Bargaining power and the structure of alliance contracts B. Taub A. - - PDF document

Bargaining power and the structure of alliance contracts

• B. Taub∗
• A. Seth†

May 2, 2010

Abstract Consider an alliance between an entrepreneurial biotech company and an established pharmaceutical company. Frequently, small biotech firms require complementary assets and services from big pharmaceutical com- panies in order to develop a promising molecule to the stage whereby they can generate cash flows. At the same time, big pharmaceutical firms need the innovations developed by entrepreneurial biotech firms to build a pipeline of promising drugs. The relationship between biotech firms and pharmaceutical firms is thus characterized by complementarities. Actually realizing the potential of the complementarities can be diffi-

• cult. In the first place, the technological possibilities are characterized by
• risk. In the second place, although there are profits from collaboration,

the partners also have divergent interests. Each firm has its own objec- tives: a big pharmaceutical firm has its own portfolio of projects that it is trying to optimize, and therefore may defect from a joint development agreement with the biotech firm if another promising molecule for the same end use becomes available. Similarly, the biotech firm has its own

• priorities. For example, it could seek to use the research efforts funded by

its big pharmaceutical partner on related projects from which it derives private benefits. Accordingly, it becomes important to consider how con- tracts can be designed to achieve cooperation between the two firms. Our theory provides a recipe for these contracts. Unlike property rights models that focus only on how cooperation may be achieved in the investment stage of a project, our theory explic- itly considers the resolution of incentive problems that arise both in the investment stage and the subsequent execution stage. Thus, it provides a richer and more complete explanation of cooperative action. Our theory quantifies the ability of each firm to extract rents from a contract; we refer to this as bargaining power. We show that excessive bargaining power can prevent an alliance from forming because of the excess incentive to defect. But investment by firms can moderate this bargaining power to overcome the defection incentive and allow alliance formation. We quantify the

∗University of Illinois. †Virginia Tech.

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resulting contract in terms of investment, the cost of that investment, payoffs, and net profits, all of which can be asymmetrically distributed across the firms.

Contents

1 Introduction 3 1.1 Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Bargaining power and the resolution of the formation problem . 4 2 Relationship to the literature 5 2.1 The prt approach . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Limitations of the prt approach . . . . . . . . . . . . . . . . . . 5 3 The model 6 3.1 Some technical details of the model . . . . . . . . . . . . . . . . . 8 3.2 The link between investment and bargaining power . . . . . . . . 8 3.3 Mechanics of the model . . . . . . . . . . . . . . . . . . . . . . . 10 4 The impact of costs 12 4.1 The consequence of positive costs . . . . . . . . . . . . . . . . . . 13 5 A basic example of joint profit maximization 14 5.1 Stage 1: Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Stage 2: Optimizing costs conditional on formation . . . . . . . . 16 5.3 The effect of costs . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.4 Increasing the initial relative bargaining power of firm 1 . . . . . 18 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 A basic example of selfish profit maximization 21 6.1 Equal costs—the selfish case . . . . . . . . . . . . . . . . . . . . . 22 6.2 Increasing firm 1’s initial relative bargaining power—the selfish case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.3 Formation is more sensitive to cost in the selfish profit maximiza- tion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 Conclusions 26 2

1 Introduction

Two pirates, Captain Flint and Long John Silver, want to row to an island that lies across a strait in order to dig up a treasure chest. There are rival pirates who are trying to get there first. Flint and Silver have a boat, but it lacks oars. They must decide whether or not to make oars cooperatively, each knowing that he cannot reach the island before the rivals if he rows alone. Flint’s and Silver’s respective investments will impact the quality of the oars they make and thereby the speed with which they cross the strait, and whether they cross it at all. If they decide to build the oars, each must subsequently also decide how hard to row across the strait. Because they are pirates, they understand that the other will have incentives to shirk when rowing. Each can pretend to row hard, but the other cannot verify effort instantaneously. However, both can

• bserve that the other has deviated from a pattern of rowing and rest.

They must therefore solve two incentive problems. They may have the in- centive to underinvest in building the oars, and they may also have the incentive to shirk as they row. These incentive problems seem unconnected, but we will show that they are linked. Flint’s and Silver’s problem is analogous to that of two firms considering whether to undertake investments in relationship-specific complementary assets in a world of incomplete contracting. For example, the terms of the 1978 con- tract between Genentech and Eli Lilly for joint research on human insulin was the subject of protracted negotiation.1 How firms solve such incentive problems is central to the theory of the firm.

1.1 Formation

To show how to solve the contracting problem we create an analytical framework that incorporates the following elements:

• The incentive to invest and the cost of investing during the oar construc-

tion phase.

• The incentive to cooperate during the rowing stage.

Unlike previous formulations that focus on one or the other of these incentive problems, our framework examines the link that arises because the pirates an- ticipate how their investments will affect their incentives when they are rowing. We denote the successful resolution of the contracting problem as formation. This resolution entails two stages: the stage in which the oars are made, which we denote the investment stage, and the rowing stage, which we denote the execution stage. In the investment stage the pirates decide not only whether to make the oars but also how big to make them. These decisions will be influenced both by the costs of making oars and by incentives in the rowing stage. Unless

1Ultimately, the world’s first biotechnology drug, Humulin, was the outcome of this alliance.

With its patent expiring in 2002, Humulin is still a major drug with 2003 sales of over $1 billion—the treasure!.

