Information Design in the Hold-up Problem∗
Daniele Condorelli†and Balázs Szentes‡
May 4, 2017
Abstract
We analyze a bilateral trade model where the buyer can choose a cumulative distribution
function (CDF) from a set of distributions supported on [0, 1]. This CDF then determines her
valuation. The seller, after observing the buyer’s choice of the CDF but not its realization, makes
a take-it-or-leave-it offer. If the buyer can choose any CDF, the price and the payoffs of both the
buyer and the seller are shown to be 1/e in the unique equilibrium outcome. The equilibrium
CDF of the buyer generates a unit-elastic demand and trade takes place with probability one.
We consider various extensions by introducing additional constraints on the buyer’s choice
set and examine the robustness of our basic results. For example, we analyse models where the
buyer’s CDF must satisfy finitely many general moment restrictions. We also study scenarios
where the buyer’s value distribution is given but she can reshape it by either adding risk or by
destroying values. In all these cases, the buyer’s equilibrium demand is shown to be unit-elastic
and trading is efficient.
∗
We have benefited from discussions with Dirk Bergemann, Ben Brooks, Jeremy Bulow, Sylvain Chassang, Eddie
Dekel, Andrew Ellis, Jeff Ely, Faruk Gul, Yingni Guo, Sergiu Hart, Johannes Horner, Francesco Nava, Santiago
Oliveros, Ludovic Renou and Phil Reny. The comments of several seminar audiences are also gratefully aknowledged.
†
Department of Economics, University of Essex, Colchester, UK. E-mail: [email protected].
‡
Department of Economics, London School of Economics, London, UK. E-mail: [email protected].
1
1
Introduction
A central result in Information Economics is that the distribution of economic agents’ private information is a key determinant of their welfare. In auctions, for example, asymmetric information
between the seller and the bidders regarding bidders’ valuations affects all agents’ payoffs. In the
context of labour markets, asymmetric information between the employers and workers regarding workers’ productivities affects workers’ wages. When analysing environments with asymmetric
information, most microeconomic models assume that the distribution of types is exogenous. However, if an agent’s payoff depends on her information rent then she is likely to take actions in order
to generate this rent optimally. This paper contributes towards this problem by reconsidering the
hold-up problem from the perspective of optimal information design.
The literature on hold-up typically focuses on environments where the buyer’s investment
decision has a deterministic impact on her valuation for the seller’s good. If the seller has full
bargaining power, he appropriates all the surplus generated by the investment. Hence, if investment
is costly, the buyer does not invest at all and ends up with a payoff of zero. In contrast, if
investments have privately observed stochastic returns, the buyer’s investment decision induces
asymmetric information at the contracting stage. Therefore, the buyer may capture some of the
surplus through receiving information rent which, in turn, could partially restore her incentives to
invest on the first place. Our objective is to analyze the buyer’s problem of optimally designing
the stochastic returns with various constraints on the available distributions.
For an example, consider a worker’s decision to invest in human capital. Prior to going to
the job market, an aspiring freelancer chooses the type and length of her education. This decision
is observed by future employers but it is conceivable that the freelancer has private information
regarding her productivity generated by her education. When the freelancer is about to be hired
for a project, the offered contract can depend on her education but not on her actual productivity.
Of course, if the choice of education perfectly reveals the worker’s productivity, the employer’s
optimal contract might hold down the worker to her reservation payoff. Therefore, this freelancer
may chose an education which is only a noisy signal of her productivity.
In our baseline model, we do not impose any restriction on the set of distributions, except
that the maximum valuation is bounded. Specifically, the buyer can choose any CDF supported
on [0, 1] which then determines her valuation for the object. After observing the buyer’s CDF, the
seller sets a price at which either the buyer trades or the game ends. The central trade-off arising
in this model can be explained as follows. If the price of the good was given, the buyer would be
better off the higher is her valuation. However, the seller who knows that the buyer’s valuation is
2
likely to be high will charge a higher price. As a result, when choosing her value-distribution, the
buyer faces a trade-off between the higher payoff she would receive from the good and the higher
price she would have to pay.
We show that this game has a unique equilibrium outcome. In each equilibrium, the price
is 1/e and the buyer’s distribution is supported on [1/e, 1], so that trade always occurs and the
seller’s payoff is 1/e. The buyer’s value-distribution is a combination of a continuous distribution
on [1/e, 1) defined by the CDF F ∗ (v) = 1 − 1/ (ev) and an atom of size 1/e at v = 1. The total
equilibrium surplus is found to be shared equally between the seller and the buyer, so that the
buyer also receives an equilibrium payoff of 1/e. The efficiency loss that results from the buyer’s
desire to generate information rent is found to be around one quarter of the first-best social surplus.
The demand function generated by F ∗ maximizes consumer surplus in a monopoly market
where the maximum willingness-to-pay cannot exceed one and production is costless. This is
because each subgame induced by the buyer’s CDF is equivalent to the problem of a monopolist
with zero marginal cost who faces a unit mass of consumers whose aggregate demand is generated
by the CDF. As a result, the buyer’s problem of choosing the payoff-maximizing CDF is identical to
the problem of identifying the demand function which maximizes consumer surplus in this market.
In Section 7, we also characterize the demand curve which maximizes consumer surplus in a market
where many firms are competing a’la Cournot.
Our baseline model provides a rationale for the endogenous emergence of unit-elastic demands. Indeed, the equilibrium CDF of the buyer, F ∗ , generates a unit-elastic demand function
on (1/e, 1), such that the probability that the buyer is willing to buy the good at price p is
1 − F ∗ (p) = 1/ (ep). When faced with this demand function, the seller is indifferent between
charging any price in [1/e, 1], since any price in this range will result in an expected profit of
p [1/ (ep)] = 1/e. We demonstrate that this result arises because the buyer can always increase
her payoff by choosing a different CDF unless the seller is indifferent between any price on the
support of the buyer’s CDF. If the latter does not hold, the buyer can move probability mass from
suboptimal prices to higher valuations without changing the seller’s incentive to set a certain price.
Our second main insight is that the equilibrium outcome is ex-post efficient, i.e., trade occurs
with probability one. As we explained above, given the buyer’s equilibrium CDF F ∗ , the seller finds
it optimal to set the price at the lowest possible valuation of the buyer. In contrast, if the buyer’s
valuation was given exogenously, the seller would often find it optimal to set a price which excludes
low-value buyers from trade. However, if a distribution induces a price above the smallest valuation
on the support, the buyer can increase her payoff by choosing the same distribution conditional on
the value being higher than the price. The reason is that, conditional on trade, the buyer’s payoff
3
is the same but she trades more often if she chooses the conditional distribution.
We consider various extensions of our benchmark model and examine the robustness of our
results. Perhaps most importantly, we analyse scenarios where the set of distributions available
for the buyer is restricted. First, we assume that these distributions must satisfy general moment
restrictions, e.g. the buyer’s expected valuation is bounded. Second, we characterize equilibria in
models where the buyer starts with a prior value distribution but she can take certain actions in
order to reshape this prior. For example, the buyer can add arbitrary risk so her final distribution
will be a mean preserving spread of the prior CDF. We also consider cases where the buyer can
stochastically destruct value, that is, her chosen distribution must be first-order stochastically
dominated by the original distribution. We then introduce a cost associated to the buyer’s choice
of a CDF. We show that, in most of these cases, the buyer’s demand is unit elastic and the trading
decisions are ex post efficient. Finally, we discuss how the equilibria of our baseline model are
modified if there is a positive production cost, the seller does not have full bargaining power and
the buyer’s choice is unobservable.
There are papers on the hold-up problem which also consider the buyer’s ability to generate
asymmetric information. In the model by Gul (2001) and Lau (2008), the buyer can generate
asymmetric information by randomizing over imperfectly observable investment decisions. Gul
(2001) shows that if investment is unobservable, the hold-up problem is as severe as in the fully
observable case. In the unique mixed-strategy equilibrium, the efficiency gain from the buyer’s
randomization over positive investments is fully offset by the ex-post efficiency loss due to bargaining
disagreement. Lau (2008) assumes that the seller observes the buyer’s investment perfectly with a
given probability, and receives an uninformative signal otherwise. Her main result is that efficiency
is maximized when this probability is strictly between zero and one. Nevertheless, the buyer
does not earn information rent and ends up with a payoff of zero. Hermalin and Katz (2009)
consider the hold-up problem incorporating asymmetric information after the buyer’s investment
decision. In particular, a larger investment results in a first-order stochastic shift in the buyer’s
value distribution. The authors consider several scenarios in which the degree of observability of the
investment is varied and find that the buyer is made worse off if her investment is unobservable.1
Similarly, we demonstrate that in our model, the buyer’s equilibrium payoff is zero if the seller
is unable to observe the buyer’s CDF. Finally, Skrzypacz (2004) also emphasizes, as we do, that
the desire to generate information rent can shape the buyer’s incentives to invest. However, the
author’s analysis focuses on dynamic bargaining, where Coasian dynamics imply that the buyer
1
The consequences of the observability assumption in hold-up problems were also explored in Tirole (1986) and
Gul (2001) under various bargaining protocols.
