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We analyze noncontractible investments in a model with shading. A seller can make an investment that affects a buyer’s value. The parties have outside options that depend on asset ownership. When shading is not possible and there is no contract renegotiation, an optimum can be achieved by giving the seller the right to make a take-it-or-leave-it offer. However, with shading, such a contract creates deadweight losses. We show that an optimal contract will limit the seller’s offers, and possibly create ex post inefficiency. Asset ownership can improve matters even if revelation mechanisms are allowed.

In the last twenty five years or so a formal literature on asset ownership and firm boundaries—known as property rights theory—has developed. This theory is based on the idea that parties write incomplete contracts ex ante and that the allocation of asset ownership influences how contracts are completed ex post. In the standard model, parties reach an ex post efficient outcome through renegotiation once the state of the world is realized. However, the ability to exercise residual control rights increases the incentive of an asset owner to make noncontractible relationship-specific investments, and, as a consequence, it is optimal to assign asset ownership to those whose investments are most important^{1}. Empirical support for the idea that noncontractible investments are influenced by asset ownership can be found in a number of papers, including Woodruff [

As is now well known, one weakness of the property rights theory is that in the standard model only a particular class of contracts is considered. Specifically, revelation mechanisms of the Moore-Repullo-Maskin-Tirole type (in combination with third parties and/or lotteries) can do better and indeed often achieve the first-best. Such mechanisms are not observed in reality but they pose a challenge to the theory.

Partly to overcome this weakness, an alternative approach has recently been developed based on the idea that contracts are reference points^{2}. According to this approach a contract, negotiated under (relatively) competitive conditions, circumscribes or delineates parties’ feelings of entitlement. Parties do not feel entitled to outcomes outside the contract, but, possibly as a result of self-serving biases, may feel entitled to different outcomes within the contract. In the simplest version each party feels entitled to the ^{3}. When a party does not receive his entitlement, he feels aggrieved and retaliates by shading in noncontractible ways, hurting the other party, and creating deadweight losses^{4}. Under these assumptions, a more open-ended contract leads to more aggrievement and shading, and there is an optimal degree of contractual flexibility. Revelation mechanisms do not achieve the first-best because they depend on the existence of many possible outcomes, and therefore cause shading. In Hart [

So far, the literature on contracts as reference points has focused on ex post inefficiency and has downplayed noncontractible investments. An open question is whether the newer approach can incorporate such investments and whether it is consistent with the empirical findings, referred to above, that asset ownership influences such investments. The purpose of the present paper is to provide a positive answer to this question.

We develop a simple model in which a seller invests in the quality of a good or service provided to a buyer (thus the investment is a “cross-investment” in the sense of MacLeod and Malcomson [

In our model, in contrast to much of the literature on contracts as reference points, specific performance is allowed. Therefore, in the absence of investment the first-best could easily be achieved with a contract that mandates trade at level one at a fixed price. Specific performance generates no shading since the contract is rigid and contains only one outcome. In such a setting, asset ownership would have no role.

However, a specific performance contract does not achieve the first-best in the presence of investment. The seller has no incentive to invest since she receives the same price whether value is high or low. In order to encourage investment the parties will choose a more flexible contract; for example, they might allow the seller to make a take-it-or-leave-it offer to the buyer. In this paper we will rule out renegotiation and so in the absence of shading such a contract would achieve the first-best: the seller receives all the surplus and so has the socially correct investment incentives. However, in the presence of shading such a contract does not achieve the first-best since the buyer will feel aggrieved when the seller chooses a high price. We will show that the first-best is generally no longer attainable, and that the parties face a trade-off between providing investment incentives, achieving ex post efficiency, and avoiding shading.

