Dewey
HB31
.M415
orking paper
department
of economics
SEARCH IN THE LABOR MARKET:
INCOMPLETE CONTRACTS AND GROWTH
Daron Acemoglu
94-23
June 1994
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
SEARCH IN THE LABOR MARKET:
INCOMPLETE CONTRACTS AND GROWTH
Daron Acemoglu
94-23
June 1994
0CT21199't
June, 1994
SEARCH IN THE LABOR MARKET, INCOMPLETE CONTRACTS AND GROWTH
Daron Acemoglu
Department of Economics
Massachusetts Institute of Technology
Keywords: Search, Human
Capital,
JEL
Wage
Determination, Incomplete Contracts, Growth
Classification: J31,
O40
Abstract
This paper shows that search in the labor market has important effects on accumulation decisions.
In a labor market characterized by search, employment contracts are naturally incomplete and this creates
wedge between the rates of return and marginal products of both human and physical capital. As a result,
when a worker invests more in his human capital, he increases the rate of return on physical capital.
a
Provided that these factors are complements in the production function,
of investment for firms. Then, because physical capital
return
on
all
human
in
human
to
accumulate
capital goes up.
Thus
capital.
Applying
this
and
These pecuniary increasing returns lead
to the possibility of multiple equilibria.
They
accumulation and shows that when technology choice
wage formation
am
externality
which may lead
grateful to Paul Beaudry,
marginal product, the rate of
argument conversely, the presence of pecuniary increasing
has an important impact on growth. Finally, the paper derives
capital
its
model there are pecuniary increasing returns to scale
the more human capital there is, the more profitable it is
returns in physical capital accumulation also follows.
inefficiencies
not being paid
desired level
in this
capital accumulation in the sense that
human
is
this will increase the
to amplified
also imply that factor distribution of
new
is
lii±s
income
between unemployment and human
endogenized, search introduces a negative
to excessively fast diffusion of
new
technologies.
Roland Benabou, Olivier Blanchard, Ricardo Caballero, Russell Cooper,
Diamond, David Frankel, Per Krusell, Thomas Piketty, Andrew Scott, Robert Shimer and seminar
participants at Boston University, Federal Reserve Bank of Minneapolis, MIT, NBER Macroeconomic
Complementarities Group, Paris and Universitat Pompeu Fabra, Barcelona for helpful comments
I
Peter
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/searchinlabormarOOacem
.
1)
Introduction
Progress of knowledge
that
undoubtedly the most important engine of growth. Yet, despite the fact
is
most of the new productive knowledge can quickly spread across countries, we
significantly different long-run
growth performances. This leads many
to believe that the incentives to
acquire and apply this knowledge, therefore, the rewards to physical and
economies. Numerous factors ranging from social regard
(e.g.
observe
still
human
capital, differ across
Sawyer (1949), Baumol (1990), Cole
(1991)) to coordination across sectors (e.g. Rosenstein-Rodan (1943),
Murphy
et al (1989)) are
et al
obviously
important in this process. But economic historians also emphasize the role of institutions (e.g.
Mokyr
(1990), North (1981)) and in particular, the importance of the relations between labor and capital (e.g.
Bean and Crafts (1993), Eichengreen (1993))
to various skills
and by implication the
extended by a society
we need
rate of return
how much
market. Therefore to understand
as determinants of long-run
on physical
economic performance. Rewards
capital are
determined in the labor
of the available stock of knowledge will be exploited and
to study the organization
of the labor market and the instimtions governing
trade within productive units. If trade necessary for productive relationships does not generate a high
enough return
capital
to capital, sufficient investment will not be forthcoming.
be accumulated
This paper
various
if
starts
from
workers and firms have
to
skills are
produce. Our main argument
is
will sufficient
is
an important feature of most labor markets; both
in costly search activities
when
they are looking for a parmer to
that the presence of search will create a
wedge between
the marginal
product of labor and the wage rate (and also between the marginal product of physical capital and
of return) and that
this will
capital accumulation.
The
introduce important external effects in the process of
role of
human
not appropriately rewarded.
the premise that search
engage
But neither
human and
human and
its
rate
physical
physical capital externalities as a cause of divergent growth
performance has been emphasized by Lucas (1988,1990) and by Romer (1986). Lucas assumes
a worker increases his education,
all
exists a technological externality in
human capital
that
when
other workers also experience increased productivity, hence there
accumulation. The seminal paper by Romer, on the other
hand, assumes that technological social increasing returns are present in physical capital accumulation'
Social increasing returns as formulated
will
be low compared to the
an agent invests
'
less,
Other strands of the
first-best,
by Lucas and Romer do not only suggest
that the level of
growth
but they also lead to the amplification of the inefficiencies:
when
everyone else's output and productivity will be lower and they too will be induced
on Romer (1990) who relies on monopolistic competition and increasing
on Rebelo (1991) who takes linear accumulation rules and constant returns to scale.
literature build
returns at the firm level or
1
words, the presence of this type of interactions imply that each agent's optimal level
to invest less. In other
of investment
is
increasing in an
Cooper and John
the sense of
these models
is
that
is
it
economy wide average
(1988)).
(or that there exist strategic complementarities in
However, a possible objection
to the
mechanisms put forward by
not clear what underlies such technological externalities and also in
situations the importance of technological externalities within
capital externalities" across generations appear
more
many
a time period seem limited (though "human
plausible). This paper will
show
that
even when such
technological externalities are absent, search in the labor market will introduce pecuniary social increasing
returns to scaled within a time period. That
do not
is,
as
workers invest more in
affect others' productivity, they increase the rate
same argument
Further, the
their
human
capital,
of return on other workers' human
applies to firms' physical accumulation decisions
of Lucas and
to
capital.
and pecuniary increasing
returns in physical capital accumulation will be present as well. Therefore the externalities
do not only lead
though they
we emphasize
lower than optimal growth but they also imply similar equilibrium strategies to those
Romer and
they similarly lead to the amplification of the inefficiencies in the accumulation
process.
The
general,
effects of the increasing returns, amplification
development,
influencing growth,
potentially
However
organizational forms-'.
this
paper
is
only a
first
and complementarities that
locational
choices,
we propose can be
business cycles and even
attempt at suggesting that labor market related
imperfections and the presence of two-sided investments will lead to this type of complementarities and will
therefore try too illustrate
them by means of a very simple model. Suppose
parmership of a worker and an entrepreneur and that both parties need
that output
to
is
produced by a
undertake some ex ante
investment; the worker in
human
output will be produced
both parties are paid their marginal product and in practice there are two ways
of ensuring
this: (i)
if
human and
capital
and the entrepreneur in physical
capital.
The
efficient level of
physical capital can be traded in a competitive (Walrasian) market;
(ii)
ex
ante complete contracts can be written to determine the rewards to the different factors of production.
However, when trade
^
We
in the labor
market
is
not regulated by the Walrasian auctioneer but requires bilateral
suggest the term "pecuniary increasing returns" because the logical alternative, pecuniary externality,
often used
when
externalities,
there
is
no market
failure but only distributional effects.
As
is
in the case of pecuniary
our effects do not originate from missing markets. However, because prices are not always equal
market mediated interactions still lead to increasing remms and significant
to marginal cost in all markets,
inefficiencies.
See the recent survey by Matsuyama (1994) on the wide range of potential applications of the
complementarities (or of "circular and cumulative causation" as called by Mydral (1954)) introduced by firm
level decreasing average costs and monopolistic competition.
^
most workers cannot
search, both of these solutions run into problems. First, since search implies that
move from one
costlessly
firm to another, wages and the rate of return on physical capital will be
determined by bargaining on the quasi-rents created by
anonymity; workers do not
them and
know who
this leads to a natural
The implication
is
that
their future
this
employers will be, as a
wages and
rates of return
on
also increase.
human
Now
that
human and
capital that a firm expects to
and firms find
Consequently, the average level of physical capital also increases and
by investing
in
not paid
its
human capital,
the rate of return
on
into operation only
comes
employ
physical capital are complements, an increase in the average
capital increases the average product of physical capital
is
the rest of the story
to both factors are positively related to average products, entrepreneurial profits will
Next provided
physical capital
cannot contract with
be equal to the marginal
capital will not
worker invests more, he increases the average human
and because the return
result, they
incompleteness of contracts.
products of these factors but instead will vary with the average product.
together: as a
Second, search introduces
immobility.
the
marginal product), the rate of return on
the
human
when
now by
worker has improved the
firms respond to the
initial
more
the
human
rate of return
capital investments of other
it
profitable to invest.
same argument
capital goes up. Therefore
on physical
capital
workers (indirectly because
investment).
We
thus end
(i.e. that
and
indirectly,
this effect
comes
up with increasing returns
but these are pecuniary not technological. The same argument naturally applies to physical capital
investments, and pecuniary increasing returns in physical capital accumulation are also present.
the decision rules in our
economy resemble
those of Lucas (1988) and
Romer (1986)
As
a;id lead to
a result
amplified
inefficiencies.
