Review of Accounting Studies, 5, 27–56 (2000) c 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Intrafirm Trade, Bargaining Power, and Specific Investments TIM BALDENIUS [email protected]Graduate School of Business, Columbia University, New York, NY 10027 Abstract. This paper compares the performance of standard-cost with negotiated transfer pricing under asymmetric information. Negotiated transfer pricing generally achieves higher expected contribution margins, as this method tends to be more efficient in aggregating private information into a single transfer price. Standard-cost transfer pricing confers more bargaining power to the supplier and therefore generates better incentives for this division to undertake specific investments. The opposite holds for buyer investments. If a corporate controller has disaggregated information about divisional costs and revenues, then the firm can improve upon the performance of standard-cost transfer pricing by setting a centralized transfer price equal to expected cost plus a suitably chosen mark-up. Keywords: Transfer pricing, asymmetric information, specific investments, hold-up problem Intracompany transfers of intermediate products are often conducted under conditions of asymmetric information. The manager of the supplying division frequently has private information about the cost of producing the intermediate good, whereas the manager of the buying division has better knowledge regarding the net revenues from selling the final product. Accounting textbooks tend to advocate negotiated transfer pricing as a mecha- nism that permits divisional managers to incorporate their local information into transfer pricing and quantity decisions (Kaplan and Atkinson (1998, 461)). At the same time, the economics literature has established that negotiations under asymmetric information entail inefficiencies: profitable transactions are forgone as the parties strive for more favorable prices by overstating cost and understating revenues (Myerson and Satterthwaite (1983)). For the theory of transfer pricing, it does not seem to be well-understood how any inefficien- cies under negotiated transfer pricing compare with those under commonly-used alternative mechanisms. This paper conducts a performance comparison of negotiated and standard-cost transfer pricing in a setting of asymmetric information and incomplete contracting. 1 In our binary quantity model, the transfer pricing scheme serves two incentive purposes. First, divisional managers should transfer the intermediate good whenever this is profitable from a firm-wide perspective. The second purpose is to provide incentives for the divisions to undertake relationship-specific investments. For instance, the selling division may acquire equipment up-front in order to reduce its variable production cost or the buying division may invest in marketing activities. If a division is engaged in many different transactions, then investments can generally not be linked to certain products and hence are not contractible. In such a setting, negotiations have been shown to suffer from underinvestment—or “hold-up”—
Intrafirm Trade, Bargaining Power, and SpecificInvestments
TIM BALDENIUS [email protected] School of Business, Columbia University, New York, NY 10027
Abstract. This paper compares the performance of standard-cost with negotiated transfer pricing under asymmetricinformation. Negotiated transfer pricing generally achieves higher expected contribution margins, as this methodtends to be more efficient in aggregating private information into a single transfer price. Standard-cost transferpricing confers more bargaining power to the supplier and therefore generates better incentives for this divisionto undertake specific investments. The opposite holds for buyer investments. If a corporate controller hasdisaggregated information about divisional costs and revenues, then the firm can improve upon the performance ofstandard-cost transfer pricing by setting a centralized transfer price equal to expected cost plus a suitably chosenmark-up.
Keywords: Transfer pricing, asymmetric information, specific investments, hold-up problem
Intracompany transfers of intermediate products are often conducted under conditions ofasymmetric information. The manager of the supplying division frequently has privateinformation about the cost of producing the intermediate good, whereas the manager ofthe buying division has better knowledge regarding the net revenues from selling the finalproduct. Accounting textbooks tend to advocate negotiated transfer pricing as a mecha-nism that permits divisional managers to incorporate their local information into transferpricing and quantity decisions (Kaplan and Atkinson (1998, 461)). At the same time, theeconomics literature has established that negotiations under asymmetric information entailinefficiencies: profitable transactions are forgone as the parties strive for more favorableprices by overstating cost and understating revenues (Myerson and Satterthwaite (1983)).For the theory of transfer pricing, it does not seem to be well-understood how any inefficien-cies under negotiated transfer pricing compare with those under commonly-used alternativemechanisms.
This paper conducts a performance comparison of negotiated and standard-cost transferpricing in a setting of asymmetric information and incomplete contracting.1 In our binaryquantity model, the transfer pricing scheme serves two incentive purposes. First, divisionalmanagers should transfer the intermediate good whenever this is profitable from a firm-wideperspective. The second purpose is to provide incentives for the divisions to undertakerelationship-specific investments. For instance, the selling division may acquire equipmentup-front in order to reduce its variable production cost or the buying division may invest inmarketing activities. If a division is engaged in many different transactions, then investmentscan generally not be linked to certain products and hence are not contractible. In such asetting, negotiations have been shown to suffer from underinvestment—or “hold-up”—
28 BALDENIUS
problems: the investing division bears the entire investment costs while the other divisionreceives a share of the returns.2
In a recent study, Baldenius, Reichelstein, and Sahay (1999) develop a symmetric infor-mation model with continuous quantities to compare the performance of standard-cost andnegotiated transfer pricing. In their setting, negotiated transfer pricing is in many cases thedominant mechanism, primarily because it always yields efficient trade. The present paperreexamines this performance comparison under the assumption that divisional managershave private information. We show that trade distortions arising from bargaining underasymmetric information tend to “compound” the underinvestment problem. The reasonis that investment returns are determined by the probability that trade occurs. Negotiatedtransfer pricing therefore biases the divisional investments for two reasons: the total returnsfrom investing are suboptimal, and the investing party anticipates that it will have to sharethese returns in the negotiation process.
Under standard-cost transfer pricing, if the supplying division has discretion in reportingcosts, then this division essentially gains monopoly power and tends to report a transferprice well above its cost.3 This monopolistic behavior induces trade distortions, which, inturn, lower investment incentives along the lines described in Baldenius, Reichelstein, andSahay (1999). While, in practice in the present paper, the selling division may be con-strained in its ability to exaggerate cost, the model setup captures the essential notion thata standard-cost mechanism grants the seller a first-mover advantage and thereby confersmore bargaining power to this division, as compared with negotiations. As in the symmetricinformation model referred to above, we find that a system of standard-cost transfer pric-ing avoids hold-up problems for the supplier. At the same time, buyer investments sufferfrom severe hold-up problems since the transfer price set by the supplier will reflect anyinvestment undertaken by the buyer.
A central new insight provided by this paper is that trade is in many cases less efficientunder standard-cost transfer pricing than under negotiations. Standard-cost transfer pricingsuffers from the principal disadvantage that the price reflects only the seller’s information.Under negotiations, in contrast, the payment is responsive to both divisions’ information.One scenario where standard-cost transfer pricing entails fewer distortions than negotiationsoccurs when the supplier is “fairly knowledgeable” about the buyer’s revenues. Then thequantity traded under standard-cost transfer pricing approaches first-best, while negotiationsstill suffer from inefficiencies.
If only the seller invests, then the standard-cost mechanism approaches optimal perfor-mance in case of a fairly knowledgeable seller. However, if the buyer’s revenues are highlyuncertain to the seller, then the firm faces a tradeoff between higher seller investments un-der standard-cost transfer pricing versus more efficient trade under negotiations. In settingswhere only the buying division invests, the relative performance of standard-cost transferpricing degrades as the negotiation mechanism generally tends to dominate along bothdimensions—trade efficiency and investment incentives.
The analysis departs in several respects from existing literature. Most of the recent work ontransfer pricing has adopted a mechanism design approach: a central office designs optimalrules for allocating resources and compensating divisions based on their messages.4 Thepresent paper takes a more applied—and decentralized—view of intrafirm trade: the center
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 29
is confined to choosing generic transfer pricing rules that are essentially interpreted asmeans to allocate bargaining power among the divisions. The two candidate mechanismsare modeled as special cases of a general bargaining procedure with the only differencebeing that standard-cost transfer pricing confers more bargaining power to the supplier.Our analysis abstracts entirely from compensation issues. We assume throughout thatdivision managers seek to maximize their expected divisional income without modelingany underlying moral hazard problems.
Our results suggest the following pattern of how the firm should optimally allocate bargain-ing power among the divisions. To minimize hold-up problems, bargaining power shouldreside with the investing division: if the supplier invests, then in some cases standard-costtransfer pricing achieves higher expected firm profit; however, negotiated transfer pricingtends to dominate if the buyer invests. On the other hand, to minimize trade distortions,bargaining power should be given to the division that has “more private information.” Toverify this intuition, we investigate limit scenarios where only one division has privateinformation. We also show that private information partly shields a division’s investmentfrom hold-up problems: the other division has to bargain more cautiously, not knowing theinvesting party’s reservation price.
In the main part of the analysis, the central office does not have access to informa-tion necessary for playing an active role in setting the transfer price; instead it only as-signs the divisions certain rights and obligations. We also consider a model extensionwhere the corporate controller receives noisy signals about the division’s environments,which allows it to compute a centralized transfer price. It then turns out that the firmcan improve upon the performance of standard-cost transfer pricing by centrally settingthe transfer price equal to expected cost plus a suitably chosen mark-up. However, ifboth divisions have private information, the firm faces a tradeoff between negotiated andcentralized transfer pricing: negotiations yield more efficient trade while the central-ized mechanism avoids hold-up problems and hence generates better investment incen-tives.
Section 1 of this paper describes the basic model. In Section 2, the transfer pricingschemes are introduced. Sections 3 and 4 contain the performance comparison for settingswhere either the supplying or the buying division invests, respectively. In Section 5, we askwhether the central office should actively engage in setting the standard-cost transfer price,if it has some prior cost estimate. Section 6 concludes the paper.
