Slack in the Intensive Margin: The UK and Its Underemployment Radek ’auer

∗

Goethe University Frankfurt

†

Deutsche Bundesbank February 20, 2017

Abstract This paper investigates how time-related underemployment varies over the business cycle.

I introduce underemployment into a fair-wage model where I

explicitly consider hours and labor force. I estimate the model using UK data and explain why underemployment substantially increased during the Great Recession. The increase in underemployment was caused by weak demand.

Expansionary

monetary policy prevented an even bigger increase.

JEL Classication: Keywords:

E24, J22

time-related underemployment, hours, fair wages, eort, UK

[email protected] The views expressed in this paper are those of the author and do not necessarily reect the views of the Deutsche Bundesbank. ∗

†

Recently the phenomenon of underemployment has received a lot of attention and has entered the economic-policy debate (Blanchower and Posen, 2014).

In the ex-

isting literature the term underemployment tends to have dierent meanings. Here I always refer to the time-related underemployment. This concept of underemployment expresses how much slack the labor market exhibits in the intensive margin. It captures people who are employed but desire to work more hours. The typical representatives are involuntary part-time workers.

However, other groups of workers become under-

employed as well. Voluntary part-timers who wish to slightly increase their hours also suer from underemployment.

Similarly, full-time workers who want more hours are

underemployed as well. Since the Great Recession underemployment has helped policy makers to assess the state of the economy (BoE, 2014; Weale, 2014). Generally speaking, the policy makers face a lot of uncertainty. They are, for instance, uncertain about potential output or natural rate of unemployment. They want to acquire additional information from other measures and make better, more informed decisions.

The most pronounced example

in this respect is the Fed and its dashboard approach (FOMC, 2016).

Occasionally

the unemployment rate doesn't completely describe the situation on the labor market (Yellen, 2014). Therefore, the Fed prefers to look at a variety of indicators which the time-related underemployment belongs to. Table 1 gives the degree of underemployment for several countries. It reports the underemployment rate constructed according to ILO standards (ILO, 1998). The underemployment rate expresses how many percent of employees want to work more hours. Some countries, like Spain or France, experience substantial underemployment with rates close to 10%. Evidence on the US underemployment is provided by Blanchower and Levin (2015). Statisticians collect data on underemployment, and the policy makers use the data as additional information on the labor market. But surprisingly, economics lacks any theoretical foundation of the time-related underemployment.

It is unexplored how

underemployment changes over the business cycle, how it reacts to dierent shocks, or whether expansionary monetary policy eliminates underemployment. My paper delivers answers to these questions and oers a structural analysis of underemployment. The analysis I present is purely positive. Nevertheless, underemployment isn't just an abstract measure of labor-market slack. Psychologists show that underemployment negatively impacts people's well-being and can result into depressions (Dooley, Prause and Ham-Rowbottom, 2000).

2

Country

Unemployment Rate

Underemployment Rate

Australia France Germany Greece Italy Japan Spain UK

6.1 9.9 5.0 26.5 12.7 3.6 24.4 6.1

8.6 9.4 4.6 8.6 5.1 4.6 10.8 7.3

Table 1: Unemployment and Underemployment Rate in 2014. Age 15+. Labor Force Surveys. Source: ILOSTAT.

I analyze underemployment in three steps. First, I explain why the United Kingdom represents an ideal country for analyzing underemployment.

I describe how the UK

underemployment has changed over the years and point to the substantial increase in underemployment during the Great Recession. Second, I set up a macroeconomic model with fair wages. Literature on fair wages in the past (i.a., Danthine and Kurmann, 2004; de la Croix, de Walque and Wouters, 2009) concentrated solely on employment and refrained from modeling other dimensions of the labor market. I additionally consider hours and labor force. The hours relax the Solow (1979) condition of constant eort. The eort comoves with hours per worker and hence behaves procyclically, which concurs with the empirical ndings of Burda, Genadek and Hamermesh (2016). Both the extensive and the intensive margin of the labor market enter the wage equation. Not only lower unemployment but also higher hours per worker lead to higher hourly wages. Underemployment in the model occurs as a dierence between desired and actually worked hours. Third, I estimate the model with eight observables. I nd that the rise in the UK underemployment during the Great Recession was primarily caused by weak demand. The weak demand led rms to lay o workers and to shorten hours. However, workers, who experienced high unemployment and low consumption, desired to work longer hours.

Underemployment, the dierence between desired hours and actually worked

hours, inevitably occurred.

The Bank of England helped to mitigate the increase in

underemployment by expansionary monetary policy.

3

1

Underemployment and the UK

In the UK, the Oce for National Statistics collects worldwide unique data on underemployment (ONS, 2016).

In its Quarterly Labour Force Survey, the Oce for

National Statistics acquires data on the level of underemploymenthow many people are underemployedand on the volume of underemploymenthow many extra hours people would like to work. In addition to underemployment, the Labour Force Survey contains information on the opposite phenomenonthe overemployment. The Labour Force Survey hence knows how many people would like to work less hours even if it meant less pay, and how many fewer hours people would like to work. To be counted as underemployed according to the Labour Force Survey, people have to belong to one of the following three groups. 1) They are looking for an additional job; 2) they are looking for a replacement job with longer hours; 3) they are not looking for a dierent job but would prefer to work longer hours at their current basic rate of pay. Analogous groups exist for the overemployed. The overemployed are people 1) who are either looking for a replacement job with shorter hours and are prepared to accept less pay or 2) who are not looking for a dierent job but would prefer to work shorter hours even if it meant less pay. A very natural way to present the slack in the intensive margin is the underemployment index by Bell and Blanchower (2013).

The underemployment index contains

information on the level and the volume of underemployment. It is dened as

U DIt =

U nemployedt AverageHourst + Extrat − F ewert 100. (Employedt + U nemployedt ) AverageHourst

The core of the index is the unemployment rate

U nemployedt . Employedt + U nemployedt However, the index translates the unemployment rate into hours.

The index multi-

plies the numerator and the denominator of the unemployment rate by average weekly hours. Then the index adds the net underemployment tor.

Extrat

Extrat − F ewert

to the numera-

represents the aggregate desired increase in hourshow many extra hours

per week all underemployed workers together desire.

F ewert

stands for the aggregate

desired decrease in hourshow many fewer hours per week all overemployed workers

4

together desire. If the aggregate desired increase equals the aggregate desired decrease (Extrat

F ewert ),

the underemployment index equals the unemployment rate. If underemploy-

ment dominates overemployment (Extrat lies above the unemployment rate. (Extrat

=

< F ewert ),

> F ewert ),

the underemployment index

If overemployment dominates underemployment

the underemployment index lies below the unemployment rate.

Figure 1 presents how underemployment and overemployment have changed over the years in the UK. During the rst half of the 2000s the UK suered from light

11 Unemployment Rate Bell-Blanchflower Underemployment Index

10 9

%

8 7 6 5 4

Figure 1:

2002

2004

2006

Unemployment Rate vs.

2008

2010

2012

Underemployment Index.

2014 Age 16+.

The UK.

2001:Q22015:Q3. Seasonally adjusted. Source: Haver and The Work Foundation.

overemployment.

But from 2006 onward the UK has experienced underemployment.

At rst underemployment slightly exceeded overemployment. The dominance became signicant in the Great Recession. Walling and Clancy (2010) discuss the level and the volume of underemployment in more detail. Tam (2010) adds data on overemployment and reports demographics of underemployment and overemployment.

2

The Model

Figure 2 intuitively explains how my model generates periods of underemployment and periods of overemployment.

The x-axis depicts the state of the economy, which

I summarize for simplicity by output

Yt

and employment

Nt .

The y-axis represents

hours per worker. The gray solid line stands for the true hours per worker

5

hm t how

many hours people actually work in a given state of the economy.

h

The blue dashed

m t how many hours people would like to

line represents the desired hours per worker work in a given state of the economy.

Hours per Worker True Hours per Worker hm t

Underemployment

Overemployment

Desired Hours per Worker hm t (Yt , Nt ) Y

low

,N

low




(Y, N )

Y

high

,N

high




Figure 2: Intuition behind the Model

In the steady state

(Y, N ),

when the economy operates at an average level, workers

are satised with the hours they work. In other words, their desired hours equal their truly worked hours (

hm = hm).

In an economic downturn

Y low , N low




, rms face lower demand.

