partial ownership arrangements and the potential for collusion

Partial Ownership Arrangements and the Potential for CollusionAuthor(s): David ReitmanSource: The Journal of Industrial Economics, Vol. 42, No. 3 (Sep., 1994), pp. 313-322Published by: WileyStable URL: http://www.jstor.org/stable/2950573 .

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THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821 $2.00 Volume XLII September 1994 No. 3

PARTIAL OWNERSHIP ARRANGEMENTS AND THE POTENTIAL FOR COLLUSION*

DAVID REITMAN

Firms can form partial ownership arrangements by purchasing claims to competitor's profits in order to commit to less aggressive competition. These arrangements can increase profits for all firms in the industry even in the absence of synergies. Using a conjectural variations model, we show that with more than two symmetric firms engaged in quantity competition or with more cooperative conjectures, partial ownership arrangements are never individually rational for all participants. Conversely, with conjectures that lead to more rivalrous equilibria than Cournot, there exist individually rational partial ownership arrangements with any number of firms in the industry.

I. INTRODUCTION

WHEN one firm makes capital, technology, or other forms of expertise available to another firm, it often is compensated through partial ownership of the receiving firm, rather than through, for example, a fixed payment or royalty. Such partial ownership arrangements (POAs) may be beneficial to society if they encourage firms to exchange expertise or assets that would otherwise not be made available. However, a society that permits POAs in order to exploit these potential gains faces the possibility that some POAs will be formed without the motivation of exploiting synergies, possibly resulting in larger profits without any counterbalancing increase in efficiency. Just as firms can frequently claim that horizontal mergers are motivated by efficiency gains even when the primary motivation is to gain market power,1 one can also question whether some partial ownership arrangements may arise primarily to increase market power.

This paper examines whether, in the absence of bilateral gains from complementary technology or expertise, firms may still have an incentive to form POAs. Firms are assumed to have symmetric costs and technology, which removes any basis for synergies. Thus the essential remaining feature of POAs is that firms have joint claims to the profits of their products. In general, having an interest in the profits of other products leads to higher industry profits overall; the question is whether this results in increased profits to the firms that participate in the POA.

* I would like to thank Reiko Aoki, Bill Boal, Paul Evans, John Hillas, Yair Tauman, and Steve Turnbull for helpful discussions, and the editor and two referees for suggestions that have significantly improved the paper.

' Fisher [1987] elaborates this point.

? Basil Blackwell Ltd. 1994, 108 Cowley Road, Oxford OX4 IJF, UK and 238 Main Street, Cambridge, MA 02142, USA.

313



314 DAVID REITMAN

As Reynolds and Snapp [1986] have shown in the case of quantity competition, joint ownership will result in more collusive outcomes among non cooperative firms. To take the simplest example, suppose symmetric Cournot duopolists each own 50%O of the profits from its competitor's product. In choosing its own strategy, each firm's objective will be to maximize the sum of the two firms' profits, and will choose the collusive output level in equilibrium. A similar result holds with N firms in the industry, as long as each firm retains only a 1/N share of its own profits. In general, POAs allow firms credibly to commit to competing less aggressively in the product market, resulting in higher prices and higher aggregate profits.

What is unclear is whether enough of the higher industry profits accrue to those firms forming the POA to make it worthwhile. As it turns out, the results for POAs among Cournot firms with symmetric costs are quite similar to those obtained by Salant, Switzer, and Reynolds [1983] for mergers. With three or more firms, no POAs are formed in equilibrium. As with mergers, there is a positive externality bestowed on other firms when a POA is formed, because the POA raises the market price; the primary beneficiaries are those firms that do not participate.2 The same result holds under more collusive market conjectures.

However, with conjectures that result in more rivalrous competition than the Cournot case, firms may be willing to form POAs solely to decrease the level of competition and increase profits. In general, for any number of firms in the market, there exists an individually rational equilibrium with POAs if the market conjectures are sufficiently close to price competition and if the total shares of ownership transferred are sufficiently large.