1. The Model
We analyze a firm consisting of two profit centers. The supplying division (Division 1)manufactures an intermediate good and ships it to the buying division (Division 2), whichuses it as an input in its production process and ultimately sells a final product externally.The model involves four dates. At date 0, the firm commits to a transfer pricing scheme. Atdate 1, the managers decide upon their respective investment levels: the supplying divisionchoosesI1 from the interval [0, I 1], and the buying division choosesI2 ∈ [0, I 2]. Theseinvestments result in divisional fixed costs, denoted bywi (Ii ), i = 1,2. Investments areassumed observable to both divisional managers. At date 2, the managersprivately learn
30 BALDENIUS
Figure 1. Timeline.
their respective valuations (types): Manager 1 observes his production cost typeθ1, andManager 2 observes his net revenue typeθ2. At date 3, the divisional managers agreeon whether to trade the intermediate good,q ∈ {0,1}, and on the corresponding transferpayment,t ∈ R+.
The supplier’s costs of manufacturing the good areC(θ1, I1) = θ1 − I1. The buyer’snet revenues from trading areR(θ2, I2) = θ2 + I2. The random variablesθ1 andθ2 areindependently distributed with cumulative distribution functionsF1(θ1) and F2(θ2), bothof which are common knowledge among the divisions and have strictly positive densities,fi (θi ), over the respective supports2i ≡ [θ i , θi ]. We denote1i ≡ θi − θ i , θ ≡ (θ1, θ2)
and I ≡ (I1, I2). In the main part of the paper, neither investmentsIi nor the supports2i
are assumed verifiable to a corporate controller. This specification seems descriptive fordivisions that undertake many different transactions and investments.
Both managers are assumed risk neutral and motivated to maximize the expected profits oftheir own division.5 Expected divisional profits in our model consist of expected contributionmargins,Mi , less the fixed cost caused by the investment:
51 = M1(·)− w1(I1) =∫21
∫22
[t (θ, I )− (θ1− I1)] ·q(θ, I )d F2(θ2)d F1(θ1) − w1(I1)
52 = M2(·)−w2(I2) =∫21
∫22
[(θ2+ I2)−t (θ, I )] ·q(θ, I )d F2(θ2)d F1(θ1) − w2(I2)
for the selling and the buying division, respectively. The functions〈q(·), t (·)〉 are deter-mined by the transfer pricing scheme in place.
Clearly, from the viewpoint of the firm as a whole, the intermediate good should be tradedwhenever the buyer’s valuation exceeds the seller’s cost:
q∗(θ, I ) = 1 if and only if θ2+ I2 ≥ θ1− I1. (1)
Equation (1) characterizes the efficient trading rule chosen by the firm in case of completeinformation and centralized decision making. Denoting byEθ [·] the expectation operatorwith respect toθ , we shall assume that 0< Eθ [q∗(θ, I )] < 1, for all I , to avoid trivialintrafirm transfers. This is equivalent to assuming that the relevant cost and revenue supportsintersect for all investments.
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 31
Suppose that only the seller can undertake specific investments. The relevant costs andrevenues then are given byC(θ1, I1) = θ1− I1 andR(θ2) = θ2. Let first-best expected firmprofits as a function of seller investments be denoted by
whereM∗(·) ≡ M∗1(·)+M∗2(·) andq∗(θ, I1) = 1 wheneverθ2 ≥ θ1− I1. As a benchmark,we derive the first-best investments by differentiating (2) with respect toI1. By the EnvelopeTheorem, the necessary first-order condition is
w′1(I∗1 ) = Prob{q∗(θ, I ∗1 ) = 1}. (3)
The seller should optimally undertake investments up to the point where the marginal costof investment equals the expected marginal cost savings.
Likewise, if only the buying division invests, then the efficient trading rule,q∗(θ, I2),calls for trade if and only ifθ2 + I2 ≥ θ1. Efficient buyer investments are determined in asimilar fashion as in (2) and (3), withI2 substituting forI1,w2(·) for w1(·), and5∗(I2) for5∗(I1). The resulting first-order condition isw′2(I
∗2 ) = Prob{q∗(θ, I ∗2 ) = 1}.
(A1) Suppose that only Divisioni, i ∈ {1,2}, invests. Then the function5∗(Ii ) is assumedto be single-peaked with an interior maximizerI ∗i ∈ [0, I i ].
2. The Transfer Pricing Mechanisms
Standard-cost Transfer Pricing
Under standard-cost transfer pricing, Manager 1 reports a standard cost number,ts, after hehas observed his cost parameterθ1.6 The transfer price then equals this cost report whichis assumed not to be audited by a corporate controller. The buying division will accept theoffer wheneverθ2 + I2 ≥ ts, which happens with probability 1− F2(ts − I2). Assumingthat Manager 1 is not constrained at all, he will act as a profit-maximizing monopolist whofaces a customer with an uncertain valuation for the good:
Differentiating this expression with respect tot yields the first-order condition7
θ1− I1 = φ(ts(θ1, I ), I2), (4)
whereφ(t, I2) ≡ t− [1−F2(t−I2)]f2(t−I2)
.Manager 1 marks up his true production cost ofθ1− I1 byan amount that balances the tradeoff between contribution margin versus the risk of refusalby Manager 2. As is well known from adverse selection models, this amount is determinedby the inverse hazard rate of the buyer’s type distribution,[1−F2(·)]
f2(·) . The resulting tradingrule under standard-cost transfer pricing becomes:
qs(θ, I ) = 1, if and only if θ2+ I2 ≥ ts(θ1, I ). (5)
32 BALDENIUS
As in Baldenius, Reichelstein, and Sahay (1999), we note that standard-cost transferpricing generally leads to inefficient trade, asts(θ1, I ) ≥ θ1− I1. We will extensively dealwith uniform distributions,F1 andF2. The seller’s cost reporting strategy then is:
Our formulation of standard-cost transfer pricing is somewhat extreme in that the sellingdivision faces no direct constraints on its ability to exaggerate cost. However, profit centersare generally engaged in many transactions and often find ways to shift costs across productsand services, for instance by allocating overhead costs in a way that burdens the internallytransferred good. Our model captures the most common complaint against standard-costtransfer pricing—that the standards are set, or at least influenced, by an interested party ina biased way.8 Section 5 will relax this assumption.
Negotiated Transfer Pricing
As an alternative organizational mode, the firm might let the managers negotiate the terms ofthe trade. Essentially, this yields a more symmetrical allocation of bargaining power amongthe divisions. We model the bargaining process as anequal-split sealed-bid mechanism,following Chatterjee and Samuelson (1983).9 Both managers submit sealed bids, and tradeoccurs if and only if the buyer’s bid,b, exceeds the seller’s bid,s. In this case, the surplusis split equally:tn = 1
2(b+ s).Manager 1’s bidding strategy,s: 21× [0, I 1]× [0, I 2] → R+, maps “local type informa-
tion” and observable investments into bids, as does Manager 2’s strategy,b: 22× [0, I 1]×[0, I 2] → R+. In order to derive a Bayesian-Nash equilibrium in linear bidding strategies,we follow Chatterjee and Samuelson in restricting attention to uniform type distributions.The linear bidding strategies solve the following simultaneous optimization problems:
s(θ1, I ) = arg maxs
Eθ2
[(s+ b(θ2, I )
2− θ1+ I1
)· qn(θ, I )
](7)
b(θ2, I ) = arg maxb
Eθ1
[(θ2+ I2− s(θ1, I )+ b
2
)· qn(θ, I )
], (8)
where
qn(θ, I ) = 1 if and only if b(θ2, I ) ≥ s(θ1, I ). (9)
We first modify Chatterjee and Samuelson’s (1983) characterization of the Bayesian-Nashequilibrium in linear bidding strategies for the case of up-front investments.10
Lemma 1 Suppose that the divisions’ valuations are uniformly distributed and that thedivisions have invested I≡ (I1, I2). Then the equal-split sealed-bid mechanism yields the
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 33
following linear equilibrium bidding strategies:
s(θ1, I ) =
b(θ2, I )= 1
12[θ2+3θ1−3I1+9I2+8θ2], if s(θ1, I )< b(θ2, I )
s(θ1, I )= 112[3θ2+θ1−9I1+3I2+8θ1], if s(θ1, I )∈ [b(θ2, I ), b(θ2, I )]
θ1− I1, if s(θ1, I )> b(θ2, I )
(10)
b(θ2, I ) =
s(θ1, I )= 1
12[3θ2+θ1−9I1+3I2+8θ1], if b(θ2, I )> s(θ1, I )
b(θ2, I )= 112[θ2+3θ1−3I1+9I2+8θ2], if b(θ2, I )∈ [s(θ1, I ), s(θ1, I )]
θ2+ I2, if b(θ2, I )< s(θ1, I )
(11)
(All proofs are contained in Appendix B.)The upper branches of the strategies apply to situations where trade occurs with certainty.
They express the intuitive notion that, say, Manager 1 will never bid less than the lowestequilibrium bid of Manager 2, since this would only reduce the transfer payment withoutfurther raising the probability of trade. The lower branches of (10) and (11) characterizesituations where the conditional probability of trade is zero.
Both managers tend to shade their offers in thats(θ1, I ) ≥ θ1− I1 andb(θ2, I ) ≤ θ2+ I2.Hence this mechanism is not incentive compatible, and a comparison of (9) with (1) revealsthat informational asymmetries give rise to trade inefficiencies under negotiations. Thisis in stark contrast to Baldenius, Reichelstein, and Sahay’s (1999) symmetric informationmodel where negotiations are always efficient ex post.
