Theory and Decision (2009) 66:373–399 DOI 10.1007/s11238-007-9071-1
© Springer 2007
JINGANG ZHAO
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
ABSTRACT. This article provides a method for estimating the bounds of transaction costs in horizontal mergers. Consider, for example, a completed monopoly merger in linear Cournot oligopolies with 10 symmetric firms. The method shows that its transaction costs are at most 25% (78%) of total premerger profits if there is zero (100%) excess capacity. Such estimations can be extended in a straightforward manner to other mergers and other oligopoly models. The estimation is based both on the profitability precondition, and on the non-empty core precondition, which postulates that the split of a merger’s profits be in its core. The article shows that the core in linear Cournot oligopolies has a non-empty interior, and indicates that the non-empty core precondition also sheds new lights on understanding important issues such as the stylized fact that mergers are likely to occur in markets plagued by excess capacities; why profitable mergers might not be formed; and why completed mergers might break up in the future. KEY WORDS: core, estimation of transaction costs, merger precondition, oligopoly JEL CLASSIFICATION: C62, C71, L14, L41
1. INTRODUCTION
Since the seminal work of Coase (1937), transaction cost analysis has led to significant advances in both theoretical and empirical studies in business, economics, and other social science disciplines. A large group of empirical studies focused on vertical integration and long-term contracting and tested the predictions by transaction cost economics (TCE) in Williamson (1975, 1985) and Klein et al. (1978), and by property right theory (PRT) in Grossman and Hart (1986) and Hart and Moore
374
JINGANG ZHAO
(1990). Shelanski and Klein (1995) found that the empirical literature is remarkably consistent with the predictions of TCE, and although David and Han (2004) found only 47% support (among 63 articles) for TCE, they found strong support for the predicted effects of asset specificity and weak support for the predicted effects of uncertainty. Whinston (2003) suggested that PRT offers a richer set of predictions that need more empirical tests.1 Another group of empirical studies estimated the size of transaction costs in a variety of sectors, from agriculture to financial markets.2 This article continues the estimation literature by providing a computational method for estimating the transaction costs of horizontal mergers. The method helps understand how merging costs, or the transaction costs of a merger, are determined by market parameters such as market size, merger size, the rate of excess capacity, and demand and cost parameters. In linear and homogeneous Cournot oligopolies with symmetric costs, the method leads to a closed-form upper bound for each merger’s merging costs. Consider, for example, an observed or completed monopoly merger with three firms. The estimation shows that the monopoly merging costs are, at most, 21% of the premerger total profits if there is no excess capacity and 33% of the premerger total profits if there is 100% excess capacity (see Table I). With 10 firms, the estimated upper bound for monopoly merging costs increases to 25% (78%) of the premerger total profits with zero (100%) excess capacity. Such estimations can be extended to more general Cournot (or Bertrand) oligopolies in a straightforward manner. The new method for estimating merging costs is based on two merger preconditions: the well-known profitability condition and a new precondition, which postulates that the split of a merger’s total profits be in its core, or that the core of a merger be non-empty. To see the logic of the non-empty core precondition, consider the monopoly merger again and suppose its core (which coincides with the core of the market) is empty. For each proposed split of monopoly profits (i.e., a monopoly merger proposal), empty core implies
375
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
TABLE I Estimated Bounds of Monopoly Merging Costsa (as proportion of total premerger profits, determined by (11)) n\tau
0.00 0.10 0.20 0.30 0.40 0.50 0.75 1.00 1.25 1.50 1.75 2.00
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 25 100
0.13 0.21 0.23 0.24 0.24 0.24 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
0.13 0.28 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
0.13 0.33 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.35 0.35 0.35 0.35
0.13 0.33 0.42 0.42 0.42 0.42 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.40 0.40
0.13 0.33 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.45
0.13 0.33 0.54 0.53 0.53 0.53 0.52 0.52 0.52 0.52 0.52 0.51 0.51 0.51 0.51 0.51 0.51 0.50
0.13 0.33 0.56 0.67 0.66 0.66 0.65 0.65 0.65 0.65 0.64 0.64 0.64 0.64 0.64 0.64 0.63 0.63
0.13 0.33 0.56 0.80 0.79 0.79 0.78 0.78 0.78 0.77 0.77 0.77 0.77 0.77 0.77 0.76 0.76 0.75
0.13 0.33 0.56 0.80 0.91 0.91 0.90 0.90 0.90 0.90 0.89 0.89 0.89 0.89 0.89 0.89 0.88 0.88
0.13 0.33 0.56 0.80 1.03 1.03 1.02 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01 1.00
0.13 0.33 0.56 0.80 1.04 1.14 1.14 1.14 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13
0.13 0.33 0.56 0.80 1.04 1.24 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25
are only a few changes in the estimated values for 17 ≤ n ≤ 25, and 25 ≤ n ≤ 100.
a There
the existence of a blocking coalition whose members could guarantee more profits than those provided by the proposal. With profit-seeking behavior, it would be absurd for the members of this blocking coalition to accept the proposal, because they would receive less than their worst profits if they did. Since every monopoly merger proposal in empty-core markets will be blocked, a successful monopoly merger could occur only in non-empty core markets. Similarly, the non-empty core precondition should hold for each successful nonmonopoly merger. As an illustration, consider the 1998 three-way A.P.A merger whose own-
376
JINGANG ZHAO
ership was split as 44, 29, 27% between Canadian Alcan, French Pechiney, and Swiss Algroup, respectively. By the new precondition, such a split should be in the merger’s core.3 It is useful to note that the above two preconditions are independent of each other. With transaction costs, it is easy (see Section 3) to find oligopolies with an empty core and a profitable monopoly merger, and with a non-empty core and an unprofitable monopoly merger. As readers will see, the non-empty core precondition not only helps to estimate merging costs and to explain why profitable mergers might not be formed, but it also provides an understanding of possible future breakups of completed mergers (such as AOL-Time Warner). Besides, it sheds lights on both understanding the stylized fact that mergers are likely to occur in markets plagued by excess capacities and explaining the empirical finding that industries for lumpy goods, such as ocean shipping will have an empty core when demand is low (Sjostrom, 1989; Pirrong, 1992). The rest of the article is organized as follows. Section 2 describes the problem, Section 3 provides non-empty core results in Cournot oligopolies, and Section 4 provides estimations of monopoly merging costs. Section 5 extends the core results and estimation of merging costs to non-monopoly mergers. Section 6 concludes the article, and the Appendix provides all proofs.
