Essays in Labor Economics - University of Minnesota

Essays in Labor Economics

A Dissertation Submitted to the Faculty of the Graduate School of the University

of Minnesota

by Edgar Preugschat

In Partial Ful�llment of the Requirements for the Degree of

Doctor of Philosophy

Thomas J. Holmes

August 2008

c© Edgar Preugschat 2008

Acknowledgments

First, I would like to thank my advisor Tom Holmes for his continuous advice and support.

Uncountable meetings with him provided me with constant feedback and new ideas, and

moreover, they set the right incentives to get things done much faster.

For helpful comments and good suggestions I am very thankful to V.V. Chari, Larry

Jones, Timothy Kehoe, and Morris Kleiner.

Many thanks also to Javi Fernández-Blanco for frequent and insightful discussions, and

for his kindness and friendship.

I had the chance to present both in the Applied Micro Workshop and the Macro Work-

shop at the University of Minnesota. I am thankful for the feedback I received there.

Also, I am thankful to all the people who gave me comments and feedback during my

job market seminar and conference presentions.

I am grateful to the Department of Economics and the University of Minnesota for

providing me with teaching opportunities during my time as a PhD student.

Without my parents this Minnesota adventure wouldn't have been possible, I am very

thankful for their support. Finally, if it wasn't for Linde Götz I probably would never have

found this place. Ironically and sadly, I had to live here without her for the past four years

- and she in Germany without me. I thank her for staying so close to me nevertheless.

i

Abstract

The �rst essay investigates what role the interaction of �rm turnover on the one hand and

costly union organizing on the other might play in explaining the wide variation of union-

ization outcomes in the US both across sectors and states as well as over time. The paper

develops a model that combines an entry-exit framework of monopolistically competing �rms

with costly union organizing. The model is analyzed both for the case of an �e�ciently�

bargaining and a wage-setting union.

Firm turnover is a crucial determinant of the unionization rate in the US because entering

�rms are typically born as non-union and have to �rst be organized by unions. Moreover, the

union's �rm share usually diminishes only through exit of unionized �rms. In the model, the

unionization rate also depends on the equilibrium interaction of �rm entry with the union's

organizing decision: Higher union organizing deters �rm entry, and higher entry lowers the

incentives for organizing.

The results show that the steady state unionization rate is higher if 1. �rm entry costs

are higher, 2. exit rates are lower, and 3. organizing costs are lower. Further, the transition

dynamics of the model support two explanations of the long-term union decline in the US:

First, an increase in the cost of organizing, and secondly, deregulation understood as a

decrease in the cost of �rm entry.

The second essay analyzes investment in speci�c, non-substitutable skills under demand

uncertainty as a channel of labor market mismatch that doesn't rely on a matching function.

ii

iii

Education in speci�c skills such as a college program is an investment with delay. When

deciding about what skill to invest in the worker has to predict the demand prevailing at

the time of degree. A model with exogenous demand and exogenous wages is constructed

that features labor market rationing due to demand uncertainty, the investment lag and

the assumption of non-substitutability between skills. Both an individual choice problem

and an equilibrium version with endogenous entry into education types are studied. The

results indicate that longer duration of degree programs as well as lower demand persistence

increases labor market mismatch.

Contents

Contents iv

List of Tables vi

List of Figures viii

1 Unionization Rate, Organizing, and Firm Turnover 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related Literature and Empirical Findings . . . . . . . . . . . . . . . . . . . 6

1.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Agents' Static Maximization Problems . . . . . . . . . . . . . . . . 11

1.3.3 Dynamics of Entry, Exit, Unionization, and Wage Setting . . . . . . 14

1.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Steady States and Comparative Statics . . . . . . . . . . . . . . . . . . . . . 21

1.5.1 E�cient Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5.2 Unilateral Wage-Setting . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.6 Transition Dynamics within a Sector . . . . . . . . . . . . . . . . . . . . . . 36

1.6.1 Organizing Environment and Union Decline . . . . . . . . . . . . . . 37

iv

CONTENTS v

1.6.2 �Deregulation� and Union Decline . . . . . . . . . . . . . . . . . . . . 39

1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Time-To-Degree and Labor Market Mismatch 57

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.2.1 Individual Decision Problem . . . . . . . . . . . . . . . . . . . . . . . 61

2.2.2 A Partial Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . 64

2.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.2.4 Preliminary Empirical Evidence . . . . . . . . . . . . . . . . . . . . . 73

2.2.5 Possible Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Bibliography 79

List of Tables

1.1 Benchmark Parameters for E�cient Bargaining Case . . . . . . . . . . . . . 49

1.2 Comparative Statics for e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.3 Comparative Statics for r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.4 Comparative Statics for w . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.5 Comparative Statics for w̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.6 Comparative Statics for net premium 100[wnet/w̄)− 1

]. (Gross premium

= 2.63 for all (ε, η, δ)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.7 Comparative Statics for Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.8 Comparative Statics for V u . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.9 Comparative Statics for V n . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.10 Benchmark Parameters for Monopoly Union Case . . . . . . . . . . . . . . . 51

1.11 Comparative Statics for r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.12 Comparative Statics for s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.13 Comparative Statics for w/w̄ . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.14 Comparative Statics for w . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.15 Comparative Statics for e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.16 Comparative Statics for Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.17 Comparative Statics for V u . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.18 Comparative Statics for V n . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

vi

LIST OF TABLES vii

1.19 Comparison of �xed vs. endogenous entry of the elasticity of s w.r.t. η. . . 53

1.20 Benchmark Parameters (initial steady state) for transition dynamics . . . . 53

2.1 Parameter Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.2 Average employment probability for a new entrant for di�erent values of γ

and time to degree (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.3 Employment and Wage Heterogeneity (n = 0 and wA = 2wB) . . . . . . . . 72

List of Figures

1.1 Reaction Functions for Entry and Organizing . . . . . . . . . . . . . . . . . 53

1.2 Transition after change in ε or η: unionization rate r . . . . . . . . . . . . . 54

1.3 US Unionization Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.4 Transition after change in ε or η: Wage Premium . . . . . . . . . . . . . . . 54

1.6 Transition after change in ε or η: organizing rate s . . . . . . . . . . . . . . 55

1.5 US Wage Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1.7 US organizing rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.1 Relative enrollment by year, Germany, selected Subjects (Federal Statistical

O�ce Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.2 Relative number of Bachelor degrees, selected years, USA, selected subjects

(Statistical Abstract of the US, 2006) . . . . . . . . . . . . . . . . . . . . . . 75

viii

Chapter 1

Unionization Rate, Organizing, and

Firm Turnover

1.1 Introduction

This paper develops a model of union organizing dynamics in an environment of �rm

turnover to understand the large variations of private sector unionization rates in the US

both across sections and over time. The unionization rate (as well as the union wage pre-

mium) in this model is shaped by the endogenous interaction of �rm turnover on the the one

hand and costly union organizing on the other. Both the case of union with monopoly power,

and an e�ciently bargainin union are considered. Technically speaking, the paper combines

an entry-exit framework of monopolistically competing �rms with a wage-setting (either

through bargaining or monopoly power) and organizing union within a general equilibrium

model.

The fraction of private sector workers in the US belonging to a union di�ers signi�cantly

both across sectors and states, and has changed tremendously over time. The unionization

rate peaked in 1953, when about 36 % of the work force belonged to a union, and gradually

1

CHAPTER 1. UNIONIZATION RATE, ORGANIZING, AND FIRM TURNOVER 2

declined to less than 8 % in 2006 (see also �gure 1.3). Comparing di�erent industries, huge

variations of the unionization rates can be observed at all times. For instance, the current

rate in manufacturing is more than 11%, whereas in the sector of hotels and restaurants

it is less than 3 %. Moreover, di�erent states in the US (with similar workforces - but

di�erent labor legislations) di�er strongly: California has around 17 % and Texas less than

6 % unionized workers.

At the same time, as it is well-known, there is signi�cant �rm turnover in the US. For

the more recent years the aggregate rate of annually exiting (entering) establishments is

more than 11 % (13 %) of all establishments, and the corresponding rate of jobs lost due

plant closings (openings) is around 5 % (5 %).1 Firm turnover is a crucial determinant

for the unionization rate in the US. Firm entrants are typically born as non-union, so that

unions have to always organize incoming �rms.2 This process is not frictionless because

unions have to spend resources on initiating and implementing certi�cation elections, and

in addition have to overcome resistance by employers. On the out�ow side, unions rarely lose

members through union �rms that become non-union �rms. This implies that higher �rm

exit ceteris paribus lowers the rate of �rms unionized. A strength of the proposed model is

that it captures this speci�c environment of union organizing in the US. The model, however,

goes beyond the pure �ow mechanics of turnover of �rms and unions by endogenizing both

�rm entry and union organizing. In particular, the unionization rate is the outcome of a

game between the organizing union and the potential (aggregate) �rm entrant. The optimal

response of the union to higher entry is to organize less, whereas optimal entry is lower if the

union decides to organize more. Higher �rm entry both requires more organizing and thus

increases costs, and -in the case of a monopoloy union - also lowers the �rm's optimal labor

demand due to lower pro�ts. This leads to a lower optimal organizing response. From the

1See Pinker and Spletzer [2004].2There are some exceptions, for example in the auto industry, where it has happened that newly set up

plants became organized through negotiations by the unions of the mother company with its management.


entrant's perspective, higher organizing increases the threat of becoming unionized which

would lower pro�ts.

The main ingredients of the model are union organizing and entry and exit of �rms.

Unions in the model maximize net revenue, which is the number of union members times

the mark up over the non-union wage. They do so by deciding both about which wage level

to set (or to bargain for in case of the e�cient bargaining variant) and how many non-union

�rms to organize. The incentive to organize �rms is both directly given by an increase in

members, and -in case of the monopoly union - indirectly through an improvement in the

ability to set a higher wage.

This paper di�ers from the approach to unionization taken in most of the (theoretical)

literature, which has focused on the worker's demand for unions in a static setting by

weighing costs against bene�ts of union membership.34 While the purpose of this paper is

to mainly complement the standard approach to unionization, there is also evidence which

suggests that the worker's part of the unionization process is of limited relevance. First,

as mentioned above, the union status of a �rm is rarely revoked, and unions exit usually

through �rm exit. This implies that many �rms have been unionized a long time ago and

workers become unionized simply by being hired by such a �rm.5 Secondly, survey evidence

consistently shows that there is a wide gap between the percentage of workers who would

vote for a union if possible and the actual rate of workers unionized. In fact, over the past

ten years, the support for unions has even been rising, while the unionization rate continued

to decline.6

Unions in the model are separate entities with their own objectives.7 Due to reputational

3This literature started with Pencavel [1971]. See Kaufman [2002] for a summary of the literature.4In this model, from an individual perspective, the workers would always prefer to join a union.5In theory, workers can of course intentionally select themselves into a union �rm.6See Freeman [2007]. For surveys on union support see also the Gallup report that shows an approval

rate of 60 % in 2007 (Gallup poll results are available online at: http://www.gallup.com/poll/12751/Labor-Unions.aspx ).

7The model abstracts from the problem of preference aggregation. Since the model doesn't featureunemployment, individual workers always prefer to be in the union if membership is o�ered.


considerations or threats of job loss, individual workers within a �rm have low incentives to

start an organizing drive. This provides an important role for the union as an outside agent

to initiate elections to unionize. Union certi�cation in the US often is a lengthy procedure

that involves legal disputes and delay strategies on side of the employers. This is re�ected in

the model by introducing a cost function for organizing that summarizes both direct costs

of organizing workers and the indirect costs implied by countering employers resistance to

organizing. Costs are increasing in the number of �rms to be organized, and thus higher

�rm turnover will make it costlier to sustain a given unionization rate. In particular, higher

entry given �rm exit will lower the optimal organizing rate since both direct organizing

costs increase as well as a higher mass of incumbent �rms will lower pro�ts and -for the

monopoly union case - in turn will lower the intensive employment margin (worker per �rm)

and thereby union membership.

Concerning �rms, exit is exogenous so that their main decision is whether or not to

enter the market. The entrant's decision depends on the union's organizing rate in two

ways. First, a higher organizing rate increases the risk of becoming a unionized �rm with

higher labor costs. In case of the monopoly union there is a second e�ect: A higher share of

unionized �rms will lower an individual �rm's price relative to the sector average and thus

increase sales and thereby pro�ts. It can be shown that under some conditions the �rst

e�ect dominates and thus entry decreases with union organizing.

The model is applied to numerically analyze both the impact of parameter variations on

the steady state unionization rate as well as transitions following a one-time and permanent

parameter change (within a sector). The steady state unionization rate is higher if 1. �rm

entry costs are higher; 2. �rm exit rates are lower; and 3. organizing costs are lower. All of

these results are in line with the stylized facts.

The dynamics of the model allow to evaluate two channels of the observed union decline

proposed in the literature. First, it has been conjectured that in the aftermath of the Taft-


Hartley Act in 1947, organizing has become more di�cult. This seems particularly true for

the US states that adopted the so-called right-to-work laws. According to the model, a one

time change in the organizing cost implies a gradual adjustment of the unionization rate.

The model supports the hypothesis that a change in the legal environment at one point in

time can have contributed profoundly to the long-term decline of unions. A second factor

that has been emphasized is the deregulation of several industries in the US during the late

1970s and early 1980s. The path of the unionization rate following a decrease in entry costs

also follows a gradual adjustment over time. The feature of the gradual adjustment follows

from the fact that the unionization rate is based on the stocks of (non-)union �rms, and

thereby adjustments only come through changes in the �ow variables. It is interesting that

in the model not only unionization rate declines, but also the organizing rate declines due

to a one-time change in the parameters. The model shows how the long-term union-decline

can be understood as the result of a one-time change occuring right after WW II instead of

a gradual worsening of the organizing environment that has often been claimed but is hard

to show empirically.

Finally, the model also contributes to the literature on unionism and entry deterrence.

Unions in this model deter entry, and pro�ts both for a union �rm and non-union �rm (that

remains non-union) are higher if unionization is higher due to reduced entry. However, the

expected pro�t of a union �rm is always lower than that of a non-union �rm, and thus only

non-union �rms (conditional on staying non-union) would possibly support unionization.

Further, in the model the e�ect of higher entry barriers always more than compensates for

the implied higher unionization. Thus, �rms would not consider a policy that lowers entry

costs as way to reduce union power.

The remainder of the paper is organized as follows: Section 1.2 discusses the related

literature and summarizes some stylized facts. Section 1.3 and 1.4 describe the model

and the equilibrium concept. In section 1.5 the steady state and its comparative statics


are analyzed. Section 1.6 presents numerical simulations for the transition paths between

steady states in a partial equilibrium setting. Section 1.7 concludes. All proofs are in the

appendix.

1.2 Related Literature and Empirical Findings

Related Literature This paper relates to several strands of literature. First, in terms of

the model it contributes to the literature on dynamic union models (see Jones and McKenna

[1994] for an overview). Two papers are especially worth mentioning. Kremer and Olken

[2001] use ideas from epidemiology to analyze union behavior as the outcome of an evolu-

tionary equilibrium that selects those unions that only moderately extract rents from �rms

in order to reduce �rm exit and thereby union survival. Their model is similar in that it also

has organizing unions and �rm turnover. It di�ers in that exit is endogenous while entry

and organizing is exogenous. The outcome of their model that union �rms exit at a higher

rate than non-union �rms is not supported by the data (see stylized facts below). Further,

in contrast to this paper, their model neither takes into account e�ects of unions on �rm

level employment nor on the mass of incumbent �rms (which impacts pro�ts and therefore

the union's ability to extract rents). Both of these margins are important determinants for

the resulting aggregate unionization rate.

Secondly, a recent paper by Ebell and Haefke [2006] links product market competi-

tion measured by the elasticity of substitution between monopolistic-competitively supplied

goods with the support for unions by workers within a matching model. Higher competition

(due to e.g. deregulation), decreases the net gain of unionization. If competition is strong

enough, workers of any newly entered �rm will not support unions, and unions disappear

over time due to �rm turnover.8 Even though the model is similar in that it also has union

8Their measure of competition, the elasticity of substitution between goods, is claimed to be equivalentto (variation in) entry cost. In the model presented here, it turns out however, that the e�ect of highersubstitutability between goods leads to both higher entry and (weakly) higher unionization rate. This result


formation and �rm turnover, it di�ers both in that �rm turnover is exogenous and in the

way it models union formation. The model is used to explain union decline as a result of

a higher elasticity of substitution which is interpreted as deregulation. In contrast to the

proposed model here, it cannot account for the fact that at the same time unions decline

and constantly organize new �rms.