4
obtains full bargaining power as the discount factor vanishes.
Unit-elastic demands also appears in a literature on non-Bayesian monopoly pricing. Bergemann and Schlag (2008) consider a monopolist with the min-max regret criterion. The seller’s
ex-post regret is defined as the buyer’s payoff if a trade occurs and the buyer’s valuation otherwise. The authors argue that the optimal pricing policy coincides with the seller’s equilibrium
strategy in a zero-sum game played against nature in which nature chooses the buyer’s valuation
to maximize the seller’s regret. The authors show that nature’s equilibrium strategy in this case
is the same as our equilibrium CDF and the seller fully randomizes on its support.2 Neeman
(2003) considers second-price auctions with private values and addresses the problem of finding
the value-distribution which minimizes the ratio between the seller’s profit and the expected value.
The author shows that the solution generates a unit-elastic demand function. Unit-elastic demands
also play a role in models where randomization by the seller is required for the existence of an
equilibrium, since this form of demand function is required to ensure that the seller is indifferent
on the support of valuations. Renou and Schlag (2010) consider models with min-max regret and
imperfect competition and Hart and Nisan (2012) approximate the seller’s maximum revenue in a
multiple-item auction.3 Related results also appear in Brooks (2013) and in Kremer and Snyder
(2015).
Finally, our paper also relates to a recent literature on information design, see for example
Kamenica and Gentzkow (2011). Bergemann and Pesendorfer (2007) consider the seller’s problem
of designing the information structure to determine how buyers learn about their valuations prior
to participating in an auction. Bergemann, Brooks and Morris (2015) analyze a model where the
buyer’s value distribution is given and the seller receives a signal about this value. The authors
characterize the entire set of payoff outcomes that can arise from some signal structure. Unit-elastic
demand plays a crucial role in their construction. Roesler and Szentes (2015) analyze a setup where
the buyer has a given value distribution and designs a signal structure to learn about her valuation.
After observing the signal structure, the seller sets a price and the buyer trades if the expected
valuation conditional on her signal exceeds the price. The buyer in Roesler and Szentes (2015) faces
a trade-off between signal precision improving the efficiency of her purchase decision for a given
price, but potentially increasing the price since the buyer’s demand is determined by her signal.
2
Since the seller’s regret is the buyer’s payoff if there is trade, it is not a coincidence that nature chooses the CDF
which maximizes the buyer’s expected surplus.
3
In a different context, Ortner and Chassang (2014) show that, in order to eliminate collusion between an agent
and the monitor, the principal benefits from introducing asymmetric information between the agent and the monitor
by making the monitor’s wage random. The optimal wage scheme is determined by a distribution similar to our
equilibrium CDF.
5
Roesler and Szentes (2015) show that the latter determines the buyer’s equilibrium signal structure
so that the equilibrium price is the lowest across all signal structures and the buyer always trades.
As in our equilibrium, the seller is indifferent between any prices in the signal’s support.
2
Model
There is a seller who has an object to sell to a single buyer. Prior to interacting with the seller,
the buyer can choose the distribution of her valuation subject to the constraint that the valuation
is below one. Formally, the buyer can choose any F ∈ F where F is the set of CDFs supported
on a subset of [0, 1]. The seller observes the choice of the buyer and gives a take-it-or-leave-it price
offer to the buyer, p. Finally, the buyer’s valuation, v, is realized and she trades with the seller if
and only if v ≥ p.4 If trade takes place, the payoff of the seller is p and that of the buyer is v − p;
both receive a payoff of zero otherwise. The seller and the buyer are expected payoff maximizers.
We restrict attention to Subgame Perfect Nash Equilibria of this game.
We introduce a few pieces of notation. For each F ∈ F and p ∈ R, let D (F, p) denote
the demand at price p generated by F , that is, the probability of trade. Observe that D (F, p) =
1−F (p)+∆ (F, p), where ∆ (F, v) denotes the probability of v according to F .5 If the buyer chooses
a CDF F ∈ F, then the seller’s profit is Π (F ) = maxp pD (F, p).6 The seller might be indifferent
between charging different prices which result in different payoffs to the buyer. For each F ∈ F,
let P (F ) denote the set of profit maximizing prices, that is, P (F ) = arg maxp pD (F, p). Finally,
R1
let U (F, p) denote the buyer’s payoff if the seller sets price p, that is, U (F, p) = p v − pdF (v).
3
Main result
We solve our problem in two steps. First, for a given profit of the seller π, we compute the CDF
Fπ ∈ F which maximizes the buyer’s payoff subject to the constraint that the seller’s profit is π.
This step essentially reduces our search for an equilibrium to a one-dimensional problem, since the
buyer’s equilibrium CDF must be in the set {Fπ }π∈[0,1] . In the second step, we characterize the
profit π and the corresponding CDF Fπ which maximizes the buyer’s payoff. To this end, we next
define the CDF Fπ and show that it maximizes the buyer’s payoff subject to the constraint that
4
Assuming that the buyer is non-strategic in the final stage of the game and buys the object whenever her valuation
weakly exceeds the price has no effect on our results but makes the analysis simpler.
5
That is, ∆ (F, v) = F (v) − supz<v F (z). If F does not specify an atom at v then ∆ (F, v) = 0.
6
This maximum is well-defined because the buyer trades when she is indifferent.
6
the seller’s profit is π.
Equal-revenue distributions.— For each π ∈ [0, 1],


if v

 0
π
Fπ (v) =
1 − v if v


 1
if v
let Fπ ∈ F be defined as follows:
< π,
∈ [π, 1) ,
(1)
≥ 1.
Since Fπ (π) = 0, the valuation of the buyer is never below π. The function Fπ is continuous and
strictly increasing on [π, 1). On this interval, Fπ is defined by the decreasing density fπ (v) = π/v 2 .
Finally, there is an atom of size π at v = 1, that is, ∆ (Fπ , 1) = π.
Note that the seller is indifferent between setting any price on the support of Fπ , that is,
P (Fπ ) = [π, 1]. To see this, suppose first that the seller sets price p ∈ [π, 1). Then the seller’s
payoff is pD (Fπ , p) = p [1 − Fπ (p)] = p (π/p) = π. If the seller sets a price of one then his payoff
is ∆ (Fπ , 1) = π. Therefore, the seller’s payoff is π as long as he sets a price in [π, 1], that is,
Π (Fπ ) = π. As a consequence these distributions exhibit unit-elastic demand, and are also known
as equal revenue distributions.
Lemma 1 Suppose that G ∈ F and p ∈ P (G) and let π denote Π (G).
(i) Then Fπ first-order stochastically dominates G, that is, for all v ∈ [0, 1]
Fπ (v) ≤ G (v) .
(2)
(ii) Furthermore, U (Fπ , π) ≥ U (G, p) and the inequality is strict if Fπ 6= G.
Part (i) of the lemma implies that the expected value induced by Fπ is larger than that
induced by G. Recall that π ∈ P (Fπ ), that is, the seller finds it optimal to set price π in the
subgame generated by Fπ . In addition, Π (G) = π = Π (Fπ ). Therefore, part (ii) implies that
the maximum payoff the buyer can achieve subject to the constraint that the seller’s profit is π is
U (Fπ , π).
Proof. To prove part (i), note that since p ∈ P (G), it must be that for all v ∈ [0, 1],
vD (G, v) ≤ pD (G, p) .
or equivalently,
1−
pD (G, p)
+ ∆ (G, v) ≤ G (v) .
v
Since ∆ (G, p) ∈ [0, 1] and π = pD (G, p) , the previous inequality implies (2).