We find that there are two leading candidates for an optimal contract if investment is desired. In one, ex post efficient trade is achieved but the parties restrict the range of prices from which the seller can choose ex post to be as small as possible subject to the seller having an incentive to invest. This contract will lead to some shading. The second contract introduces some ex post inefficiency by allowing the seller to choose between two ex post outcomes: trade at a level of one at a high price or trade at a level below one at a low price (equivalently trade with probability less than one). The outcomes are designed so that the buyer is indifferent between the two outcomes if value is high, but strictly prefers the low trade outcome if value is low. In this contract, there is no shading. In

In

The effect of allocating assets to the buyer is more ambiguous. As with the seller, a high outside option for the buyer in the low value state increases surplus by providing higher value when trade does not take place. However, a high outside option for the buyer in the high value state reduces the amount that the seller can charge in this state, lowering the seller’s incentive to invest.

In ^{5}.

As noted, in this paper we rule out renegotiation. We do this mainly because revelation mechanisms are most powerful if parties can commit not to renegotiate and we want to show that even under these conditions asset ownership matters. However, the absence of renegotiation can also be justified on the grounds that (opportunistic) renegotiation can lead to (even) higher levels of aggrievement and shading and so parties may be discouraged from attempting it (see Hart and Moore [

The paper is organized as follows. The model is presented in

We consider a buyer and seller who can trade a good. There are three dates. At date 0 the parties meet and contract. At date ½, the seller can make a noncontractible investment that affects the quality of the good. At date 1 trade occurs.

For the moment, we suppose that the parties’ outside options are zero; we relax this assumption in _{H} in the high state and _{L} in the low state, where _{H} > _{L} > 0. The seller can affect the probability of the high state through her investment. If the seller invests, the probability that the state is high is π, where 0 < π < 1. For simplicity, we suppose that, if the seller does not invest, the state is low with probability 1. The seller incurs a nonverifiable cost

In the first-best trade would always occur at date 1 since _{L} > 0; and investment would take place as long as the social gain exceeds the social cost:
_{H} + (1 − π) _{L} − _{L}
_{H} − _{L}) >

We will be interested in the case where (2) holds. In contrast to the previous literature on contracts as reference points, we suppose that, if trade fails to occur, it is clear who is responsible and so that party can be penalized. Thus, if (2) does not hold, it is easy to achieve the first-best with a specific performance contract: trade is mandated at date 1 at a fixed price; a party who refuses to trade has to pay large damages. However, if (2) holds, a specific performance contract does not achieve the first-best since the seller has no incentive to invest: she receives the same price whether

In the second-best, a contract will consist of a set of trade-price vectors and a mechanism for choosing among them at date 1. This mechanism will be played after

We follow Hart and Moore [^{6}.

Let’s start with the case where there is no shading: _{H} in the good (high value) state, price _{L} in the bad (low value) state, the buyer will accept and the seller will capture the entire surplus. This gives the seller the socially correct investment incentives. (Lump sum transfers can be used to divide up the surplus ex ante.)

If _{H} = _{H} or _{L} = _{L} does not achieve the first-best. In the good state the seller will propose _{H}, and the buyer will accept this, but he will feel entitled to the lower price _{L} (the best outcome for him consistent with the contract), and will shade by _{H} − _{L}). The buyer will not shade in the bad state since he receives the lowest possible contractual price, and the seller will not shade in the good state since she receives the highest possible contractual price. Finally, the seller will not shade in the bad state since she does not feel entitled to an outcome in which the buyer makes a loss (^{7}.

One way the parties can reduce shading is to restrict the range of allowable prices. Suppose that they agree that the seller can choose either _{H}, where _{H} ≥ _{H} > _{L}, or _{L} = _{L}. Then the buyer’s shading in the good state is reduced to _{H} − _{L}). The seller’s payoff in this state net of shading costs is _{H}−_{H} − _{L}) ≥ 0. The condition for the seller to invest is thus: _{H} − _{H} − _{L})) + (1 − π) _{L} − _{L}
_{H} − _{L})(1 −

Shading is minimized if (4) holds with equality, and so we can solve for _{H}. We need to check that the buyer is willing to pay _{H} in the good state:
_{H} ≥ _{H}
_{H} = λ_{L} = 1, _{H} = _{L}, _{L} = _{L}, the “efficient trade” contract, since λ = 1 in both states.

