+1Jt+1
Simple Model, cnt’d• Determination of job filling rate and vacancy filling rate
— matching function:
x̃tlt−1︷ ︸︸ ︷new meetingst = σm
vtlt−1︷ ︸︸ ︷vacanciest
1−σ
×
(1−ρ)lt−1+1−lt−1=1−ρlt−1︷ ︸︸ ︷people searching for jobst
σ
.
• Job finding rate:
ft =x̃tlt−1
1− ρlt−1= σm
labor market tightness≡Γt︷ ︸︸ ︷
vtlt−1
1− ρlt−1
1−σ
= σmΓ1−σt
Simple Model, cnt’d• Vacancy filling rate:
qt =x̃tlt−1
vtlt−1= σm
(1− ρlt−1
vtlt−1
)−σ
= σmΓ−σt
• Law of motion of labor force
lt = (ρ+ x̃t) lt−1
• Resource constraint:
Ct + xtlt−1
(κ +
cqt
)= ϑtlt
• Nash sharing rule:
Jt =1− η
ηSw,t,
η share of total surplus, Jt + Sw,t, given to workers
Collecting Equilibrium Conditions• 11 equations and 11 unknowns Jt, wt, mt, Sw,t, lt, ft, qt, x̃t, Ct,
R̃t, Γt :
Jt = ϑt −wt + ρEtmt+1Jt+1
Jt =cqt+ κ
Sw,t = wt −D+ ρEtmt+1 (1− ft+1) Sw,t+1
lt = (ρ+ x̃t) lt−1, ft =x̃tlt−1
1− ρlt−1
ft = σmΓ1−σt , qt = σmΓ−σ
t
ϑtlt = Ct + xtlt−1
(κ +
cqt
)Jt =
1− η
ηSw,t
1 = Etmt+1R̃t+1, mt+1 = βu′ (Ct+1)
u′ (Ct)
Analysis• Dynamics of the system hard to analyze analytically.• Steady state much easier, and surprisingly (as we’ll see later)informative about dynamics.— Can ignore some equations, (*)
J =ϑ−w1− ρβ
, J =cq+ κ
Sw =w−D
1− ρβ (1− f )
1 = ρ+ x̃, f =(1− ρ) l1− ρl
f = σmΓ1−σ, q = σmΓ−σ
ϑl = C+ x̃l(
κ +cq
)(∗)
J =1− η
ηSw, 1 = βR̃ (∗), m = β
Core Steady State Equations• Two equations in two unknowns, Γ and w
— understood, that f = σmΓ1−σ, q = σmΓ−σ
(‘wage equation’)cq+ κ =
ϑ−w1− ρβ
→ w = ϑ− (1− ρβ)
(cq+ κ
)(‘bargaining equation’)
ϑ−w1− ρβ
=1− η
η
w−D1− ρβ (1− f )
→ w =ϑ (1− ρβ (1− f )) + (1− ρβ)
1−ηη D
(1− ρβ)1−η
η + 1− ρβ (1− f )
• Alternatively, think of these as two equations in two unknowns,l and w :
f (l) =(1− ρ) l1− ρl
, Γ (l) =(
f (l)σm
) 11−α
, q (l) = σmΓ (l)−σ
Simple Characterization of Steady StateEquilibrium
Comparative Static on Steady State
Steady State Properties of the Model
• Consider the long run (steady state) impact of a permanentshock to ϑ.
— Steady state effects (sort of) correspond, in dynamic setting,to medium run effects of persistent shock to ϑt.
— We are interested in the effects of all sorts of shocks.• In broader model, the primary avenue by which other shocksaffect labor market is via their impact on ϑt, which isendogenous in those models.
• Evaluate the effect of ϑ on market tightness, Γ :
εΓ,ϑ ≡d log Γd log ϑ
— All other labor market variables are a monotone transform onΓ.
Elasticity of Labor Market Tightness withRespect to Technology Shock
εΓ,ϑ = Υϑ
ϑ−D− τκκ,
Υ =ηρβf + (1− ρβ)
ηρβf + σ (1− ρβ)≥ 1
τκ =Υσ
1− η[ηρβf + (1− ρβ)]
Shimer Puzzle
• Shimer (AER 2005) puzzle: "for any reasonableparameterization of the model, can’t get εΓ,ϑ close to what it isin the data."
— Shimer estimates εΓ,ϑ by ratio of standard deviation of markettightness (from JOLTS data) to standard deviation of laborproductivity.
• Later, we will display stochastic simulations of the model thatprovide some support for Shimer’s approach to estimating εΓ,ϑ .
• Pissarides (2009, p. 1351), Mortensen-Nagypal (RED,2007,p.333): argue that Shimer’s estimate is too high.
— Reports that in a regression of log tightness on log laborproductivity, coeffi cient is 7.56
— Argue that 7.56 is a better estimate of εΓ,ϑ.
Hagedorn-Manovskii (AER 2008)• HM examine elasticity in standard DMP model:
εΓ,ϑ = Υϑ
ϑ−D, Υ =
ηρβf + (1− ρβ)
ηρβf + σ (1− ρβ).