Going back to (4), we can reinterpret standard-cost transfer pricing as the outcome ofanother sealed-bid mechanism, where Manager 2 is “pathologically honest” and Manager 1optimizes accordingly.11 Figure 2 depicts the bidding strategies and cost reports underthe two mechanisms, given that relevant cost and revenues are uniformly distributed overidentical supports, that is,θ1− I1 = θ2+ I2 andθ1− I1 = θ2+ I2.
The firm now faces the problem of choosing a transfer pricing mechanism that simultane-ously deals with two incentive problems: at the outset, managers should have an incentiveto undertake investments and, subsequently, they should be induced to trade the good when-ever it is profitable for the firm. The performance comparison is structured along two lines:(i) we analyze settings in which either of the divisions invests, and (ii) within each of thesesettings, we investigate scenarios where only the seller, only the buyer, or both divisionshave private information.
3. Supplier Investments
In this section, we assume that investments by the buying division are unimportant, i.e.,I2 ≡ 0. The relevant cost and revenues are given byC(θ1, I1) = θ1 − I1 andR(θ2) = θ2.Before addressing the performance comparison between the candidate mechanisms, it isinstructive to identify a scenario where one of the schemes, i.e., standard-cost transferpricing, approaches first-best performance.
34 BALDENIUS
Figure 2a.Negotiated transfer pricing.
Figure 2b.Standard-cost transfer pricing.
If only the supplier invests under standard-cost transfer pricing, then, at date 3,qs(θ, I1) =1, if and only if θ2 ≥ ts(θ1, I1), and Manager 1 issues a cost report of
ts(θ1, I1) =θ2, if θ1− I1 < φ(θ2),
φ−1(θ1− I1), if θ1− I1 ∈ [φ(θ2), θ2],θ1− I1, if θ1− I1 > θ2,
(12)
whereφ(t) ≡ t − [1−F2(t)]f2(t)
. As noted in connection with (5) above, this mechanism leadsto inefficient trade, asts(θ1, I1) ≥ θ1− I1.
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 35
With regard to investment incentives, standard-cost transfer pricing benefits from the factthat the supplying division receives a large fraction of the total contribution margin owingto its first-mover advantage. The supplier is effectively shielded from hold-up problemswhen he chooses his investment:
I s1 ∈ arg max
I1
{Eθ [(t
s(θ1, I1)− θ1+ I1) · qs(θ, I1)] − w1(I1)}.
By the Envelope Theorem, the necessary first-order condition is:12
w′1(Is1) = Prob{qs(θ, I s
1) = 1}. (13)
The seller undertakes investments up to the point where the marginal investment costsequal the expected marginal cost savings. Comparing (13) with (3), however, we notice thatthe cost savings under standard-cost transfer pricing are realized only with lower probability,sinceEθ [qs(θ, I1)] ≤ Eθ [q∗(θ, I1)], for all I1. Thus, ex-post trade distortions will diminishdivisional investment incentives.
Our first result demonstrates how the buyer’s private information impacts the performanceof the cost-based scheme. Let5s denote the expected firm profit under standard-cost transferpricing and recall that1i ≡ θi − θ i . Notice further that Proposition 1 holds for generalcost and revenue distributions.
Proposition 1 Suppose that only the supplying division invests under standard-cost trans-fer pricing. Then5s→ 5∗ as12→ 0.
Recall that the supplier’s pricing problem is determined by a tradeoff between unit con-tribution margin and the risk of over-bidding. As revenue uncertainty vanishes,12 → 0,the latter effect becomes negligible, and Manager 1 can bid more aggressively. Formally,φ(θ2)→ θo
2 ,and the middle branch in (12) evaporates. The supplying division can perfectlyextract the contribution margin, and its investments are shielded from hold-up problems.Hence, it fully internalizes the firm-wide objective. Put differently, since there are no incen-tive problems associated with the buying division, it is desirable to allocate residual profitsto the seller.13 In general, the performance of standard-cost transfer pricing degrades whenthe supplier has coarser information about the buyer’s valuation, because the transfer priceis based on the supplier’s information only and hence lacks the flexibility to reflect differentrealizations ofθ2.
The discrete nature of the trading problem is crucial for Proposition 1 to hold. In Balde-nius, Reichelstein, and Sahay (1999), in contrast, the quantity variable is continuous butthe supplier is constrained to linear pricing. Thus, standard-cost transfer pricing remainsinefficient even though the supplier knows the buyer’s willingness-to-pay.
Under negotiated transfer pricing, the bidding strategies are obtained by settingI2 = 0in Lemma 1. In particular, notice that Manager 2’s strategy,b: 22 × [0, I 1] → R+, iscontingent onI1. This will give rise to hold-up problems as demonstrated below.
For the following performance comparison, we shall confine attention to uniform pa-rameter distributions. Letθi ∼ U [θ i , θi ] denote thatθi is uniformly distributed over theinterval [θ i , θi ]. The previous discussion has shown that some transfer pricing schemesmay be “vulnerable” only with respect to certain kinds of uncertainty: for Proposition 1
36 BALDENIUS
to hold, the supplier’s cost parameterθ1 may be highly uncertain, as long asθ2 is known.To capture this, we will consider the following informational scenarios (the variableI2 forbuyer investments is included for later reference in Section 4):
(CU) One-sidedCost Uncertaintyis characterized byθ1 ∼ U [0,1] andθ2 ∼ U [θo2 , θ
o2 +
12], where12 is “small,” and22 ⊂ (0,1− (I 1+ I 2)).
(BU) Bilateral Uncertaintyis characterized byθi ∼ U [θ i , θi ] where both1i > 0, with
These scenarios are mutually exclusive, but not commonly exhaustive. Focusing onthese cases allows us to identify the main tradeoffs associated with the two transfer pricingschemes.
Cost Uncertainty describes the uniform distribution version of the scenario considered inProposition 1. Bilateral Uncertainty is a weaker condition than identical valuation supports;it holds if the valuation supports differ to a limited extent. The one-sided uncertainty casesimply that the trading problems under standard-cost transfer pricing converge to one-sidedprivate information settings where the take-it-or-leave-it offer is made by the informedmanager (Cost Uncertainty) or by the uninformed one (Revenue Uncertainty). For theremainder of Section 3, we shall again assume thatI2 = 0.
Cost Uncertainty—(CU)
If only the supplying division has private information, Proposition 1 has shown that standard-cost transfer pricing converges toward first-best. Under negotiated transfer pricing, thebidding strategies given in Lemma 1 converge to
b(θo2 , I1) = 3
4θo
2 −1
4I1 and s(θ1, I1) = max{b(θo
2 , I1), θ1− I1}, (14)
as12→ 0. Hence, in the limit, both managers knowθo2 , yet Manager 2 shades his bid, and
gains from trade are lost. Essentially, the buyer’s bargaining power results in both managerssettling on a price which is the buyer’s reservation price less a discount, and the buyer earnsa strictly positive contribution margin,θo
2 − b(θo2 , I1), if trade occurs.
Both bidding strategies under negotiations are contingent onI1. In particular, (14) impliesthat ∂b(·)
∂ I1= − 1
4. As I1 increases, the buyer captures a share of the additional contributionmargin by lowering his bid.15 The supplier’s investment problem is:
I n1 ∈ arg max
I1
{∫ b(θo2 ,I1)+I1
0
[b(θo
2 , I1)− θ1+ I1] 1
11dθ1− w1(I1)
}.
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 37
Differentiating this term with respect toI1 yields
w′1(In1 ) =
(1+ ∂b(θo
2 , I1)
∂ I1
)· Prob{qn(θ, I n
1 ) = 1}
= 3
4Prob{qn(θ, I n
1 ) = 1}. (15)
Equation (15) reveals that negotiated transfer pricing suffers from underinvestment fortwo reasons: first, since Prob{qn(θ, I1) = 1} < Prob{q∗(θ, I1) = 1}, for all I1, the costreduction is realized only with lower probability and, secondly, the hold-up problem dueto ∂b(·)
∂ I1= − 1
4 further degrades investment incentives. We denote by5n the expected firmprofit under negotiated transfer pricing.16
Proposition 2 Suppose (A1) holds and there is only Cost Uncertainty (i.e., (CU) holds). Ifonly the supplying division invests, negotiated transfer pricing is dominated by standard-cost transfer pricing. Formally,lim12→05
n < lim12→05s.
Our results pertaining to the performance comparison are summarized in Table 1 below.Negotiated transfer pricing suffers from two deficiencies. First, for any investmentI1,Mn(I1) < M∗(I1), due to the discount demanded by the buyer. For some realizations ofθ , negotiations will not lead to an intrafirm transfer even though trade would be efficient.Secondly, negotiated transfer pricing suffers from underinvestment as described above.Both these deficiencies generalize to all three informational cases: as long as there isprivate information associated with at least one division’s type distribution and the supportsintersect, negotiations suffers from trade distortions and hold-up problems.