Another way for the parties to reduce shading is to make the (λ, _{L}. Let (λ_{H}, _{H}), (λ_{L}, _{L}) denote the trade-price vectors in the good and bad states respectively. Keep λ_{H} = 1 and choose λ_{L} so that the high value buyer is indifferent between (λ_{H}, _{H}) and (λ_{L}, _{L}). This yields
_{H} − _{H} = λ_{L}_{H} − _{L}

Since the high value buyer is indifferent, he is not aggrieved and does not shade. Thus the condition for the seller to invest is now
_{H} − _{L}) ≥

Combining (8) – (9) yields

For ex post efficiency reasons we want λ_{L} to be as large as possible and so (10) will hold with equality. We continue to set _{L} = _{L}_{L}, so that the buyer breaks even in the bad state. This ensures that the seller chooses (λ_{L}, _{L}) in the bad state and that there is no seller aggrievement: the seller does not feel entitled to an outcome where the buyer would quit. The gross surplus from this contract is
_{H} = 1, λ_{L} = 1 – _{H} = _{L}_{L}, _{L} = λ_{L}_{L}, the “no aggrievement” contract.

The general analysis in

If max (_{L}, _{1} − _{2} − _{L} the optimal contract is specific performance. Net surplus is _{L}.

If max (_{L}, _{1} − _{2} − _{L} then there is an optimal contract of the following form: the seller chooses between (λ_{H}, _{H}) and (λ_{L}, _{L}) at date 1 and the buyer can reject without penalty (leading to no trade), where

If max (_{L}, _{1} − _{2} – _{1} − _{H} = λ_{L} = 1, _{H} = _{L}, _{L} = _{L} (the “efficient trade” contract). Net surplus is _{1} −

If max (_{L}, _{1} − _{2} − _{2} − _{H} = 1, λ_{L} = 1 – _{H} = _{L} _{L}, _{L} = λ_{L}_{L} (the “no aggrievement” contract). Net surplus is _{2} −

Note that in Case 2a, _{1} − _{L}, and it is easy to show that this implies (6)^{8}.

It follows immediately from Proposition 1 that, if investment is first-best efficient ((2) holds), then the second-best is strictly inferior to the first-best: in (1) there is no investment, and in (2) there is shading or inefficient trade. At the same time note that, if _{L} is close to zero, the “no aggrievement” contract approximates the first-best, since the efficiency losses from inefficient trade in the bad state are small.

We have seen in

We take the view that asset ownership affects outside options. In particular, an allocation of asset ownership at date 0 permits the seller to achieve

We suppose that the parties’ payoffs are linear in the level of trade. Thus, under the trade-price vector (λ,

An equivalent interpretation is that with probability λ the buyer trades with the seller and with probability 1 − λ the buyer trades on the outside market, but either way the buyer always pays the seller

Similarly, the seller’s payoff is

We assume

We also suppose
^{9}. Note that if the seller’s investment is purely an investment in the seller’s human capital, then ^{10}.

As in

It is useful to compute the analogous contracts to the “efficient trade” and “no aggrievement” contracts in

In (19), the buyer’s participation constraint is binding in the bad state. The buyer is aggrieved by _{L} ((15) and (18) imply that _{L}. The seller is indifferent between investing and not (see (3) with _{L} replaced by _{L}). The condition that the buyer does not quit in the good state is

Surplus from this contract is given by _{1},as in (7).

