Pre-Contractual Investment Competition in a
Large Matching Market
Seungjin Han∗
University of Toronto
December 21, 2001
Abstract
Workers and firms may underinvest due to contractual externality associated with investment. This paper considers the situation in
which the surplus depends not only on investments but also on a firm’s
type and workers value physical capital as well as monetary payment,
so contracts are two dimensional. Investment decision is more complicated in the sense that workers disagree ex post on whether or not
additional physical capital investment makes the firm more desirable.
The ex ante attribute competition makes ex ante the monetary payment decision for human capital, the part of contract. Workers compete for a better contract with human capital and firms use contracts
to compete for high human capital prior to match. In a large market,
the higher marginal value of the equilibrium contract as a whole on
high human capital internalizes the externality so the ex ante attribute
competition reaches full efficiency in a rational expectations equilibrium. The ex ante attribute competition derives efficient investments
in a situtation where the equal treatment property for investment does
not hold.
∗
Department of Economics, University of Toronto, 150 St. George Street, Toronto,
Canada M5S 3G7. Corresponding e-mail address: shan@chass.utoronto.ca. I am very
grateful to Michael Peters for his insightful guidance at every stage of the research. Comments from Hao Li and Aloysius Siow is thankfully acknowledged. All errors are mine
alone.
1
1
Introduction
In many cases a firm makes investment decision on his machinery (physical capital) and a worker does on her skill (human capital) prior to match.
When they match and write contract, capitals are sunk. So contract often
distributes the surplus such that an agent does not capture the total benefit
of his/her own investment and this leads to underinvestment when an agent
perceives this problem ex ante. This is referred to as the “hold-up” problem
in the literature.
The “hold-up” problem was one of the important issues in the literature
on contract or ownership of assets.1 Conventional institutional or contractual
approach often takes outside option as exogenously given or considers a match
in isolation: outside option and investment decision will interact each other
since one determines the other at least partly. The contractual solutions
in the literature are quite sophisticated and complex in contrast to simple
contracts observed in practice.
The hold-up problem has been projected in a matching environment
where many agents make investment decisions prior to match in order to
endogenize outside option and investment decision. Burdett and Coles [4]
endogenize pre-marital investments in the presence of search frictions with
nontransferable utility. When one side of market undergoes costly investment
so as to elicit proposals from the other side, singles become more selective
so it may create a situation where everybody has to invest much to remain
“acceptable”. If there are search frictions, then a future partner of an agent
will benefit from her attributes. Underinvestment may be prevalent due to
the uncertainty about future (Acemoglu, [1]).2
Recently, research has focussed on the possibility that the matching process
by itself might be enough to provide correct investment incentives without
the need for complicated bilateral contracts. It will be referred as “direct
matching design” in this paper.
One approach is to formulate a matching process that provides explicit
monetary rewards for worker and firm investment. In Cole Mailath and
1
Grossman and Hart [8], and Holmstorm [10] discuss the effect of ownership right on
investment decision when complete contracts are impossble. Bilateral contractual solutions
are derived in Rogerson [12], Aghion, Dewatripont, and Rey [3] and Gul [9] through
renegotiation, default option, and repeated bargaining.
2
In the framework of one-side investment decision, some of papers find the remedy to
the hold-up problem. See Acemoglu and Shimer [2] and Shi [13].
2
Postlewaite [5], [6] the matching process is formulated as a bargaining game
with transferable utility. The ex post core of assignment at the matching
stage generates monetary payments that workers and firms take into account
when making their investment decisions. When a worker, for example, thinks
about his human capital investment he is motivated by the monetary reward
that this core provides. In Felli and Roberts [7], the ex post monetary payments are generated in one side investment decision in a finite market by
carefully specifying a non-cooperative bidding game in which workers can
bid monetary rewards for firms’ investment.
Peters and Siow [11] eliminate all ex post monetary transfers and ask
whether assortative matching by itself is enough to provide correct ex ante
investment incentives in a large marriage market. The return that a male
single gets from investing in additional human capital is simply the female
single’s high human capital investment that the male eventually marries. So,
investments are pure public goods. In the first stage, all singles believe that
everybody faces the same non-stochastic return function that maps from a
female single’s investment to a male single’s investment. They show that in a
rational expectations equilibrium in which all traders understand the true relationship between their own investments and the quality of the partner they
will attract, the hold up problem can be resolved. A key part of the Peters
and Siow argument is that ex ante investment decisions are one dimensional.
A female single who invests in additional her human capital makes herself
more attractive to every male single on the market, and similarly for male
singles.
The match considered in the direct matching design is anonymous in the
sense that surplus depends on investments but not on an agent’s type and
reward for investment is one dimensional: monetary payment or a partner’s
investment. This paper considers a model in which surplus also depends on
the firm’s type. The matching design and the model in this paper makes the
firm’s ex ante decision two dimensional and the contract also two dimensional.
Workers care the overall job environment such as monetary payment and
physical capital of firms. The value of physical capital depends on the level
of workers’ human capital. A worker’s human capital investment changes her
value of a contract, so contracts cannot be ordered in a way that everybody
in the market agrees on their values. In the equilibrium described below,
workers will disagree ex post on whether or not additional physical capital
investment makes the firm more desirable. A firm’s investment again becomes
specific in the sense that it may be attractive only to certain types of worker.
3
The matching stage equilibrium in the bargaining game is characterized
by an equilibrium bargaining rule which specifies a stable bargaining outcome
in all possible sets of capitals and derives the ex-post core of assignment with
a given set of capitals. An important property of a stable bargaining outcome is the equal treatment. This means that the same level of capital gets
the same monetary payment regardless of agents’ type and match. The bargaining game assumes that in a large market there exist two market reward
schedules, one for human capital investment and the other for physical capital
investment, based on the equal treatment property. If the surplus depends
on type, then the equal treatment property does not hold in a stable bargaining outcome. The assumption is no longer valid in the more general model
this paper describes because the equal-treatment property does not hold in
a stable bargaining outcome.
One can endogenize both investment decision in the bidding game. Firms’
marginal incentives to undertake investments are determined by their outside
option that depends on the surplus of the match between the firm and the
immediate competitor to the worker the firm matches with in equilibrium.
Firms are not able to capture all the increase in the surplus by increase in
physical capital investment. Both workers and firms undertakes investments
strictly less than the efficient level regardless of market size. Suppose that
the monetary payments are a predetermined proportion of a firm’s output:
the capital investments are pure public goods. This changes the preference
itself so the matching competition does not lead to efficient investments.
We are primarily interested in the investment competition in a large market. In a small market, capital investments are discontinuous. An agent
would like to decrease capital investment, and monetary payment if he is a
firm, given other agents’ decisions as long as she can keep the relative position. A pure strategy equilibrium may not exist in some situations. Even
in a large market, where capital investment and wage are both continuous
in agent’s type, the worst agents in both sides will always find themselves
worst. So, they have the incentive to reduce capital investment or monetary
payment because, given set of other agents’ decision, they can always match
with the worst partner. The discontinuity in the bottom match is the reason
that a non-coorperative game such as the bidding game and the pure public good approach does not have a pure strategy equilibrium without belief
even in a large market. Instead of deriving a mixed strategy equilibrium
that will exist in our framework, this paper applies a rational expectations
equilibrium. In a large market, a single agent’s deviation from his/her cur4
rent investment or monetary payment decision will not change other agents’
relative position nor their partner’s quality. All agents in each side of market
believe that they face the same deterministic return on human capital. The
idea of how to make endogenous the same deterministic return is discussed
in section 7.
The ex ante attribute competition described in this paper makes ex ante
the monetary payment decision for human capital, the part of contract. A
firm makes investment and monetary payment decision and a worker makes
investment decision prior to match according to their beliefs. In equilibrium
their beliefs are fully fulfilled. Conditions are provided under which matching
considerations alone will create the appropriate investment incentives for all
parties.
When making investment decision, firms foresee the value of contract to
specific human capital level and workers does the impact of their investment
decision on the quality of match and value of specific contract. Each side of
the market competes ex ante for a better partner. After making the attribute
decision, the second stage involves only the matching process without ex post
monetization.
The incentive to meet a firm with better contract makes a worker of high
type invest more on human capital. High human capital is more valuable to
a firm of high type. Therefore, a firm of high type can offer a better contract
so a worker of high type matches with a firm of high type in equilibrium:
assortative matching. It is surprising that assortative matching is endogenous in equilibrium no matter how contracts are complicated and even if no
workers agrees on the value of contract.
The other equilibrium property is that the externality associated with
investment decision is completely internalized by the market return function.
The market return function is formed such that the marginal cost of capital
is equal to the social marginal benefit in a pair so investments are bilaterally
efficient. Bilateral efficiency is achieved through the correct valuation of the
equilibrium contract even in the absence of complete contract: the marginal
value of the equilibrium contract is equal to the marginal product of human
capital. The ex ante attribute competition makes the market value capital
investment through contract as a whole not just monetary payments alone.
The return function in the pure public good approach is adjusted until the
marginal rates of substitution of a worker and a firm in every match become
equal each other. The monetary reward schedules in the bidding game makes
the marginal increase in the monetary reward equal to the marginal increase
5
in the surplus at the matching stage. At the first stage, it makes the marginal
increase in the monetary reward equal to the marginal cost of investment.
