Learning on the job and the cost of business cycles

Learning on the job and the cost of business
cycles
Karl Walentin and Andreas Westermark
Sveriges Riksbank
Eesti Pank, April 2018
Walentin and Westermark () LotJ and the cost of business cycles Eesti Pank, April 2018 1 / 61

Cost of business cycles
Long standing question in macroeconomics: How large are the
welfare costs of business cycles?
Costs of aggregate consumption ‡uctuations are small
I Lucas (1987)
Costs of countercyclical idiosyncratic consumption risk might be
larger
I Imrohoro¼glu (1989) and many others
E¤ects of cycles on the average level of output
I Seminal idea in DeLong and Summers (BPEA, 1989)
I Hassan and Mertens (AER, 2017) - risk premium from
misperceptions
I Duprez, Nakamura and Steinsson (2017) - downward nominal
wage rigidity
I Den Haan and Sedlacek (QE, 2014) - ine¢ cient separations
I Jung and Kuester (JEDC, 2011) - non-linearity of matching
function

Our research question
How large is the employment, output and welfare cost of business
cycles?
Speci…cally, we document and quantify a new channel whereby
business cycles reduce the average level of employment and output
due to two aspects in the labor market

Our research question
How large is the employment, output and welfare cost of business
cycles?
Speci…cally, we document and quantify a new channel whereby
business cycles reduce the average level of employment and output
due to two aspects in the labor market:
1 Negative Beveridge correlation, i.e.
correlation (vacancies,unemployment) < 0
) employment is reduced by aggregate volatility
2 Learning on-the-job (LotJ) implies that the human capital
distribution is increasing in the employment rate
I General human capital stemming from learning on-the-job:
Pissarides (1992) and Ljungqvist and Sargent (1998)

Beveridge correlation + matching function
In a search model the number of new jobs m is a non-linear function
of vacancies (v) and unemployment (u)
Using a standard Cobb-Douglas matching function:
mt = ftut =
vt
ut
1 ω
ut
The employment ‡ow equation then implies that aggregate volatility
reduces the number of new jobs:
Em ¯m
δ
δ + ¯f
(1 ω)
¯f
¯v
cov (v, u)
¯f
¯u
var (u) + (Ef ¯f ) Eu
δ denotes the exogenous separation rate and
ω is the matching function elasticity
Take-away: The number of new jobs decrease with aggregate
volatility if the Beveridge correlation is negative and (Ef ¯f ) 0

Goal of exercise - what we do
Set up search and matching model with learning on-the-job and
skill loss when unemployed to capture the main mechanism
Provide credible quanti…cation by capturing the main
determinants of the size of output cost of business cycles
Use the model to quantify the cost of business cycles of
this mechanism

Preview of main result and its determinants
Our results indicate sizeable negative e¤ects of aggregate volatility on
employment, output and welfare.
Welfare costs of our mechanism = 0.52-1.49% of steady state welfare
If account for transition dynamics: 0.20-1.09% of steady state welfare
Size of the e¤ects of business cycles is mainly determined by:
1 The sensitivity of the human capital distribution to changes in
employment, and
2 The sensitivity of job creation to changes in the human capital
distribution.

Related literature, in terms of modelling (technical)
Modelling heterogenous workers and …rms, and wage bargaining
framework:
I Lise and Robin (AER, 2017)
I Cahuc, Postel-Vinay and Robin (ECMA, 2006)
Earnings loss size and persistence - models to …t the facts in
Davis and von Wachter (BPEA, 2011 )
I Burdett, Carrillo-Tudela and Coles (2015)
I Jarosch (2015)
I Jung and Kuhn (JEEA, 2016)
I Krolikowski (AEJ-Macro, 2015)
I Huckfeldt (2016) - only one of these papers with aggregate
shocks

Model overview
Workers: Random search, both on and o¤ the job
Firms face standard matching function and linear vacancy
posting costs
LotJ: Worker heterogeneity in terms of human capital, x
I Markov process πxe (πxu)
F x increases if employed, decreases if unemployed
One-worker …rms
Job (match) heterogeneity in productivity, y
I Drawn after posting vacancy y v g (y)
Utility is linear in consumption
Surplus sharing with job ladder (Cahuc, Postel-Vinay and Robin,
2006)