We see that the seller’s outside options have no effect on the surplus in the efficient trade contract given that

Turn next to the “no aggrievement” contract. The condition that the high value buyer is indifferent between the high and low value outcomes, which implies that he does not shade, becomes

Finally, the condition that the buyer’s participation constraint is binding in the bad state is

Combining (21) – (22) yields

To sum up, the “no aggrievement” contract is: λ_{H} = 1,λ_{L} = 1 − _{H} = _{L}_{L} = λ_{L}(_{L} −

As shown in

It is easy to check that the first inequality in (27) holds if and only if the “efficient trade” contract (see (19)) generates strictly higher surplus than specific performance. Thus (27) guarantees that, whenever “efficient trade” is strictly superior to specific performance, (20) holds. Note that (27) is automatically satisfied if

Assume

If max (_{L}, _{1} − _{L} the optimal contract is specific performance. Net surplus is _{L}.

If max (_{L}, _{1} − _{L}, then there is an optimal contract of the following form: the seller chooses between (λ_{H}, _{H}) and (λ_{L}, _{L}) at date 1 and the buyer can reject without penalty (leading to no trade), where

If max (_{L}, _{1} − _{1} − _{H} = λ_{L} = 1, _{H} = _{L} − _{L} = _{L} − _{1} −

If max (_{L}, _{1} − _{H} = 1, λ_{L} = 1 − _{H} = _{L}(_{L} − _{L} = λ_{L} (_{L} −

If we compare _{2} with _{2} when _{L} in (25) must fall to compensate for this; but it also decreases the effects of inefficient trade since value outside the relationship is higher. The second effect dominates. An increase in _{L} in (25) must fall to restore the seller’s investment incentives; and the increase in _{H} = 1,

In sum, seller outside options are unambiguously good for surplus (given (18)), whereas buyer options need not be (since they may discourage seller investment). However, buyer outside options are good for surplus if

We now turn to the case where

Consider the contracts in Proposition 2. Make one small change in the assumption about aggrievement. Suppose that if _{H}, _{L} are the contractual prices then the parties feel entitled to any price in the range (_{L}, _{H}), not just to _{L}, _{H}^{11}. Start with the “efficient trade” contract in (19). In the presence of outside options the buyer does not expect to pay less than _{L} − _{L} – _{L}) ≥ _{L} = _{L} − _{H} = _{H} − _{L}) ≥

Next assume that _{L} − _{L}) < _{L} = _{L} −

Given the shading in the good state equal to _{H} −

The third case is where _{L} –

The “no aggrievement” contract in Proposition 2 is unchanged. However, if _{L} − _{H} = 1 and _{H} = _{L}, _{L}) to (λ_{H}, _{H}). Now choose (λ_{L}, _{L}) so that the buyer’s participation constraint is binding in the bad state and the seller has an incentive to invest:

(32)–(33) yield

Note the role of the condition _{L} would be negative. For ex post efficiency reasons we want λ_{L} to be as large as possible, but λ_{L} cannot exceed 1. Thus maximal gross surplus from this second “no aggrievement” contract is

To emphasize again: this contract produces no aggrievement or shading since in the good state the buyer does not feel entitled to pay less than the seller’s outside option and the seller is getting the highest price; and in the bad state the buyer prefers (λ_{L}, _{L}) to (λ_{H}, _{H}), but the buyer’s participation constraint is binding and so the seller does not feel entitled to more.

Note the importance of the assumption that _{L} − _{L}, _{L}) to (λ_{H}, _{H}), the seller would choose (λ_{H}, _{H}) in the bad state, and this would not give her the right incentive to invest. To sum up:

(*) If _{H}_{H}_{L}_{L}_{L} − _{H} + (1 − π) _{L} − (1 − π) max{ _{L} −

We have found three contracts (in addition to specific performance) that are feasible for the case ^{12} However, the contracts that we have identified provide a lower bound for what can be achieved.