• HM argue that to get elasticity up and solve Shimer puzzle,have two choices, given that in practice the data do not permitmuch playing around with ρ, β, f .— Reduce responsiveness of wage to ϑ by increasing share ofsurplus going to labor, η (w→ D as η → 0)• But, η has relatively small impact. When η = 0, Υ = 1/σ, and σ
conventionally in neighborhood of 1/2. So, Υ bounded between1 (when η = ∞) and 2 (when η = 0).
— Raising D has potentially huge impact via effect onϑ/ (ϑ−D) .
• Hagedorn and Manovskii conclude that only way to solveShimer puzzle is to increase D.
Costain-Reiter
• Two objections to HM solution:
— In practice, D is not very high.— Costain-Reiter: Solutions to Shimer puzzle tend to implyunreasonably high values of du/d log D, where u = 1− l is theunemployment rate -
dud log D
= (1− ρl) l (1− σ)Dϑ× εΓ,ϑ
• Pissarides (2009, p. 1365):— Argues that in the data, du/d log D ' 0.011 : "A ten percentincrease in unemployment benefits results in a 1.1 percentagepoint increase in unemployment."
Numerical Properties of Elasticity Formula• Following graph varies η from 0.26 to 0.55 (results are onlyreported for parameterizations that imply q, f ∈ (0, 1))— first row: c = 0, κ = 0.816, σm = 0.664, σ = 0.52, ϑ = 1,
ρ = 0.9, β = 1.03−1/4, D = 0.365.— second row (standard DMP): same as first row, except
c = 0.571, κ = 0.
• Findings— Shifting meeting costs to fixed portion (first row) substantiallyboosts εΓ,ϑ.
— Consistent with HM, increasing η has tiny impact on εΓ,ϑ andpresumably achieves this via the (tiny or no, in case of firstrow) reduction in wage response.
— With fixed costs of hiring labor, εΓ,ϑ is bigger. Increased wageinertia (reducing dw/dϑ) plays no role in this.
• Consistent with Shimer, no parameterization comes close tosolving the Shimer puzzle.
0.3 0.4 0.5
4.6
4.8
5
5.2
5.4
5.6
eta
GAM elasticity
0.3 0.4 0.5
1
1
1
1
1
eta
dw/dvartheta
0.3 0.4 0.5−1
−0.5
0
0.5
1
eta
vacancy posting cost share
0.3 0.4 0.50.11
0.115
0.12
0.125
0.13
0.135
0.14
eta
du/dlogD
0.3 0.4 0.5−1
−0.5
0
0.5
1
eta
dlog(w)/dlogD
0.3 0.4 0.5
1.915
1.92
1.925
1.93
1.935
1.94
1.945
1.95
eta
GAM elasticity
0.3 0.4 0.5
0.86
0.87
0.88
0.89
0.9
0.91
0.92
eta
dw/dvartheta
0.3 0.4 0.50
0.5
1
1.5
2
eta
vacancy posting cost share
0.3 0.4 0.5
0.038
0.04
0.042
0.044
0.046
0.048
0.05
0.052
0.054
eta
du/dlogD
0.3 0.4 0.5
0.035
0.04
0.045
0.05
0.055
0.06
0.065
eta
dlog(w)/dlogD
Elasticity of Labor Market Tightness withRespect to Technology Shock
• Formula:εΓ,ϑ = Υ
ϑ
ϑ−D− τκκ
— Increase in D raises elasticity, illustrates Hagedorn andManovskii (2008, AER)
— Increase in κ raises elasticity, illustrates Pissarides (P,Econometrica, Vol. 77, No. 5, September, 2009, pp.1339-1369), and Christiano, Trabandt and Walentin (CTW,Journal of Economic Dynamics & Control, vol. 35, 2011, pages2038-2039).
— Formula similar to the one in Ljungqvist and Sargent ("TheFundamental Surplus in Matching Models", manuscript)
More Numerical Properties of ElasticityFormula
• To establish baseline parameter values:— set: β = 1.03−0.25, η = 0.9, ρ = 0.9, σ = 0.52, ϑ = 1, κ = 0.1.— select values of D, σm and c so that l = 0.945, q = 0.70,
D/w = 0.4.
• In following graph,— First row: vary value of κ, holding β, η, ρ, σ, ϑ, D fixed atbaseline and setting σm and c so that l = 0.945, q = 0.70.
— Second row: vary D/w, holding β, η, ρ, σ, ϑ fixed at baseline,c = 0, and setting σm, κ and D so that l = 0.945, q = 0.70,D/w takes on specified value.
• Of course, the elasticity is the change in market tightness withrespect to ϑ, holding all other structural parameters constant.
