Dual Transfer Prices with Unobserved Cost

Transcription

1 Dual Transfer Prices with Unobserved Cost Nicole Bastian Johnson Haas School of Business University of California, Berkeley Thomas Pfeiffer Department of Business Administration, University of Vienna May 13, 2008 We would like to thank seminar participants at U.C. Berkeley and Arizona State University for helpful comments and suggestions.

2 Dual Transfer Prices with Unobserved Cost Abstract This paper examines the effectiveness of dual (non-zero-sum) transfer prices in a decentralized firm, in terms of efficient internal trade and external price-setting incentives, when marginal costs are not observable and the upstream division has monopoly power in the external market. The main message of our paper is that when costs are not observed dual-transfer pricing is more complicated than it may appear, because the situation cannot be treated as two separate unrelated optimization problems. We would like to be able to use the information about cost embedded in the upstream external price to make the price faced by the downstream firm more efficient, but in so doing we may introduce incentives for the upstream division to distort the external price. We find that first-best internal trade and price-setting can be achieved in the special case when demand is observed by headquarters, but the optimal dual transfer prices can take a highly non-linear form. When demand is random and unobserved by headquarters, first-best internal trade is not possible and it is often optimal to use the upstream external price in the downstream transfer price because it is informative about marginal cost, even if this induces distortion in the external price.

3 1 Introduction In many decentralized firms, managers are given decision-making authority and are rewarded for maximizing divisional profits. In this type of organizational arrangement, transfer prices play an important role in coordinating economic activity within the firm. For many types of transactions, however, designing a single transfer price that induces optimal behavior by all parties involved in an internal transaction is difficult or impossible. In this paper we examine the usefulness of a dual (non-zero sum) transfer pricing system in creating optimal trading and price-setting incentives for the two divisions. Generally, transfer prices are assumed to be zero-sum, but in a dual transfer pricing system, internal transfers are priced at two different rates when calculating divisional profits for the two units involved in the transaction. Dual transfer pricing systems are often discussed briefly in managerial accounting textbooks as a potential solution to problems that arise with zero-sum prices. In particular, a transfer price must be equal to the supplying division s marginal cost in order to induce the buying division to purchase an efficient quantity. But if the supplying division is not capacity constrained, its marginal cost is equal to its marginal production cost 1, and a transfer price set equal to marginal production cost leaves the selling division with no profit and few incentives to invest in cost-reducing or revenue-enhancing technology. Some advocate dual transfer prices as a solution to this problem and suggest that the downstream division pay marginal cost, while the upstream division receives marginal cost plus a markup or a (possibly discounted) external market price. 2 Such a scheme would provide efficient trade incentives for the downstream division while allowing the upstream division s transfer price 1 If the upstream division is capacity constrained and operates in a competitive external market, the marginal cost of transferring a good internally is the marginal production cost of the internal good plus the lost profits from replacing an external sale with an internal transfer. Baldenius and Reichelstein (2006) show that when the upstream division is capacity constrained and has market power, the optimal transfer price will be the external market price minus a suitably chosen discount. 2 For example, see Horngren, Datar and Foster (2002), Eldenburg and Wolcott, (2004), and Maher, Lanen and Rajan (2006). 1

4 to be designed to achieve other objectives, such as providing investment incentives. But despite their potential to provide increased flexibility in providing efficient incentives in a decentralized firm, dual transfer prices do not appear to be used widely in practice. One explanation for this discrepancy could be that the dual transfer pricing schemes described above depend on the assumption that marginal costs are observable and contractible at the time the downstream division makes its quantity choice. In practice, however, information about realized marginal cost is often not observable to all parties in real-time. Rather, cost information is reported end-of-period accounting reports and may be measured with noise or aggregated with other information. When marginal costs are not immediately observable, it becomes difficult to set the downstream division s transfer price to marginal cost and thus induce efficient internal trade. We examine what level of efficiency can be achieved with dual prices, in terms of efficient internal trade and efficient external price-setting incentives, when marginal costs are not observable and external demand is non-competitive. The main thrust of our paper is that when costs are not observed dual-transfer pricing is more complicated than it may appear, because the situation cannot be treated as two separate unrelated optimization problems. We would like to be able to use the information about cost embedded in the upstream external price to make the price faced by the downstream firm more efficient, but in so doing we may introduce incentives for the upstream division to distort the external price. Dealing with this tradeoff is the main subject of our analysis. We start our analysis by showing that in the special case where demand functions are observed by headquarters, the upstream division s external pricing function can generally be inverted to find its marginal cost. If we set the downstream transfer price equal to the inverted price function, the upstream division may have incentives to distort its external price to increase profits from internal trade, but it s possible to get first best if the upstream transfer price is chosen carefully. However, the upstream division s demand function will take on a highly non-linear, and possibly non-monotonic form. The complexity of the optimal dual transfer prices, even in this simplified setting, may partially explain why they are not often observed in practice. 2