Obviously random matching
(as
assumed by most search models) and high costs of changing
partners (in terms of foregone earnings in the process) are extreme assumptions and to the extent that our
results
issues,
depend on
we
these, they will
formally analyze wage determination under different matching assumptions and costs of
changing partoers.
(rather than full
competition.
have limited applicability to more organized markets. To deal with these
We
show
that very small search imperfections
randomness) are sufficient for
The important
all
feature for our results
and ex ante anonymity of matching
our results as long as they rule out Bertrand type
is
that decentralized trade should
be subject to some
transaction costs.
It is
useful at this stage to relate our
main mechanism
to earlier literamre.
most notably Diamond (1982) and Mortensen (1982) have stressed
affect the probability distribution
Although such
The search
that "actions taken
literature,
now by one
agent
over future states that others experience" (Mortensen (1982, p. 968)).
externalities are theoretically appealing, their importance
is
limited. First there
is
a counter-
acting effect; as the market
becomes more crowded some agents
will also find
more
it
difficult to find
parmers. Secondly, these externalities would manifest themselves in the form of increasing returns to scale
in the
matching technology and the evidence
Pissarides (1986), Blanchard and
Diamond
is
that the
matching technology exhibits constant returns
seem
(1989)) and thus no aggregate externalities
to
(e.g.
be present.
This paper instead shows that important externalities are created by actions that affect the value offuture
matches and analyzes
this
in a general equilibrium setting.
All our effects are derived
from market
mediated interactions (thus the term pecuniary) and do not come from the properties of the matching
technology (as
would be
it
the case in
Grout (1984) pointed out
Diamond (1982)
for instance).
in a partial equilibrium setting that
underinvestment would arise when
investors could not capture the whole of the surplus they create because of ex post hold-up problems.
first
difference
from Grout
is
that instead of
constraints, all our results are derived
contracts and
makes
and
from
bilateral search
credit imperfections unnecessary.
two sided investment and search lead
inefficiency
assuming incomplete contracts and imposing severe
to the possibly to multiplicity of equilibria.
and the ensuing pecuniary increasing returns imply that
no possible allocation of property
analysis,
no
feasible contract)
rights literature (e.g.
equilibrium setting
rights
is
the finding that
to a natural amplification of the initial
Also the
in contrast to
bilateral
ex ante investment aspect
Mortensen (1982) or Hosios (1989),
rights over the surplus (and in contrast to
would
restore efficiency. In this sense, our paper
Becker (1975)'s seminal
is
related to the property
Grossman and Hart (1986), Hart and Moore (1989)) which analyzes
how
rents.
However,
this literature
does not obtain the economy-
wide increasing returns we obtain from the general equilibrium interactions, nor does
incompleteness of contracts from bilateral search. Davis (1993) and Caballero and
qualities
in a partial
incompleteness of contracts leads to underinvestment and the allocation of property
can help by changing the division of
show how
respectively
liquidity
which introduces the incompleteness of
The more important innovation
to increasing returns,
Our
the effects
emphasized by Grout (1984)
will influence the
it
derive the
Hammour
(1993)
composition of job
and the timing of job destruction and creation decisions. Finally Van Der Ploeg (1987) uses
Grout's effect in a growth context to show the possibility of suboptimal growth rates. However,
papers have one sided investment. As a
result,
all
of these
the pecuniary increasing returns, the amplification of
inefficiencies (and the possibility of multiplicity) of our paper are absent in these models.
The
3.
The
basic
rest of the
section 3(ii),
model
is
described in section 2. The main results of the paper are contained in section
paper extends our theoretical framework and obtains a number of
we show
production. Second,
we
that there exists
new
results. First, in
an optimal distribution of income across different factors of
demonstrate in section 4 that multiple equilibria are possible. Third,
we
derive a
negative
wage formation
may be
adoption
externality in the presence of technology choice
excessively
fast
due
to
this
Finally
externality.
unemployment, show how high unemployment discourages human
Appendix B contains
a further multiplicity. Section 6 concludes.
Appendix
A
offers a general equilibriimi
in
capital
section
technology
we endogenize
5,
accumulation and thus lead to
the proofs of all the propositions while
wage determination model
that gives the
wage
rule
we use
in the
on heterogeneity and
the paper as the unique equilibrium under different assumptions
main body of
how
and show
matching technology.
2)
The Basic Model and The Competitive
We consider
workers equal
to
1
Allocation
model of non-overlapping generations. Each generation
a
and a continuum of entrepreneurs also normalized
of two parts. In the
to
1
.
consists of a
The
life
they choose their investment levels. Each worker's
first,
of each agent consists
human
involves the extent to which he wants to learn and extend the stock of knowledge that
in the
economy
education). Similarly, entrepreneurs decide
(e.g.
Production takes place in the second part of each agent's
Consumption takes place
parmership
is
assumed
to
at the
life in
to scale
(1)
capital decision
is
already present
to invest in their skills.
partoerships of one worker and one firm.
end of the period and then agents
be constant return
how much
continuum of
The production function of a
die'*.
and takes the form
yrAh:'zr"
where
h, is
the
human
capital level of the
function of a worker of generation
t is
worker and
z, is
the skill level of the entrepreneur.
The
utility
given as
1+Y
where
in the
"
It
is
the
human
capital investment,
economy defined
Our economy
is
7
is
a positive parameter and
H,., is the
we
human
capital
as^
certainly very stylized but these assumptions are only adopted for simplification.
overlapping generations structure implies that entrepreneurs have to accumulate
but
stock of
will often think of
z,
as physical capital.
Also
skills rather
The non-
than physical capital
this setting requires that agents benefit
from the stock of
knowledge of their parents but again this is not a crucial assumption. If we use infinitely lived agents investing
in human and physical capital in every period (e.g. as in Rebelo (1991)) and embed this in a search framework,
we would still obtain the pecuniary increasing remms which are the key to all our results.
'
This formulation will give us endogenous balanced growth. However,
would also apply
by (3).
in
an exogenous growth context where Ht
,
all
the effects derived in this paper
grows exogenously rather than being determined
H^-rp^hUdi
(3)
where superscript
human
denotes worker
i
capital of the worker,
h„
i
and
will
be dropped whenever
this will
cause no confusion. The
given by the following equation
is
/t,=(l-/,)(l-W-,
(4)
This formulation assumes that the worker absorbs and extends the stock of knowledge of either his parents
or of the society by his
can argue
human
capital investment,
that the cost of effort
(or alternatively higher
According to
(4),
human
human
is
proportional to
Interpreting the utility function (2) in this way,
1,.
Ht.i
because the worker has to absorb
information
this
capital inherited firom the earlier generations increases the value
capital depreciates at the rate 6 if
undertaken by the worker. The
utility is
maximized subject
no further human
to (4)
we
of
leisure).
capital investment
is
and the budget constraint
c<W=w^^
(5)
where W,
is
income
the
level of the
entrepreneur has a similar
worker and w,
is
the
wage
rate per unit of
human
capital.
Each
function given by
utility
l+Y
where
at
ej., is
time
t-1
the investment of the entrepreneur and
and
the stock of entrepreneurial skills of the
Z,., is
economy
defined similarly as
is
Z,_^=j\;_^di
(7)
and also
Zr{l-e;){\-8)Z,_,
(8)
which has a similar explanation
to (4).
Each entrepreneur maximizes her
utility, (6),
subject to (8) and the
budget constraint,
c<R=r;:,
(9)
where R,
is
A
the total
complete description of behavior in
worker and firm;
and
(iii)
skill.
a
income of the entrepreneur and
wage
(ii)
payments are
hmnan
capital
and
(b)
involves;
(i)
an investment decision for each
and a
rate of return for
each level of entrepreneurial
We start with the Walrasian system which
in time.
That
the auctioneer calls out
on entrepreneurial
capital,
and
their
given the final rewards, the ex ante investment decisions are privately
optimal.
return
economy
will be in equilibrium iff (a) given the distribution of types, their allocation
in equilibrium,
is,
the return to entrepreneurial skill.
given the distribution of types of workers and firms, an allocation of workers to firms
level for each level of
The economy
this
rt is
is
frictionless
wage schedule
and
all
as a function of
and trade stops when
all
allocations take place at
human
capital levels
markets clear. Note
one point
and
rates of
first that the allocation
problem with competitive markets
is
Kremer
straightforward (e.g.