II. THE MODEL

Assume there are N firms competing in the market. The firms face a linear demand curve P = A - Q, where Q is aggregate output. Firm i produces qi, with Q = EN 1 qi. In order to focus on the market power rationale for POAs, assume that firms have identical and constant marginal costs of production, c, with c < A. It will be convenient to use notation as if the marginal costs of production were zero; for this we define a = A - c.

Following Kwoka [1992], we use a conjectural variations approach to model a range of degrees of rivalry in the product market. We need to go a step beyond Kwoka in assuming that the conjectured response of each firm to a change in a rival's output is identical. Let v = aqjlaqi for all i and j. Now define V = (N - l)v. As in Kwoka, V = Ej iaqjlaqi is the common

2This statement presumes there are no potential synergies, as studied in Farrell and Shapiro [1990a]. The externality from horizontal mergers was first discussed by Stigler [1950].

? Basil Blackwell Ltd. 1994.



OWNERSHIP ARRANGEMENTS AND COLLUSION 315

beliefs for each firm about its competitors' response to a change in output. This reduces to Cournot conjectures when V = 0. If V = -1, the model results -in the competitive equilibrium, while values of V e (- 1,0) result in equilibria "between" Cournot and Bertrand. Positive values of V yield equilibria more cooperative than the Cournot equilibrium.

Before firms choose their quantities, they may form POAs. In practice, joint ownership can lead to many different mechanisms for determining the competitive strategy of the jointly controlled product. (Bresnahan and Salop [1986] describe a number of possibilities, together with their implications for concentration measures.) Throughout this paper, only the simplest arrangement is used: each firm retains the right to set output levels for its own product, regardless of what share of its product market profits have been transferred to other firms. In other words, partial ownership confers no rights of control; this is perhaps the most natural way to capture the semantic distinction between partial ownership and a joint venture. There is no limit placed on the number of firms that can have a profit claim on any other firm. All POAs are assumed to be formed simultaneously. After any POAs are formed, firms compete in the product market.

To capture the effect of partial ownership of competitors' profits on competition, we need to add some additional notation to the model. Let a be the fraction of firm i's profit owned by firm j. Also define ci = J caij to be the total fraction of firm i's profit owned by other firms and f3i = EYaja to be the total shares of other firms' profits owned by firm i. Define S = 1i Xij to be the total shares sold in the industry.

Firm j pays firm i for the rights to a share of its profits; this transfer price is given by tij.' The claims on profits sold to other firms apply only to the short run profits earned in the final stage of the game and do not include fixed costs or any net profits from selling shares. Given all existing POAs, firm i's profit as a function of its output and the output of its competitors is

(1) 7i = (1 - ai)qi(a - Q) + E ejiqj(a - Q) + E (tij - tji) j#i j#i

The first order condition for qi can be written as

(2) ~~(a -Q) + v/3[ (a -Q) 1 (2) 4~'i = I + V + (I+V(la)l- ji qj 1+ (1+ V)(1- (xi) I j "

As long as V < 0, qi is decreasing in aji. This demonstrates the intuition that firms will produce less aggressively if they own shares in the profits of

3In practice, this transfer need not be a cash payment, but any sort of contribution to production, marketing, etc., that does not affect the variable costs of production. The results also apply if firms simply buy stock in their competitors, without actually contributing anything more than capital to production.




316 DAVID REITMAN

competing firms. The smaller the share of its own profits retained by a firm, the more it will decrease its own output when it buys shares of a competitor. This suggests that firms would prefer to sell their shares to another firm that retains a relatively small share of its own profit. The first order condition only holds with equality, of course, if qi is non-negative. If firm i owns enough shares in enough of its competitors, it will not produce anything and will just collect its share of its partners' profits. However, it can be shown that being at a strict corner solution is never optimal.