5 We then examine the more general case with random demand functions which are not observed by headquarters at the time it sets the transfer pricing rule. In this case it is generally no longer possible to invert the upstream external market price to get upstream production cost, so first-best internal trade is generally not possible. The upstream division s external market price still contains information about production cost, however, and we show that including the upstream external market price as a variable in the downstream division s transfer price will improve the efficiency of internal trade. However, if the downstream transfer price is a function of the upstream division s external market price, the upstream division will tend to distort the its external price in order to maximize the sum of internal and external sales. If no ex-post cost signal is available, the only way to get first-best price setting incentives in the upstream division is to forego the cost information in the external price and make both prices in the dual price independent of the external market price. When an ex-post accounting signal about cost is available, it is possible to choose an upstream transfer price that does not induce any price-setting distortion, but surprisingly, this is not always optimal. In fact, a distorted upstream price can sometimes be more informative about marginal cost than the first-best monopoly price. When this is the case, headquarters can improve firm profits by setting an upstream transfer price that actually induces distortion in the external market price. The optimal upstream transfer price trades off increased efficiency in internal trade with lost profits in the upstream external market. In this case, in the optimal dual price, both the upstream and downstream transfer prices will be functions of the external market price. Overall, our results demonstrate that optimal dual prices depend on several key factors of the firm s environment, including the observability of marginal costs and the characteristics of the external demand functions. In general, when marginal cost information is not observable and the upstream division has external market power, the optimal dual prices do not take on any of the forms commonly suggested in textbooks and may be complicated non-linear functions. Little work has been done on dual transfer prices and our paper is a first step in filling 3

6 this gap in the transfer pricing literature. Previous work by Cook (1955) and Hirshleifer (1956) showed that using market prices as transfer prices induce the efficient firm-wide solution if intermediate markets are competitive. Our work partially builds on Baldenius and Reichelstein (2006), who extend this framework by analyzing the effectiveness of marketbased transfer prices with non-competitive external markets and show that intracompany discounts always improve firm-wide profits relative to a transfer price equal to market price if the seller s capacity is constrained. They find ambiguous results, however, when capacity is unconstrained. The assumption of non-competitive external markets is key in our paper, since an upstream manager in a competitive market cannot distort the external price, removing the main tension in our model. We extend Baldenius and Reichelstein (2006) by showing that in a similar setting, decoupling the transfer price allows us to get first-best in the unconstrained case when demand functions are non-random, but that with random demand functions, first-best is still generally not feasible. In general, our work also adds to the larger transfer pricing literature, which spans several decades. In the absence of an external market, Hirshleifer (1956, 1957) has demonstrated that setting the transfer price equal to the marginal production costs induces the optimal centralized solution. Extending this setting, Edlin and Reichelstein (1995), Baldenius, Reichelstein, and Sahay (1999), Vaysman (1996), Baldenius (2000), Sahay (2003), Johnson (2006), and others have investigated the effectiveness of cost-based and negotiated transfer prices in providing ex-ante investment and ex-post trading incentives. Our approach follows the majority of the recent papers in the transfer pricing literature in assuming that managers seek to maximize divisional profits for exogenous reasons. Compensation contracts based on divisional profits are common in practice, but this modelling approach is a departure from the recent agency theory and mechanism design literature that allows compensation contracts to arise endogenously. The remainder of the paper proceeds as follows. In Section 2, we describe the model and in Section 3, we derive the optimal dual transfer pricing system when demand functions are non-random. In Section 4 we find conditions for the dual price that guarantee first-best 4