(1993)); the
most
be allocated to the most productive entrepreneur because the marginal willingness
worker
is
highest
when
level of capital
the
is
In other words,
highest.
skilled
to
will
be matched together when they have the same ranks in
allocation, the competitive equilibriimi
will
wage
rate
and
pay for the best
i
on entrepreneurial
all
and entrepreneur
their respective orderings^.
rate of return
will
imagine a ranking of
entrepreneurs in descending order and then a similar ranking for workers, then worker
j
worker
Given such an
capital for
each pair
be determined as follows:
Given these wage
rates, the equilibrium
accumulation decisions ("saving rates") will be given
as;
/,={(l-S)w(/i,^,)}'^
.11^
e={{\-8)r{h,^)V^^
which
set the
marginal cost of investment equal to the marginal benefit. This gives us our
like all others in this
Proposition
1:
paper
is
proved
is
unique, symmetric^ and Pareto optimal. All equilibrium
economy grows
paths converge to a unique balanced growth path along which the
and has z/h
g ^=(l+{a"(l-a)'"'M(l-5)}'^^)(l-5)-l
worker
We
fiilly efficient in
will not increase the welfare of other
worker. In particular, increasing
therefore say that this
all
model does not
the sense that
different
i,
^uj(J)=
at a
where
capital investment
by a
reduces the pay-off to the
will not
is
be Pareto improving.
growth
similar to the
given point in time by investing more because
ds
(
Js:
g*^
from Romer (1986) or Lucas (1988). In Lucas'
of the external effects and moreover, each agent wants to invest more
formally define for every worker
amount
it
exhibit social increasing returns. This
and Romer 's models, an agent would improve welfare
More
more human
workers and firms more than
is
at the rate
ratio equal to (l-a)/a.
agents' investments by a small
model of Uzawa (1965) or Rebelo (1991) and
*
which
in the appendix.
The competitive equilibrium
This economy intratemporaly
first result
when
others increase their investment
and similarly for each entrepreneur
j
.
Then
i
h,>hi
and j have the same rank iff 12^(0 = fiFG)- I" the competitive allocation (or efficient matching) i and j will be
matched together iff (i) they have the same rank or (ii) when i is unmatched and has the lowest rank (nw())
among the unmatched workers and 3j*: nvv(i) = fiFC*). andj has the lowest BpC-) among remaining entrepreneurs.
^
All workers choose the
same investment
level
and likewise for
all
entrepreneurs.
because of the technological increasing returns to scale. These features are absent in
there are
no technological
externalities (within a period)
and the labor market
current generation does not take into account the positive effect that
it
there
is
no way
that the future generations
earlier ones)
redistributions
3) Search,
make
and
it is
competitive. Yet the
the productivity of future
is still
Pareto Optimal^ because
can compensate the current one for the increased investment
each generation consumes in only one period and there
compensate
model because
on
creates
generations by investing more. Nevertheless, the competitive allocation
is
this
no asset
is
that later generations
(i.e.
can use
to
therefore not possible to increase investment and then by a series of
agents better-off.
all
Incompleteness of Contracts and Pecuniary Increasing Returns
We now
turn to an
economy
which productive partoerships are formed via anonymous random
in
matching. At the beginning of the second period of their lives
(i.e. after
the investment decisions have been
made), workers and entrepreneurs are matched one-to-one, so that no worker nor entrepreneur remains
unemployed
parmer
(see section 5,
to another will
on
this).
However, despite
be costly ("search"). Note that
investment decisions are taken
first
this lack
this
of unemployment, moving from one
economy has an
explicit temporal dimension;
and allocations and equilibrium prices are determined
later.
This
is
obviously an extreme assumption; although important educational choices are taken before workers arrive
to the labor market,
an important part of the human capital
is
also accimiulated
on
the job.
Such an
extension will not however change our main results'.
Combined with
the presence of mobility costs, the timing of investment choices will introduce a
classic hold-up situation.
Although
how much
each agent has invested
may be
contractible, they cannot
write a contract with their partner to relate their payments to this investment level because the identity of
this
parmer
costly, they
is
unknown
have
at the time
of investment. Further, since changing parmers at the search stage
to share the surplus within the partoership
of making sure that he or she will receive the return of
and entrepreneurs' returns are
*
This
is
outcomes
who
invests
more has no way
higher investment. This implies that workers'
m general related to the average product of their investment (as well as,
or
obviously not the only Pareto Optimal outcome. Yet
that can
it is the only one in die set of Pareto optimal
be achieved by a competitive price system and also the one that maximizes the welfare of the
current generation. Throughout the paper
'
this
and an agent
is
Acemoglu (1993b) shows
we
will use the
term Pareto optimum to refer to
model of training if workers and firms cannot guarantee
The main externality there is between the current employer of
that similar effects arise in a
that their relationship will continue indefinitely.
the worker and his fumre employers
who
this allocation.
also benefit
from the worker's human
8
capital.
For the main resuh of the paper we
instead of, the marginal product).
lead to a
wage determination
wages and
rates of return
rule
on
whereby
will
assume
worker obtains a proportion
the
that search imperfections
of the
jS
total surplus, thus
depend only on average product. Appendix
capital
A
offers a
wage
determination model which gives this result as the unique equilibrium under a variety of assumptions on
heterogeneity and matching technology. Although other assumptions on technology and bargaining would
imply different sharing rules,
from marginal products
(i)
Search
in the
all
our qualitative results would go through whenever wages are different
irrespective of the magnitude of this
Labor Market and Pecuniary Increasing Returns
In this section a worker
random. This implies
who
is
match with
in a
change parmers so the equilibrium allocation
is
-
output
y;
obtains Wi=|8yi and matching
naturally, in equilibrium
will not
follows that the expected income of a worker witii
of entrepreneurs
total
is
worker has an equal probability of meeting each entrepreneur and likewise
that each
for each entrepreneur irrespective of their skill levels
to
wedge.
human
be entirely random.
some agents may decide
From
on
capital h; conditional
these assumptions
it
the average investment
given as
(12)
w(h,f(z;y-''di)=pAh^fiz;y"'di
The worker
will
always obtain a proportion
|3
the entrepreneur z/ as well as his investment
similarly given
of the
h,.
total
output which will depend on the capital level of
The expected income of an entrepreneur with
capital
z, is
by
(13)
R(f(h;rdi^;)=(i-p)A2!-''{(h:rdi
Total incomes are not directly related to marginal but only to average product. Consequently, the marginal
reward
to
one more unit of investment
is
proportional to average product. This implies the following
investment rules;
i,_,Hiia(i-s)Ahr'f(z;y-diy'''
(14)
e,_^={{i-m-<^){^-8>4zrf(h,Ydiy"
The
difference between these expressions and (11)
present.
However,
in (14) they take the
investment level a worker has a
first
is
worth noting. In both, pecuniary interactions are
form of pecuniary increasing returns
to scale.
order impact on the welfare of firms in contrast to (1
By
1)
increasing
where
its
the only
impact was through the equilibrium rates of returns and thus was of second-order. Moreover, when a
worker increases
his
human capital
level of all the entrepreneurs.
investment, this also has a first-order impact
And by
the
same argimient,
on
the desired investment
the increase in the entrepreneurs' investment
decision will have a first-order impact on the welfare and desired investment level of aU workers. This
is
the channel
decisions;
which mtroduces pecuniary increasing returns and amplified
by investing
less
inefficiencies in the accumulation
workers (firms) not only create a negative impact on the welfare of other
workers and firms but also reduce
their desired
mvestment
levels.
Proposition 2: The decentralized search economy with random matching has a unique equilibrium. All
equilibria paths
converge to a balanced growth path along which the economy grows
and
g''={U{a"(l-ay-"l3"(l-py'"(l-8)Ay'''){l-8)-l
is
at the rate
g''
where
g^ The decentralized economy
less than
is
Pareto dominated by the competitive equilibrium and exhibits pecuniary increasing returns in the sense that
a small increase in
agents investment will
all
make everyone
Since part of the emphasis of this paper
increasing returns,
it is
is
on
better off.
the amplification of inefficiencies
instructive to assess the quantitative difference
in the next subsection (3= a
between
share of labor in total output,
So
if
.6.
We
take
6=0, competitive growth
widens when 5
is
generations and
A
7=
rate
is
1
model
very simple,
is
/3
(i.e.
we do
will see
this as the
base
set these equal to the
This implies that g''=0.51g''-
economy grows
this exercise
become much
2%
at
larger
in each period
when we
whereas
g' is
take configurations
not have optimal distribution of income across factors). Because our
should of course be interpreted with caution but
amplification of inefficiencies through our
(ii)
and
As we
twice as high as our decentralized equilibrium. This gap
equal to 13.5%, nearly seven times faster. These gaps
not equal to
0=a
to simplify the expression.
equal to 0.569, the decentralized
is
g''.
to pecuniary
For instance with 6 = .9 which implies limited knowledge spill-overs across
positive.
where a
and
minimizes the distortions in our economy. Therefore taking
case would minimize the difference between g" and g^ Let us then take
0.495.
g*^
due
mechanism can indeed be
it still
suggests that the
quite significant.