Suppose _{L}, _{3}

Remark: It is easy to check that

We can use Propositions 1 and 3 to see the effect of outside options on surplus. Observe that _{1}_{2}_{3}

Armed with Propositions 2 and 3, let us now turn to the effects of asset ownership. Suppose that there is a fixed set of assets at the disposal of the buyer and the seller, where these assets can be individually or jointly owned. We would expect that the more assets a party owns, and therefore can walk away with if the relationship breaks down, the higher will be that party’s outside option. We might also expect that asset ownership would affect the outside option in the good state more than in the bad state (this is the assumption made in much of the property rights literature). We therefore assume:

(**) If an asset that was previously owned by the buyer, or was jointly owned, now becomes owned by the seller then

Let’s start with the case where

One case where we can make a clear prediction is if an asset is idiosyncratic to the seller: we define this to mean that transferring the asset from the buyer to the seller increases the seller’s outside option without reducing the buyer’s outside option (whatever other assets the buyer and seller own).

Assume

Compare _{2} with

Proposition 4 implies that joint ownership of an asset (or separate ownership of strictly complementary assets) is suboptimal. An asset that is jointly owned yields a zero outside option for the buyer but would yield a positive outside option for the seller if ownership were transferred to her.

Proposition 4 does not hold generally for the buyer. It may not be optimal for the buyer to own an asset that is idiosyncratic to him (an asset is idiosyncratic to the buyer if transferring the asset from the seller to the buyer increases the buyer’s outside option without reducing the seller’s outside option), given that this may increase

Assume

_{2} with

Let’s turn now to the case where _{1}_{2}

Thus, we have

Assume (27). Start with joint ownership of all assets, where _{L} − _{3}_{L}.

It remains only to establish strictness. But this follows from the fact that, if _{L} −

Note that the conclusion that seller ownership is better than joint ownership holds also when

Proposition 6 provides some support for the idea, that, taking joint ownership as a starting point, allocating ownership of all the assets to the seller can bring us closer to the first-best. The intuition is that an increase in _{L} < 1, raising _{3} may become available (see (35)).

Note that Proposition 6 does not establish that seller ownership is optimal. It is possible that buyer ownership is even better. Indeed, we have seen that a change in asset ownership that increases buyer outside options may also increase surplus. All Proposition 6 shows is that seller ownership is better than joint ownership. One case where we can go further is when

It is interesting to compare the results on asset ownership in this section with Hart [

Of course, the result that the seller should own assets given that her investment is important and that the seller and (under certain assumptions) the buyer should own assets that are idiosyncratic to her or him is also analogous to findings in the standard property rights literature (see, in particular, Hart and Moore [

In this section we show that, for the case where the seller’s outside options are state independent—^{S}

As in Hart and Moore [_{H}, _{H}), (λ_{L}, _{L}), respectively, and suppose that they lie in ∑. (We do not allow for third parties in the contract; we doubt that they would add anything in the present context.)

Each party feels entitled to the best contractual outcome for him subject to the other party realizing some reservation payoff. In the case of voluntary trade, this reservation payoff is the party’s outside option since each party can quit without penalty. However, what is it more generally (e.g., if one party cannot quit at all or can quit only by paying some penalty or must play some other game to determine the quitting price)? Following Hart and Moore [

Since this game is zero-sum it has a unique equilibrium in payoff terms. Player 1’s reservation payoff is his equilibrium payoff in this game. (It might be − ∞.) We can carry out a similar exercise for the second player. This will be a different game since the roles are reversed: player 2 maximizes his payoff while player 1 minimizes player 2’s payoff. Player 2’s reservation payoff is his equilibrium payoff in this game.

Note that a party’s reservation payoff will generally differ from his outside option. For example, under a specific performance contract a party’s outside option is irrelevant.

Denote the seller’s reservation payoff by ^{S}^{B}^{S}

Armed with these reservation payoffs, we can compute aggrievement levels for the buyer and seller in the two states as follows:
_{1}, _{1}) , (λ_{2}, _{2}) ε∑, and (λ_{1},p_{1}) is below player 1’s reservation level while (λ_{2}, _{2}) is above, then player 2 feels entitled to a convex combination of (λ_{1}, _{1}), (λ_{2}, _{2}) such that player 1’s reservation level is satisfied in expected terms.

