Strategic transfer pricing, absorption costing, and observability

Management Accounting Research, 2000, 11, 327–348 doi: 10.1006/mare.2000.0134 Available online at http://www.idealibrary.com on

Strategic transfer pricing, absorption costing, and observability Robert F. Gox* ¨

This paper analyses the use of transfer pricing as a strategic device in divisionalized firms facing duopolistic price competition. When transfer prices are observable, both firms’ headquarters will charge a transfer price above the marginal cost of the intermediate product to induce their marketing managers to behave as softer competitors in the final product market. When transfer prices are not observable, strategic transfer pricing is not an equilibrium and the optimal transfer price equals the marginal cost of the intermediate product. As a strategic alternative, however, the firms can signal the use of transfer prices above marginal cost to their competitors by a publicly observable commitment to an absorption costing system. The paper identifies conditions under which the choice of absorption costing is a dominant strategy equilibrium. c 2000 Academic Press
Key words: transfer pricing; absorption costing; product pricing.

1. Introduction A common problem for vertically integrated firms is the coordination of activities among divisions in order to achieve an efficient allocation of resources within the organization. This task typically involves the determination of transfer prices for those goods and services that are exchanged at the divisional level. According to Hirshleifer (1956) the transfer pricing problem is solved by setting the transfer price equal to the marginal cost of the intermediate product. Although this well-known result seems widely accepted from a theoretical point of view, there is sufficient empirical evidence that firms are frequently using full cost-based transfer prices instead.1 This study offers a theoretical explanation for the existing gap between accounting theory and company *Otto-von-Guericke-University of Magdeburg, Faculty of Economics and Management, PF 4120, 39016 Magdeburg, Germany. E-mail: [email protected] Received 17 April 1999; accepted 30 April 2000. 1 See e.g. Kaplan and Atkinson (1998, p. 454), and Horngren et al. (1994, p. 872). 1044–5005/00/030327+22

$35.00/0

c 2000 Academic Press


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practice by providing a model of two competing hierarchies that gives rise to a strict preference for full cost-based transfer prices. In particular, the basic model considers two divisionalized firms facing price competition on the final product market. In this setting the optimal transfer price does not equal marginal cost because its main function is to serve as a commitment device vis-´a-vis the competitor. Namely, by charging transfer prices above marginal cost of the intermediate product both firms can commit their marketing managers to behave as softer competitors on the final product market. Accordingly, the resulting equilibrium profits strictly exceed the profits attainable under marginal cost-based transfer pricing. To understand the intuition behind this result, consider the usual incentive structure in divisionalized firms when the firms’ headquarters delegate the responsibility of pricing decisions to divisional managers and evaluate the agents’ performance by their divisional profits. Since the transfer prices are exogenous parameters of the agents’ profit maximization problems, the firms’ headquarters can commit their managers to the desired pricing strategies by adjusting the transfer prices accordingly. Moreover, since both firms have an incentive to raise their transfer prices strategically, there exists a unique non-cooperative equilibrium in which both firms charge transfer prices above the marginal cost of the intermediate product. Conversely, centralized firm could not credibly commit themselves to the agents’ equilibrium strategies because the marginal cost of the intermediate goods are exogenous parameters of the firms’ decision problems. In other words, choosing the managers’ equilibrium strategy would not be a profit-maximizing strategy for centralized firms. Also, the firms could not simply replace strategic transfer pricing by mandating final product market prices because announcing a price schedule different from the profit-maximizing prices of the centralized firm would not be an equilibrium.2 One may, however, consider secret pricing agreements between the firms as an alternative collusion device. However, since cartel contracts are illegal and tacit collusion usually provides incentives to cheat, owner-managed firms will generally be confined to act as non-cooperative players in the pricing game on the final product market.3 Thus, strategic transfer pricing also implies a strict preference for the delegation of competencies to managers over centralized decision-making, whereas both alternatives would lead to equivalent outcomes in the classical Hirshleifer setting. The basic concept of strategic transfer pricing has, however, one fundamental weakness that also applies to most of the models reviewed in the next section. As pointed out by Katz (1991) and Bagwell (1995), unobservable contracts cannot serve as credible precommitments unless they are employed for other than strategic reasons. This observation limits the direct use of strategic transfer pricing to the case of observable transfer prices. Although it may be reasonable to assume that the firms in a small industry do know their competitor’s transfer prices, it seems promising to identify strategic alternatives when the transfer prices are not common knowledge because establishing transfer prices above marginal cost would be beneficial for both firms. In the last part of this paper it is demonstrated that a publicly observable commitment to an absorption costing system may create the desired managerial incentives because 2 A formal proof for this claim is given in Proposition 2 on page 335. 3 See Jaquemin and Slade (1989, p. 417). In a multiperiod setting, firms may also be able to carry out irreversible investments to create credible precommitments (Brander and Spencer, 1983). However, these may be too costly to carry out and hence incredible, see Tirole (1988, chapter 8) for a review of related literature.

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it allows the firms to signal a deviation from marginal cost-based transfer pricing to their competitor even if the actual transfer prices are not observable.4 Moreover, we are able to identify conditions under which the mutual choice of absorption costing is a dominant strategy equilibrium between firms. This rest of this paper is organized as follows. The next section provides a short review of related literature. Section 3 outlines the model assumptions and analyses the pricing strategies of two centralized firms. Section 4 provides the solution of the strategic transfer pricing game under the assumption that transfer prices can be observed before the agents must decide on their pricing strategies. Section 5 enlarges the theme by considering the case of unobservable transfer prices and the impact of cost system choice on the firms’ strategies. Section 6 concludes the paper with a summary of results.

2. Related literature The idea behind strategic transfer pricing is closely related to a branch of the industrial organizations literature pursued by Vickers (1985), Fershtman and Judd (1987), and Sklivas (1987). Based on the pioneering work of Schelling (1960), these authors demonstrate the strategic benefits of incentive contracts as credible precommitments vis-´a-vis a competitor when managers are allowed to play the market game on behalf of the owners.5 Recent studies by Gal-Or (1993), and Hughes and Kao (1997b) have demonstrated that accounting data may serve similar purposes. Both papers consider the impact of common service cost allocations on the optimal production policies of two multiple product firms facing both duopolistic quantity competition on the final product market and production externalities within the firm.6 Although the specific shape of the ‘congestion’ cost function assumed in their models makes it difficult to derive general results, it becomes evident from numerical computations performed by Hughes and Kao (1997b) that decentralized firms are usually forcing their managers to behave as more aggressive competitors than centralized firms by employing a tidy cost allocation rule. The reason is that the tidiness requirement leads to an implicit subsidization of the firms’ products as compared with the relevant costs under centralized decision-making. As a consequence, both firms are earning strictly lower profits under decentralization except for some corner solutions where the whole congestion costs are allocated to one of the firms’ profit centers.7 Strategic transfer pricing has only recently been explored by Alles and Datar (1998) 4 Another way to cope with the observability problem would be to employ a contract designed to enhance efficient risk sharing in an agency setting. The optimal contract would not only enhance risk sharing but also create managerial incentives that are complementary with the strategic goals of a firm, see Hughes and Kao (1997a) for a recent example. A similar reasoning allows unobservable tidy cost allocation to provide strategic incentives in Hughes and Kao (1997b). 5 The typical equilibrium contract offered to the firms’ managers in this type of models is a linear combination of profit and sales, see e.g. Fershtman and Judd (1987). 6 The major difference between both models is that the second division in Gal-Or (1993) operates on a perfectly competitive market while Hughes and Kao (1997b) consider two duopolistic divisions acting on separate markets. 7 Despite its inefficiency decentralization is still a dominant strategy equilibrium in many cases because a centralized firm would be worse off when competing against a decentralized competitor. Hence, the firms are usually facing a prisoners’ dilemma (Hughes and Kao, 1997b, p. 277).