The Role of Institutions, Property Rights and Optimal Factor Shares
The growth
shared
rate of the decentralized equilibrium,
among workers and
depend on the
entrepreneurs.
institutional structure
The
g'',
depends on
/8,
the
way
that total output
is
division of surplus between the factors will in general
of the economy and also on the allocation of property rights. Although
investment and education subsidies financed by non-distorionary taxation could clearly restore efficiency,
it
will
be more instructive
to investigate
property rights as captured by
/3
(as in
to the bilateral investment aspect,
out that the externality
whether efficiency can be achieved by changing institutions and
Mortensen (1982) and Hosios (1989)). However,
such a way of achieving efficiency
we propose cannot be
easily
removed.
10
is
in
our model, due
not possible and this again points
Proposition 3: i3=a achieves the highest balanced growth. For
all
values of
the
/3
economy
is
Pareto
more
unit of
inefficient.
The
intuition
is
human capital investment and
To minimize
a
straightforward,
(1-a)
is
captures
how much
total
output increases by one
the incremental contribution of entrepreneurial investment to output.
workers should receive a proportion a of the
distortions,
get a proportion (l-a). This also
best could be achieved by
makes
making
it
total
clear that if only one of the parties
product they create and firms
had ex ante investtnents,
that party the full residual claimant of the final returns.
case since both parties have important ex ante investments, they both need to be
claimants and this
is
their
ex ante investments. Yet,
than a straightforward reallocation of property rights. In particular, since
Although a social planner could implement a system of this
this.
In summary,
market has important
which
/3
effects
in
if
our
the share of the
this requires
upon ex
sort, the decentralized
in
full residual
known who
not
it is
a pair, efficient provision of incentives requires multilateral contracts conditional
achieve
made
not possible. Expressed alternatively, efficiency can be restored
worker (and of the entrepreneur) can be conditioned upon
However
first-
will
more
form
ante investments.
equilibrium caimot easily
our model represents the institutional arrangements in the labor
on accumulation decisions but no value of
/3
restores efficiency in this
economy'".
4)
Technology Choice, Multiple Growth Paths and Negative
Wage Formation
Externality
Section 3 demonstrated the presence of pecuniary increasing returns and amplification (or strategic
complementarities). Such interactions can in general lead to a multiplicity of equilibria, yet given the set-up
of the previous model,
equilibria
negative
we
obtained a unique equilibrium. In this section
can easily arise in
wage formation
We
this setting.
externality
modify our model such
is
will also
show
that in the
will
show
that multiplicity of
presence of technology choice a
naturally introduced.
that the production function of a
parmership
is
given as
y=Ah,
(15)
so output
We
we
is
linear in
human
capital.
Entrepreneurs on the other hand have a technology choice which
determines A. At zero cost they can choose a technology that has a marginal product A, or they can incur
an additional cost
kH,.,
and obtain the higher productivity A2. The incremental cost of advanced technology
Given the wage determination stage of Appendix A, more complicated sharing rules cannot arise in
equilibrium. However, it is also straightforward to see that rules of the form Wi=f(yj) where f(.) is a nonlinear
function but does not directly depend on ex ante investments will not help at all in achieving better allocations.
'°
11
depends on the knowledge level of the society
too. Specifically this
because entrepreneurs have to absorb
this
knowledge
assumption will ensure the existence of balanced growth paths. Again in
this section
(H,.,)
no worker or entrepreneur remains unemployed.
In this economy, there will in general be firms of high and low technology and as a result workers
may want
to switch
from one firm
to another.
Appendix
A
shows
that
under some plausible assumptions,
the possibility of such switches will not change the results of the previous section.
are constrained to choose between discrete alternatives, this
may no
longer be true. Thus in contrast to the
previous section, the size of the mobility costs (the costs of changing parmers)
this
we
reason
start
low mobility costs which
will then turn to
a)
with high mobility costs which will
However, when firms
may become
illustrate the possibility
will introduce the negative
important. For
of multiple equilibria;
wage formation
we
externality.
High Mobility Costs
High mobility
costs imply that because changing partaers
firm he
matched with and wages
prohibitively costly, a
is
worker
will
be determined by bargaining between
produce with the
first
the entrepreneur
and the worker without reference to "outside options". Therefore workers again obtain
a proportion
of the
/3
is
total output, i.e.
Wj=i8yi, as in the previous section.
of firms that adopt the high productivity technology
capital h, will
have an expected wage equal to
neutral, the optimal
human
capital investment
(16)
When
will again
at
time
t
by
T;,
Now
denoting the proportion
this implies that a
j8{(1-t,)A, +TtA2}h,. Since
from
worker of human
(2), the
worker
is
risk-
given as
is
r(T,) = {^((l-T,>4i-T^2)(l-5)}^'^
all
firms are expected to use the low productivity technology, the
human
capital investment of the
workers will be given by
(17)
i,'"{oy{^^{i-s)y'y
In contrast
when
will optimally
all
firms are expected to adopt the technology with the higher marginal product, workers
choose the higher level of human
(18)
capital investment
l,'"(l)={pA^{l-8)y'^
The
profitability of the
of the workers'
human
firms possess the
new investment
capital. In turn,
for the entrepreneurs
workers are willing
new technology because
their
workers better-off and when workers invest more
By
is
obviously increasing in the level
more
to invest
average product
section, pecuniary increasing returns are present.
is
in
human
capital
when
all
higher. Therefore as in the previous
investing in the
new
technology, firms are making
in response, all firms experience increased profits.
defining, B,=(l-5)(H-{i3A,(l-6)}"^) and B2 = (l-6)(H-/3A2(l-5)}"^),
12
the
we
have;
Now
Proposition 4: Suppose 'W=l3y„ then
if
(1-|S)A,B,
<
(l-jS)A2B,
-
k and (1-|8)A,B2
<
(l-i3)A2B2
-
k, in
every period there exist two pure strategy symmetric Nash equihbria, one in which the high cost
technology
is
adopted and one in which
new technology
Intuitively, for the
terms of our model,
However, workers
will
this
not.
to
be productive, a large scale of production
corresponds to workers choosing a high level of
will only
be the case when
it is
do
all their
when
this
they expect high rewards
same forces were present,
the
required. In
capital investment.
high average products), and
(i.e.
future parmers are expected to choose the
the previous section although the
human
is
this
more productive technology.
Cobb-Douglas functional form
In
led to a
unique equilibrium. Here due to the discreemess of the technology choice, multiple equilibria can easily
arise".
b) Small Mobility Costs
With high mobility
costs, the
indirectly, a positive externality
advanced technology creates a positive externality on workers and
on fellow entrepreneurs. When
possibility of heterogeneity across firms,
The
pairs.
partoers
intuition is clear,
when
cheap, he would want to
is
it
may no
a worker
move
is
the mobility cost
is
small and there
is
the
longer be appropriate to assume that Wj=/3yj for
all
matched with a firm of low technology and changing
unless he
paid more than a proportion
is
/3
of the output of the
low productivity firm.
Let us denote the cost of changing partners by
knows
that
he
will find a
new parmer
proportion of entrepreneurs
who
if
To
e.
adopted the more advanced technology.
its
productivity slightly above Aj
unravel the multiplicity of equilibria.
between A, and Aj and thus leads
due to the non-convexity.
This
is
It is
if
that the
In particular again
If the
his expected cost will be'^ e/r. Therefore the
she could do this
at
let
worker
r be the
worker keeps moving
see the importance of the discreetness note that in the low equilibrium an entrepreneur
increase
'^
Appendix A. This implies
as in
he incurs the mobility cost
matched with a more productive entrepreneur
"
e
wage
until
rate that
would prefer
to
a small cost. If possible, such actions would
therefore the non-convexity (discreetness) that forces her to choose
to the multiplicity.
not strictly correct in general since
The negative
we must
externality of the next subsection
is
likewise
use the conditional probability that an unmatched
more productive technology and this may be different from the proportion of entrepreneurs
who have chosen the more productive technology. But we will only look at the cases where r tends to zero and
entrepreneur has the
to 1,
and in both cases (19) applies exactly. Also note
All workers will have the
same rank
same human
that this is a special case of
capital level, h, thus iiw(i)=0 for all
as the high technology firms and they should
13
all
receive /SAzh.
i.
Lemma
3 in
Appendix A.
Therefore they have exactly the
a worker with
human
capital
h receives when matched with a low productivity firm has
him not
to switch. This equation
straightforward to explain.
is
he can get in a more productive firm minus the cost of moving
of the pie he will obtain his share and
if
it
to
The outside option of the worker
such a firm.
than his share
If this is less
amount
in his initial partnership or
he switches and obtains
expected value). In the previous sections, the outside option was never binding whereas
this in
From
increase and
equation (19),
rate that
see that as a firm invests, the outside of option of
all
firms using the old technology have to pay higher prices. This looks very
all
pecuniary externality that
rate
we can
is
what
is
more, his outside option would be binding and he would
is
obtain his outside option (either he gets this
wage
be
tvj=max {^^h, I3A^--}
(19)
for
to
also present in competitive models; as the
demand
now.
it is
workers
much
will
like a
for labor increases, the
goes up. However, a pecuniary externality in a competitive labor market will increase the wage
faced by
is
all
funis thus will not bias the technology choice. In our case the outside option
is
never binding for fums that have adopted the advanced technology and wages are set with reference to
average product. Therefore as workers increase their investments, high productivity entrepreneurs make
more
profits.