Key assumptions
Job heterogeneity in productivity
+
On-the-job search
On the job search: takes the e¤ect of employed workers on job
creation into account

Key assumptions
+
On-the-job search
) Job (productivity) ladder - lower average productivity of …rst
match after unemployment than average existing match

Key assumptions
+
On-the-job search
) Job (productivity) ladder - lower average productivity of …rst
match after unemployment than average existing match
Wage ladder steeper than productivity ladder, as no
negotiation capital at …rst job after unemployment

Timing within a period
1 Shocks are realized:
I Idiosyncratic human capital, x
I Aggregate productivity, z
I A fraction ν of workers retire/exit
2 Exogenous and endogenous separations of matches
3 Firms post vacancies and workers search
4 New matches are formed, wages are set and production takes
place

Distribution of matches - law of motion
Shock realization and separations yields distribution of matches
hs (x, y)
Matching step yields:
h (x, y) = hs
(x, y)
+mass hired from unemployment
mass lost to more productive matches
+mass poached from less productive matches

Value of unemployment and total value of a match
The value of unemployment
B (x, z, Γ) = b +
1 ν
1 + r ∑
x02X
∑
z02Z
[ ∑
y02Y
f z0
, Γ0
g y0
B x0
, z0
, Γ0
+ β max P x0
, y0
, z0
, Γ0
B x0
, z0
, Γ0
, 0
+ 1 f z0
, Γ0
B x0
, z0
, Γ0
] πxu x, x0
π z, z0
,
β is worker bargaining power, ν is probability of exiting and r is the
discount rate
Γ is the endogenous state
Total value of a match, P ( ) , where p ( ) = xyz:
P (x, y, z, Γ) = p (x, y, z) + future terms

Centrality of total surplus of a match, S
Surplus
S = P B
Employer gets a share of the surplus when bargaining
) S determines the level of vacancy postings and which
matches are formed
E¢ cient separations
) S determines endogenous separations
(Note: Wages are not allocative)

Worker value - connection to wage
Worker value W ()
W () = w +
1 ν
1 + r
(future terms)

Worker wage setting
Following Cahuc, Postel-Vinay and Robin (2006)
Case 1: Workers hired out of unemployment:
Wage w0 set so that worker value W satis…es:
W w0
, x, y, z, Γ = B (x, z, Γ) + βS (x, y, z, Γ) .
Case 2: Employed workers meeting another …rm, ˜y:
If S (x, ˜y, z) > S (x, y, z) the worker switches to the new match and
w0 is set so that:
W w0
, x, ˜y, z, Γ = P (x, y, z, Γ) + β [S (x, ˜y, z, Γ) S (x, y, z, Γ)] .
else, stays with old match and w is set so that:
W w0
, x, y, z, Γ =
max fP (x, ˜y, z, Γ) + β [S (x, y, z, Γ) S (x, ˜y, z, Γ)] , W (w, .)g

Worker wage setting (part II)
Case 3: Employed workers not meeting another …rm
Value restricted by bargaining set:
B (x, z, Γ) + βS (x, y, z, Γ) 6 W (w, x, y, z, Γ) 6 P (x, y, z, Γ)
If outside this set ) set to closest boundary of set by changing w
If none of these cases, then the wage, w, is …xed

The resulting worker value function
Γ denotes endogenous aggregate states
W (w, x, y, z; Γ) = w +
1 ν
1 + r
∑
x0
∑
z0
πxe x, x0
π z, z0
2
4
s0 unemployment value
+ (1 s0)
(1 s1f (z0, Γ0)) non-poach value
+s1f (z0, Γ0) poach value
3
5
where
s0
: separation
all values are functions of x0
, y0
, z0
and Γ0
job …nding rate: f z0
, Γ0
= αθ z0
, Γ0 1 ω