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and Narayanan and Smith (1998). Both papers consider the case of cost-based transfer pricing and find that the optimal transfer price under price competition contains a markup over the marginal cost leading both firms’ managers to raise their prices on the final product market to the mutual benefit of both firms. These findings are similar to the results of the basic model presented in Section 4 of this paper. Both papers, however, do not provide solutions to the problem of creating managerial equilibrium incentives when transfer prices are not observable.8 Finally, the results of this paper also draw a parallel to recent findings from the principal–agent literature, despite the fact that we assume symmetric information within the firm.9 Following Amershi and Cheng (1990) and Vaysman (1996), the optimal transfer pricing mechanism under asymmetric information is a ‘cost plus’ type price schedule, consisting of a standard cost refund and a markup covering a profit share, an information premium and a compensation for the agent’s effort. From the strategic point of view, these contracts are not only complementary to the markup rule in the case of duopolistic price competition, but also allow firms to signal a deviation from the intermediate product’s marginal cost for other than strategic reasons. Although a comprehensive analysis of these issues seems promising, it is beyond the scope of this study.

3. Model assumptions and benchmark case

Market and cost structure Consider a model of two decentralized firms (i = 1, 2), each of them consisting of headquarters (HQ) and two divisions: the manufacturing division Ai , and the marketing division Bi . Division Bi is organized as a profit center and controlled by a manager, who is free in the operating decisions concerning his department and responsible for divisional performance. Subsequently, the marketing manager will also be referred to as agent Bi . Division Ai produces an intermediate product that serves as an input for division Bi and is organized as a cost center because it sells its output only to its internal customer. The marketing division further processes the intermediate good and sells it in the final product market. In contrast to conventional transfer pricing models, the market for the final product is assumed to be imperfectly competitive. The market is exclusively served by marketing divisions B1 and B2 , each of them offering a different brand within a larger product class. Assuming price competition between marketing divisions, the market demand for the final product of department Bi is a function of its own price pi and the competitor’s price p j . The demand function qi ( pi , p j ) is assumed

8 While Narayanan and Smith (1998) at least recognize the importance of observability, Alles and Datar (1998) do not only assume observable transfer prices but also claim ‘Each firm need not observe its competitorís choice of transfer price, and consequently, there is no necessity for each firm to commit to its transfer price. Rather, given the common knowledge that some transfer prices must be chosen, each firm will react to its predictions of what that transfer price will be.’ 9 A substantial amount of recent transfer pricing literature has dealt with various aspects asymmetric information including Ronen and Balachandran (1988), Christensen and Demski (1989), Banker and Datar (1992), Wagenhofer (1994) or Edlin and Reichelstein (1995).

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to satisfy the usual properties in partial equilibrium analysis: ∂qi < 0, ∂ pi

∂qi > 0, ∂pj

∂ 2 qi ≥ 0. ∂ pi ∂ p j

(1)

Since products are (weak) substitutes, the demand for the final product of division Bi falls with its own price pi and rises with the competitor’s price p j . Furthermore, according to the last condition in (1), the marginal demand effect of a price change by firm i is (weakly) increasing in the price of firm j.10 Suppose also that the following conditions hold ∂q j ∂qi = ∂pj ∂ pi

and

∂q j ∂qi <− . ∂ pi ∂ pi

(2)

While the first condition in (2) exhibits the symmetry of cross effects on demand, the second condition states that both firms’ products are imperfect substitutes assuring that none of the firms can capture the entire market by marginally undercutting its rival’s price.11 Suppose that both firms set a constant transfer price per unit of the intermediate good, denoted by ti , and let—for the sake of simplicity—each final product unit require one unit of the intermediate good to be finished at a constant further processing cost of ci per unit. Under these conditions, the marketing manager Bi faces the following profit maximization problem: max iB = ( pi − ci − ti ) · qi ( pi , p j ). pi

(3)

To ensure the existence of a Nash equilibrium p = ( pi∗ , p ∗j ) in the agents’ market game, the profit functions iB are assumed to be strictly concave in each agent’s own price pi . The equilibrium will be unique, if the following condition holds:12 2 B ∂ 2 iB ∂ j

∂ pi2

∂ p 2j

−

2 B ∂ 2 iB ∂ j > 0. ∂ pi ∂ p j ∂ p j ∂ pi

(4)

Given the preceding assumptions regarding internal trade, it remains to specify the cost functions of the manufacturing divisions, which are given by Ci (xi ) = vi (xi ) + Fi ,

∂vi > 0, ∂ xi

∂ 2 vi ≥ 0, ∂ xi2

(5)

where xi is the production quantity of the intermediate product, vi (xi ) are the variable costs of production, and Fi are the fixed costs of department Ai . The assumption that Ci (·) is twice differentiable and (weakly) convex ensures the existence of an interior 10 Obviously, this effect would only equal zero for a linear demand function but not for more general demand systems. 11 This problem is also known as the Bertrand paradox and occurs in the case of identical products, see e.g. Tirole (1988). 12 Although the Nash equilibrium is a static concept, the inequality (4) is also referred to as the condition for reaction function stability, because it ensures that any arbitrary price strategy p = ( pi , p j ) converges against the Nash equilibrium in a myopic adjustment process (Dixit, 1986, p. 109).

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solution of the firm’s profit maximization problem, and allows for the case of a linear cost function as well as for the case of increasing marginal costs. However, since Division Ai is organized as a cost center, its only task is to produce the required quantity of the intermediate product at the lowest possible cost. Therefore, we will henceforth assume that xi = qi ( pi , p j ). Centralized decision-making In the centralized setting HQ of firm i solves the following problem: max i = ( pi − ci ) · qi ( pi , p j ) − vi (qi ( pi , p j )) − Fi . pi

Maximizing (6) with respect to pi yields the following first-order condition:   ∂ i ∂qi ∂vi · = qi ( pi , p j ) + pi − ci − =0 i, j ∈ {1, 2}, i  = j. ∂ pi ∂qi ∂ pi

(6)

(7)

The Nash equilibrium between both firms’ HQ on the final product market is defined by the system of implicit reaction functions given by (7). The vector of equilibrium prices, p = ( pim , p mj ), can be obtained as the solution of the system, provided a closed-form solution does exist. The equilibrium condition can be rewritten as   |εi | ∂vi ∂qi pi pim = · ci + where εi = < −1, (8) |εi | − 1 ∂qi ∂ pi qi indicating that both firms mark up their marginal cost, ci +∂vi /∂qi , by a constant factor that solely depends on their own price elasticity of demand, designated by εi . Denote the equilibrium demands by qim = qi ( pim , p mj ), then the benchmark profit of firm i under centralized decision-making can be expressed as: im = ( pim − ci ) · qim − vi (qim ) − Fi .