Yet low technology firms are forced
pay higher wages because the outside option of
to
their
workers becomes binding. Consequently, as more firms invest in the new technology, a negative wage
formation externality
is
created and the low technology becomes less attractive relative to the advanced
technology.
To
analyze the interaction in
technology, for
In contrast,
positive values of
all
when
this
a high
e,
case
more
carefully, let e-»0. If
no firm chooses the advanced
workers will have no binding outside option and
number of firms choose
the
more advanced technology, from
will obtain |SAh.
(19) workers can
always obtain /SAjh (either they meet or switch to a high productivity firm or their outside option
when
they bargain with a low productivity firm).
investment equal to
technology
is
l,.i(l)
as in (18) above.
proportional to {A,B2
-
strategy equilibria,
e^,
one
productivity technology
if
in
is
(l-j8)A,B,
>
-
this level
Whereas
fiA.2^i}-
technology will be proportional to {(l-j8)A2B2
Proposition 5: As
At
Anticipating this, they choose their
of
human
human
binding
capital
capital investment, return to
the return to choosing the
low
more productive
k}.
(1-/3)A2B,
-
k and A2B2
-
k
which the high productivity technology
used.
is
>
is
A1B2 there
exist
two symmetric pure
adopted and one in which the low
The advanced technology can be adopted even when
optimal to do so.
14
it is
not socially
As
wage formation
a result of the negative
diffusion of
new
technologies.
anticipate that they will
human
the higher
When
a high
economy may experience
externality, the
number of firms
are expected to adopt this technology, firms
have to pay higher wages even when they do not possess the new technology and
capital of
workers makes the new technology sufficiently
attractive.
be borne in mind that the negative externality counteracts the positive externality
far,
thus
it
may
reduce the inefficiencies. But, since
excessively fast technological diffusion
5)
Unemployment,
number of new
may
it
has to
we have emphasized
so
also result'^.
Capital Accumulation and Multiple Equilibria
we
all
relax the assimiption that
and a new source of strategic
effects
However,
can more than offset the positive externality,
it
Human
In this section
too fast
workers find employment. This will introduce a
interactions. In order to illustrate these points
we
maintain the model of the previous section but instead of a technology choice entrepreneurs have to decide
whether to be active or not. In the previous section both sides had ex ante investments and here workers
have a human
capital investment
an "ex ante investment"
The
utility
and firms have
sufficient to give us all
of entrepreneurs
now
is
to
be active or not;
t^
only takes the values
active entrepreneurs.
worker has
or
The
cost of
to absorb the stock of
1
results.
The former denotes
.
the entrepreneur deciding to be inactive
to allocate her talents to production. E,
becoming productive depends upon
H,.,
to
market becomes
entrepreneurship are related to search
become entrepreneurs
congested'''.
is
A
at
time
''•
Young (1992)
government
the
measure of
It
also depends
upon
because of decreasing returns when
t
formulation that captures the idea that the costs to
to derive ijl(E)
from the standard matching function
(1990)). According to this interpretation, each entrepreneur incurs a set-up cost
to
is
whereas
because in the same way as the
knowledge of the society so does the entrepreneur.
number of agents who decide
the entrepreneurial
'^
this is also
given by
61=1 implies that the entrepreneur has decided
Et, the
our
ex ante whether
vXc„e,)=c-n(E,)e^,_,
(20)
where
to decide
tjq
(e.g. Pissarides
and also decides how
for instance argues that such excessively fast adoption occurred in Singapore but attributes
it
policies.
This type of congestion would follow from search, product or factor market competition. If this type of
decreasing
remms were
active and one
not present,
where none
we would have two extreme
Also
from a standard matching ftmction by
are.
as noted before, the mobility cost e
relating
require the second part of each agent's
the possibility of
life to
it
one in which all entrepreneurs are
we have used so far can be derived
equilibria,
to time lost while searching for a
be specified as a continuum) and
unemployment but we preferred not
to concentrate
15
on
it
up
new
partner (which
we would
this already implicitly
to this point.
assumed
many
When
vacancies to open.
filled falls
and each entrepreneur needs
we wUl
(1989); thus in no point
ingredient
is
of vacancies.
to
vacancies, the probability that each vacancy will get
open more vacancies. Note
capture
all
(or active employers)
workers looking for a job (which
assumption for us
is
that
v<l
vacancy getting
we can assume
these features
form where the number of jobs
is
(i.e.
The
require increasing returns in the search technology.
the natural one that the probability of a
To
in particular that according to this
matching technology can be constant returns to scale as found by Blanchard and
interpretation, the
Diamond
more open
there are
equal to
in our
1
v+^ can be
decreasing in the number
filled is
that the search technology takes a
U
given as E=V''U^ where
is
V
model) and
less than or
equal to
is
number of
the
1).
crucial
is
Cobb-Douglas
number of
the
vacancies.
The key
This implies;
^(°^=''«
(21)
lim£_i At(£)='»
The
intuition is simple.
filled is 1, thus
When
all
As
the
number of vacancies goes
an entrepreneur can become active
at
to zero, the probability that a
no additional cost except for
entrepreneurs are active, each entrepreneur needs to open a very high
order to be matched with a worker. Since in general not
all
vacancy becomes
the set-up expense
number of vacancies
r/g.
in
entrepreneurs enter the market, there will be
unemployment among workers. In particular when E enti-epreneurs
are active, the
unemployment (rate)
will
be equal to Ut^l-Ef
Random matching
entrepreneur. Since
his expected
remrn
implies that each worker has a probability (1-u,) or E, of ending
we assumed
that the
worker's
is
not productive
when he
is
unmatched,
li{l-u,)Ah,
Therefore
utility
maximization implies
The investment decisions of workers and
the presence of
unemployment
the lower will the
growth
the
growth
rate of this
in (23); in particular, the higher
rate be.
However
this is
also endogenous, so the correlation between
Acemoglu
economy
is
the
be further distorted due
will
unemployment
rate of the
to
economy,
only a partial equilibrium result since unemployment
unemployment and growth
variables differ across economies and time periods (see
(1992),
capital
is
(22)
is
human
up with an
will
depend on how
Aghion and Howitt (1992), Bean and
structural
Pissarides
(1993b)).
Next we need
to determine the
unemployment rate
of the entrepreneurs. Entry will stop only
when
in this
economy. For
return to entrepreneur ship
cost, thus equilibrium requires
16
is
this
we
equal to
turn to the behavior
its
M(£,)=(l-iS>4(l-5)(l-{/3£^(l-5)}i/^)
(24)
Inspection of (23) and (24) indicates the presence of pecuniary increasing returns to scale.
entrepreneur decides to enter the market,
will
be unemployed
invest more,
all
And
falls.
workers. This amplification of the
1
made
when
open and thus the
the loss of skill of the
unemployment remains
shows some
unemployed
high. In our
in
capital but since labor
When unemployment
is
not paid
its
is
unemployment
equilibria with different
similarity to that of Pissarides
unemployment
economy, the
unemployed but via forward looking choices of
capital investment to undertake.
human
intuition of this multiplicity
and right-hand sides of (24) are both
left
more than once and
the proportion of the long-term
level of
When workers
equation (23).
-
also leads to the possibility of a further multiplicity
shows. The two curves that denote the
increasing in E,. These curves can intersect
(1992) where
more
better off because they share the increased productivity of the
initial inefficiencies
and growth rates are possible. The
is
effect
high, fewer vacancies
does not work through
workers on
the
low, workers find
how much human
marginal product, entrepreneurs benefit from
Diamond (1982) and Mortensen (1991) where
to multiplicity of equilibria.
does not depend on such technological assumptions. Robinson (1993)
it
relates
human
higher
this
also similar
capital formation to
is
However, our
result
another paper that has a similar
unemployment
we do
as
here, but again based
based on technological externalities. Finally, Acemoglu (1993b) also obtains a complementary result
it is
that high
unemployment discourages technological innovation and human
reason
that with high
is
adopting
new
At
we
is
in
technological increasing returns (respectively in the
matching technology and the production function) lead
result, especially as
more
profitable to invest
it
investment and are more willing to be active which then reduces unemployment. The result
to
an
workers are made better off because the probability that they
in response to this they decide to invest
entrepreneurs are
of equilibria as Figure
all
When
unemployment, the
technologies
this stage
it
is
is
capital accumulation
effective cost of training that firms take into account
outcome of
in their
who
human
use their
economy. Since the
will get jobs should be educated (i.e.
capital
skills.
as the decentralized
The
and some of them, just as
crucial difference
is
is
all
purpose
this
constrained by exactly the
social planner cannot
random matching),
in the decentralized
economy. For
this
define the constrained Pareto efficient allocation where the social planner
only workers
when
higher.
instructive to also look at the efficient
same matching imperfections
but the
workers
will
make
sure that
have
to invest
outcome, will be unemployed and not
that the social planner will internalize the spillovers
and instead
of setting the ex ante investment given by (23), he would choose r(E) = {A(l-6)E}"^, thus in investing in
human
capital,
he will take into account
that entrepreneurs will also benefit
Similarly, in (24) entrepreneurs ignore the positive externalities they create
17
on
from
this
investment level.
the workers, neither
do they
>
LLI
cr
U.
o ^^
£
r
3 3
CU
"S
c
^:5.Co
*•*
C/)
cc
0)
C
CD
f—
LU
=L
<
consider the negative impact they have on fellow entrepreneurs by increasing their entry costs. The social
planner will internalize
all
these effects and replace (24) by
(25)
fi{E)^EfjLXE)=A(l-8)(lHA{l-8)Ey''^)
Proposition 6: Consider the economy described above.