0 0.05 0.12
4
6
8
10
12
kappa
GAM elasticity
0 0.05 0.1
0.97
0.975
0.98
0.985
0.99
0.995
kappa
dw/dvartheta
0 0.05 0.1
0.2
0.4
0.6
0.8
1
kappa
vacancy posting cost share
0 0.05 0.10.05
0.1
0.15
0.2
0.25
0.3
kappa
du/dlogD
0 0.05 0.1
2
4
6
8
10
12
x 10−3
kappa
dlog(w)/dlogD
0.4 0.6 0.8
5
10
15
20
25
30
Dw
GAM elasticity
0.4 0.6 0.8
1
1
1
1
1
Dw
dw/dvartheta
0.4 0.6 0.8−1
−0.5
0
0.5
1
Dw
vacancy posting cost share
0.4 0.6 0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Dw
du/dlogD
0.4 0.6 0.8−1
−0.5
0
0.5
1
Dw
dlog(w)/dDw
Messages from the Formula
• As κ increases and share of costs to meet worker due tovacancy costs declines, εΓ,ϑ increases (P,CTW,CET).
• As D/w increases, εΓ,ϑ increases.• Response of wage to ϑ shows little variation across experiments.
— Response even moves in the ‘wrong’direction in response to κ— Hall-Shimer intuition, that key to resolving Shimer puzzle liesin reducing responsiveness of wage, is called into question bythe results (see Hagedorn and Manovskii (2008, AER) andLjungqvist and Sargent for further discussion).
• Perturbations which work towards resolution of Shimer puzzle,simultaneously imply a big response of unemployment tounemployment benefits, D (see Costain and Reiter, JEDC2008).
• For intuition, see below.
Alternating Offer Bargaining• Hall-Milgrom intuition (later, called into question)
— With alternating offer bargaining, wage rate in part reflectscosts and benefits of bargaining, detaching wage a little frombroader economic conditions.
— Result: with reduced sensitivity of wage to general conditions,when a shock launches an expansion, firms enjoy a larger shareof the rents from a match and therefore have a greaterincentive to expand employment.
— Hall-Milgrom suggested that this is how alternating offerbargaining may contribute to resolution of Shimer puzzle.
— To resolve puzzle, still need to reduce c and raise κ.
• Baseline specification:— Each worker-firm pair bargains each period.— Bargain over current wage rate, taking outcome of future wagebargains given.
— ‘Period-by-Period Bargaining’.
Alternating Offers• Each quarter is divided into M equal subperiods, m = 1, .., M.
— Firm makes an opening wage offer in m = 1.— Worker may reject and make a counter offer in m = 2.— Firm may reject worker’s wage offer and make a new offer innext sub-period,...
— If there is a whole sequence of rejections, worker makes atake-it-or-leave-it offer in last subperiod M.
• If an offer is accepted in any sub period m, production beginsimmediately.
— Value of production in any subperiod is ϑt/M.
• Solution to the bargaining problem:
w1t (≡ wt) , w2
t , ..., wMt .
Firm’s Offer: round 1
• Firm offers w1t as low as possible subject to worker not rejecting
it:
utility of worker who accepts
firm offer and goes to work︷︸︸︷V1
t =
utility of worker who rejects
firm offer and intends to make counteroffer︷ ︸︸ ︷δU1
t + (1− δ)
(DM+V2
t
)where,
V1t ≡ w1
t +Etmt+1 [ρVt+1 + (1− ρ) (ft+1Vt+1 + (1− ft+1)Ut+1)]
Worker Offer: round 2• Worker proposes highest possible wage w2
t subject to firm notrejecting it:
value of firm that
accepts worker offer︷︸︸︷J2t =
value of firm that rejects worker
offer and intends to make counteroffer︷ ︸︸ ︷δ× 0+ (1− δ)
[−γ+ J3
t
]• The firm incurs cost γ to make a counter offer.
• Firm value:
J2t ≡
value of worker output in subperiods 2 to M︷ ︸︸ ︷ϑt
M− 1M
−w2t + ρEtmt+1Jt+1
Alternating Offers, Final Round
• Each bargaining round requires the wage for the next round.• If they go to last round with no agreement, the worker makes afinal, take-it-or-leave-it-offer:
value of firm that
accepts worker offer in last round︷︸︸︷JMt =
value of firm that rejects worker’s
take-it-or-leave-it offer︷︸︸︷0
or
JMt ≡
1M
ϑt −wMt + ρEtmt+1Jt+1 = 0,
or
wMt =
1M
ϑt + ρEtmt+1
=κ︷︸︸︷Jt+1
Calculations
• To determine wt ≡ w1t , firm first solves wM
t , wM−1t , wM−2
t , ...,w2
t .
• M equilibrium conditions for the M unknowns.
• Linearity of bargaining equilibrium conditions implies:— simple equation determines spot wage, wt.
Alternating Offer Sharing Rule
• Wage satisfies:
Jt = β1
=Sw,t︷ ︸︸ ︷(Vt −Ut)− β2γ+ β3 (ϑt −D) ,
where
α1 = 1− δ+ (1− δ)M , α2 = 1− (1− δ)M ,
α3 = α21− δ
δ− α1, α4 =
1− δ
2− δ
α2
M+ 1− α2,
and βi = αi+1/α1, i = 1, 2, 3.
Alternative Bargaining Arrangements• Alternative arrangement has workers and firms bargaining justonce, when they first meet. Equilibrium allocations always thesame.
