Unemployment and Business Cycles

Lawrence J. ChristianoMartin EichenbaumMathias Trabandt

Background

• Key challenge for modern business cycle models.

— How to account for observed volatility of labor marketvariables?

— Central issue going back to dawn of modern macro models,Lucas and Rapping (1969).

• Standard diagnosis

— For plausibly parameterized models, in a boom, wages rise toorapidly, limiting expansion of employment.

— Classic RBC models (Chetty), standard e¢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.,e§ects of extending unemployment benefits).

What We Do• Develop and estimate a model in which wage inertia is derivedas an equilibrium outcome.

• Build on Hall-Milgrom (2008, HM):— When workers and firms bargain, they think they’re better o§reaching agreement than parting ways.

— Disagreement leads to continued negotiations.— HM’s key insight: if negotiation costs don’t depend sensitivelyon state of economy, neither do wages.

• Our dynamic GE model embeds this source of wage inertia andaccounts for key features of the business cycle.

• Sticky wages have been an essential (and, somewhatembarrassing) feature of business cycle models— they are no longer necessary.

Empirical Results

• Estimation strategy: Bayesian impulse response matching.

— Shocks to monetary policy, neutral and investment-specifictechnology.

— Our model performs well relative to this metric.— Outperforms standard alternatives.

• Alternative strategy: focus on Shimer-type unconditionalmoments.

— Example: labor market tightness is much more volatile thanlabor productivity.

— Our model has no di¢culty in accounting for this fact.— No Shimer puzzle.

Labor Market Model

• Large number of identical and competitive firms; producehomogeneous output using only labor, l

t

.

• Firm pays fixed cost, k, to meet a worker with probability 1

(GT, GST).

— In our empirical work we also consider a standard DMP setupwhere cost of meeting a worker is increasing function of labormarket tightness.

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

.

•J

t

and m

t+1

are determined in general equilibrium.

• Free entry and zero profits dictate:

k = J

t

.

Value Functions

• Value of employment to a worker:

V

t

= w

t

+E

t

m

t+1

[rV

t+1

+ (1 r) (ft+1

V

t+1

+ (1 f

t+1

)U

t+1

)] .

where f

t+1

V

t+1

are job-to-job transitions

• Employment law of motion and job finding rate:

l

t

= (r+ x

t

) l

t1

and f

t

=x

t

l

t1

1 rl

t1

where x

t

denotes the hiring rate.

Value Functions

• Value of unemployment to a worker:

U

t

= D+ E

t

m

t+1

[ft+1

V

t+1

+ (1 f

t+1

)U

t+1

] .

where D denotes unemployment benefits.

Bargaining

• 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 O§ers• Each quarter is divided into M equal subperiods, m = 1, .., M.

— Firm makes an opening wage o§er in m = 1.

— Worker may reject and make a counter o§er in m = 2.

— Firm may reject worker’s wage o§er and make a new o§er innext sub-period,...

— If there is a whole sequence of rejections, worker makes atake-it-or-leave-it o§er in last subperiod M.

• If an o§er is accepted in any sub period m, production beginsimmediately.

— Value of production in any subperiod is J

t

/M.

• Solution to the bargaining problem:

w

1

t

( w

t

) , w

2

t

, ..., w

M

t

.

Firm’s O§er: round 1

• Firm o§ers w

1

t

as low as possible subject to worker not rejectingit:

utility of worker who accepts

firm o§er and goes to workz}|{V

1

t

=

utility of worker who rejects

firm o§er and intends to make countero§erz }| {dU

1

t

+ (1 d)

D

M

+V

2

t

where,

V

1

t

w

1

t

+E

t

m

t+1

[rV

t+1

+ (1 r) (ft+1

V

t+1

+ (1 f

t+1

)U

t+1

)]

Worker O§er: round 2• Worker proposes highest possible wage w

2

t

subject to firm notrejecting it:

value of firm that

accepts worker o§erz}|{

J

2

t

=

value of firm that rejects worker

o§er and intends to make countero§erz }| {d 0+ (1 d)

hg+ J

3

t

i

• The firm incurs cost g to make a counter o§er.

• Firm value:

J

2

t

value of worker output in subperiods 2 to Mz }| {J

t

M 1

M

w

2

t

+ rE

t

m

t+1

J

t+1

Alternating O§ers, 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-o§er:

value of firm that

accepts worker o§er in last roundz}|{J

M

t

=

value of firm that rejects worker’s

take-it-or-leave-it o§erz}|{

0

or

J

M

t

1

M

J

t

w

M

t

+ rE

t

m

t+1

J

t+1

= 0,

or

w

M

t

=1

M

J

t

+ rE

t

m

t+1

=kz}|{J

t+1

Calculations

• To determine w

t

w

1

t

, firm first solves w

M

t

, w

M1

t

, w

M2

t

, ...,

w

2

t

.