7
(3)
To see part (ii), note that
Z
Z 1
v − πdFπ (v) ≥
U (Fπ , π) =
1
Z
p
π
π
1
v − pdG (v) = U (G, p) ,
v − πdG (v) ≥
where the first inequality follows from Fπ first-order stochastically dominating G and the second
one from π = pD (G, p) ≤ p. In addition, the first inequality is strict unless Fπ = G (see Proposition
6.D.1 of Mas-Colell et al (1995)).
Figure 1: Improving on the Uniform Distribution
demand
1
D(Fπ, v)
D(G, v)
π
p
1
price
Figure 1 illustrates the statement and the proof of Lemma 1 for the case when G is the
uniform distribution, that is, G (v) = v on [0, 1]. In this case, D (G, v) = 1 − v, the seller’s optimal
price, p, is 1/2 and the seller’s profit, π, is 1/4. The consumer surplus, U (G, 1/2), is the integral
of the demand curve on [p, 1] , represented by the darker shaded triangle. If the buyer’s valuation
is determined by the CDF Fπ , the seller’s profit is π irrespective of which price he charges on the
interval [1/4, 1]. This implies that D (Fπ , p) = D (G, p) and D (Fπ , v) > D (G, v) at v 6= p. In other
words, the demand curve generated by Fπ is above that of G and the two curves are tangent to
each other’s at p. This explains part (i) of the lemma. To see part (ii), note that switching from G
to Fπ increases the buyer’s payoff for two reasons. First, since D (Fπ , v) > D (G, v), the integral of
the new demand curve on [p, 1] goes up. Second, if the seller charges π instead of p, the consumer
8
surplus is the integral of the demand on a larger range than before. The increase in the buyer’s
payoff from moving (G, p) to (Fπ , π) is the lighter shaded area on the figure.
Let F ∗ and p∗ denote the equilibrium CDF of the buyer and the on-path equilibrium price
of the seller, respectively. We are now ready to state our main result.
Theorem 1 In the unique equilibrium outcome, F ∗ = F1/e and p∗ = U (F ∗ , p∗ ) = Π (F ∗ ) = 1/e.
The seller’s equilibrium strategy is not determined uniquely off the equilibrium path. He
might charge different prices if he is indifferent after observing out-of-equilibrium CDFs.
Proof. First, we show that there is an equilibrium in which the buyer chooses F1/e and the seller
responds by setting price 1/e. Recall that P (Fπ ) = [π, 1] for all π, so the seller optimally charges
the price 1/e in the subgame generated by F1/e . Next, we argue that the buyer has no incentive
to deviate either. If the buyer does not deviate, her payoff is U F1/e , 1/e . By part (ii) of Lemma
1, any deviation payoff of the buyer is weakly smaller than U (Fπ , π) for some π ∈ [0, 1]. So, it is
sufficient to show that U (Fπ , π) is maximized at π = 1/e. Note that
Z 1
Z 1
Z
U (Fπ , π) =
v − πdFπ (v) =
vfπ (v) dv + ∆ (Fπ , 1) − π =
π
π
1
π
π
dv = −π log π,
v
(4)
where the second equality follows from fπ (v) = π/v 2 and ∆ (Fπ , 1) = π. The function −π log π
is indeed maximized at 1/e. From the equality chain (4), the buyer’s payoff is U F1/e , 1/e =
−1/e log 1/e = 1/e in this equilibrium. Finally, note that Π F1/e = 1/e.
It remains to prove the uniqueness of the equilibrium outcome. Part (ii) of Lemma 1 implies
that the equilibrium payoff of the buyer, U (F ∗ , p∗ ), is weakly smaller than U FΠ(F ∗ ) , Π (F ∗ ) . We
showed that U FΠ(F ∗ ) , Π (F ∗ ) ≤ U F1/e , 1/e in the previous paragraph, so the buyer’s payoff
cannot exceed 1/e in any equilibrium. Since 1/e is the unique maximizer of U (Fπ , π), part (ii)
of Lemma 1 also implies that in order for the buyer to achieve this payoff, she must choose F1/e
and the seller must charge 1/e. Therefore, to establish uniqueness, it is sufficient to show that the
ε be ε + F
buyer’s equilibrium payoff is at least 1/e. To this end, for each ε ∈ (0, 1/e), let F1/e
1/e
ε is constructed from F
on [p, 1) and F1/e otherwise. The CDF F1/e
1/e by moving a mass of size ε
ε then the seller strictly prefers to set
from the atom at v = 1 to v = 1/e. If the buyer chooses F1/e
price 1/e.7 In addition, the buyer’s payoff is ε-close to U F1/e , 1/e = 1/e. Therefore, the buyer
can achieve a payoff arbitrarily close to 1/e irrespective of the seller’s strategy to resolve ties and
hence the buyer’s equilibrium payoff cannot be smaller than 1/e.
7
Note that, by setting price 1/e , the seller achieves the same payoff as if the buyer chose F1/e . For any price in
(1/e, 1], the seller’s payoff is strictly smaller than after the choice of F1/e because the probability of trade is smaller
by ε.
9
From the proof of Theorem 1 it follows that the buyer’s expected valuation generated by F1/e
is 2/e. Indeed, from (4),
Z
Z 1
vdF1/e (v) =
1/e
1
Z
1
vf1/e (v) dv + ∆ F1/e , 1 =
1/e
1/e
1
1
2
dv + = .
ev
e
e
Note that efficiency requires the buyer to choose a distribution which would specify v = 1
with probability one, and the seller to quote a price which is weakly less than one. The total surplus
would be one. In contrast, the total surplus in equilibrium is only U F1/e , 1/e + Π F1/e = 2/e.
Hence, the efficiency loss due to the buyer’s desire to generate information rent is 1 − 2/e ≈ 0.26.
As mentioned earlier, requiring the upper bound of the buyer’s CDF’s support to equal one
is just a normalization. If this upper bound is k then the equilibrium price, the payoff of the buyer
and the payoff of the seller are k/e. As k goes to infinity, the payoffs also converge to infinity.
4
Moment Restrictions and Costly Investment
In our baseline model, the only constraint faced by the buyer is that the upper bound of the
valuation-support must be smaller than one. This section considers various extensions of our game
by imposing moment restrictions on the set of distributions from which the buyer can choose.
First, we consider the case where the expectation of the buyer’s CDF is bounded. Then we analyse
environments where the choice set of the buyer is the set of those CDFs which satisfy finitely many
arbitrary moment restrictions. We demonstrate that, in all these situations, trade is efficient and
the seller is indifferent between setting any price on the support of the buyer’s equilibrium CDF.
4.1
Bounded Expectation
In this section, we assume that the buyer is perfectly flexible at choosing the riskiness of her value
distribution but has limited ability to influence its expectation. To be more precise, suppose that
the buyer faces an additional constraint when choosing the distribution F , namely that the expected
value must lie in the interval I = [µ, µ] ⊂ (0, 1], that is, the buyer’s choice set is
Z 1
FI = F ∈ F :
vdF (v) ∈ I .
0
Of course, if 2/e ∈ I then the new constraint does not bind and the unique equilibrium
outcome is identical to the one identified by Theorem 1. Next, we show that, if 2/e ∈
/ I, the
equilibrium CDF of the buyer is going to be again an equal-revenue distribution which generates
an expected value of either µ or µ, depending which of these values is closer to 2/e. To this end,
10
we first argue that the range of means induced by the class of equal-revenue distributions defined
by (1) is the interval (0, 1]. Recall from the proof of Theorem 1 that the expected value of the
equal-revenue distribution Fπ is −π log π + π. This function converges to zero as π goes to zero,
its value is one at π = 1 and it is continuous and strictly increasing on (0, 1]. Therefore, for each
µ ∈ (0, 1] there exists a unique πµ ∈ (0, 1] such that the expectation of Fπµ is µ, that is,
−πµ log πµ + πµ = µ.
We now characterize the equilibrium CDF of the buyer, FI∗ , and the equilibrium price, p∗I .
Proposition 1 If the buyer’s action space is FI then, in the unique equilibrium outcome, FI∗ = Fπµ
and p∗I = πµ where



 µ if µ < 2/e,
µ=
2/e if 2/e ∈ I,


 µ if 2/e < µ.
Since the seller charges πµ , trade always occurs and U Fπµ , πµ = µ − πµ .
Proof. We only show that there exists an equilibrium outcome with the properties described in
the statement of the theorem. The uniqueness proof is essentially identical to that in the proof of
Theorem 1. If 2e ∈ I, the statement of the proposition follows from Theorem 1.