2. DESCRIPTION OF THE PROBLEM
I will lay out the method for estimating merging costs (i.e., the transaction costs of mergers) around the monopoly merger in a homogeneous Cournot oligopoly, which can be extended to other mergers and to other oligopoly models in a straightforward manner. A homogeneous Cournot oligopoly is given by an inverse demand P (xj ) and n cost functions Ci (xi ), 0 ≤ xi ≤ zi , i ∈ N = {1, . . . , n}, which is equivalent to the following normal form game:
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
= {N, Zi , πi },
377 (1)
where for each i ∈ N, Zi = [0, zi ] with zi > 0, and πi (x) = P (xj )xi − Ci (xi ), xi ∈ Zi , is its profit. I assume throughout the following assumption, called A0: A0 : (i) The inverse demand is decreasing, and each πi (x) is continuous in x and quasi-concave in xi ; (ii) there exist unique premerger and postmerger equilibria, which are interior solutions; and (iii) the capacity and cost function for each merger S ⊆ N are given by zS = j ∈S zj , and
(2)
CS (q) = Min{j ∈S Cj (xj )|q = j ∈S xj ≤ zS , xj ≥ 0, j ∈ S}.
(3)
Adding capacity not only makes our merger model more realistic, but also makes the core theory nontrivial.4 The cost function (3) represents a weak form of synergy (although it has been referred to as the “no synergy” case in Farrell and Shapiro, 1990): Each merger S could use its members’ most efficient technology up to zS = j ∈S zj , so a merger would involve both the exit of all inefficient members and an increase in the efficient member’s capacity.5 Define a monopoly merger contract as a triplet (N; x ; θ), x is the monopoly where N is the set of merging members, xj ≤ zj , πj (x) ≤ πj ( x ) for all xj ≤ zN ), and θ supply (i.e., is a split of the monopoly profits (i.e., θ ≥ 0, θi = π = πj ( x )). Adding the split θ is crucial in this study. Although such a split is the most important issue in merger negotiations,6 it has not been studied in the existing merger literature. Now, consider the core of market (1), which is the same as the core of the monopoly merger. For each S = N , its guaranteed profit is given by v(S) = Max Min i∈S πi (xS , yT ) = Min Max i∈S πi (xS , yT ), xS
yT
yT
xS
(4) / S} is the set of outsiders, (xS , yT ) = (xS , y−S ) is where T = {i|i ∈ / S ; the feasible a vector w ∈ R n with wi = xi if i ∈ S, = yi if i ∈
378
JINGANG ZHAO
regions for xS and yT are {xS ∈ R s+ |j ∈S xj ≤ zS } and ZT = j ∈T Zj . A profit vector θ ∈ R n+ is in the core of market (1) if it is a split of the monopoly profits and if it gives each coalition S no less than its guaranteed profits v(S) (i.e., if θi = π and j ∈S θi ≥ v(S) for all S = N). Denote the set of all core vectors for (1) as Core (). Note that Core () is well defined, because the functions in (4) are continuous, and the choice sets are compact and convex. In more general situations, one would have vα (S) = Max Min i∈S πi (xS , yT ) xS
yT
< vβ (S) = Min Max i∈S πi (xS , yT ), yT
xS
which implies β -core ⊆ α -core (Aumann, 1959).7 As the author has shown in an earlier study (Zhao, 1999), α -core = β -core always holds in oligopoly markets, so there is no need to make the α - and β -distinction here, and one can simply use the term core. Let x denote the Cournot equilibrium in market (1) (i.e., the premerger equilibrium at which each xi is i ’s best response to x−i = (x1 , . . . , xi−1 , xi+1 , . . ., xn )), and πj (x) be j ’s premerger profits. Proposition 1 below summarizes two preconditions for the monopoly merger. PROPOSITION 1. (Necessary conditions for the monopoly merger): Suppose firms are profit-seeking, and suppose a monopx ; θ ) has occurred. Then, the following two preoly merger (N; conditions hold: (i) θj ≥ πj (x) for all j; and (ii) θ ∈ Core (). Part (i) is the profitability precondition (i.e., incentive to merge), and part (ii) is the non-empty core precondition. If part (i) fails, at least one firm is worse off with the merger; if part (ii) fails, at least one blocking coalition is worse off with the merger. The failure of either would be absurd under profit-seeking behavior. Therefore, both must hold. Observe that the two preconditions are necessary, rather than sufficient, conditions for the monopoly merger, and they
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
379
are independent of each other. Examples 1 and 2 in the next section report two oligopolies, one with an empty core and a profitable monopoly and the other with a non-empty core and an unprofitable monopoly. The non-empty core precondition, or θ ∈ Core(), could easily be misunderstood (and incorrectly criticized) as below. Under A0, one has v(S) = Max {i∈S πi (xS , z−S )|xS }.
(5)
Since producing z−S (at the full capacity) is not a credible action by the outsiders, the core concept is problematic, so the non-empty core condition is useless. Such argument, however, derives from a misunderstanding of the core. The core does not require any S to produce z−S . Instead, it requires that total profits be maximized and divided in such a way that each S=N receives no less than its worst profits v(S). Given z−S in (5), v(S) is typically small, implying that the core is possibly large. However, the large size of the core strengthens, rather than weakens, the claim that θ ∈ Core() holds (i.e., the nonempty core precondition).