If 7/o>(l-i8)A(l-6),
there
exist either
one
equilibrium with no activity or three balanced growth path equilibria with different unemployment rates'^.
Pecuniary increasing returns are present. Equilibrium with lower unemployment Pareto dominates the ones
with higher unemployment and
6)
all
equilibria are constrained Pareto inefficient.
Concluding Remarks
The main argument of
on accimiulation
this
paper
is
that imperfections in the labor
decisions. Search and related imperfections create a
market have important
wedge between
effects
the marginal product
of factors of production and their rates of return, as a result they distort the investment incentives for
accumulable factors. Also because of search, a natural incompleteness of contracts
induced and
is
prevents the possibility of providing the right incentives at the ex ante investment stage.
only
tip
of the iceberg;
rate but, since the
wage
when workers accumulate
rate
is
less
human
capital, this not only
not equal to the marginal product of labor,
it
However
this
this is
depresses the growth
also reduces the profitability
of physical capital investment. As a result, there will be less physical investment but, by the same
argument,
this
reduces the rate of return on
Consequently the original inefficiency
is
human
capital
and
all
the
workers will also want
to invest less.
amplified and increasing returns are obtained, yet the externalities
are not technological but derived from interactions in the labor market.
The paper shows
that this type of
framework implies
that labor
market
institutions
have a direct
impact on the rate of growth and that there exists an optimal distribution of income across factors. Further,
a
number of other
interesting results naturally follow. First, a multiplicity of equilibria
is
shown
Second, the possibility of a negative wage formation externality with small mobility costs
Finally,
when unemployment
is
endogenized, a
unemployment and a new source of
This paper
is
a
first
new
link
multiplicity of equilibria
attempt
at
pointing
out
between human
is
capital
is
to exist.
illustrated.
accumulation and
obtained.
important
interactions
between search and
Note that because of our simple setting, the equilibrium at time t does not affect the likelihood of high or low
unemployment equilibrium at time t+1. Therefore, there exist non-balanced growth equilibria in which there
is high growth and low unemployment for a number of periods and low growth and high unemployment for
'^
some other
periods.
18
accumulation/investment decisions.
First,
there,
new
of the effects suggested in
this
paper require more research.
wage determination with matching and heterogenous agents needs
equilibrium
an attempt
Many
at this see
Shimer and Smith (1994)). Although the
may
effects
also be introduced.
effects
emphasized
For instance, interesting questions
to
to
be
in this
fully solved (for
paper will survive
be answered are; whether
with sufficient heterogeneity, small mobility costs create a "large" divergence between marginal products
and
rates of
remrn under
different modelling assumptions;
what the extent of equilibrium misallocation of
workers to firms will be in the presence of non-negligible mobility costs; what the effects of
matching are in the presence of unemployment (on
this see
Acemoglu
(1994)). Second, a
efficient
more forward
looking model of capital accumulation would be useful. Using such a model, the quantitative effects of the
mechanisms proposed
can be evaluated more carefully. Third, an investigation of the
in this paper
empirical relationship between factor shares and growth rate of output and of productivity will be
informative.
Finally and
most importantly,
it
should be investigated whether the
dealing with the imperfections created by the process of costly exchange.
the organization of the markets.
industrial
We know
development went hand
in
firom the accounts of
economy can
The
first
candidate
economic historians
hand with the development of markets
find
is
ways of
to
change
that the process of
(for a fascinating account see
Braudel (1982)) and that in general a number of markets have become more centralized in the process
while also certain others such as the labor market can be argued to have become more "decentralized".
will
be interesting
efficient are
is
to
and
to investigate
how
what the conditions for centralization and for
this
they interact with the process of (human) capital accumulation.
change the organizational forms of firms.
An example of such a phenomenon
is
It
process to be socially
The second candidate
analyzed in Acemoglu
(1993a) where changes in the internal organization of the firm correct "general equilibrium" inefficiencies
created by moral hazard and adverse selection.
No
such solution
is
possible in the
model of our paper, but
with more structure, the joint determination of organizational forms and market allocations can be studied.
I
hope
to
be able
to
pursue these issues in future work.
19
.
Appendix
(i)
A
A - Wage
Determination
Simple Model of General Equilibrium Wage Determination
Let us assume that once a pair is formed, both parties incur a cost equal to
e
when
they change
parmers. This can be interpreted as a monetary or non-monetary mobility cost or a flow loss because
finding a new parmer takes time'*. We will show that in our framework, even for very small values of e,
from the relevant marginal products. We also assume that
no agent can simultaneously bargain with more than one party and Bertrand type competition is ruled out.
Thus our bargaining stage will be similar to Shaked and Sutton (1984) which is an extension of
the rates of return will be completely decoupled
Rubinstein's (1982) framework.
following the
random matching
(Figure Al) where y
is
More
precisely
we assume
the following structure. After pairs are
stage, the entrepreneur (firm, F)
the total output produced if there
is
W,
and the worker,
agreement in
this
formed
play a subgame r(y)
subgame.
First, the firm
makes a wage demand (node A) which can be refused by the worker (node B). If the worker refuses, he
no cost make a counter offer (node C) or decide to leave and find a new parmer at cost e (to both
parties). If he makes an offer, the entrepreneur can refuse this (node D) and quit at cost e. If she decides
to continue with bargaining. Nature decides whether the firm or the worker wUl make the last offer (node
can
at
E)''
and
is
at this stage quitting is
denoted by
/3.
no longer permitted. The probability
This strucmre in a simple
way
We
choose
worker
will
make
the offer
captures the importance of institutional regulations that
determine the balance of power between workers and firms.
bargaining position and vice versa.
that the
/3
If
/3
is
high, the worker has a strong
as a parameter rather than use
an
infinite
horizon game
with alternating offers and equal discoimt factors because this formulation enables us to investigate later
whether a particular value of /3 (which can be also seen as a particular allocation of property rights on
returns)
would
internalize the externalities
bargaining, both find
new parmers
we
final
emphasize. In the case where one of the parties terminates
(incurring the cost e) and play a similar
subgame r(y') (where
y' is not
heterogeneity). The presence of the mobility cost
same as y if there is
e is the crucial
feature for us; it capmres in our setting that search and changing parmers are costly. Without this cost our
economy would be frictionless. As e becomes smaller, the amount of frictions are diminished; in the
standard matching framework where search costs are foregone earnings, this corresponds to the discount
necessarily the
rate
approaching
1
At any point in time, this economy will have a complicated history which describes the way agents
have matched and bargained up to that point. We are only interested in equilibria that do not depend on
this history in a complicated way or are Markovian and implies that all agents will play the same strategies
number of agents, when a worker
This would imply that switching
end bargaining, no new
would be even less attractive thus bias the result
in our favor. Since we have a continuum of agents, we can avoid this problem by assuming that there always
exists some unmatched agents so that switching partners is always possible, yet the set of unmatched agents may
be of measure zero. This modelling assumption is also a convenient substimte for the more rigorous strategy
of looking at the steady state of a bargaining market with entry and exit (e.g. Osborne and Rubinstein (1990)).
'*
With a
partners
finite
may
exist.
(or an entrepreneur) decides to
partners
We have chosen Namre to move at node E to simplify the game tree, however an alternative game form
where the worker and the entrepreneur asymmetrically alternate in making offers would do equally well as long
as there is some sort of discounting (e.g. a small probability that the game will end without agreement after a
rejection). However, applying the Nash solution irrespective of the extensive form would not be satisfactory
since this may not correspond to the equilibrium of a well specified bargaining game (see Binmore, Rubinstein
and Wolinsky (1986)). It can also be argued that in Rubinstein's (1982) framework with alternating offers, /3
will only depend on discount factors. But, in real world bargaining simations many other features seem to be
"
more important
in determining the relative bargaining strength.
20
nP>
CD
dd
O
c
.[]
-<
J}
Q
C
m
(D
-<
(D
-<
in each completely identical
thus
we
1:
r(y).