An optimal contract, which induces investment, is a choice of ∑ and a message game that maximizes the expected social surplus subject to the investment constraint; that is, which maximizes
_{j}, _{j}) is an equilibrium of the mechanism in state

The proof proceeds by relaxing various constraints and showing that the solution to the relaxed problem can be implemented in such a way that all the original constraints are satisfied. Note first that
_{H} + (1 − λ_{H}) ^{S}^{S}_{L} + (1 − λ_{L}) ^{S}^{S}_{L}, _{L}) in the good state and (λ_{H}, _{H}) in the bad state. (This is a critical step—it would not be possible if the seller’s outside options varied with the state.) It follows that

Also

Consider the relaxed problem in which we replace (36)–(37) by (41)–(42), and choose (λ_{H}, _{H}), (λ_{L}, _{L}) directly, ignoring the constraint that they must be the equilibrium of some mechanism. In other words, we maximize S subject to (39) and (41–42). It is immediate that in the relaxed problem it is optimal to set _{H} < 1, S can be raised by increasing λ_{H} and _{H} such that _{H} − λ_{H}r^{S} stays constant, since this increases λ_{H} (_{H} − _{H} and hence reduces the right-hand side of (41), and does not disturb (39). Hence λ_{H} = 1. In addition, we always want _{H.} Hence (39) holds with equality.

We can thus consider the further relaxed problem:
_{L} _{L}, _{H}.

Here _{L} a little, and adjusting _{H} to satisfy (44), increases _{L} = 1. It follows from the definition of Case 1 that _{H} < _{L}, but then (44) can hold only if _{H} to keep (44) satisfied increases

Again

S is decreasing in λ_{L} by (16) and so it is optimal to reduce λ_{L} to the point where either _{L} = 0. But (15) and (17) imply that _{L} = 0 .

Hence λ_{L} will be such that

Now _{L} and

Moreover,

Suppose

Hence λ_{L} = 0 is optimal. However, substituting λ_{H} = 1, λ_{L} = 0 in (43) shows that surplus is weakly lower, and strictly lower if _{H} = 1, λ_{L} = 0,

We are left with the case where _{L}, either λ_{L} = 0 or λ_{L} = 1 is optimal (or all 0 ≤ λ_{L} ≤ 1). We have already argued that λ_{H }= 1, λ_{L} = 0 is (weakly) dominated by the “no aggrievement” contract in Case 2. All that remains is λ_{H} = λ_{L} = 1. But this is the other contract considered in Proposition 2.

We are left with two candidates for an optimal cntract with investment:

Let _{H} = _{L} − _{L} − _{L} = λ_{L}(_{L} − _{H}, _{H}) and (λ_{L}, _{L}) at date 1 with the buyer able to say no, _{L} = λ_{H} = 1 outcome in (53) is strictly superior to specific performance, then the first inequality in (27) holds. Hence by (27), (20) does indeed hold.

Thus, we have found a mechanism that implements the solution of the relaxed problem. Moreover, it satisfies all the constraints of the original problem, (38)–(39). Therefore, it must solve the original problem. We have thus proved Propositions 1 and 2.

Remark: It is worth considering why this argument does not apply when _{L}, _{L}) may be below the seller’s reservation level and so the buyer may not feel entitled to it. However, following our earlier discussion, if (λ_{H}, _{H}) provides the seller with strictly more than her reservation payoff in the good state, the buyer will feel entitled to a convex combination of (λ_{H}, _{H}) and (λ_{L}, _{L}) such that the seller receives her reservation payoff in expected turns. Convexification destroys the linearity that made the above analysis relatively simple, and I have been unable to characterize an optimum in this case. Among other things it is possible that an optimal contract will now consist of more than two trade-price vectors (This may also be a feature of an optimal contract when (27) fails to hold.).