3. CORE EXISTENCE RESULT
The non-empty core precondition heightens the need to check core existence in merger studies. For a coalitional transferable utility (TU) game = {N, v}, Bondareva (1962) and Shapley (1969) had shown that its core is non-empty if and only if it is balanced, and the author recently showed (Zhao, 2001) that its core is non-empty if and only if its minimum no-blocking payoff (MNBP) is below v(N), where MNBP is defined as8 MNBP = Min {i∈N θi |θ ∈ R n+ ; i∈S θi ≥ v(S) for all S = N}. (6) The MNBP method characterizes both the core and the core’s interior: the core has a non-empty (relative) interior if and only if MNBP < v(N) holds. Examples 1 and 2 below
380
JINGANG ZHAO
illustrate an empty core and a non-empty core using the MNBP method. EXAMPLE 1. n = 3, P = 6 − xj , Ci (xi ) = 0.8xi , 0 ≤ xi ≤ 1.5, i = 1, 2, 3. The premerger and monopoly profits are πi ≡ 1.69, πm = 6.76. Assume that the monopoly merging cost is MCM = 1.65. Then, v(123) = πm − MCM = 5.11 > πj = 5.07, so the profitability condition holds. The core is empty because v(123) = 5.11 < MNBP = 5.13. EXAMPLE 2. n = 3, P = 6 − xj , Ci (xi ) = 0.5xi , 0 ≤ xi ≤ 2, i = 1, 2, 3. The premerger and monopoly profits are πi ≡ 1.89, πm = 7.56. Assume that the monopoly merging cost is MCM = 2. Then, v(123) = (πm − MCM) = 5.56 < πj = 5.67, so the profitability condition fails. The core is non-empty because v(123) = 5.56 > MNBP = 4.59. Although the core of the oligopoly (1) can be legitimately criticized for being large, its general existence so far has not been fully understood. The author showed (Zhao, 1999) that the core of the oligopoly (1), where v(S) is given in (4), is non-empty if parts (i) and (iii) of A0 hold and if all πi (x) are concave. Note that concavity is a strong condition because it does not apply in linear oligopolies (i.e., πi (x) = (a − xj − ci )xi is not concave), and therefore it remains as an open problem to show core existence in oligopolies beyond the concavity assumption. In particular, it is nontrivial and useful to check core existence in previous nonlinear merger models with capacity added.9 Now, consider a linear oligopoly, or a vector (a, c, z) ∈ R 2n+1 ++ with c = (c1 , . . . , cn ), z = (z1 , . . . , zn ), where demand and cost functions are: P (xj ) = a − xj , Ci (xi ) = ci xi , 0 ≤ xi ≤ zi , all i . Without loss of generality, assume c1 ≤ · · · ≤ cn . Then, parts (ii–iii) of A0 become: cS = Min {cj |j ∈ S}, CS (q) = cS q, 0 ≤ q ≤ zS = j ∈S zj , and (a − c1 )/2 ≤ z = zj . (7)
Proposition 2 below shows that in all linear oligopolies, the core is not only non-empty, but also has a non-empty interior.
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
381
PROPOSITION 2. Consider the linear Cournot oligopoly given by (a, c, z) ∈ R 2n+1 ++ . Assume parts (ii–iii) of A0. Then, its core has a non-empty relative interior. The non-emptiness of the core’s interior should provide insights for future empirical work on cartel and merger agreements. In the event of small shocks to the market, a core with a non-empty interior remains non-empty, whereas a nonempty core with an empty interior could become empty. Therefore, a long-lived cartel suggests that the core has a non-empty interior. However, a short-lived cartel suggests that either the core is empty or that it is non-empty with an empty interior, which seems to be a probable explanation for the reported negotiations to break up the 6-year-old AOL-Time Warner merger. Proposition 2 is proved by the MNBP method. The strategy in the proof is to first obtain a closed-form MNBP in symmetric linear oligopolies (see (19)), then extend the result to asymmetric models. In a symmetric linear oligopoly, ci ≡ c and zi ≡ z for all i . For simplicity, let the same c and z denote and the scalar marginal cost the vectors in (a, c, z) ∈ R 2n+1 ++ and capacity in ci ≡ c and zi ≡ z. This should cause no confusion, because the meaning is clear in the context. Let MC(S) denote the merging cost for each S ⊆ N . Assume, for simplicity, MCM = MC(N ) > 0 and MC(S) = 0 for all S = N . As shown below, MNBP leads to a closed-form upper bound (the smallest) of monopoly merging cost, below which the core is non-empty. PROPOSITION 3. Let the core of the monopoly merger in (a, c, z) ∈ R 2n+1 be Core(). Assume MCM > 0, and that all ++ solutions are interior. (i) In symmetric linear oligopolies with ci ≡ c and zi ≡ z, all i, Core() = ∅ if and only if (a − c)2 /4 − {n(a − c − z)2 /[4(n − 1)]} ≥ MCM;
(8)
(ii) in asymmetric linear oligopolies given by (a, c, z), Core() = ∅ if (a − c1 )2 /4 − {n(a − c1 − zmin )2 /[4(n − 1)]} ≥ MCM.
(9)
382
JINGANG ZHAO
In part (ii), c1 = Min{ci |i ∈ N} and zmin = Min{zi |i ∈ N} are the minimal marginal cost and capacity, respectively. The core results in (8)–(9) imply two conclusions that are consistent with known stylized facts and empirical evidences. First, (8) or (9) implies that a smaller merging cost or larger capacity will strengthen the non-empty core precondition, and thus make the monopoly merger more likely to be formed. This conclusion is consistent with the stylized fact that mergers are likely to occur in markets plagued by excess capacities. Second, (8) or (9) implies that a smaller intercept a (i.e., low demand) will weaken the non-empty core precondition,10 and thus make the monopoly merger less likely to be formed. This conclusion is consistent with empirical studies on ocean shipping (Sjostrom, 1989; Pirrong, 1992) that suggested that industries for lumpy goods with avoidable costs will have an empty core when market demand is low.
4. ESTIMATION OF MONOPOLY MERGING COSTS
Since lowering merging costs will strengthen both preconditions, it stands to reason that there are low merging costs in sectors with strong merger activities and high merging costs in sectors with weak merger activities. Such empirical implication allows us to estimate the bounds of merging costs for observed and unobserved monopoly mergers. π be monopoly Let πj (x) be firm j ’s premerger profit, profits, and MNBP be given in (6). Proposition 4 below provides an upper bound of merging costs for each observed monopoly merger, which also can be understood as a lower bound for the merging costs of unobserved monopoly mergers. Recall that MCM denotes the monopoly merging costs and MC(S) denotes the merging cost for each S = N . PROPOSITION 4. Consider the oligopoly in (1), assume MC(S) = 0, all S = N . Let MCM denote the monopoly merging costs, and let MCM∗ be defined as MCM ∗ = π − Max{πj (x), MNBP }.
(10)
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
383
(i) Suppose the monopoly merger is observed. Then, MCM ≤ MCM ∗ . (ii) Suppose the unobserved monopoly merger is prevented by one of the two preconditions. Then, MCM > MCM ∗ .