Another connection to the literature is concerned with the IO side of the model. First,

this paper is built on a simpli�ed version of the Hopenhayn [1992] model of industry dy-

namics. The main di�erence is that here there are no productivity shocks and thus exit is

exogenous. Secondly, this model relates to the literature on unions and entry deterrence

that was initiated by the work of Williamson [1968].9 In the model here, unions deter �rm

entrance in part because the presence of unions lowers the expected pro�ts of an entrant. As

mentioned in the introduction, the support for unions by �rms di�ers, however, depending

on their union status.

Finally, this paper contributes to the recent attempts to identify the reasons for the

long-term union decline in the US. This emerging interest is partly motivated by the con-

jecture that this decline might have caused the rise in wage inequality. Besides the already

mentioned paper by Ebell and Haefke which links the union decline to deregulation, the one

by Acemoglu et. al. [2001] is particularly interesting. They propose a model where work-

ers di�er by skill and where unions �atten the skill-wage pro�le. Skill-biased technological

change implies that high-skilled workers are less willing to form a coalition with the low-

skilled workers and therefore the unionization rate decreases. While this argument certainly

plays a role, it is limited by the fact that the group of workers that has most bene�ted from

skill-biased technological change has never been unionized to a large extent. Moreover the

rents that unions capture are not only taken from better-skilled employees but also from

�rm owner's pro�ts.

is not central to the paper here and available on request.9For a brief summary of the literature see the article by Robin Naylor in Addison and Schnabel [2003].


Stylized Facts The following summarizes a set of stylized facts about unions and �rm

turnover in the private sector of the US. Facts 1 to 6 are features of the data the model

attempts to capture, whereas facts 7 and 8 are used to motivate important assumptions of

the model.10

1. Industries with higher �rm entry rates have lower unionization rates (Chappell et. al.

[1992]). Relatedly, unionization rates are positively correlated with industry concen-

tration (Ebell and Haefke, 2006)

2. Unionization rates are ceteris paribus lower in US states that have adopted so-called

right-to-work laws (Ellwood and Fine [1987], Holmes [1998]).

3. Unionization rates are higher in industries with lower �rm exit rates (Kremer and

Olken [2001]).

4. The unionization rate has declined steadily from 1953 to today (see �gure 1.3 and the

source cited there).

5. The gross union membership growth rate has been positive at all times. The average

yearly rate for the last decade is about 2.4 % (Holmes and Walrath [2007]; see also

�gure 1.7 for the long-term development).11

6. The union wage premium has been relatively constant with a slight downward trend

(Blanch�ower and Bryson, 2002).12

10A source of disaggregated data estimated from the CPS is provided by Hirsch and Macpherson. Thesedata can easily be accessed at http://unionstats.com/.

11This number is an upper bound since it also includes gains from poaching, i.e. unions attracting membersof other already existing unions.

12The union wage premium is the percentage of the union wage over the non-union wage. Estimating thepremium is usually done with a Mincer-type regression but in general has problems such as endogeneity ofthe selection into union �rms etc. In addition, a major problem is the availability of data on bene�ts inaddition to wages. The study cited here seems to give a consensus estimate but is not supported by all ofthe literature.


7. The aggregate rate of union decerti�cation is close to zero (e.g. Farber and Western,

2001).

8. Exit rates for �rms do not di�er signi�cantly by union status (DiNardo and Lee [2002],

Dunne and Macpherson [1994], Freeman and Kleiner [1999])

1.3 Model

The exposition of the model is focused on the monopoly union case and the necessary

changes for the e�cient bargaining variant are given within the text.

1.3.1 Environment

The model combines a Hopenhayn-style entry-exit framework of monopolistically compet-

itive �rms with unions that either set or bargain for wages and organize �rms.13 The

economy is populated by a continuum of ex-ante identical workers of constant mass L. Out-

put markets are structured by a continuum of sectors j ∈ [0, 1], each of which produces

a continuum of goods of endogenous measure µj . Goods within sectors di�er from goods

across sectors by having a higher elasticity of substitution. Moreover, entry costs, εj , and

exit rates, δj may be di�erent across sectors. In each sector there is one union.14 Thus,

the union is small vis-à-vis the aggregate economy, and therefore takes aggregates and the

behavior of other unions as given. However, a union is big in relation to �rms within

its sector, which in turn take the union's decisions as given (or bargain with the union).

This constellation can be interpreted as an intermediate position between centralized and

13Firms (or synonymously plants) are therefore either union free or completely unionized. This is asimpli�cation of the US case, where some plants are partially organized (especially in states where the unionshop is prohibited). However, the di�erence between union members and union-covered workers is small.On average the di�erence is around one percentage point (see the cited website by Hirsch and Macphersonfor data).

14This abstracts from the fact, that in the US unions sometimes organize workers that are (completely)outside their sector.


decentralized bargaining, which approximates the situation observed in the US.15 On the

aggregate level there are markets for the aggregate good, Qt, and non-union labor lnuj,t , which

clear every period. These economy-wide markets determine the non-union wage level wnut ,

which is uniform across sectors, and the aggregate output price, P , which will be used as

a numeraire. The given multi-sector structure with CES demands has several advantages.

First, on a theoretical level it allows to separate general equilibrium e�ects of the union's

behavior on the aggregate wage and output from the union's maximization problem. This is

important because in this class of models the comparative statics of aggregates is sensitive

to parameter assumptions.16 Further, it allows having unionized �rms coexisting with non-

uion �rms. Moreover, as mentioned above, it avoids strategic interactions between unions.

For empirical purposes, with this richer structure the model can be easily adjusted to the

cases of cross-sector heterogeneity (e.g. with respect to entry costs or exit rates). This bet-

ter facilitates the accounting for the aggregate unionization rate and union wage premium

given the pronounced heterogeneity of union outcomes across sectors.

The environment for �rm turnover and the dynamics of the union status of �rms is the

following: Firms who enter pay an up-front entry cost εj . All �rms exit at a �xed rate δj .

Moreover, each period non-union �rms and entrants can become unionized. Entrants always

start out as non-union. However, union �rms cannot change back to non-union. This is

asymmetry is motivated by the stylized facts.17

15The study by Katz [1993] �nds that union bargaining in the US is a mixture between multi-companyand plant level bargaining. He claims that there is a trend in direction of more decentralization. Marshalland Merlo [2004] state that for the more recent time period the percentage of pattern bargaining (unionscoordinate their wage bargaining across many �rms) is still about 25% of all bargaining.

16See the discussion in the third paragraph of section 1.5.3.17Farber and Western [2001] report that the decerti�cation rate, that is the rate by which unionized �rms

lose their union status, is positive but insigni�cant.


1.3.2 Agents' Static Maximization Problems

This paragraph explains the decisions of consumers, specialized and aggregate producers

made within a given period. Output is produced and consumed every period. Since there

is no savings market18, both each worker's utility maximization problem and each �rm's

pro�t maximization problem is static. The resulting pro�t and labor demand functions

(πijt, lijt) of the specialized producers are then used in the formulation of the dynamic

decision problems of entrants and unions given in the next subsection. Time indices are

omitted in this section.

Workers/Consumers Workers in �rm i ∈ µj and sector j ∈ [0, 1] earn wages and receive

pro�t shares, and decide about consumption each period. Labor supply is inelastic. The

representative worker solves each period:

max QC (1.1)

s.t. PQC ≤∫j∈[0,1]

∫i∈µj

(wijlij + πnet

ij

)didj

where the price index is given by:19

P ≡

(∫j∈[0,1]

(∫i∈µj

pρ1

ρ1−1

ij di

)ρ2(ρ1−1)ρ1(ρ2−1) dj

)ρ2−1

ρ2 . (1.2)

The set µj contains all union and non-union �rms in sector j. In each period (time index

omitted) and sector there is a mass uj of identical union �rms and a mass nj of identical

18In the part about the model dynamics below, �rms who enter pay a sunk cost, which have to be paidby future pro�ts. This implicitly assumes a credit market. As usual in this kind of models I ignore thesavings market. To avoid inconsistencies, this arrangement could be formally justi�ed by the assumptionthat workers, in contrast to �rm owners, are discriminated with respect to their ability to borrow and lend.

19All the derivations of the CES demands and resulting price formulas are standard despite the addeddimension of j sectors and therefore omitted.


non-union �rms, which are both determined endogenously as explained further below. The

symbol πnetij denotes pro�ts net of entry costs.20

Final Good Production Intermediate goods qi,j provided by monopolistic producers

(see next paragraph) are assembled into a �nal good Q each period by the following constant

returns production function:

Q = (∫

j(∫

i∈µj

qρ1ij di)

ρ2ρ1 dj)

1ρ2 (1.3)

It is assumed that 1 > ρ1 > ρ2 > 0, which implies that goods are more substitutable

within than across sectors .

The demand functions for the intermediate goods qij resulting from cost minimization

of the aggregate producer are given by:

qij(pij) = p1

ρ1−1

ij

(∫i∈µj

pρ1

ρ1−1

i,j di

) ρ1−ρ2ρ1(ρ2−1)

Q̂ (1.4)

where Q̂ ≡ QP1

1−ρ2 .21

Monopolistic Firms The following is for the monopoly union case. Given the demand

function from above and wages (which depend on the union status), the producer of spe-

cialized good i solves the problem of a monopolistic competitor:22

maxpij

pijqij(pij)− wijF(qij(pij)) (1.5)

20See also remark 1 in appendix C.21Note that the production of the aggregate good doesn't require any labor input. Adding labor to the

production process would not, however, change any of the results of the model and is omitted for simplicity.22Note, that for the e�cient bargaining speci�cation, a unionized �rm cannot freely choose employment.


Technology is given by the labor input requirement function: F(q) = κqα, with α ≥ 1,

and κ > 0. I allow for decreasing returns, which could be justi�ed by the presence of the

�xed factor implied by the entry costs.

The pro�t of an individual �rm can be derived as a function of wages:

πij(wij) = wij

ρ1ρ1−α m

α(ρ2−ρ1)ρ1(α−ρ2) (

ρ1

ακ)

ρ2α−g Q̂

α(1−ρ2)α−ρ2 (1− ρ1

α) (1.6)

where i ∈ {u, n} is indicating the union status of the �rm, and m = ujwj

ρ1ρ1−α + njw̄

ρ1ρ1−α ,

with wj being the union wage, w̄ being the economy-wide non-union wage, and uj and nj

denoting the masses of of union and non-union �rms in sector j.

The term mα(ρ2−ρ1)ρ1(α−ρ2) (note, that it is taken to a negative power), expresses a sectoral

demand e�ect implied by the relation of the �rm's own price (which is a function of the

wage) to an index of the sectoral price. In case of the monopoly union, this sectoral price

index is increasing in the share of unionized �rms, denoted by r̃. It will turn out that in

that setting it follows - leaving the total mass of �rms within a sector constant, and taking

the aggregates w̄ and Q̂ as given - that a higher share of unionized �rms (which have higher

wages) leads to higher pro�ts and thereby higher labor demand (for both unionized and a

non-union �rms).

In case of e�cient bargaining, the pro�t maximization problem for a non-union �rm is

exactly as described above. In section 1.5.1 it is shown that a union �rm will set employment

equal to that of a non-union �rm, i.e. lu = ln(thus, the output prices are also the same,

pu = pn) whereas the wage is set above the non-union wage to appropriate a share of the

pro�ts: wu =(1 + σ(α−ρ1

ρ1))

wn, where σ ∈ [0, 1] is the bargaining power parameter.


1.3.3 Dynamics of Entry, Exit, Unionization, and Wage Setting

This subsection describes the dynamic aspects of the model. First I will detail the sequence

of moves within each period and the laws of motion . Then, I will describe the potential

entrant's decision problem. The last part explains the union's wage and organizing choice

problem.

Timing For each sector j the timing within every period t is as follows (the monopoly

union case is given in parentheses):

1. Potential entrants decide whether or not to enter and pay entry costs εj if they enter.

2. The union chooses the organizing rate sjt (union sets monopoly wage wjt).

3. A fraction sjt of both entrants (ejt) and non-union incumbents (njt) is unionized.

4. The union and �rm bargain over wjt and ljt23

5. Firms, both entrants and incumbents, exit at an exogenous rate δj .

6. Incumbent �rms demand labor lijt and produce output qijt.

Laws of Motion The states of the model are the masses of union and non-union �rms in

each sector. Given the environment and the timing of the decisions and events, the states

evolve according to the following laws of motion:24

uj,t+1 = (1− δj) [ujt + sjt(njt + ejt)] (1.7)

23Notice, there is a slight di�erence in timing for the e�cient vs. monopoly case. The latter simultaneouslydecides about wage and organizing, whereas the former �rst sets s and then later bargains about w. Themain reason is that it simpli�es the problems considerably in each case so that analytic results can beobtained.

24Note that I am following the convention that xt , x ∈ {u, n}, denotes the mass of �rms determined lastperiod, thus xt+1 is the stock of �rms at the end of period t.


nj,t+1 = (1− δj)(1− sjt)(njt + ejt) (1.8)

ejt is the total mass of entrants in sector j determined by a zero pro�t condition introduced

below.

As was emphasized in the description of the environment, in this set-up unions can gain

market share from both entrants and incumbents, but can lose only through the exit of union

�rms. An immediate consequence of this is that older �rms have a higher likelihood to be

unionized, since the (steady state) probability of being non-union in period T conditional on

surviving to that period is (1− s)T+1, which goes to zero for T →∞ as long as s > 0. This

is a feature also reported in empirical work.25 Note, that this result rests on the assumption

(supported by empirical studies) that union and non-union �rms don't di�er with respect

to exit rates.

Value of Firms and Entry Decisions Each period there is in�nite or su�ciently large

supply of potential entrants who upon entry have to pay an up-front sunk entry cost. Once

�rms have entered they don't make any intertemporal decisions. Their discounted pro�ts

are simply the discounted sum of their static pro�ts, given the sectoral aggregate mass of

entrants ej,t∗ , the time path of the union's choices of organizing {sjt∗}t∗≥t, the negotiated

wages jt∗}t∗≥t, and the path of the economy-wide aggregate wage w̄jt∗and output Qt∗ for

all t∗ ≥ t. Given all these future values, it is then possible to formulate the value of each

�rm as a function only of the states and the time period. The value function of a unionized

�rm is:

V ujt(ujt, njt) = πujt(ujt, njt) + βf (1− δj)V u

j,t+1(uj,t+1, nj,t+1) (1.9)

25See for example Freeman and Rogers [2006], p. 67 and Brown and Medo� [2003]. The latter study �ndsthat the correlation is relatively weak however.


where δj is the exogenous �rm exit rate.

The value function of a non-union �rm is given by:

V njt(ujt, njt) = πn

jt(ujt, njt) +

βf (1− δj)[sj,t+1Vuj,t+1(uj,t+1, nj,t+1) +(1− sj,t+1)V n

j,t+1(uj,t+1nj,t+1)] (1.10)

With probability sj an entrant will become unionized, with probability (1 − sj) it will

stay non-union this period. The value of an entrant is therefore given by:

V ejt(ujt, njt) = sjt(1− δj)V u

j,t(ujt, njt) + (1− sjt)(1− δj)V nj,t(ujt, njt)− Ptεj (1.11)

where εj are real entry costs in sector j. In order to make their decisions, potential entrants

have to anticipate what the future path of both the union's choices within the sector, what

the future equilibrium mass of entrants will be and how the aggregate output and the

non-union wage will evolve.

In equilibrium the mass of entrants ejt is determined by the following zero-pro�t condi-

tion:26

V ejt ≤ 0, = 0 if ejt > 0 (1.12)

The equilibrium entry response as a function of the states will be denoted by fejt(ujt, ujt).

