Lecture 3: Employment and Unemployment

Lecture 3: Employment and Unemployment

Anna Seim and Paul Klein, Stockholm University

November 11, 2019

Contents

• Di�erent kinds of unemployment.

• Labour market facts and developments.

• Models of wage rigidity.

• The bathtub model of unemployment.

• Search and matching.

• Literature: Jones (2014), Chapter 7; Pissarides (2000).

Employment and Unemployment

• Employment: a state in which an individual has a paid job oris self-employed (operating a business).

• Unemployment: a state in which someone who would like towork is actively searching for a job but is not employed.

• Labour force: the sum of the employed and the unemployed.

• Employment rate: the number of people who are employed asa share of the working-age population (typically individualsaged 15-64).

• Unemployment rate: the number of people who areunemployed as a share of the labour force.

Di�erent Kinds of Unemployment

• The natural rate of unemployment: the (hypothetical)unemployment rate that prevails if the economy is neither in aboom nor in a recession. But see Rogerson: Theory Ahead ofLanguage, 1997.

• Frictional unemployment: unemployment due to workerstransitioning between jobs.

• Structural unemployment: unemployment due to(geographical-, skill-) mis-match and labour marketinstitutions.

• Cyclical unemployment: unemployment due to the businesscycle.

Di�erent Kinds of Unemployment cont'd.

• Actual unemployment = natural unemployment + cyclicalunemployment.

• Natural unemployment = structural unemployment +frictional unemployment.

Extracted from: Jones (2014).

The unemployment rate in Sweden 2005-2016

Source: SCB.

Extracted from: EEAG (2016).

Long term unemployment (> 6 months)

Extracted from: Swedish Fiscal Policy 2016, Report of the Swedish Fiscal Policy Council.

Long term unemployment (> 6 months)

Labour market �exibility in some OECD countries

Note: In�ow and out�ow rates are monthly.Source: Elsby, Hobijn & �ahin (2008).

The Swedish Labour Market, 1980-2010


The Swedish labour market, 1980-2010


Fraction of adult population (18-64) in labour force

Source: SCB.

Unemployment rate

Source: SCB.

Harmonized versus registered unemployment

Source: FRED, OECD.

Note: �Registered� (according to the OECD de�nition) is not the same as �inskrivenpå Arbetsförmedlingen�. Instead it seems to be �öppet arbetslösa�, i.e. unemployed not enrolled in alabour market program. �Harmonized�, on the other hand, seems to be taken from SCB'sArbetskraftsundersökningar (AKU), a labour market survey (enkätundersökning).

Models of the Labour Market

1. Models of wage rigidity.

2. Models of labour-market �ows:

- The bathtub model of unemployment.

- The search model by Pissarides (2000).

Models of Wage Rigidity

• When wages are rigid, they fail to adjust in response tochanges in labour supply and demand.

• The (New) Keynesian view: wage rigidity may cause large�uctuations in employment.

• Wage rigidity may refer to insu�cient �exibility, but also tothe wage being determined by factors other than supply anddemand.

Extracted from: Jones (2014).

Causes of Wage Rigidity

• Collective bargainingI It takes time to renegotiate.I The union may exercise monopoly power and restrict supply to

maximize total wage bill.

• Minimum-wage lawsI It takes time to legislate in response to shocks.I May fail to equate supply with demand in any case.I But may o�set monopsony power and increase employment.I Empirical evidence is mixed. See the work of David Card and

Alan Krueger.I Albrecht and Axell: too many unproductive �rms in equilibrium

• E�ciency wagesI Wages may need to be kept high to incentivize e�ort and

honesty.

The Bathtub Model of Unemployment

• Simple model of labour-market dynamics.

• Notation:

L: the (�xed) labour force.

Et : the number of people employed at time t.

Ut : the number of people unemployed at time t.

s: the separation rate.

f : the job-�nding rate.

The Bathtub Model of Unemployment

• Model consists of two equations.