Markov
are only interested in
Lemma
is
subgame
We
would also want
all strictly
positive values of
Proof: Consider the extensive form in Figure
entrepreneur by V^^ which
we
subgame
perfect,
Perfect Equilibria (see for instance Fudenberg and Tirole (1991)).
With homogenous agents, random matching and
the unique equilibrium for
the equilibrium to be
Al and
let
the
subgame r(y)
as described above,
W|=(8y
e.
us denote the
will try to determine. If the
supremum expected remrn of
game reaches node
(l-B)y. Next consider node D; since
the
E, the expected remrn of
worker is By and that of the firm is
same game, the maximum the entrepreneur can expect from changing parmers is V^^-e. On the other hand,
she can say no and stay in, reach node E and obtain (l-6)y. Since we are looking for the supremum of her
equilibrium pay-offs, let her return at D be given by the maximum of these two amounts. Therefore at node
C the worker has to offer the firm max{V^E-f,(l-B)y}. Now move to node B, the worker can say no and
continue with bargaining in which case the game will proceed to node C and he will obtain y-max{V^£-e,(l6)y}. Alternatively, he can say no and leave to obtain his outside option. Since we are after the supremum
of the pay-off set of the firm, let us suppose that the worker gets his infimum, denoted by V'^. It then
the
follows that at node A, the
supremum of
(Al)
Now
all
the pay-off set of the entrepreneur
pairs are playing the exact
is
F/=y-max{Fj,-e,y-max{F/-e,(l-/3)>'}}
noting that V^£-I-V',^,=y, this equation has a unique solution which
We
can
now
it is
This
not equal to zero) in
lemma
tells
is
a unique equilibrium irrespective of the value of
which the worker receives By.
us that even with very small search frictions, there
that
when he
left
would get her marginal product,
equilibrium, when the worker leaves, he
if
the firm he meets
is
the pair
would
The inmition
simple. If the
is
e.
However,
in
will meet another firm and enter a very similar bargaining
exactly the
credible) and bargaining will take place over the
defmed
wedge between
a large
just bargain over the surplus,
same
as the
one he
expect anything better and hence his outside option will not be binding
wages
may be
the firm, he would get his marginal product and the entrepreneur
likewise
simation. In fact,
e (as
QED
the marginal product of factors of production and their rates of return.
worker could ensure
V^E=(1-B)y.
repeat the above argument with the infimum of the pay-off set which will imply that
V'E=(1-B)y. Thus given the extensive form, there
long as
is
is
bargaining with, he cannot
(i.e.
the threat to quit
whole of the pie and the "inside" option
is
not
will determine
remrn from r(y) without the right to exercise the outside option).
(1971)'s famous result and to Shaked and Sutton (1984). It
follows from this results that if we augment the model of the paper with the wage determination game
outlined here, the result of Proposition would be the unique symmetric equilibrium. The analysis of the
("inside" option
This intuition
is
is
very similar
to
as the
Diamond
next subsection shows that in some other cases
The
we can
limitations of this result should also
actually also rule out
be noted.
First,
of heterogenous agents because the worker (or the entrepreneur)
will return to this point in the next subsection
where we
will
it
non-symmetric
may be moving
show
to a better partner.
same
results if e
competition so that a worker
becomes
is
arbitrarily small.
Second,
we
We
that with efficient matching, the
presence of heterogenous agents does not change our results and with random matching,
exactly the
equilibria.
does not always hold in the presence
we
again obtain
are not allowing Bertrand type
never able to bargain with two firms simultaneously. If
we allow
a firm
meet two workers (or vice-versa) with a certain probability, then there will be a closer relationship
rates of return and the marginal products but the competitive outcome will not be achieved
unless we remove the mobility costs completely. It is natural that in our context that two workers would
never simultaneously meet an entrepreneur and vice versa. If a worker is bargaining simultaneously with
to
between the
21
two firms they
who
Thus ex ante a firm
will both get zero surplus.
or employed by,
already in negotiation with,
is
will
have no incentive
worker
to contact a
another firm. Third, although
I
believe this
formulation, where each party has the right to exercise the outside option, captures the most important and
we know from the literature surveyed and analyzed by Osborne and Rubinstein (1990) that
when Namre decides when bilateral bargaining will end and parties will be forced to take their outside
realistic effects,
options, bargaining markets converge to the competitive equilibrium as the discount factor tends to
1,
e.g.
Gale (1987)'^
(ii)
Robustness of Wage Determination: Heterogeneity and Efficient Matching
The divergence between marginal products and rates of return plays a crucial part
in
our
results.
The first is random matching; the second is to
homogenous
agents
in Lemma 1) and the third is that multilateral
(i.e.
before the investment stage. Relaxing the last assumption would take us away
Three assumptions have been used
to obtain this result.
concentrate on symmetric equilibria
contracts cannot be written
from
we
the investigation of the decentralized equilibrium" (but see concluding remarks). In this section
will relax the first two.
Agents certainly do not randomly run into each other when they want
distribution of characteristics influence
To
results?
investigate this
efficient, initial
we
who
that the highest skilled
the highest capital. In other words, the initial
foomote
6).
i
whom.
to trade. Prices
random matching
Is
and the
the source of our
define the polar case; efficient matching. If the matching technology
matches are organized so
in section 2; for instance
will trade with
and
j
match
will
worker
is
matched
is
to the entrepreneur with
be the same as the competitive allocation analyzed
will put together in the initial
match
if
they have the same rank (see
This implies that in the presence of efficient matching, more investment increases not only the
average product but also the likelihood of ending up with a good job. Such a matching technology
may
in
general take us nearer to a competitive allocation where wages not only distribute the final returns across
different factors of production but also allocate the "right"
Lemma
Let us assume that matching
2:
determination stage. Denote the
equilibrium
we have
efficient
i
entrepreneurs and workers).
parties switches, they will
and entrepreneur
We
move
j
j
to the "right" jobs.
and agents are potentially heterogenous
output of worker
Wi=/3yij where entrepreneur
Proof: Consider worker
'*
total
is
workers
i
with entrepreneur
and worker
i
j
by
yy.
Then
at the
in the
wage
unique
have exactly the same ranks.
such that 12^(0 =0 and fip(j)=0
(i.e.
highest skilled
want to establish that Wij=jSyjj for this pair. Suppose one of the
subgame r(y) where y<yij. Suppose for concreteness that the worker
first
to
The
closest strucmre to our paper is in Bester (1989) which has a spatial competition model where moving
between locations is costly. He shows that as this cost (which corresponds to our mobility cost, e) tends to zero
and discounting disappears, the equilibrium converges to the competitive outcome. However, this is due to the
assumption that at the beginning of each subgame Namre decides who will make the first offer and there is
discounting within the bargaining subgame. Thus with the cost of moving going to zero, any buyer
the first to
make
the offer can quit
and eventually find a match where he
will be the first
one
to
who
is
not
do so and vice
versa for the seller. This implies that with the mobility cost arbitrarily close to zero, the outside option cannot
be worse than the inside option which can only be true in the competitive equilibrium. In our case, after a
separation, agents find themselves at an identical node and thus even for very small values of the mobility cost,
the inside option may be preferred to the outside one.
''
Also
this
would require the relaxation of
(1990) show that
when
this
assumption
is
the ex ante anonymity assumption but Rubinstein
and Wolinsky
relaxed there exists no limit theorems for the decentralized equilibrium
converging to the competitive outcome.
22
meets a firm j* with
Zj.
can only get more than
of the
match,
initial
< Zj
Wy
off for
all
if
i
an equality the argument of Lemma
the entrepreneur gets
But
/Syj.j..
1 applies). Obviously, the worker
below her inside option when matched with the worker
y^. <yij also implies that the loss that entrepreneur j* is
worker
larger than the gain of
equilibrium, worker
(if this is
i.
Thus
worker
if
i
wanted
to switch,
making must be
she would too. Therefore in
cannot get a higher wage in any subgame r(y) and hence would end-up
positive values of
The same argument
e.
applies to entrepreneur
j.
Now
worse
strictly
given that the highest
ranked workers and entrepreneurs choose not to switch, the same argument applies to the next highest
ranked and they too can only move to a worse match, so the outside option
Thus
products
is
this
lemma demonstrates
that the divergence
between
is
not binding, etc.
rates of returns
QED
and the marginal
not due to our random matching assumption but caused by the presence of transaction costs of
new parmers). Next we can also state
but only when e is arbitrarily small. Yet
decentralized trading (costs of breaking-up parmerships or finding
a similar result for
random matching with heterogenous agents
to characterize the fiill equilibriimi when e is
we have been unable
at the
wage determination
Lemma
stage and matching
assume
3: Let us
that
matching
is
is
non-negligible, agents are heterogenous
random;
random and agents
are potentially heterogenous at the
wage
determination stage. Denote y^ as the total output of worker i with entrepreneur j. Then as e-^, the unique
equilibrium is Wi=/3yij where entrepreneur j has exactly the same rank as worker i.
