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Keywords:

  • D23;
  • L22

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

We extend the property-rights framework to allow for a separation of the ownership rights of access and veto and for sequential investment. Parties investing first do so before contracting is feasible. It is possible, however, that parties investing second can share (at least some of) their investment costs. Along with this cost-sharing effect, the incentive to invest is affected by a strategic effect generated by sequential investment. Together these effects can overturn some of the predictions of the property-rights literature. For example, the most inclusive ownership structure might not be optimal, even if all investments are complementary.

Propriété, accès et investissement séquentiel. On développe le cadre d'analyse des droits de propriété en permettant une séparation des droits de propriété, d'accès et de véto, et l'investissement séquentiel. Les agents investissant d'abord le font avant que la contractualisation soit possible. Il est cependant possible pour les agents investissant en second lieu de partager (en partie tout au moins) leurs coûts d'investissement. En plus de cette possibilité de partage de coûts, l'incitation à investir est affecté par un effet stratégique engendré par l'investissement séquentiel. Ensembles ces deux effets peuvent entraîner le renversement de certaines prédictions dans la littérature sur les droits de propriété. Par exemple, la structure de propriété la plus inclusive peut ne pas être optimale, même si tous les investissements sont complémentaires.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

When trade requires relationship-specific investments, incomplete contracting can lead to inefficiencies (see Williamson 1975; Klein, Crawford, and Alchian 1978; Grout 1984). By altering bargaining power, a clever reassignment of property rights can (partially) alleviate the hold-up problem by protecting investors from expropriation at renegotiation. The key finding of the property-rights approach – Grossman and Hart (1986), Hart and Moore (1990) and Hart (1995) – is that when contracts are incomplete, the residual rights of control that come with ownership over critical assets become important in determining the bargaining power (and claim on surplus) of the agents. Under these conditions asset ownership is designed to optimize the investment incentives of agents who make key relationship-specific investments by protecting them from opportunism when renegotiation occurs.

Demsetz (1996) and Rajan and Zingales (1998) have criticized the model of Hart and Moore (1990) for its broad definition of ownership. Others, such as Bolton and Scharfstein (1998), have pointed out that the original property-rights model is at best a theory of the entrepreneurial firm, as it does not fit the picture of large modern-day corporations where ownership by shareholders is often separated from the day-to-day control of managers.

It follows that a more nuanced definition of property rights is needed to give a clearer understanding of real-world firm structures. Bel (2014) unbundles ownership into the right to access and use an asset and the right to veto access to an asset.1 The possibility of unbundling access and veto rights has far reaching consequences for the optimal allocation of property rights because the requirement that rights of access and veto always be transferred together is often too restrictive to solve complex incentive problems. For example, when asset ownership is non-rivalrous and all investments are complements, Bel (2014) found that it is optimal for all agents to have access to every asset and that no veto powers should be allocated at all. Thus, there should be a kind of communal access to resources. This ensures that the hold-up problem is minimized because nobody can threaten to withhold these assets from another party.

A second restriction of the standard property-rights model is that specific investments are made simultaneously. A related literature focuses on the timing of investments as a way to overcome holdup (see, e.g., Neher 1999; Noldeke and Schmidt 1998; Smirnov and Wait 2004a; and Smirnov and Wait 2004b).2 One of the key insights of this literature is that sequential investment can alleviate holdup by allowing investment to occur when the contracting environment is more complete.

This paper combines the more refined notion of ownership and the possibility of making sequential investment to construct a new model of the optimal allocation of property rights.

With simultaneous investment, as in the standard model, all agents invest ex ante, before contracting is possible. For instance, a group of scientists and a large manufacturer of video game consoles are collaborating to develop and bring to market a new graphics processor unit that is to be included in the next generation of consoles. The two tasks (development of a new graphics processor, completed by the computer scientists, and the establishment of the production process, undertaken by the manufacturer) might need to be completed at the same time to ensure that the product is ready for the start of the new season. Once both relationship-specific investments have been sunk, the project becomes tangible and the parties renegotiate, where each party's bargaining power depends on the assets they own.

Rather than investing in the two tasks at the same time, it could be the case that the scientific investment must be made first. This situation could arise when it is not possible to start establishing a manufacturing process before the exact nature of the graphics processor is known. For instance, the initial stage might require the scientists to develop the new technology necessary for the graphics processor. While none of these investments could have been adequately described in a contract ex ante, as the research proceeds, information about the project and the next stage of investment required could become available. Given the project is already underway, this new information could affect the relative bargaining power of the scientist and the manufacturer, potentially resulting in a sharing of some ex post investment costs.3 Anticipating this ex ante, both the scientist and the manufacturer take this information into account when they invest.

We analyze the incentives to invest in these alternative timing regimes, extending Smirnov and Wait (2004a) by allowing for an arbitrary number of investors, alternative cost-sharing rules and the possibility of investments to be complementary at the margin.4 Given this structure, we identify three features that affect both ex ante and ex post investment. (1) Cost-sharing. If followers are not required to pay their full investment costs, they will have an enhanced incentive to invest, possibly even an incentive to overinvest relative to the surplus-maximizing level, given ex ante investment. (2) Internalization effect. As the followers observe ex ante investments before making their own, there is a Stackelberg-type strategic effect that is not present with simultaneous timing; given complementarity, this strategic effect enhances the incentive to invest for the leaders. It is even possible that ex ante investors have an incentive to overinvest relative to the surplus-maximizing level, given equilibrium investment by the followers. (3) Discounting future payoffs. If an investment is sequenced, ex ante investors need to wait longer to receive their payoffs, dampening their incentive to invest.5 These issues are explored in Section 'Investment Incentives'.

The timing of investment has implications for the optimal allocation of property rights. Specifically, when we allow for sequencing of investment and the possibility of cost sharing, the standard predictions of the property-rights model do not necessarily hold. We explore these issues in Section 'Property Rights with Sequential Investment'. As noted, Bel (2014) finds that with non-rival asset ownership, all parties in the grand coalition should have access to all assets, and no one should be able to veto any other party's access (called the most inclusive ownership structure). This is not necessarily true if investments need to be made sequentially. We show that with sequential investment, the most inclusive ownership structure might not be optimal, even with complementary investments, when, owing to cost-sharing, an increase in the followers' ownership rights reduces the leaders' incentive to invest, or when additional follower ownership rights encourage them to (further) overinvest.