Proposition 5 below uses Proposition 4 to obtain more accurate estimations of monopoly merging costs in symmetric linear oligopolies. PROPOSITION 5. Given a linear symmetric oligopoly (a, c, z) ∈ R 3++ , assume MC(S) = 0 for all S = N . Let MCM ∗ be given in (10), and let τ ≥ 0 denote the rate of excessive capacity as defined in z = (1 + τ )(a − c)/(n + 1). Then ∗ MCM ⎧ √ n(a −c)2 (n−1)(n+1)2 −n(n−τ )2 ⎪ ⎪ ⎪ if τ < τ1 = n−2 n−1; ⎨ (n+1)2 4n(n−1) = 2 2 ⎪ √ n(a −c) (n−1) ⎪ ⎪ ⎩ if τ ≥ τ = n−2 n−1. 1 (n+1)2 4n (11)
Table I shows the values of (11) as a proportion of premerger total profits (i.e., MCM* divided by n(a−c)2 /(n+ 1)2 ). These estimated bounds for monopoly merging costs have three notable features. First, each row (for a fixed number of firms) shows how excess capacity increases the estimated monopoly merging costs. With n = 10 firms, for example, the estimated bound for MCM increases from 25 to 78% and to 125% of premerger total profits when excess capacity increases from 0 to 100% and to 200%. Second, each column (for a fixed level of excess capacity) shows that the estimated bound initially increases and then decreases as the number of firms increases. Finally, most of the estimated values are based on the non-empty core precondition (i.e., πj (x) < MNBP); only those values at the right side of the bolded entries in the first five rows (i.e., n ≤ 6) are based on the profitability precondition (i.e., πj (x) ≥ MNBP, or τ ≥ τ1 ). The above estimations could be refined either by reducing the upper bounds of merging costs for observed monop-
384
JINGANG ZHAO
oly mergers or by increasing the lower bounds of merging costs for unobserved monopoly mergers. There are three areas in which future studies could achieve such improvements. First, by assuming that the unobserved monopoly merger is prevented by both preconditions, the estimated lower bound π − for its merging costs can be increased to MCM0 = ∗ Min{πj (x), MNBP} > MCM , where MCM ⎧ 0 n(a−c)2 (n−1)2 ⎪ ⎪ ⎪ ⎨ (n + 1)2 4n = ⎪ n(a−c)2 (n−1)(n + 1)2 −n(n−θ )2 ⎪ ⎪ ⎩ (n + 1)2 4n(n−1)
√ if θ < θ1 =n−2 n−1; √ if θ ≥ θ1 =n−2 n−1;
(12) which will lead to significantly larger values than the lower bounds given in Table I. Second, coalition formation (or core refinement) theory has increased each coalition’s value in (4) or (5) and hence MNBP in (6), which leads to a lower upper bound of merging costs for observed monopoly mergers. In such studies, the value v(S) in (5) is increased by replacing z−S with outsiders’ credible actions, three of which are: (i) they break up into singletons and behave as individual players, called γ - or complete-breakup belief; (ii) they stay as a coalition and behave as a joint π -max unit, called δ - or loyal-relation belief; and (iii) they choose the better of (i) and (ii), called e− or efficientbelief. Rajan (1989) studied the γ - and δ -beliefs in linear symmetric Cournot model with n ≤ 4; Yong (2001) studied the efficient-belief in linear symmetric Cournot and Bertrand models. It remains an open problem to find more general existence of such core refinements.11 Finally, the above estimation could be refined empirically or could be tested in real sectors in four steps: (i) estimate the sector’s demand and cost functions; (ii) compute the premerger Cournot (or Bertrand) equilibrium; (iii) compute v(S) in (4) and MNBP in (6); and (iv) compute MCM* in (10) as an upper (lower) bound of merging costs for an
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
385
observed (unobserved) monopoly merger. Steps i and ii could be skipped in sectors whose Cournot (or Bertrand) equilibrium has been estimated, and steps iii and iv could be completed with widely available optimization software. Although the merging costs of an observed monopoly merger could be alternatively estimated (or collected) by the accounting method, it seems infeasible to use this method to estimate the merging costs of unobserved monopoly mergers, because such costs include both direct merging costs (e.g., document and legal fees and labor costs) and indirect merging costs (e.g., campaign donations needed for antitrust approval), most of which are unobservable.
5. EXTENSION TO NONMONOPOLY MERGERS
Given a merger S = N with s members (2 ≤ s < n), let ( xS ·
+1 πS = x−S ) ∈ R + be the unique postmerger equilibrium and x − c ) x be the merger’s postmerger profits, (a − xS − j ∈S / j S S xS is the merger’s best response to x−S , and where t = n − s, x−S = { xi |i ∈ / S} ∈ R t+ is the vector of outsiders’ outputs (i.e., x−i ). xi , i ∈ / S , is i ’s best response to each xS ; θS ) of Define a merger contract for S as a triplet (S; the set of merging firms, the merger’s outputs, and a split of πS ).12 Splitting their joint its total profits (i.e., θS ≥ 0, j ∈S θi = profits is equivalent to playing the following normal-form TU game cooperatively: S ( x−S ) = {S, Zi , πi (xS , x−S )},
(13)
where for each i ∈ S, πi (xS , x−S ) = (a − j ∈S xj − j ∈S xj )xi − ci xi / has parameters set by the outsiders’ fixed supply x−S . The merger’s core or the core of (13) is similar to the core of (1), except that the player set is S and each payoff function now x−S . has a parameter It is important to emphasize that the core of a merger S = x−S , because the N is only defined for the fixed outside supply problem of splitting its joint profit will not exist if the outside supply is not fixed. This assumes that merging members
386
JINGANG ZHAO
can neither talk to nor negotiate with outsiders. If the members could negotiate with outsiders, the problem of dividing their joint profits would be moot. Allowing for such possibility is related to the sufficient conditions for a merger, which are beyond the scope of this study. Proposition 6 below lists the preconditions for a nonmonopoly merger S , which is an extension of Proposition 1. PROPOSITION 6. (Necessary conditions for a merger S = N ): x−S ) ∈ R +1 xS · Let x ∈ R n+ and ( + be the unique pre- and postmerger equilibria for a merger S = N . Suppose firms are profit-seekxS ; θS ) has occurred. Then the following ing and a merger (S; two claims hold: (i) θj ≥ πj (x) for all j ∈ S ; and (ii) θS is in the x−S ) given in (13). core of S ( The proof for Proposition 6 is similar to that for Proposition 1. Violating either precondition will contradict the profitseeking assumption. Since the allocation θ S is in the core of (13), the postmerger equilibrium is a hybrid equilibrium (Zhao, 1992).13 By the core result in linear oligopolies in Proposition 2, the core of each S = N in linear oligopolies has a non-empty interior. Proposition 7 below extends the upper bound of merging costs for a non-empty core in Proposition 3 to each S = N . It is simple to show that the postmerger equilibrium and outsiders’ total outputs in (a, c, z) ∈ R 2n+1 ++ are: xS = [a − (t + 1)cS + j ∈N\S cj ]/(t + 2); xi = [a − (t + 1)ci + cS + j ∈N\S,j =i cj ]/(t + 2), i ∈ N\S; and y−S = j ∈N\S xj = [t (a + cS ) − 2j ∈N\S cj ]/(t + 2),