They react by

m oering shorter hours ht to the workers. Higher unemployment causes losses in housem hold incomes. The workers would like to counteract and work longer hours t . The

h

h

m m desired hours exceed the true hours ( t > ht ); underemployment arises.
high In an economic upturn Y , N high , the rms require longer hours hm t to satisfy the increased demand for their products.

Lower unemployment results into bigger

household incomes. The workers would prefer more leisure and therefore desire shorter hours

hmt. The true hours exceed the desired hours (hmt > hmt); overemployment arises.

I set up the model in two steps. First, in section 2.1 I formalize the UK economy. I model rms, government, central bank, and a representative household. I assume that the hours per worker

hm t

are set by the rms and taken as given by the household. I

receive a full description of the economy.

I learn, for example, the hourly wage, the

6

unemployment rate, or the true hours per worker 2, I obtain the state of the economy

(Yt , Nt )

hm t .

In terms of the intuitive gure

and the true hours per worker

Second, in section 2.2 I recover the desired hours per worker

hm t .

hmt for a given state of

the economy. I gure out how many hours per worker the household would like to choose if it got a chance to decide. The desired hours have no feedback eect on the economy. The representative household only answers a survey question about its desired hours while it observes the state of the economy from section 2.1 (wages, unemployment, prices, etc.). I then dene underemployment as the dierence between the desired and the true hours per worker. In what follows, I assume there is no heterogeneity of workers. All workers in the model are identical. Because I focus on the time-related underemployment of an average worker, the assumption of identical workers should not raise any serious concerns. If I was interested in skill-related underemploymenta situation in which high-skilled people stay in low-skilled jobs, it would be crucial to model heterogeneity of workers and jobs. To learn more about the stylized facts and a possible model of skill-related underemployment, I refer to the recent work by Barnichon and Zylberberg (2016).

2.1

The Economy

2.1.1 The Household A continuum of identical people

[0; 1]

lives in a representative household which maxi-

mizes expected utility with respect to a budget constraint.

max

∞

{Cτm ,Cτn ,H˜ τm ,H˜ τn ,Lτ ,bτ }τ =t

Et

∞ X



1 ln [ζ (Cτm )σ + (1 − ζ) (Cτn )σ ] − σ τ =t  1+ϕ  1+ϕ   m ˜ ˜n  H H
τ τ m κτ − κn exp βτ 1+ϕ 1+ϕ   β

τ −t

s.t.

Cτn = Anτ (1 − Lτ ) ˜ m = Lτ uτ hu eu + Lτ (1 − uτ ) hm em H τ τ τ ˜ τn = (1 − Lτ ) hn en H

7

ΞL + bτ + 2

Cτm



2 Lτ 1 + iτ −1 − 1 + Υτ = Lτ (1 − uτ ) hm bτ −1 τ wτ + Lτ uτ q + dτ + Lτ −1 Πτ

The household derives the utility from market consumption

Ctm

and home consumption

Ctn . To experience the market consumption, the household buys the nal good for the Pt .

price

At home the individuals who stay out of the labor force produce goods for the

home consumption with the xed hours

hn ,

xed eort

Ant = exp A t

n

en ,

and home productivity

Ant .




The well-being of the household is negatively impacted by eective hourshours worked multiplied by eort. I distinguish between eective hours on the market

˜ n . The labor-supply shock tive hours at home H t

˜m H t

and eec-

κm t changes how much pain the hours

on the market cause.

κ κm t = κm exp (t ) The household observes the unemployment rate the labor force. The remaining

1 − Lt

ut

and sends

Lt

individuals into

individuals stay at home in the home produc-

tion. I want to replicate the small volatility of labor-force participation; therefore, the household encounters labor-force adjustment costs, which I express in terms of the nal good. The individuals wage

wt .

Lt (1 − ut ) become employed and work hm t

The hours per worker and the hourly wage are set by rms. The workers accept

the employment contract and deliver the eort tions.

The individuals

benets

hours for the real hourly

q.

Lt ut

em t

according to behavioral considera-

become unemployed and receive the real unemployment

To claim the benets, the unemployed have to contact the employment

agency. This requires

hu

hours and eort

eu .

The household also decides on the real bonds

bt ,

which earn the real gross interest

(1 + it ) /Πt+1 , where it represents the nominal interest rate and Πt+1 the gross ination rate (Πt = Pt /Pt−1 ). Additionally, the real lump-sum taxes Υt and the real β dividends dt enter the budget constraint. The variable t denotes a demand shock. rate

βt = ρβ βt−1 + σβ νtβ

8

2.1.2 Workers' Eort Every worker employed by an initial rm

j

supplies eort

em t (j)

according to the

fair-wage hypothesis (e.g., Akerlof, 1982; Danthine and Kurmann, 2004; de la Croix, de Walque and Wouters, 2009). The workers minimize the distance to a social norm

gt (j). 2 min [em t (j) − gt (j)]

em t (j)

The eort-supply function then takes the form:

 em t (j) = gt (j) = φ0 + φ1 where

Wt (j) Pt

1−γw

1 − γw

−1

 − φ2

Wt−1 1−ω Ptω Pt−1

1−γw

−1

1 − γw

+ φ3

u u1+γ t + et , 1 + γu

(1)

φ0 R 0, φ1 > 0, φ2 > 0, φ3 > 0, γw ∈ (0; 1), γu > 0, ω ∈ [0; 1].

The workers deliver more eort if they receive a higher hourly wage (wt (j)

= Wt (j)/Pt )

or observe a higher unemployment rate. The workers deliver less eort if their reference wage increases.

The reference is the past average hourly wage.

All workers in the

e economy have the same stochastic component t in their eort supply.

et = ρe et−1 + σe νte In most of the eort functions found in the literature the workers deate the past average nominal wage

Wt−1

by the past price level

considers deating by today's price level

Pt .

Pt−1 .

In contrast, Vaona (2013)

I use an average price level

1−ω Ptω Pt−1

instead which allows for the two stricter specications. Also notice the workers' eort depends here on the hourly wages.

In standard fair-wage models, where hours per

worker implicitly stay constant, the quarterly wages drive the eort. I don't impose downward rigidity on nominal wages. Nickell and Quintini (2003) nd that in the UK the downward rigidity is macroeconomically insignicant.

2.1.3 The Bundler A representative bundler combines intermediate goods and produces the nal good

Yt

Yt (s)

from intermediate rms

using the Dixit-Stiglitz aggregator. The bundler maxi-

9

mizes its prot as follows.

max Yt ,Yt (s)∀s∈[0;1]

Pt Y t −

Z

1

Z Pt (s)Yt (s) ds

s.t.

Yt =

0

1

(Yt (s))

θ−1 θ

θ  θ−1 ds

0

2.1.4 Intermediate Firms Every intermediate rm

s ∈ [0; 1]

transforms the initial good

one to one into an intermediate good

Xt (s)

from initial rms

Yt (s).

Yt (s) = Xt (s) The initial good costs With a probability

γ

Vt (vt = Vt /Pt ).

The intermediate rms set prices à la Calvo.

they cannot reoptimize and fully index their prices to past and

trend ination. They also need to pay a revenue tax

µt .

" −θ −θ #  ∗  ∗ I I P P λ P t τ t τ |t t τ |t γ τ −t β τ −t max Et (1 − µτ ) Pt∗ Iτ |t Yτ − V τ Yτ Pt∗ λ P P P t τ τ τ τ =t ∞ X

Iτ |t =

τY −1

It|t = 1 Πξk Π1−ξ

for

τ = t + 1, t + 2, . . .

k=t

In the stochastic discount factor, the symbol

λt

stands for the shadow price of real

wealth.

2.1.5 Initial Firms The initial rms hire labor and produce the homogeneous initial good, whose price they take as given. An initial rm

j ∈ [0; 1]

maximizes prot knowing its production

function and the reaction of workers' eort. The rm decides on the hourly wage the hours per worker

hm t (j),

max m

and the workforce

∞ {wτ (j),em τ (j),hτ (j),Nτ (j),Xτ (j)}τ =t

Et

∞ X

β τ −t

τ =t

10

Nt (j). λτ [vτ Xτ (j) − wτ (j)hm τ (j)Nτ (j) λt  2 #
ΞN Nτ (j) − − exp N τ 2 Nτ −1 (j)

wt (j),

s.t. (1)

Xτ (j) =

Am τ



 δ2 m δ1 m α 2 3 (hτ (j)) − (hτ (j)) [em τ (j)Nτ (j)] 2 3 δ1 > 0, δ2 > 0, 0 < α < 1

The prot is diminished by workforce adjustment costs. For a specic size of the workforce the rm has to ne-tune the production process. The rm has to rethink its division of tasks, replan shifts, buy new or eliminate idle machines. Therefore, the rm bears the quadratic adjustment costs, which I express in terms of the nal good. The shock

N t , N N N t = ρN t−1 + σN νt ,


exp N t ,
exp N t . Such

causes that in some quarters it is costless to increase the workforce at the rate and in some quarters it is costless to decrease the workforce at the rate

a shock captures changes in the labor-market regulation which make it either easier or more dicult to hire and re. Additionally, the shock represents a counterpart of the household's demand shock. The rm spontaneously alters the demand for workers. The production function is s-shaped in the hours per worker and strictly concave in the eective workforce

em t (j)Nt (j).