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the incentive problems at both stages are resolved, there could be sub-optimal

• investment. In the extreme, formation might not even occur: the oars are not

made, leaving Flint and Silver stranded.

1.2 Bargaining power and the resolution of the formation problem

In our framework, Flint and Silver’s relative bargaining powers will influence the

• utcome, i.e., the investments made by each, and their payoffs. Their profits are

comprised of the treasure itself, which they must decide how to split in advance, adjusted by the effort each expends in getting to the island. Suppose Silver is the stronger rower—or has the bigger and more effective

• ar. If he rows diffidently while Flint rows hard, they might never get to the
• island. Silver’s relative bargaining power is higher than Flint’s, because he has

a greater influence on their speed, and on whether they get to the island at all. This will translate into his getting a greater share of the payoffs. Our framework allows us to quantify this bargaining power and to show how it is influenced by investment. First, we observe that if the combined bargaining power of the two pirates is initially too high—in our model the relevant combined bargaining power is the product of the individual bargaining powers—formation cannot occur without investment. Specifically, we will show how investment by one pirate reduces the relative bargaining power of the other. This in turn reduces the combined bargaining power sufficiently to achieve formation. The weakening of one prirate’s bargaining power via the partner firm’s in- vestment can enable formation to occur. But the exact combination of invest- ments by the two pirates that yields formation depends on their initial bargain- ing power. Our theory allows us to quantify these investments in relation to initial bargaining power and the other parameters of the model. Whether the investments in oars occur in sufficient measure to achieve forma- tion, and the relative sizes of the oars, will be dictated by the pirates’ oar-making skills, and their skills might be very different. We translate these differences in skills as differences in their costs. If their costs are very different, the oars they make will be different sizes, and consequently their bargaining power as they row will be different. Viewing the pirates as representing firms contemplating alliances, our theory thus not only rationalizes the formation of alliances, it predicts the potentially asymmetric investments that the participating firms undertake, and it also spec- ifies exactly how profits are split, which also can be highly asymmetric. More-

• ver, the requirement that the contract induce cooperation causes inefficiency

in general, and we quantify this inefficiency. We therefore have a positive theory

• f alliance contracts.

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2 Relationship to the literature

Since the seminal work of Ronald Coase [3], the theory of the firm has rep- resented a central issue of interest for research in numerous fields including economics, strategy, law, finance and accounting. In its essence, such a theory must consider the question of how to create incentives for entities that possess complementary assets, but can enjoy private gains at the expense of the other, to cooperate in activities that yield joint benefits. Property rights theory is the prevailing approach to analyzing this problem.

2.1 The prt approach

The recent literature has used Property Rights Theory (prt) to approach this problem formally. The prt formulation of the boat-building problem is as fol-

• lows. Flint owns a boat but only one oar: he needs another oar. Will Silver

contract with Flint to make it for him or will Flint make it himself? Silver is concerned that even if he did enter into a contractual agreement with Flint to make an oar, Flint might ex post find an alternative supplier of oars and then try to renegotiate or walk away from this agreement: this is the classic holdup problem. prt models predict that decisions about asset ownership will focus on re- solving this holdup problem. This often implies ownership of all assets by one party even though this is not the first-best solution. More generally, as Hart and Moore [6] show in their canonical study of a buyer-seller relationship in- volving specific investments: if ex post renegotiation cannot be prevented by the parties, the holdup problem characterized by underinvestment results. prt considers firms as defined by the group of assets they own. Ownership

• ver assets confers control rights, so that a firm can specify exactly how the

asset will be used. The value of these control rights is a function of the outside

• pportunities of the assets, i.e., the value in alternative uses. The strength of

the outside opportunities determines the threat points (N¨

• ldeke and Schmidt,

[5]) or prices at which the parties trade (Hart and Moore, [6]). So, the value

• f control rights over their respective assets creates bargaining levers for firms

and hence influences how the payoffs from investment in relationship-specific complementary assets will be divided. In effect, the bargaining power of the firms is associated only with the owner- ship of assets. Any differences in bargaining power that are not related to asset

• wnership are explicitly assumed away by utilizing a perfect Nash bargaining

solution as a way of dividing the surplus earned above the threat points of the individual parties (Hart and Moore, [6]).

2.2 Limitations of the prt approach

While prt models have developed valuable insights on the resolution of the holdup problem via the control rights associated with asset ownership, perhaps their most serious limitation is an inability to take into account the second kind 5

• f incentive problem described in our formulation of the problem of cooperation:

that of inducing appropriate effort, and the bargaining power considerations that influence this problem. As Holmstrom and Roberts [4] note, “...power derives from other sources than asset ownership and other incentive instruments than ownership are available to deal with the joint problems of motivation and coordination.” They conclude that the prt literature is unable to explain a wide range of governance forms where aspects like principal-agent problems, reputation, monitoring and measurement problems and knowledge transfer play a crucial role. prt models focus only on how cooperation may be achieved in the investment stage of a project. At the same time, these models downplay the problem of cooperation in the execution stage. In that the theory of the firm that we attempt to develop incorporates both stages, it provides a richer and more complete explanation of cooperative action.