Let us refer back to the example above in which a group of scientists collaborate with a video-game console manufacturer. Suppose that property rights are reallocated so that some critical assets that were previously accessible only by the scientists are now also accessible by the manufacturer (i.e., a more inclusive ownership structure is implemented). This clearly increases the incentives of the manufacturer because the scientists can no longer threaten to withhold these assets. However, this change may decrease total surplus if it causes the manufacturer to overinvest. In addition, this change, via the internalization effect, could reduce the scientists' investment incentives, thus further reducing total surplus.

2. The Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

The model has two periods, Date 0 (ex ante) and Date 1 (ex post). There is no discounting. The economy is populated by a finite set of n risk-neutral agents. The grand coalition is denoted by N and can be divided into two mutually exclusive but collectively exhaustive subsets inline image and inline image (inline image and inline image), such that all agents who invest ex ante are members of inline image and all agents who invest ex post are members of inline image. inline image and inline image are determined exogenously. There are J ex ante agents and I ex post agents, where inline image. The set of productive assets inline image contains a finite number of m assets.

Each agent can make a (human capital) investment inline image that costs inline image, where inline image, inline image is twice differentiable as well as strictly increasing and strictly convex in inline image, where inline image is a scalar lying in inline image.6 Thus, the marginal cost of investment is increasing with the level of investment, as summarized in Assumption 1.

Assumption 1. The cost function inline image is non-negative, twice differentiable, strictly increasing in inline image and strictly convex; that is inline image, inline image, inline image for inline image with inline image and inline image.

We adopt the framework of Grossman and Hart (1986): Date 0 investment decisions cannot be specified in a contract, as they are too complex or nebulous; ex ante investments are made non-cooperatively; and it is not possible to write a contract specifying the item to be traded ex post or to write a cost-sharing or profit-sharing contract at Date 0.

Let inline image. Consider a coalition S of agents that control a subset of assets inline image.7 A coalition, given their assets and the investments that they make, generates some value v that we represent here as the function inline image.

Definition 1. Let inline image be the value function of a coalition inline image in control of the subset of assets inline image, where x is the vector of investments by all agents in S.

The marginal return to investment of agent i in coalition S for a given vector of investments x is denoted by

  • display math(1)

The value generated by a coalition S depends on (i) the agents in S; (inline image) the assets controlled by S; and (inline image) the human-capital investments of the agents in S. We make the following standard assumptions.

Assumption 2. The value function inline image and inline image where ∅ is the empty set. inline image is twice differentiable in x and strictly increasing in x; that is, inline image for inline image. inline image is also concave in x.

Assumption 3. inline image.

Assumption 4. inline image.

Assumption 5. inline image.

Assumption 6. inline image.

Assumption 2 says that investments increase value but at a decreasing rate. Assumption 3 means that i's investment affects only coalitions of which she is a member. Assumption 4 indicates that investments are complementary at the margin. The superadditivity of the value function is captured by Assumption 5; assets and agents are always (weakly) complementary, which in turn implies the grand coalition must always produce the largest surplus. Assumption 6 says agents and assets are always complementary at the margin. As in Hart and Moore (1990), Assumptions 5 and 6 together imply that the marginal and total values are positively correlated.

2.1. Property Rights

There are many situations where the different aspects of ownership can be separated and granted to different parties. To capture this our model follows Bel (2014) in assuming that asset ownership can be unbundled into the right to access an asset and the right to veto others' access to an asset. This section formalizes the definitions of deterministic access, veto and control.

Consider first the rights to access an asset. Rights of access are essentially the right to use a particular asset – that is, put it to productive use. To describe this, let inline image be the subset of assets that coalition S can access at Date 1. By definition, if a sub-coalition (inline image) can access an asset, then the coalition to which it belongs (S) must also be able to access the asset. It follows that the grand coalition can access all assets inline image. This discussion is summarized in the following definition.

Definition 2. Let the mapping γ from the set of subsets of N to the set of subsets of inline image be defined as the access structure of the economy. The mapping γ satisfies

  • display math(2)

Now consider veto rights. A veto, when exercised, allows a party to stop someone else from using a particular asset. Specifically, coalition S has veto rights with respect to asset inline image if it can prevent a party who is not a member of S from using it.8 inline image is the subset of assets that the coalition S has veto rights over at inline image. Following Bel (2014), it is also assumed that if a subset inline image of coalition S can veto the use of an asset, then the use of that asset can also be vetoed by the whole coalition. The structure of veto rights is defined below.

Definition 3. Let the mapping χ from the set of subsets of N to the set of subsets of inline image be defined as the veto structure of the economy. The mapping χ satisfies

  • display math(3)

A party's outside option includes only the assets that it can access without the threat of being vetoed by someone else. This idea is captured by a coalition's control rights over an asset. Specifically, given the structure of access and veto rights, a coalition of agents S is said to control an asset a if and only if S has access to the asset and no coalition outside of S has veto over a. The control structure of the economy is important in determining the investment incentives of the agents because a coalition can only assume that it can put an asset to productive use (and derive surplus from it) if it controls the asset. Formally:

Definition 4. A control structure is a mapping β from the set of subsets of N to the set of subsets of inline image, such that inline image. The control structure satisfies

  • display math(4)

As noted above, this definition of ownership allows for a greater range of ownership possibilities.9 For example, a coalition could have access rights to a particular asset or a set of assets, but no veto rights; this is equivalent to being a renter of the asset or a tenant who has the rights only to use the asset. Alternatively, a coalition could have veto rights but no access rights to an asset, as would be the case with a landlord. The other two cases are equivalent to the control structures in Hart and Moore (1990): a coalition with both access and veto rights and a coalition with neither the rights of access or veto to a particular asset.