In this paper, we have studied a model where the purpose of a long-term contract is to encourage a seller to make a quality-enhancing investment, as well as to achieve ex post efficiency and to avoid shading. We have shown that, if contracts are reference points, the first-best cannot be achieved even when the parties can commit not to renegotiate. We have also shown that asset ownership can increase efficiency.

One obvious question to ask is, what happens if we suppose instead that parties can always renegotiate. This will change the analysis in a number of ways. First, inefficient ex post outcomes, where λ < 1, will no longer be sustainable. Second, certain kinds of (opportunistic) renegotiation may lead to (even) higher levels of aggrievement and shading. Third, the possibility of renegotiation will make it harder to allocate surplus to the seller: even if the seller has the right to make take-it-or-leave-it offers, the buyer can reject the seller’s offer and renegotiate. As Moore and Repullo [

The model studied in this paper is obviously quite restrictive. We have focused on the case where only one party invests and where the investment is a cross-investment. The reason for the cross-investment assumption is simple. If the investment is a self-investment (the seller’s investment affects her cost but not the buyer’s value) the first-best can be achieved with a specific performance contract since this guarantees efficient trade and ensures that the seller internalizes fully the benefit from her investment. Introducing a second cross-investment by the buyer—the buyer’s investment affects the seller’s cost—would complicate the analysis since we would require more than two states of the world but we suspect that our results—for example, some version of Proposition 5—would generalize. However, analyzing this case must await further research.

I would like to thank Philippe Aghion for stimulating conversations that led me to write this paper. I am also grateful to Maija Halonen, Richard Holden, John Moore, and Philipp Weinschenk for helpful comments, to Cathy Barrera for research assistance, and to three anonymous referees for useful suggestions. Financial support from the U.S. National Science Foundation through the National Bureau of Economic Research is gratefully acknowledged.

The author declares no conflict of interest.

See Grossman and Hart [

See Hart and Moore [

One does not need to go this far, however. See Halonen-Akatwijuka and Hart [

Shading is a form of negative reciprocity, as studied in the social preferences literature. See, Fehr and Schmidt [

In a recent paper, Herweg and Schmidt [

For a further discussion of shading, including examples of buyer and seller shading, see Hart and Moore [

Note that we are assuming that the buyer’s entitlement is defined with respect to prices rather than surplus. A buyer might feel that it is reasonable that he pays a higher price if the value of the good is high. It would be interesting to allow for this possibility in extensions of the model. We conjecture that some version of our results will continue to hold as long as the buyer’s perceived fair price does not increase one to one with the value of the good. Of course, if the increase is one to one, that is, the buyer is willing to accept a constant amount of surplus, independent of the quality of the good, the seller becomes the residual claimant and the first-best can be achieved.

One important assumption that we are making implicitly is that the buyer is not aggrieved about the seller’s investment decision per se. If he were then a specific performance contract might cause some shading as a result of the buyer being disappointed that the seller has not invested. One might even imagine that the seller would invest to forestall the buyer becoming angry and shading. Investigating situations where aggrievement is a result of ex ante as well as ex post actions is an interesting topic for future research. See also footnote 7 for a related point.

In the absence of (17), revelation schemes in combination with third parties and/or lotteries may be required to achieve the first-best even when

Note that we suppose that the parties cannot restrict the impact of outside options on payoffs in (12) - (13) except though the choice of λ, e.g., they cannot write exclusive dealing contracts.

In other words, we convexify things.

As an example, suppose that _{H} = 20, _{L} = 14, _{L} − _{H} = 1, λ_{L} = 1, _{H} = 10, _{L} =10 − 2c, where the buyer chooses between the two at date 1 and the seller can quit. There is no longer any buyer aggrievement in the good state since the seller’s participation constraint is binding (and there is no buyer aggrievement in the bad state since the buyer gets the lowest possible price). There is no seller aggrievement in the good state because the seller gets the highest possible price. However, there is seller aggrievement in the bad state: the seller receives 10−2_{H} + (1 − π) _{L} - π _{H} − _{L}) = π _{H} + (1 − π) _{L} −