3 The model

We use the prisoner’s dilemma to capture the complementarity between Flint and Silver in the rowing stage (i.e., between firms in the execution stage).2 It is well known that in a static prisoner’s dilemma, cooperation cannot be

• achieved. By playing the game repeatedly and over an infinite horizon, though,

cooperation is possible. However, in the standard repeated prisoner’s dilemma game, a continuum of cooperative equilibria arise. By adding a realistic modification of the repeated prisoner’s dilemma, we collapse this continuum to a single point that represents a unique equilibrium. In the rowboat setting, suppose the strait that Flint and Silver want to cross has unpredictable currents that affect the progress of the boat; also recall that the ri- val pirates are pressing to get to the treasure first, but their progress is unknown. If either Flint or Silver deviates from the equilibrium pattern of rowing-and-rest, then the other too ceases to cooperate as punishment. The boat is then swept

• ff course and they fail to arrive before the rival pirates. This is analogous to

the situation wherein, if firms deviate from the pattern of cooperation that they initially contracted to, they precipitate the legal termination of the contract. We capture this situation in the repeated prisoner’s dilemma with the fol- lowing device (detailed in Taub and Kibris [8]): deviations from a fixed mixed strategy that is chosen at the initiation of the contract trigger termination of the game. With sufficient patience, the equilibrium of the game is a unique

2We emphasize that we assume no technological complementarity between the firms’ invest-

ments, nor do we assume any cost complementarity. That is to say, if firm 1 invests, firm 2’s payoffs are unaffected by that investment. This is distinct from the technological complemen- tarities arising from their actions: if firm 1 acts cooperatively, firm 2’s payoffs are enhanced, due to the prisoner’s dilemma structure of the game, so there is endogenous complementarity driven by the incentives in the rowing stage. The prt literature does assume technologi- cal complementarities. However, incorporating technological complementarities here would

• bscure the complementarities arising from incentives.

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Pareto-optimal equilibrium point.3 This equilibrium point is determined by the relative bargaining power of the two players, which is in turn determined by the structure of the payoffs in the underlying game. It is also true, and crucial to our argument here, that if there is insufficient patience (or equivalently an excessively high probability of random termination) the only equilibrium is the static non-cooperative outcome. In the literature on the repeated prisoner’s dilemma, it is standard to characterize the equilibria of the game as function of the discount factor. Iin our model, we depart from [8] and from the standard approach, in which payoffs are fixed. Instead we hold the discount factor fixed and vary the pay-

• ffs of the game. We show how modifying those payoffs alters the bargaining

powers of the players, making it possible to achieve the unique Pareto-optimal equilibrium. The changes in payoffs are achieved via investments, which in our model are

• endogenous. Each firm’s investments increases its own payoffs. The intuition is

straightforward: if Flint builds a bigger oar, he can go faster. In our model, contracting on investment has a specific meaning. Such a contract has a legal implementation, but the legal implementation does not alone solve the incentive problem. The contracts are actually self-enforcing, in the sense that in principle a firm could deviate by deviating from a fixed mixed strategy but chooses not to, not because of legal strictures per se, but out of the fear of the lost surplus this would precipitate. The change in payoffs stemming from investment alters bargaining power. Specifically, one firm’s investment increases its payoffs, but at the same time reduces the bargaining power of the other firm. If done in sufficient measure, this increases the propensity to act cooperatively, thereby effecting formation.4 If Flint builds a much bigger oar at his expense, then his payoff increases (although not his marginal gain from defection, holding Silver’s action fixed). To forestall this temptation to defect, Silver must compensate him with reduced shirking. This represents a reduction of his bargaining power. The formal definition of bargaining power is as follows (See [8], p. 456): The bargaining power βi of firm i is i’s marginal gain from switching from coop- eration to defection, relative to the loss this change induces in the opponent’s payoff, holding the action of i’s opponent fixed. The cost of investment matters: each firm’s investment is costly for that

• firm. But cost minimization is not the sole goal of investment. Rather, it is

to maximize net payoffs. Thus, there are interactions between investments and incentives via payoffs and costs.

3One must add the assumption that the Pareto optimal point is always chosen from the

set of equilibria. In the setting that we employ, if there is a single equilibrium on the Pareto frontier of the game, then that point is the unique Pareto dominant equilibrium. In more general settings, equilibria on the Pareto frontier of the game might not Pareto dominate all

• equilibria. See Conley, Chakravarti and Taub (1996) for details.

4The outside opportunities of each firm are incorporated into the payoff function at the

(0, 0) point. So, the payoffs we consider are net of the outside opportunities and stem only from the interactions of the firms.

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What can we conclude? It is possible for a firm which has higher costs to nevertheless make the greater investment because of the larger impact of that investment on the bargaining powers in equilibrium.

3.1 Some technical details of the model

Our model allows us to quantify bargaining power. This in turn enables us to determine the ratio of payoffs between the two firms. As demonstrated in Taub and Kibris [8], if β1 is the bargaining power of Firm 1 and β2 the bargaining power for Firm 2, the equilibrium ratio of their payoffs is

• β1

β2 The simplicity of this formula stems from our central assumption: that the payoffs are structured so that the game frontier is a parallelogram (ibid); see Figure 1. A parallelogram structure is equivalent to requiring that the gain from defection is independent of the actions of the other player. As developed in Taub and Kibris [8], bargaining power is determined by the slopes of the facets of the game frontier: the steeper the slope of a facet of the game frontier, the less is the bargaining power for the player associated with that facet. For example, in Figure 1, Player 1’s bargaining power is linked to the slope of the right facet; his bargaining power is low relative to that of player

• 2. The reason is that his marginal gain from defection is low relative to that of

player 2.