If µ ≥ 2/e then µ = µ, so we need to show that Fπµ , πµ constitutes an equilibrium. If the
h
i
buyer chooses Fπµ , then the seller optimally responds by setting price πµ since P Fπµ = πµ , 1 .
We have to show that the buyer has no incentive to deviate. Let G ∈ FI , p ∈ P (G) and let π
denote Π (G). It is enough to show that U (G, p) ≤ U Fπµ , ππµ . First, observe that
Z
Z
vdFπµ (v) = µ ≤
Z
vdG (v) ≤
vdFπ (v) ,
where the equality is implied by the definition of πµ , the first inequality follows from G ∈ FI and
the second one follows from part (i) of Lemma 1. Since the expectation of Fπ is increasing in π,
the previous inequality chain implies that πµ ≤ π. Finally, note that
U (G, p) ≤ U (Fπ , π) ≤ U Fπµ , ππµ ,
where the first inequality follows from part (ii) of Lemma 1. The second inequality follows from
the observations that 1/e ≤ πµ (since 2/e ≥ µ) and that U (Fπ , π) is strictly decreasing in π on
[1/e, 1] (see (4)).
11
Suppose now that µ < 2/e. If the buyer chooses Fπµ , then the seller optimally responds
by setting price πµ since P Fπµ = [πµ , 1]. We have to show that the buyer has no incentive to
deviate. Let G ∈ FI , p ∈ P (G) and let π denote Π (G). We show that U (G, p) ≤ U Fπµ , πµ .
Note that
1
Z
Z
1
vdG (v) − pD (G, p) ≤ µ − π,
v − pdG (v) =
U (G, p) =
p
p
where the inequality follows from G ∈ FI and π = pD (G, p). If π ≥ πµ then this inequality chain
implies U (G, p) ≤ U Fπµ , πµ because U Fπµ , πµ = µ − πµ . If π ≤ πµ then
U (G, p) ≤ U (Fπ , π) ≤ U Fπµ , πµ ,
where the first inequality follows from part (ii) of Lemma 1. The second inequality follows from
the observations that πµ < 1/e (since 2/e ≥ µ) and that U (Fπ , π) is strictly increasing in π on
(0, 1/e] (see (4)).
In the special case where µ = µ = µ, the previous proposition implies that the CDF Fπµ is
optimal for the buyer among the CDFs with mean µ. Next, we show that Fπµ is also the CDF
which minimizes the seller’s profit among those CDFs which generate an expected valuation of
exactly µ.
Remark 1 If
R1
0
vdG (v) = µ then Π (G) ≥ Π Fπµ = πµ .
This result is also reported in Neeman (2003) and Kremer and Snyder (2016).
Proof. Let π denote Π (G). Note that
Z
1
Z
vdFπ (v) ≥
0
1
Z
vdG (v) = µ =
0
1
vdFπµ (v) ,
0
where the inequality follows from part (i) of Lemma 1, the first equality is the hypothesis of this
R1
remark, and the second equality follows from the definition of πµ . Since 0 vdFp (v) is strictly
increasing in p, the previous inequality chain implies that π ≥ πµ .
Again, if the upper bound of the CDF’s support is required to be k instead of one, the
∗
equilibrium price and the profit would be kπµ/k
and the equilibrium payoff of the buyer would be
∗
∗
kU Fπµ/k
, πµ/k
. What happens if k goes to infinity for a fixed µ?
∗
∗
∗
Remark 2 For all µ ∈ R++ , limk→∞ kU Fπµ/k
, πµ/k
= µ and limk→∞ kπµ/k
= 0.
12
This remark implies that if the upper bound k can be arbitrarily large, the buyer extracts
the full surplus, µ, by choosing a CDF which induces a price arbitrarily close to zero.
∗ log π ∗
∗
Proof. By Proposition 1, −πµ/k
µ/k + πµ/k = µ/k or equivalently,
∗
+1=
− log πµ/k
µ
∗ k.
πµ/k
∗
Since πµ/k
goes to zero as k goes to infinity, the left-hand side converges to infinity. Therefore,
∗
the right-hand side must also go to infinity. Hence, the price kπµ/k
must converge to zero, which
∗
∗
implies that the seller’s payoff goes to zero and the buyer’s payoff, kU Fπµ/k
, πµ/k
, goes to µ.
4.2
General Moment and Variance Restrictions
In this section, we assume that the buyer faces constraints on the variance and general moments of
her value distribution. Previously, we have shown that if the buyer faces only constraints regarding
the expectation of her valuation, she still chooses an equal-revenue distribution, Fπ , in equilibrium.
Recall that the upper bound of the support of this distribution is the largest possible valuation,
which we normalized to be one. In addition, the CDF Fπ specifies an atom of size π at one.
Therefore, Fπ generates relatively large higher moments. Furthermore, if π is small then Fπ also
has a sizable variance. One way to tailor an equal-revenue distribution Fπ to reduce its variance
and higher moments is to truncate it at a threshold value which is smaller than one, that is, to
move all the probability mass from above the threshold to the threshold itself. Below, we show that
the buyer’s equilibrium distribution will be in this class of truncated equal-revenue distributions.
In what follows, we introduce the additional constraints and formally define the action space
of the buyer. With a slight abuse of notations, for each F ∈ F, let V ar (F ) denote the variance of
the random variable which is distributed according to the CDF F , that is,
Z 1
V ar (F ) =
(v − µ)2 dF (v) ,
0
where µ denotes the expectation of F , that is,
R1
0
vdF (v). Next, we introduce general moment
restrictions. To this end, let αi : [0, 1] → [0, 1] be a continuous, strictly increasing, and convex
function for all i = 1, ..., n. We normalize αi (0) to be zero and αi (1) to be one. We refer to the
function αi as a moment function. In addition, let γi ∈ (0, 1) for i = 1, ..., n. The CDF F ∈ F
satisfies the ith moment restriction if
Z
1
αi (v) dF (v) ≤ γi .
0
13
For example, if αi (v) = v i for i ∈ N then the ith constraint is a restriction on the ith moment
of the distribution. In this section, the buyer is required to choose a CDF which satisfies all the
moment restriction and a constraint on the variance of the distribution. To be more precise, the
choice set of the buyer is
Z
2
Fα = F ∈ F : V ar (F ) ≤ σ and
1
αi (v) dF (v) ≤ γi for all i = 1, ..., n .
0
The special case where the variance is not restricted (σ 2 = ∞) and there is only a mean constraint,
i.e., n = 1 and α1 (v) = v, was discussed in the previous section.
Next, we define a set of distributions and show that the buyer’s equilibrium choice is in this
class. For each π ∈ [0, 1] and B ∈ [π, 1] let



 0
B
Fπ (v) =
1−


 1
if v ∈ [0, π),
π
v
if v ∈ [π, B),
if v ∈ [π, 1].
Note that the support of FπB is [π, B] and it specifies an atom of size π/B at B. Again, the seller is
indifferent between charging any price on its support. Notice that Fππ is a degenerate distribution
which specifies an atom of size one at π and Fπ1 = Fπ .
The next lemma states that for each CDF G there is an element in this class which secondorder stochastically dominates G and makes the buyer weakly better off while generating the same
profit to the seller as G.
Lemma 2 Suppose that G ∈ F, p ∈ P (G) and let π denote pD (G, p). Then there exists a unique
B ∈ [π, 1] such that
(i) G is a mean-preserving spread of FπB and
(ii) U (G, p) ≤ U FπB , π .
Proof. See Roesler and Szentes (2016).
We now characterize the equilibrium CDF of the buyer, Fα∗ , and the equilibrium price, p∗α .
Proposition 2 Suppose that the buyer’s action space is Fα . Then, in the unique equilibrium
∗
outcome, the buyer chooses Fα∗ = FπB∗ and the seller sets price p∗α = π ∗ where
(π ∗ , B ∗ ) = arg max
π∈(0,1)
B∈[π,1]
U FπB , π : FπB ∈ Fα .
14
(5)
∗
Note that the buyer’s equilibrium CDF, FπB∗ , again generates a unit elastic demand. In
addition, since the seller charges price π ∗ , trade occurs with probability one.
Proof. We only show that there exists an equilibrium outcome with the properties described in
the statement of the theorem. The uniqueness proof is essentially identical to that in the proof of
Theorem 1.