(9)

It is easy to verify that the same solution would be obtained if the firms set their transfer prices equal to the marginal cost of the intermediate product. Hence, the centralized setting is equivalent to the classical Hirshleifer solution of the transfer pricing problem. The next section will, however, demonstrate that the existence of a competitor in the final product market allows firms to improve their overall profitability by delegating the pricing decisions to their marketing managers.

4. Strategic transfer pricing In this section the strategic transfer pricing game between both firms’ headquarters is analysed under the assumption that both managers can observe their competitor’s transfer price before selecting their pricing strategies for the final product. The consequences of unobservable transfer prices will be discussed in Section 5. Given the observability assumption, the firms’ maximization problems become a two-stage game with perfect information. On the first stage of the game both firms’ headquarters simultaneously choose their transfer prices ti and t j . Once transfer prices have been chosen by headquarters, both managers observe the outcome of the first stage before they simultaneously determine their pricing strategies on the second stage of the game.

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The solution of the two-stage game is obtained by backward induction starting with the managers’ pricing decisions on stage two. Accordingly, each marketing manager maximizes his divisional profit function (3) with respect to his own price pi for a given pricing choice p j of agent B j . The resulting pair of first-order conditions ∂ iB ∂qi = qi ( pi , p j ) + ( pi − ci − ti ) · = 0, ∂ pi ∂ pi

i, j ∈ {1, 2},

i  = j,

(10)

implicitly defines the Nash equilibrium of the second stage market game between the marketing managers. For a given demand system the equilibrium solution p ∗ = ( pi∗ , p ∗j ) is uniquely determined by both firms’ transfer prices. Accordingly the equilibrium strategies can be written as pi∗ (ti , t j ) = arg max iB ( pi , p ∗j , ti ).

(11)

pi

Expression (11) shows that the agents’ equilibrium strategies are functions of both firms’ transfer prices from the perspective of their headquarters. Hence, the distinguishing feature of transfer pricing vis-´a-vis a competitor is the fact that the transfer price of firm i does not only affect the pricing decision of agent Bi but also the pricing strategy of agent B j . In other words, the transfer price has a strategic effect. This effect, however, affects the profit of agent Bi only indirectly. To see this, consider a small change of ti . Initially a change of the transfer price will affect the profitability of the marketing department Bi and cause the manager to change his price pi . Then, although the profit of the competitor’s marketing department B j does not directly depend on ti , the manager of department B j will react to his competitor’s transfer pricing policy. In particular, he will react to a change of ti because the shift in agent Bi ’s pricing strategy induced by the new transfer price ti has a direct impact on his profit. Lemma 1 formally states how a change of ti affects the agents’ equilibrium strategies: Lemma 1 The equilibrium prices chosen by the agents in the market game are strictly increasing in both firms’ transfer prices: ∂ pi > 0, ∂ti

∂pj > 0. ∂ti

Proof See the Appendix. ✷ Consider first the sign of the direct effect in Lemma 1. From the agent’s first-order condition (10) it is easy to check that an increase of ti is equivalent to an upward shift of the marketing department’s marginal cost curve. Since the marginal revenue schedule remains unchanged, agent Bi must increase his price pi to equate both curves at a lower output quantity for a given price of his competitor. The best response of agent B j to this change of his competitor’s price is to raise his own price p j as well. This is due to the fact that, in general, prices are strategic complements while quantities are strategic substitutes as observed by Bulow et al. (1985). A more formal statement of this observation is given by the following condition ∂ 2 i > 0, ∂ pi ∂ p j

(12)

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indicating that the marginal profit of firm i increases when firm j raises its price.13 As a consequence, the intensity of competition is reduced and the market outcome moves closer to the cartel solution. Moreover, since condition (12) holds for both firms, the strategic effects will reinforce each other to the mutual benefit of both competitors when both firms increase their transfer prices. From the preceding analysis we are now prepared to derive the first-stage transfer pricing equilibrium between both firms’ headquarters. Given the optimal pricing and production choices of the divisional managers, both headquarters will maximize the sum of divisional profits, i∗ = iB∗ + ti · qi ( pi∗ (ti , t j ), p ∗j (ti , t j )) − vi (qi ( pi∗ (ti , t j ), p ∗j (ti , t j ))) − Fi ,

(13)

with respect to their transfer prices, where iB∗ := iB (ti , t j ) is given by iB (ti , t j ) = ( pi∗ (ti , t j ) − ci − ti ) · qi ( pi∗ (ti , t j ), p ∗j (ti , t j )).

(14)

The notation iB (ti , t j ) for the equilibrium profit of division Bi in (14) emphasizes that not only the equilibrium prices, but also the equilibrium profits, are functions of the transfer prices from the perspective of HQ. The necessary conditions for a Nash equilibrium between both firms’ headquarters are obtained by differentiating total profit i∗ with respect to the transfer price ti , yielding:    ∗ ∂ i∗ ∂ iB (ti , t j ) ∂qi ∂ pi∗ ∂vi ∂qi ∂ p j · =0 = + qi∗ + ti − + ∂ti ∂ti ∂qi ∂ pi ∂ti ∂ p j ∂ti i, j ∈ {1, 2}, i  = j.

(15)

The partial derivative of the marketing department’s profit with respect to the transfer price is given by ∗ ∂ iB (ti , t j ) ∂ iB ∂ pi∗ ∂ iB ∂ p j ∂ iB = + + ∂ti ∂ pi ∂ti ∂ p j ∂ti ∂ti ∗ ∂qi ∂ p j = 0 + ( pi∗ − ci − ti ) · − qi∗ , ∂ p j ∂ti

(16)

where the first term must be zero from (10) because we are looking for the impact of transfer pricing on the agents’ equilibrium strategies. The remaining two terms correspond to the indirect and direct effects of transfer pricing (as explained above). The last term ∂ iB /∂ti captures the direct reduction of divisional profit by an increase in marginal costs which is simply −qi∗ according to Hotelling’s Lemma. This effect would, however, also occur in a non-oligopolistic environment. The characteristic term for strategic transfer pricing is the second expression in (16). It is strictly positive from Lemma 1 and the assumptions about the demand system in (1), and captures the 13 Observe that the expression ∂ 2 i ∂qi = + ∂ pi ∂ p j ∂p j

 pi − ci −

∂vi ∂qi

 ·

∂ 2 vi ∂qi ∂qi ∂ 2 qi − · · ∂ pi ∂ p j ∂qi2 ∂ pi ∂ p j

is strictly positive from the assumptions regarding the demand system in (1) and the cost function in (5).

