lecture 3: employment and unemployment
TRANSCRIPT
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).
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)
Long term unemployment (> 6 months)
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
Extracted from: Swedish Fiscal Policy 2009, Report of the Swedish Fiscal Policy Council.
Extracted from: Swedish Fiscal Policy 2015, Report of the Swedish Fiscal Policy Council.
Extracted from: Swedish Fiscal Policy 2015, Report of the Swedish Fiscal Policy Council.
Extracted from: Swedish Fiscal Policy 2015, Report of the Swedish Fiscal Policy Council.
Extracted from: Swedish Fiscal Policy 2015, Report of the Swedish Fiscal Policy Council.
The Swedish labour market, 1980-2010
Extracted from: Swedish Fiscal Policy 2009, Report of the Swedish Fiscal Policy Council.
Extracted from: Swedish Fiscal Policy 2011, Report of the Swedish Fiscal Policy Council.
Extracted from: Swedish Fiscal Policy 2016, Report of the Swedish Fiscal Policy Council.
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.