7 upstream external price setting when demand is random and examine the trade-off between first-best external price-setting and efficient internal trade. Section 5 concludes. 2 Model In this section we describe the model assumptions that we use throughout the rest of the paper. Consider a decentralized firm with two divisions. The manager of each division makes pricing and quantity decisions for his own division and is evaluated on the basis of ex-post divisional profit. The upstream (or selling) division produces a good that is sold to external customers and is also transferred internally to the downstream (or buying) division. The two divisions act independently to maximize their own profits and the two external markets are independent, in the sense that changing the price in one market does not affect demand in the other market (i.e. the cross-price elasticities are zero.) We assume that capacity in the upstream division is unconstrained. Initially, we assume that the divisional revenue and cost functions are parameterized by a (multidimensional) state variable, θ = {θ u, θ d } Θ, where each component θ i is drawn from a compact support. Given an external price, p u, the upstream division faces an external demand function, q u (p u, θ u ). Given an internal transfer of q d units, the downstream division s net revenue is R(q d, θ d ), where R q > 0 and R qq < 0 for all θ Θ with subscripts denoting partial derivatives. The upstream division s production cost, c(θ u ), is linear in the quantity produced. Prior to decision-making, the upstream division observes the state variable θ and the downstream division observes θ d. In other words, the upstream division observes all cost and demand parameters, while the downstream division knows only the revenue associated with selling the final product. For ease of exposition, we will drop the subscripts on θ u and θ d. Headquarters does not observe θ, but does know its probability distribution. In addition, headquarters may receive an end-of-period accounting report in Date 3. This report, which is received after all trade and price-setting decisions have been made, allows headquarters to observe ĉ = c + ɛ, a noisy signal about the upstream division s production cost. The 5

8 random variable, ɛ, is mean-zero and is uncorrelated with all other model variables. In the analysis that follows we examine the outcomes that can be achieved for both the case when ĉ is available for contracting and the case in which headquarters does not receive any signal about cost. Headquarters sets a transfer pricing policy at Time 0 that defines how internal trade will impact each division s profit. These transfer prices may be functions of the chosen upstream price p u, as well as of ĉ (if available). In particular, the upstream division will receive t u for each unit transferred internally and the downstream division will pay t d for each unit it purchases internally, resulting in divisional profits of: π u = (p u c(θ))q u (p u, θ) + t d (q d c(θ)) (1) π d = R(q d, θ) t d q d (2) If t u = t d = t, then we refer to the transfer price, t, as a zero-sum price. If t u t d, then we refer to the pair of transfer prices, {t u, t d }, as a dual transfer price. Clearly, the sum of the divisional profits will not equal total firm profit if a dual price is used. The time line of events in the model is as follows: transfer pricing rule set by headquarters θ realized p u, q d chosen, trade takes place ĉ (possibly) observed by headquarters Figure 1: Time Line 2.1 First Best In the a first best, full-information setting, p u and q d are jointly chosen to maximize firm profit. The firm s objective function is: max π f = (p u c(θ))q u (p u, θ) + R(q d, θ) c(θ)q d (3) p u,q d 6

9 and assuming interior solutions, the first best levels of p u and q d solve the first order conditions: q u (p u, θ) + (p u c(θ)) q u(p u, θ) = 0 R q (q d, θ) c(θ) = 0 (4a) (4b) When first best is achieved, the upstream division sets the external price to the firstbest external monopoly price, p u, and the downstream division chooses the efficient internal trade quantity. It is immediately apparent that in the decentralized firm, (4b) will only be achieved if the downstream transfer price t d is equal to c(θ). In this setting the transfer pricing problem faced by headquarters is how to get t d as close as possible to c(θ) while causing as little distortion as possible in the external market price p u. 3 Dual Transfer Prices and Non-Random Demand Functions We begin by examining the question of whether it is possible for headquarters to set the transfer price functions to achieve the first-best outcomes in the special case when the external and internal demand functions q u (p u ) and q d (t d ), but not the cost c, are known to headquarters before it sets the transfer pricing rule. This assumption may be somewhat unrealistic in practice, but it is interesting to examine how dual transfer prices work and how upstream and downstream divisions interact even in this simple setting. It turns out that in this case possible to achieve first best, as long as the upstream monopoly price can be inverted to find the cost. If so, we can choose the upstream transfer pricing function such that the upstream division still has the incentive to choose the profitmaximizing external price. However this requires setting an upstream transfer price t u that is non-linear in the upstream price p u. (The downstream price t d may or may not be non-linear.) We make the following assumptions: 7

10 Upstream demand is such that the external monopoly price p u is a strictly increasing function of cost. Denote this function as p u = g(c). Downstream demand will always be positive if t d is set to equal cost, for all possible costs in the cost distribution. Downstream demand is differentiable. In this case we state the following proposition: Proposition 1 When the demand functions are known to headquarters, the following dual transfer pricing system yields first-best firm profits: t d (p u ) = g 1 (p u ) (5) t u (p u ) = g 1 (p u ) + A q d (c) dc c=g 1 (p u) q d (g 1 (p u )) (6) where A is an arbitrary constant. Proof: See Appendix In equilibrium, this form of t u causes the upstream division to choose p u, generating downstream transfer price g 1 (p u) = c, providing efficient trade incentives to the downstream division. Substituting c for g 1 (p u ) in (6) allows us to rewrite t u as t u = c + A q d (c) dc q d (c) (7) The second term in equation (7) has a natural interpretation. The q d (c) dc term is (up to an additive constant) equal to the consumer surplus enjoyed by the downstream division when it faces a transfer price equal to c. In fact we could make this term equal to the consumer surplus by replacing the indefinite integral by the definite integral c q d (s) ds, assuming this integral is finite. If we multiply through by q d (c), we find that the total value of transfer payments to the upstream division is given by q d (c) c + c q d (c) dc + A. This payment has three 8