The monetary reward schedules are adjusted until the marginal surplus of
investment are equal to the marginal cost in every match. Even in the ideal
Arrow-Debreu equilibrium, the bilateral efficiency condition is derived with
referring the price schedules for investments. The ex ante attribute competition generates the bilateral efficiency condition by competition for a better
matching quality without referring the return on human capital.
Assortative matching together with bilateral efficiency is the sufficient
condition for pareto optimality among all possible matchings.
2
Model
There is a continuum of firms and workers. Let Ω = [θ, θ] and Ψ = [δ, δ] be
sample space of a firm’s type and a worker’s type, where θ > 0 and δ > 0.
For A ∈ Ω and B ∈ Ψ, F (A) and W (B) denote the measures of the sets
of firms and workers whose types are in A and B respectively. We assume
F (Ω) ≤ W (Ψ). The firm of type θ chooses physical capital, k, at cost rk and
the worker of type δ chooses human capital, h, at cost e(h, δ). A firm’s payoff
function and a worker’s payoff function are given as follows:
Π(k, h, w, θ) = f (k, h, θ) − w − rk
U(k, h, w, δ) = u(k, h) + w − e(h, δ)
where r is the rental rate of physical capital taken as given, and w is monetary
payment given to a worker. A worker’s payoff depends on not only monetary
payment and human capital but also physical capital. u(k, h) + w makes the
contract two dimensional. A firm’s output depends not only on capitals but
also on his type.
Π and U are assumed to be C 2 . They satisfy the usual assumptions:
fk > 0, fh > 0, fθ > 0, uk > 0, uh > 0, eh > 0, eδ < 0. All second derivatives
of the payoff functions are assumed as usual: fkk < 0, fhh < 0, fθθ < 0,
ukk < 0, uhh < 0, −ehh < 0, −eδδ < 0. In addition, the marginal payoff of one
capital in the firm’s production function is assumed to be increasing in the
other and his own type. The marginal payoff of physical capital to a worker
is increasing in her own human capital but investment on human capital is
more costly to a worker of low type.
6
Assumption 1 f, u, and −e are strictly supermodular in all arguments:
0
0
0
0
0
0
0
0
0
(a) f ((k, h, θ)∧(k , h , θ ))+f ((k, h, θ)∧(k , h , θ )) > f (k, h, θ)+f (k , h , θ )
0
0
0
for (k, h, θ) and (k , h , θ ) in which at least two elements are different
0
0
0
0
0
0
(b) u(k, h) + u(k , h ) > u(k , h) + u(k, h ) for k > k and h > h
0
0
0
0
0
(c) −e(h, δ) − e(h , δ ) > −e(h , δ) − e(h, δ ) for h > h and δ > δ
0
Workers enjoy their partner’s capital on top of wage. Suppose that there
0
0
0
are two contracts, (k, w) and (k , w ), and that two workers provide h and h
respectively. Given workers’ human capital, two different workers may value
0
0
(k, w) and (k , w ) in different order because the payoff of physical capital
depend on a worker’s human capital. In general, we cannot find a unilateral
ordering of (k, w) that every worker agree because it depends on individual’s
human capital as well. Workers will disagree ex post on whether or not
additional physical capital investment makes the firm more desirable, so a
firm’s investment becomes specific in the sense that it may be attractive only
to certain types of worker.
Now consider pareto optimal allocations. Suppose that agents in a match
make investment decision and monetary payment agreement in isolation. At
any given level of monetary payment, investment will be decided in a way
that the private marginal benefit of investment is equal to the marginal cost
of investment:
fk (k, h, θ) − r = 0
uh (k, h) − eh (h, δ) = 0
This investment will be referred to as the Nash investment. The investment decisions are not efficient because a worker and a firm can be both
better off by increasing both capitals. Bilaterally efficient investments in a
pair are determined at a point where there does not exist another pair of investments that makes both agents in the match better off at the same time.
The necessary and sufficient condition for bilateral efficiency in a match is
uk (k, h) + fk (k, h, θ) − r = 0
fh (k, h, θ) + uh (k, h) − eh (h, δ) = 0
The bilaterally efficient levels of human capital and physical capital are
unique in a match although monetary payment is undetermined. This is
7
the special feature of the quasilinear payoff function. When there are more
than one agent in each side of market, bilateral efficiency alone is not the
sufficient condition for efficiency among all possible matchings. Let ϕ3 be a
matching function which specifies the type of a firm with whom a worker of
type δ matches and M be the set of all matching functions. The concept of a
bilateral efficient outcome identifies the set of pareto optimal outcome subject
to a given matching function. Even if a set of capitals and monetary payments
are pareto optimal given matching function, it is possible that there exists
another matching function and subsequent capitals and monetary payments
which make all agents weakly better off and some agent strictly better off.
We need stronger concept to extend the concept of pareto optimal outcomes
among all possible matching functions, M.
Definition 1 A set of capitals and wages and a matching function {k(ϕ(δ)),
w(ϕ(δ)), h(δ)| ϕ ∈ M} is globally pareto optimal if it is bilaterally efficient
and there does not exist other matching function and subsequent capitals and
0
0
0
wages {k(θ) , w(θ) , h(φ(θ)) | φ ∈ M} such that for all θ ∈ Ω
0
0
0
0
(a) f (k(θ) , h(φ(θ)) , θ)−w(θ) −rk(θ) ≥ f (k(θ), h(ϕ(θ)), θ)−w (θ)−rk (θ)
0 0
0
0
0
0
0
0
(b) u(k(θ ) , h(φ(θ )) ) + w(θ ) − e(h(φ(θ )) , φ(θ )) ≥ u(k(θ), h(ϕ(θ))) +
w (θ) − e(h(ϕ(θ)), ϕ(θ))
0
where ϕ(θ) = φ(θ ).
A globally pareto optimal outcome has no other set of matching function,
capitals, and wages which weakly increases all agents’ payoff and strictly increases some agents’. Whether a capital and monetary payment allocation is
globally pareto optimal therefore depends on capital and monetary payment
allocations in other matches. It is the assortative matching property that is
important to identify a globally pareto optimal outcome.
0
0
Definition 2 A matching is assortative if ϕ(θ) > ϕ(θ ) for θ > θ .
If payoff functions are transferable, assortative matching maximizes social net surplus by assumption 1. Any non-assortative matching is not globally pareto optimal regardless of bilateral efficiency with transferable payoff
3
When F (Ω) < W (Ψ), W (Ψ) − F (Ω) is the measure of workers who do not match. We
implicitly assume that they matches with firms that always have zero phyical capital, zero
output.
8
function. Assortative matching with bilateral efficiency however serves as
sufficient condition for globally pareto optimal outcome as lemma 1 proves.
Lemma 1 Any assortative bilateral efficient outcome is globally pareto optimal.
Proof. See Appendix A.
3
3.1
Direct Matching Designs
Bargaining game
Cole, Mailath, and Postlewaite [5],[6] formulate a bargaining procedure at
the matching stage taking investment decisions as given with a transferable
payoff function. In addition, the surplus generated in a match depends only
on capitals but not on types. An equilibrium bargaining rule specifies a
stable bargaining outcome in all possible sets of capitals. A stable bargaining
outcome makes nobody find a new partner with whom he/she can get higher
monetary payment, so it identifies the ex post core of assignment in which
the total social surplus is maximized with a given set of capitals.
Assuming that every agent can foresee the impact of their investment
decision on the matching stage, they formulate a non-coorperative game for
investment decision at the first stage. Therefore, whether a set of capitals
can be in an equilibrium depend on how a bargaining rule specifies the offequilibrium stable bargaining outcomes. In a finite game, a set of capital
can be an equilibrium or not, depending on off-equilibrium stable bargaining
outcomes while one can always construct an efficient equilibrium.
An important property of a stable bargaining outcome is the equal treatment property. This means that the same level of capital provided by different agents gets the same monetary payments regardless of agents’ type
and their partners. If the surplus does depend on type, then the equal treatment property does not hold in a stable bargaining outcome by Remark 1 in
appendix B. Consider two matches in which workers have the same human
capital and firms have the same physical capital but firms’ types are different.
A stable bargaining outcome requires no waste condition: shares of partners
in a match is equal to the surplus. If the equal treatment is satisfied, then
9
either the shares in the match with the firm of low type is not feasible or the
match with the firm of high type wastes some of the surplus.
An agent’s deviation would not change other agents’ relative position nor
their partner’s capital in a large market, so the matching externalities would
disappear. They assume that in a large market there exists an single market
monetary payment schedule for investment in each side based on the equal
treatment property, capturing the idea that in a large market an agent who
mimics another agent will receive the same monetary payment as the imitated
agent. If the surplus is type-dependent, their assumption is no longer valid
because the equal-treatment property does not hold in a stable bargaining
outcome. In general, a bargaining procedure cannot support the idea that
everybody faces the same market monetary payment schedule for investment
in a large market.
3.2
Bidding game
Felli and Roberts [7] also consider the ex-post monetization for investments.
They formulate a bidding game in which every worker bids monetary payments to every firm at the matching stage. They assume that matching occurs sequentially, so there are T substages, where T is the number of firms.
At each substage, only one firm decides which bid to accept. Any equilibrium
match is the ex-post core of assignment given investment decisions by Remark 2 in appendix C. If the surplus is type dependent, then the ex-post core
of assignment does not imply the assortative match between human capital
and physical capital.
Denote (kt , θt ) as the equilibrium physical capital and the type of the firm
who chooses a worker at substage t. Let ht be the equilibrium human capital
of a worker whose bid is the highest among bids submitted to the firm of
type θt and hr(t) be the equilibrium human capital of a worker whose bid is
the second highest. Now consider the last substage at the matching stage.