— negotiate over wage rates in each date and state of natureassociated with the duration of their match.
— they do not care about the precise pattern of wage payments,only the present discounted value (PV).
— many patterns are possible, including the pattern in theperiod-by-period bargaining assumed in the paper.
• one pattern: worker receives fixed nominal wage as long asshe’s with firm.
• Wages of new hires more volatile than wages of incumbents.
• Key issue associated with PV bargaining: commitment.— no need to address these issues in period-by-period bargaining.
Effect of Alternating Offer Bargaining onEquilibrium Conditions
• Replace Nash Sharing rule:
Jt =1− η
ηSw,t
with alternating offer sharing rule:
Jt = β1Sw,t − β2γ+ β3 (ϑt −D) .
• With this sharing rule, the steady state elasticity,d log Γ/d log ϑ is altered.
Elasticity of Market Tightness with Respectto Productivity
• elasticity formula now has γ in there...higher γ, higherelasticity...
εΓ,ϑ = Υϑ
ϑ−D− τκκ − τγγ
Elasticity formula: the pieces
ψ ≡ ρβf + σ (1− ρβ) (1+ β1)
ρβf + (1− βρ) (1+ β1),
Υ =β1 + β3 (1− ρβ (1− f ))
ψa
τκ =(1+ β1) (1− ρβ) + ρβf + ρβf (σ−1)
ψ
a,
τγ =
(1− βρ (1− f ) + ρβf (σ−1)
ψ
)β2
a
a = β1 + β3
(1− βρ (1− f ) +
ρβf (σ− 1)ψ
)
Following Figure
• Top and bottom rows constructed as in earlier figure which isreproduced here.
0 0.05 0.12
4
6
8
10
12
kappa
GAM elasticity
0 0.05 0.1
0.97
0.975
0.98
0.985
0.99
0.995
kappa
dw/dvartheta
0 0.05 0.1
0.2
0.4
0.6
0.8
1
kappa
vacancy posting cost share
0 0.05 0.10.05
0.1
0.15
0.2
0.25
0.3
kappa
du/dlogD
Two things in top row: (i) Nash (star) to AOB has big effect on market tightness elasticity(ii) Search (kappa = 0) to hiring (kappa > 0) has big effect, but has bigger impact on AOB than NashBottom row: (i) rise in D/w has huge effect on labor market volatility but also raises impact of benefits
(ii) going from Nash to AOB involves a (small) rise in wage inertia
0 0.05 0.1
2
4
6
8
10
12
x 10−3
kappa
dlog(w)/dlogD
0.4 0.6 0.8
5
10
15
20
25
30
Dw
GAM elasticity
0.4 0.6 0.8
1
1
1
1
1
Dw
dw/dvartheta
0.4 0.6 0.8−1
−0.5
0
0.5
1
Dw
vacancy posting cost share
0.4 0.6 0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Dw
du/dlogD
0.4 0.6 0.8−1
−0.5
0
0.5
1
Dw
dlog(w)/dDw
Message of Formula
• Parameter values:
β = 1.03−0.25, σ = 0.52, ρ = 0.9, ε = 6, ρa = 0.95,Nash︷ ︸︸ ︷
η = 0.9,
Alternating offer bargaining︷ ︸︸ ︷γ = 0.01, δ = 0.005, M = 60,
• Going to alternating offer bargaining (AOB) increases elasticityof tightness with respect to ϑ.
— However, it simultaneously increases the impact of permanentchanges in unemployment benefits.
• Wage inertia plays quantitatively little or no (second row) rolein results (in first row, moves in ‘wrong’direction with κ).
Some Intuition About Elasticity Formula• The wage equation with q expressed in terms of Γ :
cσm
Γσ + κ =ϑ−w1− ρβ
.
• Differentiate:
εΓ,ϑ =1σ
ϑ (1− dw/dϑ)
ϑ−w− κ (1− ρβ).
• Two ways to raise εΓ,ϑ :— Numerator: wage inertia (reduce dw/dϑ), but this exhibits toolittle variation to play an appreciable role
— Denominator: reduce steady state profits, net of amortizedfixed cost of meeting worker.
• Reducing profits:— Rise in κ directly reduces (relevant measure of) profits.— Rise in D or γ shifts bargaining equation (i.e., sharing rule) upand, given downward-sloped wage equation with c > 0, wrises, reducing profits.
Impact of Bargaining Parameters
• Baseline model parameters:— set: β = 1.03−0.25, ρ = 0.9, σ = 0.52, ϑ = 1, κ = 0.098,
δ = 0.005, γ = 0.01, M = 60, D = 0.396.— select values of σm and c so that l = 0.945, q = 0.70.
• Change γ (top row) and δ (bottom row)
— always adjust σm, c to fix q, l, but keep all other parameters atbaseline.
• Results:— Reducing δ and/or raising γ raises εΓ,ϑ.— Change in wage inertia very small, but, significantly, moves in‘wrong’direction.
— du/d log (D) is mirror image of εΓ,ϑ (Costain-Reiter).