11 parts: a reimbursement for the cost of producing for the downstream division, the amount of consumer surplus earned by the downstream division on the transfer, and an arbitrary lump sum payment. Thus the payment has the effect of internalizing the downstream benefit in the upstream pricing decision. Note that the optimal upstream transfer price is not a single function, but a family of functions, where the value of the arbitrary constant will determine the expected value of payments that go to the upstream division. These functions, far from being simple additive or multiplicative discounts on p u, may be highly non-linear, and depending on the choice of A may even be non-monotonic over the relevant region. Further, first-best will in the general case not be implementable if either the upstream or downstream demand functions are random, even if the probability distribution is known to headquarters. A random downstream demand function generally prevents us from determining the consumer surplus earned by the downstream division given c, which makes it impossible to fully internalize the benefits to the upstream division. A random upstream demand function in general makes it impossible to invert the external monopoly price to find cost. 3 We explore random demand functions in the following sections. 4 Dual Transfer Prices with Random Demand We now turn to the case where demand is allowed to be random. Also, we allow transfer prices to be functions of ĉ, if available. First examine the upstream division s price-setting incentives. In particular, we seek to understand how the upstream division s transfer price affects its incentives in setting the external market price. The upstream division chooses ˆp u to maximize its expected profit function, conditional on the information about cost and demand observed by the upstream division and encap- 3 An exception is a family of demand functions with constant elasticity and with a multiplicative random demand factor. 9

12 sulated in θ: ˆp u = argmax E[(p u c(θ))q u (p u, θ) + (t u (p u, ĉ) c(θ)) q d (t d (p u, ĉ)) θ] (8) p u The equilibrium upstream price, ˆp u, is the solution to: q u (p u, θ) + (p u c(θ))q u(p u, θ) + (E[t u (p u, ĉ) θ] c(θ)) q d t d t d p u + E[t u(p u, ĉ) θ] p u q d = 0 (9) First, note that since q d is a random demand function, we cannot set E[t u ] such that p u = p u for all q d unless (E[t u (p u ) θ] c(θ)) q d t d t d p u + E[t u(p u ) θ] p u q d = 0. (10) Proposition 2 The upstream division will set p u = p u for all values of θ if and only if one of the following conditions is met: 1. E[t u (p u ) θ] = c(θ) 2. t u(p u ) p u = t d(p u ) p u = 0 Proof: See Appendix The first condition can only be met by setting t u = ĉ. In this case, the upstream division is exactly reimbursed for production costs in expectation and cannot change the profits from internal trade by manipulating the external market price. The second condition says that the upstream division will not deviate from setting the first-best external monopoly price if both the upstream and the downstream divisions transfer prices are independent of p u. The requirement that t u be independent of p u is fairly intuitive - if the upstream transfer price is a function of p u, then the upstream division will maximize its divisional profits by setting p u to maximize the sum of internal and external profits, instead setting the external price to the first-best monopoly price. To understand why t d must also be independent of p u in the second part of the proposition, notice that even if the upstream division can t manipulate its own transfer price 10

13 by changing p u, it can still affect q d by changing p u as long as the downstream division s transfer price is a function of p u. In this case, shifting p u away from p u will change the quantity demanded by the downstream division and again, the upstream division will set p u to maximize combined profits from internal and external trade. Note as a corollary that if we do not observe a signal ĉ, it will be impossible to meet condition 1 of Proposition 2. In this case the only way to achieve no distortion in the external market price will be to choose t u and t d such that neither one is a function of p u. Textbooks that advocate dual pricing often argue that the upstream transfer price should be market-based (market price or discounted market price) or cost-plus (variable cost plus a markup or full cost). Proposition 2 shows that a market-based upstream price will always result in distorted external price-setting incentives. A cost-plus upstream transfer price (ĉ plus an additive or multiplicative markup or some function of E[c]) will result in distorted external price setting incentives if t d is a function of p u, but not if t d is cost-based. If both t u and t d are cost-based (i.e. neither one is a function of p u ), however, we give up the opportunity to use the cost information in p u to improve the downstream transfer price and potentially increase the efficiency of internal trade. We explore the trade-off between efficient price-setting in the upstream division and using the external price to provide cost information to the downstream division in the next section. 4.1 Downstream Transfer Price We know turn to the downstream division and examine the role of the upstream market price in providing information about the upstream division s marginal production costs. We showed in the previous section that in order to induce the upstream division to always set p u = p u, it is necessary to satisfy one of the two condition in Proposition 2. In this section, we set t u = ĉ (Condition 1 in Prop. 2) and derive the optimal downstream transfer price. Note that as long as ĉ is available for contracting, setting t u = ĉ will weakly dominate a dual price that satisfies Condition 2 in Prop. 2 because both dual prices induce first-best external price setting, but the first condition does not restrict the form of the downstream transfer price. 11