3.2 The link between investment and bargaining power

We equate investment by firm 1 to an outward movement of the right facet, corresponding to an increase in firm 1’s payoffs when firm 2 is cooperating. So, if firm 1 invests I1, then its payoffs increase by I1 when firm 2 is cooperating. We assume that payoffs when firm 2 is not cooperating are unaffected by that investment, reflecting the necessity of the interaction of the firms to obtain the payoffs. This investment leaves the slope of the right facet of the game frontier unaffected—it is a parallel shift, thereby leaving firm 1’s bargaining power unaf-

• fected. Moreover, the resulting game frontier is still a parallelogram. However,

the slope of the upper and lower facets of the game frontier is reduced, so that firm 2’s bargaining power is reduced. If the pirates make their oars and they begin rowing to the island, at each instant some outcome can occur. They can get to the island, which happens at a random time because of random currents and winds, go to the site of the treasure, and dig up the treasure. They can also get to the site of the treasure

• nly to find it gone, the other pirates having gotten there first. Indeed, Flint

and Silver might see the other pirates pull up to the shore of the island before 8

Figure 1: Parallelogram payoff structure

4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

they even arrive. Finally, they might simply continue to row, with none of these

• ther outcomes.

In the game representation of these possibilities, we quantify the potential for the game to end by δ, and correspondingly the probability that it continues until the next round with 1 − δ. As is evident from the pirate story, the outcome for Flint and Silver might be good (they get the treasure) or bad (the other pirates get there first). When the game is modeled with this structure, the probability that the game will continue to the next round, 1−δ, corresponds to a standard discount factor, which in intuitive terms is the patience of the players, and δ can thus be viewed as their impatience. As in the standard repeated prisoner’s dilemma, there is an upper bound on the impatience firms can have in order for cooperation to occur in this game. Thus, δ∗ represents the maximum degree of impatience enabling cooperation in the initial game. Figure 2 includes the ray that constitutes the equilibrium set if the actual value of δ is equal to δ∗, and the shaded area is a measure of the degree to which impatience exceeds that needed to attain cooperation if the actual δ exceeds δ∗. The initial parallelogram has an implicit δ∗, determined by the initial bar- gaining powers as follows (ibid, p.456): δ∗ = 1 −

• β1β2

9

Figure 2: The equilibrium set with δ∗

∆ 0.646447 4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

However, the δ that determines the equilibrium of the game is given exogenously, might not equal δ∗, and cannot be changed. It is determined by technological factors (such as unobservable currents in the strait that speed up or slow down the progress of the boat randomly). We assume that this δ (which we denote δF ) exceeds the δ∗ in the pre-investment situation and therefore, without further modification, no cooperation is possible. Investment achieves this modification. In particular, investment by firm 1 reduces the bargaining power of firm 2. This in turn increases the implied δ∗. Similarly, firm 2 can invest, lowering β1 and thereby increasing δ∗. With sufficient investment by one or both firms, it is possible that δ∗ attains the level of δF , then enabling cooperation. This is formation. The influence of the incentives in the investment stage can result in asym- metry: one firm will invest more than the other, and at higher cost, in order to achieve formation. Also, this firm will achieve payoffs proportionally greater than the payoffs of the other firm. Nevertheless, firms voluntarily hew to this arrangement, because it enables them to cope with the incentive problems and achieve cooperation.

3.3 Mechanics of the model

In our initial experiment, we alter the initial state of the game. Specifically, we move the right facet of the parallelogram in a particular way, namely, rightward 10

in a parallel way and we thus maintain the parallelogram properties. This corresponds to investment by firm 1. We illustrate this in Figure 3. Figure 3: Parallelogram before and after investment by firm 1

∆ 0.646447 4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

Figure 3 shows that because firm 2’s bargaining power has been diluted by the movement of the frontier, the equilibrium ray that could be achieved if δF were small enough has tilted in firm 1’s favor. A mathematical consequence of this structured movement is that the possible payoff combinations that just achieve formation are restricted to a rectangle, and this rectangle is not altered by the change in the initial parallelogram. We denote this rectangle the formation rectangle. This rectangle appears in Figure 4. We divide the procedure for determining the equilibrium into two conceptual

• stages. The first stage is to determine how much investment by firm 1 is needed

to attain the formation rectangle. The second stage is deciding which point on the rectangle is chosen. If the equilibrium payoff combination lies on the horizontal segment of the rectangle, it indicates that the game payoff for firm 2 is fixed. Correspondingly, if the equilibrium payoff combination lies on the vertical segment of the rectangle, it indicates that the game payoff for firm 1 is fixed. Initially we focus on the vertical segment cases. By firm 2 increasing its investment, with firm 1 decreasing its investment so as to remain on the rectangle, then firm 2’s own game payoff can be increased without affecting firm 1’s game payoff. However, firm 2 increases its costs by doing this. 11

Recall that formation is characterized by the equilibrium set being a ray rather than a cone. The point where this ray intersects the formation rectangle is the combination of equilibrium game payoffs. Investment by firm 2 increases the slope of this ray, and therefore increases its own payoff. If firm 2 were acting selfishly, it would continue this process until the marginal increase in its game payoff was just equal to the marginal increase in its costs. If the firms are jointly maximizing profits, as firm 2 increases its investment, it incurs costs. As firm 2 increases its investment, firm 1 must shrink its in- vestment in order to continue to exactly attain the formation rectangle. Firm 2’s investment will continue until the marginal profit for firm 2 (i.e., marginal game payoff minus marginal cost) equals the marginal profit of firm 1 (marginal payoff of firm 1—which is zero because the game payoff for firm 1 is fixed in the vertical segment of the formation rectangle—minus its marginal cost). Because firm 1 is disinvesting, its marginal profit is also positive.