Tightness
Total search e¤ort, L ∑
x2X
us (x) + s1 ∑
x2X
∑
y2Y
hs (x, y)
us (x) denotes distribution of unemployed workers and
hs (x, y) denotes distribution of matches
Cobb-Douglas meeting function, M = αLωV1 ω
Optimality condition for vacancy posting: c0 (v) = qJ
q : Probability of a …rm (vacancy) meeting a worker
J : Expected value of a new match to an employer
) Aggregate labor market tightness (using symmetry, V = v):
θ (z, Γ)
V
L
=
αJ (z, Γ)
c0
1
ω

Expected value of a new match to an employer
J =
1
L ∑
x2X
∑
y2Y
us
(x, z) max f(1 β) S (x, y, z, Γ) , 0g g (y)
+
s1
L ∑
x2X
∑
y2Y
∑
˜y2Y
hs
(x, ˜y, z)
max f(1 β) (S (x, y, z, Γ) S (x, ˜y, z, Γ)) , 0g g (y)

Solution algorithm (part I of II)
Solving for B and P requires use of value function iteration
) Need to put state variables on a grid
State includes endogenous aggregate distributions hs (x, y) and
us (x) for the next period
I These distributions matter only through next period labor
market tightness, θ0
I θ itself fully determined by L and J
I L proportional to m1 = ∑x us
(x)
I J adds two moments representing its two terms:m2, m3
I Aggregate endogenous state Γ captured by 3 moments
I To compute next period values of these moments we assume a
linear relationship to today’s moments:
m0
m = Hm m1, m2, m3, z0

Solution algorithm (part II of II)
Obtain θ0
by:
I Simulating allocations including θ and mi and then running
regressions
I Predict the 3 moments fmi g using 1-period lag of fmi gi=1,2,3,
z and regression coe¢ cients
I Compute implied θ0
= Θ (m0
1, m0
2, m0
3; z) (R2 > 0.995)
)The large-dimensional distributions can be replaced by mi ,
State vector becomes
0
B
@ w, x, y
| {z }
idiosyncratic
; z, m1, m2, m3
| {z }
aggregate
1
C
A
Using the functions Hm and Θ we can compute values B and P
I ...and then the entire allocation
Solve for wages w residually, given the expected future values for
the worker: w = W (.) future value

Calibration approach
1 Well established parameter values set outside model
2 Following 7 parameter values set by matching 7 moments of the
model
I Matching function productivity
I Relative search intensity of employed
I Human capital dynamics
I Unemployment payo¤
I Distribution of initial match-speci…c productivity g
I Volatility of (exogenous) TFP
I Worker bargaining power

Parameters set outside model
Monthly frequency
Explanation Value Source
ω Matching function elasticity 0.5 Pissarides (2009)
δ Exogenous match sep. rate 0.030 JOLTS & Fujita-Ramey
c0 Vacancy posting cost 0.06375 Fujita-Ramey
ν Retirement rate 1/(40 12) 40 year work life
ρ TFP shock persistence 0.960 Hagedorn-Manovskii
r Interest rate 1.051/12 1 Annual r = 5%

Moments matched
Moment Target value (data) Model value
U2E transition rate, mean 0.340 0.357
J2J transition rate, mean 0.0320 0.0290
Unemployment, std.dev. 0.107 0.0973
GDP, std.dev. 0.0136 0.0136
Wage disp: Mean-min ratio 1.50 1.70
Wage elasticity wrt productivity 0.449 0.445
Return to experience 0.0548 0.0518

Unusual moments matched - detailed de…nitions
1 Wage dispersion: Mean-min ratio (50th percentile/10th
percentile)
2 Return to experience: Buchinsky et al. (2010)
I The wage increase in the 3rd year for workers who works at
least three years for the same employer

Parameters set through moment matching
Parameter Explanation Main identifying moment
α Matching function productivity U2E transition rate, mean
s1 Search intensity of employed J2J transition rate, mean
xup Human capital gain, probability Return to experience
b Unemployment payo¤ Unemployment, std.dev.
β Bargaining strength of workers Wage elasticity wrt prod.
σy Match-speci…c productivity disp Wage disp: Mean-min ratio
100σz TFP shock std.dev. GDP, std.dev.