Union Organizing and Wage Setting In each period the union of sector j decides

about the rate sjt at which to organize non-union �rms and wages wjt. I analyze the two

26Note, the numerical simulations for the transition dynamics currently use a modi�ed entry condition.See remark in Appendix C.


most common approaches to union wage determination: The �rst is the so-called e�cient

bargaining, and means that every period the union and each �rm in the sector Nash-bargain

over both wage and employment to maximize the weighted product of their net pay-o�s.

The second is the so-called right-to-manage approach in which the union unilateral sets the

wage and the �rm is free to choose employment.27 As noted above, for technical reasons I

adopt a slightly di�erent timing for each case: in the right-to-manage case the union sets

s and w simultaneously, whereas in the bargaining scenario the union �rst decides about s

and then bargains with the �rm.

In the theoretical union literature following the work by Pencavel [1971] union formation

has mostly focused on the worker's decision on the costs and bene�ts of joining a union.

Here, I complement this view by focusing on the costly organizing process carried out by

the union as an autonomous agent. Modeling both the wage and the organizing decision

simultaneously has not been explored much in the literature.28

The organizing decision is modeled as a trade-o� between employment gains on the

extensive margin (additional �rms) and the cost of organizing C(s).

The union's Bellman equation is in case of the monopoly union:29

Wjt(ujt, njt) = maxsjt∈[0,1],wjt

{U(ujt, njt, sj,t)− PtCj(ujt, njt, sj,t) + βWj,t+1(uj,t+1, nj,t+1)}(1.13)

with period payo�30: U(.) = (wjt − w̄t)ujtlujt(ujt, njt, wjt, sjt).

In case of the e�cient bargaining union, we have instead:

27The evidence on the union's objective function is mixed at best. See Kaufman [2002] for a summary ofthe literature.

28With the exception being the partial equilibrium model in Chezum and Garen [1997].29Note, that for (mostly technical) reasons to be explained later I allow the discount factor of �rms (βf )

to be di�erent than that of unions (β).30For the formula for labor supply see equation (1.17) below.


Wjt(ujt, njt) = maxsjt∈[0,1]

{U(ujt, njt, sj,t)− PtCj(ujt, njt, sj,t) + βWj,t+1(uj,t+1, nj,t+1)}

(1.14)

with U(.) = (webjt − w̄t)ujtl

ebujt, where {web, lebu } is the solution to the bargaining problem

further described in section 1.5.1 below.31

The maximization is subject to the laws of motion and the time path of the equilibrium

entry response jt∗}t∗≥t and the aggregates w̄t and Qt. Further, it is assumed that the union

cannot borrow so that the period payo� has to be non-negative. Denote the union's policy

functions for the wage and for organizing by fwjt(ujt, njt) and fs

j,t(ujt, njt) respectively.32

Considering the observed practice that unions as institutions get a (�xed) percentage of

their members' wages, a straightforward interpretation of U is that the union maximizes its

revenues net of organizing costs.33

Organizing costs are in terms of the aggregate output34 and are speci�ed by:

Cj(ujt, njt, sj,t) = (njt + ejt)ηjsγjt,

where cost parameter ηj > 0, γ > 1 and entry is a function of the states: ejt = fejt(ujt, njt).

The cost function is strictly convex and increasing in s (for some of the results, more

stringent conditions on γ are imposed). The costs for a given organizing rate s are propor-

31For the di�erence in the maximization problems see footnote 23 above.32In terms of the data, the union's income through the dues seems to be even more restricted since dues

typically are not more than 2 or 3% of the wage, which is much below the average wage premium of about15%.

33Alternatively, the union's objective can be understood as the payo� of its end-of-period members. Oneproblem with this interpretation is, however, that the model is silent about how individual workers areallocated to �rms after a �rm exits (or its labor force is reduced after a wage increase). Thus, this secondinterpretation would have to impose further assumptions on where workers can go after leaving a �rm topin down the exact value of union membership for a worker.

34On the aggregate, organizing costs are QUt =

∫jCj(.) · uj,tdj). This part of the output is subtracted

from output for consumption. For the remainder of the model I will leave this cost implicit, i.e. will onlydeal with gross aggregate consumption QC (but I report the wage premium net of the organizing cost inthe section with the numerical results.


tional to the total mass of �rms, njt + ejt, that can potentially be organized. This means

that costs are independent of �rm size (implying that organizing larger �rms yield a higher

return to unions). The costs of organizing in this model can be interpreted as a summation

of several factors. First, there are actual costs which have to be paid to union employees for

the organizing drive.35 Secondly, the success of organizing depends on the �rm behavior.

Firms can (illegally) dismiss workers joining the union organizers. Moreover, there are many

possibilities to delay the organizing procedure. Once organized, �rms can further delay the

bargaining process, which is supported by the fact that a signi�cant portion of certi�cations

do not achieve a wage agreement. All of the �rm's counter measures make it harder for

the union to organize.36 The model in this paper does not attempt to map these obstacles

in an explicit manner and treats a �rm's union avoidance as part of the organizing cost

function.37 Rather, the model is used to show how this friction is qualitatively relevant in

the context of �rm turnover.

1.4 Equilibrium

The equilibrium of the economy is de�ned in three steps. The �rst two describe the industry

equilibrium and the union's response in each sector. The third step is concerned with the

aggregate decisions of the representative consumer and the aggregate goods producer, and

the market clearing conditions for the aggregate good and labor given the outcomes of the

industry equilibrium.

The equilibrium chosen here is one where the union can commit to a sequence of future

organizing decisions (which imply a sequence of wage and entry decisions, since each entry

behaves competitively). A more natural concept in this context that also avoids time in-

35See Voos [1984] as the only paper known to me that presents data on union organizing expenditures.36See Kleiner [2001] for more details and further literature about management resistance.37For an empirical investigation of how organizing cost and anti-union resistance together determine the

unionization outcome see Abowd and Farber [1990].


consistency would have been a Markov-Perfect equilibrium. In order to get some analytic

results and hence better intuition of the model implications the commitment case has been

chosen. My numerical simulations indicate that there is not much of a di�erence in the

results for either equilibrium concept.

The following is the for the case of a monopoly union. The necessary modi�cations for

the e�cient bargaining case are straightforward and therefore omitted.

De�nition. Normalize Pt = 1 for all t. Given some initial state vector {uj0, nj0}j∈[0,1],

an equilibrium consists of prices{{pujt, pnjt, wjt}j∈[0,1] , w̄t

}t≥0

, quantities{{ujt, njt}j∈[0,1] , Qt, Q

Ct , Qε

t, Qπnett ,

}t≥0

, demand functions {qjt(.), lujt(.), lnjt(.)}j∈[0,1], se-

quences of

{{e∗jt, s

∗jt, w

∗jt

}j∈[0,1]

}t≥0

value functions V ujt, V n

jt , V ejt, Wjt, policy functions fw

jt

and fsjt and the equilibrium entry response function fe

jt such that for all t ≥ 0:38

1. Given the aggregate variables {w̄t∗ , Qt∗}∞t∗=t for each sector j:

(a) For each specialized goods producer, given the demand function (1.4) , the max-

imand of (1.5) is given by the pro�t function (1.6), and the labor demands give

the corresponding optimal input demands.

(b) Given the value function of an entrant as de�ned by (1.9)-(1.11), and given{e∗jt∗+1, s

∗jt∗ , w

∗jt∗

}t∗≥t

the potential entrant solves maxenter,don′t

{V e

jt, 0}.

(c) Given fwjt and fs

jt, fejt gives the total mass of entrants ejt such that the zero pro�t

condition (1.12) holds.

(d) fejt = ejt for all j and all t.

2. Given the aggregate variables {w̄t∗ , Qt∗}∞t∗≥t , the laws of motion for ujt and njt, and

{e∗jt∗}t∗≥tfor all t, for each sector j:

(a) Wjt solves the Bellman equation (1.13).

38The time index of the value and policy functions is understood as a time argument of these functions.


(b) The policy functions fwjt and fs

jt attain the RHS of (1.13).

(c) fwjt = w∗

jt and fsjt = s∗jt for all j and all t.

3. Given {wjt, ujt, njt}j∈[0,1] , {pujt, pnjt}j∈[0,1], and labor demands{lujt(.), lnjt(.)}j∈[0,1]

as determined in 1. and 2., w̄t and{{qjt(pijt)}j∈[0,1]

}i∈{u,n}

, QCt , Qε

t,Qπnett , and Qt

are such that:

(a) QCt solves the consumers problem in (1.1).

(b) The demand functions for specialized goods in (1.4) solve the �nal goods pro-

ducer's cost minimization problem for a given price pijt , i ∈ {u, n}, and output

Qt.

(c) The goods market clears: Qt = QCt + Qπnet

t + Qε (where Qt is de�ned in (1.3),

the aggregate entry costs are Qεt ≡

∫j εj · [uj,t+1 + nj,t+1]dj, and Qπnet

t is the

consumption from aggregate pro�ts net of entry costs).

(d) The labor market clears: L =∫j [uj,t+1lujt(w̄t, .) + nj,t+1lnjt(w̄t, .)]dj (where the

LHS is the time invariant and inelastic total labor supply).

Note, that the equilibrium is not well-de�ned if the unions were able to completely unionize

the economy, since then the labor market would not clear. This case could be avoided by

either assuming that at least some sectors have high enough organizing cost, ruling out the

possibility s = 1 or that some positive mass of sectors by default doesn't have a union.

1.5 Steady States and Comparative Statics

This section analyzes the comparative statics of steady states for the two versions of union

wage determination. First, I consider the case where the union bargains every period with

each �rm individually. The wage then depends on the exogenous bargaining strength of the

unions. It turns out that �rm level employment will then equal to that of a non-union �rm,


i.e. the union only takes part of the pro�ts without changing the labor market allocation.

Secondly, the case of a wage setting union is analyzed. Here, employment is chosen by the

�rm in a pro�t maximizing way. In this setting both union wages and �rm level employment

depend on the share of unionized �rms. Union wages exhibit a positive relationship with

union organizing (and the unionization rate).

For both cases I analyze the steady state comparative statics with respect to entry cost,

union organizing cost, and the �rm exit rate.

1.5.1 E�cient Bargaining

In the following I will analyze the problem assuming that the union in each sector bargains

with each �rm over both wages and employment. The problem is solved by axiomatic Nash

bargaining with share parameter σ. The union's net payo� is the di�erence in wages times

employment (that is the outside option is to take the competitive wage), whereas the �rm's

net payo� is simply pro�ts (the outside option is to not produce in that period). This

set-up abstracts from features of real world bargaining like strikes, hold-outs and hiring of

replacement workers.

Given the bargaining strength of the union, σ ∈ [0, 1], each period (and in each sector)

wage and employment are chosen to maximize:

maxw,luσln {(w − wn)lu}+ (1− σ)ln {p(q(lu))q(lu)− wlu}

The �rst order conditions yield the following result.

Lemma 1. The bargaining solution is the union choosing employment equal to the non-

union employment: lu = ln. The wage is given by: wu = σpnl(1/α)−1n + (1 − σ)w̄ =[

1 + σ(α−ρ1

ρ1)]w̄.

Thus, the union doesn't alter employment and simply takes a share σ of the �rm's pro�ts


(which is easy to see for the case where note, that if α = 1).

Given the solution to wages and employment, the union's optimal (interior) choice for s

is given by the solution to the FOC w.r.t. s of the corresponding sequential version of the

union's problem in (1.14)39, where the laws of motions for u and n have been substituted

in and steady state has been imposed:

[1 + β(1− δ)(1− s)] (wu − wn)(1− δ)(n + e)lu(e) = η(n + e)[γ − β(1− δ)s]sγ−1

This expression, roughly speaking, equates marginal bene�ts of organizing with marginal

costs. Solving for the indirect response function (e as a function of s) we get (using the

steady state expression for lu, u and n):40

gu(s) =

(η[γ − β(1− δ)s]sγ−1

[1 + β(1− δ)(1− s)]σ α−ρ1

ρ1(1− δ)

(1−δ

δ

)x2w̄x1(1+x2)KL

)1/x2

(1.15)

If e increases, marginal cost of organizing go up whereas there are two countervailing

e�ects on marginal bene�ts. Marginal costs go up, because more �rms have to be organized

for given s. Marginal bene�ts go up for the same reason. However, bene�ts also decrease,

since more entrants mean lower pro�ts per �rm and thereby fewer workers per �rm. In

addition, there is an intertemporal e�ect: Higher s today means that the union needs only

a lower s tomorrow in order to maintain a constant unionization rate, which implies that

the optimal s for a given e is lower in the dynamic compared to the single period case. In

order for the reaction function to be decreasing it is required that marginal costs go up by

more than the bene�ts when increasing e. It can be shown, that the total e�ect is negative

39The proof for the equivalence of the sequential problem and the value function formulation is omitted.40Further details and the shorthand symbols x1, x2, x3, KL and Kπ used here and later are de�ned in the

appendix. The �rst three are terms are consisting only of parameters, whereas the latter two also containaggregate output and aggregate price.


if γ > 1 + β2(1− δ)2, that is if the organizing cost function is su�ciently convex. Note, for

s → 0 the function tends to ∞, whereas for s = 1, it takes on some �nite positive value. A

su�cient condition for strict convexity of the response function is y > 2.41

Turning to the potential entrant's problem, given the formulas for V u and V n, and the

steady state expressions for u and n, the steady state value for an entrant is:

V e =1− δ

1− βf (1− δ)(1− s)

(s

1− βf (1− δ)πu + (1− s)πn

)− Pε

where πn is the steady state version of (1.6), and πu = (1−σ)πn . Solving this for entry we

get

ge(s) =

(ε

(1− δ)(

1−δδ

)x2w̄x1(1+x2)Kπ

sβ(1− δ) + 1− β(1− δ)s[β(1− δ)− σ] + 1− β(1− δ)

)1/x2

(1.16)

Note, that at the endpoints of this function take on �nite values with ge(0) > ge(1).

The function is strictly decreasing in s, because, a higher organizing rate increases the

probability of becoming a less pro�table �rm (higher wage). A su�cient condition for strict

convexity is βf (1− δ) > σ, which is not very restrictive from an empirical point of view.

The following proposition establishes the existence of a unique sector equilibrium.42

Proposition 1. If ε ≤ 1−σσ η(γ − β(1− δ)) holds, a sector equilibrium for given aggregates

w̄ and Q exists. Further, if both γ > 2 and βf (1 − δ) > σ the equilibrium is unique. If in

addition βf = 1 and all sectors are identical then a general equilibrium exists and is unique

for interior values.

The existence proof can be extended to the case of heterogeneous sectors, but this case

41This condition seems to be more restrictive than necessary, my numerical tests indicate that for γ >1 + β2(1− δ)2 the function is both decreasing and convex.

42The aggregate equilibrium requires that there is a solution to the equations for aggregate entry, outputand the non-union wage. The solutions to s and r do not depend, however, on the aggregate variables.


is omitted here.

In the following, I study the comparative statics with respect to entry costs (ε), organiz-

ing costs (η) and exit rates (δ) across sectors, i. e. I take the aggregate variables as given.43

A discussion of these results follows in section . Note, however, since organizing s and

the unionization rate r do not depend on the aggregates, all the comparative static results

regarding these variables hold true for the aggregate economy (which is not necessarily the

case for �rm entry).44

Changes in Entry Costs The �rst result states that higher entry costs (ε) for entering

�rms lead to more organizing and a higher unionization rate.

Proposition 2. Given an interior solution and the conditions for a unique equilibrium, if

�rm entry costs are higher, entry e is lower, the organizing rate s and the unionization rate

r are higher.

Higher entry costs obviously lower entry. At the same time each entrant will have

higher pro�ts and therefore a greater number of workers. This implies that the bene�ts

of organizing per �rm increase (whereas the costs per �rm stay constant), and thus it is

optimal to increase organizing. The unionization rate r is de�ned by rjt ≡ Lujt

Lujt+Lnjt. Using

the formula for the sectoral labor demand:

Li(wi) = i (wn)α

ρ1−α

[u(w)

ρ1ρ1−α + n(w̄t)

ρ1ρ1−α

]α(ρ2−ρ1)ρ1(α−ρ2)

κ( ρ1

ακ

) αρ1−ρ2 Q̂

α(1−ρ2)α−ρ2 (1.17)

where i ∈ {u, n}, andmaking use of the steady state versions of the laws of motions (given

in the appendix) one can write the steady state unionization rate as:

43I analyze the same exercise numerically for the aggregate economy. Table 1.3 and the following onecon�rm that the same results go through for the aggregate economy (in the given examples it is assumedthat all sectors are identical).

