This article incorporates an information structure with partial
information into the canonical hold-up problem. The optimal information
structure balances the tradeoff between ex ante efficiency (the
"information rent" effect) and ex post efficiency (the
"bargaining disagreement" effect). With one-shot bargaining,
it occurs at an intermediate level of information asymmetry; when there
is repeated bargaining, it is attained with perfect asymmetry.
Asymmetric information, the parameter that is frequently ignored in the
literature, turns out to be an important welfare instrument for the
hold-up problem. Our results therefore provide a basis for institutional
design regarding the optimal control of information flow.
1. Introduction
* In their influential paper, Klein, Crawford, and Alchian (1978)
explain vertical integration in terms of reducing transactions costs for
"post-contractual opportunistic behavior." Since this seminal
contribution, there has been an extensive literature attempting to
account for existing economic institutions based on a fundamental
phenomenon: the hold-up problem. When one party transacts with another,
it often involves some relationship-specific investment. Because
contracts are incomplete, these parties have to rely on bargaining to
divide the surplus of investment. Typically, however, the agent who
makes this ex ante investment is not its residual claimant because she
does not have all the bargaining power at the ex post bargaining stage.
Knowing that her sunk investment cost will not be fully compensated, she
therefore underinvests. Despite the widespread study of this classic
problem and the presence of information asymmetry in virtually every
economic situation, "[a]symmetric information has played a very
limited role in the analysis of the hold-up problem" (Hart, 1995).
In the bulk of the existing incomplete-contract literature initiated by
Grossman and Hart (1986) and Hart and Moore (1988), all the variables of
interest are assumed to be observable at the bargaining stage.
Therefore, ex post efficiency is automatically guaranteed and any
inefficiency comes from ex ante underinvestment. Such a simplifying
assumption is particularly problematic considering the emphasis of human
capital investment in the literature. Even if investment is purely
physical, its degree of specificity may still be private knowledge.
Recently, asymmetric information has started to appear in the
literature, most notably Gibbons (1992) and Gul (2001). (1) They study
the classic hold-up problem in which the investor has no bargaining
power in the one-shot bargaining game that follows. As expected, this
complete lack of bargaining power translates into a complete lack of
investment incentives. They then ask what happens when the investment
decision is not observable. In fact, the joint surplus is exactly the
same with or without observability. Like other papers incorporating
asymmetric information into the hold-up problem, however, they both
assume either full information or no information.
In contrast, this article studies the hold-up problem under partial
information, which is a more realistic description of most situations.
Our model consists of a risk-neutral buyer who is interested in buying a
single unit of a product. She makes a costly investment that
deterministically increases her value of the item. With some
probability, the investment is observed by the seller, who is also risk
neutral. The seller then makes a take-it-or-leave-it offer. By varying
the probability of observability, we characterize explicitly the various
effects of asymmetric information on the hold-up problem and the
sensitivity of its equilibrium outcome. It might be tempting to
generalize from the results of Gibbons (1992) and Gul (2001) that any
degree of information asymmetry necessarily leads to the same welfare
level. We show that the parties' joint surplus in fact varies
non-monotonically with the degree of information asymmetry. The key
observation is that asymmetric information introduces two
counterbalancing forces. It increases disagreement at the bargaining
stage but, because the information rents strengthen investment
incentives, underinvestment is reduced. Within a wide range of
information structures, the joint surplus is strictly greater than that
under either full information or no information. Put differently, the
optimal information structure occurs at the interior. Therefore, it can
be misleading to just look at the two polar cases.
Indeed, in some situations, bargaining does not end after a
one-shot attempt. Yet, repeated bargaining does not improve the joint
surplus at all, as long as there is full observability. More strikingly,
Gul (2001) goes further to show that unobservability, together with
repeated bargaining, does solve the hold-up problem completely: the
parties attain the first-best outcome in the limit, as the time between
successive offers goes to zero. Recognizing the requirement of
unobservability may be unrealistic in this context, he further
conjectures that "a small amount of asymmetric information between
the buyer and the seller regarding the buyer's investment level may
be sufficient" to achieve this socially efficient outcome.
We extend our partial-information hold-up model to infinite-horizon
bargaining by letting the seller make repeated offers. This extension
also allows us to look at the interactions between the negative
"bargaining disagreement" and the positive "information
rent" effects in a dynamic context. With repeated bargaining,
"bargaining disagreement" is replaced by "bargaining
delay." Recall the seller always extracts the whole surplus in one
shot whenever the true investment is observed. As the time between
successive offers goes to zero, any delay is essentially killed off also
when investment is not observed. The buyer therefore becomes the
"information-truncated" residual claimant to her investment.
Consequently, her investment incentives and hence the joint surplus
improve unambiguously with a higher degree of information asymmetry.
Nevertheless, the first-best outcome can be achieved only under the
extreme case of perfect information asymmetry.
As a byproduct, our results demonstrate that the possibility of
repeated bargaining is always desirable.
We further endogenize the information structure by allowing the
seller to carry out costly information acquisition. In the more
realistic scenario when he cannot commit to a particular information
structure, there is always overacquisition relative to what is socially
optimal. Nevertheless, an intermediate level of information asymmetry
emerges endogenously in equilibrium, confirming the robustness of our
previous results.
The central message of this article is that asymmetric information,
the parameter that has been frequently ignored in the literature, turns
out to be an important welfare instrument for the hold-up problem. Its
implications are twofold. First, our results provide a basis for
institutional design regarding the optimal control of information flow.