2.2. The Timing of Investment

There are two alternative investment timing regimes. With simultaneous investment all parties invest ex ante before contracting is possible, as in the standard property-rights model – that is, inline image and inline image. At the end of the ex ante period all relationship-specific investments have been made. At this stage, the parties bargain over the surplus (detailed below). The timing is summarized in Figure 1, but it is important to note that investments are not contractible ex ante and a surplus sharing rule is never feasible (i.e., surplus is never verifiable).

image

Figure 1. Simultaneous Investment

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The alternative is sequential investment. We assume that the subset of agents inline image invest ex ante. Having observed this, the ex post agents make their investments. Following Smirnov and Wait (2004a) and Chen and Chiu (2010), we allow for the possibility of a cost-sharing arrangement regarding ex post agents' costs. For example, it could become evident after the completion of the initial phase of a research project what type of additional investments need to be made, even if the final product is still nebulous and unverifiable. In this case, the cost of the new laboratory or the next stage of research could be shared among both ex ante and ex post investors. A similar situation could arise with an author writing a book, a screen writer writing a movie script, or a musician making a recording. After the initial investment, the nature of the product becomes more tangible. The anticipation of this ex ante could give rise to expectations about the sharing of the costs of the second-stage investments. Note that even if some cost sharing is possible, it may not be feasible to write a contract directly on investment, as the investment could involve a considerable human-capital element and the quality of the investment might be non-verifiable.10

Once all investments have been made, the parties can negotiate over the distribution of surplus. Figure 2 summarizes the timing of the sequential model.

image

Figure 2. Sequential Investment

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2.3. Bargaining over Surplus and Costs

In the simultaneous investment model the agents bargain over the allocation of surplus in the ex post period. We follow convention and use the Shapley value. Letting inline image be agent i's share of gross surplus, the following equality must hold:

  • display math(5)

Equation (5) says that the surplus allocated to all agents sums to the total surplus generated by the grand coalition. As in Hart and Moore (1990), the Shapley value inline image is defined as follows:

Definition 5. Agent i's share of gross surplus inline image is given by the Shapley value

  • display math(6)

where inline image.

Now consider sequential investment. Again, agent i's share of gross surplus inline image is given by the Shapley value. Sequential investment introduces the possibility of sharing ex post investment costs among all the agents. To capture this let λ be the set of exogenous sharing rules detailing how ex post investment costs will be shared among all agents, where inline image. Note that, since inline image denotes the proportion of ex post agent i's investment cost paid by agent l, we make the additional requirements that inline image inline image and that inline image. Consequently, the following equality holds:

  • display math(7)

Equation (7) shows that the sum of all the individual ex post investment costs borne by the different parties is equal to the total ex post investment costs incurred.

To provide some intuition, consider again the research project example from the Introduction. If all investments are made simultaneously, each investor pays his own costs. With sequential investment, however, λ determines how much of the cost of the secondary research phase is paid for by ex ante investors, and how many of these costs are borne by the ex post investors themselves. As in other incomplete-contract models, agents bargain over ex post surplus, and they have some expectation ex ante about the share they will receive. A similar logic applies to the sharing of the ex post investment costs. After the completion of ex ante investment, it is possible that the relative bargaining strengths of the two parties have been somewhat altered, perhaps through the revelation of some new information. Anticipating this change ex ante, each agent expects to pay the share of ex post costs given by λ. Note also that this general setup allows for inline image inline image, such that each ex post investor pays for all of his/her own investment costs. This also generalizes the cost-sharing rule in Smirnov and Wait (2004a), in which inline image. In principle, the solution concept of this game could also be the Shapley value.11

3. Investment Incentives

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

Different incentives will apply, depending on whether investment will occur simultaneously or sequentially. In this section we compare investment incentives of the two timing regimes. To provide a benchmark for these comparisons, we first outline the investment necessary to maximize total surplus.

3.1. First-Best Incentives

The grand coalition N generates the largest surplus for any given investment (Assumption 5). As in Hart and Moore (1990, 1127–8), the first-best maximization problem is:

  • display math(8)

From this, the welfare-maximizing vector of investments inline image solves

  • display math(9)

The solution for inline image exists and is unique, given Assumptions 1 and 2. We now turn our attention to the investment outcomes under the simultaneous and sequential timing regimes, taking for the moment the allocation of property rights (the control structure β) as given. First we analyze simultaneous investment, as illustrated in Figure 1. Second, we examine sequential investment, as shown in Figure 2.

3.2. Simultaneous Investment

With simultaneous investment all agents invest ex ante. Attempting to maximize their surplus, the first-order condition for each agent is

  • display math(10)

where, from Definition 5 and Assumption 3, agent i's marginal return is

  • display math(11)

The solution for each agent's investment choice exists and is unique, given Assumptions 1 and 2. We will denote the vector of equilibrium investment choices as inline image.

3.3. Sequential Timing

With sequential investment we consider ex post and ex ante investments in turn.

3.3.1. Ex Post Investments

Ex post parties make their choices after observing ex ante investments. Taking ex ante investment as given, an ex post agent i's maximization problem is

  • display math(12)

where inline image is the vector of ex ante investments and inline image is the vector of ex post investments. The first term in equation (12) is agent i's share of gross surplus. The second term is i's share of other ex post investment costs, while the last term is i's own investment cost. If inline image, agent i does not pay the full cost of investment.12 The vector inline image of ex post equilibrium investments solves the first-order conditions:

  • display math(13)

From condition (13), ex post equilibrium investment is determined by the control structure of the economy β, the set of cost-sharing rules λ and the level of ex ante investment inline image. Following this, we say that the vector of ex post equilibrium investments be governed by the implicit function inline image, where inline image. The next lemma shows how ex post investment responds to changes in ex ante investment.

Lemma 1. For a given control structure and set of sharing rules, an increase in any ex ante investment, ceteris paribus, (weakly) increases equilibrium investment of all ex post agents; that is, inline image and inline image.

Proof. See the Appendix.

Lemma 1 relies on the assumption that all investments are complementary. If an ex ante agent increases her investment, the marginal productivity of all ex post agents also increases, encouraging higher ex post equilibrium investment.