44See the discussion about the general equilibrium e�ects below.


r∗j =

1 +

[1δj

s∗j1− s∗j

]−1−1

(1.18)

Thus, higher s implies a higher unionization rate r. Note that for the e�cient bargaining case

the rate of �rms unionized, r̃, and the unionization rate are identical because employment

is the same across union and non-union �rms.

Changes in Organizing Costs Next, I turn to the e�ect of lowering organizing costs,

measured by the scaling parameter η.

Proposition 3. Given an interior solution and the conditions for a unique equilibrium,

lower organizing costs η increase s and therefore r, but lower entry e.

Higher organizing costs will increase costs relative to bene�ts of organizing, thereby

decreasing the optimal s for any given value of entry e. By the same argument as before

this will also decrease r.

Changes in Exit Rates The e�ect of a change in the exit rate δ is not clear-cut. A change

in δ moves both curves similarly. However, δ also directly a�ects the �rm unionization rate

r̃ (=r) through the laws of motion for u and n, because over time higher exit rates lead to

less accumulation of union �rms (see also equation 1.18). It turns out that this direct e�ect

is dominating so that the ambiguous outcome of the union-entry equilibrium doesn't a�ect

the overall result very much. Table 1.3 and the following present numerical results for a

range of parameters which con�rm that a higher exit rate leads to a lower unionization rate.

The dominating e�ect is that a higher exit/turnover rate makes it harder for the union to

accumulate �rms.

Changes in the Bargaining Power The last result is concerned with variations of the

exogenous bargaining power parameter.


Proposition 4. Given an interior solution and the conditions for a unique equilibrium,

higher bargaining power for the union σ increases s and therefore r, but lowers entry e.

Higher σ lowers the payo� of an entrant, therefore lowers entry for any given s, but for

the union, it increases marginal bene�t without changing marginal cost, therefore, allows

for a higher e given any s. Thus both e�ects lead to a higher equilibrium value of s. In the

next section, wages will be set by the union. It turns out that here the causality runs the

other way round: Higher organizing will imply a higher wage due to a union share e�ect

that says that the higher the union share in the output market, the higher is the sector-wide

average output price, and thus the higher pro�ts and optimal labor demand.

1.5.2 Unilateral Wage-Setting

In this section the union is assumed to set the wage for the whole sector, whereas each

�rm decides about the optimal employment level (the �rm has the �right to manage� once

the wage is set). In contrast to the previous set-up, here both the wage as well union �rm

employment exhibit a share e�ect: An economy with a higher percentage of �rms unionized

(or with a higher organizing rate) c.p. will have higher pro�ts, higher employment, and

higher union wages. This implies that the (worker) unionization rate r now di�ers from the

�rm unionization rate r̃ by the intensive margin, i.e. the di�erence in workers per �rm. If s

moves up both r̃ and the union wage w, the e�ect on r is ambiguous in principle. It turns

out, however, that the direct e�ect of s is always stronger, i.e. higher s will lead to higher

r.

The union's problem is now to simultaneously set the wage and the optimal organizing

rate. I will �rst turn to wage determination. Wages do not directly involve an intertemporal

trade-o�. In steady state the FOC for the union's wage can be written as:

x3 +x1x2

1 + δ(1s − 1)(w

w̄ )−x1+

11− (w

w̄ )−1= 0 (1.19)


The following lemma shows that the wage is increasing in organizing, and that the wage

is bounded.

Lemma 2. Assume an interior solution. The union wage is increasing in s. Moreover, the

wage is bounded: ww̄ ∈ [ α

ρ1, α

ρ2].

To see where this share e�ect comes from, it is useful to rewrite the pro�t of a union

�rm function given in (1.6) as:

πu(w) = wρ2

ρ2−α µα(ρ2−ρ1)a(α−ρ2)

(r̃ + (1− r̃)k

ρ1ρ1−α

)α(ρ2−ρ1)ρ1(α−ρ2) ·Kπ

where it is assumed that k = w̄w < 1 (i.e. the union wage is bigger than the competitive

wage) , and µ = u+n denotes the total mass of �rms in sector j, and Kπ is a term containing

aggregate output. Thus, increasing the �rm unionization rate r̃ will increase the share of

the high price union �rms and therefore lower the relative price of each �rm, which will

increase their demand. A further implication is that the elasticity of labor demand will be

lower for higher r̃ (or s) as well, allowing for a higher wage for any given employment level.

The result rests on the fact that each sector is small vis-a-vis the aggregate economy, so

that the indirect output e�ect does not matter.

Next, I turn to the union's (indirect) entry response for s, which is, again, derived from

the FOC of the corresponding sequential formulation of the union's problem:

g̃u(s) =

( [ηγsγ−1 − η(1− δ)βsγ

][1− β(1− δ)]x2 [ s

δwx(1− s)w̄x1 ]x2(1− δ)−(1+x2)

[1 + β(1− δ)(1− s)]KL(w(s)− w̄)wx3 [ sδ (1 + x2)wx1 + (1− (1 + x2

δ )s)w̄x1 ]

)1/x2

where w(s) is the optimal wage implied by the FOC w.r.t. the wage given above. Unfortu-

nately, the shape of the response function - besides the results that its left endpoint tends

to in�nity, whereas its right endpoint takes on a �nite positive value - cannot easily be

constrained by simple conditions on the parameters. The complication comes from the fact


that now the marginal bene�t is not only a�ected by s through an increase in the number

of unionized �rms but also by two indirect e�ects: First, higher s raises the sectoral price

index as explained above, thereby increasing the marginal bene�ts of higher s. Secondly,

it increases the optimal wage, which in turn has an ambiguous e�ect on the union's utility.

On the one hand it raises the bene�t directly, on the other hand, a higher wage also implies

lower �rm level employment.

The �rm entry response function can be solved from the steady state version of the value

of an entrant:

g̃e(s) =(

ε

Kπ

)1/x2

(1− δ)−x2−1

x2

([1− βf (1− δ)(1− s)][(1− (1− δ)(1− s)]x2

[ s1−βf (1−δ)

w(s)x1 + (1− s)w̄x1 ][ sδx1 + (1− s)w̄x1 ]x2

)1/x2

The response function in this case is also more complicated due to the share e�ect. However,

it can be shown that for the case of βf = 1, g̃e is strictly decreasing in s. 45

It is still possible to show existence in this setting. However, it is di�cult to give simple

conditions for uniqueness and the comparative statics. Therefore, all comparative statics

are analyzed numerically.

Proposition 5. A steady state sector equilibrium exists if

εη

(1−β(1−δ))(γ−β(1−δ))

(ρ1

α

)x1+1 ((1 + x2)(ρ1

α

)x1 − x2

)≤ 1

The more complicated response functions make it hard to guarantee uniqueness. All

comparative static results in this section are therefore explored numerically, where only

such parameter constellations are considered that exhibit unique equilibria.

It should be noted, that in the case of the monopoly union the comparative static

e�ects on the unionization rate r are not only dependent on s but also on the relative

�rm employment levels of union and non-union �rms. Thus, the unionization has both an

45It is possible to generalize the result for values of βf < 1. However, since for all the computationalresults it is assumed that the discount factor equals unity, the proof is omitted.


intensive margin (workers per �rm) and an extensive margin (�rms unionized). In contrast

to the e�cient bargaining scenario the corresponding formula for the unionization rate is:

r∗ =

1 +

((w∗

w̄∗

) αρ1−α

[1δ

s∗

1− s∗

])−1−1

(1.20)

In this case, higher s will directly increase r, but through the positive e�ect of s on the

union's wage,w, (see lemma 2) it decreases r. The following lemma shows that the direct

e�ect is always dominating the indirect e�ect.

Lemma 3. An increase in the organizing rate s implies an increase of the unionization rate

r.

An example for the comparative static result is presented in table 1.11 and the following.

The numerical example con�rms all the results that have been established for the e�cient

bargaining case. Higher entry costs, lower organizing costs, and lower exit rates all imply

higher unionization rates. In addition, the wage premium is positively correlated with

unionization rate, as has been shown in Lemma 2. 46

1.5.3 Discussion of Results

The previous two sections have shown that both versions of the model are able to rationalize

the observed facts discussed in the introductory chapter: Sectors with a higher level of entry

due to 1. higher entry costs, 2. lower exit rates, and 3. lower cost of organizing have a

higher organizing and unionization rate. The numerical results indicate that all the results

also hold when taking the general equilibrium e�ects into account.47 I will brie�y discuss

each of the comparative statics result in turn.

1. Higher entry cost imply lower entry and thus fewer �rms. Typical examples of

more more concentrated industries with a low entry are the still relatively high unionized

46The results have been tested for a broad range of parameters; additional results are available on request.47See also the discussion about the general equilibrium outcomes further below.


industries of automobiles and aircraft manufacturing. The interpretation given by this model

complements the ones frequently given in the literature, namely that in higher concentrated

industries there is a higher chance of union �contagion� and/or that high �xed costs create a

hold-up problem that is exploited by unions. Here instead, the higher unionization directly

comes from the interaction with entry. That is, both low entry and high unionization have

the common cause of high entry costs.

2. As was discussed in section 1.5.1, the impact of δ on the equilibrium interaction

between entry and organizing is ambiguous and likely to be small. The exit rate does

however have also a direct impact on the �ow movements: Everything else constant, a higher

exit rate increases the (out)�ow from union �rms relatively to the stock of incumbents,

thereby directly lowering the �rm unionization rate. This direct impact on the �ows is the

driving force for the total e�ect. Kremer and Olken [2001] regress sectoral unionization

(coverage) rates on �rm exit rates, controlling for several industry characteristics and �nd

that a 1 percentage point increase in the exit rate implies a 3.4 percentage point decrease

in the unionization rate. Regarding the example in the introduction, in manufacturing the

(recent) job loss rate due to plant closings is .7%, whereas in the service subsector of hotels

and restaurants it is about 2%. 48

3. As expected, lower organizing costs increase the share of �rms unionized r̃ and the

unionization rate r. The direct e�ect is that lower costs make it more worthwhile to organize

more. In addition, there is also an induced e�ect that higher organizing deters entry, which

in turn increases optimal organizing. The di�erences in the legal environment across states

imply di�erences in the organizing costs. In particular, most of he southern states in the

US adopted the so-called right-to-work legislation have lower unionization rates on average

compared to states without such a legislation.

48The numbers for �rm exit go in the same direction. Data are available online at http://www.bls.gov.Looking at low-skill service jobs compared to goods producing jobs, an additional factor to the higher �rmturnover rates is that the job turnover rates for existing �rms are also higher, making it even more di�cultto organize for unions.


To sum up, costly union organizing together with �rm entry and exit is an important

mechanism that potentially drives unionization outcomes in the US. Future work is aimed

at investigating its quantitative relevance. The next paragraphs focus on other interesting

implications and aspects of the model.

Wage Outcomes and Union Share E�ects The two formulations of the model di�er

in the way wages are determined. In the e�cient bargaining formulation the mark-up over

the non-union wage depends directly on the parameters, in particular bargaining power σ.

In contrast, as lemma 2 shows, the wage of the monopoly union is itself a function of the

organizing rate (and therefore of the �rm unionization rate). In this case the model exhibits

a union share e�ect. If the union has control over a bigger part of the sector's �rms it can

demand a higher wage due to the fact that competition from non-union �rm with lower

prices is less severe (that is the di�erence of the union �rm's price to the sector average is

smaller). Even though intuitive and often cited by union leaders, it is di�cult to isolate

such an e�ect in the data, not last because it is not always easy to �nd sectors with clear

boundaries (especially if competition also comes from foreign �rms). A study by Coggins

and Johansson [2002] is able to con�rm this share e�ect for the case of grocery stores, which

compete only locally.

In case of the e�cient bargaining approach there is a �xed mark-up over the competitive

wage, and thus the model does not have any share e�ect as in the monopoly formulation. It

is interesting however, as lemma 4 shows, that in this case the causality runs the other way

round - higher exogenous bargaining power allows the union to capture more �rms, because

it is more bene�cial to do so. Thus the same correlation between the wage premium and

the union share holds but for di�erent reasons.

So far the focus has been on gross wages for union members, because this is usually

the wage used to estimate union wage premia. Union members in the model (implicitly)

pay, however, union dues to cover the organizing expenses. The numbers for the net wage


numbers are reported in tables 1.6, and suggest that union wages are still higher than non-

union wages (note however, that due to the general equilibrium feedback e�ects, the level

of the union wage is higher if organizing costs are higher).

Induced E�ects of Endogenous Entry Firm entry and union organizing are endoge-

nously determined in this model. As it has been shown, the union's organizing response

is decreasing in the level of entry, as well as entry is decreasing in union organizing. One

implication is that changes that a�ect union organizing, in particular changes in union or-

ganizing cost η, will be ampli�ed by the endogenous entry response (see diagram 1.1 for an

illustration). Compared to a model where entry was �xed, the model here has an induced

e�ect of entry. Consider a decrease in the cost of organizing. Unions will increase their

organizing activity. Since higher organizing depresses the value of an entrant and thereby

the level of entry, it will in turn make organizing even more bene�cial because each �rm now

makes higher pro�ts. Table 1.19 presents an example comparison between the endogenous

entry and the �xed entry case (for the e�cient bargaining scenario). The numbers give the

elasticities of a change in η and their ratio.49 The elasticity is higher if entry is endogenous.

The di�erence in this example is relatively small (the elasticity for the endogenous case is

between .5% to 2.5% higher). One reason for this is that the benchmark parameters set the

bargaining power to a relatively low value, σ = .1. The examples for the monopoly union

version that I inspected show a much higher curvature of the entry response function and

thus a bigger impact of the endogeneity of entry.

Unions as an Entry Deterrent A second implication of the endogenous interaction

between entry and organizing is that unions act as a deterrent to potential entrants. In

both speci�cations of the model, a higher rate of organizing will decrease the number of

entrants through reduced expected pro�ts (the pro�ts of the incumbents, however, can

49To compute the elasticity for the �xed entry case I take the entry from the endogenous case and keepit constant when increasing η. This means that for each number, entry is di�erent.


increase). Since non-union �rms coexist with union �rms in this model, the support for

or opposition to unions by incumbent �rms can be con�icting. Both types of �rms gain

from higher unionization due to reduced entry50 (see tables 1.8, 1.9, 1.17, and 1.18 for

numerical examples). However, non-union �rms always have (signi�cantly) higher pro�ts

than unionized �rms. Thus, while unions and non-union �rms (conditional on staying non-

union this period) always gain from lower organizing cost, unionized �rms are only better

of given that they would have to stay union. Therefore, unionized �rms would not support

unionization, whereas non-union �rms (after the organizing drive has taken place) might

not oppose lower organizing cost (given the risk neutrality assumption). This suggests that

political pressure against unions is high if the unionization rate is high, because a majority

of �rms could gain from deunionization. At low unionization rates in contrast, most pressure

against unions is expected to come from individual �rms that are a�ected by organizing.

This is in accordance with the events in the US. The main legal change in favor of �rms

occurred in the late 1940s when unionization was at its peak. In the 1970s and 1980s unions

had to su�er from increased employer resistance, as some authors suggest, but no major

legal changes occurred.51

Another related question is whether �rms would support a reduction of the barriers

to entry in order to reduce union power. In this model, all the numerical results suggest

that the direct negative e�ect of lower entry costs on pro�ts is outweighing the gains from

lower unionization.52 This is in contrast to the result in Naylor [2002], where �rms have an

interest in increasing entry in order to lower the impact of unions on labor cost.

50And in the monopoly union case also through the union share e�ect, which increases both types of �rms'pro�ts.

51As emphasised above, the di�erence in the evaluation of the impact of unionization for union versusnon-union �rms only occurs in this way at the end of the period, i.e. when organizing has already takenplace. This means that the comparison with respect to s is not really meaningful within the model, sinces cannot be changed at this point. However, think of a model, where some �rms believe that it is hard toorganize them, i.e. there is some heterogeneity in union resistance. Then the comparison who would favorwhich policy towards unions would apply in a more straightforward manner.