where t = (n − |S|) = (n − s) is the number of outsiders. Assume for simplicity that MCS = MC(S) > 0 and MC(T) = 0 for all T = S and T ⊆ S . PROPOSITION 7. Given S = N in (a, c, z), let the postmerxS ; x−S ) ∈ R t+1 ger equilibrium be ( ++ , its core be Core(S ) = x−S )), y−S = j ∈N\S xj , and all solutions be interior. Core(S ( (i) Assume ci ≡ cS , zi ≡ zS , all i ∈ S . Then, Core(S ) = ∅ if and only if
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
387
(a − y−S − cS )2 /4 − {s(a − y−S − cS − zS )2 /[4(s − 1)]} ≥ MCS; (14) (ii) with asymmetric merging members, Core(S ) = ∅ if (a − y−S − cS )2 /4 − {s(a − y−S − cS − zS−Min )2 /[4(s − 1)]} ≥ MCS. (15)
In (15), cS = Min{ci |i ∈ S} and zS−Min = Min{zi |i ∈ S} are the minimal marginal cost and capacity of S , respectively. So far we have characterized the non-empty core precondition for a single merger. It is simple to define the core for each merger in a set of simultaneous mergers and to extend Proposition 7 to such multiple mergers. I will omit the equivalent statement of Proposition 4 for nonmonopoly mergers (i.e., the upper and lower bounds of merging cost) in nonlinear models, because such extensions are straightforward. Proposition 8 below estimates the bounds of merging costs for each nonmonopoly merger in symmetric linear oligopolies. PROPOSITION 8. Given a symmetric linear oligopoly (a, c, z) ∈ R 3++ with n firms. For a merger S = N with s members, 2 ≤ s < n, let MCS = MC(S) > 0 be its merging costs, MC(T ) = 0 for all T ⊂ S , and MCS∗ be given by MCS ∗ =
s(a − c)2 (n + 1)2 − s(n − s + 2)2 . (n + 1)2 s(n − s + 2)2
(16)
(i) Suppose the merger S is observed. Then, MCS ≤ MCS ∗ . (ii) Suppose the unobserved merger S is prevented by one of the two preconditions. Then, MCS > MCS ∗ .
The above MCS* does not depend, unlike MCM* in (11), on the rate of excess capacity, because the involved MNBP for S ( x−S ) is always below the merger’s premerger total profits (i.e., j ∈S πj (x) > MNBP(S )). Table II below shows the values of MCS* as proportion of each merger’s premerger total profits (i.e., (16) divided by s(a − c)2 /(n + 1)2 )) for profitable S = N . Note that there is only one profitable S = N (i.e., s = n − 1) in
388
JINGANG ZHAO
TABLE II Estimated Bounds of Merging Costs for Merger S = N (as proportion of its total premerger profits, determined by (16)) s\n
5
s\n
6
s\n
7
s\n
8
s\n
9
3 4 s\n 8 9
0.00 10 0.49
4 5 s\n 16 17 18 19
0.09 20 0.04 0.53 1.58
5 6 s\n 25 26 27 28 29
0.19 30 0.03 0.42 1.15 2.68
6 7 s\n 35 36 37 38 39
0.29 40 0.30 0.82 1.76 3.79
s\n 53 54 55 56 57 58 59
60 0.08 0.38 0.85 1.61 3.01 6.01
s\n 63 64 65 66 67 68 69
70 0.23 0.58 1.12 2.01 3.63 7.12
s\n 72 73 74 75 76 77 78 79
80 0.11 0.39 0.79 1.40 2.41 4.26 8.23
s\n 81 82 83 84 85 86 87 88 89
90 0.01 0.23 0.54 0.99 1.67 2.81 4.88 9.34
7 8 s\n 44 45 46 47 48 49 s\n 91 92 93 94 95 96 97 98 99
0.39 50 0.18 0.57 1.21 2.39 4.90 100 0.11 0.35 0.70 1.19 1.95 3.21 5.51 10.45
markets with 6 ≤ n ≤ 10. With 10 firms, for example, the profitable nonmonopoly merger is the merger of nine firms, whose merging costs are at most (at least) 49% of its premerger profits if it is observed (if it is unobserved and prevented by one of the preconditions). Similar to the monopoly case, future studies could increase the estimated lower bound of merging costs for each unobserved S=N by assuming that it is prevented by both preconditions,14 could reduce the estimated upper bound of merging costs for each observed S = N by applying new results in coalition formation (or core refinement) theory, and could refine
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
389
the estimations by empirical investigation or by extension to more general models such as with multi-products or with nonlinear demand and costs or with differentiated goods.
6. CONCLUSION
The non-empty core precondition requires that the split of a merger’s profits be in its core. The above analysis indicates that this new precondition not only leads to a method for estimating merging costs, but also provides new understandings of several important issues: (i) why profitable mergers might not be formed; (ii) why completed mergers might break up in the future; and (iii) the stylized fact that mergers are likely to occur in markets plagued by excess capacities. Such benefits of the new precondition suggest the need to find more general core results beyond Cournot oligopoly; beyond the known concavity condition for non-empty core in Cournot oligopoly; or beyond the new result that the core in linear Cournot oligopolies has a non-empty interior. It also suggests the need to advance coalition formation theory and core refinement theory, which will improve the estimation of merging costs and could lead to sufficient conditions for horizontal mergers. Since the estimation method can be extended in a straightforward manner to differentiated multi-product nonlinear Cournot (or Bertrand) oligopolies, readers are left with a rich source of future empirical studies on estimating merging costs in particular sectors. The author hopes readers will be encouraged to conduct such empirical studies or to apply the new precondition to other binding contract problems, such as joint ventures and legal cartels. In particular, it would be useful to use the new precondition to predict possible future mergers (or breakups) and to explain past merger activities. For example, it seems that the observed merger waves around the turn of the 21st century could be partially explained by reductions in merging cost due to changes in accounting regulations.15
390
JINGANG ZHAO
APPENDIX
Proof of Proposition 1. Discussions following the proposition
form a proof. Our strategy in proving Proposition 2 is to first establish core existence in symmetric markets, and then extend it to asymmetric markets. The proof is completed by establishing the following four lemmas. Given a game = {N, v}, recall that its minimum no-blocking payoff (MNBP) is defined in (6), and that its core is denoted as Core(). LEMMA 1. Given a game {N, v}, let = {N, v } be a new game such that v (S) ≤ v(S) for all S = N . Then mnbp( ) ≤ mnbp(). Proof of Lemma 1. Since v(S) is reduced to v (S), the feasible region for given in (6) is enlarged to that for . Thus, the minimum value of the same objective function in (6), i∈N xi ,
is reduced from mnbp() to mnbp( ). We first establish core existence using the following four assumptions: xS = (a − cS − j ∈S A1 : For each S = N, / zj )/2 ≤ zS . 2 xS ) >0, where v(S) is given by (4). A2 : For each S=N, v(S)=( A3 : cj = c > 0 for all j ∈ N . A4 : zj = z > 0 for all j ∈ N . With A1, the unconstrained optimal supply of S in finding its v(S) is an interior solution (i.e., less than or equal to its capacity). With A2, the price facing any S is always greater than its marginal cost cS , this guarantees that each coalition produces a positive amount of output. With A3 and A4, firms are symmetric in marginal cost and in capacity. For any 1 ≤ k ≤ n, let S(k) = {T ⊆ N ||T | = k} denote the set of coalitions who have precisely k members. By A1 − A4, the guaranteed profit for each S ∈ S(k) is equal to v(k) = (a − (n − k)z − c)2 /4(= v(S) = vα (S) = vβ (S)).