Parameters set through moment matching
Parameter Explanation Value
α Matching function productivity 0.686
s1 Search intensity of employed 0.426
xup Human capital gain, probability 0.0427
b Unemployment payo¤ 0.374
β Bargaining strength of workers 0.848
σy Match-speci…c productivity disp 0.259
100σz TFP shock std.dev. 0.698
Unemployment payo¤/output in best possible match=0.60

Human capital dynamics
The estimated human capital parameters imply, on average:
1.41% loss of human capital per month of unemployment
0.207% gain of human capital per month of employment
I Our estimate for employed workers is in between Huckfeldt
(2016) and Jarosch (2015),
I Our estimate for unemployed workers is about as large as in
these papers.
Note: only learning on-the-job subset of total human capital -
excludes formal schooling

Welfare measure
Linear utility in consumption
)aggregate welfare=aggregate consumption
Two ways of interpreting unemployment payo¤ b
1 Pecuniary transfer: C=GDP-vacancy costs
2 Home production (util of leisure): C=GDP-vacancy costs+b u
Presumably, the truth is an intermediate case:
b consists of both home production and transfers
I We report upper and lower bounds for the welfare cost

Gains from eliminating business cycles
We solve our model with and without aggregate shocks to
compute the cost of business cycles
Gains of eliminating bc (%)
Welfare, b transfer 1.49
Welfare, b home prod 0.52
GDP 1.45
Employment 1.34

Importance of human capital dynamics
Baseline No human capital dynamics
Welfare, b transfer 1.49 0.26
Welfare, b home prod 0.52 0.02
GDP 1.45 0.25
Employment 1.34 0.34

E¤ects from human capital on job creation
Job creation determined by
J (z, Γ)
= ∑
x2X
∑
y2Y
1
L
us
(x, z) max f(1 β) S (x, y, z, Γ) , 0g g (y)
+ ∑
x2X
∑
y2Y
∑
˜y2Y
s1
L
hs
(x, ˜y, z)
max f(1 β) (S (x, y, z, Γ) S (x, ˜y, z, Γ)) , 0g g (y) .
L is total search e¤ort of workers
us and hs are the distributions over unemployed and employed
workers
First term recruitment from unemployment - E (x u ( ))
Second term recruitment from other …rms - E (x h ( ))

E¤ect of eliminating business cycles on human capital
distribution:
E¤ects of eliminating bc (%)
E (x u ( )) 4.36
E (x h ( )) 0.18
Elasticity of J wrt E (x u ( )) is
dJ
dE (x u ( ))
E (x u ( ))
J
= 1.27
Elasticity of J wrt E (x h ( )) is
dJ
dE (x h ( ))
E (x h ( ))
J
= 0.39
=) E¤ects through the unemployed dominate

Robustness of gains from eliminating business
cycles
Model version Welfare, b transfer (home pr) GDP Empl
Baseline 1.49 (0.52) 1.45 1.34

cycles
Baseline 1.49 (0.52) 1.45 1.34
Wide human cap range 1.94 (0.94) 1.89 1.43

cycles
Baseline 1.49 (0.52) 1.45 1.34
Wide human cap range 1.94 (0.94) 1.89 1.43
β = 0.50 1.78 (0.56) 1.83 1.42

Welfare gains, accounting for transition
Welfare, b home prod Welfare, b transfer
Steady state 0.52 1.49
Transition 0.20 1.09
Two reasons for gains being lower when accounting for transition:
1 Discounting of future - half-time of GDP transition is 4.5 years
2 Cost of reaching higher employment - extra vacancy posting
needed

Indicative empirical evidence of our mechanism
1 Hairault et al. (2010) …nd signi…cant positive e¤ects of TFP
volatility on average unemployment.
2 Ramey and Ramey (AER, 1995) and Luo et al. (2016) …nd
signi…cant negative relationship between volatility of output and
the growth rate of output
Direct (non-earnings based) evidence of general human capital loss is
scarce.
One exception: Edin and Gustavsson (2008) - Intellectual ability
loss from unemployment
I Being out-of-work for a year implies losing skills equivalent to
0.7 years of schooling.
Evidence of high local unemployment a¤ecting future "employability"
Yagan (2017)