• Workers in the labour force either employed or unemployed:

L = Et +Ut . (1)

• The change in unemployment is given by:

∆Ut+1 ≡ Ut+1−Ut = sEt − fUt . (2)

The Bathtub Model of Unemployment cont'd.

• Steady state when Ut+1 = Ut or ∆Ut = 0.

• Setting ∆Ut+1 = 0 in (2) and using (1) implies

s(L−Ut) = fUt .

• Divide by L to obtain s(1−ut) = fut , where ut = Ut/L is theunemployment rate.

• Solving for u, we obtain:

u =s

s + f. (3)

Persistence and Stability

• Equation (2) implies that unemployment evolves according to:

Ut+1 = sL+ (1− f − s)Ut . (4)

• The term (1− f − s) captures unemployment persistence.

• The dynamic process (4) is stable and tends to return to thesteady state if 0< (1− f − s) < 1.

A Search Model of the Labour Market

• Developed by Dale Mortensen and Chris Pissarides in the1970s.

• Unemployed workers and �rms search for each other in thelabour market. The search process is costly.I It takes time for workers and employers to �nd a jobI Recruiting requires resources

• Workers and �rms consider the implications of their actions bycalculating the Present Discounted Value (PDV) associatedwith di�erent states.

• Unemployment arises because �rms are hit by exogenousshocks that trigger job separations, i.e. breakups of existingmatches.

Notation

• L: the (�xed) labour force.

• u = U/L: the unemployment rate.

• v = V /L: the number of vacancies as a share of the labourforce.

• m: the number of matched worker/employer pairs as a shareof the labour force.

• M: the matching function.

• θ ≡ v/u: labour-market tightness.

• λ : rate at which �rm/worker-pair breakups occur.

• p: the value of output associated with one job.

• p · c : the cost of hiring.• r : the real interest rate.

• w : the real wage.

• z : the real return to unemployment.

• β : the relative bargaining power of workers.

The Matching Function

• Only the uL unemployed workers search for jobs.

• There are vL vacancies posted by �rms.

• Workers and jobs that are successfully matched are randomlydrawn from the sets, uL and vL, respectively.

• Workers and �rms are matched according to a technologycaptured by the matching function:

mL = M(uL,vL) (5)

• The function M is increasing in both arguments (uL and vL),concave, and homogenous of degree 1.

Job Creation and Job Destruction

• Incidentally: in the data, unemployment falls and rises as thejob creation falls and rises, not so much as the job destructionrate falls and rises.

• Think of mL as a �ow rate per unit of time; it's a timederivative.

• Job creation occurs when a �rm and a searching worker meetand agree to form a match at a bargained wage.

• A match lasts until a �rm-speci�c, negative shock, re�ectingchanges in technology or demand, causes job separation.

• The worker-�rm pairs that are hit by shocks are randomlyselected.

Matching and Labour Market Tightness

• Labour market tightness, θ ≡ v/u, measures the relativenumber of searchers in the market.

• The rate at which a vacant job is �lled is given by

q(θ) =M(uL,vL)

vL= M

(uv,1), (6)

• where the last equality follows from the homogeneity of M.

Unemployment Dynamics

• Consider a small time interval, ∆t.

• The mean number of workers who enter unemployment during∆t is

λ (1−u)L∆t. (7)

• The mean number of workers who exit from unemploymentduring ∆t is

mL∆t. (8)

Unemployment Dynamics cont'd.

• It will prove useful to rewrite (8) in terms of u rather than m.

• Equation (6) suggests that M(u,v) = vq(θ). This implies thatthe out�ow from unemployment, (8), can be re-written as

mL∆t = vq(θ)L∆t = uθq(θ)L∆t. (9)

• When ∆t→ 0, the change in unemployment, du/dt, is givenby the mean in�ow into unemployment, (7), minus the meanout�ow from unemployment, (9):

u̇ ≡ du

dt= λ (1−u)L−uθq(θ)L. (10)

The Beveridge Curve

• In the steady state u̇ = 0, so that

λ (1−u)L = uθq(θ)L. (11)

• Solving for u in (13), we obtain the Beveridge curve:

u =λ

λ + θq(θ). (12)

• Unemployment persists in the steady state because of thematch-speci�c shocks causing job separations and hence aconstant �ow into unemployment.