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profit impact of the competitor’s reaction to a change in division Bi ’s cost situation. Substituting (16) into (15) and solving for ti yields the following equilibrium solution of the transfer pricing game:   ∂vi ∗ α · pi − ci − ∗ ∂vi ∂qi ∂ pi∗ ∂qi ∂ p j ∂qi ti∗ = − > 0, β = < 0. (17) , α= ∂qi β ∂ p j ∂ti ∂ pi ∂ti Proposition 1 The optimal transfer price exceeds the marginal cost of the intermediate product. Proof From Lemma 1 and (1) α > 0 and β < 0. Therefore we can conclude that the second term in (17) is negative. ✷ Substituting the optimal transfer price into the manager’s first-order condition (10) and solving for pi yields the new equilibrium prices in the decentralized setting: pi∗ =

|εi | · (ci + ti∗ ) > pim . |εi | − 1

Since ti∗ > ∂vi /∂qi , these are obviously higher than in the centralized game as given by equation (8). Moreover, from (12) both firms must also earn strictly higher profits than in the centralized setting. Thus, strategic transfer pricing can be regarded as an implicit form of tacit collusion because the overall intensity of competition on the final product market is lower than under centralized competition. However, the new equilibrium can only be achieved in the decentralized setting. While both firms’ headquarters can commit their marketing managers to act as ‘softer’ competitors on the product market by charging a transfer price above the marginal cost of the intermediate product, headquarters could not credibly commit themselves to the same strategy because the intermediate products’ marginal cost are exogenous parameters of the firms’ profit functions. Hence, any deviation from the Nash strategy in the centralized setting would not be credible because it would not be profit maximizing. One may, however, consider the possibility of replacing strategic transfer pricing by mandated final product market prices. To check the feasibility of this policy, assume that both firms simply announce a price schedule, denoted with pi , to their marketing managers. Under these conditions the only task of the managers would be to communicate the price schedule to the firms’ customers. Thus, the profit-maximizing price schedule of firm i can be found by solving the following maximization problem max i = ( pi − ci ) · qi ( pi , p j ) − vi (qi ( pi , p j )) − Fi ,

(18)

pi

where p j denotes the competitor’s mandated pricing rule. From the solution of the firm’s maximization problem in (18) we readily obtain the following proposition: Proposition 2 The final product market equilibrium induced by strategic transfer pricing cannot be replicated by a mandated price schedule. Proof The first-order condition of problem (18) is given by   ∂ i ∂qi ∂vi · = qi ( pi , p j ) + pi − ci − =0 i, j ∈ {1, 2}, ∂ pi ∂qi ∂ pi

i  = j.

(19)

From (19) and (7) it is obvious that the equilibrium price schedule equals the equilibrium strategy of the centralized firm. ✷

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R1(p2)

π1K

R1(p2, t*1) R2(p1, t*2)

K R2(p1)

π1N N

π2N

π2K

(p1) Figure 1. Strategic transfer pricing equilibrium.

It follows immediately from Proposition 2 that both firms strictly prefer decentralization rather than centralization because there is no way to achieve the collusive market outcome by a centralized decision rule. A graphical analysis of the strategic transfer pricing equilibrium for the case of a linear demand system is depicted in Figure 1. The analysis starts at point N , the Nash equilibrium under centralized decisionmaking, which is determined by the intersection of the reaction functions R1 ( p2 ) and R2 ( p1 ). Note that these reaction functions would coincide with those of the managers when the intermediate product would be transferred at marginal cost. The corresponding equilibrium profits are given by the isoprofit curves π1N and π2N , respectively. When both headquarters adopt strategic transfer pricing policies, the managers’ reaction functions are shifted away from the origin to the new equilibrium point K where either firm’s isoprofit curve has a tangency point with its competitor’s reaction function. The new equilibrium is determined by the intersection of the reaction functions R1 ( p2 , t1∗ ) and R2 ( p1 , t2∗ ) with both firms charging higher prices than in the original equilibrium, and attaining higher profit levels which are depicted by the isoprofit curves π1K and π2K . 5. Unobservable transfer prices Game structure and equilibrium strategies So far it has been assumed that the managers observe their competitor’s transfer price before deciding on their pricing strategies for the final product. This may be a reasonable assumption for a small industry with similar products and relatively stable production and demand conditions where transfer prices are only subject to changes within longer periods.14 Since strategic transfer pricing is beneficial for both firms it may also be reasonable to suppose that the duopolists will disclose their transfer pricing practices 14 Empirical evidence on transfer pricing practices of German firms indicate that a substantial amount of firms are revising their transfer prices on an annual basis only (Coenenberg, 1992, p. 471).

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voluntarily. This does not necessarily require that transfer pricing methods have to be reported in financial statements which would barely be consistent with common company practice. As will be shown in the latter part of this section, a proper choice of the firms’ accounting systems can serve similar purposes. However, in order to stress the importance of information about the transfer pricing practice of the competitor, we will first consider the extreme case of unobservable transfer prices. Like the game with observable actions, the game with unobservable actions has two stages. The first stage consists of the two headquarters simultaneously choosing their transfer prices ti and t j . In contrast to the original game with perfect information, the agents are asymmetrically informed about the outcome of the first stage. Each agent can only observe the transfer price of his own firm, not the transfer price of the competitor, when deciding on his pricing strategy. Hence, neither firm’s headquarters can expect the other firm’s manager to react directly to its transfer price since agent Bi cannot observe t j . Nevertheless, each firm can still be influenced on the pricing strategy of its own manager by deviating from the intermediate product’s marginal cost. Such manipulations must, however, be an equilibrium strategy for the firm’s headquarters because otherwise a deviation from the marginal cost of the intermediate product would lead the marketing manager to suboptimal pricing decisions and thereby to a reduction of the firm’s profit. Since the only proper subgame of the modified game is the entire game itself, backward induction cannot be applied to solve for the equilibrium. Rather, the equilibrium strategies of the four players must be based on rational conjectures about the other players’ strategies as in a simultaneous game. Let us first consider the market game between both managers. As in the observable transfer price game, both agents first maximize divisional profits with respect to prices. Since divisional profits are not functions of the competitors’ transfer prices, there is no difference between the firstorder conditions of the managers’ maximization problems in the cases of observable and unobservable transfer prices. However, the reaction functions given by (10) are no longer sufficient to compute the equilibrium strategies because neither manager knows the other firm’s transfer price. While agent Bi can derive his own reaction function pi∗ ( p j , ti ) from (10) as a function of the other agent’s price p j and his own transfer price ti , he can only predict the other agent’s profit-maximizing price strategy by forming a rational conjecture tˆj about the transfer price of his competitor. Given the presumed transfer price, manager Bi can compute his optimal pricing strategy by substituting the conjectured reaction function p ∗j ( pi , tˆj ) into his own reaction function and solving for pi . An equilibrium in the managers’ market game is then given by the following pair of prices: pi∗ ( p ∗j ( pi∗ , tˆj ), ti )

for

i, j ∈ {1, 2},

i  = j.

(20)

It should be immediate from (20) that the equilibrium price of manager Bi is no longer a function of his competitor’s transfer price t j as in the original game with observable transfer prices. Moreover, for an equilibrium of the whole game the managers’ conjectures must be consistent with the profit-maximizing transfer prices. These are obtained as the solution of the following problem faced by both firms’ headquarters max i = iB ( pi∗ ( p j , ti ), p j , ti ) + ti · qi ( pi∗ ( p j , ti ), p j ) − vi (qi ( pi∗ ( p j , ti ), p j )) − Fi . ti

(21)

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In contrast to the observable transfer pricing game of the previous section, the profit expression in (21) is a function of agent Bi ’s reaction function pi∗ ( p j , ti ) and the competitor’s price p j because H Q i can still manipulate the pricing strategy of agent Bi but not the pricing strategy of agent B j by the choice of its transfer price. Since agent B j cannot observe ti , the game between H Q i and B j is equivalent to a simultaneous move game, although the transfer price is actually set in advance.15 Hence, the first-order condition for the firm’s optimization problem is given by:   ∂ iB ∂ pi∗ ∂ iB ∂ i ∂qi ∂ pi∗ ∂vi · = + + qi∗ + ti − = 0. (22) ∂ti ∂ pi ∂ti ∂ti ∂qi ∂ pi ∂ti Since the strategic effect of the original game in (16) vanishes, the first term must be zero from the agent’s first-order condition. Moreover, since ∂ iB /∂ti = −qi∗ according to Hotelling’s lemma, the optimality condition (22) simplifies to tim =

∂vi . ∂qi

(23)

As in the traditional Hirshleifer model, the optimal transfer price equals the marginal cost of the intermediate product. A deviation from this policy would only result in a suboptimal pricing strategy of manager Bi without offering any strategic benefits. Moreover, the only conjectures consistent with this result are tˆi = tim .