14 The optimal t d associated with t u = ĉ will be a function of p u. Surprisingly, however, we are then able to show that firm profits can sometimes be improved by setting an upstream transfer price that induces the upstream division to deviate from p u because the resulting external price can be more informative about the production cost, c, than the first-best price. We show this result by deriving the optimal downstream transfer price when t u = ĉ and then showing that the use of a zero-sum transfer price of the form t = t u = t d = p u + δ will result in higher firm profits for some parameter values. (Note that this zero-sum price is not a constrained version of the dual price, which requires t u = ĉ.) To simplify the analysis, we assume that demand functions are linear. The upstream and downstream division s external demand functions are, respectively: q u (p u ) = a u b u p u q d (p d ) = 2(a d b d p d ). For the downstream division, it will be more convenient to use the inverse external demand function in the analysis that follows: p d (q d ) = 2a d q d 2b d. The demand intercepts, a u and a d, are random variables that are observed only by the managers of the two divisions, but not by headquarters. The slope coefficients are constant and observed by all parties. To facilitate discussion, define the firm s per-unit contribution margin for the external upstream and downstream markets, respectively, as (d u q u /b u ) and (d d q d /(2b d )) where d i = a i /b i c for i {u, d}. To avoid a corner solution, we assume that d u > 0 and d d > 0 (i.e. the optimal trade quantity in both markets is positive). When demand functions are linear, first best is achieved when p u and q d are chosen to solve: ( ) 2ad q d max π f = (p u c)(a u b u p u ) + c q d (11) p u,q d 2b d 12

15 and first best levels of p u and q d are: p u = a u + b u c = c + d u (12a) 2 b u 2 qd = a d b d c = b d d d (12b) Substituting the first best price and quantity choices from (12a) and (12b) into the firm s objective in (3) yields the expected first-best profit level: E[π f ] = b u E[d 2 u] 4 + b d E[d 2 d ] 2 (13) The divisional profit functions for the two divisions, given transfer prices {t u, t d }, are: π u = (p u c)(a u b u p u ) + (t u c) q d (14) ( ) 2ad q d π d = t d q d (15) 2b d In making its quantity decision, the downstream division maximizes its expected profit function in (15) by choosing the following quantity: q d = a d b d t d = q d + b d(c t d ). (16) A comparison of (12b) and (16) shows that the downstream division will choose the efficient transfer quantity if and only if t d = c. However, c is not observable at the time of the quantity transfer decision. Headquarters could set t d = E[c], but as we show below, it will be optimal to include the cost information in the upstream market price in the downstream division s transfer price as well. Let t u = ĉ. Then if we plug (16) and (12a) into the firm s objective in (11), the downstream portion of expected firm profit given the transfer price t d is: E[πd dual ] = b d E[d 2 d ] b d E[(t d c) 2 ] 2 2 (17) This profit function is equal to the downstream portion of first-best profit in (13) minus a mean-squared error term: MSE = b d E[(t d (p u ) c) 2 ]. (18) 2 13