For simplicity, assume there are T +1 workers. In the last substage, there are
only two workers unmatched. Denote hT and hr(T ) to be high and low human
capital between the two. The reward to low human capital is ww (hr(T ) ) ≡
u(0, hr(T ) ) if the worker with low human capital is eventually unmatched.
Therefore, the bid, b(hr(T ) , θT ), of the worker with hr(T ) is not higher than
f (kT , hr(T ) , θT )+u(kT , hr(T ) )−ww (hr(T ) ).4 In general, the monetary payments
4
In the last substage, the unique equilibrium bids are therefore, b(hT , θT ) =
10
for ht and (kt , θt ) for t = 1, ..., T are
wf (kt , θt ) = f (kt , hr(t) , θt ) + u(kt , hr(t) ) − ww (hr(t) )
ww (ht ) = f (kt , ht , θt ) + u(kt , ht ) −
f (kt , hr(t) , θt ) − u(kt , hr(t) ) + ww (hr(t) )
(1)
(2)
Given above monetary payments, a firm’s payoff and a worker’s payoff are
Π = f (kt , hr(t) , θt ) + u(kt , hr(t) ) − ww (hr(t) ) − rkt
U = f (kt , ht , θt ) + u(kt , ht ) − f (kt , hr(t) , θt ) −
u(kt , hr(t) ) + ww (hr(t) ) − e(ht , δt )
(3)
(4)
One can construct an equilibrium where the competition is at its peak
and assortative matching and increasing capitals in type are satisfied. This
section focuses on the most competitive equilibrium with firms’ choice order
which coincides with their type. Suppose that a firm chooses a worker at
substage t if his type is the tth highest among all firms. If an equilibrium
shows the assortative matching and increasing capitals, then the firm of type
θt matches with the worker of type δt at substage t and her human capital is
the tth highest. Furthermore, the second highest bidder to the firm of type
θt is the worker of type δt+1 , so hr(t) = ht+1 . Define S(k, h, θ) ≡ f (k, h, θ) +
u(k, h).5
The firm of type θt chooses physical capital in equilibrium such that
Sk (kt , ht+1 , θt ) − rkt = 0
(5)
and physical capital is increasing in θ if the equilibrium matching is assortative by Remark 3 in appendix D. Next question is the investment decision
by workers. It is straightforward to show that human capital is increasing in
workers’ type because of assumption 1.(c). If physical capital is assortative,
then the ex post core of assignment implies that the equilibrium matching is
b(hr(T ) , θT ) = f (kT , hr(T ) , θT ) + u(kT , hr(T ) ) − ww (hr(T ) ). The monetary payment to
the firm of θT , wf (kT , θT ), and the monetary payment to the worker with hT , ww (hT ), are
wf (kT , θT ) = f (kT , hr(T ) , θT ) + u(kT , hr(T ) ) − ww (hr(T ) ) and ww (hT ) = f (kT , hT , θT ) +
u(kT , hT ) − f (kT , hr(T ) , θT ) − u(kT , hr(T ) ) + ww (hr(T ) ).
5
A firm’s monetary payment and a worker’s monetary
payment are derived by the rePT
cursive substitution: wf (kt , θt ) = S(kt , ht+1 , θt ) − i=t+1 [S(ki , hi , θi ) − S(ki , hi+1 , θi )] −
PT
ww (hr(T ) ) and ww (ht ) = i=t [S(ki , hi , θi ) − S(ki , hi+1 , θi )] + ww (hr(T ) ). A firm’s payoff
and worker’ payoff are Π = wf (kt , θt ) − rkt and U = ww (ht ) − e(ht , δt )
11
in fact assortative. Given this matching condition in equilibrium, the worker
who matches with the firm of θt makes the human capital investment decision
at 6
Sh (kt , ht , θt ) − eh (ht , δt ) = 0
(6)
The positive level of hT +1 is critical to the existence of an equilibrium.
Suppose that a worker cares only monetary payments:w − e(h, δ). Then, any
worker who does not match with a firm does not invest at all, so hT +1 = 0.
In equilibrium, the payoff of the firm at substage T is f (kT , hT +1 , θT ) − rkT .
Therefore, the firm will not invest at all as well. Then, the worker who
matches with a firm at substage t also does not invest at all. One cannot find
an equilibrium in which every agent undertakes a positive level of investment.
To get around the existence problem, Felli and Roberts assume that capital
is a function of investment and type. Therefore, a worker who does not invest
at all can have positive level of human capital.
Firms underinvest even in the assortative match since their marginal incentives to undertake investments are determined by their outside option that
depends on the surplus of the match between the firm and the immediate
competitor to the worker the firm matches with in equilibrium which yields
a strictly lower surplus than the equilibrium one. The bidding game makes
each worker residual claimant of the surplus produced in her equilibrium
match. Therefore, the worker is able to appropriate the marginal returns
from her investment. Her investment decision is constrained-efficient given
underinvestments of firms in the assortative equilibrium match. (5) and (6)
however implies that both workers and firms underinvest compared to the
globally pareto optimal investment level regardless of the market size.
3.3
Pure public good approach
Peters and Siow [11] eliminate all ex post monetary transfers and ask whether
assortative matching by itself is enough to provide correct ex ante investment
incentives in a large marriage market. The reward for investment is the
partner’s investment so investments are pure public goods. A man who invests
6
One can show that the payoff of the worker of type δt is monotonic decreasing in
any interval to the right of the (ht+1 , ht−1 ) and increasing in any interval to the left
(See proposition 4 in Felli and Roberts [7]). Therefore, this payoff has a unique global
maximum.. Hence, the worker of type δt has no incentive to deviate and change her
investment decision.
12
in additional human capital makes himself more attractive to every woman
on the market, and similarly for women. The return that a single gets from
investing in additional human capital is simply the higher human capital
investment of the spouse that a single eventually marries.
In a rational expectations equilibrium, every single believes that everybody faces the same non-stochastic market return function that maps a single’s investment to the spouse’ investment. They show that in a rational
expectations equilibrium in which all singles understand the true relationship between their own investments and the quality of the partner they will
attract, the hold up problem can be resolved.
They propose the extension of their model into labour market. A firm and
a worker match each other and a firm transfers a positive proportion of the
firm’s surplus to a worker. This makes every agent’s payoff depend on only
investments so investments become pure public goods. Define the market
return function of human capital: k = g(h). Then, the payoff functions
become
U = u(g(h), h) + αf (g(h), h, θ) − e(h, δ)
Π = (1 − α)f (k, g−1 (k), θ) − rk
High physical capital is more attractive to every worker in the market and
high human capital is more attractive to every firm in the market. g(h) is
expected to be increasing in h. Since it is costly for workers and firms to
supply capital beyond the Nash level, workers and firms will be willing to
supply more capital only if they can get a return for their investment. A
increasing g(h) will imply the assortative matching in equilibrium.
The return function k = g(h) is a rational expectations equilibrium if
there is an interval [k, k] and [h, h] such that for every k ∈ [k, k] and for
every h ∈ [h, h] there exist type level θ(k) and δ(h) such that
(a) F (A) = W (B) where A = [θ(k), θ(k)] and B = [δ(h), δ(h)] for every
k ∈ [k, k] and for every h ∈ [h, h]; θ(k) = θ and θ(k) = θ; δ(h) = δ and
δ(h) = δ ∗ , where F (Ω) = W ([δ ∗ , δ]
(b) h ∈arg max {u(g(b
h), b
h) + αf (g(b
h), b
h, θ(k)) − e(b
h, δ(h))}
b
h
(c) k ∈arg max {(1 − α)f (b
k, g−1 (b
k), θ(k)) − rb
k}
b
k
13
Condition (a) specifies the measure condition for the assortative match
in equilibrium. In every match, k = g(h), the worker and the firm solves the
problem (b) and (c) in equilibrium. Therefore, the maximization conditions
are
[uh (g(h), h) + αfh (g(h), h, θ(k)) − eh (h, δ(h))] +
0
[uk (g(h), h) + αfk (g(h), h, θ(k))] g (h) = 0
£
¤
(1 − α) fk (k, g −1 (k), θ(k)) − r +
0
(1 − α) fh (k, g−1 (k), θ(k))g −1 (k) = 0
(7)
(8)
From (7) and (8), the equilibrium return function, k = g(h), equates
−
(1 − α) fh
αfk + uh − eh
0
= g (h) = −
uk + αfk
(1 − α) fk − r
(9)
The equilibrium is not globally pareto optimal. A bilateral efficient outcome
satisfies the usual tangency condition which is fk + uk − r = 0 and fh +
uh − eh = 0. Since α is exogenously given and the same in every match, it
distorts agents’ preference itself. An equilibrium is bilaterally efficient with
the distorted preferences as (9).
4
Benchmark: Arrow-Debreu Equilibrium
In this section, we consider the ideal world in which the market is complete
and derive the first-best outcome as benchmark in the framework of the
Arrow-Debreu equilibrium. The different level of capital is a different good.
There are infinite number of markets in which firms and workers make transactions of these public goods. Let m(k, h) be the market in which agents
make transaction of one unit of h level of human capital and one unit of k
level of physical capital. Each worker provides only one unit of h and each
firm buys only one unit of h. Each firm provides only one unit of k and each
worker buys only one unit of k. There are prevailing market prices for h
and k respectively: p(k) and q(h). Given market prices, a firm decides how
much physical capital to sell and how much human capital to buy. A worker
decides how much physical capital to buy and how much human capital to
sell.