Moreover, because the optimal information structure depends crucially on
whether bargaining is one shot or repeated, use of this instrument
should depend on the situation considered. In particular, if the good
being traded is perishable or short-lived and there is no room for
repeated bargaining, then the optimal institution will involve an
intermediate degree of information asymmetry. On the other hand, where
repeated bargaining is viable, it will be desirable to ensure its
feasibility and limit information flow to the greatest possible extent.
One theoretical advantage of our model is that it concerns
information flow between firms. In contrast, the property rights models
based on the authoritative work of Grossman and Hart (1986) and Hart and
Moore (1990) propose solving the hold-up problem using integration or
disintegration, while ignoring the (potentially large) impacts such a
restructuring might have on the internal organization and incentives
within each firm. In a world in which most firms consist of multitiers
and agency problems are abundant, it is hard to justify that their
internal organization and incentives will remain intact after a merger
or an acquisition. This point has been emphasized by Bolton and
Scharfstein (1998) and Holmstrom and Roberts (1998).
In practice, the control of information flow between firms can be
achieved, for example, by adjusting the relevant disclosure laws. In
some sense, our results also contribute to the disclosure literature. As
Dye (2001) points out, several important classes of disclosure models
give extreme predictions regarding the optimal level of disclosure. This
article, in contrast, provides a rationale for the optimality of partial
disclosure, which is more consistent with empirical observations.
The rest of this article is organized as follows. Section 2
describes the main model, followed by its analysis in Section 3.
Sections 4 and 5 modify the basic model to accommodate repeated
bargaining and information acquisition, respectively. Finally, Section 6
concludes this article. All proofs are relegated to the Appendix.
2. The partial-information hold-up model
* A risk-neutral seller (he) owns an indivisible good that is of
value to a risk-neutral buyer (she). Before any trade takes place, the
buyer can choose a level of relationship-specific investment I [member
of] [0, [bar.I]] to undertake by incurring a sunk cost of C(I), which
will potentially increase her return of the good to R(I). The seller
then makes a take-it-or-leave-it offer to sell the good at price p
[greater than or equal to] 0, which the buyer may either accept or
reject. The overall payoffs of the buyer and the seller are thus given,
respectively, by the following, depending on whether they reach
agreement to trade or not:
[U.sup.B] = R(I) - p - C(I) and [U.sup.s]=p if trade takes place;
[U.sup.B] = -C(I) and [U.sup.s] = 0 otherwise.
The novelty of our model is that it incorporates an information
structure q into this canonical model. More precisely, after the buyer
has made her investment choice I and before the seller chooses his price
offer, the latter observes an imperfect signal [eta] [member of] {I, X}
sent by Nature so that with probability (1 - q) [member of] [0, 1], he
observes the true investment level whereas with probability q, he learns
nothing. Mathematically, [for all] I,
Pr[[eta] = I|I] = 1 - q;
Pr[[eta] = X|I] = q.
We will call any [eta] = I [member of] [0, [bar.I]] an informative
signal and [eta] = X the uninformative signal. Intuitively, q is a
measure of the degree of information asymmetry between the parties: as q
increases, it becomes less likely that the seller is informed of the
buyer's investment choice and there is a bigger gap in terms of the
information each possesses. Notice that information of this format is
regarded as "hard information." (2) This exogenous information
structure can be interpreted as the information flow due to the existing
institution, or that set deliberately by policymakers.
We assume that both R(I) and C(I) behave "nicely": they
are strictly increasing and twice differentiable; R(I) is concave
whereas C(I) is convex, with one of them being strict. It is also
standard to assume that the social surplus function, R(I) - C(I), has an
interior optimum (denoted by [I.sup.*]), that is,
R'(0) - C'(0) > 0; R'([bar.I]) - C'([bar.I])
< 0.
Moreover, R(0) > 0 and C(0) = 0, so that the good is valuable to
the buyer even without any investment. Finally, we assume
R'(0)/C'(0) < [infinity]. (3) The cost and return functions
here encompass those in Gibbons (1992) and Gul (2001), therefore their
models are our special cases when q = 0 and 1.
[GAMMA](q) will represent this static-bargaining game with
exogenous information structure q. Throughout, we will use sequential
equilibrium as the solution concept, which is standard in the analysis
of dynamic games with nontrivial information sets. The equilibrium
payoff of the buyer, that of the seller, and the equilibrium joint
surplus will be denoted by [U.sup.B](q), [U.sup.s](q), and W(q),
respectively.
3. Interior optimal information structure
* This section analyzes the equilibrium outcome of the main model.