Proof: Since the mobility cost
workers to entrepreneurs
to a better partoer
-
is
as in the proof of
Lemma 2
to the outside option. Therefore, for all
Both
efficient
small, agents will switch partoers until the best allocation of
e is arbitrarily
achieved. But in this allocation, no worker (or entrepreneur) can be
i
and
j
-
and since
e is still positive, the inside
of the same rank, Wjj=/3yjj.
(i.e.
the
most
skilled
is
preferred
QED
matching and random matching with small mobility costs lead
heterogenous agents are matched in the "right" way
option
moving
where
to a situation
worker with the most
skilled
entrepreneur). But given this allocation, the outside options will never be binding because no party can be
moving
to
a better match;
entrepreneurs.
Now we
can
thus the
option determines the wages and rates of return to
"inside"
state;
e^,
the equilibrium of
economy even when agents
are allowed to use
Corollary to Proposition 2: With efficient matching or with random matching as
Proposition 2
is
non-symmetric
the unique equilibrium of the fully specified
strategies.
Proof: Consider a case in which two workers of
the first
above
worker has higher
h, (by continuity)
utility
and
hj
> hj,
human
capital h,
and hj have different
utility levels. If
then the second worker can increase his investment to slightly
and get a better rank thus the same job and the same
utility (or arbitrarily
close
it). If h, <h2, this time he can reduce his investment to slightly above h, and the same argument applies.
Hence a contradiction and all workers have to obtain the same level of utility and the same applies to all
entrepreneurs. Given that the problem we have is strictly convex, this can only be possible, if they all
choose the same level of ex ante investment and this gives exactly the same outcome as Proposition 2 as
to
the unique equilibrium
The
3
still
intuition
of
even considering non-symmetric outcomes.
this corollary is straightforward.
QED
The wage determination
rules of
Lemmas
give concave pay-off functions to each agent which leads to a situation in which they
23
all
2 and
choose the
"
same
and in
levels of investment ex ante
Appendix
B
-
same
this case the result is exactly the
as Proposition 2^°.
Proofs of Propositions 1-6
Proof of Proposition
1: (11) defines
we
optimal decision rules. Substituting from (12)
get
——-=—a K^—
//!+/,)
(A2)
for all pairs. Balanced
growth
is
e,il-e,)
l-aH,_,
when
growth
only possible
the
rate of Z;
and that of H, are equal which
implies investment levels
^^^^
l=e={a''(l-ay'"A{l-8)}
(A3) makes sure that both Hj and Z, grow
Equation (A2) also implies that
capital
is
z/h ratio
if
wUl be accumulated, and vice versa
same
at the
and output
than (l-a;)/a,
is less
the ratio
if
rate
is
more
more than
will also
grow
at this rate
entrepreneurial capital than
(l-a)/a.
Thus
the balanced
g^
human
growth path
globally stable.
Pareto optimality follows from the observation that this allocation maximizes the welfare of
generation
t
given
H(.,
and
Z,.i
and
that there is
no
possibility of redistribution welfare across generations,
so increasing the investment of current generation to improve the welfare of future generations will not be
a Pareto improving action.
Proof of Proposition
their
problem
is
2:
QED
Equation (14) implies that
concave, thus they will
argument applies
to entrepreneurs.
all
Imposing
this
'^
(A4)
for all pairs. Balanced growth
is
all
workers face the same distribution of returns and
choose the same level of human capital investment and the same
only possible
condition
'
obtain
^
=
when
we
the
^
growth
rate of Z,
and
that of H, are equal
which
implies
2
y^^)
7_„_/„,a/i
and therefore the economy grows
z/h ratio
^
„,\l-aQa/i
o\l-a
l=e={a%l-ay-''P''(l-py-''A(l-8)}''
is
at the rate
g**
Efficient matching
human
is
not always innocuous.
By
as given in the text.
different than the one consistent with balanced growth,
When
the
same argument
i.e. Q;j8/(l-a)(l-(8),
as above, if the
transitory
dynamics
workers face the risk of being unemployed (as in section
workers to increase their likelihood of being employed and this may lead to
an "education race". The intuition is that with unemployment, there is ex post heterogeneity and workers will
5), higher
capital enables
change their ex ante investment decisions to influence their likelihoods of ending-up in different groups.
the model of the section 5 we change the matching technology to efficient matching, an equilibrium fails to
(details available
from the author). These
issues are further investigated in
in the presence of non-convexities, small values of
e
will lead to
24
new
Acemoglu
effects
(1994). Also
which we study next.
If in
exist
similarly,
economy back to balanced growth.
Both on and off the balanced growth path,
will bring the
levels, all agents
prove
to
is
make everyone
be proportional
better-off.
is inefficient.
The
a small increase in
last thing
all
agents
Consider such a change. The impact on the welfare of a worker
{l-S)(l-a){l-p)Ah"z,'"dz^H(l-S)a^hrW'yi'}dh,
coefficient of dh,
worker
will
better-off.
all
Therefore the dynamic equilibrium
off.
to
(A6)
The
a Social Planner imposes the competitive investment
that pecuniary increasing returns are present in the sense that
investment will
will
can be made better
if
is
be positive.
equal to zero by the first-order condition and thus the effect on the welfare of the
A
similar term applies for the welfare of the entrepreneurs
Thus a small increase
in the investments of all current agents
future agents) better-off and the
Proof of Proposition
equilibrium
is still
economy
3: Straightforward
Pareto dominated for
is
current agents (and so,
all
subject to pecuniary increasing returns.
maximization of
all
makes
and they too are made
values of
/3.
g**
QED
gives 13= a but by the above argument the
QED
Proof of Proposition 4: If an entrepreneur chooses the advanced technology when all others are expected
to choose the low technology, workers will only invest 1,(0), thus the entrepreneur will make {(1-/3)A2B2k}Ht., and if this is less than {(l-i3)A,B,}H,.i, there exists an equilibrium in which the advanced technology
is
not adopted. Conversely, the profit to adoption
when
others are anticipated to adopt
all
is
{(1-|S)A2B2-
workers now choose 1,(1)); if this greater than {(l-i8)A,B2}H,.,, then there exists an equilibrium
which the new technology is adopted. Since both conditions can be satisfied simultaneously, there exists
k}H,.i (since
in
multiple equilibria.
QED
Proof of Proposition 5: First consider the case in which e is positive and t tends to zero (i.e. all firms
choose the low productivity technology). This implies that the worker will have to switch infinitely many
parmers before finding a firm with the high technology and thus his outside option is not binding and he
always receives /3A,h. This gives us the same value of
1,(0)
and the same condition as
in Proposition
4 for
investment in the less productive technology to be an equilibrium.
Next consider the case with
t-^1
and e-^. Equation (19) for the wage
rate implies that a firm
without the more productive technology has to pay exactly the same wage rate as the more productive
firms.
Thus
the return to the
advanced technology
is
low productivity technology
as before {(l-/3)A2B2-k}H,.i.
is
{A,B2
-
/SA2B2}H,.,
The comparison of
whereas the return
these
to the
two amounts gives
the
new technology to be adopted as A2B2- k > AjBj.
contrast for the new technology to be adopted in the first best we need;
2
j_
A^{1HA2{1-S)}'')~k-AUC>A,(U{A,(1-S)}y)
condition for the
In
(A7)
A^^-k-^UC>A^B^
where AUC.H,.,
(A7)
is
more
is
the utility cost of higher investment for each worker.
restrictive than
A2B2-k> A,B| even when
AUC
technologies that would not be adopted in the first-best can
if e is
small enough.
is
still
It is
straightforward to see that
set equal to zero. Therefore,
be adopted
in the decentralized
advanced
economy
QED
Proof of Proposition 6: Random matching again implies that all workers are facing the same convex
maximization problem and they will all choose the same level of investment. Thus the intersections of the
two curves denoting the right and left-hand sides of (24) characterize all the equilibria. When no
25
human capital at all. If rjQ is greater than (l-/3)A(l-5), an
who deviates and decides to enter would not make enough profits to cover her start-up costs
human capital investments of the workers. However, we also know from (21) that if all firms
entrepreneur enters, workers do not invest in
entrepreneur
given the
want
to
be active,
this is infinitely costly for the last firm, thus in the
neighborhood of
1, fi(E) is vertically
to active entrepreneurship. Therefore, if the two curves will intersect
must intersect at least two more times. The presence of pecuniary increasing returns can be
demonstrated by exactly the same method as in the proof of Proposition 2. To analyze the inefficiency,
denote the efficient allocation as (£',1') such that E' entrepreneurs become active and each worker chooses
above the curve that denotes returns
at all they
investment
1'. It
decentralized
follows from comparing (24) and (25) that
economy
is
constrained Pareto inefficient.
26
if
QED
E=E', then
1
caimot be equal to
I',
thus the
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