Summary
Described and quanti…ed new mechanism that makes business
cycles costly in terms of average level of output and consumption
I Learning on-the-job makes ampli…es business cycle cost
substantially
Welfare cost of business cycles from new mechanism is large:
I 0.52-1.49% di¤erence between steady states
I 0.20-1.09% including transition dynamics
Policy implication: Stabilizing unemployment raises the average
level of output
I Rationalizes an unemployment stabilization mandate for policy
makers (e.g. central banks)

Future research
Can apply model to other questions regarding cyclical
employment, wages or earnings
I Hysteresis
I Speci…c follow-up project: Quantify how much potential output
has fallen due to Great Recession, due to lower employment
2009-2012
F Lower aggregate human capital
F Higher mismatch / lower match-speci…c productivity (fell down
the job ladder)

If you prefer hiring costs...
The mechanism yielding a negative impact of aggregate volatility on
employment still stands:
Eu ¯u
1
δ + ¯f
f(cov (f , u)) + (Ef ¯f ) Eug
Under the condition: corr (f , u) < 0

Key assumptions to get main result - discussion
1 Negative Beveridge correlation, i.e
corr (vacancies,unemployment) < 0 and a matching function,
or corr (f , u) < 0
I Additional mechanisms with similar implications exist:
Convexity of vacancy posting costs, capital adjustment costs,
other convex costs

Simplifying assumptions made
No physical capital
I can be relaxed
Utility linear in consumption
I harder to relax
Only one aggregate shock, “exogenous TFP”- catch all

Full expression for match value P
P (x, y, z, Γ) = p (x, y, z) +
1 ν
1 + r ∑
x02X
∑
z02Z
[(1 (1 δ) po
P>B ) Bs
f ∑
˜y02Y
s1f z0
, Γ0
P x0
, y, z0
, Γ0
+ β max P x0
, ˜y0
, z0
, Γ0
P
+ 1 s1f z0
, Γ0
P x0
, y, z0
, Γ0
g]πxe x, x0
π z, z0
where ˜y0 denotes the match quality of the poaching …rm and the
indicator for non-separation is:
po
P B = 1 Ps
x0
, y, z0
, Γ0
Bs
x0
, z0
, Γ0
.

Wage distribution - law of motion
Non-trivial task to keep track of wage distribution, hw (w, x, y, z; Γ)
Γ denotes aggregate endogenous state vector
From previous period have hw (w, xold , yold , zold ; Γold )
1 Shock realization and separations yields hw
+ (w, x, y, z; Γ)
2 Matching and wage setting step monstrous:
hw
(w, x, y, z; Γ) = hw
+ (w, x, y, z; Γ)
+mass hired from unemployment
+mass poached from less productive matches
mass lost (+won) due to wage changes within match

Mechanic illustration of why business cycles are
costly (in terms of earnings losses)
Comparing a non-stochastic world to a two-state world
Expansion Recession Non-stoch
# Displaced workers 1 3 2
Earnings loss/worker 1 year 2 years 1.5 years
Aggregate earning losses 1 year 6 years 3 years
Avg aggregate 3.5 years > 3 years

Mechanic illustration (cont’d)
Aggregate earnings loss=earnings loss per worker*number
workers displaced
Both factors are increasing in unemployment rate
If neither factor too concave in unemployment rate:
Aggregate earnings loss a convex function of
unemployment rate
) Business cycles are costly in terms of earnings loss
E¤ect is on the level of aggregate output
I Not about idiosyncratic risk and how that is shared (Krebs,
2007, AER)