The market productivity

A Am t = exp t m

m

m

Am t

is purely exogenous.




A A = ρAm A t t−1 + σAm νt

m

S-shaped production functions are present in economics since Turgot (1767). The specic s-shaped relation between output and hours per worker is discussed by Cahuc, Carcillo and Zylberberg (2014).

At low levels of hours the workers suer from set-

up costs and have to warm up. So the production rises slowly. Afterward the work is routinized, and the production speeds up. At a certain point workers inevitably become tired, and the increases in production slow down. I choose the simplest s-shaped functional form one can think ofa third-order polynomial. For every level of productivity and eective workforce, the production function has an inection point at

hm t (j) = δ1 / (2δ2 )

and a maximum at

prot-maximizing rm, of course, never crosses the maximum.

11

hm t (j) = δ1 /δ2 .

A

For a better illustra-

tion, I depict the production function in gure 3.

Steady State 2

Maximum Production: hm t (j) =

1.8

Inflection Point: hm t (j) =

1.6

δ1 δ2

δ1 2δ2

1.4 1.2 Xt (j)

1 0.8 0.6 0.4 0.2 0

1

0.8

0.6

0.4

0.2

0

0.2

0.1

0

em t (j)Nt (j)

Figure 3:

0.3

0.5

0.4

0.6

hm t (j)

The Production Function. Am t = 1.

Parameters

δ1 , δ2 ,

and

α

are recovered from

steady-state relations.

The rst-order conditions with respect to the hourly wage

m worker ht (j), and the workforce

αvt Am t



Nt (j)

wt (j),

the hours per

take the following form.

 δ2 m δ1 m 2 3 α−1 (h (j)) − (ht (j)) [em φ1 (wt (j))−γw [Nt (j)]α = hm t (j)] t (j)Nt (j) 2 t 3  m 2 m m vt Am δ h (j) − δ (h (j)) [et (j)Nt (j)]α = wt (j)Nt (j) 1 2 t t t   δ2 m 2 3 α α−1 m m δ1 (ht (j)) − (ht (j)) [em αvt At + t (j)] [Nt (j)] 2 3  
Nt+1 (j) λt+1 Nt+1 (j) m N Et β ΞN − exp t+1 2 = wt (j)ht (j)+ λt Nt (j) (Nt (j))  
ΞN Nt (j) N − exp t Nt−1 (j) Nt−1 (j)

(2)

(3)

(4)

2.1.6 Workers' Eort in Equilibrium I combine (2) with (3). I drop the index obtain:

em t

=

δ1 m ht αφ1 2 m δ1 ht

j −

−

12

because all initial rms are identical and

δ2 3

2 (hm t )

2 δ2 (hm t )

wt1−γw .

(5)

The workers' eort

em t

varies over the business cycle. The Solow condition doesn't hold

for two reasons. First, the relatively general eort function (1), in which the logarithmic specication is nested, causes the eort to comove with the hourly wage. Second, the explicitly modeled hours per worker enter the eort condition. More hours per worker then imply higher eort.

2.1.7 The Wage Equation I insert the eort condition (5) into the eort function (1). I again drop the index

j

and get an expression for the real hourly wage.

" wt1−γw

2 δ1 m h − δ32 (hm φ1 t ) 2 t − αφ1 m 2 1 − γw δ1 hm t − δ2 (ht )

# =

−φ0 (1 − γw ) + φ1 − φ2 1 − γw

φ2 1−γw −ω(1−γw ) wt−1 Πt 1 − γw φ3 u − u1+γ − et t 1 + γu

(6)

+

The hourly wage is persistent; it depends on its past value (2013), higher ination

Πt

lowers the real hourly wage.

wt−1 .

Similarly to Vaona

Because I explicitly model

hours, both labor marginsintensive and extensiveinuence the hourly wage. More hours per worker

hm t

and lower unemployment rate

ut

increase the hourly wage. Ceteris

paribus positive eort shocks negatively impact the hourly wage.

2.1.8 The Workforce Adjustment Costs Move the Hours per Worker Suppose the initial rms don't pay any workforce adjustment costs (ΞN

= 0).

In such

a case, the rms only change their eort, workforce, and hourly wage over the business cycle. But the hours per worker stay constant. They lie under the maximum-production point and well above the inection point.

3δ1 3δ1 (2 − α) δ1 < hm < t = 4δ2 2δ2 (3 − α) δ2 If the production function gets closer to being linear in the eective workforce, the hours per worker decrease.

∂hm −3δ1 t = <0 ∂α 2δ2 (3 − α)2 13

The rms change the hours per worker only if they want to avoid the costly workforce adjustment (ΞN

> 0).

Under zero workforce adjustment costs the equilibrium eort only depends on the hourly wage:

1−γw em . t = φ1 wt The constant hours per worker also simplify the workforce and the wage equation.

"

3 (δ1 )2 (2 − α) αφα1 α(1−γw )−1 vt Am Nt = t wt 2 4δ2 (3 − α)

wt1−γw =

1 # 1−α

−φ0 (1 − γw ) + φ1 − φ2 φ2 1−γw −ω(1−γw ) + w Π φ1 γw φ1 γw t−1 t φ3 (1 − γw ) 1+γu 1 − γw e u −  − (1 + γu ) φ1 γw t φ1 γw t

2.1.9 The Relation between the Workforce and the Hours per Worker Workforce and hours per worker approximately add up to total hours worked on the market

Ht (Ht = Nt hm t ):

ˆ m. ˆt = N ˆt + h H t

Stylized facts for the majority of OECD countries reveal that total hours

ˆm ˆt , and that hours per worker h marily driven by workforce N t

ˆt H

are pri-

are relatively stable (King

and Rebelo, 1999; van Rens, 2012). In the UK, especially since the 1990s, the driving forces behind the total hours have been more balanced.

In gure 4, I plot the HP-ltered workforce, total hours, and

hours per worker. During 1971:Q12015:Q4 the contemporaneous correlation between

ˆ m ) equaled 0.54. The workforce ˆt ) and the hours per worker (100h the workforce (100N t and the hours per worker exhibited standard deviations of 0.96 and 0.56.

However

notice that in 1995 the volatility of workforce started to resemble the volatility of hours per worker. Due to the workforce adjustment costs, the model features enough exibility to produce hours per worker which are either less or more volatile than the workforce. I demonstrate this feature of the model by combining the linearized versions of (3) and

14

4

ˆt 100H ˆ 100hm

3

t

100Nˆt 2

%

1 0 −1 −2 −3

1975

1980

1985

1990

1995

2000

2005

2010

2015

ˆ t ), Hours per Worker Figure 4: HP-Filtered Natural Logarithm of Total Hours (100H ˆ m ), and Workforce (100N ˆt ). λ = 1600. 1971:Q12015:Q4. Source: Haver (100h t (MGRZQ@UK, YBUSQ@UK).

(4):

ˆt = N

1 ˆ β ˆ m + 1 − βρN N . ˆt+1 + (2 − α) (3 − α) lsY h Nt−1 + Et N t 1+β 1+β α (1 + β) ΞN 1+β t

The equation makes clear how moves in the hours per worker cause moves in the workforce. A crucial role plays the parameter of the workforce adjustment costs

ΞN .

If

ΞN

is small, moves in the hours per worker lead to large moves in the workforce. The hours per worker are then less volatile than the workforce. If

ΞN

is large, moves in the hours

per worker lead to negligible moves in the workforce. The hours per worker are then more volatile than the workforce. Let's analyze two limiting cases.

Case 1:

ΞN = 0

In the absence of workforce adjustment costs, the hours per worker

stay constant, and only the workforce moves.

