Lecture 5: Unemployment Theories
Patrick Macnamara
ECON60111: Macroeconomic Analysis University of Manchester
Fall 2020
Introduction
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Unemployment
• Unemployment is one of the central subjects of macroeconomics
• Existence of substantial levels of unemployment in normal times raises several questions:
• What are the reasons labor markets differ from Walrasian markets in ways that cause significant unemployment?
• What are the welfare consequences of normal unemployment?
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Key Facts About the Labor Market
• Key facts:
• Both total hours and employment are highly correlated
(positively) with output
• Real wages shows a low correlation with output
• This is consistent with the view that the labor market is described well by a supply and demand model with flexible wages only if labor supply is quite elastic.
• However, little support for the hypothesis of highly elastic labor supply in micro-empirical evidence.
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Outline
We study two models of the labor market with frictions:
1. Efficiency wages – Shapiro and Stiglitz (1984)
• Asymmetric information – workers choose how much effort to supply, but firms have limited ability to monitor worker.
• Unemployment occurs in equilibrium, high wages create incentives for workers not to “shirk.”
2. Search and matching – Mortensen and Pissarides (1994)
• Workers and firms meet up in a de-centralized process.
• Because of frictions matching workers to firms, there is
unemployment in equilibrium.
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Efficiency Wages
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Shapiro-Stiglitz Model
• The efficiency-wages model that we will consider is due to Shapiro and Stiglitz (1984)
• Asymmetric information between workers and firms: • Workers choose level of effort
• Firms have limited ability to observe effort
• There is unemployment in equilibrium
• Higher wage makes workers more willing to exert effort.1
• If workers were to “shirk” (i.e., zero effort), they could be
caught and fired – and then they become unemployed.
• With no unemployment, fired workers could immediately find a new job. (Hence, no reason to exert effort.)
1Hence, the term “efficiency wages,” as the high wages increase productivity or “efficiency.”
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Workers
• There are L ̄ workers
• Representative worker’s lifetime utility is
∞ −ρt
U= e u(t)dt, ρ>0
0
• Instantaneous utility:
w(t) − e(t) if employed
u(t) = 0 if unemployed
where w = wage, e = worker’s effort.
• Twopossiblevaluesfore: e=0ore=e ̄.
• Therefore, at any time, worker is in one of 3 states:
• E: employed, exerting effort
• S: employed, not exerting effort or “shirking” • U: unemployed
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Firms
• There are N firms
• Firm profits at time t are
π(t)=F(e ̄L(t))−w(t)[L(t)+S(t)], L(t) = number of workers exerting effort
F′ >0,F′′ <0
S(t) = number of workers shirking
• Firms choose w and L, being sure to set w high enough so
that workers choose not to shirk.2
• First order condition for L:
e ̄F′(e ̄L) = w (1) ⇒ ensures full employment without imperfect monitoring
2We will soon be precise about exactly what the wage needs to be to guarantee this.
• Assume e ̄F′(e ̄L ̄/N) > e ̄
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Transition Between States
• Workers can transition between the 3 states:
• Employed workers lose jobs at exogenous rate b
• Unemployed workers find jobs at endogenous rate a • Shirking workers are detected at exogenous rate q
• These transitions are modeled using a Poisson process • Suppose worker starts job at time t0:
Pr(emp. in same job from t0 to t) = P(t) = e−b(t−t0) Pr(emp. from t0 to t + τ|emp. from t0 to t) = P(t + τ) = e−bτ
P(t)
• Probability worker is still employed time τ later is e−bτ , regardless of how long the worker has been employed.
• b = hazard rate for job breakup, where P′(t)/P(t) = −b.
• Shirking detection and job finding rate modeled in same way
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Values of E, U and S
• We use “dynamic programming” to analyze the model.
• Let Vi denote the “value” of being in state i ∈ {E,S,U}.
• Vi = expected value of discounted lifetime utility from the present moment forward of a worker in state i.
• Due to Poisson assumption, Vi doesn’t depend on how long worker has been in state i.
• Vi will be constant over time in steady state • We will derive expressions for each Vi
• To do so, we will divide up time into intervals of length ∆t
• We will analyze how the value today depends on values of
being in different states in the future.