∗
If the buyer chooses FπB∗ then the seller optimally responds by setting price π ∗ since P (Fπ∗ ) =
[π ∗ , 1]. We have to show that the buyer has no incentive to deviate. Let G ∈ Fα , p ∈ P (G) and let
∗
π denote Π (G). It is enough to show that U (G, p) ≤ U FπB∗ , π ∗ . By part (i) of Lemma 2, there
exists a B such that G is a mean-preserving spread of FπB . Next, we show that FπB ∈ Fα . First,
note that
V ar FπB ≤ V ar (G) ≤ σ 2 ,
where the first inequality holds because G is a mean-preserving spread of FπB and the second one
follows from G ∈ Fα . In addition, since αi is convex and G is a mean-preserving spread of FπB , it
follows that for each i = 1, ..., n
Z
1
αi (v) dFπB
Z
(v) ≤
0
1
αi (v) dG (v) ,
0
see Proposition 6.D.2 of Mas-Colell et al (1995)). Furthermore, since G ∈ Fα ,
Z
1
αi (v) dG (v) ≤ γi .
0
The previous three displayed inequalities imply that FπB ∈ Fα .
Finally, note that
∗
U (G, p) ≤ U FπB , π ≤ U FπB∗ , π ∗ ,
where the first inequality follows from part (ii) of Lemma 2 and the second one from FπB ∈ Fα and
∗
(5). This inequality means that choosing G instead of FπB∗ is not a profitable deviation.
5
Costly Investment
In this section, we characterize equilibria in models where the buyer’s investment is costly. We
show that the buyer’s problem of choosing a costly CDF is the dual of her problem when choosing
a costless CDF with a single moment restriction. Then we use the arguments developed in the
previous section to show that the equilibrium CDF of the buyer again generates a unit elastic
demand.
15
We now assume that the buyer has to incur an additive cost C (F ) if she chooses F . That is,
if the buyer chooses F and her valuation is v then her payoff is v − p − C (F ) is she trades at price
p and −C (F ) otherwise. We assume that each particular valuation has a deterministic cost and
the cost of a CDF F is the expected cost generated by F .8 To be more precise, let c : R+ → R+
be an increasing, convex function such that c (0) = 0 and c (1) = 1. Then the cost of F is given by
Z
C (F ) = c (v) dF (v) .
(6)
Since c (1) = 1, the buyer only chooses CDFs supported on [0, 1].
The problem of the buyer here is the dual of her problem considered in the previous section
with a single moment restriction in the following sense. Suppose that the buyer’s optimal choice is
G. Then G must remain optimal even if investment is costless but the buyer is restricted to choose
from the set of those distributions which are cheaper than G, that is,
Z
FC = F ∈ F : c (v) dF (v) ≤ C (G) .
To see this, note that if the buyer’s payoff induced by a CDF F ∈ FC was higher than that by G
then F would generate a higher payoff to the buyer even if the cost is taken into account because
F is cheaper than G. Observe that the buyer’s problem of choosing from FC at no cost is analyzed
in the previous section. Indeed, the constraint defining FC is a convex moment restriction with
α1 ≡ c and γ1 = C (G). Proposition 2 then suggests that the equilibrium choice of the buyer is a
truncated equal-revenue distribution.
Proposition 3 Suppose that the cost of each CDF F ∈ F is given by (6). Then, in the unique
∗
equilibrium outcome, the buyer chooses FπB∗ for some π ∗ ∈ (0, 1) and B ∗ ∈ (π ∗ , 1] and the seller
sets price p∗ = π ∗ .
(π ∗ , B ∗ ) = arg max
π∈(0,1)
B∈[π,1]
U FπB , π − C FπB : FπB ∈ Fα .
(7)
Proof. We only show that there exists an equilibrium outcome with the properties described in
the statement of the theorem. The uniqueness proof is essentially identical to that in the proof of
Theorem 1.
∗
If the buyer chooses FπB∗ then the seller optimally responds by setting price π ∗ since P (Fπ∗ ) =
[π ∗ , 1]. We have to show that the buyer has no incentive to deviate. Let G ∈ Fα , p ∈ P (G) and let
∗
∗
π denote Π (G). It is enough to show that U (G, p) − C (G) ≤ U FπB∗ , π ∗ − C FπB∗ . By part (i)
8
In Section 8, we consider other possible specifications of the buyer’s cost function.
16
of Lemma 2, there exists a B such that G is a mean-preserving spread of FπB . Next, we show that
FπB is cheaper than G, that is, C FπB ≤ C (G). Note that
C FπB =
Z
1
c (v) dFπB (v) ≤
0
Z
1
c (v) dG (v) = C (G) ,
(8)
0
where the inequality follows from c being convex and G being a mean-preserving spread of FπB (see
Proposition 6.D.2 of Mas-Colell et al (1995)). Therefore,
∗
∗
U (G, p) − C (G) ≤ U FπB , π − C FπB ≤ U FπB∗ , π ∗ − C FπB∗ ,
where the first inequality follows part (ii) of Lemma 2 and (8), and the second one from (7).
6
Reshaping a Given Prior
In this section, we consider scenarios where the buyer’s valuation is given by a CDF H and she
can take certain actions to tailor this distribution. We maintain the assumptions that the buyer’s
decision is observable and that there is no asymmetric information at the moment when this action
is taken. In other words, the prior distribution H is common knowledge and the buyer takes an
action before observing its realization. We analyze various environments depending how the buyer
can reshape the prior H.
First, we consider a situation where the buyer can add arbitrary amount of risk to the
distribution H. Second, we analyse environments where all the buyer can do is stochastically
destroy (or create) value. In neither of these cases, we explicitly model the set of actions which
the buyer can take in order to modify the CDF H. Instead, we identify the buyer’s action space
with set of distributions that can arise from tailoring H. For example, if the buyer can add risk
then this space is going to be set of those CDFs which are mean-preserving spreads of H. If the
buyer can destroy (or create) value, she is required to select a CDF that is first-order stochastically
dominated by (or first-order stochastically dominates) H.
6.1
Designing Riskiness
Here, we characterize the buyer’s problem of optimally adding risk to her prior value distribution
H. We do not impose any restriction on the type of risk the buyer can induce but we continue to
assume that the maximum valuation of the buyer cannot exceed one. If the buyer adds risk to H
then the resulting distribution will be a mean-preserving spread of H. Therefore, we assume that
17
the buyer can choose any CDF which is a mean-preserving spread of H.9 Formally, we assume that
the choice set of the buyer is
H=
Z
F ∈F :
Z
x
Z
F (v)dv ≥
vdF (v) = µ and
where µ denotes
H(v)dv ∀x ∈ [0, 1] ,
0
0
R
x
vdH (v).
Next, we introduce a class of distributions and then show that there always exists an equilibrium where the buyer chooses a CDF from this class. For each π ∈ (0, 1] and l ∈ [π, 1] define
Fπ,l (v) =



 1−
1−


 1
π
l
π
v
if v < l,
if v ∈ [l, 1).
if v ≥ 1
The CDF Fπ,l is a convex combination of an atom at zero and the equal-revenue distribution Fl .
Indeed, if the buyer’s valuation is zero with probability 1 − (π/l) and it is determined by Fl with
probability π/l then the resulting CDF is exactly Fπ,l . Of course, when the seller decides what price
to set, he conditions on the event that the buyer’s value is determined by Fl . Hence, P (Fπ,l ) = [l, 1]
and Π (Fπ,l ) = (π/l) l = π. Observe that Fπ,π = Fπ and the CDF Fπ,1 specifies an atom of size π
at one and places the rest of the probability mass at zero.
The next lemma shows that for each CDF G there is an element in this class which is a
mean-preserving spread of G, induces the same profit for the seller and a weakly higher payoff to
the buyer.
Lemma 3 Suppose that G ∈ F, p ∈ P (G) and let π denote pD (G, p). Then there exists a unique
l ∈ [π, 1] such that
(i) Fπ,l is a mean-preserving spread of G,
(ii) U (Fπ,l , l) = µ − π and
(iii) U (G, p) ≤ U (Fπ,l , l) and the inequality is strict if D (G, p) < 1 − G (0) .
Part (i) states that the valuation distributed according to Fπ,l is more risky than the one distributed according to G. Recall that Π (Fπ,l ) = π, so part (iii) says that this more risky distribution
generates the same payoff to the seller and a higher payoff to the buyer.