14
Definition 3 Suppose that (k, h) is the equilibrium one unit of physical capital and human capital in m(k, h) such that k = k(δ) = k(θ) and h = h(δ) =
h(θ),where k(δ) and k(θ) are the demanded and supplied levels of physical
capital respectively and h(δ) and h(θ) are the supplied and demanded level of
human capital respectively.{k = k(δ) = k(θ), h = h(δ) = h(θ), p(k), q(h)} is
a Arrow-Debreu equilibrium if
(a) Worker’s Maximization:
(k(δ), h(δ)) ∈arg max [u(k, h) − p(k) + q(h) − e(h, δ)]
k,h
(b) Firm’s Maximization:
(k(θ), h(θ)) ∈arg max [f (k, h, θ) + p(k) − q(h) − rk]
k,h
0
0
0
0
0
0
(c) F [θ : (k(θ ), h(θ )) ≥ (k, h)] = W [δ : (k(δ ), h(δ )) ≥ (k, h)] for all
(k, h).
Definition 3.(c) specifies the condition analogous to the market clearing
condition. A worker who demands k and supplies h makes transactions only
with a firm who exactly supplies k and demands h given market prices. In
equilibrium, the measure of firms who sell physical capital and buy human
capital no less than k and h must be equal to the measure of workers who
buy physical capital and human capital no less than k and h.
Lemma 2 Any Arrow-Debreu equilibrium satisfies bilateral efficiency, increasing capital in type, and assortative matching.
Proof. Given p(k) and q(h), the worker’s maximization implies the following
first order conditions.
0
uk = p (k)
0
uh − eh = q (h)
(10)
(11)
The firm’s maximization has the following first order conditions.
0
fk − r = −p (k)
0
fh = q (h)
15
(12)
(13)
(10) to (13) implies that
0
−[fk − r] = p (k) = uk
0
fh = q (k) = −[uh − eh ]
(14)
(15)
It is obvious that above two equations implies bilateral efficiency.7
It is straightforward to show that human capital is non-deceasing by
assumption 1.(c). Now suppose that the worker of type δ ∈ [δa , δb ] has
the same human capital, say h. The firms who match with one of those
workers have physical capital increasing in his type because of (14). Then
0
0
fh (k(θ), h, θ) 6= fh (k(θ ), h, θ ). It contradicts (13). Therefore, human capital
must be increasing in the worker’s type.
Suppose that equilibrium matching is not assortative. In equilibrium, the
firm of type θ make transactions of (k, h) with the worker of type δ and the
0
0
0
0
firm of type θ does transactions of (k , h ) with the worker of type δ , where
0
0
θ > θ and δ < δ . By definition 3,
0
0
0
0
f (k, h, θ) + p(k) − q(h) − rk ≥ f (k , h , θ) + p(k ) − q(h ) − rk
0
0
0
0
0
0
0
f (k , h , θ ) + p(k ) − q(h ) − rk ≥ f (k, h, θ ) + p(k) − q(h) − rk
0
The sum of the two inequalities is
0
0
0
0
0
0
f (k, h, θ) + f (k , h , θ ) ≥ f (k , h , θ) + f (k, h, θ )
0
0
0
θ > θ and h < h . If k ≤ k , then above inequality contradicts assumption
0
1.(a). Therefore, k > k . The demanded physical capital by the worker of
type δ given her supplied human capital h satisfies
0
uk (k, h) = p (k)
0
Substituting h with h ,
0
0
uk (k, h ) > p (k)
0
7
0
0
0
The demand of physical capital and the supply of human capital, k(p , q : δ), h(p , q :
δ), are derived from (10) and (11). (12) and (13) specifies the supply of physical capital
0
0
0
0
and the demand of human capital, k(p , q : θ), h(p , q : θ). The equilibrium prices are
0
0
0
0
0
0
0
0
derived from k(p , q : δ) = k(p , q : θ) and h(p , q : θ) = h(p , q : δ). The equilibrium
quantity of physical capital and human capital are derived by substituting the prices into
either demand or supply of capitals as usaul. However, we can derive the equilibrium
quantity of capitals directly from (14) and (15) without reference to the prices because of
the quasilinearity of payoff functions.
16
0
This implies that if a worker supplies h > h, her demanded physical capital
0
must be higher than k. This contradicts k > k . Therefore, the matching
must be assortative in equilibrium. If the match is assortative and human
capital is increasing in the worker’s type, then (14) implies physical capital
is increasing in the firm’s type in equilibrium.
If a worker does not make transactions of capitals in the market, she will
do home production. The worker doing home production has the following
maximization
Max u(0, h) − e(h, δ)
h
Consider the market where the firm of the worst type makes transactions of
physical capital and human capital. Lemma 2 implies that the worker’s type
who makes transactions with the firm is δ ∗ such that W [δ ∈ [δ, δ ∗ ]] = F [Ω].
The prices in the bottom market must make demand equal to supply in
the market. All the workers of higher type than δ ∗ will not enter the bottom
market because of lemma 2. If there is neither excess supply of human capital
nor excess demand of physical capital in the bottom market, then the worker
whose type is slightly below δ ∗ is just indifferent between entering the bottom
market and doing home production. In the limit,
u(k, h, q(h) − p(k)) − e(h, δ ∗ ) = u(0, h, 0) − e(h, δ ∗ )
(16)
where h = arg max[u(0, h, 0) − e(h, δ ∗ )], k = k(δ ∗ ) = k(θ), and h = h(δ ∗ ) =
h(θ)
(14), (15), and either (10),(11) or (12), (13) with p(k) and q(h) from
(16) completely characterize the Arrow-Debreu equilibria. As long as either
(10),(11) or (12), (13) satisfy the usual Lipshitz condition, we can derive the
full path of p(k) and q(h) given any p(k) and q(h) satisfying (16). Since
there is the unique market value for capital in each equilibrium, a firm and
a worker are able to write a complete contract, which specifies the level of
human capital and corresponding monetary payment. Monetary payment is
simply q(h) − p(k). Lemma 2 implies that the allocation of human capital
and physical capital is efficient in equilibrium. Although there are multiple
equilibria with different price levels, they are all identical in the sense that
capital allocation is the same and monetary payment in a market is the same
in all equilibria.
17
5
Ex-ante Attribute Competition
In this section, we consider the incomplete market where market prices for
human capital and physical capital do not exist so that complete contract is
impossible.
Investment/monetary payment decision and subsequent matching are formalized by a two-stage non-cooperative game. In the second stage, every
worker chooses simultaneously at most one firm as potential partner and
then every firm chooses simultaneously at most one worker among those who
chose the firm. In the second stage, each agent’s strategy is who he/she
matches with given investment/monetary payment decision. Physical capital, human capital, and monetary payment are all public information in the
second stage. We assume that, in the first stage, every agent can fully foresee the impact of their decision on matching. In the first stage, every worker
makes investment decision and every firm makes investment and monetary
payment decision. There is no ex post monetary payment decision. Only the
matching process is involved in the second stage.
In a small market, the choices of capitals and monetary payments will
be discontinuous in this game. Given other agents’ decisions, an agent will
have the incentive to reduce capital or monetary payment as long as the
relative position does not change because she can still meet the partner with
the same relative position in the other side. Or an agent could be willing to
increase the investment on capital because of discrete increase in payoff by
slight increase in the quality of a partner. We may not have a pure strategy
equilibrium in a small market in some situations. The large market does not
improve the situation as section 7 describes.
Instead of deriving a mixed strategy equilibrium that will always exist in
this game, this paper considers the case in which an agent has the belief about
the return on human capital investment. In a large market, a single agent’s
devation will not change other agents’ relative position nor their parner’s
quality. When agents make ex ante decision, they believe that every agent
faces the same deterministic return on human capital. We do not attempt to
restrict the structure of the market return in belief. In a rational expectations
equilibrium we describe below, the expected market return on human capital
is exactly realized as they expected.
We first consider the second stage given investment and monetary payment decisions. Define S = XSθ as product of the set of strategy for two
attributes for firms, where Sθ = {(k, w) ∈ <2 } and T = XTδ as product of
18
the set of strategy for workers, where Tδ = {h ∈ <}. Human capital that a
firm with (k, w) gets in equilibrium is defined as
η(k, w; s, t) = h
where s ∈ S and t ∈ T. A worker with high human capital is always preferable
to firms, so she can get a better contract than a worker with low human
capital does. So, h, k, and w satisfy the following condition: for all h
0
0
0
0
0
0
0
0
W [δ : h(δ ) ≥ h] = F [θ : u(k(θ ), h(δ ))+w(θ ) ≥ u(k, h(δ ))+w for some δ ]
(17)
Since a firm always hires a worker with the highest human capital among
those who applied to the firm, workers with higher human capital than h
apply to the firm with better contract than (k, w) for any h. In equilibrium,
there is no more than one worker who applies to a firm. One-to-one matching between firms and workers in the matching stage must satisfy (17) in
equilibrium.
Now look at the first stage. Given η, define a return function on human
capital
v(h) = u(k, η(k, w; s, t)) + w
= u(k, h) + w
where v ∈ V . V is the set of all possible return functions on human capital.