• Then we will take the limit as ∆t → 0.
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Value of E
• Divide time into intervals of length ∆t.
• Let VE (∆t) and VU (∆t) be the values of employment and unemployment at the beginning of an interval (given ∆t).
• Consider value of employment:
∆t −bt −ρt
e e (w − e ̄)dt
0
utility during interval (0,∆t)
+ e−ρ∆t e−b∆t VE (∆t) + (1 − e−b∆t )VU (∆t) (2)
Prob. in state E Prob. in state U tomorrow tomorrow
Assume worker who loses job during interval (0,∆t) can’t get new job until beginning of next interval.3
VE (∆t) =
3This approximation will become irrelevant as we let ∆t → 0.
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Value of E
• Next compute utility over interval (0,∆t):
t=∆t ∆t −bt −ρt e−(ρ+b)t(w −e ̄)
e e (w−e ̄)dt=
0 −(ρ+b)
• Substitute back into (2)
1 − e−(ρ+b)∆t (w − e ̄) ρ+b
VE(∆t) =
=
t =0
1 − e−(ρ+b)∆t (w − e ̄)
ρ+b
utility during interval (0,∆t)
+ e−ρ∆t e−b∆t VE (∆t) + (1 − e−b∆t )VU (∆t) (3)
• Then we can solve for VE (∆).
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Value of E
• Solving (3) for VE (∆):
VE(∆t) = w −e ̄ + e−ρ∆t(1−e−b∆t)VU(∆t) (4) ρ + b 1 − e−(ρ+b)∆t
• Take limit as ∆t → 0.
• VE(∆t) → VE and VU(∆t) → VU. • Use L’Hôpital’s rule:4
lim
∆t→0
e−ρ∆t (1 − e−b∆t ) = lim −ρe−ρ∆t + (ρ + b)e−(ρ+b)∆t
1 − e−(ρ+b)∆t
∆t→0 (ρ + b)e−(ρ+b)∆t
ρb∆t =lim1− e
∆t→0 ρ+b =1−ρ=b
ρ+b ρ+b
4According to L’Hôpital’s rule, limx→c f (x)/g(x) = limx→c f ′(x)/g′(x).
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Value of E
• Taking the limit of (4) as ∆t → 0:
V E = w − e ̄ + b V U ρ+b ρ+b
• Re-arrange:
ρ V E = ( w − e ̄ ) − b ( V E − V U )
This is the typical formulation of the value in continuous time.
• To interpret (5), treat VE as if it’s the value of an asset • ρVE is the return to being in state E
• w − ̄e is the “dividend”
• VE − VU is the “capital loss” which arrives at rate b
(5)
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Value of S and U
• Analogously, we can determine value of shirking: VS
(6) Dividend is w per unit time, capital loss of (VS − VU ) arrives
at rate (b + q)
• Value of unemployment: VU:
ρVS =w−(b+q)(VS −VU)
ρVU =a(VE −VU)
Divided is zero, capital gain arrives at rate a
(7)
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No-Shirking Condition (NSC)
• Firmsetswagew sothatVE ≥VS.
• Ensures it’s optimal for workers to choose e = ̄e
• No reason for firms to pay more than what’s needed to
guarantee workers do not shirk, so VE = VS . • Equating (5) and (6) and imposing VE = VS :
(w−e ̄)−b(VE −VU)=w−(b+q)(VE −VU)
ρVE ρVS
V E − V U = e ̄ q
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No-Shirking Condition (NSC)
• Next subtract (7) from (5)
(5): ρVE =(w−e ̄)−b(VE −VU)
(7): ρVU =a(VE −VU)
⇒ ρ(VE −VU)=(w−e ̄)−(a+b)(VE −VU)
• Using VE − VU = e ̄/q from previous slide, this implies:
w =e ̄+(a+b+ρ)e ̄ (8)
decreasing in q.
• Next we need to determine a (which is endogenous)
q
The wage needed to induce effort is increasing in e ̄, a, b, ρ,
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No-Shirking Condition (NSC)
• Movements in and out of unemployment must be equal
b× NL =a×(L ̄−NL) ⇒ a=
NLb L ̄−NL
num. emp.
• Substitute into (8):
num. unemp.