3.3.2. Ex Ante Investments

Consider next the maximization problem for an ex ante agent. Given the Stackelberg-timing of investment, ex ante investors incorporate the (implicit) ex post best-response functions inline image into their maximization problems. A representative ex ante agent inline image solves

  • display math(14)

The first term of (14) gives j's share of gross surplus, while the middle term specifies how much of the ex post investment costs are paid for by j; when inline image, ex post agents can contract on – and redistribute – some of their costs to other parties. The last term on the right gives the cost of investing inline image. As shown in Smirnov and Wait (2004a), if an ex ante investor anticipates a negative return, she will not invest, potentially accentuating the hold-up problem. However, to focus on other issues, we consider the case when inline image; that is, all ex ante investors anticipate a non-negative return.

The vector of ex ante equilibrium investments inline image solves the first-order conditions:

  • display math(15)

inline image, where inline image is the marginal return to investment and inline image is the investor's marginal cost.

Again, to focus on the issue at hand, we make an additional assumption to guarantee that there exists a unique solution to the system of equations (13) and (15). Specifically, the following condition on the objective function (14) is sufficient to ensure both existence and uniqueness.13

Assumption 7. The objective function (14) of an ex ante agent inline image is concave in inline image and strictly concave in inline image.

Assumption 7 requires that the indirect effect that arises as a result of changes in others' equilibrium investment (in response to a change in a j's investment) is sufficiently small that it does not materially alter the relative concavity of the investor's benefits compared with her costs. Assumption 7 allows us to make the following statement.

Lemma 2. The solution to the system of equations (13) and (15) exists and is unique.

Proof. See the Appendix.

Comparing conditions (10) and (15) reveals that the incentives of the ex ante investor change, owing to the additional term, which we label inline image, which is equal to

  • display math(16)

This new term inline image arises under sequential investment because ex ante agents internalize the effect that their investment choices have on ex post investment – we refer to this as the internalization effect.

To analyze the internalization effect, note that inline image (Lemma 1). Moreover, the term inside the summation bracket on the right-hand side of equation (16) can be separated into two parts. First, consider inline image. An increase in ex post equilibrium investment has two opposing effects: (a) it (weakly) increases the values of coalitions that ex ante player j and ex post player i are members, increasing the share of surplus (Shapley value) for j; and (b) it (weakly) increases the values of coalitions that contain i but not j, reducing the Shapley value claim on surplus of ex ante player j. However, from Assumption 6, the positive effect of (a) weakly dominates the negative effect of (b). To see this, note that from Definition 5, inline image. In addition, from Assumption 6, inline image; the overall sign is non-negative.

The second term inside the summation brackets in (16), inline image, denotes the portion of ex post costs i paid by ex ante agent j. It follows that any inline image will make this term weakly positive. Consequently, as inline image and inline image, the sign of the internalization effect is ambiguous. However, if inline image, the second effect would be arbitrary small, meaning the internalization effect is non-negative. Before outlining the first proposition, we introduce the following definition to aid in its exposition.

Definition 6. inline image is the minimum value such that, for a given control structure and set of sharing rules inline image where inline image, all internalization effects for ex ante agents are non-negative (inline image inline image).

We are now in a position to provide sufficient conditions to ensure sequential investments are higher than their simultaneous counterparts.

Proposition 1. For a given control structure and set of sharing rules, provided inline image, all sequential investments will (weakly) increase relative to investments with the simultaneous timing regime.

Proof. See the Appendix.

The intuition for this proposition is as follows. First, assume there is no cost sharing (inline image). As in the standard Stackelberg model, the followers make their choices after having observed the leaders' actions. Knowing this, and because of complementarity, the leaders increase their respective investments relative to the simultaneous case, anticipating that the ex post investors will also increase their investments. As a result, with no cost-sharing effect (inline image), complementarity and sequential investment (weakly) increase both ex ante and ex post investment levels relative to the simultaneous timing regime. Importantly, this result is not knife-edged; as all the functions are continuous, one can slightly decrease all inline image, and still have an increase in all investments relative to the simultaneous case. That is, provided all values of inline image sufficiently high – so the imposition of follower costs on ex ante agents is relatively small – all investments weakly increase. Our assumption of complementarity between investments drives this result.

If, however, inline image is small enough for some i, it could be that some ex ante investors reduce their investment relative to their simultaneous choice because of the potential increase in costs they face. Furthermore, if cost sharing is too high, ex post agents can overinvest relative to the surplus maximizing level, given ex ante investment. It is also possible that ex ante agents overinvest, relative to the surplus maximizing level, given ex post investment. Note that overinvestment is not possible under the simultaneous regime. We highlight the intuition for these results by considering a model of bilateral trade.

3.3.3. Investment in a Bilateral Trade Model

Assume there are two agents, where agent 1 invests ex ante, while agent 2 invests ex post. For a given control structure β, the coalition containing only the first agent generates a surplus of inline image, while the coalition containing only the second agent generates a surplus of inline image. The coalition containing both agents generates a surplus of inline image. The following Shapley values for both players can be derived: inline image and inline image. The system of equations (13) and (15) in this case yields

  • math image(17)

Given that inline image, and inline image, the ex post agent's full marginal contribution to surplus minus her marginal cost can be represented as

  • display math

The second term on the right-hand side is the ex post investor's first-order condition, taking into account cost sharing and holdup. From the second equation of (17), this will be will be zero in equilibrium. Consequently, we can write the internalization effect as

  • display math(18)

where inline image is described by the second equation of system (17). This representation of the internalization effect is useful, as it describes agent 1's incentives in terms of how agent 2's incentives differ from the first best. Note that, for a given level of x1, the ex post agent has the (conditional) surplus-maximizing incentives when the internalization effect for 1 is exactly zero. It follows that if the internalization effect is positive (negative), the ex post agent underinvests (overinvests) relative to the (conditional) surplus-maximizing level.

Let us now introduce another definition to help with our exposition of the results.

Definition 7. The ex ante (ex post) agent's investment is general if inline image inline image inline image. In contrast, the ex ante (ex post) agent's investment is specific if inline image inline image inline image.

The marginal return for a general investment is the same in the presence of the other party as without it. The return on a specific investment is greater inside the relationship than outside, meaning that higher effort increases the Shapley value for the other party.