Bilateral Uncertainty—(BU)
When the firm adopts negotiated transfer pricing under Bilateral Uncertainty, then onlythe lower two branches of the bidding strategies in (10) and (11), respectively, can occur.Again, by (11), Division 2 will lower its bid, the higher the seller’s investments:∂b
∂ I1= − 1
4,
for eachθ2, I1, provided the conditional probability of trade is strictly positive. Manager 1then choosesI n
1 so as to maximize∫ b(θ2,I1)+I1
θ1
∫ θ2
b−1(s(θ1,I1),I1)
[b(θ2, I1)+s(θ1, I1)
2−θ1+ I1
]d F2(θ2)d F1(θ1) − w1(I1),
whereb−1(s(θ1, I1), I1) = θ1 + 14(θ2 − θ1 − 3I1), is the lowest realization ofθ2 such that
trade occurs, conditional onθ1 and I1.17 The transfer payment, as the average of the twobids, is a function ofI1. The necessary first-order condition for an optimumI n
1 is
w′1(In1 ) =
[1+ 1
2
(∂b(·)∂ I1+ ∂s(·)
∂ I1
)]· Prob{qn(θ, I n
1 ) = 1}
−∫ b(θ2,I n
1 )+I n1
θ1
[∂b−1(·)∂ I1
· (s(θ1, I n1 )− (θ1− I n
1 ))1
1112
]dθ1. (16)
38 BALDENIUS
After some simplifications, this condition becomes
w′1(In1 ) =
3
4Prob{qn(θ, I n
1 ) = 1}. (17)
Under negotiated transfer pricing, we find that the supplier’s investment incentives arereduced by the hold-up termdtn
d I1= 1
2[ ∂b∂ I1+ ∂s
∂ I1] = − 1
2.However, now there is an additionalpositive first-order effect arising from an expansion of the set{θ |qn(θ, I1) = 1}, as expressedin the second line in (16). The reason for this is that for the cutoff values−1(b(θ2, I1), I1)
the supplier would derive a strictly positive contribution margin if trade occurred.We are now in a position to compare the performance of the transfer pricing schemes
given that both divisions have private information.
Proposition 3 Suppose that only the supplying division invests under Bilateral Uncertainty(i.e., (BU) holds). Then;
i Negotiated transfer pricing achieves higher expected contribution margins: Mn(I1) >
Ms(I1), for all I1;
ii Standard-cost transfer pricing generates better investment incentives: Is1 > I n
1 .
Proposition 3 implies that the overall profitability comparison is indeterminate: the firmfaces a tradeoff between higher investments under standard-cost transfer pricing versusmore efficient trade under negotiations. The second part of the result is driven by the factthat the seller’s marginal return from investing is uniformly higher under standard-costtransfer pricing. Formally, a comparison of (13) with (17) reveals that, for allI1,
Prob{qn(θ, I n1 ) = 1} > Prob{qs(θ, I n
1 ) = 1} > 3
4Prob{qn(θ, I n
1 ) = 1}.
The first part of Proposition 3 is worth relating to Myerson and Satterthwaite’s (1983) workon optimal trading mechanisms. Myerson and Satterthwaite observe that the symmetricalsealed-bid mechanism is outcome-equivalent to an optimal revelation mechanism, if thetypes are uniformly and symmetrically distributed. Essentially, a mechanism is optimal,if the least favorable types just break even in expectation, and if both divisions’ objectivesare given equal weight.18 Since the standard-cost transfer payment is not contingent on thebuyer’s type realization, the latter symmetry requirement cannot be met by this scheme. Wedemonstrate this lack of flexibility by showing how the transfer payments under BilateralUncertainty are affected by changes in the types:
dtn
dθ1= ∂tn
∂s
∂s
∂θ1= 1
3<
dts
dθ1= 1
2,
while
dtn
dθ2= ∂tn
∂b
∂b
∂θ2= 1
3>
dts
dθ2= 0.
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 39
The negotiated transfer payment is equally contingent upon both type realizations, but lesssensitive than under cost-based pricing with respect to changes inθ1. Our result shows that,although negotiated transfer pricing leads to suboptimal trade if the supports differ, thissystem still outperforms a regime where one division makes a take-it-or-leave-it offer. Putdifferently, even thoughθ2 enters the negotiated transfer payment in a biased way, the firmstill wants this information to be incorporated.19
Revenue Uncertainty—(RU)
If the supplying division’s cost is “almost” known to the buyer, as postulated by (RU), thenthe bidding strategies derived in Lemma 1 converge tos(θo
1 , I1) = 14 + 3
4θo1 − 3
4 I1 andb(θ2, I1) = min{s(θo
1 , I1), θ2}, as11→ 0. The supplying division willmark upits actualcost by an amount equal tos(θo
1 , I1) − (θo1 − I1) = 1
4(1− θo1 + I1). This represents its
contribution margin if trade occurs. The seller’s investment problem now becomes
I n1 ∈ arg max
I1
{1
4(1− θo
1 + I1) · [1− F2(s(θo1 , I1))] − w1(I1)
}.
The corresponding first-order condition is
w′1(In1 ) =
1
2Prob{qn(θ, I n
1 ) = 1}. (18)
Thus, the supplier’s investment incentives under negotiated transfer pricing are particu-larly weak as11→ 0. Comparing (15) and (17) with (18) yields an important qualitativeobservation that is reflected in Table 4 in Appendix A:Hold-up problems tend to diminishwith private information on the part of the investing division.
Comparing the performance of the two transfer pricing schemes, we encounter a similartradeoff as in Proposition 3: standard-cost transfer pricing induces higher investmentsbut less efficient trade compared with negotiated transfer pricing. For the simple case of aquadratic investment cost function, the trade effect can be shown to dominate the investmenteffect.
Proposition 4 Suppose (A1) holds,w1(I1) = β1
2 I 21 and there is one-sided Revenue Un-
certainty (i.e., (RU) holds). If only the supplying division invests, then negotiated transferpricing dominates standard-cost transfer pricing, as11→ 0.
The intuition for this result is that, with a quadratic cost function, the differenceI s1 − I n
1is too small to make the investment effect dominate the trade effect.20
4. Buyer Investments
We begin this section by assuming that seller investments are unimportant, i.e.I1 = 0,while the buying division may raise its net revenues by undertaking specific investments.Accordingly, we will speak of the induced “relevant revenue distribution” defined over22×
40 BALDENIUS
[0, I 2] with support [θ2+ I2, θ2+ I2] for given I2. Under both transfer pricing mechanisms,the supplying division’s bidding, or cost reporting, strategies will now depend onI2. Thisgives rise to hold-up problems underbothmechanisms. As demonstrated in the previoussection, hold-up problems are driven by the distribution of bargaining power. Whereas theseller’s investments were perfectly shielded under standard-cost transfer pricing, we wouldnow expect this regime to entail more severe hold-up problems than negotiated transferpricing. This intuition will indeed be confirmed below.
Under negotiations, the divisions’ bidding strategies are obtained by settingI1 = 0 inLemma 1. Manager 1’s cost report issued under cost-based transfer pricing equals
ts(θ1, I2) =
θ2+ I2, if θ1 < 2θ2− θ2+ I2,
12(θ2+ θ1+ I2), if θ1 ∈ [2θ2− θ2+ I2, θ2+ I2],
θ1, if θ1 > θ2+ I2,
(19)
which is obtained by rewriting (12) for the case of uniform type distributions. The respectivetrading rules areqn(θ, I2) = 1 wheneverb(θ2, I2) ≥ s(θ1, I2), andqs(θ, I2) = 1 wheneverθ2+ I2 ≥ ts(θ1, I2).
Cost Uncertainty—(CU)
If Division 2’s revenues become known to the supplying division, then the latter divisioncan extract the entire surplus under standard-cost transfer pricing. Obviously, this makesthe buying division unwilling to invest at all, so thatI s
2 = 0. This drastically illustrates ourprevious observation that private information plays a crucial role as an investment shield.
Under negotiated transfer pricing, the buying division’s investment incentives resemblethe seller’s incentives under Revenue Uncertainty as given in (18). Hence, the firm facesa tradeoff between positive investment under negotiations versus ex-post efficiency understandard-cost transfer pricing. Depending on the functional form ofw2(I2), either effectmay dominate.
Bilateral Uncertainty—(BU)
Under standard-cost transfer pricing, Manager 1 will charge a transfer payment ofts(θ1, I2)=12(θ2 + θ1 + I2) at the trading stage, provided Bilateral Uncertainty prevails. We noticethat ∂ts(·)
∂ I2= 1
2. In light of this severe hold-up problem, the buyer’s investment satisfies thefirst-order condition
w′2(Is2) =
1
2Prob{qs(θ, I s
2) = 1}. (20)
Under negotiated transfer pricing, the buying division’s investment incentives are deter-mined in a similar fashion as in (17), which yields the first-order conditionw′2(I
n2 ) =
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 41
34 Prob{qn(θ, I n
2 ) = 1}. Comparing this with (20) suggests thatI n2 > I s
2. To conduct theprofit comparison, however, we require a technical condition on the profit function undernegotiated transfer pricing. We define the following:
5n(I2)≡Mn(I2)−w2(I2)=∫21
∫22
[θ2+ I2−θ1] ·qn(θ, I2)d F1(θ1)d F2(θ2)−w2(I2).
(A2) The function5n(I2) is single-peaked with an interior maximizerI n2 ∈ (0, I 2).21
Proposition 5 Suppose (A1) and (A2) hold and there is Bilateral Uncertainty (i.e. (BU)holds). If only the buying division invests, then negotiated transfer pricing dominatesstandard-cost transfer pricing.
Proposition 5 reflects that negotiated transfer pricing yields both higher investments andgreater expected contribution margins for all investments. This result and the underlyingintuition extend to situations of one-sided Revenue Uncertainty.