L ̄e ̄ w=e ̄+ ρ+L ̄−NLb q
This is the no-shirking condition (NSC)
Wage needed to deter shirking increases with employment.
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Determining the Equilibrium
• Aggregate labor demand NL, from (1):
e ̄(NL) N
Due to assumptions on F, this is a conventional labor
demand curve where NL decreases with w.
• Laborsupplyis0forw
• Without imperfect monitoring, equilibrium occurs at NL = L ̄,
where labor supply intersects with labor demand
(assumption e ̄F′(e ̄L ̄/N) > e ̄ needed for this)
• With imperfect monitoring, equilibrium occurs where labor
demand intersects with NSC
• There is then unemployment in equilibrium
• There is excess labor demand, but unemployed workers
e ̄ F ′
= w
cannot bid the wage down
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Equilibrium With No Frictions
Full employment in equilibrium, shifts in labor demand would have no effect on employment
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Equilibrium With Frictions
Equilibrium at E, unemployment in equilibrium, wage is higher than the Walrasian wage
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Higher Probability of Detecting Shirking
Higher q moves economy closer to the Walrasian equilibrium –
firms can decrease the wage because it’s easier to detect shirkers.
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Decrease in Labor Demand
NSC curve quite vertical for low levels of unemployment – so there still can be large effect on the wage.
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Welfare in Shapiro-Stiglitz Model
• Since e ̄F′(e ̄L ̄/N) > e ̄, first-best allocation is for everyone to be employed and exert effort.
• Is the equilibrium constrained efficient? That is, could a planner do better, while still being subject to the asymmetric information problem?
• Equilibrium still is inefficient, employment is too low
• Suggests policy could improve welfare
e.g., Shapiro and Stiglitz find wage subsides could improve welfare.
This would shift labor demand to the right, increasing the wage and employment along the NSC.
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Shapiro-Stiglitz: Summary
• Elegant micro-founded way to explain involuntary unemployment
• May be unrealistic in that workers are “rational cheaters” only – only wages and punishment motivate them.
• It introduced into labor economics a set of tools which feature prominently in search models
• Less important than search and matching models today
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Search and Matching: Assumptions
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Mortensen-Pissarides Model
• The search and matching model that we will consider is due to Mortensen and Pissarides (1994)
• Key friction is that it is difficult to match workers to firms • Firms post vacancies, unemployed workers search for jobs • Number of matches determined by a “matching function”
• Once workers and firms meet, they bargain over the surplus • Wages determined by Nash bargaining
• Due to frictions in the search and matching process, it results in some unemployment in equilibrium.
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Basic Assumptions
• Model is set in continuous time.
• Economy consists of workers and jobs.
• Workers can be either employed (E) or unemployed (U). • Jobs can be either filled (F) or vacant (V).
• Each job can have at most one worker: F=E
F = number of filled jobs, E = number of employed workers • Total number of workers is fixed at 1:
E+U=1 U = number of unemployed workers
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Workers
• Worker’s utility:
u(t) = b if unemployed
w(t) if employed
Worker is risk-neutral, receives wage w(t) in employment,
and receives b ≥ 0 in unemployment.
• Worker’s discount rate is r – this is fixed and is exogenous
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Jobs
• Profits per unit time from a job:
y −w(t)−c if filled
π(t) = −c if vacant • Output from filled job is y per unit time
Labor costs are w(t)
• Both filled and vacant jobs involve a constant, exogenous
maintenance cost c > 0
• Assume y > b + c, so a filled job produces positive value
• Vacant jobs can be created freely – thus, the number of jobs is endogenous
• Jobs end at rate λ (exogenous)
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Matching Function
• Frictions prevent unemployed workers and vacant jobs finding each other immediately
• Unemployment and vacancies yield a flow of meetings between workers and jobs:
M(t) = M (U(t),V(t)), MU > 0,MV > 0 M(t) is the number of meetings per unit time.
• Since all meetings lead to hires:
E ̇ (t ) = M (U (t ), V (t )) − λE (t )
• If M(U,V) exhibits…
• increasing returns to scale ⇒ thick-market effects • decreasing returns to scale ⇒ crowding effects
• We assume that M(U,V) exhibits constant returns to scale. 32/50
Matching Function Under CRS
• Let θ(t) ≡ V(t)/U(t) be labor market tightness:
M (U(t), V (t)) = U(t)M (1, V (t)/U(t)) ≡ U(t)m (θ(t))
where m′(·) > 0, m′′(·) < 0.