A worker’s payoff function is then defines as
U(h, δ; v) = v(h) − e(h, δ)
A firm’s payoff function is defined as
Π(k, w, θ; v) = f (k, h, θ) − w − rk
The payoff functions incorporate the belief that everybody faces the same
deterministic return (v) on human capital. An ex-ante attribute game with
belief, Λ, is the collection, {Ω, Ψ, S, T, V, U (h, δ; v), Π(k, w, θ; v)}. We define
a rational expectations equilibrium.
Definition 4 Given Λ, a rational expectations equilibrium (REE) is (s, t, v)
= (Xsθ , Xtδ , v), where sθ = (k (θ) , w(θ)) ∈ Sθ , tδ = h(δ) ∈ Tδ , and v ∈ V
such that
19
0
0
(a) U (h (δ) , δ; v) ≥ U(h , δ; v) for all δ ∈ Ψ and h ∈ Tδ
0
0
0
0
(b) Π(k (θ) , w (θ) , θ; v) ≥ Π(k , w , θ; v) for all θ ∈ Ω and (k , w ) ∈ Sθ
(c) For all h(δ) = η(k (θ) , w (θ) ; s, t), v(h(δ)) = u(k(θ), η(k (θ) , w (θ) ; s, t))
+w (θ)
In this section, the necessary conditions for a REE are derived. Next
lemma proves that regardless of matching property, human capital is increasing in the worker’s type and firms have the right incentive to invest the
efficient level given the matched worker’s human capital.
Lemma 3 In a REE, human capital is increasing in the worker’s type and
every firm’s investment decision satisfies
fk (k, h, θ) + uk (k, h) − r = 0
(18)
where h is the matched worker’s human capital.
Proof. See Appendix E.
The non-decreasing property of human capital comes only from assumption
1.(c). It is more costly for a worker of low type to have high human capital.
If some worker has high human capital, then she cannot be of low type in
equilibrium. Increasing property of human capital is implied by the fact that
there is no interval of worker’s type where every worker provides the same
level of human capital. In this interval, every worker must be indifferent
between matching with the current firm and deviating by slightly increasing
human capital to a new firm who matches with a worker with the same
human capital, so is the worker of the worst type in this interval. Because of
assumption 1.(b) and 1.(c), a worker of the highest type in this interval will
always find it profitable to deviate by slightly increasing her human capital
to the firm who matches with a worker of the worst type in this interval .
(18) in lemma 3 is quite intuitive. First, note that the firm of type θ has
the trade-off between monetary payment and physical capital in his payoff
and worker’s. The firm of type θ has to offer a contract to a worker at least
as good as the market return on her human capital. This constraint always
binds in equilibrium.
u(k, h) + w = v(h)
20
Therefore, his payoff is
f (k, h, θ) + u(k, h) − rk − v(h)
(19)
(19) implies that a firm is able to capture all the increase in surplus given
market return on human capital. Firms are the residual claimant of the
surplus. Therefore, they have the right incentive to invest the efficient level
with the matched worker’s human capital.
Lemma 4 Equilibrium match is assortative.8
Proof. Suppose that the equilibrium match is not assortative. Then, there
exist two matches such that the firm of type θ matches with the worker of
0
0
type δ and the firm of type θ matches with the worker of type δ , where
0
0
0
θ > θ and δ < δ . The firm of type θ has (k, w) and the firm of type θ has
0
0
0
0
(k , w ). The worker of δ has h and the worker of type δ . h < h because
the human capital is increasing in the worker’s type. In equilibrium, it is not
0
0
0
profitable for the firm of type θ to deviate and offer (k , w ) to get h . The
0
firm of type θ also is not better off by deviating and offering (k, w) to get h.
Therefore,
0
0
0
f (k, h, θ) − w − rk ≥ f (k , h , θ) − w − rk
0
0
0
0
0
0
f (k , h , θ ) − w − rk ≥ f (k, h, θ ) − w − rk
0
The sum of the two inequalities is
0
0
0
0
0
0
f (k, h, θ) + f (k , h , θ ) ≥ f (k , h , θ) + f (k, h, θ )
0
0
0
θ > θ and h < h . If k ≤ k , then above inequality contradicts assumption
0
1.(a). Therefore, k > k . The firm of high type invests more on physical
capital even in non-assortative match. In equilibrium, monetary payment
makes a firm’s constraint binding such that
u(k, h) + w = v(h)
8
When there are search frictions, assortative matching requires not only the production
function be supermodular but also its log first- and cross-partial derivatives with transferable utility (Shimer and Smith [14]) or production function is log supermodular with
nontransferable utility (Smith, [15]).
21
Therefore, the equilibrium payoff of the firm of type θ is
f (k, h, θ) + u(k, h) − rk − v(h)
Define S(k, h, θ) ≡ f (k, h, θ) + u(k, h). Consider the firm of type θ deviates
0
0
0
e such that u(k, h ) + w
e = v(h ). In
to the worker of type δ by offering (k, w)
equilibrium, this deviation is not profitable:
0
0
S(k, h, θ) − rk − v(h) ≥ S(k, h , θ) − rk − v(h )
0
(20)
0
≈
Consider the firm of type θ deviates to the worker of type δ by offering (k , w)
≈
0
such that u(k , h)+ w= v(h). In equilibrium, this deviation is not profitable:
0
0
0
0
0
0
0
0
S(k , h , θ ) − rk − v(h ) ≥ S(k , h, θ ) − rk − v(h)
(21)
The sum of (20) and (21) is
0
0
0
0
0
0
S(k, h, θ) + S(k , h , θ ) ≥ S(k, h , θ) + S(k , h, θ )
0
0
(22)
0
Since k > k , θ > θ , and h > h, assumption 1.(a) and 1.(b) implies
0
0
0
0
0
0
S(k, h, θ) + S(k , h , θ ) < S(k, h , θ) + S(k , h, θ )
(23)
(22) contradicts (23). Therefore, the equilibrium matching is assortative.
The assortative matching implies that a firm of high type matches with a
worker of high type, so high capital. The assortative matching and increasing
human capital in the worker’s type imply that physical capital is also increasing in the firm’s type by (18). The increasing capital with respect to type
and the assortative matching is the necessary conditions for any REE. The
next section fully characterize the unique REE with the bilateral efficiency.
It is important that the increasing capital and the assortative matching are
derived without referring the ex-post core of assignment.
6
Existence of Rational Expectations Equilibrium
The previous section shows that the assortative match holds and capital is
increasing in type in equilibrium. The firm is able to capture all the increase
22
in the surplus with the matched worker’s capital, so his investment decision
is efficient given human capital. In this section, first consider the worker’s
incentive to invest on human capital. Since we know that the equilibrium
matching is assortative, it is convenient at this stage to use the equilibrium
matching function. Define ϕ (θ) as the equilibrium matching function which
specifies the type of worker the firm of type θ matches with in equilibrium.
0
Since the equilibrium matching is assortative at the second stage, ϕ (θ) > 0.
Let {k(θ), w(θ), h(δ), ϕ (θ)} be an equilibrium capital and monetary payment
allocation and the matching function, where h(ϕ (θ)) = η(k(θ), w(θ)). When
the worker of δ invests on human capital by h(b
δ), her payoff is
δ)), h(b
δ)) + w(ϕ−1 (b
δ)) − e(h(b
δ), δ)
U(b
δ, δ) = u(k(ϕ−1 (b
If the worker of type δ invests h(δ) in equilibrium, then she has no incentive
to deviate her investment to match with another firm in the neighborhood
of δ. In equilibrium, ∂U
= 0 at b
δ = δ:
∂b
δ
0
0
uk (k(ϕ−1 (δ)), h(δ))k (ϕ−1 (δ))ϕ−1 (δ)
0
(24)
0
0
+[uh (k(ϕ−1 (δ)), h(δ)) − eh (h(δ), δ)]h (δ) + w (ϕ−1 (δ))ϕ−1 (δ) = 0
When the firm of type θ invests on physical capital by k(b
θ) and makes monb
etary payment equal to w(θ), his payoff is
Π(b
θ, θ) = f (k(b
θ), h(ϕ(b
θ)), θ) − w(b
θ) − rk(b
θ)
If the firm of type θ makes investment and monetary payment decision equal
to (k(θ), w(θ)), then he has no incentive to deviate his current decision to
= 0 at b
θ = θ:
match with another worker. In equilibrium, ∂Π
∂b
θ
0
[fk (k(θ), h(ϕ(θ)), θ) − r]k (θ)
0
0
(25)
0
+fh (k(θ), h(ϕ(θ)), θ)h (ϕ(θ))ϕ (θ) − w (θ) = 0
(24) and (25) tells only that every agent does not have the incentive locally.
Lemma 5 in appendix F proves that in fact these two conditions makes every
agent have no incentive to deviate anywhere. The sum of (24) and (25) in
every match, ϕ(θ) = δ, is
0
0
[fh (k(θ), h(ϕ(θ)), θ) + uh (k(ϕ(θ))), h(ϕ(θ))) − eh (h(ϕ(θ)), ϕ(θ))]h (ϕ(θ))ϕ (θ)
23
0
+[fk (k(θ), h(ϕ(θ)), θ) + uk (k(ϕ(θ))), h(ϕ(θ))) − r]k (θ) = 0
(26)
We know that
fk (k(θ), h(ϕ(θ)), θ) + uk (k(ϕ(θ))), h(ϕ(θ))) − r = 0
(27)
from lemma 3. Human capital is increasing in the worker’s type and the
matching is assortative in equilibrium. Therefore,
fh (k(θ), h(ϕ(θ)), θ) + uh (k(ϕ(θ))), h(ϕ(θ))) − eh (h(ϕ(θ)), ϕ(θ)) = 0
(28)
because of (26). Every match in equilibrium is bilaterally efficient by (27)
and (28).