52Dewatripont [1988] discusses the related issue whether or to what degree the entry deterring e�ect ofsunk capital has to be traded o� with the simultaneous increase in the union's bargaining power.


Sensitivity of the General Equilibrium E�ects This section discusses some of the

general equilibrium implications of the model. I limit myself to the case of identical sectors

and focus mainly on the case of e�cient bargaining.53

First, inspecting the response function for the union 1.15 and aggregate entry 1.16 it

is noteworthy that by setting them equal to determine s all the terms involving aggregate

variables cancel out. Table (1.7) and the following ones54 present an example of the com-

parative statics for the aggregate variables Q and w̄. In the given parametric example, Q is

increasing in η and decreasing in ε. The appendix shows that Q is increasing in e but also

increasing in s. This comes from the fact that due to the CES speci�cation, the factor shares

are constant and that all current sector pro�ts are spent on entry costs if the �rms' βf = 1.

Since the aggregate �rm pro�t share decreases if there are more union �rms (through higher

s) total output has to be scaled up for a given level of entry e to cover the �xed entry costs.

Thus, higher entry costs ε will lead to two countervailing e�ects: First lower entry will lower

Q. Indirectly, also s increases and thus Q. In fact there exist parameter values for which

the indirect e�ect via s is dominant and thus e and Q go in opposite directions.

Furthermore, the results concerning the aggregate variables are sensitive to the param-

eters for the within (ρ1) and across sector elasticities of substitution (ρ2), as well as the

returns to scale of the intermediate goods production (α). There are cases possible where

higher entry cost lead to higher entry on the aggregate. The ambiguity comes from the fact

that the entry response function is decreasing in the non-union wage w̄ but increasing in

aggregate output Q. However, both w̄ and Q are increasing in e, so that the total general

e�ect how entry is a�ected by itself depends on which e�ect is stronger.

As it is the case in the standard monopolistic competition model, statements about the

aggregate variables are parameter dependent. Therefore, an evaluation of welfare e�ects has

to be based on empirically supported parameter values. A strength of this model, however,

53All the technical details are in the appendix in the section �Computation of the Steady State Aggregates�.54Table 1.16 and following for the monopoly union.


is that the unionization rate does not depend on the aggregate variables. Moreover, the

multisector structure allows to analyze the model for given aggregate variables and focus

on the interaction of �rm turnover and organization within a sector without the described

ambiguities. Sectors can be compared within one equilibrium, so that all sectors face the

same aggregate variables and the within sector results are thus not (as much) a�ected by

the ambiguity of the aggregate e�ects.

1.6 Transition Dynamics within a Sector

This section studies numerical simulations of the sectoral transition dynamics after a one-

time change in either the entry or the organizing costs for given (and constant) aggregate

output and non-union wage. I will limit myself to the case of a wage setting union. Here, I

also use a Markov-Perfect equilibrium instead of the commitment case that was used in the

previous section for analytical convenience.

Unions in the US have been in a long-term decline since the early 1950s. Figure 1.3

shows the paths of the aggregate unionization rate and membership numbers. The popular

perception is that the major cause is the structural change from manufacturing to services.

Several authors, however, have long recognized that from a simple accounting perspective

the role of this structural change and other changes in the composition of the workforce

is limited.55 While it is true that services have been growing and traditionally have a

lower unionization rate than goods producing industries, it is also the case that in many

goods producing sectors the rate of the decline has been much stronger, which diminishes

the relative importance of the structural change. For example, comparing durable goods

manufacturing with retail trade for 1983 and 2006, the aggregate unionization rate (for

these 2 sectors) drops from 17.7% to 7.8 %. If we keep the employment weight of the

manufacturing sector constant it drops to 8.0 %. This small di�erence comes from the fact

55See e.g. Farber [1990].


that while the durable goods sector employment share is lowered by about 10 %, the drop

in the durable goods unionization rate is from 29.2 % to 11.9 %, whereas in the retail trade

sector it is only from 8.6% to 5%.56

Other causes have to come from either organizing (including the worker's willingness to

join that is not modeled here), or �rm turnover, or di�erences in the growth rates of union

versus non-union �rms. The �rst two channels are discussed in more detail in the following

sections. Regarding the possibility of di�erent growth rates, the (little) evidence given in

the literature suggests that this is quantitatively not a very important factor (see Bronars

and Deere [1993]).

1.6.1 Organizing Environment and Union Decline

The union's costs of organizing depend both on the direct expenditures for union organizers

and indirectly on the legal environment for union organizing and bargaining. In particular,

they depend on the implied possibilities for employers to counter union organizing.57 The

most important modi�cation of the National Labor Relations Act from 1935 was the Taft-

Hartley Act in 1947. The main changes a�ected the strike rules and the possibility for

individual states to enact so-called right-to-work (RTW) laws, that prohibit union shops

(i.e. unionized plants where union membership is mandatory). This prohibition creates

a free rider problem because workers can bene�t from unions without contributing. Even

though there is no consensus of how exactly the Taft-Hartley Act diminishes union power,

both the event study by Ellwood and Fine [1987] and the comparison of union outcomes

across state borders by Holmes [1998] support the hypothesis that unions are less successful

in the presence of RTW laws.

This paragraph studies the dynamic implications of a permanent increase in organizing

56The numbers are estimates from the CPS, provided by Barry Hirsch and David Macpherson on theirwebsite http://www.trinity.edu/Hirsch/unionstats/.

57These union avoidance strategies have been discussed extensively in the literature. See for exampleKleiner [2001].


costs (scale parameter η). I will look at the transition path from a low η to a high η while

keeping the aggregates constant. For the example it is assumed that the initial steady state

has 50 % lower (marginal) organizing costs than the �nal steady state. The out-of-steady

state unionization rate for given states ut and nt is given by:

rjt =

1 +

((wjt

w̄t

) αρ1−α

[ujt

(1− sjt) (njt + ejt)+

sjt

1− sjt

])−1−1

(1.21)

Thus, factors that a�ect the entry level e not only impact r through s and the wage w,

but also directly. The direct e�ect of e is absent in the steady state version of this formula

given in equation (1.20), since the entry rate is solely determined by the exogenous exit rate

δ.

Starting at the values of the state variables for the lower cost case, I simulate the

transition path using the policy functions for s, w, and e. Figure 1.2 shows the transition

path for the unionization rate. The model implies a slow transition. The new steady state

is reached only after 50 periods, whereby half of the total di�erence in the unionization rate

is already reached after 10 periods.

The US data of the unionization rate are given in �gure 1.3. The decline is gradual and

has taken several decades (and is still continuing). The decline doesn't follow immediately

after the enactment of Taft-Hartley, which could partly be explained by the fact that one

important provision, the �right-to-work� was only adopted gradually across the US states.58

Moreover, the southern states which are the main adopters, increased their employment

shares only gradually - although steadily - over time.59 The next �gures compare the

58Eleven States (mostly southern states) enacted the RTW during the 1940s right after the Taft-HartleyAct. Five more states enacted the law in the 1950s. For each decade from 1960-1989 on state was added.During the 1990s two states and in 2001 one state enacted the RTW legislation. Data are provided by theUS Department of Labor.

59In addition, the southern states which were the main adopters of the RTW legislation, also were thestates that had relative low unionization rates initially. A unionization drive in the in the second half ofthe 1940s and early 1950s (�Operation Dixie�) failed, most likely because of the political economy of theJim Crow laws. Therefore, unions never gained much strength in the southern states due to a more di�cultorganizing environment.


wage premium and the organizing rate in the model with the data. Interestingly, also the

organizing rate follows a path of gradual decline that can also be seen in the data. About the

wage premium in �gure 1.4 it is noteworthy that even though it follows the same pattern as

the unionization rate, the absolute movement is relatively small: whereas the unionization

rate in this example falls from 58 % to 11%, the wage premium decreases from 41 % to 24

%. This feature is interesting because it could help to understand why the observed trend in

the data is relatively �at in spite of the long term decline in the unionization rate (see Figure

1.5). Finally, looking at the organizing rate, both the trend and the pattern of the decline

from the simulation �ts with the data (see �gures 1.6 and 1.7).60 Another explanation of

how changes in organizing costs might have contributed to the long-term decline could be

that during the years of WW II unions were not resisted as much in order to avoid labor

unrest that could disrupt defense operations. According to the data in Freeman [1997]

the aggregate unionization rate in 1940 was 26%, whereas in 1945 it was 34%. Thus, one

could argue that during wartime organizing costs were lower, and that after returning to

a normal state of the economy with higher organizing cost the achieved unionization rate

was no longer sustainable. The model therefore has an explanation of how a change in the

environment a long time ago could have triggered the long-term decline. Thus, the model

does not rely on a continuous worsening of the organizing environment that has often been

claimed but which is hard to substantiate with facts.

1.6.2 �Deregulation� and Union Decline

Several authors claim that deregulation of entry barriers is an important reason for the

decline of unions in the US. Wachter [2006] in a descriptive study interprets the post-war

economic history as one that moved from a �corporatist� to a �competitive� environment,

where the corporatist regime entails not only barriers to entry in certain industries but

60The graph of the organizing rate is taken from Farber and Western [2001]. They use data from NLRBelections which don't include all of the organizing activity.


also general price controls that were enacted e.g. during the Korean war. More speci�c

deregulations occurred in the late 1970s and early 1980s in trucking, telecommunications,

construction, utilities, and the airline industry. In the case of the trucking industry union-

ization as well as the wage premium decreased rapidly after the Motor Carrier Act from

1980 (see e.g. Clark et. al. [2002]). In the airline industry on the other hand, unionization

didn't su�er as much (ibid.), which could be the result of that even though more smaller

carriers exist now and thus on average there are about 25% more airlines operating on a

given route, the number of bigger carriers decreased substantially.61

In the model of this paper we can study the impact of deregulation by permanently

decreasing the cost of entry in a sector.62 The model delivers a similar response to the

one discussed in the previous paragraph where organizing costs were increased: Both the

unionization rate and the organizing rate decline along a gradual transition path. The wage

premium also declines, but again at a very low rate.63

Assessing the overall impact of deregulation it can be argued within this model that

since only a relatively small fraction of sectors experienced an e�ective change of the regu-

latory environment, the impact of deregulation on the aggregate unionization rate is small.

However, deregulation could potentially have contributed to the acceleration of the decline

observed during the late 1970s and early 1980s.

1.7 Conclusions

This paper developed a rich general equilibrium model of unions that decide about organiz-

ing and wages in an environment with �rm turnover. The model can account for the fact

61For a summary of the outcomes of the Airline Deregulation Act see the Encyclopedia of Economicsarticle by Alfred E. Kahn [http://www.econlib.org/library/Enc/AirlineDeregulation.html].

62The paper by Ebell and Haefke [2006] which also studies deregulation as a cause for the union de-cline models deregulation as an increase of the elasticity of substitution between specialized goods. Theirmechanism for the unionization choice is however very di�erent from the one proposed here.

63Graphs are omitted for this case.


that unions gain and lose �rms simultaneously. Two speci�cations of the union utility func-

tion commonly used in the literature have been analyzed. An important di�erence between

the monopoly union approach and the e�cient bargaining approach is that in the former

a higher unionization rate allows for higher wages, thereby giving another motivation for

union organizing.

The model helps to account for the large observed variations of unionization rates across

industries, states, and over time. In accordance with empirical �ndings, the steady state

unionization rate is higher if 1. entry costs are higher, 2. exit rates are lower, and 3. if

organizing costs are lower. Further, due to the stock-�ow approach to the unionization rate,

one time changes in the parameters imply gradual adjustments in the unionization rates.

The paper supports two possible explanations of the long-term union decline in the US:

First, a change in the legal environment in the late 1940s that is likely to have increased the

cost of organizing, and secondly the deregulation of barriers to entry in several industries

right before and after 1980. Moreover, the paper models the union wage premium explicitly.

Due to the general equilibrium structure the indirect e�ect on the non-union wage is also

accounted for. In case of the monopoly union approach, the adjustment path of the wage

premium following for instance changes in the organizing costs exhibits only relatively small

absolute changes in wages for large changes in the unionization rate, which is in line with

relatively �at wage premium that has been estimated from the data.

Regarding the union decline in the US it is true that more recently most other indus-

trialized countries have experienced such a downward trend. However, the US shows the

strongest decline as well as the one that started the earliest. Since the proposed mecha-

nism in this model is (almost) unique to the US case, an explanation for the cross country

timing di�erence could be that even though there is a general trend of declining demand

for union services, the mechanism based on the organizing friction and �rm turnover on the

unionization rate only applies for the US case.


In order to use this model to quantitatively account for the data, it would be desirable

to integrate three further aspects. First, concerning the union organizing process, also the

worker's side plays a role, even though - as has been argued in this paper - this role seems to

be limited. In order to incorporate worker's decisions, the model should explicitly include

unemployment so that workers have to balance union wage gains with unemployment costs.

Further, the organizing cost function used in the model only implicitly accounts for union

avoidance measures on the �rm side. One important strategy on the �rm side that has

been recognized in the literature is the so-called threat e�ect, which means that the �rm

strategically increases wages to avoid unionization (see e.g. Lazear [1983] and Dickens

[1986]). Third, the model only partly captures the reallocation of �rms following say a

change in organizing costs in some part of the economy (e.g. due to adoption of right-to-

work laws) since there is no growth in the model. If the economy grows then changes in

the di�culty to organize become even more ampli�ed. This is important in light of the

fact that most of the manufacturing growth in the second half of the 20th century has

taken place in the Southern states, which are predominantly right-to-work states. A further

possible variation is allowing for endogenous exit. The additional gain would be that then

predictions about unionization and �rm productivity would be possible. Exploring these

extensions as well as a quantitative application is left for future research


Appendix

A. Proofs of the Propositions

Preliminaries To simplify the arguments I introduce some additional notation:

x1 ≡ ρ1

ρ1−α ;

x2 ≡ αρ1

(ρ2−ρ1)(α−ρ2) ;

x3 ≡ αρ1−α ;

Kπ ≡ (QP1

ρ2−1 )α(1−ρ2)

α−ρ2 ( ρ1

ακ)ρ2

α−ρ2 (1− ρ1

α );

KL ≡ (QP1

ρ2−1 )α(1−ρ2)

α−ρ2 Ao( ρ1

ακ)α

α−ρ2 ;

Note that given the assumptions on the parameters, we can infer that:

x1, x2, x3 < 0 , x2 > −1, x3 < −1,and x1 − x3 = 1.

For future reference, the steady state values of u and n are given by: u = s(1−δ)δ(1−(1−δ)(1−s))

and n = (1−s)(1−δ)1−(1−δ)(1−s) .

Proofs

Proof of Lemma 1. From the FOCs it directly follows that ddlu

p(q(lu))q(lu) = w̄, i.e. marginal

revenue equals the competitive wage as is the case for the non-union �rm. Thus, the union

�rm has the same amount of employment as the non-union �rm. The wage can be expressed

as a mark-up over the non-union wage by combining the solutions to the monopolist's op-

timal choice for (as given in 1.17 with wu = w̄), the production function (q(l) = κlα),

and the solution for the monopolist's optimal price (given that both types set employment

compatible with the competitive wage):

pn = (w̄)ρ1−1ρ1−α

[(u + n)(w̄)

ρ1ρ1−α

] (1−α)(ρ2−ρ1)ρ1(α−ρ2) ( ρ1

ακ

) ρ2−1α−ρ2 Q̂

(α−1)(1−ρ2)α−ρ2 .