5 10
x 10−3
4
5
6
7
8
9
10
11
gamma
GAM elasticity
5 10
x 10−3
0.955
0.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
gamma
dw/dvartheta
5 10
x 10−3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
gamma
vacancy posting cost share
5 10
x 10−3
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
gamma
du/dlogD
5 10
x 10−3
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
gamma
dlog(w)/dlogD
5 10
x 10−3
6
7
8
9
10
11
12
delta
GAM elasticity
5 10
x 10−3
0.975
0.98
0.985
0.99
0.995
delta
dw/dvartheta
5 10
x 10−3
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
delta
vacancy posting cost share
5 10
x 10−3
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
delta
du/dlogD
5 10
x 10−3
1
2
3
4
5
6
7
8
9
10
x 10−3
delta
dlog(w)/dlogD
Why Wage Inertia Seems to Move in‘Wrong’Direction with Market Tightness• Note from the previous slide that dw/dϑ appears to comovewith εΓ,ϑ.
• This is a complete contradiction to the Hall-Shimer intuitionthat to get volatility up you need to have wage inertia.
• Following is the analytic formula for dw/dϑ :
dwdϑ=
β1(w−D)βρ(1−σ)fϑ(1−βρ+βρf )2
εΓ,ϑ +1
1−ρβ − β3
β11−βρ+βρf +
11−ρβ
— Note that, for given f and w, this is an increasing function ofεΓ,ϑ.
— It was verified numerically, that an increase in γ or decrease inδ raises w, reinforcing positive association between dw/dϑ andεΓ,ϑ.
Role of Wage Inertia and Effects ofAlternating Offer Bargaining
• Conventional intuition:— rise in γ or reduction in δ ‘insulates’wage from business cycle— with greater wage inertia, labor market variables respond moreto shocks.
• Conventional wisdom contradicted in previous numerical results.• Could the results reflect the calibration of c and σm in previousresults?
• Next Figure: vary γ and δ holding all other structuralparamerers fixed at baseline.
— Bottom row: results unaffected; top row: impact of γ on εΓ,ϑreversed.
9 10 11
x 10−3
11
11.5
12
12.5
13
13.5
14
gamma
GAM elasticity
9 10 11
x 10−3
0.9975
0.998
0.9985
0.999
gamma
dw/dvartheta
9 10 11
x 10−3
0.015
0.02
0.025
0.03
gamma
vacancy posting cost share
9 10 11
x 10−3
0.28
0.29
0.3
0.31
0.32
0.33
0.34
0.35
gamma
du/dlogD
9 10 11
x 10−3
4
5
6
7
8
9
10
x 10−4
gamma
dlog(w)/dlogD
5 10
x 10−3
9
9.5
10
10.5
11
11.5
12
delta
GAM elasticity
5 10
x 10−3
0.9984
0.9984
0.9985
0.9985
0.9986
0.9986
delta
dw/dvartheta
5 10
x 10−3
0.02
0.022
0.024
0.026
0.028
0.03
0.032
delta
vacancy posting cost share
5 10
x 10−3
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
delta
du/dlogD
5 10
x 10−3
5.4
5.6
5.8
6
6.2
6.4
x 10−4
delta
dlog(w)/dlogD
Understanding Previous Results• Following graph repeats:
εΓ,ϑ =1σ
ϑ (1− dw/dϑ)
ϑ−w− κ (1− ρβ),
from the previous graph.• In addition, display
"εΓ,ϑ due to wage inertia"=1σ
ϑ (1− dw/dϑ)
[ϑ−w− κ (1− ρβ)]baseline,
where [ϑ−w− κ (1− ρβ)]baseline represents profits in thebaseline parameterization.
• Also, display
"εΓ,ϑ due to profits"=1σ
ϑ [(1− dw/dϑ)]baseline
ϑ−w− κ (1− ρβ).
• The figure shows how the numerator and denominatorcontribute to the previous results.
8.5 9 9.5 10 10.5 11 11.5
x 10−3
8
10
12
14
16
18
20
22
gam
Ela
stic
ity o
f Mar
ket T
ight
ness
w.r
.t. v
arth
eta
elasticity
elasticity, due to wage inertia
elasticity, due to profits
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
x 10−3
8
9
10
11
12
13
14
delta
Ela
stic
ity o
f Mar
ket T
ight
ness
w.r
.t. v
arth
eta
elasticity
elasticity, due to wage inertia
elasticity, due to profits
Previous Results
• Results for increase in γ :
— Rise in γ reduces εΓ,ϑ.— Consistent with earlier intuition, focusing on profits alone leadsto implication that εΓ,ϑ rises.
— Elasticity actually falls because rise in dw/dϑ dominates εΓ,ϑ.— Hall-Milgrom focus on wage inertia justified, but their intuitionthat an increase in γ reduces dw/dϑ is contradicted.
• Results for decrease in δ :
— Leads to increase in εΓ,ϑ.— Hagedorn-Manovskii intuition that focuses on level effect onprofits is vindicated.
— Hall-Milgrom intuition that focuses on wage inertia iscontradicted.