• The job-finding rate is
a(t)= M(U(t),V(t)) =m(θ(t)) U(t)
• The vacancy-filling rate is
α(t)= M(U(t),V(t)) = m(θ(t))
V(t) θ(t)
• Assume the matching function is Cobb-Doublas:
M(U,V) = kU1−γVγ ⇒ m(θ) = kθγ with k > 0 and 0 < γ < 1.
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Wage Determination
• Wage must be high enough that the worker wants the job, and low enough so that the firm wants to hire the worker.
There exists a range of wages meets these requirements.
• Therefore, to pin down the wage, we assume Nash bargaining.
• The surplus from the match is split so that ...
• a fraction φ of the surplus goes to the worker. • a fraction 1 − φ of the surplus goes to the firm.
φ is the worker’s bargaining power. φ is exogenous and 0 ≤ φ ≤ 1.
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Equilibrium Conditions
• Dynamic programming is used to analyze the model.
• Let Vi(t) be the value of being in state i ∈ {E,U,F,V} at time t, which now can vary with time.
Workers:
VE (t ) = value of employment VU(t) = value of unemployment
Jobs:
VF (t ) = value of filled job VV (t) = value of vacancy
• Flow value will have the following form:
rVi (t ) = dividend + V ̇ i (t ) + capital gain (loss)
Interpretation: “return” to being in state i has 3 elements: • dividend, or, direct instantaneous benefit or cost
• V ̇i(t), a change in value when not in the steady-state
• capital gain or loss, due to switching to another state with
some probability
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Equilibrium Conditions: Flow Values
• The flow value of being employed is
rVE (t ) = w (t ) + V ̇ E (t ) − λ [VE (t ) − VU (t )]
• The flow value of being unemployed is
rVU (t ) = b + V ̇ U (t ) + a(t ) [VE (t ) − VU (t )]
• The flow value of a filled vacancy is
rVF (t ) = [y − w (t ) − c ] + V ̇ F (t ) − λ [VF (t ) − VV (t )]
• The flow value of a vacancy is
rVV (t ) = −c + V ̇ V (t ) + α(t ) [VF (t ) − VV (t )]
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Further Equilibrium Conditions
• Nash bargaining implies
VE (t ) − VU (t ) = φ [(VE (t ) − VU (t )) + (VF (t ) − VV (t ))]
total surplus from match
⇒VE(t)−VU(t)= φ [VF(t)−VV(t)] 1−φ
• Impose free entry condition:
VV (t) = 0 for all t
• Take E(0) as given.
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Search and Matching: Solving the Model
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Steady-State Equilibrium
• Atthesteadystate,E ̇(t)=0andV ̇E =V ̇U =V ̇F =V ̇V =0. • Dropping t, we have the following equations:
rVE =w−λ(VE −VU) (9) rVU =b+a(VE −VU) (10) rVF =y−w−c−λ(VF −VV) (11) rVV =−c+α(VF −VV) (12)
VE −VU =[φ/(1−φ)](VF −VV) (13) kU1−γVγ =λE (14) a = kU1−γVγ/U (15) α = kU1−γVγ/V (16) U=1−E (17) VV =0 (18)
• 10 equations, we can solve for the 10 unknowns (VE,VU,VF,VV,U,V,E,a,α,w)
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Solving for the Steady-State Equilibrium
• To solve the model, solve for VV as a function of E. Then, impose that VV (E ) = 0.
• First, let’s solve for the wage w implied by Nash bargaining. (9): rVE=w−λ(VE−VU)
(10): rVU =b+a(VE −VU)
⇒ VE−VU= w−b (19)
a+λ+r
(11): rVF =y−w−c−λ(VF −VV)
(12): rVV =−c+α(VF −VV)
⇒ VF−VV= y−w (20) α+λ+r
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Nash Bargaining Wage
• (19) and (20) into the Nash bargaining condition, (13):
w−b φy−w a+λ+r = 1−φ α+λ+r
VE −VU
• Solve for w:
w=b+ φa+(1−φ)α+λ+r φ(y−b) (21)
(a + λ + r)
• For intuition, first suppose a = α. In this case:
VF −VV
w − b = φ(y − b) and y − w = (1 − φ)(y − b)
• Now suppose a > α,
workers find new jobs faster than firms find new workers: w − b > φ(y − b) and y − w < (1 − φ)(y − b)
• When a < α, the reverse occurs.