Our primary goal is to see how the introduction of information asymmetry
might affect the nature of the hold-up problem and hence the join
surplus of the parties, when writing a contract is formidable. (4) An
essential step is to understand the buyer's investment incentives
in equilibrium. For any information structure q, define the
information-truncated return as
[R.sup.T](I; q) = q x R(I) - C(I). (1)
Intuitively, because the seller is going to extract the whole
surplus upon receiving any informative signal [eta] = I [member of] [0,
[??]], part of the return--namely, (1 - q) x R(I)--is lost for certain
from the buyer's perspective. As a consequence, when evaluating the
costs against benefits of investment, the buyer never takes into account
this lost return. Correspondingly, the optimal information-truncated
investment [??](q) is defined to be the investment level that maximizes
the information-truncated return,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)
Where there is no confusion, we will frequently suppress the
dependence of [??] on q. When there is no information flow (i.e., q =
1), the information-truncated return and the social surplus function
coincide and so the optimal information-truncated investment is simply
the first-best level,
[??](q) = [I.sup.*] iff q = 1. (3)
With too much information (i.e., q below some critical value), on
the other hand, the optimal information-truncated investment is 0. More
precisely, it is straightforward to check that this critical information
structure [??] occurs at C'(0)/R'(0),
[??](q) = 0 iff q [less than or equal to] [??] =
C'(0)/R'(0). (4)
Clearly, the buyer will never be motivated to invest whenever q
[less than or equal to] [??]. The reason is obvious: the gains from
investment simply cannot cover the sunk costs even if she receives all
the surplus at the uninformative signal [eta] = X. As a result, the
parties receive the trivial equilibrium payoffs given in Proposition 1.
Proposition 1. When q [less than or equal to] [??],
[U.sup.B](q) = 0,
[U.sup.s](q) = R(0).
Will the buyer ever be motivated to invest? The answer is positive,
as long as there is enough information asymmetry. We will show that for
q > [??], the equilibrium turns out to be of mixed strategies. We
know that the seller will always quote the price R(I) upon receiving
[eta] = I. In this mixed strategy equilibrium, therefore, the buyer
randomizes her investment choice and the seller randomizes his price
offer only upon receiving [eta] = X. Let us first introduce the
following notation regarding the sequential equilibrium of [GAMMA](q)
for q > [??]:
[G.sup.q](I) = buyer's investment distribution;
[I.sup.h](q) = upper bound of investment distribution's
support;
[I.sup.l](q) = lower bound of investment distribution's
support;
[F.sup.q](p) = seller's pricing distribution at [eta] = X.
The detailed derivation of this equilibrium can be found in the
Appendix. An important step is to argue that the buyer randomizes her
investment over the connected interval [0, [??](q)]. Recall the optimal
information-truncated investment [??](q) is strictly positive whenever q
> [??]. To see why it is the upper bound of the investment
distribution's support, notice that the buyer would never invest
more that [??](q), as her maximum possible marginal gains could never
compensate for her marginal costs. It is also impossible that
[I.sup.h](q) [??](q), because the seller would never offer a price above
R([I.sup.h]) and the buyer would become the
"information-truncated" residual claimant on any investment
just above [I.sup.h] (q). Furthermore, the lower bound of her investment
distribution's support is exactly 0. Suppose [I.sup.l](q) > 0;
then because the seller would never charge a price below R ([I.sup.l]),
the buyer's payoff at [I.sup.l] (q) would be negative.
[FIGURE 1 OMITTED]
Having established its support, it takes one further step to derive
[G.sup.q](I) (and similarly for [F.sup.q](p)) as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
To obtain a concrete idea of the buyer's investment decision,
we now plot her equilibrium investment distributions for two distinct
information structures using an arbitrary pair of cost and return
functions. Figure 1 compares [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII] against [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where [q.sup.h] > [q.sup.l] > [??]. As shown in the diagram,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] overlap up to [??]([q.sup.l]), the
upper bound of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
support, at which point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] jumps to 1. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] continues to increase smoothly until it reaches [??]([q.sup.h]),
the upper bound of its support, until it finally jumps to 1.
Clearly, a higher degree of information asymmetry shifts the
investment distribution up by first-order stochastic dominance.
Intuitively, with more information asymmetry, it becomes less likely
that the seller will charge his price contingent on investment. This,
coupled with the wider support of the pricing distribution, means that
the buyer will receive more information rents to compensate for her sunk
costs. As a consequence, her investment incentives are enhanced. We will
refer to this positive impact on joint surplus as the information rent
effect. At the same time, it also becomes more probable that the parties
will disagree over the terms of trade and fail to realize the gains from
trade. We will call this negative impact the bargaining disagreement
effect. For any equilibrium investment level I chosen by the buyer in
F(q), let [T.sup.q](I) denote the expected gains from trade over R(I).
In other words, [T.sup.q](I) tells us how much of the return from I will
actually be realized. Proposition 2 below characterizes the two
counterbalancing effects formally.
Proposition 2. For any [q.sup.l], [q.sup.h] with [q.sup.h] >
[q.sup.l] > [??]:
(i) (Information rent effect)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (I) first-order
stochastically dominates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII]t (I);
(ii) (Bargaining disagreement effect)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We recall Gibbons (1992) and Gul (2001) have shown that the parties
always jointly receive R(0) with either no information or full
information. This naturally leads us to suspect that the negative
bargaining disagreement effect and the positive information rent effect
always cancel each other out. In Proposition 3, we compute explicitly
the equilibrium payoff of each party and show that this extrapolation is
actually false. In particular, except at q = 1, the joint surplus is
always higher than R(0).
Proposition 3. When q > [??],
[U.sup.B]S(q) = 0,
[U.sup.S](q) = [(1-q) x ln R([??])/R(0) + 1] x R(0)
To see why this result is true, the key observation is that R(0) is
in the support of the seller's pricing strategy at the
uninformative signal, which will be accepted for sure. In other words,
the seller obtains R(0) on average whenever he does not observe the true
investment. However, he almost always quotes a price strictly higher
than R(0) at the informative signals, which again will always be
accepted. Therefore, the seller's overall expected payoff must be
higher than R(0). In the extreme case of perfect information asymmetry,
however, the seller always receives the uninformative signal and his
overall payoff is exactly R(0). This special case is also confirmed by
our formula: when q = 1, (1 - q) x ln R([??])/R(0) = 0 and so [U.sup.S]
(q) = R(0).