•M equilibrium conditions for the M unknowns.

• Linearity of bargaining equilibrium conditions implies:

— simple equation determines spot wage, w

t

.

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 as he’swith 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.

Alternating O§ers in a Simple Macro Model

• Competitive final goods production: Y

t

=

2

41Z

0

Y

1

l

f

j,t

dj

3

5l

f

.

•j

th input produced by monopolistic ‘retailers’:

— Production: Y

jt

= exp(at

)hj,t

.

— Homogeneous good, h

j,t

, purchased in competitive markets forreal price, J

t

.

— Retailers prices subject to Calvo sticky price frictions (no priceindexation).

• Homogeneous input good h

t

produced by the firms in our labormarket model, ‘wholesalers’.

A Simple Macro Model ...

• Representative household:

E

0

•

Ât=0

b

t

ln C

t

P

t

C

t

+ B

t+1

W

t

l

t

+ P

t

D (1 l

t

) + R

t1

B

t

+ T

t

• Household SDF, m

t+1

= bC

t

/C

t+1

.

A Simple Macro Model ...

• Key log-linearized equilibrium conditions:

ˆ

C

t

= E

t

ˆ

C

t+1

ˆ

R

t

p

t+1

p

t

= bE

t

p

t+1

+ (1q)(1bq)q

real marginal costz }| {ˆ

J

t

a

t

C

ˆ

C

t

+ kxl

ˆ

x

t

+ ˆ

l

t1

= Y

ˆ

Y

t

ˆ

R

t

= a

ˆ

R

t1

+ (1 a)hf

p

p

t

+ f

y

ˆ

l

t

i+ #

R,t

Calibration/Parameterization

Parameter Value DescriptionPanel A: Parameters

b 1.03-0.25 Discount factorx 0.66 Calvo price stickinessl

f

1.2 Price markup parameterr

R

0.7 Taylor rule: interest rate smoothingr

p

1.7 Taylor rule: inflation coe¢cientr

y

0.1 Taylor rule: employment coe¢cientr 0.9 Job survival probabilityd 0.005 Prob. of bargaining session break-upM 60 Max bargaining rounds per quartert 0.95 Roots for AR(1) technology

Panel B: Steady State Values400(p 1) 0 Annual net inflation rate

l 0.945 Employmentkxl/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 (Jt

) associated with worker rises.— Wholesalers hire more workers, raising probability thatunemployed worker finds a job.

• Workers’ disagreement payo§s rise.— Increase in workers’ bargaining power generates rise in realwage.

• Alternating o§er bargaining mutes rise in real wage.— Allows for large increase in employment, substantial decline inunemployment, small rise in inflation.

Alternating O§ers: Intuition

• Wages are relatively insulated from general economicconditions.

• To gain some intuition, it’s useful to see how bargainingparameters influence responsiveness of wage to generaleconomic conditions.

• Consider bargaining session between worker and firm in partialequilibrium.

— They take all variables outside their control as given.

• Consider bargaining session between a single worker and a singlefirm after a rise in U

t

experienced idiosyncratically by that pair.

Intuition...

• Suppose we’re in nonstochastic steady state.

— All aggregate shocks are fixed at their unconditional means,aggregate variables are constant

— Ongoing idiosyncratic uncertainty at the worker-firm level.

w

U

d log w

d log U

=U

w

dw

dU

•U and w denote value of unemployment and equilibrium wagein non-stochastic steady state.

• Derivative treats rise in U as something experiencedidiosyncratically by one worker-firm bargaining pair.

Lower Break-up Probability

• Consider extreme case where d = 0.

— There’s no chance that workers and firms are thrown to theiroutside option during negotiations.

— Here the value of unemployment, U, doesn’t enter directly intoindi§erence conditions governing worker and firms o§ers.

— So we don’t expect the real wage to depend much on outsideconditions (shocks).

• By continuity, larger values of d raise importance of U inworker’s disagreement payo§, make real wage more sensitive toshocks.

Lower Firm Negotiation Costs

• Decrease in g raises disagreement payo§ of the firm, puttingworker in weaker bargaining position.

— Other things equal, this leads to decrease in w

i: dw

i

/dg > 0

— But decrease is same regardless of U

i

, so dw

i

/dU isindependent of g: d

dw

i

/dU

/g = 0.

•d(dw

i

/dg)/dU = 0

• E§ect of change in g on elasticty w

U

operates entirely throughits e§ect on steady state U/w.

Lower Firm Negotiation Costs

• Zero profit condition of firms implies w is independent of g.