Bilateral Investments
How do the above findings generalize to situations where both divisions have investmentopportunities? Clearly, the divisional investment decisions are interdependent: high valuesof I1 raise the probability of trade and hence raise Division 2’s incentive to invest, aswell. This might suggest that if, say,I n
2 (I1) À I s2(I1), for all I1, then this strategic
complementarity could also induce the seller to invest more under negotiated than understandard-cost transfer pricing. For the case of quadratic investment costs, however, thisstrategic effect turns out to be too weak to overcome the hold-up problem, so thatI s
1 > I n1
and I s2 < I n
2 can be shown to hold in equilibrium.To examine this point in more detail, notice that the dual role of private information—as
source of trade distortions and as investment shield—has countervailing investment effects.Under one-sided Revenue Uncertainty, for instance, trade efficiency considerations call fornegotiated transfer pricing since Prob{qn(θ, I ) = 1} > Prob{qs(θ, I ) = 1} for all I . Thatis, the firm-wide marginal investment returns are higher under negotiated transfer pricing. Incomparison with the cost-based scheme, however, negotiations shift bargaining power fromManager 1—who is very vulnerable to hold-up problems—to Manager 2 whose investmentsare partly shielded by private information. Thus total hold-up problems are exacerbatedwhile the probability of trade increases. The net effect generally remains ambiguous.22
Table 1 summarizes the results obtained for the one-sided investment cases and highlightsthe main themes of this paper. Recall thatM denotes expected firm-wide contributionmargin,5 denotes expected profit and the superscriptsn and s denote negotiated andstandard-cost transfer pricing, respectively.
By pair-wise comparison of the last two cells in each column, we find that the perfor-mance of standard-cost transfer pricing degrades when investments by the buying divisionare essential. For all informational scenarios considered in our analysis,I s
1 > I n1 and
I s2 < I n
2 both hold. This confirms the intuition obtained from symmetric information
42 BALDENIUS
Table 1.Performance of the transfer pricing schemes.
models that hold-up problems will be mitigated if the investing division is given more bar-gaining power (Grossman and Hart, 1986). Secondly, by comparing the cells in any ofthe rows in Table 1, we find that bargaining power should be conferred to the division thathas “more” private information in order to reduce trade distortions. In agency-theoreticterms: if only one division has private information, then this division should be madethe principal—i.e., given all contracting power—rather than letting both divisions negoti-ate.
5. Centralized Transfer Pricing
The preceding analysis was built on the assumption that the center for a corporate controllerin general has less information about Divisioni ’s operations—on a disaggregated productline level—than has Divisionj which is trading this product with Divisioni . When choosinga transfer pricing scheme, the center was assumed to know only which of the divisions had aninvestment opportunity and whether the setting was one of cost and/or revenue uncertainty.In such a setting there is a natural demand for decentralization simply due to lack of centrallyheld information. While this seems descriptive for large diversified companies, in thissection we shall modify this scenario by assuming that the corporate controller knows theseller’s cost support,21. The question then arises whether the controller should intervenein the standard setting process in a centralized fashion, thereby avoiding monopolistic sellerbehavior.23
Suppose only the supplying division has an investment opportunity. To avoid hold-upproblems, suppose also that the central office can commit to a transfer price,t , beforeinvestments are undertaken. For anyt, I1 andθ , the trading rule becomes
q(t, θ, I1) = 1, if and only if θ2 ≥ t ≥ θ1− I1.
If the center is assumed to know21, then it is reasonable to assume that it also knows22.Settingt equal to expected cost then turns out to be suboptimal. The optimalcentralizedex-ante transfer price,tc, under Bilateral Uncertainty (BU) instead solves:
Program I:
tc ∈ arg maxt
max{t+I1,θ1}∫
θ1
θ2∫min(t,θ2}
[θ2− θ1+ I1] f2(θ2) f1(θ1)dθ2dθ1− w1(I1)
, (21)
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 43
subject to
I1 ∈ arg maxI1
max{t+ I1,θ1}∫
θ1
θ2∫min(t,θ2}
[t − θ1+ I1] f2(θ2) f1(θ1)dθ2dθ1− w1( I1)
. (IC)
The transfer price maximizes the expected firm profit, given that the supplying division willsubsequently choose its investment so as to maximize its own expected divisional profit,as expressed by the investment constraint (IC). As a technical prerequisite, we impose acondition that ensures that all investment problems are concave:
(A3) For all I1, w′′1(I1) >
121112
(θ2− θ1+ I1).
In the Appendix, it is shown that the solution to Program I can be expressed astc =12[θ1− I1(tc)+ θ2], with I1(tc) denoting the solution to (IC), givent = tc. Sinceθ2 ≥ θ1
holds by (BU), the optimal centralized transfer price takes the form ofstandard-cost-plus-mark-up: tc = [E[θ1] − I1(tc)]+m, where the mark-up is given bym= 1
2(θ2−θ1+ I1(tc)).While mark-ups are commonly observed in practice, our model provides a new rationalefor this practice: in a discrete quantity setting with bilateral private information, the optimalmark-up over expected cost trades off the risks of refusal by the two divisions and actuallyenhances trade efficiency.24
We first conduct a performance comparison between this centralized mechanism andstandard-cost transfer pricing, as defined in previous sections, for the case of seller invest-ments. Let the expected firm profit achieved under Program I be denoted by5c.
Proposition 6 Suppose (A3) and (BU) hold and the corporate controller knows the pa-rameter ranges{21,22}. If only the supplying division invests, then centralized transferpricing dominates standard-cost transfer pricing:5c > 5s.
A similar result can be established for buyer investments where the case for centralizationis even more pronounced due to avoided hold-up problems. According to Proposition6, the firm can indeed improve upon the performance of cost-based transfer pricing bycentralizing the standard-setting process and employing a suitable mark-up, given that thecorporate controller is endowed with sufficient information.
The natural question now is whether such a centralized mechanism outperforms negotiatedtransfer pricing, as well. Proposition 7 demonstrates that this performance comparisonremains indeterminate (withMc(I1, t) denoting the contribution margin component of theobjective function in (21), for any arbitrary valuesI1 andt).
Proposition 7 Suppose (A3) and (BU) hold and the corporate controller knows the pa-rameter ranges{21,22}. If only the supplying division invests, then:
i Negotiated transfer pricing achieves higher expected contribution margins than cen-tralized transfer pricing: Mn(I1) > Mc(I1, t), for all t and I1;
ii Centralized transfer pricing generates higher investments: Ic1 > I n
Mi Mn < min{Ms,Mc} → M∗ Ms ≤ Mc < Mn < M∗ Ms < Mn < Mc→ M∗
A similar result holds for buyer investments. Due to the flexibility with regard to bothdivisions’ parameter realizations, negotiations still achieve the highest trade efficiency ofall mechanisms considered, despite the fact that divisions bid strategically. The centralizedmechanism, on the other hand, avoids hold-up problems and, therefore, induces higherinvestments.
The optimal mark-up,m, proves to be a flexible instrument facilitating first-best tradeefficiency whenever one of the type distributions converges. We demonstrate this for thecase ofI1 = I2 = 0: if 11 → 0 (condition (RU)), then the firm setsm = 0 yieldingtc = θo
1 ; whereas under one-sided cost uncertainty (CU),m= θo2 − E[θ1] so thattc = θo
2 infact becomes a net-revenue-based transfer price (technically,m may be negative). In boththese cases,t is chosen so as to shift residual profits to that division whose parameter issubject to uncertainty. Table 2 summarizes these findings.
It is worth stressing that the profit-enhancing role for centralizing the transfer pricingprocess crucially hinges on the central office’s knowing the parameter ranges21 and22.This assumption is much stronger than the one underlying our basic model where we haveassumed that the center is even more “remote” from Divisioni’s operating activities, on aproduct line level, than is Divisionj.
6. Concluding Remarks
In the preceding analysis, private information and the need for up-front investments jointlydetermine a firm’s preference for alternative transfer pricing mechanisms, given that thecenter lacks precise information to set a realistic transfer price itself. One empirical impli-cation of our results is that standard-cost transfer pricing should be observed in cases wherethe supplier makes cost-reducing investments and faces little uncertainty with respect to thebuying division’s revenues. In most other cases, however, negotiated transfer pricing tendsto dominate, either because more efficient intrafirm transfers offset the disadvantage fromhold-up problems, or because negotiated transfer pricing is superior in both dimensions:investment and intrafirm transfers.
Our model is stylized in several respects. First, negotiations in practice proceed sequen-tially, rather than as a sealed-bid mechanism. Trade inefficiencies may then be reduced atthe cost of interdivisional haggling. Also, under standard-cost transfer pricing, the supplierwill generally be constrained in his reporting behavior as the accounting system will beable to detect over-reporting if it exceeds some bound. Such partial verifiability of standardcosts will improve the realized contribution margin for given investments at the expense ofdiminished investment incentives for the seller (Sahay, 1997; Baldenius, Reichelstein andSahay, 1999).
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 45
Throughout our analysis, we have viewed the managers’ information as exogenous.Through organizational design, however, the firm may be able to affect the informationalenvironment. For instance, job rotations across divisions will enhance inter-divisional in-formation transmission. Taking investment incentives into account, this will not always bedesirable. For instance, if only the buyer invests under standard-cost transfer pricing, thenone can construct examples where the firm prefers the buyer to have private information,despite the attendant trade distortions. This is because private information ensures positiveinvestment. In fact,bothdivisions may benefit from private information on the part of thebuyer. So, even if the seller could observe the buyer’s revenues, he may want to commitnot to use this information in order to induce the buyer to invest. Such commitment power,however, seems difficult to achieve.25
Our performance comparison has been confined to negotiated and standard-cost transferpricing. Frequently, the intermediate good can be traded in imperfectly competitive externalmarkets, and the firm may then consider market-based transfer pricing as a third alternative.The presence of external markets also affects negotiated transfer pricing since the bargainingoutcome may depend on the divisions’ “outside options.” It would advance the appliedtheory of transfer pricing if future research could assess the relative strength of all differentmechanisms that are reported to be commonly used by companies.