(24)

Together the conditions (20), (23), and (24) define the unique Nash equilibrium of the whole game. Hence, given the information structure of the game it can neither be optimal for one of the firms to use a strategic transfer price nor can it be rational for one of the agents to conjecture a strategic deviation from the intermediate product’s marginal cost. Absorption cost-based transfer pricing as a strategic device The introduction of a new accounting system usually requires substantial investments into software, employee training, or consulting services. Accordingly, the choice of an accounting system possesses the character of a long-term commitment and is very likely to be observed by the competitors in a small market. Moreover, empirical studies indicate that a remarkable number of firms are using absorption cost systems for various purposes such as product costing, performance evaluation and also for transfer pricing.16 Thus, it seems reasonable to assume that the cost system choice also determines the nature of transfer prices utilized by a firm. In particular, this should be true for firms using standardized cost accounting software like SAP which is usually based on a firm-wide database that makes it difficult to use different cost figures for different purposes. Suppose that both competitors must make a publicly observable commitment to an accounting system before determining their transfer prices. After deciding on the 15 Indeed the fact that a subsequent player cannot observe his predecessor’s action in a sequential game is equivalent to playing simultaneously, see Fudenberg and Tirole (1991, p. 70). 16 According to Horngren et al. (1994, p. 872), between 37 and 41 per cent of U.S. firms are using transfer prices based on full costs for their internal transactions.

Strategic Transfer Pricing, Absorption Costing, and Observability

Firms choose accounting system

Managers observe accounting systems

Firms set transfer prices

339

Managers set product prices

Figure 2. The accounting system choice game.

nature of the accounting system the firms compute the transfer prices according to the rule specified by the accounting system, and communicate them to their managers. Finally, the managers decide on the final product market prices based on their own transfer price for the intermediate product and their knowledge about the competitor’s accounting system. Thus, we consider a three-stage game consisting of the accounting system choice on stage one and the well-known transfer pricing and final product market choices on stages two and three as exhibited in Figure 2. Equilibrium under complete information Assume there are only two cost system alternatives under consideration: absorption costing and variable costing. Accordingly, the set of actions on the first stage contains only two elements, i.e. the firms’ strategies are si ∈ {ai , m i } where si = ai if firm i chooses absorption costing, and si = m i if firm i chooses variable costing. Suppose further, that each accounting system contains a unique rule for the determination of transfer prices as stated in (25): ti (m i ) := tim = vi (qi∗ (tim , t sj )),

ti (ai ) := tia =

Ci (qi∗ (tia , t sj )) qi∗ (tia , t sj )

,

(25)

where qi∗ (·) is the equilibrium demand for product i on the final product market, and ti (si ) := t sj is the competitor’s transfer price under cost system si . It should be obvious that the firms’ second-stage decisions are solely determined by the outcome of the first-stage game under these conditions. Moreover, since tim  = tia except for special cases, the accounting rule will most likely have an impact on the managers’ equilibrium pricing strategies on stage three. In particular, if tia > tim , both managers will charge higher prices on the final product market under absorption costing than under marginal costing. In light of the preceding results regarding strategic transfer pricing, absorption costing may therefore be beneficial for firms facing duopolistic price competition, whereas conventional wisdom suggests that the use of full costbased transfer prices is not optimal for divisionalized firms. If tia < tim , however, there will be no strategic advantage because any transfer price below the marginal cost of the intermediate product will force the marketing managers to behave more aggressively in the market game by lowering their final product prices. As a consequence, the resulting firm profits would fall below the benchmark profit under centralized decision-making. Thus, a necessary condition for the optimality of absorption costing is tia =

vi (qi∗ (tia , t sj )) + Fi qi∗ (tia , t sj )

> tim .

(26)

It should be obvious that condition (26) is satisfied by a linear cost function but it will also be satisfied in the case of a cost function with increasing marginal costs if the

340

R. F. Gox ¨

fixed costs are sufficiently large which seems to be a very reasonable assumption if we consider the cost structure of modern manufacturing industries.17 To emphasize the difference between the original two-stage game and its modified three stage version, assume for now that, as before, all parameters of the firms’ profit functions are common knowledge. Although this assumption will be relaxed in the next subsection, it makes it easier to understand the basic trade-off driving the firms’ cost system choices. Since the outcome of the first stage game becomes known before the market stage of the game, the managers can infer the corresponding transfer price rule (25) from their knowledge about the other firm’s accounting system and perfectly predict its transfer price even if it is not observable per se. Nevertheless, the situation is different from the observable transfer pricing equilibrium in Section 4 because once the cost system is fixed, the firms have no degree of freedom in adjusting their transfer prices according to strategic requirements. Rather, the firms restrict themselves to make an exclusive choice between tim and tia by their commitment to a particular accounting system. To determine a unique equilibrium of the first-stage game, we must identify conditions under which firm i would prefer absorption costing over variable costing regardless of firm j’s accounting system choice. Hence, we are looking for a subgameperfect dominant strategy equilibrium of the modified game. One may also consider the weaker requirement of a subgame-perfect Nash equilibrium, but a dominant strategy equilibrium seems more appropriate in light of robustness. Obviously, absorption costing is the dominant system choice when the following condition holds: i (ai , s j ) ≥ i (m i , s j )

for

s j ∈ {a j , m j }.

(27)

Thus, we require that both i (ai , a j ) ≥ i (m i , a j ), and i (ai , m j ) ≥ i (m i , m j ). Fortunately, we can restrict the subsequent analysis to the latter condition because it is easy to demonstrate that it implies the former. To verify the claim, we first utilize the fact that the accounting system decision of firm i implicitly determines a unique final product market price for a given accounting system of firm j, allowing us to formulate the following necessary condition for our claim i ( pia , paj ) − i ( pim , paj ) ≥ i ( pia , p mj ) − i ( pim , p mj ), or equivalently 

pia pim

∂ i ( pi , paj ) ∂ pi

 d pi >

pia pim

∂ i ( pi , p mj ) ∂ pi

dpi ,

(28)

where pia denotes the price under absorption costing, and pim denotes the price under variable costing which is equivalent to the optimal price under centralized decisionmaking. Obviously, pia must exceed pim if (26) holds because the marketing manager faces a higher marginal cost curve under absorption costing than under variable costing. Given that pia > pim , it should be evident that (28) is indeed true because prices are strategic complements (i.e. ∂ i ( pi , p j )/∂ pi ∂ p j > 0) from (12). Thus, we can draw the following conclusion. If firm i prefers absorption costing while its competitor relies 17 In particular, we can equivalently determine a lower bound for the fixed cost by rearranging terms in (26): Fi ≥ tim · qi∗ (tia , t sj ) − v(qi∗ (tia , t sj )).