I Notice that Equation (14) is not explicit because θ = v/u.

The Beveridge Curve

• The above derivation is standard, but unnecessarilycomplicated.

• Evidently∆U = λ (L−U)∆t−M(U,V )∆t. (13)

• Dividing by ∆t and letting ∆t→ 0, we get

U̇ = λ (L−U)−M(U,V ). (14)

• In a steady state, we have

λ (1−u) = M(u,v)

where we have divided by L and exploited the linearhomogeneity of M. No, this is not an explicit expression for u,but then neither is Equation (14).

The theoretical Beveridge curve

• M(U,V ) = UαV 1−α

• α = 0.63

The theoretical Beveridge curve

• How does the curve shift if the unemployment rate isincreasing or decreasing?

The Beveridge Curve

• The Beveridge curve implies that, for a given λ and θ (or,equivalently, vacancy rate v) there is a unique equilibriumunemployment rate.

• The parameter λ is given, but v is endogenously determined.

• To close the model, two more equations are needed: ajob-creation condition and a wage curve.

• Key ideas in that context:I Wage bargaining; splitting the match surplusI Free entry; vacancies are posted until posting another

generates zero pro�ts

Digression on Hagedorn, Karahan, Manovskii & Mitman

• In the MP model, high unemployment bene�ts are associatedwith high unemployment

• The reason (at least in the above version of the model) is notthat it makes individual workers lazier in their search behaviour

• The reason is that it enhances the bargaining position ofworkers, leaving less surplus for the employer, reducing thenumber of vacancies

• This is a �macro� e�ect�it's not about the behaviour of anyindividual unemployed person, it's about how many workersput upward pressure on wages and hence downward pressureon vacancies

Digression on Hagedorn, Karahan, Manovskii & Mitman

• Hagedorn, Karahan, Manovskii & Mitman (2016) establishthat this �macro� e�ect exists empirically.

• After the recent �nancial crisis in the United Sates, bene�tduration increase from 26 weeks to up to 99 weeks.

• The paper looks at bordering counties in distinct states.

• More generous bene�ts in a state generates higherunemployment there via higher wages and fewer vacancies.

• Meanwhile, there is no evidence of the �burning platform� viewthat increased bene�ts reduce search intensity.

Firms

• Each �rm has one vacancy that it seeks to �ll by searching forworkers in the market.

• When the vacancy is �lled, the �rm produces output p > 0,sold in competitive markets.

• When the vacancy is open, the �rm faces a �xed search costp · c > 0 per unit of time.

• The number of jobs, v is endogenous and determined by pro�tmaximization. When each �rm only has one vacancy, thiscorresponds to all pro�t opportunities from new jobs beingexploited, so that V = 0.

Firms cont'd.

• Let J and V be the PDVs of expected pro�t from an occupiedjob and a vacant job, respectively.

• V satis�es the following Bellman equation:

rV =−pc +q(θ)(J−V ). (15)

• Imposing V = 0 on (15) yields:

J =pc

q(θ). (16)

• Equilibrium labour market tightness ensures that the expectedpro�t from a new job equals the expected cost of hiring aworker.

The Job Creation Condition

• To derive the job-creation condition, we need to eliminate theasset value of an occupied job, J, in (16).

• J satis�es the following equation:

rJ = p−w −λJ. (17)

• Using (17) to eliminate J in (16), we obtain the job-creationcondition:

p−w − (r + λ )pc

q(θ)= 0. (18)

• Equation (18) corresponds to a marginal condition for thedemand for labour.

Workers

• The labour force is �xed and each worker's search intensity isgiven.

• When employed, the worker earns the real wage w , determinedin wage bargaining with the hiring �rm.

• When unemployed, the worker searches for employment andenjoys the real return z , notably comprising unemploymentbene�ts.

• The worker's PDV of employment and unemployment play akey role in wage bargaining and are derived below.