Acknowledgments
I would like to thank Regina Anctil, Richard Frankel, Jack Hughes, Rick Lambert (the edi-tor), Dilip Mookherjee, DJ Nanda, Roy Radner, Bente Villadsen, two anonymous refereesand seminar participants at Berkeley, Carnegie Mellon, Chicago, Columbia, Duke, LondonBusiness School, LSE, Michigan, Minnesota, Northwestern, NYU and Vienna for helpfulcomments. In particular, I am indebted to Stefan Reichelstein for numerous discussions.
Appendix
A Bidding Strategies and Investment Incentives
In Table 3 we summarize the divisions’ bidding strategies under the transfer pricing schemes,provided one-sided uncertainty prevails with1i → 0. Table 4 contains the marginal invest-ment returns for all informational scenarios and illustrates the role of private informationas an investment shield.
B Proofs
Proof of Lemma 1: We follow Chatterjee and Samuelson (1983) in deriving the linear-strategy Bayesian-Nash equilibrium under negotiated transfer pricing.
For given I , let τ1 ≡ θ1 − I1, andτ2 ≡ θ2 + I2 denote the relevant divisional costsand revenues. The corresponding support boundaries areτ 1 ≡ θ1 − I1, τ1 ≡ θ1 − I1,
τ 2 ≡ θ2 − I2, andτ2 ≡ θ2 − I2. This induces (uniform) distributionsFi (τi ), defined over[τ i , τi ]. Now restate the maximization problems as given in (7) and (8):
s(τ1, I ) = arg maxs
∫ b(τ2,I )
s
[1
2(b+ s)− τ1
]dG2(b, I ),
b(τ2, I ) = arg maxb
∫ b
s(τ 1,I )
[τ2− 1
2(b+ s)
]dG1(s, I ).
The bid distribution functionsGi are induced by (i) the underlying type distributionsFi (τi ),and by (ii) the bidding strategiess andb, whereG1(ξ, I ) = F1(s−1(ξ, I )) andG2(ξ, I ) =F2(b−1(ξ, I )). The first-order condition for the seller is
1
2[1− G2(s(·), I )] − (s(·)− τ1) · g2(s(·), I ) = 0,
with gi denoting the density function toGi . Definingx ≡ b−1(s, I ), we haveg2(s, I ) =f2(x)/b′(x, I ), τ1 = s−1(b(x, I ), I ), andG2(s, I ) = F2(x), and the first-order conditioncan be restated as
s−1(b(x, I ), I ) = b(x, I )− 1
2b′(x, I )
1− F2(x)
f2(x).
Proceeding in a similar fashion for the buyer yields
b−1(s(y, I ), I ) = s(y, I )+ 1
2s′(y, I )
F1(y)
f1(y),
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 47
wherey ≡ s−1(b, I ). A Bayesian-Nash equilibrium now is a solution to these two linkeddifferential equations. Restricting attention to linear bidding strategiess(τ1, I ) = α1(I )+β1(I ) · τ1 andb(τ2, I ) = α2(I )+ β2(I ) · τ2, we have
s−1(b(τ2, I ), I ) = b(τ2, I )− 1
2β2(I ) · (τ2− τ2),
b−1(s(τ1, I ), I ) = s(τ1, I )+ 1
2β1(I ) · (τ1− τ 1).
By differentiation, we can determine the slopes of the strategies, which turn out to beindependent ofI : β1 = β2 = 2
3. It follows that the intercept terms areα1 = 14 τ2 + 1
12τ 1
andα2 = 112τ2+ 1
4τ 1. Re-scaling both divisions’ valuations in terms ofθi = τi ± Ii yieldsthe linear strategies
s(θ1, I ) = 1
12[3θ2+ θ1− 9I1+ 3I2+ 8θ1] and
b(θ2, I ) = 1
12[θ2+ 3θ1− 3I1+ 9I2+ 8θ2],
given that the first-order conditions are necessary and sufficient.The boundary conditions follow from the fact that, say, the buyer will never submit a
bid greater than the seller’s highest equilibrium bid. In cases where one division’s typerealizationθi is very unfavorable such that even for the most efficientθj -realization inequilibrium there will be no trade, we make the assumption that divisioni submits its truevaluation. For an extensive discussion of the boundary conditions, the reader is referred toChatterjee and Samuelson (1983).
Proof of Proposition 1: According to (1),q∗(θ, I1) = 1 if and only ifθ1− I1 ≤ θ2. Undercost-based transfer pricing, the supplying division quotes a cost report according to (12),and trade occurs if and only ifθ1− I1 ≤ φ(θ2) ≡ θ2− (1− F2(θ2))/ f2(θ2).
The first-best expected firm profit and that under standard-cost transfer pricing can bewritten as functions of12 ≡ θ2− θ2:
5∗(12) ≡∫ θo
2+12
θo2
∫ θ2+I ∗1 (12)
0[θ2− θ1+ I ∗1 (12)] d F1(θ1)d F2(θ2)−w1(I
∗1 (12)),
5s(12) ≡∫ θo
2+12
θo2
∫ φ(θ2)+I s1 (12)
0[θ2− θ1+ I s
1(12)] d F1(θ1)d F2(θ2)− w1(Is1(12)),
where I ∗1 (12) and I s1(12) are determined according to (3) and (13). Now, since
lim12→0 φ(x) = x, it follows immediately that
lim12→0
5s(12) =∫ θo
2+lim12→0 I s1 (12)
0
[θo
2 − θ1+ lim12→0
I s1(12)
]d F1(θ1)
= lim12→0
5∗(12)
since lim12→0 I s1(12) = lim12→0 I ∗1 (12).
48 BALDENIUS
Proof of Proposition 2: To show that the standard-cost-based mechanism dominates as12→ 0, we only need to prove that negotiated transfer pricing does not approach first-bestperformance. We first show that, for allI1, lim12→0 Mn(I1,12) < lim12→0 M∗(I1,12),where
Mn(I1,12) ≡∫21
∫ θo2+12
θo2
[θ2− θ1+ I1] · qn(θ, I1)d F2(θ2)d F1(θ1),
andM∗(·) is defined similarly withq∗(·) substituted forqn(·). Secondly, it is demonstratedthat lim12→0 I n
1 (12) < lim12→0 I ∗1 (12).As12→ 0, we haveb(θo
2 , I1) < θo2 for all I1, by (14). Comparing (1) with (9), there is
a non-empty set ofθ ’s for which negotiated transfer pricing does not result in trade eventhoughq∗(θ, I1) = 1. This proves the first requirement.
Similarly, by comparing (3) with (15), we find that, for allI1, the marginal investmentreturns under negotiated transfer pricing are less than first-best as12→ 0:
Mn′1 (I1) = 3
4Prob{qn(θ, I n
1 ) = 1} < Prob{q∗(θ, I n1 ) = 1} = M∗
′(I1).
Now suppose that, contrary to our claim,I n1 ≥ I ∗1 would hold. By revealed preference,
A necessary condition forI n1 > I ∗1 then is thatM∗
′(I1) ≤ Mn′
1 (I1) for someI1, which yieldsa contradiction. As shown in Edlin and Shannon (1998), (A1) rules out thatI s
1 = I ∗1 , whichcompletes the proof of Proposition 2.
Proof of Proposition 3: Part i. Given Bilateral Uncertainty prevails as defined by (BU),then the expected contribution margins for givenI1 under the two schemes are:
Mn(I1) =∫ b(θ2,I1)+I1
θ1
∫ θ2
b−1(s(θ1,I1),I1)
[θ2− θ1+ I1] d F2(θ2)d F1(θ1), (22)
Ms(I1) =∫21
∫ θ2
ts(θ1,I1)
[θ2− θ1+ I1] d F2(θ2)d F1(θ1), (23)
where (22) uses the fact thats−1(b(θ2, I1), I1) = b(θ2, I1)+ I1. The ex-post comparisonthen is a matter of straightforward algebra:
Mn(I1) =∫ 1
4 (3θ2+θ1+3I1)
θ1
∫ θ2
θ1+ 14 (θ2−θ1−3I1)
[θ2− θ1+ I1] d F2(θ2)d F1(θ1)
= 9
641112(θ2− θ1+ I1)
3
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 49
> Ms(I1) =∫21
∫ θ2
12 (θ2+θ1−I1)
[θ2− θ1+ I1] d F2(θ2)d F1(θ1)
= 1
81112[(θ2− θ1+ I1)
3− (θ2− θ1+ I1)3],
where the inequality holds due toθ2− θ1 ≥ 0, by (BU). This proves part i.Part ii. It remains to be shown thatI s
1 > I n1 . As in the proof of Proposition 2, a
sufficient condition for this to hold is that the marginal investment returns are uniformlygreater under cost-based pricing. The investment incentives again are determined by thesupplier’s marginal contribution margin as given by (17) and (13):
Mn′1 (I1) = 3
4Prob{qn(θ, I1) = 1} and
Ms′1 (I1) = Prob{qs(θ, I1) = 1}.