16 The best headquarters can do is to choose the function t d to minimize MSE, given the available contracting variable, p u. MSE is minimized when t d (p u ) = E[c p u ]. For tractability and consistent with practice, we restrict t d to be a linear function. In this case, MSE is minimized when t d is chosen to be the best linear predictor of E[c p u ]. Note that if the joint distribution of c and a u is bivariate normal, the best predictor of E[c p u ] is linear. Proposition 3 When t u = ĉ, the optimal linear downstream transfer price is given by: t d = (1 α )E[c] + α (p u δ) where and α = Cov(c, p u) V ar(p u ) = 2b 2 u V ar(c) + 2b u Cov(a u, c) V ar(a u ) + b 2 u V ar(c) + 2b u Cov(a u, c) δ = E[d u] 2 Proof: See Appendix Proposition 3 tells us that the upstream division s transfer price is set to induce p u = p u, the optimal downstream transfer price is a linear combination of expected cost and the additively-adjusted market price. Since the adjustment term, δ, is equal to the monopoly markup, E[p u δ] = E[c]. Then (16) implies E[q d ] = E[q d ], since E[t d] = E[c] for all α. If the per-unit cost is constant across all states of the world, then Cov(c, p u ) = 0 and the weight on the adjusted market price will be zero (α = 0). Conversely, if the upstream division s profit margin, d u, is constant then p u δ = c and all of the weight in the optimal transfer price will fall on the discounted market price (α = 1). In either case, the transfer price will be actual cost and first-best profits will be achieved. 4 Finally, notice that although the optimal dual transfer price looks like a weighted average, α is not constrained to be between zero and one. 4 Note that when the variance of d u is zero, then Cov(a u, c) = 1 2b u V ar(a u) + bu V ar(c) and V ar(au) = 2 b 2 uv ar(c), yielding the desired result. 14

17 When the variances of c and d u are both non-zero (so that first best is not achievable), expected profits will deviate from expected first-best profits by dual = Π fb Π dual, with dual = b d [V ar(c) Cov(c, p u) 2 ] 2 V ar(p = b d V ar(c) (1 ρ 2 c,p u) 2 ) (19) u where ρ 2 c,p u is the squared correlation coefficient and p u = au+bu c 2 b u is the first-best external monopoly price. The deviation from first best profits is minimized when V ar(c) = 0 or when c and p u are perfectly correlated (i.e. ρ 2 c,p u {1, 1}), so that p u is a perfect signal about c. The deviation reaches it s maximum when c and p u are uncorrelated (ρ 2 c,p u = 0), so that p u provides no information about actual costs, c. 4.2 Upstream Price Distortion and Internal Trade Efficiency So far, we have derived the upstream transfer price that induces first-best external price setting and given this price, we derived the efficient downstream transfer price. In this section, we show that there is a trade-off between providing the upstream division with efficient external price-setting incentives and creating a downstream transfer price that incorporates all available information about the marginal cost of the internal good in order to motivate efficient internal trade. In particular, we show how firm profits can be improved by inducing the upstream division to deviate from p u if doing so makes the external market price a better signal about the firm s true cost. Using the distorted external market price in the downstream transfer price instead of the first-best external price can improve the efficiency of internal trade and the firm would chose the transfer prices to trade off the increased profit from more efficient internal trade with the decrease in profit associated with setting an external upstream price that is not the first-best monopoly price. To highlight the intuition behind this observation, we compare the dual transfer price described above in which t u = c and t d is calculated according to Proposition 3, with a zerosum discounted market price of the form t z = p u δ where δ is a suitably chosen discount and t u = t d = t z. We choose this functional form because it is common in practice and because it provides a tractable framework to build intuition for our results. It s important to note that the zero-sum transfer price is not a constrained version of the dual transfer 15

18 price because t u = ĉ under the dual scheme and t u = p u + δ under the zero-sum scheme. As we will show below, the discounted market transfer price, t z, provides incentives for the upstream manager to distort the external market price in order to maximize his divisional profits. In fact, Baldenius and Reichelstein (2006) show that in a general setting, a first-best outcome is not achievable when the zero-sum transfer price is a multiplicative or additive function of the discounted external upstream market price and upstream capacity is unconstrained. Given the transfer price t z = p u δ, the downstream division will choose a quantity, q d, to solve the following problem: max q d π d = The resulting transfer quantity is (( ad q ) ) d (p u δ) q d. (20) b d 2b d q z d = a d b d (p u δ). (21) The upstream division s transfer price is now a function of the external price, so it will set the external price to maximize the sum of internal and external profits, choosing p u to solve the following problem: max p u π u = (p u c)(a u b u p u ) + (p u δ c)(a d b d (p u δ)). (22) The resulting external price is: p z u = 1 2 ( ) au + a d + c + b d δ (23) b u + b d b u + b d The zero-sum price, p z u, is a weighted average of the internal and external monopoly prices that the upstream division would choose if it could perfectly price discriminate between the two markets. 5 5 Imagine that the upstream division could perfectly price discriminate between the two markets, choosing an external price, p m u, and an internal price p m i δ. Then the upstream division would set: p m u = p u = au + buc 2b u, p m i δ = p i = a d + b d c 2b d. (24) The zero-sum price, p z u, has the following form: p z u = (1 γ) p m u + γ p m i where γ = b d /(b u + b d ). 16