Strategic Transfer Pricing, Absorption Costing, and Observability

341

on marginal costing, it will also prefer absorption costing if the competitor switches to absorption costing as well. However, since tia is solely determined by the firm’s cost structure by means of the rule specified in (25), it is not assured that absorption costing will always increase the firm’s profit. In particular, we can make the following observation: Proposition 3 If the cost structure of firm i is such, that tia (determined according to the rule specified in (25)) lies in the interval T = (tim , ti+ ), where ti+ solves + m m m m m i ( pi (ti+ , t m j ), p j (ti , t j )) = i ( pi (ti , t j ), p j (ti , t j )),

absorption costing is a dominant strategy equilibrium. Furthermore, if i (·) is not only concave but also symmetric, ti+ = 2ti∗ − tim . Proof See the Appendix. ✷ The intuition behind Proposition 3 is easily explained: observe first, that (27) trivially holds, if tia = ti∗ because in this case the full cost-based transfer price simply replicates the optimal solution of the original two-stage game in Section 4. Whereas under variable costing, the firm would only attain the profit of a centralized firm. Since ti∗ > tim from (17), it should be obvious that firm i would also prefer absorption costing if its competitor transferred its intermediate product at the marginal cost. Furthermore, since the firms’ profit functions are strictly concave in their own transfer prices, condition (27) must hold for all transfer prices between the marginal cost of the intermediate product and the strategic transfer price ti∗ given by (17). However, the concavity of the profit function also guarantees that condition (27) will hold for some transfer prices tia > ti∗ because the attainable profit will still be higher than under marginal costing. However, there exists a critical full cost transfer price ti+ > ti∗ for which the profit under absorption costing exactly equals the profit under marginal costing. Accordingly, ti+ is obtained as one of the two solutions of + m m m m m i ( pi (ti+ , t m j ), p j (ti , t j )) = i ( pi (ti , t j ), p j (ti , t j )),

(29)

where the other solution is obviously tim . Finally, if the profit function is not only concave but also symmetric ti∗ − tim = ti+ − ti∗ , or equivalently, ti+ = 2ti∗ − tim . The preceding arguments are illustrated in Figure 3. Consider the situation of firm i when firm j relies on marginal costing and sets the transfer price t j = tim . The profit of firm i for this case is depicted by the curve i (ti , t m j ). m Since the profit of firm i is strictly concave in its own transfer price and ti is lower than the optimal transfer price at the peak of the curve, a shift towards the right from tim up to any absorption cost-based transfer price tia between tim and ti∗ will obviously increase the firm’s profit. Moreover, even if tia > ti∗ , the profit is still higher than under variable costing unless tia exceeds the critical value of ti+ which is found by solving (29). Once ti+ is found, we can also determine an upper bound for the fixed costs of firm i that ensures the existence of a dominant strategy equilibrium. In particular, rearranging the rule for the computation of tia in (25) yields: ∗ + m ti+ · qi∗ (ti+ , t m j ) − v(qi (ti , t j )) ≥ Fi .

(30)

342

R. F. Gox ¨ i (ti,tj )

i (ti,t aj )

i (ti,t mj )

tim

ti+

ti++

(ti)

Figure 3. Admissible full cost-based transfer prices.

Thus, if the cost function of firm i satisfies (30), tia ≤ ti+ , and firm i will prefer absorption costing even if firm j employs variable costing. As demonstrated above, this implies that firm i will also prefer absorption costing if firm j also switches to absorption costing. The profit of firm i when firm j uses absorption costing is depicted by the curve i (ti , t aj ) in Figure 3. Since prices are strategic complements, the profit curve of firm i will incur an upward shift if firm j employs absorption costing. Thus, the critical transfer price for which firm i is indifferent between absorption costing and marginal costing will also move to the right, where the critical transfer price under symmetric absorption costing , ti++ , is obtained as one of the two solutions of i ( pi (ti++ , t aj ), p j (ti++ , t aj )) = i ( pi (tim , t aj ), p j (tim , t aj )). However, despite the fact that both competitors would attain higher profits by charging transfer prices tia ∈ (ti+ , ti++ ), absorption costing is only a dominant strategy equilibrium if tia ≤ ti+ for i = 1, 2. Since (27) is violated for t sj = t m j there exists no dominant strategy + ++ a equilibrium for ti ∈ (ti , ti ). Under these conditions, the best response to variable costing is variable costing. However, since this observation is also true for absorption costing, the accounting system choice game has two Nash equilibria, namely (si = ai ) and (si = m i ) for tia ∈ (ti+ , ti++ ).18 Although one may argue that the firms would prefer to play the pareto superior full cost equilibrium, it is generally not clear which equilibrium will be chosen when the cost system choice is leading to transfer prices within the range ti+ and ti++ . 18 In particular, for t a ∈ (t + , t ++ ) the following conditions are satisfied i i i i (ai , a j ) ≥ i (m i , a j )

and

i (m i , m j ) ≥ i (ai , m j ).

Thus, if both firms choose absorption costing, no firm has an incentive to deviate but the same is true if both firms choose variable costing.

Strategic Transfer Pricing, Absorption Costing, and Observability

343

Equilibrium under incomplete information So far we have assumed that both firms know each otherís profit functions with certainty making the prediction of the cost-based transfer prices a fairly trivial task for the marketing managers. To make things interesting, we will now drop this assumption and assume that neither firm knows the precise amount of the other firm’s fixed cost. In what follows, we will therefore assume that F˜i is a random variable from the perspective of firm j that can take values from the finite set Fi = {Fi1 , . . . , Fik , . . . , Fim } with probability φik > 0, where Fi1 < Fi2 < · · · < Fin . Furthermore, we assume that both firms have common priors, i.e. we suppose that the distributions of fixed costs are common knowledge, and that each marketing manager can perfectly predict the expectations of its competitor. The players now face a game of incomplete information that can be solved using standard techniques by treating each firm with a different cost type k as a separate player.19 Thus, we need to define a separate strategy for each player of type k in order to solve for the Bayesian equilibrium of the modified game. Accordingly, the expected profit of division B in firm i with cost type k, and under cost system si is given by B s ˆ ik = ( pik − ci − tik )·

n 

φil · qik ( pik , p jl ),

(31)

l=1 s indicate that each cost type where the price pik , the quantity qik , and the transfer price tik k raises a different profit maximization for the marketing manager of firm i. Fortunately, we can simplify the analysis somewhat because there is no uncertainty about the variable m = t m for all k if both firms prefer variable cost of production. Thus, pik = pim and tik i costing, implying that the profit of firm i is still given by i (m i , m j ) := i ( pim , p mj ). To address the effect of cost uncertainty on the managers’ equilibrium strategies, we will next consider the critical case of firm i employing absorption costing when firm j prefers variable costing. Due to the lack of uncertainty over the profit function of firm j the managers are facing the following profit maximization problems: B a ˆ ik max = ( pik − ci − tik ) · qik ( pik , p j ), pik

ˆ Bj = ( p j − c j − t m max j )· pj

n 

φik · q j ( p j , pik ).