4 The impact of costs

We begin our specific analysis with the case where costs are zero. Our discussion centers around the situation in which firms are maximizing joint profits, but the results also hold when firms selfishly maximize profits. As firm 2 increases its investment, firm 1 must shrink its investment in order to continue to exactly attain the formation rectangle. Firm 2’s investment will continue until the marginal profit for firm 2 (i.e., marginal game payoff) equals the marginal profit

• f firm 1 (marginal payoff of firm 1—which is zero because the game payoff for

firm 1 is fixed in the vertical segment of the formation rectangle). As illustrated in Figure 4, the firms will move up the formation rectangle until they attain its apex. At that apex, as is also evident in Figure 4, the full cooperation point of the parallelogram generated by that combination of investments is attained. One might conjecture that by altering the combination of investments, that the payoff for one firm would be increased at the expense of the other in such a way that the joint payoffs might be increased. But this cannot happen: for incentive reasons, the equilibrium payoffs are constrained to lie on the formation rectangle, and the apex of the rectangle therefore is most efficient equilibrium point (in the sense that the combined payoffs are maximized); alternatively one could note that it is Pareto dominant. We also note that the apex of the formation rectangle is the only point

• n the rectangle at which efficiency is attained in the conventional sense as

well: at other points on the rectangle, the parallelogram for which that point is the equilibrium has a full-cooperation point that is outside the rectangle and therefore unattainable in equilibrium. Thus, full cooperation will not occur at any other equilibrium point.5

5We emphasize that the notion of efficiency we are using in describing the full cooperation

point is that the maximum of the joint payoffs is attained; all the equilibria that lie on the game frontier are of course efficient in the sense of Pareto optimality even though they do not

12

Figure 4: Zero cost equilibrium for joint profit maximization example

∆ 0.75 4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

The apex equilibrium point is determined entirely by the bargaining powers

• f the firms once the appropriate investments have taken place. The sharing

rule of the payoffs is then determined by the ratio of the bargaining powers. We thus have a benchmark theory of contracts. Proposition 4.1 If costs of investment are zero then if firms invest so as to just achieve cooperation, they will invest so that they attain the apex of the formation

• rectangle. At that apex, they play full cooperation in the corresponding game.

Thus, the equilibrium is efficient and Pareto dominant.

4.1 The consequence of positive costs

In reality investment is costly. In investing as they strive to attain the apex of the formation rectangle, the firms’ costs increase. If those costs are convex, the firms will reach a point where their marginal benefits from improved payoffs just equal the marginal costs of investing, and that is the equilibrium. Our main point is that this equilibrium is not at the apex of the formation rectangle, and therefore is inefficient in both of the senses we earlier described. In the extended examples that follow, we explore the details of this situation. We assume that costs are convex, and in our specific examples we assume that costs are quadratic, with values aiI2

i for each firm, and we allow the two

attain this maximum.

13

firms’ cost parameters a1 and a2 to differ. Our first observation is if costs are so high that the marginal cost exceeds the marginal benefit from achieving formation, it will not occur. We do not view this as a flaw of our theory. On the contrary, it provides us with an explanation for why there is not a single mega-firm for the whole economy—some alliances do not form. We next begin our detailed analysis of specific examples. If Flint and Silver were beholden to a more powerful pirate, Blackbeard, just as divisions of a firm are beholden to headquarters, they would be told to make oars that took account of each other’s costs, so as to maximize profits jointly, conditional on their subsequent selfish rowing. We designate this the joint profit maximization case, and examine it first in order to generate a benchmark efficiency outcome. Subsequent to that we consider the situation in which Flint and Silver make

• ars and also row in an entirely selfish way, mirroring how firms contemplating

alliances behave. We denote such situations selfish profit maximization.

5 A basic example of joint profit maximization

We now work out an example of joint profit maximization in which formation

• ccurs. Both firms invest, but asymmetrically and with asymmetric costs; prof-

its are also distributed asymmetrically, and the contract is inefficient in the first best sense. More concretely, firm 2 has more initial bargaining power, but it ends up investing more even though its marginal cost of investment is higher, and yet its profits are higher than firm 1’s profits. While it is true that firm 2

• btains the preponderance of the profits, firm 1 also obtains positive profits.

Figure 5 shows an initial parallelogram. The dotted ray is the equilibrium set that would prevail if the value of δF were low enough, namely the δ∗ value of .65. The surrounding cone expresses the distance needed to attain the equilibrium ray. The parameters for this model are given in Table 1. Table 1: Parameters for the initial example A1CC A2CC A1CD A2DC A1DC A2CD δF a1 a2 3 3

• 1

6 4

• 3

.75 .001 .1 In this example, the initial bargaining powers are .17 and .75, so that firm 2 has more bargaining power initially. The initial value of δ∗—the value of δ that would exactly achieve formation with no other influences—is .65—smaller than the actual value of δ, δF . Therefore no cooperation is possible in the absence of any changes. 14

