Employment, Unemployment and Turnover
D. Andolfatto
June 2011
Introduction
• In an earlier chapter, we studied the time allocation problem
max { ( ) :  =  +   +  = 1}
• We usually assume an “interior solution;” i.e., 0  ( )  1
• In reality, many people appear be at a “corner solution;” i.e., ( ) = 0
• People with  = 0 (over the previous 4 weeks) are labeled nonemployed
• People with   0 (over the previous 4 weeks) are labeled employed
• Over short periods of time, many people make transitions into and out of
employment
— in Canada, roughly 1.5 million workers per month
— relative to an average employment level of 17 million workers
• these gross flows of workers largely cancel over time (relative to population
growth), but exhibit both cyclical and seasonal variation
— e.g., in a recession, E ⇒ N flows (job loss) increase and N ⇒ E flows
(job finding) decrease
— implies net decrease in employment (with net changes much smaller
than gross flows)
Modeling employment turnover
• For simplicity, assume that time is indivisible; i.e.,  = 0 or  = 1
• Then the time allocation problem is simple: choose the action that yields
the highest utility payoff
• Payoff to employment ( = 1) is   = ( +  0)
• Payoff to nonemployment ( = 0) is   =  ( 1)
• There exists a number () that satisfies  ( +  0) =  ( 1)
• Optimal employment decision takes the following form:
 =
(
1 if  ≥ ()
0 if   ()
• () is called a person’s reservation wage
— the minimum wage that would induce a person to work (a measure of
“choosiness”)
— theory suggests that reservation wage is an increasing function of 
(nonlabor income or wealth)
• People with different ( ) characteristics will make different employment
choices
— high  low  individuals more likely to be employed
— low  high  individuals more likely to be nonemployed
— note: theory can be extended to include differences in the valuation of
leisure
• Turnover (gross flows of workers into and out of employment) can be
generated by assuming that people experience idiosyncratic shocks to ( )
over time
• Large gross flows of workers consistent with even a constant level of employment
Policy implications
• In this model economy, people cannot save or insure themselves
• In reality, insurance markets hampered by asymmetric information problems
(an insurance company may have difficulty in ascertaining the true value
of  and  for individuals, and individuals have an incentive to misreport
these values)
• In this model, the lack of insurance results in too much employment (people
are less able to afford home production)
• In principle, a tax-financed consumption-insurance program could be welfareimproving
Why does average labor productivity sometimes increase in a recession?
• Unlike other G7 economies, labor productivity actually rose in the U.S. in
the last recession (see Figure 4, Andolfatto and Williams)
• One explanation is that low-skilled workers are more likely to exit employment during a recession (especially in a “flexible” labor market, like the
U.S.)
• So even if everyone’s productivity falls, the average productivity of those
who remain employed may increase
• This is sometimes referred to as “composition bias”
Unemployment
• About 60% of the adult (16+) population is employed; about 40% are
nonemployed, or “jobless”
• Nonemployment is not the same thing as unemployment
• Standard labor force surveys ask people whether they’ve done any paid
work in the previous 4 weeks
• If answer is no, they then ask questions relating to job search activities
• Nonemployed persons who are not looking for work are classified as nonparticipants
Job vacancies
• A job vacancy corresponds to an “unemployed job” from the perspective
of a firm
• At any given time, job vacancies coexist with unemployed workers
• Why do firms with vacant positions not hire unemployed workers until
unemployment rate falls to zero, or supply of vacant positions is exhausted?
• Modern view is that labor markets operate more like decentralized marriage
markets, as opposed to centralized commodity markets
• Jobs and workers need to be properly matched; do not want to match with
first partner that comes along
• Time and resources must be consumed in finding the right partner
• Some luck is involved, and it generally does not make sense to wait for the
perfect partner
• The typical optimal strategy is to set a reservation quality level (reservation
wage is an example)
— if the partner you contact meets or exceeds a minimum set of criteria,
then accept any proposal
A dynamic model of unemployment
• Imagine a world populated by firms and workers, with time  = 0 1 2  ∞
• There are many firms, each with a single job that must be occupied by a
worker if output is to be produced
• At any point in time, firms are either active, vacant, or idle
• Workers have one unit of time; they use it to work (if employed) or search
(if nonemployed)
— i.e., workers do not value leisure
• An employed worker and active firm jointly produce output 
• Wages and profits are determined by a bargaining rule that divides the
joint output in some manner; e.g.
 = 
 = (1 − )
for some exogenous 0    1
• Job-worker relationships breakdown with probability 0    1 in each
period
• Firms discount expected future profits by factor 0    1
• Let  denote the value of an active firm at date 
 =   + (1 − ) +1 +  2(1 − )2 +2 + · · ·
 =   + (1 − )+1
• Since   = (1 − ) it follows that  = +1 =  (constant); solving...
=
"
#
1−