Proof of Proposition 1. In order to show existence given aggregates, we have to insure that

an intersection point of the two reaction functions exists. First note that both functions

are continuous. Furter, at s = 0, the union's value for e is ∞ whereas the entrant's value is

positive and �nite. Thus, it is required to have ge(1) ≥ gU (1). Using the formulas for ge and

gu one can show that this is the case if and only if ε ≤ 1−σσ η(γ − β(1− δ)), i.e. entry costs

have to be small enough relatively to organizing cost (note, if the union has all bargaining

power, and s = 1 then all �rms are unioinized in steady state and no pro�ts are generated

on the aggregate so that entry costs would have to be zero in that case). Uniqueness follows

if we impose the su�ciency criteria for strict convexity of both functions. Since two convex

functions can intersect at most twice, having ge(0) < gU (0) and ge(1) ≥ gU (1) insures

uniqueness. For existence of the general equilibrium, I refer to the discussion in the section

about computation of the aggregates. There it is argued that s doesn't depend on the

aggregates, so that {e,Q, w̄} can be determined for a given s. It is also shown that both

w̄ and Q can be written as functions of e only. Thus, an equilibrium exist if a solution to

ge(e) − e = 0 exist. Given the result in the mentioned section, the entry reaction fucntion

can be written as g(e) = Aex, where x ≤ ifρ22

αρ1

α−ρ1

(α−ρ2)(1−ρ2) ≥ 1. If it is less than zero, then

a solution is guaranteed, since ge(e) tends to ∞ at e = 0 and goes to zero for e →∞, so it

has to intersect with f(e) = e. In case x = 0, ge(e) will be some positive constant, again

guaranteeing an intersection point. Finally, if x is positive, ge is either concave or convex

and ge(0) = 0. At e = 0 there will be a trivial steady state (since all �rms are of measure

0, there will no demand even if an individual �rm would enter and employ workers). In

both the strict convex and the strict concave case there has to be an intersection with the

45-degree line. Thus a non-trivial steady state exists. Disregarding the trivial case, all the

equilibria are unique.

Proof of Proposition 2. I will employ the following informal argument: Both reaction func-

tions are downward sloping given the conditions for a unique equilibrium. Entry costs only


a�ect the entrant's reaction function. Higher ε moves the curve down. This implies that

the intersection point moves to the right, i.e. s increases. Since union and non union �rms

have the same amount of employment the �rm unionization rate and unionization rate are

identical and r increases with s according to (1.20).

Proof of Proposition 3. Lower η moves up the union's reaction function, and thus shifts the

intersection point to the right, and to a lower e. This further implies that r increases.

Proof of Proposition 4. Higher σ lowers the payo� of an entrant, therefore lowers entry for

any given s, but for the union, it increases marginal bene�t without changing marginal cost,

therefore, allows a higher e for any s. Geometrically, the union's curve shifts up, whereas

the entrant's curve shifts down. Both shifts imply higher s and lower e, thus the total e�ect

must be higher s and lower e. This in turn implies higher r.

Proof of Lemma 2. The FOC can be solved for s =(1 + x3+x1x2−(x3+x1x2+1)w

w̄

δx1(ww̄

)1−x1−δx3(ww̄

)−x1

)−1. The

result follows from the assumptions on the parameters and taking the �rst derivative of s

w.r.t. ww̄ . The bounds then follow from substituting in s = 0 and s = 1 into to original FOC

given above.

Proof of Proposition 5. From g̃e and g̃u and the results about the optimal wage, it follows

that at s = 0 the union's response function tends to ∞ whereas the entrant's response

function is a �nite positive number. The given condition on the parameters holds if and

only if g̃e(1) ≥ g̃u(1). Since both response functions are continuous for s ∈ (0, 1) this insures

an equilibrium.

Proof of Lemma 3. The proof is based on taking the derivative of ?? w.r.t. the wage ratio,

k = w/w̄, where the FOC given in 1.19 has been solved for s and substituted in. Given the

assumptions on the parameters algebra shows the result.


C. Computational Algorithm

Computation of Steady State Aggregates The following lemma gives a conditions

which simpli�es the computation of the steady state variables considerably. If the �rm's

discount factor equals unity, then the sum of the sector's pro�ts in a given period equal to

the sum of the factual entrant's expected pro�ts.64

Lemma 4. If the �rm's discount factor is βf = 1, then in steady state the sum of all

entrants' pro�ts equal to the sector's aggregate period pro�ts.

Proof. The result follows from applying the formulas for the steady state equations of u and

n to the sum of the period pro�ts:

eV egross = e (s(1− δ)V u + (1− s)(1− δ)V n)

= e(

(1−δ1−β(1−δ)(1−s)

[s

1−β(1−δ)πu + (1− s)πn

])= e

((1−δ

1−(1−δ)(1−s)

[sδπu + (1− s)πn

])= uπu + nπn

In the following I discuss the solution to the steady state aggregates for the e�cient

bargaining case and further assume for simplicity that all sectors j are identical. The

monopoly union case is similar.

The aggregates are determined by the following market clearing conditions:

QS = QC + Qσπ + Qε

and

L = ulu(w̄) + nln(w̄)

Consider �rst the goods market clearing condition. Using the optimal labor demand 1.17,

the de�nition of the price index 1.2, and the relationship PQC =∫i wilidi it can be shown

64Note, that technically even with βf = 1 there is still discounting of pro�ts, since �rm's exit at rate δ.This assumption has been employed previously by Melitz [2003].


that due to the CES assumption 1.3 that the factor shares are constant QC = ρ1

α QS . In

the e�cient bargaining version, the unionized workers also get a share σ of the (non-union)

�rms' pro�ts, so that the additional income is Qσπ = σ uu+n(1− ρ1

α )QS .

Given assumption of the Lemma 4, all the �rms period pro�ts equal to the expected

discounted value of the entrants which in turn equal to sum of the entry costs given the zero

pro�t condition. This means that Qs = ρ1

α QS +σ uu+n(1− ρ1

α )QS +eε or QS = eε(1− ρ1

α)(1− σu

u+n)

(in case of the monopoly union this reduces to QS = eε1− ρ1

α

since here workers are not

taking part of the pro�ts). Thus aggregate output is linear in e (for given s in the e�cient

bargaining version).

Turning to the labor market clearing condition, using the optimal labor demand 1.17

(note that in the e�cient bargaining case both types of �rms have the same labor demand),

we can solve for the competitive wage:

w̄ =

(L(u + n)−(x2+1)Q

−α(1−ρ2)α−ρ2 κ−1(

ρ1

ακ)−

αα−ρ2

)1/(x3+x1x2)

Thus w̄ is increasing in e.

Now, to compute all the aggregates, �rst note by setting the union's and the entry

reaction function equal to each other all terms with aggregate variables cancel out so that s

doesn't depend on them. Taking for example the entry reaction function, using the formulas

for QS and w̄ given above, one can show that

ge(e) = Ae

− 1x2 (x3 + x1x2)︸︷︷︸

<0

x1(1 + x2)2︸︷︷︸<0

+α(1− ρ2)α− ρ2︸︷︷︸

>0

where A is a term involving parameters and s. The exponent is ≤ 0 if and only if

ρ22

ρ1

α−ρ1

(α−ρ2)(1−ρ2) ≥ 1. The sign comes from countervailing e�ects of Q and w̄ on entry. Higher


wage decreases entry, whereas higher output increases it. Therefore the total general equi-

librium e�ect depends on both the marginal rates of substitution, ρ1 and ρ2 as well as the

returns to scale α. In particular, the condition always holds if α = 1.

Once we have found e from ge(e) − e = 0, we can compute Q and w̄ from the given

equations.

Computation of the Transition Dynamics The following describes the steps of com-

putation of the value functions used for the transition dynamics for the case of a Markov-

Perfect equilibrium. I use a grid for the states (u, n), but I interpolate the future values

bilinearly.

Remark: Currently I modify the entry part in the following way to avoid kink points in

the entry policy function. First, I assume that each period there is a unit mass of potential

entrants which face an idiosyncratic i.i.d. cost draw from an exponential distribution with

parameter λ. I identify entry costs now with the mean value, i.e. ε = 1/λ. This new

assumption implies that there is always some positive mass of entrants since gross pro�ts

are always greater than zero for any �nite mass of �rms. The marginal �rm then has zero

expected pro�ts, but all inframarginal have positive pro�ts.

1. Compute the steady state values of w̄ and output Q.

2. For a grid of (u, n), guess fw(u, n) and fs(u, n).

3. For a grid of (u, n), guess fe(u, n).

4. Compute V u(u, n) and V n(u, n) using linear interpolation for future states.

5. Solve for fe(u, n) from V e = 0.

6. Solve for fw(u, n) and fs(u, n) given new fe(u, n).

7. Iterate on 4. - 6. until policy functions converge


8. Repeat for new steady state and compute transitions using the policy functions.

D. Numerical Results

E�cient Bargaining

Parameter β βf ρ1 ρ2 α κ γ σ L θ

Value 1.0 1.0 .95 .8 1.2 1.0 2.5 .1 1000 1

Table 1.1: Benchmark Parameters for E�cient Bargaining Case

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 208.18 158.41 128.16 261.74 198.76 160.44 318.73 241.75 194.83

.5 225.28 172.31 139.84 282.82 216.58 175.90 343.09 263.04 213.82

2.5 234.86 180.78 147.48 292.92 225.69 184.27 353.46 272.53 222.65

Table 1.2: Comparative Statics for e

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 0.5567 0.6288 0.6866 0.4842 0.5572 0.6178 0.4223 0.4934 0.5543

.5 0.2443 0.2871 0.3243 0.1972 0.2341 0.2667 0.1626 0.1943 0.2227

2.5 0.0922 0.1104 0.1268 0.0720 0.0866 0.0998 0.0580 0.0700 0.0808

Table 1.3: Comparative Statics for r

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 0.7428 0.6836 0.6410 0.7596 0.6991 0.6554 0.7577 0.6974 0.6539

.5 0.7496 0.6905 0.6479 0.7660 0.7057 0.6622 0.7634 0.7035 0.6602

2.5 0.7529 0.6940 0.6516 0.7687 0.7087 0.6654 0.7657 0.7060 0.6629

Table 1.4: Comparative Statics for w


δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 0.7237 0.6661 0.6245 0.7401 0.6812 0.6386 0.7383 0.6795 0.6371

.5 0.7304 0.6728 0.6313 0.7463 0.6876 0.6453 0.7438 0.6855 0.6433

2.5 0.7335 0.6763 0.6349 0.7490 0.6805 0.6483 0.7461 0.6879 0.6459

Table 1.5: Comparative Statics for w̄

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 1.991 1.999 2.006 1.878 1.886 1.894 1.783 1.791 1.798

.5 1.969 1.971 1.973 1.854 1.856 1.859 1.759 1.762 1.764

2.5 1.962 1.962 1.963 1.846 1.847 1.848 1.751 1.752 1.753

Table 1.6: Comparative Statics for net premium 100[wnet/w̄)− 1

]. (Gross premium = 2.63

for all (ε, η, δ))

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 365.67 336.56 315.56 467.43 430.22 403.35 559.54 515.02 482.86

.5 369.03 338.96 318.96 471.35 434.30 407.53 563.76 519.52 487.56

2.5 370.63 341.69 320.78 473.04 436.14 409.48 565.43 521.36 489.54

·

Table 1.7: Comparative Statics for Q

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 1.1913 1.6606 2.0132 1.1350 1.5250 1.9185 1.1746 1.5779 1.9850

.5 1.1532 1.5443 1.9379 1.1017 1.4745 1.8493 1.1436 1.5297 1.9177

2.5 1.1355 1.5167 1.8991 1.0878 1.4526 1.8181 1.1316 1.5106 1.8903

Table 1.8: Comparative Statics for V u

δ = .04 δ = .05 δ = .06η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 1.2795 1.7114 2.1448 1.2305 1.6472 2.0658 1.2831 1.7186 2.1567

.5 1.2625 1.6864 2.1113 1.2121 1.6192 2.0274 1.2624 1.6865 2.1118

2.5 1.2547 1.6741 2.0940 1.2044 1.6070 2.0101 1.2544 1.6737 2.0935

Table 1.9: Comparative Statics for V n


Unilateral Wage Setting

Parameter β βf ρ1 ρ2 α κ γ L θ

Value 1.0 1.0 .95 .8 1.2 1.0 2.5 1000 1

Table 1.10: Benchmark Parameters for Monopoly Union Case

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 0.1344 0.3806 0.6638 0.1248 0.3610 0.6577 0.1174 0.3455 0.6551

20 0.0527 0.1562 0.3198 0.0488 0.1452 0.3012 0.0458 0.1267 0.2866

80 0.0208 0.0613 0.1287 0.0192 0.0568 0.1195 0.0180 0.0533 0.1123

Table 1.11: Comparative Statics for r

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 0.0203 0.0894 0.2868 0.0231 0.1000 0.3273 0.0257 0.1097 0.3653

20 0.0070 0.0245 0.0669 0.0081 0.0278 0.0749 0.0090 0.0308 0.0823

80 0.0026 0.0083 0.0193 0.0030 0.0095 0.0219 0.0034 0.0106 0.0244

Table 1.12: Comparative Statics for s

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 1.2857 1.3346 1.4037 1.2840 1.3303 1.4020 1.2827 1.3270 1.4013

20 1.2717 1.2896 1.3215 1.2710 1.2876 1.3177 1.2705 1.2861 1.3147

80 1.2665 1.2731 1.2847 1.2662 1.2723 1.2830 1.2660 1.2718 1.2818

Table 1.13: Comparative Statics for w/w̄

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 2.3910 1.4172 0.9768 2.2462 1.3339 0.9158 2.1328 1.2688 0.8680

20 2.4572 1.5109 1.0578 2.3042 1.4199 0.9956 2.1850 1.3487 0.9470

80 2.4850 1.5589 1.1195 2.3285 1.4622 1.0516 2.2066 1.3867 0.9984

Table 1.14: Comparative Statics for w


δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 508.20 63.01 15.48 476.68 59.07 14.49 452.10 56.01 13.72

20 515.77 64.45 15.49 483.40 60.46 14.52 458.18 57.35 13.78

80 519.22 65.53 15.85 486.40 61.42 14.86 460.87 58.22 14.10

Table 1.15: Comparative Statics for e

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 2439.4 1512.2 1114.6 2288.1 1417.7 1043.3 2170.1 1344.2 988.1

20 2475.7 1546.9 1115.0 2320.3 1451.1 1045.8 2199.3 1376.4 991.9

80 2492.2 1572.6 1141.1 2334.7 1474.1 1070.3 2212.2 1397.3 1015.0

Table 1.16: Comparative Statics for Q

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 0.5075 3.3023 12.6245 0.5071 3.2742 12.6987 0.5079 3.2588 12.808

20 0.4592 2.6033 9.3283 0.4617 2.5971 9.2495 0.4648 2.5986 9.208

80 0.4407 2.3209 7.5617 0.4444 2.3319 7.5574 0.4484 2.3464 7.573

Table 1.17: Comparative Statics for V u

δ = .04 δ = .05 δ = .06η\ε 1 5 15 1 5 15 1 5 15

5 1.0528 5.3955 16.8319 1.0655 5.4843 17.2931 1.0785 5.5729 17.770

20 1.0458 5.2737 16.0763 1.0574 5.3393 16.3193 1.0693 5.4056 16.563

80 1.0433 5.2324 15.7836 1.0545 5.2912 15.9742 1.0659 5.3510 16.167

Table 1.18: Comparative Statics for V n


δ = .04 δ = .05 δ = .06

η\ε .3 .4 .5 .3 .4 .5 .3 .4 .5

.1 end. e -.6603 -.6552 -.6498 -.6620 -.6583 -.6543 -.6640 -.6615 -.6589

�. e -.6438 -.6379 -.6321 -.6458 -.6406 -.6356 -.6485 -.6441 -.6399

ratio 1.0256 1.0271 1.0280 1.0251 1.0276 1.0294 1.0238 1.0270 1.0295

.5 end. e -.6691 -.6688 -.6684 -.6681 -.6681 -.6677 -.6677 -.6676 -.6675

�. e -.6599 -.6583 -.6568 -.6604 -.6604 -.6589 -.6611 -.6598 -.8586

ratio 1.0140 1.0160 1.0177 1.0118 1.0138 1.0155 1.0100 1.0119 1.0135

2.5 end. e -.6684 -.6686 -.6688 -.6678 -.6679 -.6680 -.6674 -.6675 -.6676

�x. e -.6646 -.6641 -.6637 -.6648 -.6643 -.6639 -.6650 -.6646 -.6642

ratio 1.0057 1.0068 1.0077 1.0045 1.0054 1.0062 1.0037 1.0044 1.0051

Table 1.19: Comparison of �xed vs. endogenous entry of the elasticity of s w.r.t. η.