19 Then Headquarters will choose the discount, δ, to optimize total ex-ante expected firm profits by solving the following problem: ( max π f = (p z u c)(a u b u p z ad δ u) + + a d b d (p z ) u δ) c (a d b d (p z u δ))), (25) b d 2b d The optimal value of δ is: where λ = b d b u+2b d. ˆδ = E[d u] 2 λ E[d d], (26) 2 Straight-forward algebra reveals that when ˆδ < 0, E[p z u] is greater than au b u. To avoid a corner solution in which the upstream division refuses to participate in the external market, we assume that λe[d d ] < E[d u ], so that δ will always be positive. 6 At first glance, it may appear that the dual transfer price derived in the previous section would always dominate the discounted market price, since the dual price aggregates cost and market information and does not induce distorted external price setting by the upstream division. We find, however, that this intuition is not always correct, as suggested by the following proposition: Proposition 4 There exist parameter values for the distributions of c, a d and a u such that the transfer price t u = t d = p u + δ outperforms the dual transfer price of the form t u = ĉ, t d = (1 α )E[c] + α (p u δ), in terms of expected firm profit. We reemphasize that the zero-sum transfer price is not a constrained version of the dual transfer price, but rather the two prices put different constraints on the upstream transfer price, t u. The dual transfer price requires that t u = ĉ, the final reported cost, while the zero-sum transfer price requires that t u = t d = p u δ, a discounted market price. 6 Baldenius and Reichelstein (2006) show that when demand curves are linear and the transfer price is of the form t = p u δ, the optimal adjustment is always a discount rather than a markup, but with more general demand functions, a markup can be optimal. 17

20 To build intuition, recall that the external market price in the dual and zero-sum cases are, respectively, p u = a u + b u c 2b u, p z u = 1 2 [ au + a d + (b u + b d ) c b u + b d ] + b d b u + b d δ (27) These prices play two roles in our model. First, they determine profits in the external market. First-best upstream profits are achieved in the dual case, since p u is the external monopoly price. In the zero-sum case, pu z is distorted relative to p u by a u, a d and δ. This distortion moves upstream profits away from first best and in expectation, increases the expected upstream market price relative to p u: E[p z u ˆδ] = E[p u] + λ E[d d]. (28) 2 The second role that the two prices in (27) play in our model is to transmit information about c to the downstream division through the transfer price. The information about cost in p u is distorted by variation in a u. For p z u, the information about cost is distorted by a u to a lesser extent, but is also distorted by a d. Thus when V ar(a u ) is high relative to V ar(a d ), p z u can be more informative about cost than p u and can be used in the downstream transfer price to induce a more efficient quantity decision and higher downstream profits than would be obtained by basing the transfer price on p u. Then the firm faces a tradeoff between choosing a transfer pricing policy that induces first-best upstream profits and choosing a transfer price that induces a distortion in the upstream price, but has the potential to improve the signal about cost in the downstream price. The optimal policy depends on the relative parameter values in the cost and demand distributions. An important point to note is that this result holds true even if the cost and demand variables are uncorrelated with each other. Consider the following special case in which b u = b d = 1, V ar(a d ) = 0, a u, a d and c are mutually independent, and V ar(c) = V ar(a u ). Then α = 1 and dual z = 1 96 [3V ar(a u) 8E[d d ] 2 ] (29) 18

21 In this case, the zero-sum transfer price can outperform the dual transfer price as long as V ar(a u ) (or equivalently, V ar(c)) is large relative to E[d d ] 2. 5 Conclusion In decentralized firms, transfer prices play an important role in allocating profits among divisions and influencing the price and quantity decisions of divisional managers who are engaged in inter-firm transactions. In this paper we examine what level of efficiency can be achieved using dual transfer prices when the upstream division s production costs are unobservable and the external markets are non-competitive. We find that when external demand is non-random, firstbest is achievable with a carefully chosen dual price. When demand is random, first-best internal trade is generally not possible, but it is possible to construct a dual transfer price that induces the upstream division to charge the first-best monopoly price externally while setting the downstream transfer price as a function of the upstream external price to take advantage of the cost information it contains. We find, however, that this transfer price is not always optimal. Rather, firm profits can be improved by allowing the upstream division to distort the external price because the distorted external price provides a better signal about cost to the downstream division than the undistorted monopoly price. In general, the optimal dual prices that we derive are not of the form suggested by many textbooks and can be highly non-linear. Further, they entail trade-offs that are not immediately obvious. This may explain, in part, why dual prices are not widely used in practice. 19