Let [Q.sup.*] be the set of optimal information structures in
[GAMMA](q), that is,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
Corollary 1 below is immediate from Propositions 1 and 3.
Corollary 1. (Non-monotonicity and interior optimum) The
equilibrium joint surplus of [GAMMA] (q) as a function of information
asymmetry is as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Therefore, [Q.sup.*] subset] ([??], 1).
Corollary I says that within a wide range of information structures
([??] < q < 1), the overall impact due to the information rent
effect always outweighs that due to the bargaining disagreement effect,
relative to the full-information benchmark (q = 0). As a result, the
optimal information structure occurs at an intermediate level of
information asymmetry. (5)
[FIGURE 2 OMITTED]
Our results thus far have said nothing about the shape of the joint
surplus function. One curious question is whether it is single-peaked.
Without further restrictions, however, we are not able to offer a
definite answer. Nevertheless, it can be readily checked that with
linear costs, the joint surplus function W(q) is strictly concave over
([??], 1) for the following two standard classes of return functions:
(i) R(I) = [alpha] x ln(I + 1 + [beta]);
(ii) R(I) = [alpha] x [(I + [beta]).sup.[gamma]],
where [alpha], [beta] [greater than or equal to] 0 and [gamma]
[member of] (0, 1). (6)
We conclude this section by plotting the joint surplus function
using Example (E1) below.
R(I) = [square root of I + 1/8; C(I) = I. (E1)
Figure 2 depicts W(q) as the solid line. The critical information
structure occurs at [??] [approximately equal to] 0.7 and so W(q) is
flat at R(0) [approximately equal to] 0.354 for [less than or equal to]
< 0.7. This example is just a special case of (ii) above, hence W(q)
is strictly concave for q [member of] (0.7, 1) and peaks at [q.sup.*]
[approximately equal to] 0.85.
4. Repeated bargaining
* The driving force behind our main results is the tradeoff between
the positive information rent and the negative bargaining disagreement
effects. The latter arises because there is no scope for the parties to
reach agreement once an offer is rejected. What happens if they always
have a chance to bargain over again? In this section, we modify the
basic model to allow for infinite-horizon bargaining. More precisely,
the seller can make a new take-it-or-leave-it offer in every period
whenever no agreement has been reached. The payoffs of the agents are
modified accordingly as follows, depending on the time t = k x [DELTA]
at which trade takes place:
[U.sup.B.sub.[DELTA]] = [R(I) - p] x [e.sup.-r x t] - C(I);
[U.sup.S.sub.[DELTA]] = p x [e.sup.-r x t],
where k = 0, 1, 2, ..., indexes the period number; [DELTA] is the
length between successive periods; and r > 0 is the interest rate. We
will represent this dynamic game with information structure q as
[[GAMMA].sub.[DELTA]](q). Any other notation N will be replaced
analogously by [N.sub.[DELTA]].
As complicated as it might appear at first glance, this
dynamic-bargaining game turns out to be similar to its static
counterpart in a number of aspects. It is a standard result in the
bargaining literature that under this environment, the seller again
extracts all the surplus whenever he observes the true investment. As a
consequence, whenever q [less than or equal to] [??], the buyer again
never has incentives to invest and each party again obtains the lowest
possible payoff in equilibrium. Proposition [1.sup.*] is the dynamic
analog to Proposition 1.
Proposition [1.sup.*]. When q [less than or equal to] [??],
[U.sup.B.sub.[DELTA]](q) = 0;
[U.sup.s.sub.[DELTA]](q) = R(0)
Using arguments analogous to those establishing the corresponding
results previously, it can be shown that whenever q > [??], any
equilibrium again must be of mixed strategy and the bounds of the
buyer's investment distribution are again given by 0 and [??](q).
Implicit in this result is the conclusion that the buyer again receives
0 in equilibrium. What about the seller's payoff?. In this dynamic
setting, bargaining continues even if an offer is rejected. However, the
parties might forgo part of their joint surplus due to discounting. In
other words, "bargaining disagreement" is replaced by
"bargaining delay." What are the interactions between the
original information rent and this new bargaining delay effects? Lemma 1
below is central in addressing this question. Lemma 1 (No expected
delay; Gul, 2001). Fix q > [??]. For every [epsilon] > 0, there
exists [bar.[DELTA]] > 0 such that whenever [DELTA] <
[bar.[DELTA]], in any sequential equilibrium of [[GAMMA].sub.[DELTA]]
(q), the probability that the parties reach agreement by time e is at
least 1 - [epsilon], in the event that [eta] = X is generated.
We already know that there is no delay whenever an informative
signal is generated. Lemma 1 says that when bargaining becomes
infinitely frequent, the parties can almost always reach agreement
almost immediately also when the uninformative signal is generated. Put
differently, the bargaining delay effect is essentially eliminated! This
lemma is adapted from Proposition 5 in Gul (2001), where immediate
agreement is shown to be essentially achieved when investment is never
observable (i.e., in the game [[GAMMA].sub.[DELTA]] (1), using the
language of our model). The arguments establishing this no expected
delay result make heavy use of the standard results from the well-known
"Coase conjecture" concept, first suggested by Coase (1972)
and later formulated rigorously and proved formally by Fudenberg,
Levine, and Tirole (1985) and Gul, Sonnenschein, and Wilson (1986).