Now consider the bottom match. In the bottom match, the lower option
to the worker is not to work. Therefore, the following bottom match condition
holds in equilibrium.
u(k(θ), h(ϕ(θ))) + w(θ) − e(h(ϕ(θ)), δ ∗ )
(29)
=Max [u(0, h) − e(h, δ ∗ )]
h
∗
where ϕ(θ) = δ such that F (Ω) = W (Ψ) − W (B) where B = [δ, δ ∗ ). The
worker of type δ ∈ B does not match a firm in equilibrium because it is too
costly to invest human capital high enough to match with a firm. Therefore,
they will do home production, their human capital investment is made such
that
Max [u(0, h) − e(h, δ)]
(30)
h
The rational expectations equilibrium in the ex ante attribute competition
is fully characterized by proposition 1.
Proposition 1 The set of physical capital, human capital, monetary payment and matching function, {k(θ), w(θ), h(δ), ϕ(θ)}, is a REE if and only
if
(a) ϕ (θ) = δ if θ ∈ (θ, θ) such that F (Ω) − F (A) = W (Ψ) − W (B) where
A = [θ, θ) and B = [δ, δ); ϕ (θ) = δ if θ = θ; ϕ (θ) = δ ∗ if θ = θ such
that F (Ω) = W (Ψ) − W (B) where B = [δ, δ ∗ ).
(b) k(θ), h(δ), w(θ) satisfy (24), (25), (27), and (28) for all θ ∈ [θ, θ] and
all δ ∈ Ψ − B and h(δ) satisfies (30) for all δ ∈ B.
24
(c) k(θ), h(δ ∗ ), w(θ) satisfies (29)
Proof. Lemma 3, Lemma 4, and Lemma 5.
Since payoff functions of workers and firms are a concave function in k
and h, the sum of the two functions, Π + U, is the concave function in k and
h. Therefore, the bilaterally efficient levels of human capital and physical
capital are uniquely determined. Lemma 6 provides not only the existence
of the REE but also monetary payment determination.
Lemma 6 If (31) and (32) satisfy the usual Lipshitz condition, the rational
expectations equilibrium exists and change in wage with respect to θ is given
by (33).
0
∂k
(fhh + uhh − ehh )fkθ + (fkh + ukh )(−fhθ + ehδ ϕ (θ))
=
>0
(31)
∂θ
(fkh + ukh )2 − (fkk + ukk )(fhh + uhh − ehh )
µ
¶
0
∂h
−(fkk + ukk )(−fhθ + ehδ ϕ (θ)) − (fhk + uhk )fkθ
0
=
/ϕ (θ) > 0 (32)
2
∂δ
(fkh + ukh ) − (fkk + ukk )(fhh + uhh − ehh )
∂k
∂h 0
∂w
+ uk
= fh ϕ (θ)
∂θ
∂θ
∂δ
(33)
Proof. See Appendix G.
From (33), change in monetary payment with respect to θ is the linear combination of change in physical capital and human capital. Monetary payment
schedule is unique because (29) specifies the level of monetary payment in
the bottom pair.
The REE is very illustrative of how the globally pareto optimal allocation
is decentralized in the absence of complete contracts. If competition induces
assortative matching and bilateral efficiency, competition can produce the
efficient allocations. Assortative matching is generated by supermodular assumption with respect to capitals. In order to reach bilateral efficiency given
assortative matching, competition must generate the correct valuation of capital investments. As (33) shows, the marginal value of the equilibrium contract is equal to the marginal product of the firm’s output of human capital.
Therefore, the market correctly values the capital investment in equilibrium
25
through contract as a whole not just monetary payments. (24) and (25) can
rewritten with (33) as follows
0
0
fh h (δ) = [e(h(δ), δ) − uh (k(ϕ−1 (δ)), h(δ))]h (δ) (34)
0
0
(35)
fk (k(θ), h(ϕ(θ)), θ)k (θ) = [r − uk (k(θ), h(ϕ(θ)))]k (θ)
Given correct valuation of capital investments, a worker makes her investment
decision on human capital at the point where the marginal value of the
equilibrium contract is equal to the marginal net cost of human capital. This
makes again the marginal product of human capital equal to the marginal
net cost. A firm does not need to make monetary payments equal to the
marginal product of human capital since the contract is two dimensional.9
So, the marginal net cost of physical capital is the left hand side of (35). A
firm’s capital investment is decided at the point where the marginal product
of physical capital is equal to the marginal net cost. The conditions for
investment decision are exactly identical to bilateral efficiency. The correct
valuation of contract on capital investment makes an agent’s self-interested
behavior effectively maximize the social benefit of investments. If an agent
reduces capital investment, it is not optimal because it does not maximize
own payoff in the current match and furthermore he/she loses the current
partner.
The return function, k = g(h), in the pure public good approach is adjusted until the marginal rates of substitution of a worker and a firm in
every match become equal each other as (9) describes. The monetary reward
schedules in the bidding game makes the marginal increase in the monetary
reward equal to the marginal increase in the surplus at the matching stage.
At the first stage, it makes the marginal increase in the monetary reward
equal to the marginal cost of investment. The monetary reward schedules
are adjusted until the marginal surplus of investment are equal to the marginal cost in every match. If the surplus does not depend on type, then
Sk (k, h) =
∂y(k)
= ck (k, θ)
∂k
9
Monetary payment is not always increasing in θ and workers of sufficiently low
type may be willing to pay wage premium to a firm.If the payoffs are transferable, i.e.
u(k, h) = 0, ∀(k, h), then the solution is summarized by fk − r = 0, fh − eh = 0, ∂w/∂θ =
0
fh ∂h/∂δϕ (θ) and w(ϕ−1 (δ ∗ )) − e(h(δ ∗ ), δ ∗ ) = 0. Since a worker does not enjoy physical
capital of a firm, the monetary payment is always increasing in type and the change in
monetary payment is equal to the marginal product of human capital with respect to the
worker’s type.
26
Sh (k, h) =
∂x(h)
= eh (h, δ)
∂h
where c is the firm’s cost of physical capital investment. Even in the ideal
Arrow-Debreu equilibrium, the bilateral efficiency condition is derived with
referring the price schedules for investments (see (14) and (15)). The return
on human capital, v(h), in the model this paper describes only makes a firm
a residual claimant of the surplus who captures increase in the surplus by
increase in his own investment. The ex ante attribute competition generates
the bilateral efficiency condition by competition for a better matching quality
without referring v(h) as (27) and (28) shows.
The rational expectations equilibrium is globally pareto optimal because
the equilibrium is assortative match and bilaterally efficient. Furthermore,
bilateral efficiency uniquely determines human capital and physical capital
and monetary payment schedule is unique. Therefore, the REE is unique. In
fact, the REE is identical to the Arrow-Debreu equilibrium because capital
allocation is the same and monetary payment in the REE is the same as
q(h) − p(k) in the Arrow-Debreu equilibrium. Despite of the absence of
complete markets, the ex-ante attribute competition to get a better matching
quality in the REE fully reproduce the outcome in the ideal world.
7
Discussion
In general, a bargaining procedure cannot support the idea that everybody
faces the same market reward schedule for investment in a large market. If
the surplus does depend on type, then a stable bargaining outcome must
make the share of the surplus depend on both investment and type. Even in
a large market, a bargaining procedure must assign different market rewards
for investment to different agents whose type affects the surplus. This would
be a very difficult coordination task.
Every firm and every worker underinvest regardless of the market size
in the bidding game. The ex ante attribute competition in a large market
makes the market correctly value capital investments through contract as a
whole not just monetary payments in a situation in which the equal treatment property does not hold in the bidding game. It is important that the
efficiency is achieved even in the complex situations: workers disagree ex post
on whether or not additional physical capital investment makes the firm more
27
desirable, so a firm’s investment becomes specific in the sense that it may be
attractive only to certain types of worker.
An equilibrium of the bidding game does not exist in a large market.
The human capital path at the worker of the worst type among matched
has discrete jump from the unmatched worker’s human capital. (5) implies
that the firm of the worst type undertakes investment as if he maximizes
the surplus with the unmatched worker’s human capital and the firm of the
second worst type undertakes investment to maximizes the match surplus
with the human capital of the worker whose type is the lowest among matched
workers. Since these two human capitals have a distinct distance, the path
of physical capital is discontinuous at θ.
The bilateral efficiency condition in the bottom match explains why a
pure-straetgy equilibrium without belief does not exist in a large market.
They imply that workers and firms make their investment decision at the level
where private marginal benefit of capital is less than the marginal cost. They
would reduce capital investment in a match if they could. The firm and the
worker in the bottom pair know that they are worst among those who matches
in the market, which means that they can actually reduce the investment
without changing their relative position. Therefore, the investment level will
be determined by
fk (k, h, θ) − r = 0
uh (k, h) − eh (h, δ ∗ ) = 0
(36)
(37)
and monetary payment will make the worker just indifferent with not being
matched and doing home production because the worker is in the long side
of the market.
u(k, h) + w − e(h, δ ∗ ) = Max[u(0, h) − e(h, δ ∗ )]
(36) and (37) make the path of human capital and physical capital discontinuous from the bottom. So, one cannot hope the existence of a pure strategy
equilibrium in the ex-ante attribute competition even in a large market. The
bottom match problem is the typical reason that a non-coorperative game
such as the bidding game and the pure public good approach does not have
a pure strategy equilibrium without belief even in a large market.
