A formal model of firm boundaries and haggling

Economics Letters 156 (2017) 15–17

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A formal model of firm boundaries and haggling Yusuke Mori Department of Economics, Tokyo International University, 1-13-1 Matoba-kita, Kawagoe, Saitama, 350-1197, Japan

highlights • We provide a formal model in line with Williamson’s haggling theory. • In our model, the identity of the third-party arbitrator defines firm boundaries. • Integration economizes bargaining costs but employs fiat too often.

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Article history: Received 7 March 2017 Accepted 31 March 2017 Available online 11 April 2017 JEL classification: D23 L22 M21

abstract This study provides a formal model of Williamson’s haggling theory in which transacting parties engage in ex post contract renegotiation. We show that integration can economize bargaining costs but suffers from too much third-party arbitration (i.e., fails in selective intervention). © 2017 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Transaction cost Haggling Fiat Firm boundaries

1. Introduction Transaction cost economics (TCE; e.g., Tadelis and Williamson, 2013) points out that market transactions suffer from ex post costly contract renegotiations between locked-in transacting parties, which entails bargaining delay and breakdown. TCE then asserts that such haggling can be avoided if transacting parties are integrated into a single firm because internal organizations can use fiat to settle conflicts. Further, higher complexity and uncertainty make ex ante contracts more incomplete, and hence integration more likely. However, hierarchies do not always work better than markets because of their failure to provide ‘‘selective intervention’’ (i.e., fiat is exercised too often). Despite its empirical success, TCE’s arguments are relatively informal. This study thus aims to provide a formal model of haggling theory. In our model, transacting parties can call for thirdparty arbitration during the renegotiation and the only difference between non-integration and integration is its identity: court or their boss. This is a departure from existing models of firm

E-mail address: [email protected].

boundaries, but is in line with the approach taken by Tadelis and Williamson (2013). Our study is different from existing studies of ex post inefficiencies and firm boundaries in the following three ways. First, unlike Bajari and Tadelis (2001) and Hart and Holmstrom (2010), we do not focus on ex post maladaptation or coordination failure. Second, in contrast to Matouschek (2004) and Schmitz (2006), our model is not an extension of property-rights theory. Third, haggling in our model comprises neither influence activities (Powell, 2015) nor preference falsification in interim contracting (Baliga and Sjöström, 2016). The rest of the paper is organized as follows. Section 2 presents our model. Sections 3 and 4 analyze the model and present our main result. Section 5 concludes. 2. The model Two locked-in, risk-neutral parties 1 and 2 trade one unit of a good under either non-integration (M) or integration (H). Because of contract incompleteness, each may face unforeseen disturbances in trade circumstances. If a disturbance occurs, contract renegotiation, especially bargaining over how to split the

http://dx.doi.org/10.1016/j.econlet.2017.03.037 0165-1765/© 2017 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4. 0/).

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Y. Mori / Economics Letters 156 (2017) 15–17

Compte and Jehiel’s (2002) Propositions 4 and 5 show that when each party has an outside option larger than his or her concession payoff 1 − θ (i.e., Case 3), no costly reputation building takes place in equilibrium. More specifically, there is a unique perfect Bayesian equilibrium where each party (if the F type) reveals him- or herself as the F type immediately by demanding 1/(1 + δ). In equilibrium, if a party demands θ , his or her partner (if the F type) believes that he or she is of the I type with probability 1, and thus calls for third-party intervention because 1 − θ < wg holds. This fact implies that no party can be better off by mimicking the I type. Thus, the primary source of inefficiencies is costly thirdparty intervention.