Using this new definition, we now consider the ex post investor's incentives. For agent 2 to overinvest, the difference between the left-hand side and the right-hand side of the second equation of system (17) needs to be higher than inline image. Using the same logic as above, we derive the following comparison:

  • display math(19)

This condition shows the balance of two potential distortions. The right-hand side is the increase in the first party's surplus that arises from an increase in the second party's investment (this distortion is present in the simultaneous model). The left-hand side is the distortion due to the second party's not paying his full marginal costs. If the left-hand side is greater than the right-hand side of (19), there will be overinvestment. That leads us to the following result.

Result 1. An ex post agent will overinvest, relative to the conditional surplus maximizing level when ex post investment is general and inline image.

Proof. Follows from the discussion.     Q.E.D.

While agent 2's choice of investment is determined by his first-order condition, the surplus-maximizing investment also depends on the marginal benefits accruing to agent 1, and the portion of the marginal costs imposed on the agent 1. With a sufficiently high level of cost sharing, the ex post agent has an incentive to overinvest. Specifically, if ex post investment is general and the ex ante agent bears any ex post costs (inline image), there will be conditional overinvestment by ex post agent.

With sequential investment, the ex post investor can overinvest. It turns out that the ex ante agent can also overinvest compared with the surplus-maximizing level, given the ex post agent's effort. For agent 1 to overinvest, the left-hand side of the first equation of system (17) needs to be higher than inline image. This gives us the following comparison:

  • display math(20)

In a manner similar to (19), the right-hand side of (20) is the increase in the second party's surplus that arises as a result of an increase in the first party's investment. The left-hand side is the distortion due to the internalization effect.

Result 2. An ex ante agent will overinvest, relative to conditional surplus-maximizing level, when (i) the ex post agent's investment is specific; (ii) the ex ante agent's investment is general; (iii) cost sharing is not too high; and (iv) investments are strictly complementary.

Proof. Follows from the discussion.     Q.E.D.

We combine Results 1 and 2 to provide conditions for when overinvestment will occur.

Proposition 2. If either agent makes a general investment and the ex post cost imposed on the ex ante agent is not too high, there will be overinvestment.

Proof. Follows from the discussion.     Q.E.D.

The following example highlights some of the results of the previous discussion.

Example 1. Assume inline image inline image and inline image. The costs are assumed to be inline image. One can check that Assumptions 16 are satisfied. In the first-best case, investments are inline image, and inline image.

The following Shapley values for both players can be derived: inline image and inline image. From the simultaneous case FOC (10), investments are inline image, and inline image.

Now consider two different cost-sharing rules with sequential investment: (i) first let inline image and inline image; and (ii) second, assume inline image. The following system of equations describes both cases:

  • display math(21)

The solutions to (i) are inline image, inline image and inline image, indicating both investments increase with sequential investment (Proposition 1). If the ex ante player had surplus-maximizing incentives, given the anticipated x2, he would invest inline image, which means the ex ante player overinvests in the first sequential regime in comparison with the conditional surplus-maximizing level.

Similarly, the second regime results in inline image, inline image and inline image, indicating that both investments increase with the second sequential regime relative to the simultaneous alternative. Note also that the ex post agent conditionally overinvests; given the level of x1, the surplus-maximizing level of ex post investment is inline image.

To derive inline image, as outlined in Proposition 1, the internalization effect needs to be zero, so that inline image. This requires solving the system of three equations (the system given in (21) plus the equation in the previous sentence) with respect to inline image, and λ22. Calculations result in inline image. It is also possible to find values of λ22 for which the ex ante player will underinvest in comparison with the simultaneous investment, by substituting inline image in the system (21) and solving for x2 and λ22. This yields inline image, suggesting that for a low enough λ22 agent 1 shades her effort below inline image.

Note that Assumption 7 – that the objective function of an ex ante agent is strictly concave in x1 – holds in this example. Taking derivative of the first equation of system (21) with respect to x1 results in inline image, which is negative. This is because the direct effect of the first firm's investment on its benefits and costs [ − 8] is larger in absolute value than the indirect effect from a change in firm 2's investment arising from the (indirect) change in firm 1's equilibrium investment inline image. inline image

Assuming complementarity, sequential investment (without cost sharing) raises the incentive to invest for both ex post and ex ante investors. This can help alleviate some of the hold-up problem inherent in the simultaneous model. On the other hand, sequential timing can create an environment in which either the ex ante or the ex post agent, respectively, overinvest, relative to the surplus-maximizing level given the other agent's investment. In part, the incentives to (conditionally) overinvest depend on the interaction between whether a party's investment is general (or not) and the potential for cost sharing of ex post costs. However, when investments are sufficiently specific and cost sharing is not too high, there will be no overinvestment. In this case, sequential timing increases investment levels of both agents, which reduces underinvestment and increases total welfare. When the agents make relatively general investments, the impact on total welfare from having investments made sequentially, as opposed to simultaneously, is less clear. Rather, depending on the cost-sharing rules and the nature of the two agents' investments, the total welfare can either increase or decrease when we compare outcomes of the two timing regimes.14 The model, as a consequence, generalizes the results of Smirnov and Wait (2004b) that sequential investment can increase or decrease the incentive to invest and overall welfare. Moreover, our model allows us to consider the optimal allocation of property rights when there is sequential investment. We turn to this issue now.

4. Property Rights with Sequential Investment

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

Previously, the optimal allocation of property rights has been analyzed for simultaneous investments; for example, Hart and Moore (1990) examine ownership with rivalrous assets, whereas Bel (2014) investigate ownership with both rivalrous and non-rivalrous assets. We now turn our attention to the allocation of property rights when there is sequential investment. To simplify our exposition, and for the sake of space, our focus here will be on non-rivalrous assets. Note similar results and intuition apply with rivalrous assets.

In order to analyze the optimal allocation of ownership we utilize the following definition.

Definition 8. A control structure β is said to be more inclusive than control structure inline image (inline image is more exclusive than β) if and only if inline image and inline image, for at least one inline image.

Definition 8 can be interpreted as follows; if the set of assets controlled by any coalition weakly increases – ceteris paribus – the control structure is said to be more inclusive. Conversely, if the set of assets controlled by any coalition weakly decreases – ceteris paribus – the control structure is said to be more exclusive.