1 − (1 − )
• The value of a worker to a firm is: increasing in  decreasing in  increasing in  and decreasing in 
— Note:  is labor share of income (bargaining power of labor); expected
duration of employment is 1; real interest rate is  = 1 − 1
The matching process
• Unmatched workers and firms must search for each other (recall marriage
market)
• Let  denote unemployed workers and let  denote vacant jobs, and
define  ≡  (labor market tightness)
• Assume that there is an aggregate matching technology that relates aggregate search intensity ( ) to aggregate matches ; i.e.,
 = ( )
where  is increasing in ( ) and displays constant returns to scale
(CRS)
• CRS implies

=  ( 1) ≡ ()


=  (1 1) ≡ ()

• Can interpret  and  as match probabilities
•  is an increasing function of  while  is a decreasing function of 
• Note: these properties reflect congestion effects
— e.g., it becomes more difficult for firms to find workers when there are
a lot of vacancies (relative to available workers)
Recruiting intensity
• Because workers do not value leisure, nonemployed workers choose to
search for work (which is what makes them unemployed); hence,  measures aggregate search effort for workers
• Let me now describe how  is determined
• Assume that idle firms generate zero value
• An idle firm can decide to become a vacant firm at some cost  (a recruiting
cost)
• What is the expected payoff to vacancy creation?
• Contact an unemployed worker with probability ()
• Begin producing output in period  + 1
• So creating a vacancy makes sense if () ≥ 
• Assume that idle firms will continue to enter (as vacancies) until it is no
longer profitable to do so; i.e.,
() = 
• The condition above (free-entry condition) determines equilibrium  = 
•  is an increasing function of  and a decreasing function of 
• Since  is predetermined as of date  once  is determined, we can then
determine equilibrium recruiting intensity
 = 
Employment/unemployment dynamics
• Let  denote labor force;  =  + 
• The level of employment  at a point in time is predetermined (like the
capital stock)
• We can derive a stock-flow equation to describe employment/unemployment
dynamics
• Begin with stock  add flow in (job creation) ()( − ) subtract
flow out (job destruction) 
+1 = () + (1 −  − ())
• If −1  (1 −  − ())  1 then employment dynamics are stable
⇒ for any given 0 employment  converges to a steady-state ∗
• If −1  (1 −  − ())  0 then the dynamics oscillate
• If 0  (1 −  − ())  1 then the dynamics converge monotonically
• The “natural level of employment” can be solved by setting  = +1 =
∗
Ã
!
()
∗ =

 + ()
• Similarly, the “natural rate of unemployment” is
Ã
∗

!
=
Ã

 + ()
!
• ∗ depends directly on  and also on whatever parameters influence the
equilibrium 
—  depends on  and  depends on  etc.
• Easy to verify that ∗ is a decreasing function of 
• Can interpret  as any factor that influences the business sector’s expectation of the return to job creation (e.g., productivity, tax policy, etc.)
Some observations
• Unemployment here has nothing (directly) to do with wage inflexibility or
labor markets that do not clear
— unemployment is the natural byproduct of search and matching
— the nature of the wage bargain may influence the equilibrium unemployment, but is not the direct cause of unemployment
• Zero unemployment is not socially optimal
• In general, the equilibrium unemployment may be either higher or lower
than the socially optimal rate
The Beveridge Curve
• Refers to an alleged negative correlation that exists between  and  (interpreted as the byproduct of “cyclical shocks”)
• Movements off the (alleged) Beveridge curve are often interpreted as the
consequence of “structural shocks”
Cyclical asymmetry in unemployment rate dynamics
• Unemployment rates typically spike up sharply in recession, and decline
only slowly during recovery/expansion
Unemployment duration
• Most unemployment spells are relatively short
...but long duration spells have been increasing since the past recession
Current policy debate over the labor market
• Employment has slumped in the U.S. and shows little sign of recovery
• A similar phenomenon for Canada in early 1990s
• People almost universally believe that employment is “too low”
— but does anyone really know what the “socially optimal” employment
rate is?
• In any case, if employment is too low (unemployment too high), then why
is this the case?
• Keynesian interpretation is that “aggregate demand is too low”
• Why? Private sector is (irrationally) afraid to invest and recruit
• Policy prescription: government should step in and create demand
• There are many alternative interpretations
— depressed expectations are rational expectations of bad policy
— structural rearrangement of sectoral demands takes time to work through
(cannot transform construction workers into nurses)
• Warnings against Keynesian policies
— government demand largely crowds out private demand
— requires raising taxes and/or debt, providing a further drag on private
incentives to invest
• Nobody really knows for sure, so some degree of caution (and humility) is
in order