It turns out to be quite difficult in general to solve (2) for each firm so as to obtain the equilibrium output choices, given the network of POAs. Fortunately, the equilibrium profits are much simpler to obtain. We get the following result:

Proposition 1. Given c.i1 for all i, j E N, profit for firm i is

(-(1 - i + vfli)(1 + V)a2 (3) ( + 1+V S(1 -v)2 + Z (tj- tji) '

N+I + V_ ( - -ua2 joi i

Proof: Substituting (2) into (1), all the terms involving qj cancel, leaving

(4) (1 - cxi + vfli)(a -Q) + F (tJ - tji)

Aggregate profit is 1irri = (N - S(1 - v))(a - Q)2/(1 + V). Aggregate profit is just the total market output times (P - c); all the shares of profits of all the firms are accounted for exactly once. Thus we have Q(a - Q) = (N - S(1 - v))(a - Q)2/(1 + V). Solving gives (a - Q) =

a(1 + V)/(N + 1 - S(1 - v)), and substituting this in (4) gives (3). i

Each firm's profit depends on the total fraction of the profits from its own product that it sells to other firms and the total shares of other firms' profit owned, as well as the total number of shares sold in the industry. The identity of the firm from whom shares are bought or sold does not matter.4 For the symmetric case, with a = ai = fi for all i and S Na, (3) becomes zt = (1 - cx(t - v))(1 + V)a2/(N + 1 + V - Nac(l - v))2. This function is increasing in x up to x = (N - 1)/N, where profits are maximized and each firm earns an equal share of the monopoly profit. Firms can earn the collusive level of profits if each firm sells off all but a 1/N share of its own profits; it

4It should be noted that proposition 1 is essentially the same with a general demand curve, D(Q). The proof goes through exactly as with a linear demand curve up to the point of solving for the equilibrium value of Q. In equilibrium, Q satisfies (D(Q) - c)(N - S(l - v)) + QD'(Q)(1 + V)D'(Q) = 0. This is the same first order condition that characterizes market output in the symmetric equilibrium of the market without POAs and with (N - S(1 - v)) firms in the industry (ignoring integer problems). Thus the equilibrium profits for firm i in the POA game equal (1 - ai + vf5i) times the Cournot profits with (N - S(I - v)) symmetric firms. For the case of linear demand, this reduces to equation (3). ? Basil Blackwell Ltd. 1994.




does not matter if each firm sells the remainder all to one other firm or equal shares to each firm, or any other combination.

It is clear that there are aggregate gains from forming POAs, with profits ranging up to the joint profit maximizing level attainable for an appropriate combination of POAs regardless of the value of V But (3) also shows that POAs face the same problem in general as horizontal mergers among firms with symmetric linear cost functions: Forming a POA creates a positive externality accruing to firms that are not involved. Participating firms reduce their total output, raising the market price for everyone.5

With Cournot competition, Farrell and Shapiro [I990b] show that, under fairly general conditions, partial ownership will only be profitable if the acquiring firm has a smaller market share than its partner. At first glance this would seem to rule out POAs among firms with symmetric technologies. However, one effect of acquiring partial ownership of other firms is to reduce the output of one's own product. Thus it is possible that firms will form multiple POAs, each one of which is rational given the others. This network of POAs would internalize some of the externalities resulting from each POA.

Define a POA equilibrium to be a network of POAs within an industry such that, given the subsequent profits from product market competition, all POAs are individually rational for all firms, given the other POAs in the market. Here individually rational means that no firm would want to pull out of one or more of the POAs in which it is engaged either as a buyer or a seller. The rest of this section describes some circumstances under which POA equilibria can arise.

Although the profitability of a single POA for the buyer and seller of that ownership share depends on the transfer price for that share, it is not necessary to compute the transfer price to determine if the transaction is individually rational. If the joint profits of the two firms involved in the POA increase, there will be a transfer price such that both the buyer and seller come out ahead from the POA. Conversely, if joint profits fall, then regardless of the transfer price, either the buyer or the seller will be better off not engaging in that POA.