(32) (33)

k=1

Accordingly, the equilibrium conditions are given by the following system of n + 1 equations: ˆB ∂ ∂qik ik a = qik ( pik , p j ) + ( pik − ci − tik )· = 0 ∀ k ∈ {1, . . . , n}, ∂ pik ∂ pik   n  ˆB ∂ ∂q j ik a = 0. = φik q j ( p j , pik ) + ( p j − ci − tik ) · ∂pj ∂pj k=1

(34) (35)

Consider first the problem of manager j. To determine his equilibrium strategy, he will first anticipate the optimal prices of all n different manager types of firm i and substitute a ) obtained from (34) into his first-order the n resulting reaction functions pik ( p j , tik 19 This well-known ‘trick’ goes back to Harsanyi, see Fudenberg and Tirole (1991, p. 209) for details.

344

R. F. Gox ¨

condition (35). Then he will assign the corresponding probabilities φik to the resulting a )), and solve (35) for p m := p (t m , ta ), where type contingent demands q j ( p j , pik ( p j , tik j j j i a a a a ti := (ti1 , . . . , tik , . . . , tin ) denotes the vector of firm i’s type contingent transfer prices. Manager ik, in turn, anticipates j’s decision rule, substitutes the anticipated equilibrium a a a m price p j (t m j , ti ) into his first-order condition (34), and solves it for pik := pik (ti , t j ). Formally spoken, both managers’ equilibrium strategies are functions of t m j and the vector of full cost-based transfer prices tia of all possible cost types. Thus, we can rewrite the relevant case of condition (27) to obtain a necessary condition for the existence a Bayesian dominant strategy equilibrium in absorption costing:20 a m m m m m m m ˆ ik ( pik (tia , t m j ), p j (ti , t j )) − i ( pi (ti , t j ), p j (ti , t j )) ≥ 0.

(36)

a > tm To check if condition (36) is satisfied, recall that condition (26) implies tik i a m m m m m a m m m for all k. Accordingly, pik (ti , t j ) > pi (ti , t j ) and p j = p j (ti , t j ) > p j (t m j , ti ) from Lemma 1. Thus, both managers will raise their product prices even if manager j cannot perfectly predict the transfer price of manager i, or equivalently, he does not know the cost type of his opponent. Since prices are strategic complements, both firms will, in general, benefit from this situation. As in the full information case, however, firm i will prefer absorption costing only if the resulting product price, or equivalently, the transfer price under absorption costing falls in a certain range that is solely determined by the cost function properties. In particular, due to the concavity of the profit function firm i will always prefer absorption costing as long as ∗ m ∗ m pik (tia , t m j ) ≤ pi (ti , t j ), where pi (ti , t j ) denotes the equilibrium price under observable strategic transfer pricing by firm i. But as before, absorption costing will also be ∗ m preferable for firm i for some pik (tia , t m j ) > pi (ti , t j ) up to the point where the resulting m m m m m m profit equals i ( pi (ti , t j ), p j (ti , t j )). Thus, we can conclude that the basic results derived for the full information case will also hold when there is incomplete information about the properties of the cost function.

6. Summary This paper has analysed the use of transfer pricing as a strategic device in divisionalized firms facing duopolistic price competition. When transfer prices are observable, both firms’ headquarters will charge a transfer price above the marginal cost of the intermediate product to induce their marketing managers to behave as softer competitors on the final product market. Since the resulting equilibrium profits strictly exceed the profits attainable by two centralized firms, the result provides an economic rationale for the use of transfer pricing that is not available in traditional transfer pricing models. 20 It should be obvious from (28), that (36) implies that absorption costing is also preferable for firm i when firm j also uses absorption costing. Obviously, we can derive a similar condition for the incomplete information case:  pa ∂ ( p , pa ) ik ik ik j pim

∂ pik

d pik >

 pa ∂ ( p , p m ) ik ik ik j pim

∂ pik

d pik ,

where pajk := ( paj1 , . . . , pajn ) denotes the vector of type contingent full cost prices of firm j. Evidently, the modified condition must hold because prices are strategic complements.

Strategic Transfer Pricing, Absorption Costing, and Observability

345

When transfer prices are not observable, strategic transfer pricing is not an equilibrium, and the optimal transfer price equals the marginal cost of the intermediate product, as in the classical Hirshleifer model. The reason is that neither firm can manipulate the equilibrium strategy of the other firm’s manager and thus, a deviation from marginal cost would only result in a suboptimal pricing decision by the own manager. Nevertheless, as a strategic alternative the firms can signal a transfer price above the marginal cost to their competitor by a proper choice of its accounting system. This paper identifies quite general conditions under which the choice of absorption costing is a dominant strategy equilibrium. There are several other aspects of strategic transfer pricing that have not been addressed in this paper. First of all, the analysis is restricted to the case of price competition on the final product market. Obviously, the results do not carry over to the case of quantity competition because quantities are strategic substitutes, and thus, in a game with observable transfer prices the firms would rather like to subsidize the intermediate product than charge a strategic markup. Accordingly, absorption costing would not be beneficial under quantity competition. However, despite the fact that the issue of whether firms are actually using quantities or prices as strategic variables has not been resolved yet, it seems reasonable to assume that firms compete with prices. Even if this might not be true for all markets, we frequently observe companies like desktop computer producers, car manufacturers or service providers announcing product prices to their customers rather than simply fixing production quantities. Another issue that has not been considered in this paper is the existence of market alternatives for the intermediate product. However, it is easy to demonstrate that the basic results of the model would carry over to the well-known Hirshleifer case of a perfectly competitive market for the intermediate products.21 Last but not least, it might be promising for future research to analyse strategic transfer pricing in a private information setting. In particular, as mentioned in the introduction, the earlier adverse selection literature has demonstrated the optimality of ‘cost plus’ type transfer pricing schemes to mitigate agency problems. Thus, both strategic incentives and agency problems might be addressed with similar instruments. Moreover, if both firms anticipate their competitor’s agency problems, the problem of creating managerial equilibrium incentives with unobservable transfer prices could be resolved without a commitment to a particular cost system. Acknowledgements: The author acknowledges helpful comments from Alfred Wagenhofer, Rick Young, Gilad Livne, Michael Bromwich (the editor), three anonymous reviewers, and seminar participants at UC Berkeley, London Business School, University of Tuebingen, the EIASM workshop on Accounting and Economics III at the London School of Economics, and the 1999 Management Accounting Research Conference in Orlando, Florida. All remaining mistakes are the author’s responsibility.

References

Alles, M. and Datar, S., 1998. Strategic transfer pricing, Management Science, 44, 451–461. 21 An earlier version of this paper has considered this case. There it has been demonstrated that under certain restrictive assumptions, a firm may find it optimal to integrate an inefficient producing division in order to signal its competitor a transfer price above the marginal cost, see Gox ¨ (1998) for details.