To give a flavor of the arguments behind Proposition 5 in Gul
(2001) and see how they can be adapted to Lemma 1, we now provide a
concise and informal account of the "Coase conjecture." It
concerns the pricing of a durable-good monopolist facing a continuum of
consumers whose valuations v of the good are private information and are
distributed according to some distribution H(v) over the support
[[v.sup.l], [v.sup.h]]. The subgame perfect Nash equilibrium of this
game turns out to be essentially equivalent to the sequential
equilibrium of its corresponding bilateral-bargaining game with
one-sided uncertainty. In the latter game, a seller makes
infinite-horizon repeated offers to sell a good to a buyer who has
private information about her valuation, and the priors over this
valuation are distributed as H(v) over [[v.sup.l], [v.sup.h]]. The
following two assumptions are standard in this literature:
(i) the gap condition: [v.sup.l] > 0;
(ii) the Lipschitz condition at 0:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
where K < [infinity] is a constant.
With these assumptions, any such "Coase bargaining game"
can be shown to exhibit the following main properties regarding its
sequential equilibria (Gul, Sonnenschein, and Wilson, 1986):
(i) equilibrium exists and is generically unique; in particular,
all equilibria give rise to the same behaviors on the equilibrium path;
(ii) the buyer's acceptance rule is stationary in that it
depends only on the current price offer;
(iii) the seller does not randomize his price offer along the
equilibrium path after the first period;
(iv) the parties reach agreement in a finite number of periods with
probability 1.
If the buyer's valuation in this "Coase bargaining
game" is determined by her ex ante investment, then her equilibrium
investment distribution [G.sub.[DELTA]](I) is going to induce a
corresponding valuation distribution [H.sub.[DELTA]](v) with v equal to
R(I). Yet unlike the "Coase bargaining game" where H(v) is
fixed, here [G.sub.[DELTA]](I) and hence [H.sub.[DELTA]](v) are
determined endogenously in equilibrium and vary with the period length
[DELTA]. However, as long as investment is not observable, the parties
are effectively playing a "Coase bargaining game" after the
investment stage and the above properties are preserved in the
bargaining stage for any fixed [DELTA]. (7) Moreover, these properties
imply that the buyer's equilibrium acceptance rule converges as
[DELTA] approaches 0. Consequently, by moving down the
"demand" given by the buyer's actual acceptance rule
virtually without delay, the seller can extract a surplus almost
identical to the maximum possible it can achieve, namely the whole area
under the "demand" given by the buyer's limiting
acceptance rule. Put differently, "as the time between offers
becomes arbitrarily small, the seller [will] price-discriminate
arbitrarily finely in an arbitrarily small interval of real time"
with probability arbitrarily close to 1. To apply Gul's (2001) no
expected delay conclusion to our current setting (i.e., the game
[[GAMMA].sub.[DELTA]](q) with q > [??]), the crucial observation is
that after the investment stage, our "bargaining game" at the
uninformative signal [eta] = X has the same structure as Gul's
(2001) "bargaining game": in either case, the parties are
effectively involved in a corresponding "Coase bargaining
game." (8)
Recall the seller always extracts the whole surplus in one offer
whenever he observes an informative signal. With immediate agreement
being essentially guaranteed at the uninformative signal as well,
therefore, the buyer almost always becomes the
"information-truncated" residual claimant on her investment.
As a result, she almost always chooses her investment almost equal to
the optimal information-truncated level. Proposition [3.sup.*] below,
the dynamic version of Proposition 3, describes each party's
equilibrium payoff as a function of information asymmetry in the limit.
Proposition [3.sup.*]. Fix q > [??]. As [DELTA] [right arrow] 0,
that is,
[U.sup.B.sub.[DELTA]](q) = 0;
[U.sup.S.sub.[DELTA](q) [right arrow] R([??])- C([??])
Let [W.sub.0](q) denote the equilibrium joint surplus of
[[GAMMA].sub.[DELTA]] (q) as [DELTA] [right arrow] 0,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
Corollary [1.sup.*] below, a direct consequence of Propositions
[1.sup.*] and [3.sup.*], is the parallel to Corollary 1.
Corollary [1.sup.*] (Monotonicity and extreme optimum).
[W.sub.0](q) is increasing in q. Therefore, the optimal information
structure of [[GAMMA].sub.[DELTA]] (q) as [DELTA] [right arrow] 0 occurs
at perfect information asymmetry, with its joint surplus arbitrarily
close to the first-best level s R([I.sup.*]) - C([I.sup.*]).
With perfect information asymmetry, the buyer becomes the full
residual claimant on her investment and so the first-best outcome
emerges as a special case. It is nevertheless worth noting that this
socially efficient outcome is attained only under an extreme form of
information structure. Put differently, the informational requirement to
achieve the first-best outcome is severe.