Notice thatqn(θ, I1) = 1 if and only if b(θ2, I1) ≥ s(θ1, I1), while qs(θ, I1) = 1 if andonly if θ2 ≥ ts(θ1, I1). The seller’s marginal contribution margins under the respectivetransfer pricing schemes can be rewritten as:
Mn′1 (I1) = 3
4
∫ 14 (3θ2+θ1+3I1)
θ1
∫ θ2
θ1+ 14 (θ2−θ1−3I1)
d F2(θ2)d F1(θ1)
= 3
4· 9
321112(θ2− θ1+ I1)
2, (24)
Ms′1 (I1) =
∫21
∫ θ2
12 (θ2+θ1−I1)
d F2(θ2)d F1(θ1)
= 1
41112[(θ2− θ1+ I1)
2− (θ2− θ1+ I1)2]. (25)
Suppose now that, contrary to our claim, there exists some valueI1 such thatMn′1 (I1) ≥
Ms′1 (I1). Then, (24) and (25) imply that
(θ2− θ1+ I1)2 ≥
(1− 27
32
)· (θ2− θ1+ I1)
2. (26)
Under (BU), however, we know thatθ1 ≥ 14(3θ2 + θ1 + 3I1). This yields an upper bound
on the left-hand side of (26):
(θ2− θ1+ I1)2 ≤
(θ2− 1
4(3θ2+ θ1+ 3I1)+ I1
)2
= 1
16(θ2− θ1+ I1)
2.
This last expression, however, is less than the right-hand side of (26)—a contradiction. Asin the proof of Proposition 2, this impliesI s
1 > I n1 .
50 BALDENIUS
Proof of Proposition 4: Recall the first-order conditions for the seller’s investment choicesunder Revenue Uncertainty (RU) in the limit case where11→ 0:
w′1(In1 ) =
1
2Prob{qn(θ, I n
1 ) = 1} = 1
2[1− F2(s(θ
o1 , I1))] = 3
8(1− θo
1 + I1),
w′1(Is1) = Prob{qs(θ, I s
1) = 1} = 1− F2(ts(θo
1 , I1)) = 1
2(1− θo
1 + I1).
By assumption,w1(I1) = β1/2 · I 21 . It follows that
I s1 =
1− θo1
2β1− 1> I n
1 =3(1− θo
1 )
8β1− 3,
and, for anyI1,
Ms(I1) = 3
8(1− θo
1 + I1)2 < Mn(I1) = 15
32(1− θo
1 + I1)2.
Thus, the firm faces a tradeoff between trading and investment incentives.The expected profits under the two schemes are
5s = Ms(I s1)−
β1
2I s2
1 =3
8
(1− θo
1 +1− θo
1
2β1− 1
)2
− β1
2
(1− θo
1
2β1− 1
)2
= β1
2(3β1− 1)
(1− θo
1
2β1− 1
)2
,
5n = Mn(I n1 )−
β1
2I n2
1 =15
32
(1− θo
1 +3(1− θo
1 )
8β1− 3
)2
− β1
2
(3(1− θo
1 )
8β1− 3
)2
= β1
2(60β1− 9)
(1− θo
1
8β1− 3
)2
.
Now, a necessary condition for the investment effect to dominate is that
5s −5n = β1
2(1− θo
1 )2
[3β1− 1
(2β1− 1)2− 60β1− 9
(8β1− 3)2
]> 0 ⇒ β1 < 1.
We now show thatβ1 < 1 is inconsistent with interior first-best investments, as stipulatedby (A1). The first-best investments solve
By condition (RU), we haveI1 < θo1 . RequiringI ∗1 < I1 yields a contradiction since
I ∗1 =1− θo
1
β1− 1< I1 < θo
1 ⇒ β1 >1
θo1
> 1.
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 51
Proof of Proposition 5: Proceeding in a similar fashion as in the proof of Proposition 3,first note that the marginal investment returns for the buying division under the two regimesare given by
Mn′2 (I2) = 3
4Prob{qn(θ, I2) = 1} and Ms′
2 (I2) = 1
2Prob{qs(θ, I2) = 1}.
The ex-post probabilities of trade under Bilateral Uncertainty, (BU), are given by
Prob{qn(θ, I2) = 1} = 9
321112(θ2− θ1+ I2)
2,
Prob{qs(θ, I2) = 1} = 1
41112[(θ2− θ1+ I2)
2− (θ2− θ1+ I2)2].
Sinceθ2 − θ1 ≥ 0, we have Prob{qn(θ, I2) = 1} > Prob{qs(θ, I2) = 1} and, a fortiori,Mn′
2 (I2) > Ms′2 (I2), for all I2. This in turn impliesI n
2 > I s2. At the same time, modifying
Proposition 3 for the case of buyer investments yieldsMn2 (I2) > Ms
2(I2), for all I2.To complete the proof, we use ex-post dominance of negotiations for the first of the
following inequalities:
5s(I s2) = Ms
1(Is2)+ Ms
2(Is2)− w2(I
s2)
< 5n(I s2) = Mn
1 (Is2)+ Mn
2 (Is2)− w2(I
s2)
≤ 5n(I n2 ) = Mn
1 (In2 )+ Mn
2 (In2 )− w2(I
n2 ).
The last inequality holds, since, by (A2),5n(I2) is single-peaked, and
I n2 ≡ arg max
I2
{Mn1 (I2)+ Mn
2 (I2)− w2(I2)}
> I n2 ≡ arg max
I2
{Mn2 (I2)− w2(I2)}
≥ I s2,
by revealed preference. Hence,I s2 and I n
2 both fall within the region where5n(I2) ismonotone non-decreasing (by (A2)) which completes the proof of Proposition 5.
Proof of Proposition 6: Rewriting the objective function and replacing (IC) in Program Iwith its first-order condition—which is feasible by (A3)—we get the following equivalentrepresentation of Program 1:
tc ∈ arg maxt
{1
21112(θ2− t)(t − θ1+ I1)(θ2− θ1+ I1)− w(I1)
}, (27)
subject to:
I1 ∈ arg maxI1
{1
1112(θ2− t)(t − θ1+ I1)− w′1( I1)
}. (IC)
52 BALDENIUS
From (27) and (IC), it is obvious that, holding investments fixed, the same valuet thatmaximizes the objective function, also maximizes investments (which are driven by theprobability of trade). Notice further that the seller’s objective in (IC) is concave inI1 by(A3) and that the firm’s objective in (27) is increasing inI1 for all values ofI1 for whichthe selling division’s profit, as given in (IC), is increasing inI1. Thus, solving Program I isequivalent to solving
Program I ′: tc = 1
2(θ1+ θ2− I1(t
c)),
where I1(t) solves (IC) for anyt . Let I c1 , Mc and5c, respectively, denote the resulting
seller investment, expected firm-wide contribution margin and expected firm-wide profitunder Program I’.
The proof proceeds along two steps: we first show thatI s1 ≤ I c
1 , and then prove ex-post(trade) dominance of centralized transfer pricing. Under standard-cost transfer pricing, theseller’s marginal investment return for anyI1 is (see (25)):
Ms′1 (I1) = 1
41112[(θ2− θ1+ I1)
2− (θ2− θ1+ I1)2].
Under centralized transfer pricing (Program I’), given thatt = tc = 12(θ1 + θ2 − I c
1), thesupplier’s marginal investment return for anyI1 is found by differentiating
Mc1(I1, t
c) ≡∫ tc+I1
θ1
∫ θ2
tc
[tc − θ1+ I1
]d F2(θ2)d F1(θ1)− w1(I1)
with respect toI1:
∂Mc1(I1, tc)
∂ I1= 1
41112[(θ2− θ1+ I1)
2− (I1− I c1)
2].
Now, in equilibrium,I1 = I c1 , and the necessary first-order condition reads
w′1(Ic1) =
∂Mc1(I1, tc)
∂ I1
∣∣∣∣I c1
= 1
41112(θ2− θ1+ I c
1)2.
Assumption (A3) ensures that all investment problems are strictly concave inI1. Nowsuppose that, contrary to our claim,I s
1 > I c1 would hold. A necessary condition for this to
be the case is that
Ms′1 (I
c1) =
1
41112[(θ2− θ1+ I c
1)2− (θ2− θ1+ I c
1)2] >
∂Mc1(I1, tc)
∂ I1
∣∣∣∣I c1
,
which obviously yields a contradiction becauseθ2 − θ1 ≥ 0, by (BU). Also note thatI c1 < I s∗
1 , whereI s∗1 ≡ arg maxI1
{Ms
1(I1)+ Ms2(I1)− w1(I1)
}. This holds because
d(Ms1(I1)+ Ms
2(I1))
d I1
∣∣∣∣I c1
= 3
81112[(θ2− θ1+ I c
1)2− (θ2− θ1+ I c
1)2]
>1
41112(θ2− θ1+ I c
1)2 = ∂Mc
1(I1, tc)
∂ I1
∣∣∣∣I c1
,
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 53
where the inequality is derived in a similar fashion as in the proof of Proposition 3, i.e., byinvoking the lower bound onθ1 as stated in (BU). Hence,I s
1 ≤ I c1 < I s∗
1 . Thus, by (A3),we have5s(I c
1) > 5s(I s1), which yields the first of the following inequalities:
5s ≡ Ms(I s1)− w1(I
s1)
≤ Ms(I c1)− w1(I
c1)
< Mc(I c1 , t
c)− w1(Ic1) ≡ 5c.
The strict inequality follows from the fact that the expected contribution margin, evaluatedat I c
1 andtc, is strictly greater under the centralized scheme:
Mc(I c1 , t
c) =∫ tc+I c
1
θ1
∫ θ2
tc
[θ2− θ1+ I c1 ] d F2(θ2)d F1(θ1)= 1
81112
(θ2−θ1+ I c
1
)3> Ms(I c
1) =1
81112
[(θ2−θ1+ I c
1
)3− (θ2−θ1+ I c1
)3].