Workers cont'd.

• Let U and W denote the PDVs of the expected income streamof an unemployed and an employed worker, respectively.

• U satis�esrU = z + θq(θ)(W −U). (19)

• Since rU is the average expected return to human capitalduring the search process, it is the minimum compensationrequired to give up search and therefore the worker'sreservation wage.

• W satis�esrW = w + λ (U−W ). (20)

• Workers do not quit their jobs as long as W ≥ U, which holdsif w ≥ z .

Wage Determination

• In equilibrium, a successful match generates economic rentsthat are shared in wage bargaining between the �rm andworker.

• The bargained wage, wi , maximizes the weighted product ofthe worker's and the �rm's net return from the job match.

Wage Determination cont'd.

• Formally, the two parties face the following maximizationproblem:

maxwi

(Wi −U)β (Ji −V )(1−β), (21)

• where 0≤ β ≤ 1 is the relative bargaining power of workersand Ji and Wi depend on wi according to (17) and (20).

• Taking logs, the maximization problem, (21), may be written

maxwi

β ln(Wi −U) + (1−β )(Ji −V ). (22)

Wage Determination cont'd.

• The �rst-order condition (FOC) is:

β

(Wi −U)

∂Wi

∂wi+

(1−β )

(Ji −V )

∂Ji∂wi

= 0. (23)

• Since (17) and (20) imply ∂Wi/∂wi =−∂Ji/∂wi , the FOCmay be written

Wi −U = β (Wi +Ji −U−V ). (24)

The Wage Curve

• To convert (24) into a wage curve, use (16), (17), (19) and(20) to get rid of the value functions.

• By imposing V = 0 and realising that in equilibrium all �rmspay the same wage, wi = w ∀i , the wage curve can be written

w = (1−β )z + βp(1+ cθ). (25)

Equilibrium

• The steady-state equilibrium is a triple, (u,θ ,w) that satis�esthe Beveridge curve (14), the job-creation condition (18) andthe wage curve (25), repeated here for convenience:

u =λ

λ + θq(θ),

p−w − (r + λ )pc

q(θ)= 0,

w = (1−β )z + βp(1+ cθ).

• The unique equilibrium can be illustrated in two diagrams: onein the θ -w -plane and one in the u-v -plane.

Equilibrium Wages and Market Tightness

Equilibrium Vacancies and Unemployment

The Swedish Beveridge Curve 1980-2010

Extracted from: Kocherlakota, N. (2012), "Monetary Policy Transparency: Changes and Challenges",Speech at the Data Matters Forum, Rapid City, South Dakota, May 23

Dynamics in the Mortensen-Pissarides model

• An unexpected, permanent change in any of the parametersleads to an immediate jump in labour market tightness θ andthe wage w to the new steady state

• That steady state is determined by Equations (18) and (25)

• Unemployment and vacancies then converge gradually to thenew steady state

• If the matching function is Cobb-Douglas, convergence to thesteady state is governed by a linear ordinary di�erentialequation.

Dynamics in the Mortensen-Pissarides model• Suppose M(U,V ) = UαV 1−α

• Then Equation (18) becomes

w = p− (r + λ )pcθα

• Let θ be equilibrium labour market tightness

• Then, de�ning u := U/L, we have

u̇ = λ (1−u)−θ1−αu

and hence

u(t) =λ

λ + θ1−α+ exp

{−(λ + θ

1−α )t}[

u(0)− λ

λ + θ1−α

]and

v(t) = θ ·u(t)

• Notice that the speed of convergence is increasing inequilibrium tightness

Dynamics in the Mortensen-Pissarides model

I What is the half-life?

I If a(t) = e−kt then the half life t1/2 = ln(2)/k

I Hence the half life in the MP model is

t1/2 =ln(2)

λ + θ1−α≈ 0.69

λ + θ1−α

where if you want to confront this with empiricalmeasurements, you should probably specifyM(U,V ) = AUαV 1−α and derive the equilibrium equationsagain.

Lecture 3: Employment and Unemployment

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