9
We refer to Johnson and Myatt (2006) for an extensive discussion of environments, such as product design and
advertisement, where the shifts in demand are modeled as mean-preserving spreads.
18
Proof. First, we show that there exists an l ∈ [π, 1] such that the expectation of Fπ,l is µ. Note
that
Z
Z
vdFπ,1 (v) = π ≤ µ =
Z
vdG (v) ≤
Z
vdFπ (v) =
vdFπ,π (v) ,
where the second equality follows from G ∈ H, the second inequality follows from part (i) of Lemma
R
1, and the last equality from Fπ,π = Fπ . Since vdFπ,l (v) is continuous and strictly decreasing in
l on [π, 1], the Intermediate Value Theorem implies that there exists an l such that
Z
vdFπ,l (v) = µ.
(9)
Next, we prove that there exists a z ∈ (0, l) such that Fπ,l (v) ≥ G (v) if v ≤ z and Fπ,l (v) ≤
G (v) if v ≥ z. To this end, let
z = inf {y ∈ [0, 1] : Fπ (v) ≤ G (v) for all v ≥ y} .
We have to show that Fπ,l (v) ≤ G (v) if v ≥ z. Note that if v ≥ l then
Fπ,l (v) = Fπ (v) ≤ G (v) ,
where the inequality follows from part (i) of Lemma 1. Therefore, z ≤ l. Suppose that y < z and
Fπ,l (y) < G (y). Since Fπ,l is constant on [0, l] and G is increasing, it follows that G (v) > Fπ,l (v)
for all v ∈ [y, z] ⊂ [y, l] which contradicts to the definition of z. We conclude that Fπ,l (v) ≥ G (v)
on [0, z].
We are ready to prove that Fπ,l is a mean-preserving spread of G. If x ≤ z then Fπ,l (v) ≥
G (v) for all v ∈ [0, x] and hence,
Z
x
Z
Fπ,l (v)dv ≥
0
If x ≥ z then
Z
x
G(v)dv.
0
Z
Fπ,l (v)dv = 1 − µ −
0
x
1
1
Z
Fπ,l (v)dv ≥ 1 − µ −
x
Z
G(v)dv =
x
x
G(v)dv,
0
where the inequality follows from Fπ,l (v) ≤ G (v) on [z, 1]. The previous two displayed inequalities
imply part (i).
To see part (ii), note that
Z 1
Z
U (Fπ,l , l) =
v − ldFπ,l (v) =
l
1
Z
vdFπ,l (v) − lD (Fπ,l , l) =
l
1
vdFπ,l (v) − lD (Fπ,l , l) = µ − π,
0
where the third equality follows the fact that Fπ,l (v) = Fπ,l (0) for all v ∈ [0, l] and the last equality
follows from (9) and π = Π (Fπ,l ).
19
To prove part (iii), note that
Z
1
v − pdG (v) = U (G, p) ,
µ−π ≥
p
where the inequality follows from the fact that the buyer’s payoff cannot exceed the first-best
total surplus,µ, minus the seller’s profit, π. The previous two displayed equations imply that
U (G, p) ≤ U (Fπ,l , l). The last inequality is strict whenever the total surplus in the outcome (G, p)
is strictly less than µ, that is, trading is ex-post inefficient. Trading is ex-post inefficient if and
only if the probability that v ∈ (0, p) is positive, that is, D (G, p) < 1 − G (0).
Suppose that there is an equilibrium where the buyer chooses G and the seller sets price p.
Applying Lemma 3 to this profile (G, p) yields the result that there is also an equilibrium where
the buyer chooses a CDF from the set {Fπ,l }. If the buyer chooses Fπ,l , it is optimal for the seller
to set price l which, in turn, provides the buyer with a payoff of µ − π (see part (ii) of Lemma 3).
The distribution which maximizes the buyer’s payoff is therefore defined by the lowest π for which
Fπ,l ∈ H for some l ∈ [π, 1]. Let us define π ∗ as this lowest value, that is,
π ∗ = min{π ∈ [0, 1] : ∃l ∈ [π, 1] such that Fπ∗ ,l ∈ H}
(10)
and let l∗ denote the unique value for which Fπ∗ ,l∗ ∈ H.
Proposition 4 If the buyer’s choice set is H then there is an equilibrium where the buyer chooses
Fπ∗ ,l∗ and the seller sets price l∗ . If there is also an equilibrium where the buyer chooses G and the
seller sets price p then
(i) U (G, p) = U (Fπ∗ ,l∗ , l∗ ) = µ − π ∗ ,
(ii) D (G, p) = 1 − G (0) ,
(iii) Π (G) = π ∗ and
(iv) Fπ∗ ,l∗ is a mean-preserving spread of G.
The distribution Fπ∗ ,l∗ again generates unit elastic demand for prices in [l∗ , 1]. However,
trade only occurs with probability π/l∗ . Nevertheless, since only those buyers don’t trade whose
valuation is zero, trading decisions are ex-post efficient.
Parts (i) and (iii) of this proposition imply that the payoffs of the buyer and seller are the
same across all equilibria. In addition, by part (ii), trading decisions are efficient. In general, the
buyer’s equilibrium CDF is not determined uniquely because there might be many CDFs which
are mean-preserving spreads of H and generate payoffs µ − π ∗ and π ∗ to the buyer and the seller,
respectively. However, according to part (iv), the CDF Fπ∗ ,l∗ is a mean-preserving spread of any
20
other equilibrium CDF. This means that Fπ∗ ,l∗ induces more risk in the buyer’s valuation than any
other equilibrium CDF.
Proof. First we show that there is an equilibrium where the buyer chooses Fπ∗ ,l∗ and the seller
sets price l∗ . We have already argued that P (Fπ∗ ,l∗ ) = [l∗ , 1], so the the seller optimally responds
to Fπ∗ ,l∗ by setting price π ∗ . It remains to argue that the buyer has no incentive to deviate. Let
G ∈ H, let p ∈ P (G) and let π denote Π (G). Part (i) of Lemma 3 implies that there exists an
l ∈ [π, 1] such that Fπ,l is a mean-preserving spread of G. Observe that
U (G, p) ≤ U (Fπ,l , l) = µ − π ≤ µ − π ∗ = U (Fπ∗ ,l∗ ) ,
(11)
where the first inequality follows from part (iii) of Lemma 3 , the first and third equality from part
(ii) of Lemma 3 and the second inequality from (10).
Suppose now that there is an equilibrium where the buyer chooses G ∈ H and the seller
sets price p. Again, let π denote Π (G). Note that equation (11) is still valid. Similarly to the
uniqueness proof of Theorem 1, it is easy to show that for each ε > 0, there is a CDF, Fε ∈ H,
close to Fπ∗ ,l∗ and pε ∈ [0, 1] such that pε = P (Fε ) and
U (Fπ∗ ,l∗ , l) < U (Fε , pε ) + ε.
(12)
Then for each ε > 0
U (Fε , pε ) ≤ U (G, p) ≤ U (Fπ∗ ,l∗ ) < U (Fε , pε ) + ε,
where the first inequality holds because otherwise Fε would be a profitable deviation, the second inequality follows from (11) and the third one from (12). This inequality chain implies that
U (G, p) = U (Fπ∗ ,l∗ ) which proves part (i) of the proposition. In addition, this implies that all
the inequalities in (11) are equalities. Hence, by part (iii) of Lemma 3, D (G, p) = 1 − G (0) ,
which is just part (ii) of the proposition. In addition, the second inequality in (11) is strict unless
π ≤ π ∗ , which proves part (iii). To prove part (iv), first note that Fπ∗ ,l∗ is a mean-preserving
spread of Fπ,l because π ∗ ≤ π and the expectation of both Fπ∗ ,l∗ and Fπ,l is µ. In addition, Fπ,l is a
mean-preserving spread of G by part (i) of Lemma 3. Since second-order dominance is a transitive
relation, it follows that Fπ∗ ,l∗ is indeed a mean-preserving spread of G.
We observe that the key result to the arguments of this section, Lemma 3, is the counterpart
of Lemma 2. Recall that for each G ∈ F, Lemma 2 identified a CDF, FπB , such that G is a meanpreserving spread of FπB and the buyer’s payoff generated by FπB exceeds that generated by G. In
contrast, Lemma 3 identifies a CDF Fπ,l ∈ F for each G such that Fπ,l is a mean-preserving spread
21
of G and the buyer is better off by choosing Fπ,l than G. Recall that Lemma 2 was used to show
that the equilibrium CDF of the buyer is in the set FπB if she faces convex moment restrictions
(see Proposition 2). Similarly, Lemma 3 can be used to show that the buyer’s equilibrium CDF is
in the class {Fπ,l } if she faces concave moment restrictions.