Parameter β βf ρ1 ρ2 α κ γ L θ δ ε η

Value 1.0 1.0 .95 .8 1.2 1.0 2.5 1000 1 .06 30 20

Table 1.20: Benchmark Parameters (initial steady state) for transition dynamics

E. Graphs

Figure 1.1: Reaction Functions for Entry and Organizing


Figure 1.2: Transition after change in ε or η: unionization rate r

Source: Union Sourcebook 1947-1983; U.S. Bureau of Labor Statistics

Figure 1.3: US Unionization Rate

Figure 1.4: Transition after change in ε or η: Wage Premium


Figure 1.6: Transition after change in ε or η: organizing rate s

Source: Blanch�ower and Bryson [2002]

Figure 1.5: US Wage Premium


Source: Farber and Western [2001]

Figure 1.7: US organizing rate

Chapter 2

Time-To-Degree and Labor Market

Mismatch

2.1 Introduction

This chapter studies investment in speci�c skills under exogenous demand uncertainty as a

possible channel for labor market mismatch.

Almost all modern models featuring equilibrium unemployment use some form of a

matching friction. Often this matching friction is embodied in a so-called matching func-

tion, which takes searching workers and searching �rms as inputs, and the resulting rate of

matches as output. Using a matching function simpli�es equilibrium unemployment theories

greatly. However, not knowing the underlying mechanism of this friction has at least two

disadvantages. First and generally, such a fundamental ingredient as the matching function

should have a solid foundation in order to make models convincing. Secondly, using reduced

forms limits the usefulness of policy experiments since it is not clear if the parameters of a

matching function can be assumed to be exogenous or constant (see Lagos [2000] for this

line of argument). Relatively early in the literature it has been suggested that the matching

57

CHAPTER 2. TIME-TO-DEGREE AND LABOR MARKET MISMATCH 58

friction could be rationalized as a coordination friction resulting from some �rms getting

no worker and some workers getting no o�er given a random urn ball process. While such

pure coordination frictions certainly can play a role, it seems unlikely that with modern

information technology such a friction would matter too such an extent that could explain

the frequently observed high degrees of unemployment and excess vacancies. Further, the

urn ball model has been shown to perform poorly on empirical grounds (see Petrongolo and

Pissarides [2001] and the references given therein). Moreover, the urn ball model is sensitive

to some crucial assumptions and has some undesirable theoretical properties.1

While coordination problems are part of the story, it seems also evident that mismatch

between demanded and supplied skills play a crucial role. Labor is obviously not a homoge-

nous good, and labor markets consist of submarkets for sometimes highly specialized skill

types. Shimer [2007] is a recent attempt to model mismatch as the outcome of rationing

within micro-labor markets and thereby proposes a way to formalize mismatch without

relying on a matching function.2 In his model, mismatch occurs due to segmented labor

markets and random mobility. The labor market is subdivided in many small markets,

in each of which the short side of the market determines the outcome (which is in con-

trast to the classical island model of Lucas and Prescott, 1974). Workers and jobs relocate

randomly between markets. An appealing property of the model is that aggregate shocks

lead to a negative co-movement between vacancies and unemployment, which is a basic

feature of the data that standard matching function models can replicate. The model is

also interesting in that it gives a simple and tractable justi�cation of a matching friction

within a general (dis)equilibrium framework (wages are market-clearing on the sub-market

level, and thereby regulate aggregate job creation). A strong and crucial assumption of the

1See Stevens [2007] for a summary. Also Stevens o�ers an improved model based on coordination frictionsthat rationalizes the often used Cobb-Douglas form of the matching function.

2In many ways, Shimmer's attempt resembles the earlier literature on disequilibrium models from the1970s and early 1980s (see Shimer [2007] for references). The model has two important distinctions however.First wages are endogenous, and second, the model can be used to analyze responses two aggregate shocks.


model is, however, the randomness of mobility between markets which in some sense mimics

the coordination failure of the urn ball process.3 Why shouldn't agents at least to some

degree target their job search given the ubiquitous availability of job market information?

Moroever, the segmented submarket story allows for at least two interpretations. First,

submarkets can be interpreted as spatially divided by moving cost as in the classic island

model. Secondly, submarkets could di�er by the speci�c skills required for the tasks. There

is an important di�erence between these two interpretations. Spatial frictions in principal

can be resolved (almost) instantaneously given that wages compensate for the moving costs.

Skill acquisition, in contrast, always takes time, and in many cases substantial amounts of

time.

This paper attempts to shed further light on the sources of mismatch by looking at a

very speci�c channel, namely the problem of educational choice given uncertainty in demand

for speci�c skills. Casual observation suggests that a frequent problem of �nding a job -

even if location choice is �exible - is a matter of quali�cation mismatch which usually cannot

easily be accommodated by lower wage demands. This experience seems particularly true

for new job market entrants. The question here is how the lag between choosing to learn

a speci�c skill and the time of degree a�ects the chances of �nding a suitable job at the

time of degree, given that the worker behaves optimally when choosing education.4 Thus,

aggregate mismatch (the simultaneity of unemployed workers and job vacancies) arises due

to rationing on the skill submarkets.

In this paper, I will look at both an individual decision problem, where mismatch is

simply measured by the expected unemployment time, and a partial equilibrium model,

3The mismatch model by Lagos [2000] in some sense does the opposite: mobility is endogenous - at leaston the demand side (here: taxi cabs), whereas the mass of jobs (taxi cabs) is �xed, since there is no freeentry. It is likely that the condition for matching frictions to occur in this model hinges on the absence offree entry.

4This problem is related to the literature on irreversible investment under uncertainty (see Pindyck [1991]for a survey). The di�erence here is that in the current model there is no option value for waiting, sincethere at any point in time there is a skill type in the best state. Thus, we only have to consider the standardexpected present value of an investment.


where returns to di�erent skills are (approximately) equalized.

As a �rst attempt I develop a very simple model where a worker at time 0 can chose

between di�erent skills knowing their respective demands and the stochastic process gov-

erning demand over time. The time to degree is given, and upon receiving the degree the

worker has to wait for demand for her speci�c skill. Given an optimal choice regarding the

education decision, the chance of �nding a suitable match when �nishing education increases

with lower time to degree. This follows from the fact that lower degree completion time

allows the worker to better target her choice according to future job market conditions. In

contrast, the persistence of the demand process has two countervailing e�ects on the prob-

ability of �nding a suitable match. On the one hand higher persistence of demand allows

for a better forecast which will increase the probability of �nding the right job. On the

other hand, however, in case of demand being low at the point of degree, higher persistence

increases the waiting time. However, if one takes into account that higher persistence also

lowers the subsequent risk of becoming unemployed, the �rst e�ect is likely to be decisive

for the overall e�ect on mismatch.

The following sections present the individual choice model, then an extension to a partial

equilibrium setting, which is followed by a discussion of the results and some considerations

about possible extensions.

2.2 Model

To formalize ideas, I �rst look at an individual decision problem and then extend this model

to a partial equilibrium setting in which returns to di�erent skills are equalized through

entry.


2.2.1 Individual Decision Problem

Environment and Choices There is one worker who has to decide between two options

for education, interpreted as 2 unsubstitutable skills (A or B). Education takes n periods

after which the worker starts looking for a suitable job. The worker can only take jobs that

�t with the education she has chosen. If at the time of degree there is no demand for her

skill she has to wait until demand for her skills occurs. Demand is observable and follows a

time homogeneous Markov chain. For simplicity demand switches between the two types of

skills, so that at any point in time only one type of job is in demand (in state 1 only type

A is in demand, in state 2 only type B). The only choice the worker makes is which type of

education to chose at t = 0 based on the current state of demand and the expected payo�s.

Demand for type A and B skills is stochastic and follows a Markov chain with initial

distribution π0 = (1, 0), and the transition matrix Π =

γ 1− γ

1− γ γ

.5 The initial

distribution implies that at the time of the agent's choice always skill A is in demand.

This assumption is just for convenience. The transition matrix is parametrized by γ which

indicates the degree of persistence of staying at a given type. For the rest of the discussion

I restrict the analysis to 1 > γ > 12 .6

Payo�s are as follows. If the worker gets a job, then she will be paid w forever, if the

worker is unemployed, payo� will be b < w. Time is discounted by factor β. I will, however

abstract from discounting during the time of education, since education duration is the same

for each type of skill and therefore would a�ect payo�s symmetrically.

Solution of the model In order to solve the model it is convenient to split the analysis

into two stages. First, the agent is in education, and then enters the job market. We start

5It is straightforward to allow for di�erent distributions for the two states. For the results, the symmetryassumption is not essential.

6This rules out the uninteresting case of non-switching demand (γ = 1) and the empirically less relevantcases of extremely volatile demand (γ ≤ 1

2).


with the job market stage. Denote the distribution of jobs at this point by πn = (π1n, π2

n).

Assuming that the agent has chosen type A, and assuming that in period n type B is in

demand, the expected waiting time for a type-A job to arrive is given by: mA = 11−γ .

7 Since

the waiting time is not necessarily an integer number I approximate to the nearest integer

to mA, denoted by nint(mA). Thus, expected payo� at the end of period n, if A is chosen

at the end of period 0 is:

Vn =t=nint(1/(1−γ))∑

t=1

βt(π1

nw + π2nb)

+∞∑

t=nint(1/(1−γ))+1

βtw

Next, I compute the distribution πn given π0. In the given simple case it is possible

to summarize the n-step transition so that the resulting distribution can be expressed as a

function of n and γ:8

πn = π0Πn = (1, 0)

1

212

12

12

+ (2γ−1)n

2

−1 1

1 −1

=(

12 + (2γ−1)n

2 , 12 −

(2γ−1)n

2

) (2.1)

.

Since the probability of reaching state A given that A is the initial state is always higher

than state B after n < ∞ periods (due to the assumption that γ > 12 ), and the payo�

of getting a suitable match is higher than being unemployed, it follows that it is always

optimal to choose the type of skill which is observed to be in demand in the initial period

0.

The following paragraphs discuss the comparative statics of the unemployment duration

with respect to the duration of education (n) and the degree of demand persistence (γ).

7This follows from the equation for the �rst passage time: mA = 1 + (1− γ)mA.8The n-step transition matrix formula can be proven by induction and can easily be extended to the case

where the transition probabilities di�er across states.


Unemployment duration and time-to-degree If a job of type A is chosen in period

0 the average unemployment duration d is given by the distribution of demands in period n

and the expected time of getting an o�er of type A conditional of being in state B in period

n.

E[d] = π1n · 0 + π2

nmA =(

12− (2γ − 1)

2

)1

(1− γ)

Since ∂E[d]∂n > 0 (for n < ∞), a higher duration of education implies a longer waiting

time for a suitable job to arrive. This result is due to an increase in uncertainty about the

state of demand at the time of degree.

Unemployment duration and persistence of shocks If persistence in the demand

process (measured by parameter γ) increases two e�ects occur: First, higher persistence

implies that it is more likely to have chosen the right skill when entering the job market,

thereby reducing unemployment duration (mismatch). Secondly, however, higher persis-

tence also implies that if demand is in the wrong state when entering the labor market, it

will stay longer in that state, therefore increasing the expected unemployment duration.

The overall e�ect is ambiguous. However, if n is high, it is hard to target the right job at

time to degree and so the e�ect of higher persistence on �nding a job becomes relatively more

important, so that an increase in γ tends to increase unemployment duration. Similarly, if

persistence is low to begin with (i.e. γ is close to 12 , then again, it is hard to target the right

demand at the time of degree, and an increase in persistence mainly a�ects the possibility

of �nding a job when in the wrong state, so that unemployment duration tends to go up.

The ambiguity result, however, crucially depends on the assumption that workers cannot

get laid o� once employed. If that was the case, the higher job �nding rate is likely to be

outweighed by the higher lay o� rate once employed, leaving the e�ect of higher uncertainty

when making the education choice. The model of the next section does allow for workers


being laid o� and indicates that higher persistence increases employment probabilities of

new entrants.

2.2.2 A Partial Equilibrium Model

This section extends the model to a partial equilibrium setting where labor market entrants

have to decide n periods before actually entering the market in what skill to specialize.

Entry in the two skill types is such that expected returns to each type are (approximatley)

equalized. Further, since entrants remain in the market, there is an additional source of

unemployment/mismatch due to layo�s later in the course of employment.

As in the model before, a distribution of job possibilities for non-substitutable types A

and B is assumed, which evolves over time according to a Markov chain (π0,Π). I now

allow for a non-degenerate distribution of jobs in each state. If state 1 occurs a fraction

δ of the jobs will be of type A and 1 − δ of type B. In state 2 the job probabilities are

reversed. Each period, new entrants decide about in which skill to train, which takes n

periods. Workers already in the market can become unemployed if demand for their skill

is lower than supply. I assume that all workers in a sector at a given time have the same

(un)employment probabilities. Thus mismatch can occur not only for new entrants but also

for incumbent workers. To keep the workforce constant over time, it is assumed that an

appropriately chosen fraction λ of all employed and unemployed workers (independent of

type) exits the market forever. The model is characterized by three distributions (besides

the exogenous state): �rst, there is the distribution of employed workers in a given period

(eA, eB). Secondly, a distribution of unemployed workers (uA, uB). And �nally, there is

a distribution of new entrants (mA,mB). The distributions of employed and unemployed

workers are induced by the exogenous demand shock and the optimal education choice by

the entrants. The timing is as follows:

1. The labor demand shock arrives, determining demand of labor for each type.


2. A fraction λ of both employed and unemployed workers exits (where λ is chosen so

that mass of exiters equals mass of entrants).

3. The �xed mass of new entrants (mA + mB = c, where c is some constant smaller

than the population size) arrives and new distributions of employed and unemployed

workers are determined through labor market rationing.

4. Young workers make education decisions for labor market entry in n periods given

current state and knowledge about entry decisions of previous generations.

Equilibrium is de�ned by the condition that given the laws of motion the expected return

from each skill type is the same, whenever possible (that is entry is such that in any case the

di�erence in returns is as small as possible).9 To simplify matters, I will assume that the

value of entrants is only taking into account the �rst period upon entry. Since no further

decisions are made after entry this is likely to a�ect the results only quantitatively but not

qualitatively. The expected returns depend on the realized state today and the previous

entry decisions up to n− 1 periods before the time of the current decision, i.e. the value at

the time of labor market entry for an entrant who makes the decision n periods ahead is a

function V it (xt−n, {mAj ,mBj}t−1

j=t−n+1), where {mAj ,mBj}t−1j=t−n+1 is the mass of entrants

between the own decision time and before entry (which is an empty set for n = 1), and

xt = ((At, Bt), (eAt,eBt), (uAt, uBt)) is the observed state in t (where I abuse notation to

denote the current state of demand for the two skill types by (A,B) ∈ {(δ, 1−δ), (1−δ, δ)} ).

In the previous section workers were employed forever after �rst time hirings. Here workers

can become unemployed every period (I treat everyone having the same probabilities of

employment and unemployment whether she is a new entrant or incumbent worker of any

age). This also implies that the ambiguous result regarding the persistence parameter γ

9The reason why it is not always possible to equalize the rates is that on the one hand there are alreadyincumbent workers that determine in part the employment probabilities, but on the other hand the numberof entrants is �xed and bounded.


is unlikely to hold in this context. In the previous version of the model higher persistence

implied a better targeted decision, but a longer waiting time to employment if initially

unemployed. In this version of the model, however, higher persistence also implies a lower

unemployment risk once employed, which is likely to cancel out the second e�ect. The

rest of this subsection analyzes the model �rst analytically for a simpli�ed case and then

compares numerical outcomes of the full model.

Simple Example The following analyzes a simpli�ed one-period case, where there is only

one generation of entrants, no current employees (or unemployed) other than the entrants,

and only one period after graduation. I look at the expected number of unemployed, given

some distribution πn of the states at job market entry. Note, that in this model both

the total mass of jobs and entrants is the same (and normalized to 1). I will solve for the

expected number of unemployed (in that last period only) given this distribution over states

and the equilibrium condition that the fraction of entrants is such that the expected return

from entry in both skill types is equalized.10 I show that as the probabilities of each state

occurring become more similar (as they do as n gets larger, see Equ. 2.1), unemployment

caused by mismatch increases.

The equilibrium condition is that the returns from each education type are the same.