The two property rights, access and veto, have conflicting effects on the control structure. Allocating access rights is said to be inclusive because increasing the set of assets accessed by a coalition potentially increases the set of assets it controls without diminishing the control rights of other coalitions. The allocation of veto rights, on the other hand, is exclusive because increasing the set of assets that a coalition has veto rights over has no effect on the set of assets it controls, but it potentially reduces the set of assets controlled by other coalitions.

In Bel (2014) under simultaneous investment the most inclusive control structure is optimal if all assets are complementary at the margin. Specifically, he finds that the property rights should be assigned so that each agent individually accesses all the assets, while only the agents of the grand coalition can jointly veto all assets; that is, formally inline image and inline image. Allocating access and veto rights in this way means that the control structure is the most inclusive because every agent controls every asset.

To capture some insights about this result, we introduce the following lemma.

Lemma 3. inline image is maximized when the control structure is the most inclusive, inline image.

Proof. From equation (11) we know that inline image Furthermore, from Assumption 6, a more inclusive control structure can only increase inline image for some coalitions, strengthening incentives for agents benefiting as a result of the more inclusive structure, while not weakening incentives for others.     Q.E.D.

Applying Lemma 3 to the simultaneous investment first-order conditions in (10) neatly captures the underlying effect driving Bel (2014) inclusivity result. Making ownership more inclusive (weakly) increases incentives to invest. Provided transfers between parties are feasible and given that there is always underinvestment in the simultaneous model, inclusivity increases not only (weakly) investment but also surplus.

Sequential investment complicates the analysis because of the potential interplay between ex ante and ex post investments. This interaction can be sufficiently great such that Bel (2014) result that the most inclusive ownership structure is (second-best) optimal need not hold with sequential timing. We now analyze ownership with sequential timing.

4.1. Independent Investment at the Margin

First, consider the case when investment by one party does not effect the marginal productivity of others' investments. For this, let Assumption 4 hold with equality. Consequently, the marginal return of an agent j is independent of the investments of all other agents – define such investments to be independent at the margin.15 It follows that with investments that are independent at the margin the internalization effect drops out of the ex ante first-order conditions. It also means that the first-order condition for each agent's investment is independent of all other investments, so that

  • display math(22)

Note that the first-order conditions for ex post investments are described by (13), as before. To simplify the exposition we introduce the following definition.

Definition 9. inline image is the minimum inline image such that no ex post agents overinvest relative to the surplus-maximizing level, given the equilibrium ex ante investment; that is, inline image.

The following proposition provides the sufficient conditions for the optimality of the most inclusive control structure on investment incentives, when all assets are complementary and investments are independent at the margin.

Proposition 3. Assume that all assets are complementary and investments are independent at the margin. If inline image, the optimal ownership structure is the most inclusive control structure.

Proof. See the Appendix.

Sequencing generates a potential interplay between ex ante and ex post investments. Assuming investment independence removes this channel. Our model, when investments are sequential and independent at the margin and cost-sharing is not too high, effectively replicates Bel (2014) results, where investments are simultaneous and complementary at the margin. Note that the second link between ex ante and ex post investments – the cost-sharing effect – has to be sufficiently small so as not to create perverse incentives.

4.2. Complementary Investment at the Margin

This section reintroduces the possibility of strict investment complementarity for some investments; that is, the inequality in Assumption 4 is allowed to hold strictly. Now, ex post investors will make their choices based on the observed levels of ex ante investment. Ex ante investors, on the other hand, will internalize their effect on ex post incentives when making their investment choices, as shown in equation (15).

Definition 10. inline image is the maximum inline image such that no ex ante agents overinvest relative to the surplus-maximizing level, given equilibrium ex post investment.

In the following proposition, we outline the sufficient conditions for both ex ante and ex post agent's surplus to be maximized when the control structure is the most inclusive.

Proposition 4. Assume that all assets are complementary and investments are complementary at the margin. The sufficient conditions for the optimal ownership structure to be the most inclusive control structure are as follows.

  1. inline image and
  2. inline image is maximized when the control structure is the most inclusive, ∀ inline image.

Proof. See the Appendix.

It is possible to show, contrary to Bel (2014), that the optimal ownership structure may not be the most inclusive. With sequential investment there can be (conditional) overinvestment by either the ex ante or ex post agents, respectively. If the ex post agents are overinvesting, a more inclusive property-rights regime can further accentuate this ex post over investment problem. A similar argument can apply if ex ante agents are overinvesting. Note also that a more inclusive ownership structure with respect to ex post agents, can reduce ex ante incentives if the change decreases the internalization effect. To provide some intuition we again consider the bilateral trading model.

4.2.1. Asset Ownership with Bilateral Trade

Now we return to the bilateral-trade model, this time taking into account asset ownership. Without loss of generality, assume the ex ante (ex post) agent controls the set of assets A1 (A2). Let us consider the comparative statics of ownership when we allow for additional assets to be controlled by either the ex ante agent or the ex post agent, respectively.

When the ex ante controls additional assets, her outside option inline image increases; as a result, anticipated share of surplus, inline image, also increases. Similar to the simultaneous investment model, incentives of the ex post investor are not affected. The only possible negative effect of the increase in the ex ante agent's outside option is that she may now have an incentive to overinvest.

On the other hand, when the ex post agent controls additional assets, due to his improved outside option (inline image), inline image increases. But this will mean that inline image; consequently, the internalization effect for agent 1 falls, weakening her incentives. However, there is a possible counter-effect; investments by agents 1 and 2 are complementary, which could lead to an increase in inline image. Overall, the combined effect on inline image is ambiguous. Note though, as with the ex ante agent, the ex post agent may overinvest. We outline some of these possible effects in the following example.

Example 2. Let us augment Example 1 by assuming there are two agents and three assets. The first agent invests ex ante and controls one asset, while the second agent invests ex post and controls either one or two assets. If the ex post agent controls only one asset, then the third asset is controlled by the coalition containing both agents. In addition to the payoffs and costs specified in Example 1, the coalition containing only the second agent generates a surplus of inline image, where α represents the proportion of the two assets controlled by the ex post agent (inline image).