Suppose that firm I sells an a12 = a share of its profit to firm 2. Let cxi and Pi1 be the shares bought and sold by firm i after firms 1 and 2 form their POA, and let S also be computed after this POA is formed. Assume that I - a, + vf1i > 0 for all i. From (3), joint profits for firms 1 and 2 are H1 + H2 = (2 - a- a2 + vfl1 + vf2)(1 + V)a2/(N + 1 + V - S(1 - v))2. If firms I and 2 do not form a POA, they receive H1 + H2 = (2 -al- a2 + vfll + vf2 + az(1 - v))(1 + V)a2/(N + 1 + V - (s - a)(1 - v))2. Comparing the two gives the following result:

5This externality is irrelevant with two firms in the industry, of course, since there are no outside firms; all POAs (up to joint profit maximization) are profitable in a duopoly.




318 DAVID REITMAN

Proposition 2. A POA in which firm 1 sells a12 to firm 2 will be mutually beneficial to firms 1 and 2 if and only if

(5) > (N+ I + V-S(I -v))(N-3+ V-S(l-v)+axl +c2-vfll-vfl2) O,12 (1- v)(2 - al- 2 + vtl) + vf2)

When N = 3, the second term on the right side of(5) reduces to V - a3 + v03, which is negative when V < 0. Thus:

Corollary 1. With N = 3 and V < 0, all POAs (up to the point of joint profit maximization) are profitable.

With V > 0, Proposition 2 implies that the first POA formed in the market will not be profitable for the participants. However, that does not eliminate the possibility that a collection of POAs could form, each of which is individually rational given all the other POAs in the industry. But this cannot occur, as the following proposition states.

Proposition 3. With N = 3 and V > 0, no set of POAs is individually rational for all firms.

Proof: If, without loss of generality, a, < a2 < a3, it can be shown that (5) can only be sastisfied for all POAs if firm 1 buys all of the offered shares of firms 2 and 3, firm 2 buys all of the offered shares of firm 1, al < a2, and a2 = x3 only if ox1 = 0. If the transfer payments are such that firm 1 does as well as is possible, given individual rationality for the other firms, the resulting profits for firm 1 are less than the profits earned if it does not participte in any POAs. Thus no combination of POAs is individually rational for all firms. U

Profits in this case are convex in S, so a second POA can be profitable after the first one has formed. But as long as individual rationality means that firms can pull out of any or all POAs simultaneously, there cannot be a POA equilibrium in an industry characterized by Cournot or more cooperative conjectural variations and more than two firms.

We now turn to the case of four or more firms. Unlike the three firm case, the first POA formed in the industry is, by itself, never profitable for any V.

Proposition 4. With N > 4, there cannot be a POA equilibrium with a single POA.

Proof: If no other POAs have formed in the industry, then two firms will find it profitable to transfer a share a of profits from one to the other if and only if

(6) 2 - oI-a(1-v) 2 (N+ 1 + V-a(1-v))2 (N+ 1 + V)2

i Basil Blackwell Ltd. 1994.




Solving for a gives

-7 ot <4(N + 1 + V) - (N + 1 + V)2 (7) a < - 2(1-v)

A necessary condition for the POA to be profitable is that N + 1 + V < 4, or N < 3- V,which is never true for N > 4 since V >-1. U

Once again, this proposition does not rule out the possibility of a POA equilibrium with a number of POAs forming simultaneously. In fact, such POA equilibria are possible with V < 0. Rather than trying to characterize all such POAs, the next several propositions illustrate the range of possibilities.

Consider first the symmetric equilibrium in which the firms in an industry form a "ring" of POAs: each firm buys an a share of the profits of one other firm, and for the same price each firm sells an a share of its profits to one other firm (the situation in which each firm buys an a share of the profits of each of the other firms gives virtually identical results). In order for such an arrangement to form a POA equilibrium, the condition in Proposition 2 must be satisfied. Moreover, each firm must prefer not to simultaneously drop out of the POA in which it participates as a buyer and the POA in which it participates as a seller.