Figure 5: Initial parallelogram for joint profit maximization example

∆ 0.646447 4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

5.1 Stage 1: Formation

To achieve formation, the value of δ∗ must be increased, and this in turn can be effected by reducing one or both of the bargaining powers of the firms, because δ∗ is determined by the product of the bargaining powers. We execute this reduction of bargaining power in stage 1 of the contract-generating process. Firm 1 invests, moving the right facet of the game frontier parallel to the right. The resulting game frontier is illustrated in Figure 6. Because the movement is parallel, firm 1’s bargaining power is unchanged, but firm 2’s bargaining power is reduced. Firm 1’s investment is chosen so that the reduction in firm 2’s bargaining power is just sufficient so that δ∗ is now .75, the same as the actual δ, δF . This achieves formation. The dashed lines indicate the equilibrium ray that would apply if δF were equal to δ∗. This ray tilts to the right as a result of firm 1’s investment, because the ratio of payoffs is determined by the ratio of bargaining powers, and firm 1’s bargaining power has fallen. Note also that the payoffs of both firms increase where the dashed lines intersect the game frontier, even though only firm 1 has invested. 15

Figure 6: Initial parallelogram for joint profit maximization example with Stage 2 parallelogram

∆ 0.646447 4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

5.2 Stage 2: Optimizing costs conditional on formation

Formation has been achieved. However, firm 1 has done all the investing, so only firm 1 incurs costs. If firm 2 invests, joint costs can be reduced. In the second stage, then, firm 2 invests, and firm 1 disinvests, just sufficiently to maintain

• formation. Because costs are convex, this reduces total costs.

The resulting new parallelogram keeps δ∗ fixed and the actual value of δ

• f .75, but the intersection of the equilibrium ray with the game frontier—the

equilibrium point—traces out a rectangle as different combinations of invest- ment are chosen. The issue then is to find the point on this rectangle that maximizes the combined payoffs of the firms less their costs. Figure 7 illustrates the resulting equilibrium. The smaller parallelogram is the initial set of payoffs, and the larger parallelogram is the final equilibrium set of payoffs. The formation rectangle is depicted, and the steeper of the rays is the final equilibrium set. The result of the stage-2 investment is that the bargaining powers of both firms have fallen, but firm 2, which had the larger initial bargaining power, gets a bigger share of the equilibrium payoffs: the equilibrium ray has tilted back

• up. Those payoffs have risen for both firms however: See Table 2. Also, firm 2

does the lion’s share of the investing, even though its costs are higher, but the resulting increase in its payoffs compensates for this. Its profits are positive as 16

a result. Figure 7: Equilibrium for joint profit maximization example

4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

Table 2: Outcomes for the initial example Initial bargain- ing power Final bargain- ing power Investment Payoff Profit Firm 1 .17 .10 .92 3.00 2.99 Firm 2 .75 .61 3.75 7.32 5.91 We emphasize at this point that formation does not completely overcome adverse incentives: if the firms could attain the cooperation point at the apex

• f the game frontier of the final parallelogram (the larger of the parallelograms

in Figure 7) which has payoffs of 3.92 and 6.75 for the respective firms, the total payoffs would increase by .35. Thus, a substantial degree of inefficiency remains. We have the following proposition: Proposition 5.1 (i) In achieving formation, the weaker firm will be made even weaker by the investment of the stronger firm. (ii) The contract will be inefficient in general: the cusp of the game is not in general attained. Corollary 5.2 The ratio of the equilibrium payoffs will tilt toward the firm with the initial greater bargaining power. 17

5.3 The effect of costs

By increasing firm 1’s cost parameter to be the same as firm 2’s cost parameter,

• ne would expect that firm 1’s investment will be reduced relative to firm 2’s

investment. That will increase firm 2’s final relative bargaining power, but because costs cannot be reduced by substituting toward the low-cost firm, overall joint profit can shrink and this is what we observe. Because firm 1’s cost parameter has increased, its relative investment shrinks and firm 2’s investment increases both relatively and in absolute terms. Because firm 2 has higher bargaining power, it shoulders more of the investment in order to shrink firm 1’s bargaining power and thus the product of the bargaining powers, to enable formation to occur; indeed, total investment must rise for this to occur. As a result, its payoffs tilt in its favor. But when costs are accounted for, its profits actually shrink in absolute terms. Figure 8 and Table 4 shows that the equilibrium payoffs are shifted only

• slightly. Most of the effects on profit work through costs: firm 2’s relative costs

have shrunk, and as a result its investment has increased. The parameters for this model are given in Table 3. Table 3: Parameters for the initial example with modified costs A1CC A2CC A1CD A2DC A1DC A2CD δ a1 a2 3 3

• 1

6 4

• 3

.75 .1 .1 Table 4: Outcomes for the initial example with modified costs Initial bargain- ing power Final bargain- ing power Investment Payoff Profit Firm 1 .17 .10 .92 3.00 2.94 Firm 2 .75 .63 4.10 7.58 5.89 It is now evident that if firm 1’s costs are too high, formation cannot be achieved. Even if firm 1 invests enough to reach the formation rectangle in stage 1, then shrinks its investment in stage 2, its equilibrium net profit will still be negative, so it will forego the contract. Proposition 5.3 There exists a threshold of costs such that formation is blocked.