The following Shapley values for both players can be derived: inline image and inline image. Consider cost sharing when inline image. From the sequential case FOC (13) and (15), we can get the following system:

  • display math(23)

When inline image, the equilibrium investments are inline image and inline image with inline image, while when inline image, the equilibrium investments are inline image and inline image with inline image. We can see that when α decreases, the ex ante investment x1 increases, while the ex post investment x2 decreases. The welfare with smaller α is higher.

From equation (18), when the incentives of the ex post agent are efficient, the internalization effect is zero. This means that inline image. In example 1 with inline image we calculated inline image; hence, inline image. To derive inline image, in addition to the system (23) use the condition that ex ante incentives are efficient, that is, inline image, and solve the system of three equations for inline image and λ22. The calculations give inline image. Consequently, for inline image there is no overinvestment.

Note that Assumption 7 – that the objective function of an ex ante agent is strictly concave in x1 – also holds in this example. For details, see Example 1.

With complementarity and simultaneous investment, the most inclusive structure is optimal. This is not the case in the example above. Instead, increasing the number of assets held by the ex post agent reduces the internalization effect, decreasing ex ante investment. There is another effect: more ownership rights for the ex post agent encourages him to invest more, worsening the overinvestment problem. Both effects reduce total surplus. Consequently, in such an environment, the most inclusive ownership structure is not optimal.

5. Concluding Remarks

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

In this paper we augment the standard property-rights approach in two ways. First, by separating the rights of access and veto, we allow for a more refined notion of ownership. Second, we consider the consequences when investment needs to be completed sequentially. The introduction of sequential timing creates three additional incentive effects not present in the simultaneous model: (1) a cost-sharing effect; (2) a strategic (Stackelberg) effect; and (3) a discounting of future payoffs effect. Focusing on the first two effects, we show that it is possible that the sequencing can increase both ex ante and ex post investments when investments are complementary and cost sharing of the followers' costs is not too high. We also show that it is possible, given the level of ex ante investments, that the ex post agents overinvest relative to the surplus-maximizing level. Similarly, it can be that ex ante agents overinvest relative to the surplus-maximizing level, given equilibrium ex post investment.

These sequential effects have implications for the optimal allocation of property rights. For instance, Bel (2014) shows that, when assets are complementary at the margin, all parties should have access to all assets and no coalition should have the right to veto anyone's access. However, with sequential investment this may no longer be the case. Specifically, it might be advantageous to dampen the incentives of ex post investors when greater asset control by ex post agents causes a fall in incentives for ex ante agents via a reduction in the internalization effect, or additional asset control encourages overinvestment by ex post parties. Consequently, Bel (2014) inclusivity result need no longer hold. Rather, it could be the case that surplus increases when ex post investors have their control rights reduced.

These issues are an important consideration in the ownership structures of real firms when projects are sequential in nature. For example, this prediction is consistent with the ownership structure chosen when Daiichi Sankyo, Japan's third-largest drug maker, bought 51% of the Indian generic drug manufacturer Ranbaxy Laboratories Ltd. Notably, it was Daiichi Sanko, the party engaged in R&D and drug invention (ex ante investments), who took a controlling stake in the generic pharmaceutical manufacturer (the party making the ex post investment), effectively reducing the ownership rights of the follower.16

A similar point can be made in relation to the predictions in Hart and Moore (1990) when sequencing and cost sharing are incorporated into the model. For instance, if it is optimal to dampen ex post investment incentives (to encourage investment by ex ante agents) it may no longer be optimal to have an asset that is idiosyncratic to an ex post agent to be held by that agent (Proposition 5); for any individual agent alone to have veto rights (Proposition 4); to have an indispensable (ex post) investor to own the asset (Proposition 6); or to have the same agent owning complementary assets (Proposition 8).

Appendix

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References

Proofs

A.1. Proof of Lemma 1

The system (13) characterizes the vector of Nash equilibrium ex post investments inline image:

  • display math

Totally differentiating gives

  • display math(A1)

Where inline image is the inline image matrix,

  • display math(A2)

By Assumptions 1 and 2, inline image is negative definite; that means inline image is invertible. By Assumption 4 and equation (11), the off-diagonal elements of inline image are non-negative. The inverse inline image is a non-positive matrix (see Takayama 1978, 393, theorem 4.D.3 [III”] and [IV”]).

Pre-multiplying both sides of (A1) by the inverse inline image gives

  • display math(A3)

The sharing rules are constant; hence, set inline image. Further, to isolate the impact of a change in investment by only one representative ex ante agent j, set inline image, which gives

  • display math(A4)

Consequently,

  • display math(A5)

By Assumption 4 and equation (11), inline image. Given that inline image is a non-positive matrix, the left-hand side of (A5) must therefore be (weakly) positive (i.e., inline image). Hence, an increase in investment by any ex ante agent increases ex post equilibrium investment of all ex post agents.     Q.E.D.

A.2. Proof of Lemma 2

From Assumption 7, we know that the objective function (14) for an ex ante agent inline image is concave in inline image and strictly concave in inline image. Moreover, the summation of all the objective functions of ex ante agents given by (14) yields a function that is strictly concave in inline image. Maximizing this summed function with respect to inline image gives the same FOC as (15), which has a unique solution, vector inline image. Reconsider the simultaneous investment problem, but for given values of inline image. Owing to Assumptions 1 and 2, there is a unique solution for inline image.     Q.E.D.

A.3. Proof of Proposition 1

(i) For a given control structure β, when investments are simultaneous the internalization effect inline image is zero by assumption (inline image) but when investments are sequential it can be negative, zero, or positive; equation (15) can be presented in the following way:

  • display math(A6)

Let us also represent equation (13) in a similar way:

  • display math(A7)

where a similar variable is introduced for ex post investments inline image. Note that by construction it is always true that inline image.

Now let us combine all ex ante and ex post variables into one set of variables k = 1,..., N and represent the system of equations (A6) and (A7) as

  • display math(A8)

We want to show that when all inline image inline image, all equilibrium investments are the same or higher than in the simultaneous investment case. Therefore, if the internalization effect for sequential investments is positive, inline image, then both ex ante and ex post equilibrium investment can only be higher under sequential investment than under simultaneous investment.