ˆm = 0 h t ˆt = vˆt + Yˆt − wˆt N

15

Case 2:

ΞN → ∞

If the parameter of the workforce adjustment costs goes to innity,

the workforce fully adjusts to the workforce shock. Only the hours per worker actively react to the economy.

h i α ˆt − N ˆt − wˆt v ˆ + Y t 2 (3 − 2α) + α2 ˆt = N ˆt−1 + N N t

ˆm = h t

2.1.10 The Government Because the government has to nance unemployment benets tax

Υt

on the household and a revenue tax

µt 1

Z qLt ut = Υt +

µt 0

q , it imposes a lump-sum

on the intermediate rms.

Pt (s) Yt (s) ds Pt

The unemployment benets equal a xed fraction

χ

of steady-state labor income.

q = χwhm The revenue tax follows a stochastic process.

µt = µ exp(µt )

2.1.11 The Central Bank The central bank sets the interest rate it according to a simple rule. The bank smoothly responds to ination

Πt

and unemployment rate

ut .

Monetary shocks

νti

tions from the strict rule.

it = ψi it−1 + (1 − ψi ) [i + ψΠ (Πt − Π) − ψu (ut − u)] + σi νti

2.1.12 The Market Clears The bonds are in zero net supply.

bt = 0

16

cause devia-

The nal good is split between the market consumption, labor-force and workforce adjustment costs.

Yt =

Ctm

The unemployment rate force

ΞL + 2 ut



2  
2 Lt ΞN Nt N −1 + − exp t Lt−1 2 Nt−1

is the ratio between the unemployed

Lt . ut =

2.2

Lt − Nt

(7)

and the labor

Lt − Nt Lt

The ONS Collects Data on Underemployment

In this section, I derive the desired hours per worker and the degree of underemployment. In order to do so, I mimic the way how statisticians collect data on underemployment. In every quarter, the Oce for National Statistics asks the household how many hours per worker it would like to choose. The household answers by solving an identical optimization problem in every quarter. In contrast to the household optimization in section 2.1, which constitutes the economy, the optimization here is set up to gain information on underemployment. The household even now maximizes expected utility. But it behaves as from today onward it could control the hours per worker

hmτ.

I can formalize the question from the ONS as follows.

max

˜m m {Cm τ ,Hτ ,hτ ,bτ }τ =t ∞

Et

∞ X τ =t

C



1 σ n σ ln [ζ ( m τ ) + (1 − ζ) (Cτ ) ] − σ  1+ϕ  1+ϕ   ˜m ˜n  H
τ τ m exp βτ κτ − κn 1+ϕ 1+ϕ  

β

τ −t

H

s.t.

Cτn = Anτ (1 − Lτ )

H˜ mτ = Lτ uτ hueu + Lτ (1 − uτ ) hmτemτ ˜ τn = (1 − Lτ ) hn en H

C

m τ

+

bτ

ΞL + 2



2 Lτ − 1 + Υτ = Lτ (1 − uτ ) Lτ −1

bt−1 = bt−1 17

hmτwτ + Lτ uτ q + dτ + 1 +Πiτ −1 bτ −1 τ

When people are asked about their desired hours, they implicitly consider readjusting their consumption and saving behavior.

Therefore, in addition to hours

household controls for desired market consumption

hmτ, the

Cmτ and desired bonds bτ . The ONS

gathers data on underemployment from employed people. When reporting on underemployment, people don't contemplate exiting the labor force. The obtained measure of underemployment is hence based on the labor-force participation which currently prevails in the economy (Lt ). To reect this feature, the representative household takes here the labor-force participation as given.

b

account its past decision on bonds ( t−1

Finally, the household has to take into

= bt−1 ).

Up to the rst order, the desired market consumption in quarter

C

ˆm t

= −ιr

reads:

 ∞  X 1 ˆ 1ˆ ˆ t + ιβ βt − ιAn ˜An , it+τ − Πt+1+τ + ιL L t t 1+i Π τ =0

E

where I list the parameters

t

ι

ιr > 0, ιL > 0, ιβ > 0, ιAn > 0.

in appendix A; I dene

˜A t

n

in appendix B. I assume that the

household uses model-consistent expectations over the real interest rates when answering the questions from the ONS. In consequence, the desired market consumption equals the actual market consumption

C

ˆ m = −ιr Et t

Cˆtm :

Cˆ mt

 ∞  X 1 ˆ 1ˆ ˆ t + ιβ βt − ιAn ˜An = Cˆ m . it+τ − Πt+1+τ + ιL L t t 1+i Π τ =0

I take the rst-order condition with respect to the desired hours per worker

hmt and

substitute for the shadow price of wealth. I obtain:

hˆ mt = ηw wˆt + ηuuˆt − ηe

m

where

n ˆm ˆ eˆm ˜κt − ηAn ˜A t , t − ηC m Ct − ηL Lt − 

ηw > 0, ηu > 0, ηem > 0, ηC m > 0, ηL R 0, ηAn > 0.

This relation reports how many hours per worker the household prefers. In appendix A, I describe the parameters

η

in more detail.

Ceteris paribus, a higher hourly wage rate implies more desired hours per worker. If the unemployment rate rises, the household wants to compensate for the shortage of labor income by the intensive margin. When the workers deliver more eort, the household desires to avoid the painful work and to reduce the hours. The household's felicity

18

function is concave in home and market consumption. When the market consumption lies above its steady state, the household consequently experiences low marginal utility of market consumption and low shadow price of wealth.

Work is hence less attrac-

tive. If more disutility arises from the market activity, the household naturally shows a smaller willingness to work. Whenever the home production becomes more eective, the market activity loses its appeal, and the desired hours decrease. The impact of labor-force participation on the desired hours is ambiguous.

On

the one hand, higher labor-force participation increases the disutility from the market activity and lowers the desired hours.

On the other hand, higher labor-force partic-

ipation raises the shadow price of wealth; the household wishes to earn more and to choose more hours. The overall net eect crucially depends on the parameter captures how convex the disutility from the market activity is. If

ηL

ϕ

ϕ,

which

is large enough,

becomes positive. The disutility eect dominates, and the desired hours drop with

higher labor-force participation. The last variable the household controls are the desired bonds. straint in

t

The budget con-

determines how many bonds the household would buy in a quarter where

for the rst time the household can choose the hours per worker. I take this budget constraint, substitute for dividends and lump-sum taxes, use the aggregate resource constraint and express the desired market consumption in terms of the actual market consumption. The budget constraint then simplies to:

b

ˆt =

The variable

zt

bt = whmN hˆ m − hˆ m = whmN z . Y

Y

t

t

Y

t

measures the degree of underemploymentby how many percent the

household would like to adjust the hours per worker. If the household is underemployed (zt

> 0), it wants to save.

3

Estimation

3.1

If the household is overemployed (zt

< 0), it wants to borrow.

Data and Observables

The model has eight shocks; correspondingly, I use eight observables. The rst three observables stand for the basic New-Keynesian variables: real GDP per capita GDP deator

DEFt ,

and interbank rate

BOEt .

the labor market: labor-force participation rate

19

GDPt ,

The remaining observables represent

LF P Rt ,

unemployment rate

U N Rt ,

average hours

HOU RSt ,

labor share

LSt ,

and underemployment index

U DIt .

I cover

the period 2001:Q22015:Q3 because the underemployment index (Bell and Blanchower, 2013) is only available from 2001:Q2 onward.

To capture at least partly the

unconventional monetary policy, I prefer the interbank rate to the simple policy rate of the BoE. In appendix C, I describe the data and its transformations in more detail. The observables match the data to the model in the following way.

  GDPt = Yˆt − Yˆt−1 100   ˆ DEFt = Π − 1 + Πt 400   BOEt = i + ˆit 400   ˆ t 100 LF P Rt = L + L U N Rt = (u + uˆt ) 100   ˆm − h ˆ m 100 HOU RSt = h t t−1 i h  m ˆ ˆ ˆ LSt = ls + ls wˆt + ht + Nt − Yt 100 U DIt = [u + uˆt + (1 − u) zt ] 100 3.2

Priors and Calibration

I calibrate several parameters of the model. In table 2, I report the calibrated values. The BoE targets 2% CPI ination; I correspondingly calibrate the steady-state gross ination

Π.

The selected discount factor

β

together with the ination

Π

produces

annual interest rate of 4% in the steady state. With

ζ = 0.5,

I put equal weights on home and market consumption in the utility

function. Concerning the price elasticity

θ,

I choose a conservative value of 5.