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Solving for the Value of a Vacancy
• Solve for VV :
(12): rVV =−c+α(VF −VV)
(20): VF −VV = y−w α+λ+r
⇒ rVV =−c+α y−w α+λ+r
• Now substitute for wage using (21) on previous slide:
rVV (E) = −c + (1 − φ)α(E) (y − b) φa(E)+(1−φ)α(E)+λ+r
Once we determine a(E) and α(E), this gives us a solution for rVE as a function of E.
• Then solve for E∗ such that rVV (E∗) = 0.
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Solving for a, V, α
• UsingM(U,V)=λE andU=1−E:
a=M(U,V) ⇒ U
a increases with E, a(0) = 0, a(1) = ∞
• Using kU 1−γ V γ = λE and U = 1 − E , solve for V :
V(E) = k−1/γ(λE)1/γ(1 − E)−(1−γ)/γ
V = 0 when E = 0,U = 1; V = ∞ when E = 1,U = 0. This implies a negative relationship between V and U. This relationship is known as the Beveridge curve.
• UsingV (fromabove)andU=1−E: kU1−γVγ
α=V⇒
α decreases with E, α(0) = ∞, α(1) = 0.
a(E)= λE 1−E
α(E) = k1/γ(λE)−(1−γ)/γ(1 − E)(1−γ)/γ
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Solving for E
• SetrVV(E)=0
rVV (E) = −c + (1 − φ)α(E) (y − b) = 0 φa(E)+(1−φ)α(E)+λ+r
and solve for E satisfying this equation.
• No closed form solution, but note that
lim rVV (E ) = y − b − c > 0 E→0
lim rVV (E ) = −c < 0 E→1
Therefore, rVV (E ) decreases with E and rVV (E ) = 0 for some value of E between 0 and 1.
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Determination of Equilibrium Employment
Equilibrium employment determined where rVV = 0. As E increases, a increases, α decreases and w increases.
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Effect of Increase in y
Increase in y ⇒ higher incentive to create vacancies, higher employment.
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Effect of increase in c
Higher maintenance costs c ⇒ smaller incentive to create vacancies, lower employment
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Issues
• Shifts in labor demand have quantitatively small effects on employment (i.e., Shimer puzzle):
• Employment effects of the shift in labor demand occur as a result of the creation of new vacancies.
• Given reasonable parameter values, the wage very responsive ⇒ small incentive to create new vacancies.
• Thus, employment effect is too small (Shimer, 2005).
• Some wage rigidity is necessary.
• Wage rigidity has the correct implications – i.e., brings the model closer to the data.
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Welfare in the Mortensen-Pissarides Model
• Equilibrium is inefficient relative to a first-best alternative (e.g., a social planner who can avoid the matching frictions).
• Is the equilibrium constrained efficient? That is, could a social planner, subject to the same matching frictions, improve welfare?
• Hosios condition: equilibrium is efficient when γ=1−φ
Elasticity of matches w.r.t. vacancies = share of surplus
that goes to firms
• Toomanyvacanciesifγ<1−φ;toofewifγ>1−φ.
• Creating a vacancy has a positive externality on unemployed
workers, negative externality on firms looking for workers.
• Higher share of surplus ⇒ more incentive to create vacancies
• When γ = 1 − φ, firms get share of surplus that equals their
contribution to match creation.
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Mortensen-Pissarides: Summary
• Search and matching models very popular • Some major extensions:
Model heterogeneity among both workers and jobs. ⇒ Meetings may not lead to a match being formed
Allow workers to continue searching while they are employed ⇒ Some workers transition directly from one job to another
Allow directed search (or competitive search)
e.g., continuum of “submarkets” indexed by w, each with its own market tightness θ(w) and matching probabilities, m(θ)
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