Denote by m ∈ (0, 1) the fraction of entrants that chose skill A. Assume w.l.o.g. that the

share of type A jobs in state 1 is δ > 12 .11 Denote the probability of state 1 occurring at

the time of job market entry by π. The values of choosing A and B are then given by:

VA = w(π ·min{ δ

m, 1}+ (1− π) ·min{1− δ

m, 1})

+b(π · (1−min{ δ

m, 1}) + (1− π) · (1−min{1− δ

m, 1})) (2.2)

10In this simpli�ed case one can show that returns can always be equalized.11The case of δ < 1

2is completely analogous. If δ = 1

2the model is trivial since both states are identical.

It is straightforward to modify the model with the fractions not being symmetric across states.


VB = w(π ·min{ 1− δ

1−m, 1}+ (1− π) ·min{ δ

1−m, 1})

+b(π · (1−min{ 1− δ

1−m, 1}) + (1− π) · (1−min{ δ

1−m, 1}) (2.3)

In each state there is a probability of employment, which depends on what fraction m

is entering skill A (implying 1−m for skill B). The payo� in case of employment is w, and

in case of unemployment b < w. Denote the di�erence by ∆ ≡ VA − VB. In equilibrium m

is such that ∆ = 0.

The goal is to show, that if π → 12 the expected unemployment u is maximized. π going

to 12 corresponds to an increase in n since the Markov chain with symmetric transition

probabilities converges to (12 , 1

2). Expected unemployment is given by:

u = π [max{m− δ, 0}+ max{(1−m)− (1− δ), 0}]

+(1− π) [max{m− (1− δ), 0}+ max{(1−m)− δ, 0}]

Unemployment is - as in the single agent decision problem above - the di�erence between

the demand and supply at the time of labor market entry (in this version, however, I do not

consider subsequent periods of possible unemployment). The following proposition shows

that if uncertainty about the the distribution after n periods increases, i.e. if π approaches

12 , unemployment is highest.

Proposition If δ 6= 12 , expected unemployment u is greatest for π = 1

2 (and decreasing in

both directions from 12).


Proof As a �rst step, I show that only the values in the restricted set m ∈ (1− δ, δ) are

relevant. Intuitively, if say π = 1, i.e. state 1 occurs with certainty, then there are exactly

δ jobs of type A and 1 − δ of type B, thus the values of entry are equalized if m = δ .

Now if uncertainty increases, i.e. π decreases towards 12 the chance that the total number

of type A jobs is lower than δ increases, thus type A jobs become less valuable relative to

type B jobs so that m has to decrease. A similar argument can be made by starting with

π = 0. Formally, assume m > δ (the case for m < (1− δ) is similar and omitted). Then the

di�erence of the values (from equ. 2.2 and 2.3) becomes:

∆ = w

(π(

δ

m− 1) + (1− π)(

1− δ

m− 1)

)+ b

(π(1− δ

m) + (1− π)(1− 1− δ

m))

= (w − b)(π(δ

m− 1) + (1− π)(

1− δ

m− 1)) < 0

Thus, for values m > δ (or m < (1−δ)) of entry in skill type A the equilibrium condition

cannot be satis�ed.

Given this result the formulas for ∆ and u can be simpli�ed. The equilibrium condition

becomes then:

∆ ≡ (w − b)[πδ −m

1−m+ (1− π)

1− δ −m

m] = 0

We are interested in how m changes if π changes, in particular, if it goes to 12 (that is

when the time to degree (n) gets large since then the probabilities of the states converge to

(12 , 1

2)). From the above condition we can solve explicitly only for π:

π(m) =(1−m)(1− δ −m)(1− δ)− 2m(1−m)

The formula of expected unemployment for the restricted domain of m is given by:


u = π(δ −m) + (1− π)(m− (1− δ))

Combining yields:

u(m) =(1− δ −m)(δ −m)(1− δ)− 2m(1−m)

The FOC w.r.t. m is:

u′(m) =(2m− 1)(1− δ)(1− 2δ)[1− δ − 2m(1−m)]2

= 0

The denominator has no zeros for m ∈ (1−δ, δ). The numerator is obviously 0 i� m = 12 .

Note, that the function u(m) is strictly convex on m ∈ (1− δ, δ) since

u′′(m) =2(1− δ)(1− 2δ)(1− δ − 2m(1−m) [−1− δ + 6m(1−m)]

[1− δ − 2m(1−m)]4< 0

This implies that unemployment is decreasing along both ways away from the maximum.

Finally, we plug in m = 12 into the expression for π(m), to get

π(12) =

12

which completes the argument.�

Numerical Simulations Due to the rationing feature the full model is hard to analyze

analytically. In particular, since total worker entry is naturally bounded given that there

are already incumbent workers in a sector there can be situations where it is not possible

to completely equalize the values of entry for both types. This section analyzes a numerical

simulation of the the full model where I compare lags of n = 0, n = 1 and n = 2 for di�erent

values of the persistence parameter γ. Since I assume that the value of entry only takes


into account the �rst period upon labor market entry, the model is easy to simulate (the

forward looking part is greatly simpli�ed). I generate sample Markov chains for the given

parameter γ for 25, 000 periods and then compute the time-averaged probability pe that

an entrant (A or B) �nds employment when entering the market.12 Since also incumbent

workers can become unemployed I also compute the average total employment rate pE to

be able to compare it to pe. Table 2.2.2 shows the results and Table 2.1 gives the values

chosen for the other parameters of the model.

Parameter w b δ Population Size mA + mB

Value 1 .5 .8 1 .25

Table 2.1: Parameter Choices

γ .5 .6 .7 .8 .9 1.0n = 0 pe 1 1 1 1 1 1

pE .820 .828 .860 .890 .936 1

n = 1 pe .7 .733 .794 .860 .928 1

pE .7 .720 .754 .804 .879 1

n = 2 pe .7 .687 .712 .776 .864 1

pE .7 .701 .712 .750 .832 1

Table 2.2: Average employment probability for a new entrant for di�erent values of γ andtime to degree (n)

The results con�rm that also in an equilibrium with entry such that both job types' val-

ues are equalized (whenever possible) a longer time to degree (n) implies higher uncertainty

and therefore a higher level of mismatch.13 Moreover, entrants have a higher employment

12Since I set population size to unity, this is the same as total employment of entrants in the period oflabor market entry.

13For the case of n = 2 there is a non-monotonicity when increasing persistence from γ = .5. Actually, asimilar non-monotonicity holds for n = 1 for a much smaller step size. When γ = .5 the average employmentpossibilities in the future period are exactly the same for each type, so that each sector attracts half of theentrants in equilibrium.


chance than incumbents due to the fact that they can (imperfectly) adapt to the demand

conditions. In case of no uncertainty for entrants (i.e. n = 0) there is no initial unemploy-

ment risk for incoming workers since they can perfectly distribute themselves across skill

types.

Heterogeneous Wages To facilitate comparison to the model in Lagos [2000], I allow for

di�erent wages for each skill type. The model in Lagos [2000] is similar to ours in that there

is a given distribution of agents on one side of the market, and an endogenous distribution

on the other. Furthemore, also in the model by Lagos, there is no endogenous entry on the

aggregate. He obtains the result that if payo�s across matches are di�erent (plus some other

model speci�c condition), then mismatch can occur on the aggregate. In Lagos' model there

is no future uncertainty since he looks at stationary states. Thus, in the following example,

in addition to the assumption that wA = 2wB, I consider the case where n = 0. As table

2.3 indicates, even under perfect foresight (i.e. γ = 1,where the initial state is assumed

to be state 1, i.e. 80% of jobs are of type A), there is unemployment /mismatch.14 This

is due to the fact, that now there is a higher incentive to invest in skill A than in skill B

due to the higher wages. Interestingly, the pattern for increasing γ is here reversed: very

low values of the persistence parameter yield higher employment rates than higher values.

The reason for that is that the more certain the future state is, the e�ect of the payo�

di�erential becomes more important, and entrants diversify less. In conclusion, our model

can also generate mismatch through a similar mechanism as in Lagos [2000]. However, the

assumption of exogenous variation of returns is not very desirable in a labor market context.

14As noted before, the initial, and therefore permanent state is A. If the state was �xed at B, thenemployment was pe = pE = .6. Thus, the high wage outweighs the low employment chances even more inthis case.


γ .5 .6 .7 .8 .9 1.0pe .871 .850 .838 .801 .763 .800

pE .760 .758 .757 .746 .729 .800

Table 2.3: Employment and Wage Heterogeneity (n = 0 and wA = 2wB)

2.2.3 Discussion

The comparative static results have shown that for identical wages across skills, �rst, a longer

time to complete a degree increases uncertainty about the state of demand at the time of

job market entry and thereby make it harder to target the education choice according to

job market conditions making it less likely to �nd a job immediately after graduation. And

secondly, lower persistence in the demand process does lead to lower job �nding rates at time

of degree (except for the case when a worker is employed forever). Lower persistence could be

one interpretation of the claimed increase in �turbulence� (see e.g. Sargent and Ljundqvist

[1998] ) that adversely a�ected continental European labor markets. Furthermore, in case

of heterogeneous wages across skill types, even with no education lag (n = 0) new entrants

have an incentive to enter a skill type even if in equilibrium there is not enough demand to

guarantee employment. This result is similar to the one obtained in Lagos [2000].

Concerning the �rst result, the proposed mechanism gives a further reason as to why

countries with longer time to degree tend to have lower college enrollment rates.15 The result

therefore supports the current EU education reform (�Bologna Process�) that shortens degree

times for many countries and makes curricula more �exible (and, of course, it supports the

current US system with multi-stage degrees and high �exibility of subject choice within a

(Bachelor) degree program).

15See OECD education at a glance 2007. The other reason is of course due to standard human capi-tal investment considerations. Longer degree times increase cost of education (but also its bene�ts), butmoreover more specialized/advanced knowledge might only bene�t people with high intrinsic ability


2.2.4 Preliminary Empirical Evidence

An important empirical check for the plausibility of the suggested channel here could be to

compare unemployment rates for graduates across countries, controlling for general unem-

ployment and other important country e�ects. Such a test is however, not straightforward.

One reason is that in almost all countries an important factor for unemployment is the

level of skill, which is not modeled here. That is, skilled workers on average have an ad-

vantage over unskilled workers. Thus, one would have to correct for di�erences in the skill

composition as well.

Another indicator whether this mechanism is relevant would be the over time variation

in demand for di�erent skills. Such an indicator unfortunately does not exist however, and

would be di�cult to obtain since there is not a straightforward mapping from learned skills

to job descriptions.

An indirect measure could be to look at the changes in enrollment across subjects over

time. Even though other factors such as interests and perhaps fashions play a role here, it

could serve as a �rst indication.

The following to diagrams give the variation of relative enrollement of a few selected

(quantitatively important) subjects in Germany (see Fig. 2.1) and in the US (Fig. 2.2).

It can be seen that there is some variation in the distribution over time, however the

year to year changes are typically not much more than 1 or two percentage points which

could either mean that demand doesn't �uctuate so much or that demand is more driven

by personal preferences. The year to year changes seem to be more pronounced in Germany

than in the US (not that only the last for bars in the US are yearly �gures, the other two are

by decade). Thus, much datawork is needed to settle the question of empirical relevance.


Figure 2.1: Relative enrollment by year, Germany, selected Subjects (Federal StatisticalO�ce Germany)

2.2.5 Possible Extensions

The simple model presented is not meant to be a realistic description but a �rst attempt

to capture the e�ect of time consuming education and imperfect substitutability of skills on

labor market matching. Before discussing several avenues of how to extend the model, it

should be noted that the basic idea of this model is more general than the speci�c interpre-

tation given here: When workers decide about a career after leaving education they again

make speci�c investments in further training, and more importantly in speci�c experience

on the job. When laid o� a lack in the skill requirements of current demand cannot easily

be accommodated without acquiring further training/education. Thus a similar mechanism

can cause labor market mismatch also at later stages in a career.

As regards to extensions, an immediate route could be relaxing some of the very strong

assumptions in the model.


Figure 2.2: Relative number of Bachelor degrees, selected years, USA, selected subjects(Statistical Abstract of the US, 2006)

Switching subjects during training or delay training: Delay of training wouldnt

change anything, since workers have no preferences over what to study and there is no

aggregate demand uncertainty, thus, delaying is never optimal.

However, it might be optimal to switch to a di�erent subject when persistence is high

enough and cost of studying are low enough (not modeled here). Introducing this feature

most likely will decrease mismatch of new entrants

Training at later stages: The current version of the model does not allow incumbent

workers to change their skill.16 Allowing for retraining will reduce mismatch. To add

this feature, it would be useful, however, to add cost of training and perhaps experience

accumulation and depreciation that would reduce the incentives of re-training.

Non-substitutability between skills: If skills were completely substitutable the model

16For indirect evidence for low substitutability across skills see the work by Kambourov and Manovskii[2008], who show that job switchers su�er much more in terms of wage losses when switching to a di�erentoccupation than within.


becomes trivial. If substitution is possible but imperfect, but payo�s are �xed, then �rms

would take in other workers as long as productivity is higher than costs, Thus, another

assumption would be that productivity of an badly matched worker is lower than its cost,

but then it would be rational for the worker to lower her wage. If that is possible then there

is no mismatch as long agg jobs=number of workers, since workers can always search on the

job. In addition to non-substitutability, we need something else, for example turnover costs

(hiring and/or �ring costs), then it is possible to have situations where a �rm would rather

wait for a suitable worker to arrive then taking in a bad match now. (A similar mechanism

could also and in addition work on the supply side: e.g. if workers can have switching costs

(e.g. due to knowledge depreciation if on the wrong job), then there is also an option value

associated with waiting for a better job to arrive).

Endogenous prices: The current model describes a dis-equilibrium since prices dont

play any allocative role (except in the case of exogenous heterogeneity in prices). One

reason for the rationing outcome is the assumed Leontie� technology of single worker �rms.

Standard multi-worker �rms, however, would eliminate all mismatch.

One other route to endogenize prices could be along the lines as in Shimer [2007], where

always the long side of the market only gets its reservation price and the short side gets all

the surplus. This would be possible to implement in this model, but would make it more

complicated (a discontinuity of the value function of skills will occur), without really chang-

ing the qualitative results since it reinforces the e�ect coming from expected employment

rates. For su�ciently persistent demand, it will amplify the supply adjustments, since there

is an additional payo� from studying the subject in higher demand. As in Shimers model,

in such a case, the results will depend on what wage to assign in case of demand exactly

equaling supply.

Another option would be to allow for endogenous �rm entry. This would make the model


also more comparable to standard labor market models.17

Free entry for �rms: In case of free entry, shocks could be introduced as stochastic

and idiosyncratic productivity draws for di�erent skills. If �rms can enter after resolution

of uncertainty, and can observe the composition of current and entering workers, entering

�rms can eliminate mismatch (to a certain extent, i.e. if the turnover is big enough relative

to the redistribution shocks). If �rms and workers have the same time-to-build restriction,

then they would anticipate the future market conditions in the same way. However, �rms

that receive an unexpected bad shock will exit, while workers have to stay. In case of an

unexpected good shock, no imbalance can arise, since entry is delayed on both sides. Thus,

an asymmetry will arise in this case.

Concerning the question of how to generate the negative comovement of vacancies and

unemployment along the business cycle (Beveridge curve), introducing an aggregate shock

is likely to produce this result, since an aggregate shock will increase �rm entry (while total

worker entry is �xed), but not (necessarily) change the relative size of the sectors.

2.3 Conclusion

The paper attempted to shed further light on mechanisms that can directly produce labor

market mismatch without relying on an exogenous matching function. The channel analyzed

here is the friction that arises when workers make education choices without knowing the

exact future state of demand and skills are not substitutable. In a partial equilibrium

model with endogenous worker entry and exogenous and inelastic labor demand it is shown

that longer times to degree as well as lower persistence of demand lead to higher mismatch

if payo�s are identical across skills. From a policy point of view the results support the

recent reforms of the university degrees in EU countries. The �Bologna Process� imposes a

17Note that no entry is an important implicit assumption for the result in Lagos [2000]


shortening and modularization of degrees for many countries (e.g. in Germany the shortest

university degree used to have a duration of 6 years, whereas now it is 3 years). From a

theoretical perspective the model lacks a good justi�cation for the inelastic labor demand,

a task that is left to future research. Furthermore, a more in-depth analysis of the empirical

plausibility is needed.

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Essays in Labor Economics - University of Minnesota