— Decrease in g places downward pressure on all worker-firm pairwages.

— Since equilibrium steady state w doesn’t respond to g, U mustchange to neutralize downward pressure on w.

— Rise in U (lower steady state unemployment) places upwardpressure on w increasing the worker’s disagreement payo§ andhis bargaining power.

• So a fall in g raises d log w/d ln(U)

Value�of�unemployment,�U

Real�wage,�w

G " w1 " *> � J � 4

Summary of bargaining: w � F�U;+,D,- 

Lower Unemployment Benefits

• Decrease in D lowers disagreement payo§ of workers, puttingfirm in stronger bargaining position.

— Other things equal, this leads to fall in w

i: dw

i

/dD > 0

— But fall is same regardless of U

i

, sod(dw

i

/dD)/dU = 0 ! d(dw

i

/dU)/dD = 0

• E§ect of change in D on elasticty w

U

operates entirely throughits e§ect on steady state U/w.

Lower Unemployment Benefits

• Steady state U rises with fall in D.

• So fall in D raises d log w/d ln(U)

— Increases response of wages, inflation to external shocks,— Decrease response of employment, unemployment to thoseshocks.

Value�of�unemployment,�U

Real�wage,�w

G " w1 " *> � J � 4

Summary of bargaining: w � F�U;+,D,- 

Fall�in�D�reducesbargaining�power�ofworkers.Raises�U.

More possible bargaining rounds• Consider extreme case where M is very large.

value of firm that

accepts worker o§er in last roundz}|{J

M

t

=1

M

J

t

w

M

t

+ rE

t

m

t+1

J

t+1

=

value of firm that rejects worker’s

take-it-or-leave-it o§erz}|{

0

w

M

t

=1

M

J

t

+ rkE

t

m

t+1

• Extreme case, of M = •, implies w

M

t

would be roughlyconstant (interest rate doesn’t move much).

• This insensitivity is inherited by w

1

t

( w

t

) , w

2

t

, ..., w

M

t

.

• More generally, we expect the real wage to be more sensitive toshocks when M is smaller.

0 1 2 3 4 50.2

0.21

0.22

0.23

0.24

0.25Real wage (%)

wss=0.989

wss=0.989

wss=0.989

wss=0.989

wss=0.989

0 1 2 3 4 5

−0.16

−0.14

−0.12

−0.1

−0.08

Unemployment rate (p.p.)

uss=0.055

uss=0.0016

uss=0.0027

uss=0.0332

uss=0.0064

0 1 2 3 4 50.22

0.24

0.26

0.28

0.3

0.32

0.34

Real consumption (%)

Css=0.936

Css=0.988

Css=0.987

Css=0.957

Css=0.984

Small Model Impulse Responses to a 0.1 Percent Technology Shock

0 1 2 3 4 5

−15

−10

−5

0

Inflation rate (ABP)

Baseline Higher δ Lower γ Lower D Lower M

Simple Macro Model Implications

• Our model is in principle capable of accounting for businesscycle facts and Shimer puzzle without exogenously sticky wages.

• Next, do a formal macro data analysis using medium-sizedDSGE model.

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.

•d : roughly 0.26% chance of a breakup after rejection.

•g : cost to firm of preparing countero§er is 1/4 of a day’sworth of production.

• Posterior mode of hiring cost as a percent of output (dependson k): 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

00.2

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

Intuition• Policy shock drives real interest rate down.

— Induces increase in demand for output of final good producersand therefore output of sticky price retailers.

— Retailers must satisfy demand, so they purchase more ofwholesale good driving up its relative price.

— Marginal revenue product (Jt

) associated with worker rises.— Wholesalers hire more workers, raising probability thatunemployed worker finds a job.

• Workers’ disagreement payo§s rise.— Increase in workers’ bargaining power generates rise in realwage.

• Alternating o§er bargaining limits rise in real wage.— Allows for large increase in employment, substantial decline inunemployment, small rise in inflation.

0 5 10−0.2

00.20.40.60.8

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

00.2

Inflation

0 5 10−0.4

−0.2

0

0.2Federal Funds Rate

0 5 10−0.2

00.20.40.60.8

Hours

0 5 10−0.2

00.20.40.60.8

Real Wage

0 5 10−0.2

00.20.40.60.8

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

00.20.40.6

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

00.2

Inflation

0 5 10−0.4

−0.2

0

0.2

0.4

Federal Funds Rate

0 5 10−0.2

00.20.40.6

Hours

0 5 10−0.2

00.20.40.6

Real Wage

0 5 10−0.2

00.20.40.6

Consumption

0 5 10−0.8−0.6−0.4−0.2

00.2

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.— 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

s(Labor market tightness)s(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.