(17)
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
391
Now, for each k = 1, . . . , n, consider the minimum value defined by MV (k) = {Mini∈N xi |x ∈ R n+ ; i∈S xi ≥ v(k), all S ∈ S(K)}. (18)
The next lemma provides a formula for each MV(k) and their relations. LEMMA 2. Under A1–A4, the following two claims hold. (i) MV (k) = nv(k)/k; and (ii) MV(k+1) > MV(k), for k=1, . . . , n − 1. Proof of Lemma 2. Part (i). (18) has (nk ) = n(n − 1) . . . (n − k + 1)/k! constraints, so v(k) on the right-hand side appears n(n − 1) . . . (n − k + 1)/k! times. Each xi on the left-hand side appears (n−1 k−1 ) = (n − 1) . . . (n − k + 1)/(k − 1)! times. By summing over all constraints, we have [(n − 1) . . . (n − k + 1)/(k − 1)!]xi ≥ [n . . . (n − k + 1)/k!]v(k), or xi ≥ nv(k)/k.
Hence, the minimum value of (18) is equal to MV (k) = nv(k)/k . Part (ii). By part (i), part (ii) is equivalent to g = v(k + 1) − v(k)(k + 1)/k > 0.
Let t = a − c, the above function g can be rewritten as g(t) = (t − (n − k − 1)z)2 − (t − (n − k)z)2 (k + 1)/k.
Because the solution of g = 0 is t ∗ = nz, and dg 2 /dt 2 = −2/k < 0, g is concave with a unique maximum t ∗ = nz. By A1 ( xS ≤ xS > 0), one has: (n − k)z < t ≤ (n + k)z. zS , |S| = k) and A2 ( Since g is ∩–shaped, and g((n − k)z) = z2 > 0, g((n + k)z) = z2 > 0,
we have g(t) > 0 for all t ∈ [(n − k)z, (n + k)z]. This proves the
lemma.
392
JINGANG ZHAO
LEMMA 3. Under A1–A4, the MNBP defined in (6) is given by mnbp() = MV (n − 1) = nv(n − 1)/(n − 1) = n(a − c − z)2 /[4(n − 1)].
(19)
Proof of Lemma 3. Let FR and FR(k) below denote the “feasible region” in (6) and (18): F R = {x ∈ R n+ |i∈S xi ≥ v(k), for all S ∈ (k), k = 1, . . . , n − 1}, F R(k) = {x ∈ R n+ |i∈S xi ≥ v(k), for all S ∈ S(K)}, k = 1, . . . , n − 1.
The above two expressions lead to F R = ∩n−1 i=1 F R(k).
(20)
Now, for each k = 1, . . . , n − 1, define F R(k)∗ = {x ∈ R n+ |i∈N xi ≥ nv(k)/k}.
From the proof of part (i) in Lemma 2, F R(k) ⊆ F R(k)∗ for each k = 1, . . . , n − 1. From (20) and part (ii) in Lemma 2, we have ∗ F R ⊆ F R ∗ = ∩n−1 i=1 F R(k) = {x ∈ R n+ |i∈N xi ≥ nv(n − 1)/(n − 1)}.
(21)
Observe that the minimum value of {Mini∈N xi |x ∈ F R ∗ } is equal to {Mini∈N xi |x ∈ F R ∗ } = nv(n − 1)/(n − 1) = MV (n − 1). (22)
Since the symmetric Min solution (i.e., xi = v(n − 1)/(n − 1), all i ) is included in FR, it follows from (21) and (22) that {Mini∈N xi |x ∈ F R} = {Mini∈N xi |x ∈ F R ∗ }, so (19) holds.
LEMMA 4. Given (a, c, z), let v(S) = vα (S) = vβ (S) be given by (4), and mnbp() = mnbp(a, c, z) be given in (6). Define a market (a, c , z ) by c = (c1 , . . . , c1 ) ∈ R n+ , and z = (zmin , . . . , zmin ) ∈ R n+ ,
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
393
where c1 = Min{ci |i ∈ N}, zmin = Min{zi |i ∈ N}, let the MNBP in this new market be denoted as mnbp (a, c , z ). Then mnbp(a, c, z) ≤ mnbp(a, c , z ) ≤ n(a − zmin − c1 )2 /[4(n − 1)]. (23)
Proof of Lemma 4. Consider the second inequality. If A1 and A2 are both satisfied in (a, c , z ), then Lemma 3 and (17) yield mnbp(a, c , z ) = n(a − zmin − c1 )2 /[4(n − 1)]. Note that v(S) will be reduced to a smaller value if either A1 or A2 or both are violated. If A1 is violated, v(S) will fall below (a − (n − s)zmin − c1 )2 /4 (where s denote the number of firms in S), because now v(S) is achieved at constrained solution. If A2 is violated, v(S) becomes zero. Thus, by Lemma 1, the MNBP will be reduced, so mnbp(a, c , z ) ≤ n(a − zmin − c1 )2 /[4(n − 1)].