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Amershi, A. H. and Cheng, P., 1990. Intrafirm resource allocation: the economics of transfer pricing and cost allocations in accounting, Contemporary Accounting Research, 7, 61–99. Bagwell, K., 1995. Commitment and observability in games, Games and Economic Behavior, 8, 271–280. Banker, R. D. and Datar, S. M., 1992. Optimal transfer pricing under postcontract information, Contemporary Accounting Research, 8, 329–352. Brander, J. A. and Spencer, B. J., 1983. Strategic commitment with R&D: the symmetric case, Bell Journal of Economics, 14, 225–235. Bulow, J. I., Geanakoplos, J. D. and Klemperer, P. D., 1985. Multimarket oligopoly: strategic substitutes and complements, Journal of Political Economy, 93, 488–511. Christensen, J. and Demski, J. F., 1989. Transfer Pricing in a Limited Communication Setting, Working Paper, Odense and Yale University. Coenenberg, A. G., 1992. Kostenrechnung und Kostenanalyse, Landsberg am Lech, MI Verlag. Dixit, A., 1986. Comparative statics for oligopoly, International Economic Review, 27, 107–122. Edlin, A. S. and Reichelstein, S., 1995. Specific investment under negotiated transfer pricing: an efficiency result, Accounting Review, 70, 275–291. Fershtman, C. and Judd, K. L., 1987. Equilibrium incentives in oligopoly, The American Economic Review, 77, 927–940. Fudenberg, D. and Tirole, J., 1991. Game Theory, Cambridge and London, MIT Press. Gal-Or, E., 1993. Strategic cost allocation, The Journal of Industrial Economics, 39, 387–402. Gox, ¨ R. F., 1998. Strategic Transfer Pricing, Absorption Costing, and Vertical Integration, Working Paper, University of Magdeburg. Hirshleifer, J., 1956. On the economics of transfer pricing, Journal of Business, 172–184. Horngren, C. T., Foster, G. and Datar, S. M., 1994. Cost Accounting - A Managerial Emphasis, 8th edition, Prentice Hall, Englewood Cliffs. Hughes, J. S. and Kao, J. L., 1997a. Strategic forward contracting and observability, International Journal of Industrial Organization, 16, 121–133. Hughes, J. S. and Kao, J. L., 1997b. Cross subsidization, cost allocation, and tacit coordination, Review of Accounting Studies, 2, 265–293. Jaquemin, A. and Slade, M. E., 1989. Cartels, Collusion and Horizontal Merger, in R. Schmalensee, R. D. Willig (eds), Handbook of Industrial Organization, Volume I, 415–473. Kaplan, R. S. and Atkinson, A. A., 1998. Advanced Management Accounting, 3rd edition, Prentice Hall, Upper Saddle River. Katz, M. L., 1991. Game-playing agents: unobservable contracts as precommitments, Rand Journal of Economics, 22, 201–228. Narayanan, V. G. and Smith, M., 1998. Impact of Competition and Taxes on Responsibility Center Organization and Transfer Prices, Working Paper, Harvard University and Duke University. Ronen, J. and Balachandran, K. R., 1988. An approach to transfer pricing under uncertainty, Journal of Accounting Research, 26, 300–314. Schelling, T. C., 1960. The Strategy of Conflict, Cambridge. Sklivas, S. D., 1987. The strategic choice of managerial incentives, Rand Journal of Economics, 18, 452–458. Tirole, J., 1988. The Theory of Industrial Organization, Cambridge and London, MIT Press. Vaysman, I., 1996. A model of cost-based transfer pricing, Review of Accounting Studies, 1, 73– 108. Vickers, J., 1985. Delegation and the theory of the firm, Economic Journal, 95, 138–147. Wagenhofer, A., 1994. Transfer pricing under asymmetric information - an evaluation of alternative methods, European Accounting Review, 1, 71–104.

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Appendix Lemma 1 The equilibrium prices chosen by the agents in the market game are strictly increasing in both firms’ transfer prices. Proof Totally differentiating the system of implicit reaction functions ∂ iB ( pi , p j , ti ) = 0, ∂ pi

i, j ∈ {1, 2},

i = j

defined in (10) yields ∂ 2 1B ∂ p12

d p1 +

∂ 2 1B ∂ 2 1B d p2 + dt1 = 0 ∂ p1 ∂ p2 ∂ p1 ∂t1

∂ 2 2B ∂ 2 2B ∂ 2 2B d p1 + d p + dt2 = 0. 2 ∂ p2 ∂ p1 ∂ p2 ∂t2 ∂ p22 Applying Cramer’s rule to the system of total differentials yields   2 B 2 B ∂ 2 iB ∂ 2 iB 1 ∂ j ∂ j d pi = dt j − dti , k ∂ p j ∂ pi ∂ p j ∂t j ∂ pi2 ∂ pi ∂ti where k=

∂ 2 1B ∂ 2 2B ∂ p12

∂ p22

−

∂ 2 2B ∂ 2 1B >0 ∂ p2 ∂ p1 ∂ p1 ∂ p2

from assumption (4). Since ∂ 2 Bj ∂ p j ∂ pi

> 0,

∂ 2 iB ∂qi =− >0 ∂ pi ∂ti ∂ pi

and

∂ 2 iB ∂ pi2

<0

from (12), (1) and the concavity of the profit function and d pi =

∂ pi∗ (ti , t j ) ∂ p ∗ (ti , t j ) dti + i dt j , ∂ti ∂t j

we obtain the desired results for the signs of the strategic effects:  ∂ pi∗ (ti , t j ) d pi  1 ∂ 2 iB ∂ 2 iB = = − >0 ∂ti dti dt j =0 k ∂ pi2 ∂ pi ∂ti  2 B 2 B ∂ pi∗ (ti , t j ) d pi  1 ∂ j ∂ j = = > 0. ✷ ∂t j dt j dti =0 k ∂ p j ∂ pi ∂ p j ∂t j Proposition 3 If the cost structure of firm i is such, that tia (determined according to the rule specified in (25)) lies in the interval T = (tim , ti+ ), where ti+ solves + m m m m m i ( pi (ti+ , t m j ), p j (ti , t j )) = i ( pi (ti , t j ), p j (ti , t j )),

absorption costing is a dominant strategy equilibrium. Furthermore, if i (·) is not only concave but also symmetric, ti+ = 2ti∗ − tim .

348

R. F. Gox ¨

Proof Suppose that (26) holds, implying that tia > tim , and observe next that m m ∗ ∂ i ( pi (ti , t m j ), p j (ti , t j )) > 0 for ti < ti < ti ∂ti < 0 for ti∗ < ti

(37)

since i (·) is strictly concave, and ti∗ is the unique maximizer of i (·). Thus, if tia < ti∗ the profit difference between absorption costing and marginal costing is given by: a m m m m m  i = i ( pi (tia , t m j ), p j (ti , t j )) − i ( pi (ti , t j ), p j (ti , t j ))  t a ∂ ( p (t , t m ), p (t , t m )) i i i i j j i j = dti > 0. m ∂t i ti

Suppose next, that tia > ti∗ then   i =

ti∗

m ∂ i ( pi (ti , t m j ), p j (ti , t j ))

∂ti

tim

 dti +

tia

m ∂ i ( pi (ti , t m j ), p j (ti , t j ))

∂ti

ti∗

dti > 0 (38)

as long as the following condition holds: 

ti∗

tim

m ∂ i ( pi (ti , t m j ), p j (ti , t j ))

∂ti

 dti > −

tia

m ∂ i ( pi (ti , t m j ), p j (ti , t j ))

∂ti

ti∗

dti .

(39)

Assume now that tia = ti∗ + ε such that (39) holds. Since the left-hand side of (39) is constant, while the right-hand side is increasing with ε, there exists a critical value ε + or equivalently a critical full cost transfer price ti+ = ti∗ + ε + for which 

ti∗

tim

m ∂ i ( pi (ti , t m j ), p j (ti , t j ))

∂ti

 dti = −

ti+

m ∂ i ( pi (ti , t m j ), p j (ti , t j ))

ti∗

∂ti

Furthermore, if i (·) is symmetric ti∗ − tim = ti+ − ti∗ ⇔ ti+ = 2ti∗ − tim .

✷

dti .

(40)