The standard VAT rate in the UK is 20%, which translates into the revenue tax 16.7%. The parameter

χ

µ of

is consistent with the net replacement rate of unemployment

benets which include social and housing assistance (OECD, 2016). Krueger and Mueller (2012) show the unemployed in the UK spend 6.9 minutes per day on job-search activities. If people have a time endowment of 100 hours per week,

hu

u equals 0.008 (h

= (6.9 · 7) / (100 · 60)).

I set the steady-state labor-force participation rate ment rate

u,

the steady-state hours per worker

20

m

h

L,

the steady-state unemploy-

, and the steady-state labor share

ls

Parameter

Value

Π β ζ θ µ χ hu L u hm ls Y em eu i α

1.005 0.995 0.500 5.000 0.167 0.500 0.008 0.633 0.062 0.320 0.582 1.000 1.000 1.000 (Π/β) − 1 (lsθ) / ((1 − µ) (θ − 1))

Table 2: Calibrated Parameters

to the arithmetic means of the data. The steady-state output of workers

em ,

and the eort of the unemployed

the steady-state interest rate

i

eu

Y , the steady-state eort

are all normalized to 1. I recover

and the parameter of the production function

α

from

the steady-state relations. A few words on the priors I use. I restrict the parameter of the felicity function

σ

to lie between 0 and 1, which ensures home and market consumption are substitutes. The Calvo parameter

γ

at the prior mean causes that on average the intermediate rms

don't reoptimize for three quarters. A priori

ξ

takes small realizations. Ination hence

exhibits low persistence, and the intermediate rms rather index to trend ination than to past ination. With the prior over

ω

the workers take into account both the past

and the present price level when they compute the real reference wage for the eort function, and ination has a direct signicant eect on the real hourly wage through the wage equation. I present all priors in table 3.

3.3

Posteriors

The estimated parameters are locally identiable according to Iskrev (2010) test.

I

draw 1000 realizations from the priors; each of them leads to a unique solution of the model. Because the Jacobian

J(T )

has a full column rank at all draws, the parameters

fulll local identiability.

21

I nd the mode by Sims (1999) csminwel optimizer and draw from the posterior by random-walk Metropolis-Hastings algorithm, both implemented in Dynare (Adjemian et al., 2011). I use two chains each with 100,000 draws and discard the rst 50,000. The acceptance rates are 30.2% and 30.1%. The table 3 presents the results of the estimation. I additionally depict the priors and the posteriors in appendix D. The price stickiness

γ

lies above the typical micro

estimates. On average, the intermediate rms reoptimize once every 10 quarters. As pointed out by Eichenbaum and Fisher (2007), the high price stickiness is a standard feature of models which have constant price elasticity and marginal costs which are independent of price setter's output. The Phillips curve is dominantly forward-looking because the intermediate rms primarily index to trend ination. The BoE strongly smooths the interest rate, and the cost-push shock

˜µt

shows low persistence. For the

sake of completeness, I smooth the structural shocks in appendix E.

4

Why Did the UK Become Underemployed?

4.1

True vs. Desired Hours

The underemployment index in gure 1 records how the UK has suered from severe underemployment since 2008. From that picture it remains unclear whether the enormous increase in underemployment was caused by changes in desired hours or rather by changes in actually worked hours. Remember the underemployment

zt

equals the

dierence between the desired and the true hours:

h

ˆm zt = ˆ m t − ht . I can disentangle the contribution of the two variables by the Kalman lter. Given the model and the data, which contains among others the growth rate of average hours and the underemployment index, I smooth the true hours per worker hours per worker

ˆ m and the desired h t

hˆ mt. In gure 5, I plot the smoothed series together.

At the beginning of the Great Recession, a fall in the true hours resulted into high underemployment. The desired hours, in contrast, oscillated around the steady state. In 2011, the true hours started to recover. However, at the same time the desired hours moved above the steady state. The underemployment therefore didn't vanish.

22

Prior

Posterior

Parameter

Distribution

Mean

StD

Mode

Mean

σ ϕ ΞL γ ξ ΞN γw ω φ2 / (φ1 γw ) (φ3 uγu ) / (em γw ) ψi ψΠ ψu ρβ ρAn ρκ ρµ ρAm ρN ρe 100σβ 100σAn 100σκ 100σµ 100σAm 100σN 100σe 100σi

beta gamma gamma beta beta gamma beta beta beta gamma beta normal gamma beta beta beta beta beta beta beta invgamma invgamma invgamma invgamma invgamma invgamma invgamma invgamma

0.500 2.000 2.000 0.667 0.150 2.000 0.800 0.500 0.700 0.500 0.600 1.800 0.300 0.700 0.700 0.700 0.700 0.700 0.700 0.700 2.000 2.000 2.000 0.500 2.000 2.000 1.000 0.250

0.150 0.700 0.700 0.100 0.100 0.700 0.150 0.200 0.150 0.300 0.150 0.200 0.200 0.150 0.150 0.150 0.150 0.150 0.150 0.150 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000

0.802 5.352 1.041 0.912 0.023 4.290 0.742 0.679 0.407 0.256 0.912 1.499 0.740 0.932 0.937 0.855 0.218 0.811 0.660 0.732 2.868 1.129 2.131 0.406 1.532 1.012 1.802 0.128

0.753 5.378 1.213 0.915 0.051 4.358 0.754 0.599 0.411 0.387 0.910 1.526 0.777 0.926 0.926 0.842 0.227 0.800 0.651 0.728 3.046 1.232 2.209 0.419 1.587 1.071 1.860 0.134

90% HPD 0.591 4.004 0.530 0.879 0.001 2.963 0.567 0.286 0.214 0.055 0.880 1.177 0.516 0.892 0.873 0.757 0.101 0.717 0.517 0.633 2.027 0.881 1.862 0.343 1.334 0.681 1.536 0.109

0.927 6.796 1.905 0.950 0.101 5.728 0.948 0.911 0.612 0.697 0.943 1.864 1.059 0.963 0.982 0.931 0.350 0.885 0.790 0.821 4.112 1.571 2.565 0.496 1.845 1.489 2.148 0.158

Table 3: Priors and Posteriors

4.2

Weak Demand as Possible Explanation

Economic intuition would suggest that weak demand could possibly explain the rise in underemployment during the Great Recession. Smoothed demand shocks

νtβ , which are

sizably negative at the end of 2008 and the beginning of 2009, point to the intuitive explanation. I present impulse responses to a negative demand shock in gure 6 and describe in the following paragraphs how weak demand leads to underemployment. The negative demand shock forces the household to reduce its market consumption

Cˆtm .

The output

Yˆt

drops because it equals the market consumption up to the rst

order. The rms decrease the workforce

ˆt N

and shorten the hours per worker

lower workforce raises the unemployment rate

23

uˆt .

ˆ m. h t

The

The central bank reacts to the higher

3 2

%

1 0 −1 −2 −3

True Hours per Worker Desired Hours per Worker 2002

2004

2006

2008

Figure 5: Smoothed True Hours per Worker 100 ˆ m t : 90% HPD Interval.

h

2010

ˆm 100h t

unemployment rate by cutting the interest rate

2012

2014

and Desired Hours per Worker

ˆit .

The wage equation translates the shorter hours and the higher unemployment into lower hourly wage

wˆt .

The workers weigh the lower wage against the higher unemploy-

ment and decide on the grounds of reciprocity to deliver less eort The labor-force participation rate

ˆt L

eˆm t .

increases. The rst-order condition with re-

spect to the labor force expresses the rationale for the increase:

   ΞL  ˆ ˆ t−1 + βΞL Et L ˆ t+1 − L ˆ t + lswˆt − δY Yˆt Lt − L LY LY n m ˆ − (1 + ϕ) lsˆ + δu uˆt − ϕlsh em ˜A κt . t t − δAn  t − lsϕ˜

ˆt = − δL L

I dene the parameters

δ in appendix A. The lower wage, which prevails in the economy

after the negative demand shock, demotivates from entering the labor force. However, the household experiences an additional eect. less painful.

Staying in the labor force becomes

The higher unemployment, shorter hours, and lower eort decrease the

disutility from the eective hours on the market

˜ m. H t

The household nds the labor

force more attractive and sends additional people on the market. The economy suers from higher unemployment, lower eort, and lower output. In such a situation the desired hours per worker

hˆ mt increase despite of the lower wage

and higher labor force. People desire more hours but in reality work less hours. The underemployment

zt

inevitably occurs.