Now consider the first inequality of (23). Assume for a moment that A1 and A2 are both satisfied in (a, c, z). By A1 and A2, the value for S in (a, c, z) satisfies 2 2 v(S) = ( xS )2 = (a − cS − j ∈S / zj ) /4 ≤ a − c1 − (n − s)zmin ) /4. (24)
By earlier arguments, the above v(S) in (a, c, z) will fall below 2 (a − cS − j ∈S / zj ) /4, if either A1 or A2 or both be violated. Since (a − c1 − (n − s)zmin )2 /4 is the value of S in (a, c , z ), (24)
and Lemma 1 lead to mnbp(a, c, z) ≤ mnbp(a, c , z ). Proof of Proposition 2. By parts (ii–iii) of A0, the monopoly π = (a − c1 )2 /4. By part (i) of Lemma 2, this is equal profit is to MV (n) defined in (18) for the new market (a, c , z ). It follows from Lemmas 3–4 and part (ii) of Lemma 2 that v(N ) = π = MV (n) > MV (n − 1) = n(a − zmin − c1 )2 /[4(n − 1)] ≥ mnbp(a, c , z ) ≥ mnbp(a, c, z) = mnbp().
Since v(N ) is greater than mnbp(), the core has a non-empty
interior.
394
JINGANG ZHAO
Proof of Proposition 3. We first prove part (i). By (17) and (19), the monopoly profit in symmetric markets without bindπ = (a − c)2 /4, and the MNBP is equal to ing capacities is mnbp(a, c, z) = n(a − c − z)2 /(4(n − 1)).
Hence, Core() = ∅ ⇔ v(N) = [ π − MCM] ≥ mnbp(a, c, z) ⇔ [(a − c)2 /4 − MCM] ≥ n(a − c − z)2 /[4(n − 1)] ⇔ (a − c)2 /4 − {n(a − c − z)2 /[4(n − 1)]} ≥ MCM.
Part (ii). Define (a, c , z ) by c = (c1 , . . . , c1 ), z = (zmin , . . . , zmin ) ∈ R n+ , where c1 = Min{ci |i ∈ N }, and zmin = Min{zi |i ∈ N }. Let the MN BP in this new market be mnbp(a, c , z ). By sufficient capacities, and by (17) and (19), one has mnbp(a, c , z ) = n(a − c1 − zmin )2 /(4(n − 1)).
By A1 and A2, the value for S in (a, c, z) satisfies 2 2 v(S) = (a − cS − j ∈S / zj ) /4 ≤ (a − c1 − (n − s)zmin ) /4.
(25)
Lemma 1 and (25) lead to mnbp(a, c, z) ≤ mnbp(a, c , z ) = n(a − c1 − zmin )2 /(4(n − 1)).
Hence, Core() = ∅ ⇔ [ π − MCM] = [(a − c1 )2 /4 − MCM] ≥ mnbp(a, c, z), which holds if [(a − c1 )2 /4 − MCM] ≥ {n(a − c1 − zmin )2 /[4(n − 1)]},
so (9) holds.
Proof of Propositions 4 and 5. By the two preconditions for an π − MCM ≥ Max{πj (x), observed monopoly merger, v(N) = MNBP } holds, which is equivalent to MCM ∗ ≥ MCM , so part (i) of Proposition 4 holds. If the monopoly merger is prevented by one of the two preπ − MCM < Max{πj (x), MNBP }, conditions, one has: v(N) = which is equivalent to MCM ∗ < MCM , so part (ii) of Proposition 4 holds. If both preconditions prevented the π − MCM < monopoly merger, one will have: v(N) = Min{πj (x), MNBP }, which leads to (12).
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
395
Substituting (19), z = (1 + τ )(a − c)/(n + 1) and πj (x) = (a − c)2 /(n + 1)2 into [MNBP − πj (x)], and evaluating the expression leads to √ [MN BP − πj (x)] > 0 ⇔ τ < τ1 = n − 2 n − 1.
Substituting this and π = (a − c)2 /4 into (10), and simplifying
leads to (11). Proof of Propositions 6–8. The proofs are similar to those of Propositions 1, and 3–5. One only needs to replace “N ” with
“S ” and “a ” with (a − y−S ). ACKNOWLEDGMENTS
I would like to thank Chengzhong Qin, Donald Smythe, and participants at Econometric Society Australian Meeting and EARIE conference for useful comments. This research was partially funded by a SSHRC grant. Some results in this paper had been reported in a working paper titled “A new precondition for horizontal mergers.” All errors, of course, are my own.
NOTES 1. See also Joskow (1988) for a survey of transaction cost studies in vertical integration, Rindfleisch and Heide (1997) in marketing, Jones (2002) in financial markets, and Boerner and Macher (2003) in social sciences. 2. For example, Wallis and North (1986) showed that U.S. GNP share of transaction sector grew from 25% in 1870 to 45% in 1970, and Dollery and Leong (1998) showed that Australian GNP share of transaction sector grew from 32% in 1911 to 60% in 1991; Park et al. (2002) showed that the transaction costs of rice and corn in China during 1988–1995 were 15–20% and 25% of their prices, respectively. See Wang (2003) for a survey of such estimations. 3. APA had sales revenues of $21.6 billion per year in 1998, passing Alcoa as the leader in Aluminum. The merger was canceled later due to antitrust objections.
396
JINGANG ZHAO
4. With no capacity constraint or zi ≡ ∞, v(S) = 0, all S = N, so any split of monopoly profits will be in the core. 5. Such weak synergy seems to exist in internet mergers, which upgrade software and combine capacities. Other synergies (like Perry and Porter, 1985) will lead to different core results and merit separate studies. 6. The 1997 MCI-WorldCom merger helps illustrate the role played by the split. WorldCom’s final ($37b) and initial (20% less) offers were two different merger contracts. Without the split θ , the rejected initial and the accepted final contracts would be the same, although they led to completely different results. 7. The α-core and β-core are defined by replacing the above v(S) by vα (S) or vβ (S). 8. This can be understood geometrically as below. First, draw the vertical axis for v(N), and fix all v(S), S=N; then, start with a large v(N ) (so its core is non-empty) and keep reducing v(N ). The core will eventually become empty after v(N) falls below a critical horizontal line, whose height is equal to the above MNBP. 9. For example, Davidson and Deneckere (1984), Perry and Porter (1985), Farrell and Shapiro (1990). 10. A0 implies z = (1 + τ )(a − c)/(n + 1), where τ ≥ 0. It is easy to show that the left-hand side in (8)–(9) is increasing in a when τ > 1/n, so decreases in a will weaken the non-empty core precondition. 11. Let vγ (S), vδ (S), ve (S) denote respectively the value of S under δ, γ - and e- beliefs; and Cγ , Cδ , and Ce the associated core refinements. Then, ve (S) ≥ vδ (S) ≥ vγ (S) > v(S) impels Ce ⊂ Cδ ⊂ Cγ ⊂ Core. Since ve (S) is the largest among the outsiders’ credible actions, one could conjecture that Ce = ∅ be a sufficient condition for the monopoly merger. See Zhao (1996) for a survey of such refinements, Ray and Vohra (1999) for an alternative refinement, and Allen (2000) for the significance of coalition formation. 12. With multiple post-merge equilibria, one could define a merger contract as a triplet (S; xS (x−S ); θS (x−S )) such that xS (x−S ) is its best response to x−S , and θS (x−S ) is a split of its joint profits. 13. Precisely, the post-merge equilibrium is a HSDR (hybrid solution with a distribution rule) for the partition = {S, i1 , . . . , it } with DR(S) = β-core, see Section 5 of Zhao (1996) for more discussions. 14. In the model of Proposition 8, this will increase the lower bound of merging costs for unobserved S = N to MCS0 = π − Min{j ∈S πj (x), MNBP(S )} =
s(a − c)2 4(s − 1)(n + 1)2 − s(n + s − (n − s + 2)θ ) (n + 1)2 4s(s − 1)(n − s + 2)2 ∗ > MSC .