Totally differentiating the above equation and rearranging gives

  • display math(A9)

Where inline image is the inline image matrix:

  • display math(A10)

By Assumptions 1, and 2, inline image is negative definite; that means inline image is invertible. By Assumption 4 and equation (11), the off-diagonal elements of inline image are non-negative. The inverse inline image is a non-positive matrix (see Takayama 1985, 393, theorem 4.D.3 [III″] and [IV″]).

Pre-multiplying both sides of (A9) by the inverse inline image gives

  • display math(A11)

Given that inline image is a non-positive matrix, the right-hand side of (A11) must therefore be (weakly) positive (i.e., inline image). Hence, a higher internalization effect of any ex ante agent increases equilibrium investment of all ex ante and ex post agents.

When inline image inline image it is clear that all inline imageinline image are non-negative, which means that in this case sequential investment leads to higher equilibrium investments of all ex ante and ex post agents. This final observation ends the proof.     Q.E.D.

A.4. Proof of Proposition 3

Define function

  • display math

The first-order conditions (13) and (22) are equivalent to inline image. Let β be a more inclusive control structure than inline image. For an ex ante agent inline image define the function

  • display math

for inline image, where inline image is exogenous and let inline image solve

  • display math(A12)

Totally differentiating (A12) and taking θ as the exogenous variable gives

  • display math(A13)

where

  • display math

by Assumptions 1 and 2, while the inequality

  • display math

follows from Assumption 6 and equation (11). Hence, inline imageinline image and inline image or inline image inline image.

For ex post agents, the proof that inline image inline image is similar and is omitted. Thus, a more inclusive control structure always increases ex ante and ex post investment incentives and consequently have higher equilibrium investment. Note that the condition that inline image guarantees that the ex post equilibrium investments are below the first-best level. Ex ante agents also underinvest because investments are independent at the margin. Therefore, using an equivalent argument as (Hart and Moore 1990, Proposition 1), the optimal control structure is the most inclusive control structure.     Q.E.D.

A.5. Proof of Proposition 4

Define function inline image such that the first-order conditions (13) and (15) are equivalent to inline image. Let β be a more inclusive control structure than inline image. For a representative ex ante agent inline image define the function

  • display math

for inline image, where inline image is exogenous and let inline image solve

  • display math(A14)

Totally differentiating (A14), and taking θ as the exogenous variable, gives

  • display math(A15)

where

  • display math(A16)

by Assumption 7, while condition (ii) specified in the proposition ensures

  • display math(A17)

Hence, inline imageinline image and inline image or inline image inline image.

For ex post agents, the proof that inline image inline image is similar and is omitted. Thus, a more inclusive control structure always increases ex ante and ex post investment incentives, yielding higher equilibrium investment. Note that the condition that inline image guarantees that both the ex ante and ex post equilibrium investments are below the first best. Using the same argument as in Proposition 3, the most inclusive control structure is optimal.     Q.E.D.

  1. 1

    Bel (2014) uses the properties of ownership – access, withdrawal, management, exclusion, and alienation – defined in Schlager and Ostrom (1992) and groups them into access and veto.

  2. 2

    Contracting can be made possible when projects progress with the accumulation of physical assets or collateral (see Neher 1999), or because the project itself becomes more tangible, as in Smirnov and Wait (2004a, 2004b). In Noldeke and Schmidt (1998) the hold-up problem is overcome by allowing ownership to be transferable between the parties as they make their investments sequentially. Also see De Fraja (1999), Che (2000), and Admanti and Perry (1991).

  3. 3

    Chen and Chiu (2010) and Smirnov and Wait (2004a) introduced a similar cost-sharing arrangement.

  4. 4

    Note that the option-to-own mechanism of Noldeke and Schmidt (1998) is not always appropriate because transfers of ownership can be costly. This could be especially problematic when there are inline image investors, as their solution would require inline image sequential transfers of ownership. With any small transaction cost or cost of delay, this mechanism is impractical when there are a large number of investors.

  5. 5

    This effect was analyzed in Smirnov and Wait (2004a).

  6. 6

    Note that inline image and it is possible that inline image.

  7. 7

    The specifics of the control structures are detailed below in Section 'Property Rights'.

  8. 8

    Note that veto rights only prevent others from using the asset independently – the potentially vetoed agent can use the asset in the presence of the agents who can exercise veto. Moreover, it is not possible for an agent to exercise a veto on another member of the same coalition.

  9. 9

    Note that once property rights have been initially allocated, it is too costly to alter this ownership structure.

  10. 10

    See, for example, Hart (1995, fn 15). Furthermore, Hart (1995, 79) argues that even if cost sharing or contracting on investment are possible, if either party can trigger renegotiation, the hold-up problem will remain, meaning the distribution of surplus will depend on the relative ex post bargaining position of the parties.

  11. 11

    While we do not explicitly model the determination of the cost-sharing rule, our reduced-form solution could be thought of as the outcome of an extensive-form bargaining process. The cost-sharing rule could arise, for instance, from the relative bargaining strengths of the parties, arising in an extensive-form bargaining game such as Rubinstein (1982) or Binmore, Rubinstein, and Wolinsky (1986).

  12. 12

    Note that, for tractability, the realized set of sharing rules λ is the same as agents' ex ante expectations.

  13. 13

    Gal-Or (1985) makes an equivalent assumption to ensure a unique solution in her Stackelberg-style game examining first- and second-mover advantages. Moreover, as noted by Gal-Or, uniquess is guaranteed here if the third and higher-order derivatives of all value and cost functions are zero.

  14. 14

    As noted above, an effect not modelled here is the discount factor, which will always tend to favour simultaneous over sequential investment (see Smirnov and Wait 2004b).

  15. 15

    To verify, let Assumption 4 hold with equality and substitute it into the proof of Lemma 1 to get inline image.

  16. 16

    As pointed out by a referee, this example is also consistent with Bel (2014) argument. It might be optimal to reduce a party's property rights if investments are substitutes at the margin because the generic company's investment may reduce Daiichi's incentive to invest in new drugs. Notably, in our model all investments are complementary at the margin and still the most inclusive structure may not be optimal. Rather than substitutes at the margin, our result relies on the sequencing of investment.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Investment Incentives
  6. 4. Property Rights with Sequential Investment
  7. 5. Concluding Remarks
  8. Appendix
  9. References
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