For the latter condition, it is sufficient to look at the profits for one firm before and after dropping out since the transfer prices of the two transactions are equal. Each firm prefers to participate in its POAs rather than drop out of both of them if and only if

(1 - a (I - v))(I + V)a2 (1 + V)a2 (N + I + V - Na(l - v))2

> (N + I + V - (N - 2)CX(1 - v))2

This will be satisfied when

N2 _3N)+NV-2V-D1 N2 _3N+NV-2V+D1

(I (-v)(N - 4N + 4) (I (-v)(N - 4N + 4)J

where D1 = 2 2V-VN-N + 3. Meanwhile, Proposition 2 will be satisfied when

(N2-3N+NV-2V-1-D2 N2 -3N+NV-2V-t+D28

) (l-v)(N2-4N + 2) ' (1-v)(N2-4N + 2) J

where D2 = y/2V2 + 8V - 2VN - 2N + 7. Substituting into (9) and (10) gives the next two propositions.

Proposition 5. For any N, there exist V and a such that the set of POAs in which every firm owns an a share of one other firm constitutes a POA equilibrium. ? Basil Blackwell Ltd. 1994.



320 DAVID REITMAN

Proof: The proof is by construction. Let V = (2-N)/N. Both (9) and (10) are satisfied when a = (N2- 4N + 3/2 - I/N)/(l - v)(N2 - 4N + 2). U

Proposition 6. For N = 4, when V is sufficiently close to -1, the set of POAs in which every firm owns an oc share of one other firm constitutes a POA equilibrium for arbitrarily small a.

Proof: Substituting V = -1 gives ao(1 - v) e (0, 1) in both (9) and (10). E

In general, a set of POAs is more likely to form a POA equilibrium as V gets closer to - 1, the total ownership shares transferred are large, and if there are relatively few firms. While the ring arrangement of POAs is convenient for proving existence of POA equilibria, it is not something that is likely to be observed in practice. A more realistic kind of arrangement is if firms pair off, with one firm owning an a share of its partner, so that no firm is both a buyer and a seller of ownership shares. The next proposition shows that arrangements of this type can also constitute POA equilibria.

Proposition 7. For N = 4, the set of POAs in which each of two firms sell an x share of its profits to one of the other two firms (so that each firm is either a buyer or a seller, but not both) will form a POA equilibrium when

11 ~ ~~ 2+ V - 1- 0V~ a )

- a 1v '1vJ

Proof: Only the condition in Proposition 2 is relevant now, since each firm only participates in one POA. Substitution in (5) gives the result. V

There will exist POA equilibria of this form as long as -1 < V < /2 - 2 -0.586. As in the previous proposition, if V is sufficiently close to - 1, a can be arbitrarily small.

III. DISCUSSION

The results for partial ownership arrangements contrast sharply with those of Kwoka [1992] for joint ventures when firms have symmetric and linear cost functions. While POAs will only be formed in markets that are more rivalrous than Cournot competition, Kwoka showed that joint ventures are more profitable (and under some assumptions about objectives, only profitable) in industries with cooperative conjectures. The key difference is that a joint venture involves the creation of a new firm whose additional profits can be sufficient to compensate for the externality induced by the joint venture. Note that the welfare implications for POAs are more stark than those in Kwoka: any POA reduces aggregate output, resulting in higher prices, lower consumer surplus, and a larger deadweight loss from monopoly power. ? Basil Blackwell Ltd. 1994.