It is difficult to fully characterize a mixed strategy equilibrium in which
every agent puts positive probability on any interval of the set of strategy.
An agent’s relative position is important to get a better matching quality. If
28
everybody mixes strategy, then the relative ranking of a worker’s human capital will follows a random distribution generated by other agents’ decisions.
If the number of agents increases, then a random distribution of the relative
ranking of any worker’s human capital will converge to the same distribution
by the law of large number. In principle, the assumption that everybody
faces the same return function on human capital can be endogenously generated by taking a mixed strategy equilibrium to the limit. This is similar to
the Bertrand competition with constrained capacity but makes both sides’
decision endogenous.
Appendix A. Proof of Lemma 1. Suppose an assortative bilaterally
efficient outcome is not globally optimal. Then, there exists a non-assortative
bilaterally efficient outcome such that φ ∈ M and ∀θ ∈ Ω
0
0
0
Π(k (θ) , h(φ(θ)) , w(θ) , θ) ≥ Π(k (θ) , h(ϕ(θ)), w (θ) , θ) (38)
³ 0 ´0
0
0
0 0
0
U(k θ , h(φ(θ )) , w(θ ) , φ(θ )) ≥ U (k (θ) , h(ϕ(θ)), w (θ) , ϕ(θ))(39)
0
0
where φ(θ ) = ϕ(θ). x is the outcome under a non-assortative matching
matching function φ, and x is the outcome under matching function ϕ. Since
an assortative bilaterally efficient outcome is not globally pareto optimal, at
least one of (38) and (39) holds with strict inequality. The sum of (38) and
(39) is
Z ³
´
0
0
0
0
0
0
f (k (θ) , h(φ (θ)) , θ) + u(k (θ) , h(φ(θ)) ) − rk (θ) − e(h(φ(θ)) , φ(θ)) dF >
ZΩ
(f (k (θ) , h(ϕ (θ)), θ) + u(k (θ) , h(ϕ(θ))) − rk (θ) − e(h(ϕ(θ)), ϕ(θ))) dF (40)
Ω
0
0
Now sort out A ≡ {k (θ) }and B ≡ {h(φ (θ)) } such that {K (θ) |K (θ) ∈
0
0
A, K (θ) ≥ K(θ ) for θ > θ } and {H(ϕ(θ))|H(ϕ(θ)) ∈ B, H(ϕ(θ)) ≥
0
0
H(ϕ(θ )) for ϕ(θ) > ϕ(θ )}. Given array of ordered capitals, assign K (θ)
to the firm of type θ and H(ϕ (θ)) to the worker of type ϕ (θ): form assortative matching given array of ordered capitals. The sum of firms’ and workers’
payoffs is
Z
(f (K (θ) , K(ϕ (θ)), θ) + u(K (θ) , K(ϕ(θ))) − rK (θ) − e(K(ϕ(θ)), ϕ(θ))) dF
Ω
29
By assumption 1,
Z
(f (K (θ) , K(ϕ (θ)), θ) + u(K (θ) , K(ϕ(θ))) − rK (θ) − e(K(ϕ(θ)), ϕ(θ))) dF ≥
ZΩ ³
´
0
0
0
0
0
0
f (k (θ) , h(φ (θ)) , θ) + u(k (θ) , h(φ(θ)) ) − rk (θ) − e(h(φ(θ)) , φ(θ)) dF(41)
Ω
From (40) and (41),
Z
(f (K (θ) , K(ϕ (θ)), θ) + u(K (θ) , K(ϕ(θ))) − rK (θ) − e(K(ϕ(θ)), ϕ(θ))) dF >
ZΩ
(f (k (θ) , h(ϕ (θ)), θ) + u(k (θ) , h(ϕ(θ))) − rk (θ) − e(h(ϕ(θ)), ϕ(θ))) dF
(42)
Ω
(K(θ), H(ϕ (θ))) and (k (θ) , h(ϕ (θ))) are both capital allocations with assortative matching function. Because (k (θ) , h(ϕ(θ))) is an bilaterally efficient
allocation with assortative matching function, (42) does not hold. Therefore,
an assortative bilaterally efficient outcome is globally pareto optimal.
Appendix B. Remark 1. If the surplus does depend on type, then the
equal treatment property does not hold in a stable bargaining outcome.
Proof. Suppose that a firm and a worker bargains over the share of surplus in
a pair with the surplus function f (k, h, θ)+u(k, h). Let x(k) ≥ 0 and y(k) ≥ 0
be the share of the surplus for a worker and a firm in a match. Suppose that
the equal treatment property hold in a stable bargaining outcome: x(k) =
0
0
0
0
x(k ) if k = k and y(h) = y(h ) if h = h .
Consider any two matches such that two workers of different types provide
the same level of human capital, say h, and the two firms different type (θ >
0
θ ) provide the same level of physical capital, say k. In a stable bargaining
outcome, the sum of the two shares in a match is equal to the surplus: no
0
waste condition. Suppose the match with the firm of θ satisfies no waste
condition
0
x(h) + y(k) = f (k, h, θ ) + u(k, h)
Then the match with the firm of θ has
x(h) + y(k) < f (k, h, θ) + u(k, h)
0
because f (k, h, θ ) < f (k, h, θ). It contradicts no waste condition. Suppose
that no waste condition holds in the match with the firm of θ.
x(h) + y(k) = f (k, h, θ) + u(k, h)
30
Then,
0
x(h) + y(k) > f (k, h, θ ) + u(k, h)
0
because f (k, h, θ ) < f (k, h, θ). The shares, x(h) and y(k), is not feasible in
0
the match with the firm of type θ . Therefore, the equal treatment property
does not hold in a stable bargaining outcome.
Appendix C. Remark 2 Any equilibrium match is the ex-post core of
assignment given investment decisions.
0
0
0
Proof. Consider any two matches (h, k, θ) and (h , k , θ ). Suppose that the
0
0
worker with h deviates and does not bid to the firm of θ . If the firm of
0
θ accepts the most preferred bid ahead of the firm of θ, then the firm of θ
will have exactly one less bids because some worker’s bid will be accepted by
0
the firm of θ ahead of the firm of θ. Define b(θ) as the most preferred bid
of the firm of θ in equilibrium. Since there are less competition for the firm
of θ by the deviation, the most preferred bid, bb(θ), cannot be higher than
0
0
0
b(θ). If (h , k , θ ) is the equilibrium match, then the deviation must not be
profitable
0
0
0
0
0
0
0
0
u(k , h ) + f (k , h , θ ) − b(θ ) ≥ u(k, h ) + f (k, h , θ) − bb(θ)
Since b(θ) ≥ bb(θ)
0
0
0
0
0
0
0
0
u(k , h ) + f (k , h , θ ) − b(θ ) ≥ u(k, h ) + f (k, h , θ) − b(θ)
(43)
0
When the firm of θ chooses the most preferred bid after the firm of θ, b(θ) =
bb(θ). (43) holds, either. Similarly, if (k, h, w) is the equilibrium match, the
following holds
0
0
0
0
u(k, h) + f (k, h, θ) − b(θ) ≥ u(k , h) + f (k , h, θ ) − b(θ )
(44)
Sum of (43) and (44) implies
0
0
0
0
0
0
0
0
0
0
u(k, h)+f (k, h, θ)+u(k , h )+f (k , h , θ ) ≥ u(k, h )+f (k, h , θ)+u(k , h)+f (k , h, θ )
(45)
This inequality implies that it is impossible to increase the total social surplus.
Appendix D. Remark 3 The firm of type θt chooses physical capital in
equilibrium such that
Sk (kt , ht+1 , θt ) − rkt = 0
31
and physical capital is increasing in θ if the equilibrium matching is assortative.
Proof. Suppose that the equilibrium matching is assortative. Because of assumption 1.(c), the increasing human capital is almost immediate. Consider
the last substage. The firm of θT does not select a worker yet and there are
two workers unmatched such that hT > hT +1 . The payoff of the firm of type
θT is
Π = S(kT , hT +1 , θT ) − ww (hr(T ) ) − rkT
His optimal investment decision satisfies Sk (kT , hT +1 , θT )−rkT = 0. Consider
substage t. Since the firm of type θt matches with the worker of δt , the firm
of type θt ’s investment decision satisfies
Sk (kt , ht+1 , θt ) − r = 0
Consider the firm of type of θt−s , s = 1, ..., t − 1. His investment decision also
satisfies
Sk (kt−s , ht−s+1 , θt−s ) − r = 0
Since ht−s+1 > ht+1 and θt > θt−s , assumption 1.(a) and 1.(b) implies
kt−s+1 > kt+1 . We still need to prove that the firm of θt does not have incentive
to deviate when a firm of θi matches with a worker of δi , where i < t Suppose
that the firm of type θt changes his physical capital in order to match with a
worker of type δt−s . Since the firm of type θt choose a worker at substage t,
he still has the same second highest bidder at substage t because the worker
of type δt will be chosen
PT at an earlier substage. Therefore, his payoff is still
Π = S(kt , ht+1 , θt ) − i=t+1 [S(ki , hi , θi ) − S(ki , hi+1 , θi )] − ww (hr(T ) ) − rkt .
Therefore, change in his physical capital to match with a worker with higher
human capital is not profitable.