5.4 Increasing the initial relative bargaining power of firm 1

Keeping the costs the same for both firms, we now alter the initial parallelogram so that both firm 1’s initial bargaining power is increased, and firm 2’s initial 18

Figure 8: Equilibrium for joint profit maximization example with modified costs

4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

bargaining power is decreased, in such a way that the product of the bargain- ing powers increases. The net result is that the “disequilibrium” cone widens, indicating that it has been made even more difficult to achieve formation; see Figure 9. The parameters for this model are given in Table 5. Table 5: Parameters for the joint profit maximization example with modified bargaining power A1CC A2CC A1CD A2DC A1DC A2CD δ a1 a2 3 3

• 1.5

6 4.5

• 3

.75 .1 .1 The equilibrium is illustrated in Figure 10, with outcomes listed in Table 6. Firm 2’s profit is lower relative to the more asymmetric bargaining power example, and firm 2 pulls a heavier load of investment relative to firm 1. The problem is that although firm 2’s bargaining power has declined in both absolute and relative terms, the product of the bargaining powers has still risen, from .1275 to .134. The consequence is that both firms must do more investing to attain formation—a visual cue is that the formation rectangle has enlarged. We summarize these observations with the following proposition: 19

Figure 9: Initial parallelogram for the joint profit maximization example with modified bargaining power

∆ 0.591752 4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

Table 6: Outcomes for the joint profit maximization example with modified bargaining power Initial bargain- ing power Final bargain- ing power Investment Payoff Profit Firm 1 .25 .14 2.06 4.5 4.08 Firm 2 .67 .46 4.98 8.23 5.76 Proposition 5.4 The higher the bargaining power of firm 2 relative to firm 1, the higher the investment of firm 2 relative to firm 1’s investment. Yet despite the adverse effect of increased total bargaining power, joint prof- its are ultimately higher than in the previous case. This is because the in- vestment can be more equally shared between the two firms, and the costs are therefore more equal. Because costs are proportional to the square of invest- ment, more evenly shared investment reduces overall cost and joint net profits are increased. We summarize this in the following proposition. Proposition 5.5 Making the relative bargaining powers more equal results in more evenly shared investment and raises joint profit. 20

Figure 10: Initial parallelogram for joint profit maximization example, with reduced firm 2 relative bargaining power

4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

5.5 Summary

In summary, in this example and its associated variations, formation could not be achieved without investment. The firm with the greater bargaining power, firm 2, undertakes most of the investment, which has the effect of weakening firm 1’s bargaining power sufficiently to achieve formation. Profits then tilt toward firm 2 as a result of this alteration of bargaining power, but this tilt is partially offset by the costs incurred by firm 2’s investment.

6 A basic example of selfish profit maximization

In this section we repeat the illustrative examples of the previous section. How- ever, we drop the assumption of joint profit maximization, and assume that each firm maximizes its own profit without regard for the profit of the other firm. We begin with the same payoffs as in the initial example illustrated in Figure

• 5. However, in stage 2 of the game, when calculating profit, we only maximize

the net profit of firm 2. The result is illustrated in Figure 11. The numerical outcomes are stated in Table 7. The outcomes are not de- tectably changed relative to the joint profit maximization case; see Table 2. This is because firm 1’s costs are virtually nil. Therefore in stage 2, changing firm 2’s investment hardly affects firm 1’s costs. 21

Figure 11: Equilibrium for selfish profit maximization example

4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

Table 7: Outcomes for the initial example—selfish optimization Initial bargain- ing power Final bargain- ing power Investment Payoff Profit Firm 1 .17 .10 .92 3.00 2.99 Firm 2 .75 .61 3.75 7.32 5.91 We have the following proposition and corollary, similar to the parallel propo- sitions 5.1 and 5.2; indeed the qualitative conclusions are identical: Proposition 6.1 (i) In achieving formation, the weaker firm will be made even weaker by the investment of the stronger firm. (ii) The contract will be inefficient in general: the cusp of the game is not in general attained. Corollary 6.2 The ratio of the equilibrium payoffs will tilt toward the firm with the initial greater bargaining power.

6.1 Equal costs—the selfish case

If we make firm 2’s costs the same as firm 1’s costs, the situation changes; Table 8 shows that firm 2’s investment decreases, and so do its payoffs and net 22

• profits. (The changes in the payoff outcomes are so minor that the plot of the

equilibrium looks virtually identical to to Figure 7.) However, total investment by both firms decreases, because firm 2 cuts back its investment (see Table 4 for comparison). Thus, firm 1 is forced to bear a greater part of the investment burden in the selfish case. In the joint profit maximization case, firm 2’s impact on firm 1’s profit was taken into account, and this caused it to invest more; here, firm 2 only accounts for the impact of investment on its own profit, and therefore invests less. Therefore total profits fall relative to the joint optimization problem—overall efficiency is therefore

• lower. In addition the distribution of profits across firms is affected: the share
• f profits tilts toward firm 2, the firm with the greater initial bargaining power.

Table 8: Outcomes for the initial example—selfish optimization Initial bargain- ing power Final bargain- ing power Investment Payoff Profit Firm 1 .17 .10 .92 3.00 2.91 Firm 2 .75 .61 3.75 7.31 5.91 We summarize this finding with a proposition: Proposition 6.3 Relative to the joint profit maximization case, under selfish profit maximization overall investment is reduced, relative investment shifts to- ward the firm with lower bargaining power, total profits decrease, and relative profits tilt toward the firm with the greater bargaining power.