Proof of Proposition 7: Part i. By slightly abusing notation, we first note that, holdingI1 fixed, tc(I1) ≡ 1
2(θ1+ θ2− I1)maximizesMc(I1, t) over allt . Hence, we only need toshow thatMn(I1) > Mc(I1, tc(I1)), for all I1. This is indeed the case as
Mn(I1) = 9
641112(θ2− θ1+ I1)
3 > Mc(I1, tc(I1)) = 1
81112
(θ2− θ1+ I1
)3.
Part ii. Assumption (A3) ensures that all investment problems are strictly concave.Thus, we only need to show that the seller’s marginal investment return, evaluated atI c
1(and attc), is greater under centralized transfer pricing than under negotiations:
∂Mc1(I1, tc)
∂ I1
∣∣∣∣I c1
= 1
41112(θ2−θ1+ I c
1)2 > Mn′
1 (Ic1)=
3
4· 9
321112(θ2−θ1+ I c
1)2.
This completes the proof of Proposition 7.
Notes
1. We implicitly assume that there is no viable external market for the intermediate product. Among cost-basedmechanisms, many firms prefer standard cost over actual cost in a multi-product setting, as actual costs for aparticular product are often difficult to verify. Furthermore, under a standard-cost system, the selling divisionhas an incentive to keep actual production cost low, while the buying division knows at the outset the amountit will have to pay. See Price Waterhouse (1984), Eccles (1985), Tang (1992) or Horngren, Foster and Datar(1997).
2. Recent contributions to the literature on incomplete contracting have demonstrated that the hold-up problemcan be overcome if the divisions enter into a contractual agreement at the outset; see, e.g., Chung (1991) orEdlin and Reichelstein (1995). We will assume, however, that the intermediate good cannot be contractuallyspecified at an early stage. Similar assumptions are employed by Williamson (1985), Grossman and Hart(1986), and Holmstr¨om and Tirole (1991). According to the Price Waterhouse (1984) survey, formal contractsbetween divisions are rarely observed.
54 BALDENIUS
3. The problem of overstating budgeted cost in multi-product divisions has been widely recognized in bothpractice and theory—see Eccles and White (1988, S27), Price Waterhouse (1984, 16), Horngren, Foster andDatar (1997, 909, “Points to Stress”), Ewert and Wagenhofer (1995, 525). For evidence on cost shifting undercost reimbursement contracts in multi-product settings, see Cavalluzzo, Ittner and Larcker (1998) and thereferences therein.
4. See Harris, Kriebel and Raviv (1982), Amershi and Cheng (1990), Mookherjee and Reichelstein (1992),Wagenhofer (1994), Vaysman (1996,1998), or Christensen and Demski (1998) for models stressing adverseselection and/or moral hazard issues. Narayanan and Smith (1998) analyze the commitment effects of taxdifferentials and organizational structure with respect to the firm’s ability to set strategic transfer prices in aduopoly setting.
5. Kanodia (1991) elaborates on the role of rent extraction in transfer pricing models. We assume that themanagers’ bonuses are functions of divisional income only. As in most incomplete contracting models withoutmoral hazard, underinvestment problems could be mitigated by basing the managers’ salaries on firm-wideprofit,5 =∑
i[Mi (·)−wi (Ii )], see Heavner (1998). However, in most cases internal transactions constitute
only “additional business” for the divisions (Tang (1993)). Moral hazard problems associated with otherprojects keep profit sharing from attaining first-best; see Anctil and Dutta (1999). Eccles (1985), Merchant(1989) and Bushman, Indjejikian and Smith (1995) observe that divisional performance measures generallyare the main drivers of division managers’ compensation.
6. Note that all results below remain valid in the case where the seller quotests before knowing his actual cost.Suppose that actual costs are given byθ1 − I1 + ε whereε is a random variable with zero mean. The sellersubmits a cost reportts after privately observingθ1 but before observingε, i.e. the seller isbetter informedthan the central office but he is not perfectly informed. This seems to correspond closely to the definition ofstandard costing found in textbooks.
7. The corner solutions corresponding to (4) are:ts(·) = θ2 + I2, if θ1 − I1 < φ(θ2, I2), and ts(·) = θ1 −I1, if θ1− I1 > θ2+ I2.We invoke the standard assumption that the inverse hazard rate(1− F2(·)) / f2(·) bemonotonic.
8. Baldenius, Reichelstein, and Sahay (1999) investigate the effect of partially verifiable cost reports issued bythe supplying division.
9. Linhart, Radner and Satterthwaite (1992) point out that sealed-bid mechanisms suffer from commitmentproblems in voluntary trading situations. While sequential bargaining may be more descriptive for intrafirmnegotiations, such models are very sensitive with respect to belief revisions under two-sided private information.Notice that a sealed-bid mechanism minimizes “haggling costs.”
10. Leininger, Linhart and Radner (1989) show that the symmetrical sealed-bid mechanism has infinitely manyequilibria, including non-differentiable ones. However, they provide references to experimental studies sug-gesting that players’ actual bidding strategies often are approximately linear. The result stated in Lemma 1reflects the only polynomial equilibrium strategies.
11. It is straightforward to construct a generalized bargaining mechanism where both divisions submit bids andt (s,b) = αs+ (1− α)b. Thenα = 1/2 (α = 1) corresponds to negotiated (standard-cost) transfer pricing.As the flip-side of standard-cost transfer pricing,α = 0 would characterize a mechanism where the paymentis based solely on a net-revenue report issued by the buyer. In Baldenius, Reichelstein and Sahay (1999),in contrast, standard-cost transfer pricing is not a limit case of negotiations because the seller, while indeedentitled to make a take-it-or-leave-it offer, is restricted to uniform pricing and quantity is a continuous variable.
12. All first-order investment conditions and bidding strategies are summarized in Appendix A.13. This finding is in line with Vaysman’s (1998) result that the performance of negotiated transfer pricing improves
if bargaining power is allocated to the privately informed division.14. As a technical condition, we require the buyer’s highest valuation to be greater than the seller’s highest cost
for all investments, and similarly for the lowest valuations. Moreover, (BU) ensures that only the lower twobranches of (10) and (11), respectively, arise with positive probability. In order to ensure that [θ1− I1, θ1− I1]
and [θ2 + I2, θ2 + I2] intersect for all(I1, I2), the sumI 1 + I 2 must not be too large.15. In contrast to previous transfer pricing models concerned with hold-up problems under symmetric informa-
tion—e.g., Holmstr¨om and Tirole (1991), Edlin and Reichelstein (1995)—we can explicitly track how thebargaining strategies of the divisions change if one of them makes specific investments.
16. As noted earlier, the equilibrium derived in Lemma 1 is not unique. If now12 → 0, then the problem ofmultiple equilibria may be perceived as more severe. However, dominance of standard-cost transfer pricing
INTRAFIRM TRADE, BARGAINING POWER AND SPECIFIC INVESTMENTS 55
may still be defended on the grounds of itsuniqueefficient subgame-perfect equilibrium, whereas undernegotiated transfer pricing there is an infinite number of inefficient equilibria in addition to the efficientone. As long as the buyer’s bargaining power allows him to achieve a strictly positive contribution margin,Proposition 2 will continue to hold.
17. Notice that, in equilibrium,s−1(b(θ2, I1)) = b(θ2, I1)+ I1.18. See Myerson and Satterthwaite (1983, 274), Theorem 2.19. In contrast to the present paper, Mookherjee and Reichelstein (1992) model negotiated transfer pricing as
an optimal Myerson–Satterthwaite mechanism. However, if there is one-sided private information, such amechanism can be shown to achieve first-best performance. Thus, the optimal mechanism coincides witha regime that grants unfettered bargaining power to the privately informed manager, with distributive issuesbeing dealt with by up-front payments from the informed to the uninformed division. While this is of eminentnormative importance, it does not seem descriptive for intrafirm trade. Under a sealed-bid mechanism, incontrast,bothdivisions benefit from incremental contribution margins, even if the valuation of one division isobservable.
20. For this case of quadratic investment costs, numerical simulations suggest that the trade effect tends to dominateunder Bilateral Uncertainty, as well.
21. A sufficient condition for this to hold is thatw′′2(I2) >2732(θ2 + I2 − θ1)/(1112), for all I2. Notice that
concavity of5∗(I2) ≡ M∗(I2)− w2(I2) does not imply (A2).22. For specific scenarios we can conduct profit comparisons. Suppose that one-sided revenue uncertainty prevails,
both divisions’ investment costs are described by symmetric functionswi (Ii ) = β/2· I 2i , and first-best expected
profits5∗(I1, I2) are concave in each argument with(I ∗1 , I ∗2 ) ∈ (0, I1) × (0, I2). Then negotiated transferpricing dominates standard-cost transfer pricing.
23. I am grateful to Rick Lambert and a referee for suggesting this scenario. A more general model is conceivable,where the center observes noisy signals [θ i + εi , θ i + εi ] about the divisional parameter supports2i . If thevariance of the random terms,εi , is sufficiently large, then, endogenously, the center would not want to exerciseits option to determinet in a centralized fashion.
24. Textbooks argue that the seller has to be granted a fair share of the total profit at the expense of reducedcontribution margins. In the symmetric information models of Sahay (1997) and Baldenius, Reichelstein andSahay (1999), the optimal mark-up over actual cost trades off investment incentives for the seller versus tradedistortions. If the seller has no investment opportunity, the optimal mark-up in these models will be zero.
25. A related idea is applied in Arya, Glover and Sivaramakrishnan (1997).
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