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
397
15. The merger waves occurred during the two-year period between April 1998 and April 2000, and produced the ten largest mergers in history (each valued at more than $58 billion): AOL-Time Warner ($163B, 1/00), Pfizer/Warner-Lambert ($90B, 2/00), Deutsche-Italia ($82B, 4/99), Mobil-Exxon ($77B, 12/98), Citicorp-Travelers ($73B, 4/98), Glaxo-SmithKline ($72B, 1/00), BTM-MTB ($69B, 4/00), SBCAmeritech ($62B, 5/98), BankAmerica-NationsBank ($60B, 4/98), and ATT-MediaOne ($58B, 5/99). Since the merger wave, only one mega merger, the $65B ATT-Bellsouth deal (3/2006), moved into the top 10, with the $58B ATT-MediaOne deal being removed form the list.
REFERENCES Allen, B. (2000), The future of microeconomic theory, Journal of Economic Perspectives 14, 143–150. Aumann, R. (1959), Acceptable points in general cooperative n-person games, in Tucker and Luce (eds.), Contributions to the Theory of Games IV, Annals of Mathematics Studies Vol. 40, Princeton University Press, Princeton. Boerner, C. and Macher, J. (2003), Transaction cost economics: an assessment of empirical research in the social science, Working Paper, McDonough School of Business, Georgetown University. Bondareva, O. (1962), The theory of the core in an n-person game (in Russian), Vestnik Leningrad. Univ. 13, 141–142. Coase, R.H. (1937), The nature of the firm, Economica 4, 386–405. David, R. and Han, S.K. (2004), A systematic assessment of the empirical support for transaction cost economics, Strategic Management Journal 25, 39–58. Davidson, C. and Deneckere, R. (1984), Horizontal mergers and collusive behavior, International Journal of Industrial Organization 2, 117–132. Dollery, B. and Leong, W. (1998), Measuring the transaction sector in the Australian economy, 1911–1991, Australian Economic History Review 38, 207–231. Farrell, J. and Shapiro, C. (1990), Horizontal mergers: an equilibrium analysis, American Economic Review 80, 107–126. Grossman, S. and Hart, O. (1986), The costs and benefits of ownership: a theory of vertical and lateral integration, Journal of Political Economy 94, 691–719. Hart, O. and Moore, J. (1990), Property rights and the nature of the firm, Journal of Political Economy 98, 1119–1158. Jones, C.M. (2002), A century of stock market liquidity and trading costs, Working Paper, Graduate School of Business, Columbia University.
398
JINGANG ZHAO
Joskow, P.L. (1988), Asset specificity and the structure of vertical relationships: empirical evidence, Journal of Law, Economics, and Organization 4, 95–117. Klein, B., Crawford, R. and Alchian, A. (1978), Vertical integration, appropriable rents, and the competitive contracting process, Journal of Law and Economics 21, 297–326. Park, A., Jin, H., Rozelle, S. and Huang, J. (2002), Market emergence and transition: arbitrage, transaction costs, and autarky in China’s grain markets, American Journal of Agricultural Economics 84, 67–82. Perry, M. and Porter, R.H. (1985), Oligopoly and the incentive for horizontal merger, American Economic Review 75, 219–227. Pirrong, S. (1992), An application of the core theory to the analysis of ocean shipping markets, Journal of Law and Economics 35, 89–131. Rajan, R. (1989), Endogenous coalition formation in cooperative oligopolies, International Economic Review 30, 863–876. Ray, D. and Vohra, R. (1999), A theory of endogenous coalition structures, Games and Economic Behavior 26, 286–336. Rindfleisch, A. and Heide, J.B. (1997), Transaction cost analysis: past, present and future applications, Journal of Marketing 61, 30–55. Shapley, L. (1967), On balanced sets and cores, Naval Research Logistics Quarterly 14, 453–460. Shelanski, H.A. and Klein, P.G. (1995), Empirical research in transaction cost economics, Journal of Law, Economics, and Organization 11, 335–361. Sjostrom, W. (1989), Collusion in ocean shipping: a test of monopoly and empty core models, Journal of Political Economy 97, 1160–1179. Wallis, J. and North, D. (1986), Measuring the transaction sector in the American economy, 1870–1970, in Engerman, S. and Gallman, R. (eds.), Long-Term Factors in American Economic Growth, University of Chicago Press, Chicago, 95–161. Wang, N. (2003), Measuring transaction costs: an incomplete survey, Working Paper, Ronald Coase Institute, University of Chicago. Whinston, M.D. (2003), On the transaction cost determinants of vertical integration, Journal of Low, Economics and Organization 19, 1–23. Williamson, O.E. (1975), Markets and Hierarchies: Analysis and Antitrust Implications. The Free Press, New York. Williamson, O.E. (1985), The Economic Institutions of Capitalism. The Free Press, New York, NY. Yong, J.-S. (2001), Horizontal monopolization via alliances, Working Paper, Department of Economics, University of Melbourne. Zhao, J. (1992), The hybrid solutions of an N-person game, Games and Economic Behavior 4, 145–160. Zhao, J. (1996), The stability and the formation of coalition structures in norm form TU games, Working Paper 96-13, Department of Economics, Ohio State University.
ESTIMATING MERGING COSTS BY MERGER PRECONDITIONS
399
Zhao, J. (1999), A β-core existence result and its application to oligopoly markets, Games and Economic Behavior 27, 153–168. Zhao, J. (2001), The relative interior of base polyhedron and the core, Economic Theory 18, 635–648.
Address for correspondence: Jingang Zhao, Department of Economics, University of Saskatchewan, 9 Campus Drive, Saskatoon, SK S7N 5A5, Canada. E-mail:
[email protected]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