trade value of size 1, takes place. If no disturbance occurs, on the contrary, the value 1 is created without renegotiation. The probability of each party facing a disturbance, denoted by p, and who faces a disturbance are both common knowledge. The renegotiation is formalized as an alternating-offer bargaining game in which party 1 becomes the proposer in odd periods. δ represents a common discount factor (δ ≈ 1 for simplicity). We assume that those who face disturbances may be of an inflexible type with probability ε ∈ (0, 1). As in the literature on bargaining and reputation (e.g., Abreu and Gul, 2000), the inflexible type always demands share θ , which is greater than 1/(1 + δ) (the equilibrium share of the complete-information, alternatingoffer game) and never accepts any smaller proportion of the value. Examples of the inflexible type include those who incur cost θ when adapting to a disturbance appropriately. We refer to the inflexible type as the I type and parties who are not of the I type as the F type. At any time in the bargaining, both parties can unilaterally call for third-party intervention (i.e., court ordering under nonintegration and fiat under integration). wg represents each party’s payoff in the intervention under governance structure g ∈ {M , H }. We assume 1 ≥ wH > wM . The first inequality implies that thirdparty intervention entails inefficiency (e.g., its time-consuming process causes opportunity costs). The second inequality reflects the severe limit of court ordering (e.g., the boss has better access to the requisite information or knowledge to resolve the conflict than court). Note that when 1 − θ > wg (resp. 1 − θ < wg ), our bargaining game corresponds to that without (resp. with) outside options. The game proceeds as follows. First, the governance structure is chosen. Second, disturbances may occur. Third, if either or both parties face disturbances, those who face them privately know their own types and the value split takes place; otherwise, the game ends. Fourth, if the value split is settled, the trade is realized; otherwise, each party obtains a zero payoff.

Since no third-party intervention occurs in equilibrium and the bargaining procedure is symmetric between the governance structures, the approximate aggregate payoff under each structure is the same:

3. The bargaining game

(1 − p)2 + 2p(1 − p) + 2p2 (1 − ε 2 )(1 − θ ).

This section shows the bargaining outcome by employing the results presented in the literature on bargaining and reputation. There are three cases under each governance structure. First, 1 − θ > wg and either party faces a disturbance. Second, 1 − θ > wg and both parties face disturbances. Third, 1 − θ < wg . We refer to these as Cases 1, 2, and 3, respectively. Compte and Jehiel’s (2002) Proposition 2 shows that in onesided uncertainty bargaining without outside options (i.e., Case 1), in any perfect Bayesian equilibrium, the party with type uncertainty (i.e., the party who faces a disturbance) and his or her partner without type uncertainty (i.e., the party who faces no disturbance) obtain approximately θ and 1 − θ , respectively. Lemma 1. Suppose only party i faces a disturbance and 1 − θ > wg and δ ≈ 1 hold. The bargaining cost is then negligible in equilibrium. Abreu and Gul’s (2000) Proposition 4 and Compte and Jehiel’s (2002) Proposition 3 show that in two-sided uncertainty bargaining without outside options (i.e., Case 2), bargaining delay occurs in equilibrium. To build a reputation for I-type parties, F -type parties concede (i.e., let the partner obtain θ ) only at the constant rate that keeps their partner indifferent between revealing him- or herself as the F type and mimicking the I type. Thus, for any perfect Bayesian equilibrium, each party’s (if the F type) expected payoff is approximately 1 − θ . Lemma 2. Suppose both parties face disturbances and 1 − θ > wg and δ ≈ 1 hold. The expected loss induced by the delay in reaching agreement is given by 2θ − 1.

Lemma 3. Suppose either or both parties face disturbances and 1 − θ < wg and δ ≈ 1 hold. Then, in equilibrium, if either party is of the I type, third-party intervention is called for, which causes loss 1 − 2wg under governance structure g ∈ {M , H }. 4. The optimal governance structure This section presents our main results. We can categorize the situation into three cases. First, wM < wH < 1 − θ : no third-party intervention is called for in equilibrium regardless of the choice of governance structure. Second, 1 −θ < wM < wH : the intervention may occur in equilibrium under both governance structures. Third, wM < 1 − θ < wH : the intervention may take place in equilibrium only under integration. We refer to these as Cases I, II, and III, respectively. 4.1. Case I: wM < wH < 1 − θ

If no party faces a disturbance with probability (1 − p)2 , the value 1 is created without any friction. If either party faces a disturbance with probability 2p(1 − p), the bargaining cost is negligible (see Lemma 1). If both parties face disturbances with probability p2 , costly reputation building occurs (see Lemma 2) and the aggregate expected payoff is approximately given by 2(1 − ε 2 )(1 − θ ). Note that if both parties are of the I type, perpetual disagreement occurs. We then find that the choice of governance structure does not matter. 4.2. Case II: 1 − θ < wM < wH From Lemma 3, the aggregate payoff under governance structure g is approximately given by

  (1 − p)2 + 2p(1 − p) (1 − ε) + 2εwg   + p2 (1 − ε)2 + 4ε(1 − ε)wg . When no party is of the I type with probability (1 − p)2 + 2p(1 − p)(1 − ε) + p2 (1 − ε)2 , the value 1 is created. When either party is of the I type with probability 2p(1 − p)ε + 2p2 ε(1 − ε), on the contrary, third-party intervention is called for. Since wM < wH , the optimality of integration is obvious. 4.3. Case III: wM < 1 − θ < wH Since wM < 1 − θ , the aggregate payoff under non-integration is the same as in Case I:

(1 − p)2 + 2p(1 − p) + 2p2 (1 − ε 2 )(1 − θ ).