What causes earnings loss? Components
1 The obvious one: Unemployment implies no earnings
I Cyclical variation in unemployment duration matters
2 Lower wage at next job
I Loss of match quality (y): immediate
I Loss of negotiation capital (the promised value W ): immediate
F Unemployment implies starting negotiation from scratch (from
B)
I Loss of human capital (x): gradual over unemployment spell
3 Higher probability of additional job separations as...
I Recently displaced workers have lower human capital and worse
match-quality

The e¤ect on the x-distribution of changing
employment (e)
A one period change in e and u yield the following number of net
increases in x:
#∆x = xup∆e xdn∆u
Note u = 1 e, yielding
#∆x = xup∆e xdn∆ (1 e)
#∆x = (xup + xdn) ∆e
For a permanent decrease in e the x-distribution (h(x) + u(x))
will keep on shifting downwards until:
(xup + xdn) ∆e =increase in number of hits of the lower bound+
decrease in number of hits of the upper bound
I Note the vicious circle on the way: as x-distribution falls, e
keeps on falling further
I No closed form way (in particular not independent of the x-grid)
of quantifying the total change in the x-distribution from aWalentin and Westermark () LotJ and the cost of business cycles Eesti Pank, April 2018 56 / 61

Comparison to Jung and Kuester (2011, JEDC)
As an extension Jung and Kuester consider impact of LotJ on
the cost of business cycles
They get very small e¤ects from LotJ ( 0.06% of welfare and
GDP) compared to us
This is due to several factors:
1 Their wage bargaining assumption minimizes the e¤ect of
cycles on employment compared to standard bargaining,
as can be seen from the larger e¤ects in Hagedorn and
Manovski
2 Their human capital loss happens both at separation and
gradually during unemployment
) human capital distribution less sensitive to changes in
employment
3 Their human capital scales match output and unemployment
bene…t equally. This minimizes the e¤ect on job creation from a
decrease in the human capital distribution
4 Absence of endogenous (cyclical) separations in their modelWalentin and Westermark () LotJ and the cost of business cycles Eesti Pank, April 2018 57 / 61

Human capital dynamics - interaction with
match-speci…c productivity
Ljungqvist and Sargent/Jung and Kuester uses di¤erent
assumption of human capital loss
Some part of x lost at separation
I Seems suitable in their model without match-speci…c
productivity
I Less suitable for our model

Numerical details
We use Tauchen and Hussey’s (1991) discretization of AR(1)
processes with optimal weights from Flodén (2008).
I This algorithm has been shown by Flodén (2008) to be accurate
for processes with high persistence.
The number of gridpoints for x, y and z are 10, 8 and 5
respectively.
The wage grid contains 15 points and the mi grid 2 points
I Linear interpolation over the moments, mi .
For the grid for human capital, x, we follow Ljungqvist and
Sargent (1998, 2008) in setting the ratio between the maximum
and minimum value of x to 2.

Labor market ‡ows - separations and shock
realizations (subperiods 1-2)
Distribution (density) of matches:
hs
(x, y, z) = ∑
x 1
∑
y 1
(1 ν) (1 δ) 1 fS (x, y, z) 0g
πxe (x 1, x) πy (y 1, y) h (x 1, y 1, z 1)
Distribution of unemployed develops analogously:
us
(x, z) = ν1 fx = xg + (1 ν) [∑
x 1
πxu (x 1, x) u (x 1, z 1)
+ ∑
x 1
∑
y 1
(1 fS (x, y, z) < 0g + δ1 fS (x, y, z) 0g)
πxe (x 1, x) πy (y 1, y) h (x 1, y 1, z 1)].

Labor market ‡ows - new matches
Distribution of matches after new matches are formed:
h (x, y, z) = hs
(x, y, z)
+s0
M
L
u+ (x, z) 1 fS (x, y, z) 0g f (y)
| {z }
mass hired from unemployment
+s1
M
L ∑
˜y
hs
(x, ˜y, z) 1 fS (x, y, z) > S (x, ˜y, z)g g (y)
| {z }
mass poached from less productive matches
s1
M
L
hs
(x, y, z) ∑
˜y
1 fS (x, ˜y, z) > S (x, y, z)g g (˜y)
| {z }

Learning on the job and the cost of business cycles

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