We now return to Figure 2, in which [W.sub.0](q) in Example (E1) is
depicted as the dotted line. We already know from Propositions 1 and
[1.sup.*] that whenever [less than or equal to] < [??], the buyer is
never motivated to invest. Therefore, their joint surplus is always
R(0), whether bargaining is static or dynamic. It is interesting to
compare [W.sub.0](q) against W(q) when the level of information
asymmetry is high enough to induce the buyer to invest. As is apparent
in the figure, the joint surplus is always higher with repeated
bargaining than with one-shot bargaining. This property is in fact not
specific to the particular example used. The explanation is simple. In
the static-bargaining game, the buyer randomizes over [0, [??](q)] and
so the joint surplus must be strictly lower than R([??])--C([??]), which
is what they receive in the dynamic-bargaining game. In addition, the
possible failure to realize the gains from trade in the
static-bargaining game makes their joint surplus depart even further
from R([??])--C([??]). In other words, infinitely frequent repeated
bargaining never makes the parties worse off. This observation is
recorded as Corollary 2.
Corollary 2 (Superiority of repeated bargaining). For any given q,
[W.sub.0](q) is at least as high as W(q):
[W.sub.0](q) = W(q) whenever q [less than or equal to] [??];
[W.sub.0](q) > W(q) whenever q > [??].
If we interpret one-shot bargaining as an "original
contract" and repeated bargaining (after the first shot) as a
"renegotiation process," then Corollary 2 also has the
interesting implication that a better outcome can be achieved when
renegotiation cannot be ruled out. In contrast, a number of papers in
the standard literature have demonstrated that under certain
conditions--in particular, when both investment and return are common
knowledge--the possibility of renegotiation can only make matters worse.
(9) These papers usually design a clever contract that uses very
inefficient trading outcomes off the equilibrium path as threats to
support the first-best investment in equilibrium. The introduction of
renegotiation means that any inefficiency from trade is negotiated away,
thus rendering the original out-of-equilibrium threats incredible and
destroying the original investment incentives. Under our
asymmetric-information environment and "original contract,"
the equilibrium trading outcome is necessarily inefficient. As a result,
renegotiation allows agents to realize more gains from trade also on the
equilibrium path, which alone improves the joint surplus. Moreover, the
no expected delay nature of our "renegotiation process" (again
under our asymmetric-information environment) means mat me
investor's investment incentives are strengthened by renegotiation,
which further raises the overall welfare.
5. Information acquisition
* Our analysis so far has assumed information structure to be
exogenous. Indeed, the seller is always tempted to find out the
buyer's true investment level and in some situations, it might not
be prohibitively costly to do so. Motivated by this observation, we now
extend the original model in Section 2 to allow for endogenous
information acquisition. More specifically, after the buyer has made her
investment decision and before bargaining takes place, the seller can
acquire a particular information structure q at a twice-differentiable
convex cost [LAMBDA](q) with [LAMBDA]'(q) < 0 and
[LAMBDA]"(q) > 0. To capture the idea that information
acquisition is not prohibitively costly, we make the following natural
assumptions:
[LAMBDA](1) = [LAMBDA]'(1) = 0. (9)
We will restrict attention to those equilibria in which the seller
chooses some q with probability 1. This extended game and any of its
endogenous equilibrium information structures will be denoted by
[[GAMMA].sup.en] and [q.sup.en], respectively.
As a natural benchmark for comparison, let [q.sup.FB] be any
socially efficient information structure; [q.sup.FB] thus balances the
marginal gains in joint surplus against the marginal costs of acquiring
information, (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)
Notice that the set {[q.sup.FB]} coincides with the set of optimal
information structures [Q.sup.*] in Section 3 if information were
costless. Moreover, if the seller were able to commit to a particular
information structure before investment takes place, he would indeed
choose an element from this set.
Consider now an equilibrium of [[GAMMA].sup.en] with endogenous
information structure [q.sup.en]. Because the expectation over
[q.sup.en] is correct in equilibrium, the investment and pricing
behaviors are still given by those in equilibrium of the original game
[GAMMA]([q.sup.en]) in Section 3. The seller's payoff is still
W([q.sup.en]). However, when the seller decides how much information to
acquire, the buyer's investment strategy is now taken as given. As
a result, any equilibrium information structure [q.sup.en] in this game
must satisfy instead
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)
where the right-hand side denotes the partial derivative with
respect to q when [??] is fixed at [??]([q.sup.en]). Proposition 4
summarizes the key properties of the equilibrium level of information
acquisition.
Proposition 4.
(i) (Interior and unique solution) [q.sup.en] [member of] ([??], l)
and is unique.
(ii) (Overacquisition) [q.sup.en] < min {[q.sup.FB]}.
When investment distribution is taken as given, the marginal
benefit of acquiring information is simply the marginal likelihood of
observing the true investment and hence extracting the whole surplus. In
contrast, when the seller could commit to a particular information
structure at the beginning (i.e., in the scenario that leads to a
socially optimal level), there is also a negative effect: the
buyer's investment distribution shifts down accordingly with more
information acquisition. In other words, without the possibility of
commitment, the seller fails to take this negative effect into account
and acquiring information imposes a negative externality on the buyer.
This explains why he tends to engage in too much information
acquisition. More importantly, Proposition 4 confirms the robustness of
our previous result: an interior information structure emerges as the
equilibrium outcome endogenously.
6. Concluding remarks
* Let us conclude by first considering the original
partial-information hold-up model. When there is too much information,
we have the classic hold-up problem: the buyer is never motivated to
invest and the parties jointly receive the lowest possible payoffs. What
happens when we introduce a significant degree of information asymmetry?