6.2
Destruction and Creation
In this section, we continue to assume that a prior CDF, H ∈ F, is given. First, suppose that the
buyer can take any action which decreases her valuation in the first-order stochastic dominance
sense. That is, the set of distributions available for the buyer is
H = {F ∈ F : F (v) ≥ H(v) ∀v ∈ [0, 1]} .
The next proposition states that the equilibrium choice of the buyer is the pointwise maximum of
her prior H and some equal-revenue distribution.
Proposition 5 If the buyer’s choice set is H then there exist an equilibrium and a π ∗ ∈ (0, 1], such
that the buyer’s equilibrium choice is max {Fπ∗ , H}.
In this case, the buyer’s euilibrium CDF is not unique, in general. Nevertheless, it is easy to
show that the equilibrium payoffs are unique. In addition, it is without loss of generality to assume
that trading is ex-post efficient because the buyer can always choose a CDF which has no mass on
(0, p), that is, only those buyers don’t trade whose valuation is zero. However, in general, trade
does not occur with probability one.
Proof. Let G ∈ H, let p ∈ P (G) and let π denote Π (G). It is enough to show that if the buyer
chooses Hπ = max {Fπ , H}, the seller still finds it optimal to set p and U (Hπ , p) ≥ U (G, p).
First, we show that Hπ first-order stochastically dominates G. Note that part (i) of Lemma
1 implies that G (v) ≥ Fπ (v) . Since G ∈ H, G (v) ≥ H (v). The previous two inequalities imply
that
G (v) ≥ max {Fπ (v) , H (v)} = Hπ (v) .
(13)
Next, we argue that p ∈ P (Hπ ). Observe that
pD (G, p) = π = pD (Fπ , p) ≥ pD (Hπ , p) ≥ pD (G, p) ,
where the second equality follows from π = Π (Fπ ), the first inequality follows from the fact that
Fπ first-order stochastically dominates Hπ , and the last inequality follows from (13). From this
22
inequality chain, we conclude that
pD (Hπ , p) = π.
(14)
Furthermore, for all v ∈ [0, 1],
vD (Hπ , v) ≤ vD (Fπ , v) ≤ π,
where the first inequality again follows from Fπ ≤ Hπ and the second one from π = Π (Fπ ). The
previous two displayed equations imply that Π (Hπ ) = π and p ∈ P (Hπ ).
It remains to show that U (G, p) ≤ U (Hπ , p). Observe that
Z
1
Z
1
v − pdHπ (v) = U (Hπ , p) ≥ U (G, p) ,
v − pdG (v) ≤
U (G, p) =
p
p
where the inequality follows from (13).
Next, we consider the case where the buyer can create value, that is, she can choose any CDF
which first-order stochastically dominates H. Formally, the action space of the buyer is defined by
the following set:
H = {F ∈ F : F (v) ≤ H(v) ∀v ∈ [0, 1]} .
The next proposition states that either this constraint does not bind and the buyer chooses
F1/e in equilibrium or she chooses FΠ(H) .
Proposition 6 If the buyer’s choice set is H then, in the unique equilibrium, the buyer chooses
Fπ∗ and the seller sets price π ∗ where π ∗ = max {Π (H) , 1/e}
Again, trade occurs with probability one in equilibrium and the seller is indifferent between
setting any price on the support of the buyer’s equilibrium CDF.
Proof. Again, we only show that there exists an equilibrium outcome with the properties described
in the statement of the theorem. The uniqueness proof is essentially identical to that in the proof
of Theorem 1.
Suppose first that Π (H) ≤ 1/e. Then for all v ∈ [0, 1]
H (v) ≥ FΠ(H) (v) ≥ F1/e (v) ,
(15)
where the first inequality follows from by part (i) of Lemma 1 and the second one from Π (H) ≤ 1/e.
This inequality chain implies that F1/e ∈ H, and hence, the new constraint does not bind and the
buyer can still choose F1/e .
23
Suppose now that Π (H) ≥ 1/e, so π ∗ = Π (H). By part (i) of Lemma 1, H (v) ≥ Fπ∗ (v) for
all v ∈ [0, 1], so Fπ∗ ∈ H. If the buyer chooses Fπ∗ , then the seller optimally responds by setting
price π ∗ since P (Fπ∗ ) = [π ∗ , 1]. We have to show that the buyer has no incentive to deviate. To this
end, let G ∈ H, let p ∈ P (G) and let π denote Π (G). We prove that that U (G, p) ≤ U (Fπ∗ , π ∗ ).
First, we argue that π ≥ π ∗ . Since G first-order stochastically dominates H, it follows that
D (G, v) ≥ D (H, v). This implies that for each v, vD (G, v) ≥ vD (H, v) and hence, π ≥ π ∗ . Since
π ∗ ≥ 1/e and the buyer’s payoff, U (Fπ , π) is decreasing in π on [1/e, 1] (see the proof of Theorem
1),
Z
1
Z
vdFπ (v) ≤
π
1
π∗
vdFπ∗ (v) .
Finally, observe that part (ii) of Lemma 1 implies that
Z 1
Z 1
vdG (v) ≤
vdFπ (v) .
π
p
The previous two displayed inequalities imply that U (G, p) ≤ U (Fπ∗ , π ∗ ).
7
Optimal Demand in Cournot Competition
As we mentioned earlier, the demand function D (F ∗ , p) characterized by Theorem 1 maximizes
consumer surplus in a monopoly market where the maximum willingness-to-pay cannot exceed one
and production is costless. In this section, we generalize this result and characterize the demand
curve which maximizes consumer surplus in a market where many firms are competing a’la Cournot.
We consider a market populated by a unit mass of consumers. There is a single homogenous
good produced by n(≥ 1) identical firms at no cost. Consumers demand a single unit of the
good and we continue to assume that the maximum willingness-to-pay does not exceed one. The
population distribution of the willingness-to-pay is given by the CDF F ∈ F. Firms compete a’
la Cournot, that is, each Firm i simultaneously decides how much to produce, qi . In order to
eliminate trivial equilibria, we require qi to be in [0, 1/n]. Our results extend to positive production
cost where this requirement becomes a result. If the market clears at price p, the payoff of Firm i
is pqi and consumer surplus is U (F, p).
We restrict attention to symmetric equilibria of this game, that is, each firm supplies the same
quantity. We say that q constitutes an equilibrium if the action profile (q, ..., q) is an equilibrium.
Our goal is to characterize the CDF F which maximizes consumer surplus. We call a CDF F
consumer-surplus maximizing if there is an equilibrium in the Cournot game defined by F where
consumer welfare is larger than that in any equilibrium generated by any CDF G ∈ F.
24
In order to define the market-clearing price, we extend the definition of demand function,
e (F, p) = {D (F, p) + α∆ (F, p) | α ∈ [0, 1]}. Note
D (F, p) , to demand correspondence, that is, D
e (F, p) = D (F, p) if there is no atom at p, that is, ∆ (F, p) = 0. Then the market-clears at
that D
P
e (F, p).10
price p defined by i=1 qi ∈ D
Next, we introduce a set of distributions and then show that welfare maximizing one is in
this class. For each π ∈ [0, 1/n] let
Fπn (v)
=



 0
1
n
if v < nπ,
π
v
−


 1
if v ∈ [nπ, 1),
if v ≥ 1.
The support of Fπn is [nπ, 1]. It is defined by the density fπn (v) = π/v 2 on [nπ, 1) and it has an
atom of size (n − 1) /n + π at v = 1. Observe that Fπ1 = Fπ .
The rest of this section is devoted to proving the following result.
Proposition 7 The consumer-surplus maximizing CDF in the Cournot market is Fπn∗ where π ∗ =
(1/n) e−n . Furthermore, in the consumer-surplus maximizing equilibrium of the game defined by
Fπn∗ ,
(i) each firm produces 1/n,
(i) the profit of each firm is (1/n) e−n and
(iii) the market-clearing price is e−n
(iv) the consumer surplus is (n − 1) /n + (1/n) e−n .