24

Output 100Yˆt

0.5

0

−0.5

−0.02 −0.04

−1.5

5

10

15

−0.06

20

ˆt Labor Force 100L

0.3

0.6

pp

0.2 0.1 0

5

10

15

−0.2

20

Unemployment Rate 100uˆt

0.1

0.4

0

0.2

−0.1

0

5

10 Effort

0.5

15

−0.2

20

100ˆ em t

0.5

%

%

5

10

15

−0.3

20

Hourly Wage 100wˆt

1.5

0

1

−0.5

0.5

0

−0.5

−1

−1

5

10

15

20

−1.5

5

10

15

20

Hours per Worker 100ˆhm t

−0.2

%

−0.1

−0.1 −0.15

%

−1

Interest Rate 100ˆit

−0.05 pp

0 pp

%

0

pp

ˆt Inflation Rate 100Π

0.02

5

10

15

20

Underemployment 100zt

0

5

10

15

−0.5

20

Figure 6: Impulse Responses to Negative Demand Shock

νtβ

5

10

15

20

(One Standard Deviation):

90% HPD Intervals.

Notice the limited deationary pressure after the negative demand shock. Ination in the model obeys the standard Phillips curve:

  ˆ t − ξΠ ˆ t−1 = β Et Π ˆ t+1 − ξ Π ˆ t + (1 − γβ) (1 − γ) Π vˆt + ˜µt . Π γ In appendix B, I transform

vˆt = wˆt + The eort

eˆm t ,

µt

into

˜µt .

The marginal costs

vˆt

behave according to:

2 (3 − 2α) ˆ m Am ˆ em ht − αˆ t + (1 − α) Nt − t . α

which in equilibrium comoves with the wage

wˆt

and the hours

ˆ m, h t

stabilizes the marginal costs and keeps ination close to the target. If I substitute for

25

eˆm t

by the eort condition (5), I obtain:

vˆt = [1 − α (1 − γw )] wˆt + Because

(1 − α) [6 − α (4 − α)] ˆ m ˆt − Am . ht + (1 − α) N t α

α lies close to 0.9, the wage together with the hours and the workforce has only

a limited impact on the marginal costs and consequently on the ination. Instead, the market productivity

4.3

A t

m

and the cost-push shock

˜µt

drive the ination in the model.

Mitigation by Monetary Policy

Economists generally believe that monetary expansion can free the labor market from slack in the extensive margin. However, it is an open question whether expansionary policy can also mitigate slack in the intensive margin.

As gure 7 demonstrates, a

negative interest-rate shock indeed eliminates underemployment. The model propagates the negative interest-rate shock shock initiates a sluggish response of the marginal costs ination

ˆt Π

νti in the following way. The vˆt . A minor reaction of the

arises. The small reaction of ination after the interest-rate shock agrees

with the VAR ndings of Christiano, Eichenbaum and Trabandt (2016). The lower interest rate

ˆit

stimulates the output

ˆ m and hire more workers hours h t rate

ˆt . N

Yˆt .

The rms extend the working

The higher workforce lowers the unemployment

uˆt .

The tighter labor market raises the hourly wage higher wage with supplying higher eort

eˆm t .

wˆt .

The workers reciprocate the

Because the workers deliver more eort and

work longer hours, the household perceives the labor market as painful. The household avoids the pain by choosing smaller labor force

ˆ t. L

The economy, which now produces bigger output, experiences lower unemployment and higher eort. These circumstances lead to a drop in the desired hours per worker

hˆ mt.

People desire shorter hours but in reality work longer hours. Overemployment

zt

appears.

4.4

Shock Decomposition

In gure 8, I carry out shock decomposition of underemployment

100zt .

The decom-

position reveals which shocks increased, and which shocks decreased underemployment during the Great Recession.

26

Output 100Yˆt

0.6

0.01

0

0

5

10

15

ˆt Labor Force 100L

0.1

0

−0.05 −0.1

−0.01

20

Interest Rate 100ˆit

0 pp

0.2

0.05

0.05

0.02 pp

%

0.4

−0.2

ˆt Inflation Rate 100Π

0.03

5

10

15

−0.15

20

Unemployment Rate 100ˆ ut

0.15

0

0.1

−0.1

0.05

5

10

15

20

Hours per Worker 100ˆhm t

%

pp

pp

−0.05 −0.1 −0.2

−0.15 5

10

15

−0.3

20

Effort 100ˆ em t

0.6

0.8

0.2

15

5

10

15

20

−0.05

20

Hourly Wage 100wˆt

0.2

0.6

0

0.4

−0.2

0.2

−0.4

0

−0.6

0 −0.2

10

%

%

0.4

5

5

10

15

20

Underemployment 100zt

%

−0.2

0

−0.2

5

10

15

20

−0.8

Figure 7: Impulse Responses to Negative Interest-Rate Shock

5

νti

10

15

20

(One Standard Devi-

ation): 90% HPD Intervals.

In essence, two shocks were responsible for the rise in underemployment: the demand shock

νtβ

and the workforce shock

νtN .

I have already shown how weak demand in the

β form of a negative νt results into underemployment. A positive workforce shock leads to underemployment as well. One can interpret the workforce shock as a taste shock. If a positive

νtN

realizes, rms prefer workforce to hours.

people desire longer hours.

The hours per worker fall;

Underemployment consequently arises.

In appendix F, I

νtN . κ The labor-supply shock νt had, in contrast, a negative eect on underemployment. κ Positive realizations of νt made the labor market more painful. In other words, the

plot impulse responses to

eective hours on the market brought more disutility. In such a situation, people desired shorter hours and felt overemployed. Appendix F contains impulse responses to

27

νtκ .

Initial Values

6

Monetary Policy νti

4

Effort νte

2

Workforce νtN

0

Market Productivity νtA

%

8

m

−2

Cost Push νtµ

−4

Labor Supply νtκ

−6

Home Productivity νtA

n

Demand νtβ

−8 2002

Figure 8:

2004

2006

2008

2010

2012

2014

Shock Decomposition of Underemployment

100zt

at the Posterior Mean.

2001:Q22015:Q3.

At the beginning of the Great Recession, the BoE reacted by monetary expansion. The loose monetary policy translated into negative interest-rate shocks

νti ,

which

curbed the increase in underemployment. Later on, in 2011 and 2012, the three-month interbank rate was rising back to 1%.

This development came along with positive

interest-rate shocks, which kept underemployment high.

5

Robustness Check: Hours Adjustment Costs

In addition to workforce adjustment costs, rms could suer from hours adjustment costs.

The initial rms would pay quadratic adjustment costs whenever they would

choose hours per worker dierent from the previous quarter. An initial rm concretely solve the following optimization problem.

max

∞ m {wτ (j),em τ (j),hτ (j),Nτ (j),Xτ (j)}τ =t

Et

∞ X

β τ −t

τ =t

28

λτ [vτ Xτ (j) − wτ (j)hm τ (j)Nτ (j) λt  
2 Nτ (j) ΞN N − exp τ − 2 Nτ −1 (j)  2 # Ξh hm τ (j) − −1 2 hm τ −1 (j)

j

would

s.t. (1)

Xτ (j) =

Am τ



 δ2 m δ1 m α 2 3 (hτ (j)) − (hτ (j)) [em τ (j)Nτ (j)] 2 3

After I introduce hours adjustment costs into the optimization problem, I have to consequently replace four equilibrium conditions. The rst-order condition with respect to hours (3) has to now read

vt Am t

 m hm ht+1 (j) t+1 (j) −1 m ht (j) (hm (j))2  m t  ht (j) Ξh −1 . = wt (j)Nt (j) + m ht−1 (j) hm t−1 (j)

 m λt+1 2 m α δ1 ht (j) − δ2 (hm Ξh t (j)) [et (j)Nt (j)] + Et β λt



The eort condition (5) and the wage equation (6) have to reect the new adjustment costs:

em t

=

δ1 m h αφ1 2 tm δ1 ht

−

−

δ2 3

2 (hm t )

w1−γw m 2 t δ2 (ht )

  m  1 ht Ξh  m  m α−1 α m + −1 m 2 vt Am Nt ht−1 hm t−1 t δ1 ht − δ2 (ht ) (et )  m   m ht+1 ht+1 λt+1 −Et β Ξh −1 2 , m λt ht (hm t ) " wt1−γw

2 δ1 m h − δ32 (hm φ1 t ) 2 t − αφ1 m 2 1 − γw δ1 hm t − δ2 (ht )

# =

−φ0 (1 − γw ) + φ1 − φ2 1 − γw

φ2 1−γw −ω(1−γw ) wt−1 Πt 1 − γw φ3 u − u1+γ − et t 1 + γu 1   m α−1 α + m m 2 vt Am Nt t δ1 ht − δ2 (ht ) (et )   m  Ξh ht × m −1 ht−1 hm t−1  m  m  ht+1 ht+1 λt+1 −Et β Ξh −1 2 . m λt ht (hm t )

+

29

Finally, the hours adjustment costs create additional demand for the nal good in (7).