Now the information rents created provide the buyer sufficient
incentives to invest anything between zero and the optimal
information-truncated level and this positive effect acts to raise the
joint surplus. At the same time, it becomes more likely that the parties
disagree over the terms of trade and this negative effect acts to lower
the joint surplus. Overall, the "information rent" effect
dominates the "bargaining disagreement" effect and the
resulting joint surplus improves. Next, what happens when we further
allow for repeated bargaining? As the time between successive offers
goes to zero, repeated bargaining guarantees no expected delay and hence
all the gains from trade are essentially realized in the limit. The
buyer in turn shifts her investment toward the optimal
information-truncated level, giving an overall joint surplus arbitrarily
close to the optimal information-truncated level.
Appendix
* This appendix contains all proofs and the derivation of the
sequential equilibrium of [GAMMA](q) for q > [??]. Where there is no
confusion, we will frequently suppress the dependence of various
expressions on q.
Some preliminary observations. Let us first make a few simple
observations regarding the behavior and expected payoff of each party in
any sequential equilibrium of [GAMMA](q). We will utilize these
observations without explicitly referring to them in the proofs.
* After investing I, the buyer accepts any price p if and only if p
[less than or equal to] R(I). (11)
* Upon receiving any informative signal [eta] = [??] [member of]
[0, [??]], the seller's posterior beliefs over the buyer's
investment choice are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
He therefore optimally quotes the price p = R([??]).
* Upon receiving the uninformative signal [eta] = X, the
seller's posterior beliefs over the buyer's investment choice
coincide with the actual investment distribution. His price offer p
therefore must satisfy R([I.sup.l]) [less than or equal to] p [less than
or equal to] R([I.sup.h]), where [I.sup.l] and [I.sup.h] are the upper
and lower bounds of the investment distribution's support.
* The buyer's expected payoff from investing I is therefore
given (with a slight abuse of notation) by
[U.sup.B] (I) = q x Pr[p [less than or equal to] R(I)] x [E.sub.p
[less than or equal to] R(I)) [R(I) - p] - C(I), (A1)
where [E.sub.y] [*] denotes expected value conditional on event Y.
* The seller's expected payoff from quoting p at [eta] = X is
therefore given by
[U.sup.S] (p) = Pr[R(I) [greater than or equal to] p] x p. (A2)
Proof of Proposition 1. The buyer's payoff from investing any
I > 0 is given by
[U.sup.B] (I) = q x Pr[p [less than or equal to] R(I)] x [E.sub.p
[less than or equal to] R(I)) [R(I) - p] - C(I).
Because p [greater than or equal to] R(0),
[U.sup.B] (I) [less than or equal to] q x [R(I) - R(0)] - C(I)
= q x R(I) - C(I) - q x R(0).
Also, because q [less than or equal to] [??], it follows from (4)
that
q x R(I) - C(I) - q x R(0) < q x R(0) - C(0) - q x R(0) = 0.
In other words, the buyer never chooses a strictly positive
investment level.
To sum up, the unique sequential equilibrium of [GAMMA](q) with q
[less than or equal to] [??] is as follows.
* The buyer invests 0; after investing I [member of] [0, [??]], she
accepts any price p if and only if p [less than or equal to] R(I).
* The seller's posterior beliefs after any informative signal
[eta] = [??] [member of] [0, [??]] are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
whereas those after the uninformative signal n = X are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
* The seller's pricing rule is therefore
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Q.E.D.
Derivation of sequential equilibrium of [GAMMA](q) for q > [??].
Recall the following notation:
[G.sup.q](I) = buyer's investment distribution;
[I.sup.h](q) = upper bound of investment distribution's
support;
[I.sup.l](q) = lower bound of investment distribution's
support;
[F.sup.q](p) = seller's pricing distribution at [eta] = X.
We first establish that the buyer randomizes her investment over
the connected interval [0, [??](q)] using successive claims.
Claim A1. [I.sup.l](q) = 0.
Proof Assume to the contrary that [I.sup.l] > 0. Then the
buyer's payoff from investing [I.sup.l] is given by
[U.sup.B] ([I.sup.l]) = q x Pr[p [less than or equal to]
R(I.sup.l])] x [E.sub.p [less than or equal to] R([I.sup.l]))
[R([I.sup.l]) - p] - C([I.sup.l]).
Because p [greater than or equal to] R(([I.sup.l]),
[U.sup.B]([I.sup.l]) = -C([I.sup.l]) < 0.
But this means the buyer would rather not invest than to invest
[I.sup.l]. Q.E.D.
Claim A2. [I.sup.h](q) = [??](q).
Proof Assume to the contrary that [I.sup.h] [not equal to] [??](q).
Because p [less than or equal to] R([I.sup.h]),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
But this means the buyer would rather invest [??](q) than to invest
[I.sup.h] (q). Q.E.D. Claim A3. [G.sup.q] has no gaps in its support.
Proof. Assume to the contrary that supp[[G.sup.q]] = [0, [I.sup.a]]
[union] [[I.sup.b], [??](q)], with [I.sup.a] < [I.sup.b]. Clearly the
seller will never quote any price p [member of] (R([I.sup.a]),
R([I.sup.b])). Let [??] = [E.sub.p [less than or equal to] R([I.sup.a])
[p] and let [??] = [F.sup.q](R([I.sup.a])), then
[U.sup.B] ([I.sup.a]) = q x [??] x [R([I.sup.a]) - p] -
C([I.sup.a])
[U.sup.B] ([I.sup.b]) = q x [??] x [R([I.sup.b]) - p] -
C([I.sup.b]).