Note that for the case of a monopoly (n = 1), this proposition implies that the CDF which
maximizes consumer welfare is F1/e and both the profit and consumer surplus are 1/e. That is, for
n = 1, the statement of this proposition corresponds to that of Theorem 1.
Our main insights from the monopoly case also carry forward to Cournot competition. First,
if demand is given by Fπn∗ then total production is one, so each consumer is served. In other words,
trading is efficient. Second, as demonstrated later, in equilibrium the residual demand faced by
each firm is unit elastic.
The next lemma shows that 1/n constitutes an equilibrium if the market demand is generated
by Fπn .
10
In order to guarantee the existence of a symmetric equilibrium for all G ∈ F, we assume that market clears at
the largest price if there are multiple prices at which supply equals demand.
25
Lemma 4 If demand is given by Fπn then 1/n constitutes an equilibrium in the Cournot market.
Furthermore, in this equilibrium, the profit of each firm is π and the market-clearing price is nπ.
Proof. We have to show that Firm i maximizes profit by producing 1/n if the aggregate production
of the other firms is (n − 1) /n. Note that the demand correspondence generated by Fπn (v) is
described by
e (Fπn , p)
D
=



 1
if p < nπ,
n−1
π
n + p


 0, n−1 + π n
if p ∈ [nπ, 1),
if p = 1.
If Firm i is producing qi ∈ [π, 1/n] then total production is weakly larger than (n − 1) /n + π, so
the market-clearing price, p, is determined by
n−1
n−1 π
+ = qi +
.
(16)
n
p
n
This equation implies that the profit of Firm i is π irrespective of its production on [π, 1/n]. In
addition, note that if Firm i is producing π the market-clearing price is one. Since the marketclearing price remains one for any qi ∈ [0, π), Firm i strictly prefers to produce a quantity [π, 1/n]
to any quantity below π. We conclude that 1/n indeed constitutes an equilibrium and, in this
equilibrium, the profit of each firm is π. Finally, note that by (16) with qi = 1/n, the marketclearing price is nπ.
Next, we show that our search for the consumer-surplus maximizing CDF can be restricted
to the class {Fπn }π .
Lemma 5 Suppose that q constitutes an equilibrium if demand is given by G and the profit of each
firm is π. Then
(i) Fπn ≤ G and
(ii) U (G, π/q) ≤ U (Fπn , nπ) and the inequality is strict if G 6= Fπn or q < 1/n.
Part (i) of the lemma says that the CDF Fπn first-order stochastically dominates the CDF G.
Note that since the profit of each firm is π and each supplies q, the market-clearing price must be
π/q. So, U (G, π/q) is the consumer surplus if demand is generated by G and each firm produces
q. By Lemma 4, U (Fπn , nπ) is the consumer surplus if demand is generated by Fπn and each firm
produces 1/n. Therefore, part (ii) says that consumer welfare is larger if demand is given by Fπn
than by G.
Proof. Let P (qi ) denote the market-clearing price if Firm i produces qi and the aggregate production of the other firms is (n − 1) q, that is, for all qi ∈ [0, 1/n]
e (G, P (qi )) .
qi + (n − 1) q ∈ D
26
(17)
Since q constitutes an equilibrium, it must be that for all qi ∈ [0, 1/n]
π ≥ qi P (qi ) .
Observe that
(18)
π
1 − [qi + (n − 1) q] ≤ G (P (qi )) ≤ G
,
qi
e and the second inequality follows from
where the first inequality follows (17) and the definition of D
(18) and the fact that G is increasing. Denoting π/qi by v, the previous inequality chain implies
that for all v ≥ nπ
G (v) ≥ 1 − (n − 1) q −
π
1 π
≥ − = Fπn (v) ,
v
n v
where the second inequality follows from q ≤ 1/n. Since Fπn (v) = 0 for v ≤ nπ, part (i) of the
lemma follows.
To see part (ii), note that
Z
1
π
U (G, π/q) =
v − dG (v) ≤
q
π/q
Z
1
π
v − dFπn (v) ≤
q
π/q
Z
1
v − nπdFπn (v) = U (Fπn , nπ) ,
nπ
where the first inequality follows from part (i) of this lemma and the second one from q ≤ 1/n.
Finally, note that the second inequality is strict if q < 1/n and, if q = 1/n, the first inequality is
strict unless G = Fπn .
We are ready to prove the main result of this section.
Proof of Proposition 7.
{Fπn }π .
By Lemma 5, the consumer-surplus maximizing CDF lies in the set
In addition, if market demand is determined by Fπn , then 1/n constitutes the consumer-
surplus maximizing equilibrium and the profit of each firm is π (see Lemma 4 and part (ii) of Lemma
5). In this equilibrium, the market clearing price is nπ. Then consumer surplus, U (Fπn , nπ), can
be expressed as
Z 1
Z
n
v − nπdFπ (v) =
nπ
1
vfπn (v) dv
+
∆ (Fπn , 1)
Z
1
− nπ =
nπ
nπ
= −π log (nπ) + (1 − n) π +
n−1
π
dv +
+ π − nπ
v
n
(19)
n−1
.
n
The last expression is maximized at π ∗ = (1/n) e−n , so the consumer surplus maximizing CDF is
indeed Fπn∗ . Parts (i)-(iii) of the proposition follow from Lemma 5. Finally, plugging π = (1/n) e−n
into (19) yields part (iv).
27
8
Discussion
To conclude, we discuss various assumptions of our baseline model and describe equilibria under
alternative assumptions.
Investment Cost.— It is possible to introduce costs associated with the buyer’s choice of distribution
different from the one considered in Section 5. Suppose, for example, that if the buyer chooses a
distribution and the upper bound of its support is k then she has to pay an additively separable cost
c (k). Recall that we have shown that the buyer’s payoff is k/e if she can choose any distribution
supported on [0, k] at no cost. So, when determining the upper bound of the support, the buyer
solves maxk k/e − c (k). If, for example, c (k) = k 2 /2, the optimal choice is k = 1/e.
Alternatively, one could assume that the cost of choosing a CDF is determined by its expectation. That is, if the buyer chooses the CDF F with mean µ then she has to incur an additively
separable cost c(µ). Recall that Proposition 1 states that if the buyer is restricted to select a
distributions with mean µ, she chooses Fπµ∗ and her equilibrium payoff is µ − πµ∗ . Therefore, if the
investment cost only depends on µ, the buyer solves the following maximization problem:
max µ − πµ∗ − c(µ).
µ
The solution to this problem defines the optimal mean, µ∗ , and the buyer chooses Fπµ∗ ∗ in equilibrium.
Production Cost.— We have implicitly assumed that the seller’s production cost is zero. It is
straightforward to generalize Theorem 1 to the case where the seller has to pay a cost c ∈ (0, 1)
if trade occurs. One can follow the same two-step procedure to solve the problem as in Section
3. Given that the seller’s profit must be π, the distribution which maximizes the buyer’s payoff is
defined by the continuous CDF 1 − π/ (v − c) on [π + c, 1) and an atom of size π/ (1 − c) at v = 1.
This distribution makes the seller indifferent between setting any price on [π + c, 1). The profit
which maximizes the buyer’s payoff is (1 − c) /e + 2c.
Multiple Buyers.— If there is more than one buyer and the seller is restricted to set a single price
then there are multiple equilibria. In each of these equilibria, the price is one and at least two
buyers choose distributions which specify sufficiently large atoms at one. There is an equilibrium
in which the value of at least one buyer is always one, so full efficiency is achieved. In this sense,
competition eliminates the inefficiency due to the hold-up problem.
Bargaining Power.— In our baseline model, the seller has all the bargaining power. Let us now
assume that, after the buyer has chosen the CDF, a bargaining game with random proposer ensues.
28
With probability 0 < α < 1, the buyer that makes a take-it-or-leave-it offer to the seller; while
with probability 1 − α the seller makes the offer. It is easy to see that, in choosing the CDF, the
buyer maximizes αµ(F ) + (1 − α)U (F, p) where p = min{p ∈ P (F )}. Since we know from section
4.1 that, for a fixed µ, the optimal F is a truncated Pareto distribution, we can conclude that, also
in this case, the optimal CDF will be a truncated Pareto distribution.
Unobservable Distributions.— If the seller is unable to observe the buyer’s choice, the price cannot
depend on the chosen distribution. Hence, the buyer always prefers stochastically higher valuations. As a consequence, the seller sets a price of one in each equilibrium and the buyer chooses a
distribution which specifies a large enough mass at one.
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