The model here is biased against the formation of POAs by the assumption that conjectural variations, even between partners in a POA, are unchanged when- a POA is formed. An alternative possibility is that firms anticipate more cooperative outcomes in the product market when they are linked together through partial ownership. This issue is discussed in Kwoka [1992] in the context of joint ventures and would lead to similar results in the context of POAs: if forming POAs induces firms to compete less aggressively by changing expectations beyond the lessening of competition induced by joint ownership, then POAs would be profitable under a broader range of circumstances.

The symmetric linear cost structure assumed here also biases the model against the formation of POAs. Firms with asymmetric constant marginal costs will sometimes find it profitable to form mergers in a Cournot oligopoly, even without merging to a near monopoly (see, for example, Levin [1990]). In these circumstances, POAs can also be profitable. In addition, Perry and Porter [1985] show that, with an increased marginal cost technology, Cournot firms may have an incentive to merge that is overlooked in linear cost models. Introducing increasing marginal costs will also in some cases make POAs among Cournot firms profitable.6

POAs have several advantages over mergers as a tool for increasing market power that are not captured in simple oligopoly models. In particular, multiproduct firms that overlap in some but not all of their various markets may be unwilling to merge their entire product lines. POAs enable such firms to collect much or all of the benefit from merging in just these overlapping products without the radical restructuring implied by a merger. Furthermore, managers who wish to profit from the benefits of mergers but fear the loss of control that may ensue may prefer to retain the autonomy of their firms by forming POAs rather than mergers. Thus, to the extent that market conditions are favorable to mergers, having the option to form POAs may result in a larger decrease in competitiveness in the market.

DAVID REITMAN ACCEPTED DECEMBER 1993

Department of Economics, 410 Arps Hall, Ohio State University Columbus, OH 43210-1172, USA.

6 Suppose V = 0, and consider the extreme case in which marginal cost becomes infinite above the Cournot level of output (this is the short run technology used in Kreps and Scheinkman [1983]), if two firms in a three-firm industry consider forming a POA, they will be in essence duopolists facing the residual demand after the third competitor has produced the Cournot quantity and, since there is no longer a positive externality from reducing output, will in equilibrium serve 1/2 rather than 2/3 of the residual demand through a POA.




322 DAVID REITMAN

REFERENCES

BRESNAHAN, T. and SALOP, S., 1986, 'Quantifying the Competitive Effects of Production Joint Ventures,' International Journal of Industrial Organization, 4, pp. 155-175.

FARRELL, J. and SHAPIRO, C., 1990a, 'Horizontal Mergers: An Equilibrium Analysis,' American Economic Review, 80, pp. 107-126.

FARRELL, J. and SHAPIRO, C., 1990b, 'Asset Ownership and Market Structure in Oligopoly,' Rand Journal of Economics, 21, pp. 275-292.

FISHER, F. M., 1987, 'Horizontal Mergers: Triage and Treatment,' Journal of Economic Perspectives, 1, pp. 23-40.

KREPS, D. M. and SCHEINKMAN, J., 1983, 'Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes,' Bell Journal of Economics, 14, pp. 326-337.

KWOKA, J. E., 1992, 'The Output and Profit Effects of Horizontal Joint Ventures,' Journal of Industrial Economics, 40, pp. 325-338.

LEVIN, D., 1990, 'Horizontal Mergers: the 50 Percent Bench-Mark,' American Economic Review, 80, pp. 1238-1245.

PERRY, M. and PORTER, R., 1985, 'Oligopoly and the Incentive for Horizontal Merger,' American Economic Review, 75, pp. 219-227.

REYNOLDS, R. and SNAPP, B., 1986, 'The Competitive Effects of Partial Equity Interests and Joint Ventures,' International Journal of Industrial Organization, 4, pp. 141-153.

SALANT, S., SWITZER, S. and REYNOLDS, R., 1983, 'Losses Due to Merger: The Effect of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium,' Quarterly Journal of Economics, 98, pp. 185-199.

STIGLER, G., 1950, 'Monopoly and Oligopoly by Merger,' American Economic Review, 40, pp. 23-34.




partial ownership arrangements and the potential for collusion

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