Appendix E. Proof of Lemma 3.
Suppose that the worker of δ ’s human capital is h in equilibrium. Then, it
must satisfies
0
v (h) = eh (h, δ)
0
Consider the investment decision of the worker of δ > δ. Because of assumption 1.(c),
0
0
v (h) > eh (h, δ )
0
Therefore, the worker of δ invest more in human capital than the worker of
δ does.
32
Suppose that the firm of θ matches with the worker with h in equilibrium.
The firm of θ offers a contract with k and w, so he has the following choice
problem given human capital fixed within a match
(k, w) ∈arg max f (k, h, θ) − w − rk
k,w
s.t. u(k, h) + w = v (h)
That is, if there is other (k, w) which make the worker of δ indifferent but
the firm of θ better off, then (k, w) is not the equilibrium offer. The first
order conditions are
fk − r + λuk = 0
−1 + λ = 0
Therefore,
fk (k, h, θ) + uk (k, h) − r = 0
Appendix F. Lemma 5 No agent has incentive to deviate from a current
strategy if and only if ,at b
θ = θ for all θ and at b
δ = δ for all δ,
0
0
0
δ)), h(b
δ))k (ϕ−1 (b
δ))ϕ−1 (b
δ)+[uh (k(ϕ−1 (b
δ)), h(b
δ))−e(h(b
δ), δ)]h (b
δ)
(a) uk (k(ϕ−1 (b
0
0
δ) = 0
+w (ϕ−1 (δ))ϕ−1 (b
0
0
0
0
(b) [fk (k(b
θ), h(ϕ(b
θ)), θ)−r]k (b
θ)+fh (k(b
θ), h(ϕ(b
θ)), θ)h (ϕ(b
θ))ϕ (b
θ)−w (b
θ) =
0.
Proof. The first order condition holds for all δ at b
δ = δ.
δ)
δ)
∂ϕ−1 (b
∂ϕ−1 (b
0
0
0
+ w (ϕ−1 (b
+ (uh − eh ) h (b
uk k (ϕ−1 (b
δ))
δ))
δ) = 0
∂b
δ
∂b
δ
(46)
Take the derivative of (46) with respect to δ
∂ 2U
∂ 2U
∂2U
0
+
− ehδ (h(b
δ), δ)h (b
δ) = 0
=
2
2
∂b
δ∂δ
∂b
δ
∂b
δ
∂ 2U
0
= ehδ (h(b
δ), δ)h (b
δ) at b
δ=δ
2
b
∂δ
33
(47)
By (c) in assumption 1 and the increasing property of h, (47) is positive at
all b
δ for a worker of δ. Therefore, every worker’s payoff function is a concave
function with respect to b
δ: nobody has incentive to deviate to other partner
if (a) in Lemma 4 holds.
Consider the deviation of the firm of θ to match with the worker of ϕ(b
θ)
b
b
such that |θ − θ| < ² for ² > 0. The payoff by deviation to the worker of ϕ(θ)
is
h
i
θ)), θ) − w∗ − rk∗ ≡ Max f (k, h(ϕ(b
θ)), θ) − w − rk
f (k∗ , h(ϕ(b
s.t. u(k, h(ϕ(b
θ))) + w = u(k(b
θ), h(ϕ(b
θ))) + w(b
θ)
θ)), θ)−w∗ −rk∗ = f (k∗ , h(ϕ(b
θ)), θ)+ u(k∗ , h(ϕ(b
θ)))−u(k(b
θ), h(ϕ(b
θ)))−w(b
θ)−rk∗
f (k∗ , h(ϕ(b
The current payoff is f (k(θ), h(ϕ(θ)), θ) − w(θ) − rk(θ).
Π(b
θ, θ) − Π(θ, θ)
θ)), θ) − f (k(θ), h(ϕ(θ)), θ) − r(k ∗ − k (θ)) + u(k∗ , h(ϕ(b
θ)))
= f (k∗ , h(ϕ(b
−u(k(b
θ), h(ϕ(b
θ))) − (w(b
θ) − w(θ))
= f (k(b
θ), h(ϕ(b
θ)), θ) − f (k(θ), h(ϕ(θ)), θ) − r(k(b
θ) − k(θ)) − (w(b
θ) − w(θ))
∗
∗
∗
f (k , h(ϕ(b
θ)), θ) − f (k(b
θ), h(ϕ(b
θ)), θ) − r(k − k(b
θ)) + u(k , h(ϕ(θ))) − u(k(b
θ), h(ϕ(b
θ)))
As b
θ → θ,
=
Π(b
θ, θ) − Π(θ, θ)
b
θ−θ
f (k(b
θ), h(ϕ(b
θ)), θ) − f (k(θ), h(ϕ(θ)), θ) − r(k(b
θ) − k(θ)) − (w(b
θ) − w(θ))
b
θ−θ
∗
θ)), θ) − f (k(b
θ), h(ϕ(b
θ)), θ) − r(k∗ − k(b
θ)) + u(k∗ , h(ϕ(θ))) − u(k(b
θ), h(ϕ(b
θ)))
f (k , h(ϕ(b
+
b
θ−θ
The definition of (k ∗ , w∗ ) implies
θ)), θ) + uk (k∗ , h(ϕ(b
θ))) − r = 0
fk (k∗ , h(ϕ(b
Therefore, as b
θ → θ,
f (k∗ , h(ϕ(b
θ)), θ)−f (k(b
θ), h(ϕ(b
θ)), θ)−r(k∗ −k(b
θ))+u(k∗ , h(ϕ(θ)))−u(k(b
θ), h(ϕ(b
θ))) → 0
34
Applying l’Hopital’s Law, the third line becomes
0
θ)
(fk (k (θ) , h(ϕ(θ)), θ) + uk (k (θ) , h(ϕ(θ))) − r) k (b
This is equal to zero by lemma 3. Therefore, in order to have no incentive
to deviate locally
´ 0
³
´ 0
∂Π(b
θ, θ) ³
0
0
b
b
b
b
b
= fk (k(θ), h(ϕ(θ)), θ) − r k (θ) + fh (k(θ), h(ϕ(θ)), θ) h (ϕ(b
θ))ϕ (b
θ) − w (b
θ)
b
∂θ
0
+ (fk (k (θ) , h(ϕ(θ)), θ) + uk (k (θ) , h(ϕ(θ))) − r) k (b
θ) = 0
at b
θ = θ. [fk (k (θ) , h(ϕ(θ)), θ) + uk (k (θ) , h(ϕ(θ))) − r] is zero by lemma 3.
Therefore,
´ 0
³
´ 0
³
0
0
θ), h(ϕ(b
θ)), θ) − r k (b
θ)+ fh (k(b
θ), h(ϕ(b
θ)), θ) h (ϕ(b
θ))ϕ (b
θ)−w (b
θ) = 0
fk (k(b
Let’s take the derivative of above equation with respect to θ
∂ 2Π
∂ 2Π
∂ 2Π
0
0
0
=
+
+ 2fkθ k (b
θ) + fhθ h (ϕ(b
θ))ϕ (b
θ) = 0
2
2
∂b
θ∂θ
∂b
θ
∂b
θ
∂ 2Π
0
0
b
b 0 b
2 = −2fkθ k (θ) − fhθ h (ϕ(θ))ϕ (θ)
∂b
θ
2
0
θ) ≥ 0, then ∂ bΠ2 < 0 for all b
θ for every θ ∈ Ω . Therefore, the payoff
So, if k (b
∂θ
by deviating to elsewhere is not greater than that from the current match.
Appendix G. Proof of Lemma 6. Taking derivatives of (27) and (28)
with respect to θ, we get two systems of equations:
¸ ·
·
¸
¸ · ∂k
−fkθ
fkk + ukk fkh + ukh
∂θ 0
=
0
∂h
fhk + uhk fhh + uhh − ehh
−fhθ + ehδ ϕ (θ)
ϕ (θ)
∂δ
Let Γ be defined as
Γ≡
·
fkk + ukk fkh + ukh
fhk + uhk fhh + uhh − ehh
35
¸
In fact, Γ is the Hessian of Π + U. Since Π + U is concave in k and h, Γ is
negative definite. Because Γ−1 exists, the solutions to ∂k/∂θ and ∂h/∂δ are
0
∂k
(fhh + uhh − ehh )fkθ + (fkh + ukh )(−fhθ + ehδ ϕ (θ))
=
∂θ
(fkh + ukh )2 − (fkk + ukk )(fhh + uhh − ehh )
µ
¶
0
∂h
−(fkk + ukk )(−fhθ + ehδ ϕ (θ)) − (fhk + uhk )fkθ
0
=
/ϕ (θ)
2
∂δ
(fkh + ukh ) − (fkk + ukk )(fhh + uhh − ehh )
(48)
(49)
Denominator in (48) and (49) are negative because Γ is negative definite.
The numerators in (48) and (49) are negative because of assumption 1 and
negative second partial derivative. (48) and (49) are the ordinary differential
equations and the unique solutions are exists as long as the right hand sides
of (48) and (49) satisfy the usual Lipshitz condition in k and h. Furthermore,
they are increasing in agents’ type.
∂k
∂h
> 0 and
> 0.
∂θ
∂δ
Because Lemma 4 holds in equilibrium, changes in wage is:
∂w
∂h 0
∂k
= fh ϕ (θ) − uk
∂θ
∂δ
∂θ
Therefore, wage is also uniquely determined.
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37