Yt =

Ctm

ΞL + 2



 2   2
2 Ξh hm Lt ΞN Nt t N + −1 + − exp t −1 Lt−1 2 Nt−1 2 hm t−1

Appendix G shows what the hours adjustment costs mean for the linearized model. I take the model with the new adjustment costs to the data. I calibrate the same parameters and set the same priors like in the model without hours adjustment costs. Table 4 and gure 9 reveal that the new parameter of hours adjustment costs eectively equals zero.

The other estimated parameters resemble the estimation results from

Prior Ξh

Posterior

Distribution

Mean

StD

Mode

Mean

gamma

2.000

0.700

0.071

0.094

90% HPD 0.057

0.132

Table 4: The Parameter of Hours Adjustment Costs

Ξh 18

Prior Posterior Mode

16 14 12 10 8 6 4 2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 9: The Parameter of Hours Adjustment Costs

section 3. In table 5, I compare log marginal data densities of two modelswith and without hours adjustment costs. The model without hours adjustment costs clearly outperforms

30

Model

Log Marginal Data Density

without hours adjustment costs with hours adjustment costs

−573.259 −599.520

Table 5: Log Marginal Data Densities (Modied Harmonic Mean Estimator)

the model with hours adjustment costs by 26 log points.

The model without hours

adjustment costs describes the reality more appropriately than the model with hours adjustment costs.

6

Distribution of Dissatisfaction

Because I concentrate on the net slack in the intensive margin, I don't model heterogeneity of workers. All workers supply the same number of hours to supply the same number of hours

h

hm t ;

all workers desire

m t . Either everybody is underemployed or every-

body is overemployed. So the model eectively describes the situation of an average worker. Of course, in reality underemployed and overemployed workers coexist. Some workers are satised with the hours; some workers want to slightly adjust; others desire a sizable change. My model doesn't specify the distribution of dissatisfaction among the workers. I entirely focus on the net dissatisfactionwhether underemployment or overemployment dominates, and how large the dominance is. I leave the distribution of dissatisfaction for future research.

7

Conclusion

In this paper, I investigate time-related underemploymenta situation in which people are employed but desire longer hours.

I use a macroeconomic model which fully

describes the labor market with all its margins. The model explicitly considers labor force, employment, hours, and eort. I estimate the model with UK data and present how underemployment responds to dierent shocks. The paper also explains why the United Kingdom experienced a sizable increase in underemployment during the Great Recession. The main reason for the increase was weak demand. Firms were forced to lay o workers and to shorten hours. But workers, who faced high unemployment and

31

experienced low consumption, desired to work longer hours. Underemployment had to consequently arise. A mitigating eect came from the monetary expansion of the Bank of England, which prevented an even bigger increase in underemployment during the Great Recession.

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34

speech

at

Yellen, Janet L.

2014.

Stronger Job Market.

What

the

Federal

Reserve

Is

Doing

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Promote

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speech at the 2014 National Interagency Community Rein-

vestment Conference, Chicago, Illinois, March 31, 2014.

35

A

Parameters ιr ιL ιβ

ιAn ηw ηu ηem ηC m ηL ηAn δL

δY

δu δAn

ζY σ + (1 − ζ) (1 − L)σ = ζY σ + (1 − σ) (1 − ζ) (1 − L)σ σ (1 − ζ) (1 − L)σ−1 = ζY σ + (1 − σ) (1 − ζ) (1 − L)σ ζY σ + (1 − ζ) (1 − L)σ = ζY σ + (1 − σ) (1 − ζ) (1 − L)σ ζY σ + (1 − ζ) (1 − L)σ = ζY σ + (1 − σ) (1 − ζ) (1 − L)σ uhu eu + (1 − u)hm em = ϕ(1 − u)hm em hm em − hu eu = (1 − u)hm em uhu eu + (1 − u)hm em =1+ ϕ(1 − u)hm em [ζY σ + (1 − ζ) (1 − L)σ (1 − σ)] [uhu eu + (1 − u) hm em ] = [ζY σ + (1 − ζ) (1 − L)σ ] ϕ (1 − u) hm em  ϕ [ζY σ + (1 − ζ) (1 − L)σ ] − (1 − ζ) σ (1 − L)σ−1 L [uhu eu + (1 − u) hm em ] = [ζY σ + (1 − ζ) (1 − L)σ ] Lϕ (1 − u) hm em uhu eu + (1 − u) hm em = ϕ (1 − u) hm em (1 − ζ)2 (1 − L)2σ−2 L (1 + ϕ) = ζY σ [ζY σ + (1 − ζ) (1 − L)σ ]   (1 − ζ) (1 − L)σ−1 (1 − u) L (1 − L)−1 (1 + ϕ − σ) − ls (1 − u + χu) σ + (1 − u) [ζY σ + (1 − ζ) (1 − L)σ ] lsuhu eu ϕ + lshm em [ϕ (1 − u) (1 − L) − χϕuL] + hm em (1 − u) L (1 − L) ls (1 − u + χu) [ζY σ + (1 − ζ) (1 − σ) (1 − L)σ ] = (1 − u) [ζY σ + (1 − ζ) (1 − L)σ ] (1 − ζ) σL (1 − L)σ−1 − ζY σ + (1 − ζ) (1 − L)σ ls [(χ + ϕ) hm em − (1 + ϕ) hu eu ] = (1 − u) hm em ls (1 − u + χu) (1 − L) + L (1 − u) = (1 − u) (1 − L)

36

B

The Linearized Model

Shadow Price of Wealth ζY σ + (1 − σ) (1 − ζ) (1 − L)σ ˆ σ (1 − ζ) (1 − L)σ−1 ˆ n β ˆ λt = − Yt + ˜A σ σ Lt + t −  t σ σ ζY + (1 − ζ) (1 − L) ζY + (1 − ζ) (1 − L)

Euler Equation

ˆ t = Et λ ˆ t+1 + β ˆit − 1 Et Π ˆ t+1 λ Π Π

Labor Force     ˆt − L ˆ t−1 + βΞL Et L ˆ t+1 − L ˆ t + lswˆt − δY Yˆt ˆ t = − ΞL L δL L LY LY n m ˆ + δu uˆt − ϕlsht − (1 + ϕ) lsˆ em ˜A κt t − δAn  t − lsϕ˜

Production Function Am ˆ m + αˆ ˆ Yˆt = αh em t t + αNt + t

Hours per Worker 2 (3 − 2α) ˆ m Am ˆt + αˆ ht = vˆt − wˆt − (1 − α) N em t + t α

Workforce    ΞN ΞN  ˆ ˆm ˆ ˆ ˆ ˆ Nt − Nt−1 = β Et Nt+1 − Nt + vˆt − wˆt − (1 − α) h (1 − α) Nt + t lsY lsY ΞN N Am + αˆ em + (1 − βρN )  t + t lsY t

Eort Condition eˆm ˆt + t = (1 − γw ) w

(3 − α) (2 − α) ˆ m ht α

Wage Equation wˆt =

(3 − α) (2 − α) ˆ m φ2 −ω(1−γw ) φ2 −ω(1−γw )−1 ˆ φ3 uγu 1 ht + Π wˆt−1 −ω Π Πt − m uˆt − m et αγw φ1 γw φ1 γw e γw e γw

37

Phillips Curve   (1 − γβ) (1 − γ) Π ˆ ˆ ˆ ˆ vˆt + ˜µt Πt − ξ Πt−1 = β Et Πt+1 − ξ Πt + γ