6.2 Increasing firm 1’s initial relative bargaining power— the selfish case

Next, we maintain the assumption of equal costs but increase the initial bar- gaining power of firm 1. The outcomes are presented in Table 9. Table 9: Outcomes for the selfish profit maximization example with modified bargaining power Initial bargain- ing power Final bargain- ing power Investment Payoff Profit Firm 1 .25 .15 2.88 4.5 3.67 Firm 2 .67 .41 3.75 7.31 5.91 Figure 12 illustrates this example. We first note that if the bargaining powers

• f the two firms are initially relatively equal, profits increase significantly for

the weaker firm in both the joint and selfish optimization cases: this a simple extension of Corollary 5.2. 23

We are now interested in comparing the impact of greater equality of initial bargaining powers in the selfish versus the joint case. Again, investment by both firms decreases relative to the joint profit maximization case, especially that of firm 2—compare Table 9 with Table 6. This reduces its payoffs, but increases its profits. However, total profits fall relative to the joint optimization case, because firm 1’s profits shrink more than firm 2’s profits increase. We thus have an extension of Proposition 5.5. Proposition 6.4 Making the relative bargaining powers more equal results in more evenly shared investment and raises total profit. Relative to the joint profit maximization case, investment is more even across firms but total profits are lower. Figure 12: Initial parallelogram for selfish profit maximization example, with reduced firm 2 relative bargaining power

4 2 2 4 6 8 10 Payoff 1 4 2 2 4 6 8 10 Payoff 2

There are several effects that explain the contrast between Table 9 and Table 6, as also illustrated in Figures 10 and 12. First, regardless of bargaining powers, investment by firm 2 expands the payoff frontier—the parallogram—in its favor. Even in the selfish case, this expansion also benefits the other firm. But in addition, the investing firm’s relative bargaining power increases. This tilts the equilibrium ray in its favor. The tilt of the equilibrium ray benefits firm 2 while firm 1 is harmed, moderating and potentially reversing the beneficial impact of the expansion of the payoff set on firm 1. 24

However, firm 1 can now respond with investment of its own. When costs are equal and bargaining power is relatively equal, firm 1’s investment also expands the equilibrium set so as to benefit both firms, but drives the equilibrium ray back in its own favor. In the end, the payoff set has been expanded by both firms, but the equilibrium ray is little affected. All of these compound effects occur in both the joint and selfish cases, but In the joint profit maximization case, the impacts of investment on the other firm are internalized, whereas in the selfish case they are not, leading to a different, and less efficient outcome.

6.3 Formation is more sensitive to cost in the selfish profit maximization case

Finally, we note that because incentives cannot be diluted by pooling profits in the selfish case, formation will be harder to achieve, that is, the threshold

• f costs at which formation is blocked is lower. We summarize this with the

following proposition: Proposition 6.5 The threshold of cost at which formation is blocked is lower under selfish profit maximization than under joint profit maximization. This has implications for the theory of firm formation. First of all, we remind the reader that the distinction between the joint and selfish cases has to do with the investment stage: in the selfish case, each firm invests so as to maximize its own profits in the rowing stage, and in the joint profit maximization case, the firms acknowledge their incentives to shirk in the rowing stage, but invest to maximize joint profits conditional on this shirking. For very low costs, formation can always occur. For very high costs, no formation can occur whether profits are shared or not. But there is an interme- diate range of costs where formation can occur under joint profit maximization, but not under selfish profit maximization. In this range, an organizational ar- rangement that induces joint profit maximization can result in formation. We explore potential such arrangements below. We observe that joint profit maximization always produces total profits that are at least as high as the profits from selfish optimization, because it is feasible for the joint profit maximization to attain the selfish optimization. However, the distribution of the profits in the joint case might be such that one of the firms would be harmed relative to the selfish profit it could otherwise attain, and would therefore refuse to enter into a joint profit maximization contract. It is certainly possible for that firm to be offered an initial side payment to compensate it ex ante for this anticipated harm. But if firms can renege on the investment that is promised in return for the side payment, this strategy will fail, and a merger or buyout is the only remaining way to achieve formation.6

6Bertrand and Mullainathan (2001) note that this selfish optimization occurs in the context

• f CEO compensation: compensation contracts are designed and executed that assume that,

because of monitoring or other problems, pay cannot be keyed to performance. In our context,

25

7 Conclusions

We have a theory of why firms sometimes cooperate with each other, an expla- nation for why they might fail to agree to cooperate at all, and the inefficiency that can accompany cooperation when it does occur. The theory rests on in- centives, and resides fully in a dynamic framework that is necessary to account for those incentives. Our theory segments firm cooperation into three categories which are delin- eated by parameters of cost and payoff structure. When costs are low enough, firms will form alliances in which they act selfishly. Our evidence is that the alliances will in general be inefficient in the sense that they do not maximize joint output, even though there is no physical impediment to this maximiza-

• tion. For costs that are high enough firms will fail to form alliances. But if

they maximize joint profits, as they would after a merger, they can overcome the adverse incentives and achieve formation. Finally, costs might be so high that formation is impossible: neither an alliance nor a merger can be achieved. In the real world firm alliance contracts have significant asymmetries in investment and in the division of profits. Our theory rationalizes these asym- metries and the diversity of contract structure. It has tight predictions about contract structure: it prescribes the investments required of each firm to achieve formation, and dictates a clear—and not necessarily symmetric—division of pay-

• ffs.

The empirical correlates of our model can be measured. It is straightforward to measure costs, profits and investment. Any empirical test must also have measures of bargaining power, which is at the heart of the model. Bargaining power is determined by the structure of the parallelogram, which in turn reflects the payoffs stemming from the actions of the firms. These payoffs can also be measured: they are the gains and losses from cooperative and noncooperative

• actions. We are therefore confident that our theory can be put to the test.

the issue is distinguishing the actions of the two firms (one from the other) in the rowing stage: if this is difficult, the payoffs cannot be appropriately redistributed.

26

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