Role of Wage Inertia and Effects ofAlternating Offer Bargaining
• Baseline parameter values in previous computations:— set: β = 1.03−0.25, ρ = 0.9, σ = 0.52, ϑ = 1, κ = 0.098,
δ = 0.005, γ = 0.0082, M = 60, D = 0.396.— select values of σm and c so that l = 0.945, q = 0.70(c = 0.0014, σm = 0.6638).
— υc = 0.02.
• Conventional wisdom about role of wage inertia in getting εΓ,ϑup contradicted in previous numerical results.
• Is this due to low value of υc?— new baseline: set κ, l, q so that l = 0.945, q = 0.70, υc = 0.98(κ = 0.002, c = 0.069, σm = 0.6638).
• Next figure: vary γ and δ holding all other structuralparamerers fixed at baseline.— Results unaffected.
9 10 11
x 10−3
5.6
5.8
6
6.2
6.4
6.6
gamma
GAM elasticity
9 10 11
x 10−3
0.962
0.963
0.964
0.965
0.966
0.967
0.968
0.969
0.97
gamma
dw/dvartheta
9 10 11
x 10−3
0.976
0.977
0.978
0.979
0.98
0.981
0.982
0.983
0.984
gamma
vacancy posting cost share
9 10 11
x 10−3
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
gamma
du/dlogD
9 10 11
x 10−3
0.012
0.0125
0.013
0.0135
0.014
0.0145
0.015
gamma
dlog(w)/dlogD
5 10
x 10−3
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
delta
GAM elasticity
5 10
x 10−3
0.9665
0.9665
0.9666
0.9666
0.9667
0.9667
0.9668
0.9668
0.9669
0.9669
delta
dw/dvartheta
5 10
x 10−3
0.98
0.981
0.982
0.983
0.984
0.985
delta
vacancy posting cost share
5 10
x 10−3
0.1
0.11
0.12
0.13
0.14
0.15
0.16
delta
du/dlogD
5 10
x 10−3
0.0133
0.0133
0.0133
0.0133
0.0134
0.0134
0.0134
delta
dlog(w)/dlogD
Analysis of Dynamics• Must use computer to do model simulation (e.g., perturbation,projection, extended path)
• When κ large compared to c, steady state equilibrium isindeterminate.
— Can be shown analytically for the case, mt+1 = β, c = 0 (seebelow).
• With price setting frictions (and η large enough, in the Nashsharing case), steady state equilibrium determinate.
• Do the analysis with sticky prices.— With c = 0 and alternating offer bargaining, model performswell relative to data.
— No Shimer puzzle.— This is so, despite the absence of exogenous stickiness inwages.
Indeterminacy of Steady State with HiringCosts
• Equilibrium conditions (sum of firm and worker surplus valuefunctions; law of motion of labor; job finding rate; sharing rule):
Sw,t + κ = ϑt −D+ ρβ [(1− ft+1) Sw,t+1 + κ]
lt = (ρ+ x̃t) lt−1,
ft =x̃tlt−1
1− ρlt−1,
κ =1− η
ηSw,t,
for t = 0, 1, 2, .... (mt+1 = β, for analytical convenience).• Steady state equilibrium: constant values of Sw,t, lt, x̃t, ft, thatsatisfy equilibrium conditions for ϑt = ϑ, all t ≥ 0.
• Steady state equilibrium is unique, and corresponds toSw,t = Sw, lt = l, x̃t = x̃, ft = f for all t ≥ 0, l−1 = l.
Indeterminacy of Steady State Equilibrium
• Let mt+1 = β, c = 0.• The steady state equilibrium is indeterminate: given l−1 = l,one can find another sequence of values of Sw,t, lt, x̃t, ft,t = 0, 1, 2, ..., that satisfy the equilibrium conditions and whichremain arbitrarily close to the steady state equilibrium.
— Sharing rule implies Sw,t = Sw for all t ≥ 0.— Total surplus equation implies ft+1 = f for t = 0, 1, 2, ...— Choose a value for x̃0 6= x, and set f0 = x̃0l/ (1− ρl) 6= f ,
l0 = f0 + ρ (1− f0) l 6= l.— Set lt = f + ρ (1− f ) lt−1, t ≥ 1.— Note that all these x̃t, ft, lt paths:
• converge to steady state (because 0 < ρ (1− f ) < 1)• satisfy the equilibrium conditions,• can be made arbitrarily close to steady state equilibrium bysetting x̃0 arbitrarily close to x̃.
Outline
• Integrate bargaining model into sticky price NK model.
• Evaluate the model in two ways:— Shimer style comparison to unconditional moments.— Impulse response matching.
• Conclude— Alternating offer bargaining and c = 0 solve Shimer puzzle.
Households and Final Good Firms• Household intertemporal condition:
1Xt
= Et1
Xt+1R∗t+1
Rt
π̄t+1,
R∗t+1 ≡1β
exp (at+1 − at)
Xt ≡Ct
exp (at), π̄t = 1+ πt.
• Final good firms solve
max PtYt −∫ 1
0Pi,tYi,tdj, s.t. Yt =
[∫ 1
0Y
ε−1ε
i,t dj] ε
ε−1
.