Taylor Rule ˆ t − (1 − ψi ) ψu uˆt + σi ν i ˆit = ψiˆit−1 + (1 − ψi ) ψΠ Π t

Unemployment Rate uˆt =

1−uˆ ˆt Lt − (1 − u) N L

Desired Hours

hˆ mt = ηw wˆt + ηuuˆt − ηe

m

Underemployment

n

ˆ ˆ eˆm ˜A ˜κt t − ηC m Yt − ηL Lt − ηAn  t −

h

ˆm zt = ˆ m t − ht

Shocks βt = ρβ βt−1 + σβ νtβ n

n

A ˜A = ρAn ˜A t t−1 + σAn νt

n

˜κt = ρκ ˜κt−1 + σκ νtκ ˜µt = ρµ ˜µt−1 + σµ νtµ m

m

A A = ρAm A t t−1 + σAm νt N N N t = ρN t−1 + σN νt

et = ρe et−1 + σe νte

38

m

νtβ ∼ N (0, 1) n

νtA ∼ N (0, 1) νtκ ∼ N (0, 1)

νtµ ∼ N (0, 1) m

νtA ∼ N (0, 1) νtN ∼ N (0, 1) νte ∼ N (0, 1)

νti ∼ N (0, 1)

Redened Shocks σ (1 − ζ) (1 − L)σ An  σ ζY σ + (1 − ζ) (1 − L) t uhu eu + (1 − u) hm em κ t ˜κt = ϕ (1 − u) hm em (1 − γβ) (1 − γ) µΠ µ t ˜µt = γ (1 − µ) n

˜A = t

39

C

Data ID

Description

Transformation

Observable

IHXWQ@UK

real GDP per capita

GDPt

YBGBQ@UK

GDP deator

UNAMIJ@UK MGWGQ@UK MGSXQ@UK YBUVQ@UK

3-month sterling interbank rate economic-activity rate: age 16+ unemployment rate: age 16+ average actual weekly hours worked

IHXPQ@UK IHXQQ@UK

compensation of employees as % of GDP mixed income as % of GDP Bell-Blanchower underemployment index

log-dierenced x100 demeaned log-dierenced x400 quarterly average none none log-dierenced x100 sum none

U DIt

none

Table

6:

Data.

2001:Q22015:Q3.

Seasonally

adjusted.

DEFt BOEt LF P Rt U N Rt HOU RSt LSt

I

downloaded

http://www.theworkfoundation.com/Datalab/ The-BellBlanchflower-Underemployment-Index. All the remaining data comes from

the underemployment index from Haver.

40

D

Priors and Posteriors σ

4

ΞL

ϕ

ΞN

1

0.6

0.6

0.8

3 0.4

0.4

0.2

1

0.4

0.6

2

0.2

0.2

0

0 0.2

0.4

0.6

0.8

1

1.2

0 2

4

6

8

0

φ2/(φ1 γ w )

γw

0

10

2

4

6

2

2

1

1

0

0 0.2

0.4

0.6

0.8

1

1.2

γ

20

0.2 0.4

1

1

0.5

0.5

1

0

0.5

ξ

15

15

0.6 0.8

0 −0.5

1

0

0.5

10

8

10

1

1.5

2

2

2.5

0.8

1

ψΠ

ψi

10

6

1.5

0 0

4

2

1.5 2

2

(φ3uγu )/(em γ w)

3

3

0

ω

20

2

15

1.5

10

1

5

0.5

5

5 0

0 0.4

0.6

0.8

1

0 0

0.1 0.2 0.3 0.4 0.5

ψu

0 0.2

0.4

0.6

ρβ

0.8

1

0.5

1

1.5 ρ An

ρ Am 8

2.5 2

15

1.5

10

4

5

2

10

6

1

5

0.5 0

0 0

0.5

1

1.5

0

2

0.4

0.6

ρe

0.8

0.6

4 2 0 0.8

4

3

3

2

2

1

1

0

0 0

0.2

0.4

100σi

0.6

0.8

15

0.6 ρκ

4 2 0 0.2

0.4

0.6

0.8

1

0.4

0.6

2.5

2

2

1.5

1.5

0.4

0.8

1

100σAn

100σAm

0.6

20

0.4

6

100 σβ

25

1

5

4

1

0.8

ρN

5

0.6

0.4

ρµ

6

0.4

0

1

1

1

10

0.2

5 0

0 0.2 0.4 0.6 0.8

1

1.2 1.4

0 2

4

2

8

1.5

6

1

4

0.5

2

0

0 2

3

6

8

0

10

2

4

100 σµ

100σe

1

0.5

0.5

4

5

6

8

10

100σN

1.5

2

2.5

6

8

10

8

10

100σκ

1.5

1

1

0.5

0.5

0 1

4

2

1.5

0.5

2

0 0

2

4

6

8

10

2

4

6

Figure 10: Priors and Posteriors. Grey lines depict priors; black lines depict posteriors; red vertical lines depict modes.

41

E

Smoothed Shocks

2

n

Demand νtβ

4

0

2

−2

0

−4

−2

−6

4

2005

2010

Labor Supply

−4

2015

νtκ

4

2

2

0

0

−2

−2

−4

4

2005

2010

Market Productivity

−4

2015 Am νt

4

2

2

0

0

−2

−2

−4

2005

4

2010 Effort νte

−4

2015

5

2

Home Productivity νtA

2005

2010

2015

2005

2010

2015

Cost Push νtµ

Workforce

2005

νtN

2010

Monetary Policy

2015

νti

0

0 −5

−2 −4

2005

2010

−10

2015

2005

2010

Figure 11: Smoothed Shocks: 90% HPD Interval.

42

2015

F

Additional Impulse Responses Output 100Yˆt

0

0.05

−0.2

ˆt Inflation Rate 100Π

0.06

0

Interest Rate 100ˆit

0.04 pp

%

pp

−0.4 −0.05

0.02

−0.6 −0.1

0.3

5

10

15

−0.15

20

ˆt Labor Force 100L

0.4

pp

pp

0.2 0.1 0 −0.1

5

10

15

20

15

−0.02

20

Unemployment Rate 100uˆt

0.4

0.2

0.2

0

0

−0.4

Effort 100ˆ em t

1

0

5

10

15

−0.4

20

Hourly Wage 100wˆt

1.5

5

10

15

20

−2

15

20

Hours per Worker 100ˆhm t

5

10

15

20

Underemployment 100zt

0.5

−1

−1

10

1

0

−0.5

5

−0.2

%

%

10

−0.2

0.5

−1.5

5

%

−1

0

%

−0.8

0

5

10

15

20

Figure 12: Impulse Responses to Positive Workforce Shock tion): 90% HPD Intervals.

43

−0.5

νtN

5

10

15

20

(One Standard Devia-

Output 100Yˆt

0.5

0.01

ˆt Inflation Rate 100Π

0.06

0 0

0.04 pp

%

pp

−0.01

0

−0.03

0

0.02

−0.02

−0.5

−1

Interest Rate 100ˆit

5

10

15

−0.04

20

ˆt Labor Force 100L

0.2

−0.1

5

10

15

−0.02

20

Unemployment Rate 100uˆt

0.05

5

10

15

20

Hours per Worker 100ˆhm t

0

0 −0.2

%

pp

pp

−0.05 −0.2

−0.1 −0.3 −0.4

−0.4

5

10

15

−0.6

20

Effort 100ˆ em t

0.2

0.2

−0.2

10

15

−0.4

−0.4

−0.6

−0.6

−0.8

5

10

15

20

−0.2

20

Hourly Wage 100wˆt

0

−0.8

5

10

15

20

Underemployment 100zt

−0.5 %

−0.2

5

%

0

%

0

−0.15

−1 −1.5

5

10

15

20

−2

5

10

Figure 13: Impulse Responses to Positive Realization of Labor-Supply Shock

15

νtκ

20

(One

Standard Deviation): 90% HPD Intervals.

G

Robustness Check: Hours Adjustment Costs

Hours per Worker   2 (3 − 2α) ˆ m Ξh ˆ m ˆ m  Ξh ˆm − h ˆ m + vˆt − wˆt − (1 − α)N ˆt ht + ht − ht−1 = β Et h t+1 t α lsY lsY Am + αˆ em t + t

Eort Condition eˆm ˆt + t = (1 − γw )w

  Ξh ˆ m ˆ m  Ξh (2 − α)(3 − α) ˆ m ˆm − h ˆm ht + ht − ht−1 − β Et h t+1 t α lsY lsY 44

Wage Equation   Ξh ˆ m ˆ m  Ξh (3 − α) (2 − α) ˆ m m m ˆ ˆ ht + ht − ht−1 − β Et ht+1 − ht wˆt = αγw γw lsY γw lsY φ2 −ω(1−γw )−1 ˆ φ3 uγu 1 φ2 −ω(1−γw ) Π wˆt−1 − ω Π Πt − m uˆt − m et + φ1 γw φ1 γw e γw e γw

45