Consider [??] = [gamma][I.sup.a] + (1 - [gamma])[I.sup.b] with
[gamma] [member of](0, 1),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
because q x [??] x R(I) - C(I) is strictly concave in I. As a
result,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
because [I.sup.a] [I.sup.b][member of] supp[[G.sup.q]]. But this
means the buyer would prefer 7 to either [I.sup.a] or [I.sup.b]. Q.E.D.
We next derive [G.sup.q](I) and [F.sup.q](p) using the fact that
both players must be indifferent among any choices in the supports of
their respective distributions.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Using integration by parts on [[integral].sup.R(I).sub.R(0)]
pd[F.sup.q](p), the above can be rewritten as
[[integral].sup.R(I).sub.R(0)] [F.sup.q](p)dp = C(I)/q.
Differentiating both sides with respect to I,
C'(I)/q = [F.sup.q](R(I)) * R'(I)
C'([R.sup.-l)(p))/q = [F.sup.q](p) x R'([R.sup.-1](p)).
Clearly [F.sup.q](p) is strictly increasing over [R(0), R([??])).
Therefore [for all]p [member of] [R(0), R([??])),
[U.sup.S](p) = [U.sup.S](R(0))
[1 - [G.sup.q]([R.sup.-1](p))] x p = R(0)
[1 - [G.sup.q](I)] x R(I) = R(0).
To sum up, the unique sequential equilibrium of [GAMMA](q) for q
> [??] is as follows.
* The buyer randomizes her investment according to [G.sup.q](I);
after investing I [member of] [0, [??}], she accepts any price p if and
only if p [less than or equal to] R(I).
* The seller's posterior beliefs, [for all]n = [??] [member
of] [0, [??]], are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
whereas those at [eta] = X coincide with [G.sup.q](I).
* The seller quotes the price p = R([??]), [for all][eta] = [??]
[member of] [0, [??]], and he randomizes his price offer according to
[F.sup.q](p) at [eta] = X. Q.E.D.
Proof of Proposition 2.
(i) By (5), any two investment distributions are exactly the same
up to the upper bounds of their supports. To rank any two
[G.sup.q](I)'s by first-order stochastic dominance, therefore, it
suffices to compare these upper bounds. By definition,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Because C is convex and R is concave, C'(I)/R'(I) is
strictly increasing in I. As a result, [??]([q.sup.l]) <
[??]([q.sup.h]). Q.E.D.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
which is strictly decreasing in q. As a result, [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII]. Q.E.D.
Proof of Proposition 3. By (5),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Moreover, because 0 is in the support of [G.sup.q](I), [U.sup.B]
(q) = 0. Q.E.D.
Proof of Proposition [3.sup.*]. The buyer's expected payoff
from investing I [member of] [0, [??] is given by
[U.sup.B.sub.[DELTA]](I) = q . E[(R(I) - p(t)) x [e.sup.-r x t]] -
C(I),
where t is the time at which she expects to reach agreement at
[eta] = X. By Lemma 1, we know that as A becomes arbitrarily small, the
buyer is maximizing an expected payoff arbitrarily close to the
information-truncated return
[R.sup.T](I;q) = q x R(1) - C(I)
with probability arbitrarily close to I, and so he chooses an
investment level arbitrarily close to the optimal information-truncated
investment [??](q).
The above conclusion, coupled with the fact that there is no
expected delay in the overall game, means that the expected joint
surplus approaches R([??]) - C([??]) as [DELTA] [right arrow] 0. Also, 0
is in the support of the buyer's investment distribution and so her
expected payoff must be 0. Q.E.D.
Proof of Proposition 4. Before proceeding, let us first argue
formally that both (10) and (11) are valid. Observe first that
[q.sup.FB] > [??]. Suppose otherwise, then the seller's payoff
would be R(0) - [LAMBDA] ([q.sup.FB]); but he can guarantee R(0) by
committing to q = 1 and always quoting p = R(0). Also, [q.sup.FB] < 1
by (9). The proof that [??] < [q.sup.en] < 1 is completely
analogous and is omitted here. Q.E.D.
(i) [for all]q [member of] [[??], 1], let M(q) denote the
seller's marginal net gains of information acquisition,
M(q) = [LAMBDA]'(q) + R(0) x ln R([??])/R(0).
Then
M(I) = R(0) x ln R([I.sup.*])/R(0) > 0;
M([??]) = [LAMBDA]]'([??]) < 0.
Also,
M'(q) = [LAMBDA]"(q) + R(O).
R'([??]).[??]'(q)/R([??])>0.
Clearly, M(q) is continuous over [[??],1]. Therefore, by the
Intermediate-Value Theorem, them exists a unique [q.sup.en] [member
of]([??], 1) such that M([q.sup.en]) = O. Q.E.D. (ii) [for all]q [member
of] ([??], 1),
W'(q) = R(O).ln R([??])/R(O).(-1)+(1-q).R(O).
R'([??]).[??]'(q)/R([??]) >-R(O).ln R([??])/R(O)=[partial
derivative]/[[partial derivative].sub.q]W(q,[??]).
It follows that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Because M(q) is strictly increasing, we conclude that [q.sup.FB]
> [q.sup.en]. Q.E.D.
I am particularly indebted to Dirk Bergemann and Stephen Morris for
their generous advice and encouragement. I than