• Demand curve for ith monopolist (‘sticky price retailers’)
Yi,t = Yt
(Pt
Pi,t
)ε
.
Intermediate Good Firms
• Production function:
Yi,t = exp (at) hi,t, at = ρaat−1 + εat ,
where hi,t is a homogeneous input (not labor!) withafter-subsidy nominal price, (1− ν) ϑtPt.
• Calvo Price-Setting Friction:
Pi,t =
{P̃t with probability 1− θPi,t−1 with probability θ
.
Price-setting equilibrium conditions
p̃t =Kt
Ft, p̃t ≡
P̃t
Pt, Ft =
Yt
Ct+ βθEtπ̄
ε−1t+1Ft+1
Kt =ε
ε− 1st
Yt
Ct+ βθEtπ̄
εt+1Kt+1
p̃t =
[1− θπ̄ε−1
t1− θ
] 11−ε
→ Kt
Ft=
[1− θπ̄ε−1
t1− θ
] 11−ε
st =(1− ν) ϑt
exp (at)
p∗t =
(1− θ)
(1− θπ̄
(ε−1)t
1− θ
) εε−1
+ θπ̄ε
tp∗t−1
−1
Resource Constraint and Monetary Policy
• Resource constraint:
Ct +
(cqt+ κ
)x̃tlt−1 = Yt
p∗t lt exp (at) = Yt.
• Clearing in demand and supply, input good to intermediategood producers:
ht = lt.
• Taylor rule:
Rt
R= exp {φπ (π̄t − 1)} , φπ = 1.5.
Labor Market Equations• Value of worker to firm; firm free entry condition; recursiverepresentation of household surplus, Sw,t; law of motion of lt;labor market tightness, Γt; finding rate; vacancy filling rate andsharing rule:
Jt = ϑt −wt + ρEtmt+1Jt+1
Jt =cqt+ κ
Sw,t = wt −D+ ρEtmt+1 (1− ft+1) Sw,t+1
lt = (ρ+ x̃t) lt−1, Γt =x̃tlt−1
1− ρlt−1
ft = σmΓ1−σt , qt = σmΓ−σ
tJt = β1Sw,t − β2γ+ β3 (ϑt −D)
Usefulness in Steady State ElasticityFormula for Understanding Dynamics
• Surprisingly good at predicting model dynamics (θ = 0.75).
d log Γd log ϑ
std(log Γ)std(a)
Nash Sharingκ = 0 (‘pure search costs’) 1.8 1.8c = 0 (‘pure hiring costs’) 4.0 4.3
Alternating Offer Bargainingκ = 0 6.0 6.1c = 0 11.7 12.2c = 0, δ = 0.001 47.1 41.4
Calibration/Parameterization
Parameter Value DescriptionPanel A: Parameters
β 1.03-0.25 Discount factorξ 0.66 Calvo price stickinessλf 1.2 Price markup parameterρR 0.7 Taylor rule: interest rate smoothingrπ 1.7 Taylor rule: inflation coeffi cientry 0.1 Taylor rule: employment coeffi cientρ 0.9 Job survival probabilityδ 0.005 Prob. of bargaining session break-upM 60 Max bargaining rounds per quarterρa 0.95 Roots for AR(1) technology
Panel B: Steady State Values400(π − 1) 0 Annual net inflation rate
l 0.945 Employmentκxl/Y 0.01 Hiring cost to output ratioD/w 0.4 Replacement ratio
Intuition• Policy shock drives real interest rate down.
— Induces increase in demand for output of final good producersand therefore output of sticky price retailers.
— Latter must satisfy demand, so retailers purchase more ofwholesale good driving up its relative price.
— Marginal revenue product (ϑt) associated with worker rises.— Wholesalers hire more workers, raising probability thatunemployed worker finds a job.
• Workers’disagreement payoffs rise.— Increase in workers’bargaining power generates rise in realwage.
• Alternating offer bargaining mutes rise in real wage.— Allows for large increase in employment, substantial decline inunemployment, small rise in inflation.
Medium-Sized DSGE Model
• Standard empirical NK model (e.g., CEE, ACEL, SW).
— Calvo price setting frictions, but no indexation— Habit persistence in preferences.— Variable capital utilization.— Investment adjustment costs.
• Our labor market structure
Estimated Medium-Sized DSGE Model
• Estimate VAR impulse responses of aggregate variables to amonetary policy shock and two types of technology shocks.
• 11 variables considered:— Macro variables and real wage, hours worked, unemployment,job finding rate, vacancies.
• Estimate model using Bayesian variant of CEE (2005) strategy:— Minimizes distance between dynamic response to three shocksin model, analog objects in the data.
— Particular Bayesian strategy developed in Christiano, Trabandtand Walentin (2011).
Posterior Mode of Key Parameters
• Prices change on average every 2.5 quarters.
• δ : roughly 0.26% chance of a breakup after rejection.
• γ : cost to firm of preparing counteroffer is 1/4 of a day’sworth of production.