22 Appendix Proof of Proposition 1 Assume that dual transfer pricing functions exist which implement the first-best outcomes, and denote these upstream and downstream transfer price functions as t u (p u ) and t d (p u ) respectively. For the first best to be achieved, two conditions must hold: (1) The upstream division must choose p u (c) equal to the monopoly price p u = g(c) for every possible value of c, and (2) Downstream transfer price t d (g(c)) must equal c for every value of c. The upstream division maximizes (p u c)q u (p u ) + t u (p u (c))q d (c) + (t u (p u ) c)q d (p u ) The first-best assumption that p u = p u implies that the first part of the first order condition is equal to zero. Therefore the first order conditions for the downstream portion must equal zero when p u = p u. f u(p u)q d (p u) + (t u (p u) c)q d (p u)f d (p u) = 0 (30) We know t d (p u(c)) = c differentiating both sides with respect to c and rearranging gives f d (p u(c)) = 1/g (c). Define h(c) = t u (g(c)). Substituting into (30) gives: h (c) g (c) q d(c) + (h(c) c)q d (c) g = 0 (31) (c) h (c)q d (c) + (h(c) c)q d (c) = 0 (32) The solution to this first order differential equation is given by h(c) = A cq d (c)dc q d (c) where A is an arbitrary constant. Integrating by parts and rearranging gives h(c) = c + A q d (c)dc q d (c) To find the transfer price that this implies, we invert the g(c) function to get back the value of p u. This gives us finally the optimal transfer price equations: (33) (34) t d (p u ) = g 1 (p u ) (35) t u (p u ) = g 1 (p u ) + A q d (c)dc c=g 1 (p u) q d (g 1 (p u )) (36) 20

23 Proof of Proposition 2 The upstream division chooses ˆp u to maximize its expected profit function, conditional on the information about cost and demand observed by the upstream division and encapsulated in θ: ˆp u = argmax E[(p u c(θ))q u (p u, θ) + (t u (p u, ĉ) c(θ)) q d (t d (p u, ĉ)) θ] (37) p u Taking the first order condition and solving for p u gives us the equilibrium upstream price: q u (p u, θ) + (p u c(θ))q u(p u, θ) + (E[t u (p u, ĉ) θ] c(θ)) q d t d t d p u + E[t u(p u, ĉ) θ] p u q d = 0(38) Comparing this expression with (4a), we see that the upstream division will choose ˆp u = p u when (E[t u (p u, ĉ) θ] c(θ)) q d t d t d p u + E[t u(p u, ĉ) θ] p u q d = 0. (39) Since θ is not known to headquarters, this expression will only hold for all realizations of c(θ) if E[t u(p u, ĉ) θ] = 0 and one of the two following conditions holds: t d / p u = 0 or p u E[t u (p u, ĉ) θ] c(θ) = 0. In our model, the only way to get E[t u (p u, ĉ) θ] c(θ) = 0 is to set t u = ĉ since E[ĉ] = c(θ). And in this case, E[t u(p u, ĉ) θ] = 0 then holds automatically. p u Proof of Proposition 3 When the following condition expectation is linear, E[c p u ] = α 0 + α 1 p u, (40) then the parameter values can be derived using the standard 2-variable regression equations: α 1 = Cov(c, p u) V ar(p u ) (41) α 0 = E[c] α 1 E[p u ] (42) 21

24 Plugging (41) and (42) back into (40) and rearranging terms gives us the result in the proposition. Proof of Proposition 4 The expected deviation from first best expected profit using the optimal dual transfer price from Proposition 3 is: dual = 1 [V ar(c) Cov(c, p u) 2 ] = 1 2 V ar(p u ) 2 [V ar(c) ( 1 4b 2 u V ar(a u ) V ar(c) + 1 2b u Cov(a u, c)) V ar(c) + 1 2b u Cov(a u, c) ] (43) The expected deviation from first best firm profits using the zero-sum transfer price is: z = 2b u(b u + b d ) 2 (E[b d d d ] 2 + V ar[b d d d ]) + b d V ar[b u (b u + 2b d )d u b u b d d d ] 8b u (b u + b d ) 2 (b u + 2b d ) (44) When the dual z > 0, the zero-sum transfer price produces a smaller expected deviation from first-best firm profit than the optimal dual transfer price. One set of parameters that would produce this result is: V ar(c) = V ar(a u ) = 100, V ar(a d ) = Cov(a u, a d ) = Cov(c, a d ) = Cov(c, a u ) = 0, b u = b d = 1 and E[d d ] 2 = 10, resulting in dual = 25 and z =

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