Y. Mori / Economics Letters 156 (2017) 15–17

Since 1 − θ < wH , the aggregate payoff under integration is the same as in Case II:

(1 − p)2 + 2p(1 − p) {(1 − ε) + 2εwH }   + p2 (1 − ε)2 + 4ε(1 − ε)wH . Comparing these aggregate payoffs yields the following result: Proposition 1. Suppose wM < 1 − θ < wH and δ ≈ 1 hold. Integration should be chosen when 2p(1 − p)ε (1 − 2wH ) < p2 (1 − ε)2 (2θ − 1) + 2p2 ε(1 − ε)

× {(2θ − 1) − (1 − 2wH )} .

(1)

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Since p can be interpreted as the level of complexity or uncertainty, Corollary 1 implies that high complexity and uncertainty make integration optimal, which is consistent with the main assertion of TCE. 5. Conclusion This study provides a formal model in line with Williamson’s haggling theory. We show that integration economizes bargaining costs but entails too much intervention and point out that failure in selective intervention may be caused by employees’ payoffmaximization behavior in ex post renegotiations. Acknowledgments

The RHS of Condition (1) represents the benefit of integration: fiat serves as an outside option, and hence costly reputation building is avoided. The LHS of Condition (1), on the contrary, represents the cost of integration: fiat is employed too often. Suppose either party faces a disturbance. Since wM < 1 − θ < wH , the party who faces no disturbance soon concedes under nonintegration (see Lemma 1), but may call for costly intervention under integration (see Lemma 3). This result provides a formal explanation of why integration economizes bargaining costs but fails in selective intervention. It also points out that too much intervention is triggered not only by the lack of top management’s commitment as pointed out in the literature on TCE (e.g., Tadelis and Williamson, 2013) but also by subordinates’ pursuit of better payoffs. Solving Condition (1) with respect to p (> 0) yields the following result: Corollary 1. There is p∗ (< 1) such that integration should be chosen for all p ∈ (p∗ , 1], where 2ε(1 − 2wH ) (1 − ε)2 {1 − 2(1 − θ )} + 2ε(1 − ε){2wH − 2(1 − θ )} + 2ε(1 − 2wH ) (< 1).

p∗ ≡

I thank Ricard Gil, Hideshi Itoh, and the participants at the Contract Theory Workshop, SIOE 2016, and EARIE 2016 for their helpful comments. I gratefully acknowledge the grant from the Japan Society for the Promotion of Science (13J06819). An earlier version of this paper received the Kanematsu Fellowship from the Research Institute for Economics and Business Administration, Kobe University. Any errors are my own. References Abreu, D., Gul, F., 2000. Bargaining and reputation. Econometrica 68, 85–117. Bajari, P., Tadelis, S., 2001. Incentives versus transaction costs: A theory of procurement contracts. Rand J. Econ. 32, 387–407. Baliga, S., Sjöström, T., 2016. A Theory of the Firm Based on Haggling, Coordination and Rent-Seeking. Mimeo. Compte, O., Jehiel, P., 2002. On the role of outside options in bargaining with obstinate parties. Econometrica 70, 1477–1517. Hart, O., Holmstrom, B., 2010. A theory of firm scope. Quart. J. Econ. 125, 483–513. Matouschek, N., 2004. Ex post inefficiencies in a property rights theory of the firm. J. Law Econ. Organ. 20, 125–147. Powell, M., 2015. An influence-cost model of organizational practices and firm boundaries. J. Law Econ. Organ. 31, 104–142. Schmitz, P.W., 2006. Information gathering, transaction costs, and the property rights approach. Amer. Econ. Rev. 96, 422–434. Tadelis, S., Williamson, O.E., 2013. Transaction cost economics. In: Gibbons, R., Roberts, J. (Eds.), The Handbook of Organizational Economics. Princeton University Press, New Jersey, pp. 159–189.