• Posterior mode of hiring cost as a percent of output (dependson κ): 0.54% of GDP.
Posterior Mode of Key Parameters
• Replacement ratio is 0.62.
— Defensible based on micro data (Gertler-Sala-Trigari,Aguiar-Hurst-Karabarbounis).
• Gertler, Sala and Trigari (2008) : plausible range forreplacement ratio is 0.4 to 0.7.
— Lower bound based on studies of unemployment insurancebenefits
— Upper boundary takes into account informal sources ofinsurance.
0 5 10
−0.2
0
0.2
0.4
GDP
0 5 10
−0.2
−0.1
0
0.1
0.2
Unemployment Rate
0 5 10
−0.2
−0.1
0
0.1
0.2
Inflation
0 5 10−0.8
−0.6
−0.4
−0.2
0
0.2
Federal Funds Rate
0 5 10
−0.2
0
0.2
0.4
Hours
0 5 10
−0.2
0
0.2
0.4
Real Wage
0 5 10
−0.2
0
0.2
0.4
Consumption
0 5 10
−0.2
0
0.2
0.4
Rel. Price Investment
0 5 10−1
0
1
Investment
0 5 10−1
0
1
Capacity Utilization
0 5 10−1
0
1
Job Finding Rate
Medium−Sized Model Impulse Responses to a Monetary Policy Shock
0 5 10−2
0
2
4
Vacancies
Notes: x−axis: quarters, y−axis: percent
VAR 95% VAR Mean Alternating Offer Bargaining Model
0 5 10−0.2
0
0.2
0.4
0.6
0.8
GDP
0 5 10
−0.2
−0.1
0
0.1
0.2
Unemployment Rate
0 5 10
−0.8
−0.6
−0.4
−0.2
0
0.2
Inflation
0 5 10
−0.4
−0.2
0
0.2
Federal Funds Rate
0 5 10−0.2
0
0.2
0.4
0.6
0.8
Hours
0 5 10−0.2
0
0.2
0.4
0.6
0.8
Real Wage
0 5 10−0.2
0
0.2
0.4
0.6
0.8
Consumption
0 5 10
−0.5
0
0.5Rel. Price Investment
0 5 10
−1
0
1
2Investment
0 5 10
−1
0
1
2Capacity Utilization
0 5 10
−1
0
1
2Job Finding Rate
Medium−Sized Model Impulse Responses to a Neutral Technology Shock
0 5 10
−2
0
2
4Vacancies
Notes: x−axis: quarters, y−axis: percent
VAR 95% VAR Mean Alternating Offer Bargaining Model
0 5 10
−0.2
0
0.2
0.4
0.6
GDP
0 5 10
−0.2
−0.1
0
0.1
0.2
Unemployment Rate
0 5 10
−0.8
−0.6
−0.4
−0.2
0
0.2
Inflation
0 5 10−0.4
−0.2
0
0.2
0.4
Federal Funds Rate
0 5 10
−0.2
0
0.2
0.4
0.6
Hours
0 5 10
−0.2
0
0.2
0.4
0.6
Real Wage
0 5 10
−0.2
0
0.2
0.4
0.6
Consumption
0 5 10
−0.8
−0.6
−0.4
−0.2
0
0.2
Rel. Price Investment
0 5 10
−1
0
1
2Investment
0 5 10
−1
0
1
2Capacity Utilization
0 5 10
−1
0
1
2Job Finding Rate
Medium−Sized Model Responses to an Investment−specific Technology Shock
0 5 10
−2
0
2
4
Vacancies
Notes: x−axis: quarters, y−axis: percent
VAR 95% VAR Mean Alternating Offer Bargaining Model
Comparison With Two Other Models
• Standard DMP setup:— Firms post vacancies and meet workers probabilistically(c > 0, κ = 0).
— Workers and firms split surplus using a Nash-sharing rule.
• Standard New Keynesian sticky wage model followingErceg-Henderson-Levin (2000).
— No wage indexation.
• Embed labor market models in CEE-style empirical model.— Calvo price rigidities, but no price indexation.
Model Comparisons
• Marginal likelihood:— strongly prefers our model over standard DMP and NK stickywage models by about 24 and 54 log points, respectively.
• Also, other models have relatively extreme parameter estimates.
— For example, standard DMP formulation (Nash-sharing plussearch), posterior mode of replacement ratio is 0.97.
Cyclicality of Unemployment and Vacancies• Similar to Shimer (2005), we simulate our model subject to astationary neutral technology shock only.
— Fixed parameter values.
Standard Deviations of Data vs. Models
σ(Labor market tightness)σ(Labor productivity)
Data 27.6
Standard DMP Model 13.6
Our Model 33.5
• Estimated DMP models also do well here.
Conclusion
• We constructed a model that accounts for the economy’sresponse to various business cycle shocks.
• Our model implies that nominal and real wages are inertial.— Allows to account for weak response of inflation and strongresponses of quantity variables to business cycle shocks.
• Model outperforms sticky wage (no-indexation) NK in terms ofstatistical fit.
• Given limitations of sticky wage model, there’s simply no needto work with it.
