Transfer pricing in a multidivisional firm: A cooperative game analysis

Abstract

We consider the transfer pricing decision for a multidivisional firm with an upstream division and multiple downstream divisions. The downstream divisions can independently determine their retail prices, and decide on whether or not they will purchase from the upstream division at negotiated transfer prices. To allocate the firm-wide profit between upstream and downstream divisions, we construct a cooperative game, show the convexity of the game, and then compute the Shapley value-based transfer prices for the firm.

Introduction

The transfer pricing problem is of significant importance to multidivisional firms which need to consider the allocation of firm-wide profit between an upstream division and multiple downstream divisions. In such transfer pricing problems, all divisions of a firm can independently make their decisions as if they were operating in a decentralized setting. This means that all downstream divisions can determine their own retail prices, and decide on whether or not they will buy from the upstream division at negotiated transfer prices. For details regarding the transfer pricing decision in multidivisional firms, see online Appendix A.

In this paper, we consider the transfer pricing decisions of a multidivisional firm where the upstream and downstream divisions negotiate the transfer price that results in a fair allocation of the maximum system-wide profit surplus among three or more divisions. The downstream divisions then determine their retail prices. In this paper, to reflect the fact that customers are sensitive to retail prices, we assume that the demand in the market a downstream division serves is dependent on the division’s retail pricing decision. We learn from [5] that most transfer pricing publications assumed the demand to be independent of the retail price, i.e., a constant, and we find that only a few recent publications (e.g., [1], [2]) used a linear, deterministic, price-dependent demand function for transfer pricing problems.

We assume that the upstream division’s unit production cost is not a constant but a decreasing, convex function of the production quantity—i.e., the downstream division’s order quantity, or the demand faced by the downstream division. Such a modeling approach renders our model and analysis more realistic, because it is consistent with the wide existence of “economies of scale,” see, e.g., [9]. The quantity-dependent cost function also distinguishes our paper from other transfer pricing papers.

In reality, the upstream division of a multidivisional firm usually sells its intermediate products to multiple downstream divisions that are located in different marketing areas. In Section 2, we analyze the transfer pricing decisions for such a system by using a cooperative game theory. We believe that this is an appropriate methodology for our transfer-pricing analysis because all downstream divisions of a multidivisional firm are “free” to determine whether or not they will buy from the upstream division at negotiated transfer prices. Specifically, each division is able to decide on whether it will trade with the upstream division in a non-cooperative setting or in a cooperative setting. In the non-cooperative setting, the upstream and downstream members make their transfer prices and retail prices in Stackelberg equilibrium, respectively. In the cooperative setting, the downstream members choose the globally-optimal retail prices that maximize the system-wide profit, and negotiate transfer prices with the upstream member. In order to guarantee that the downstream members are willing to adopt the globally-optimal retail prices, we will use the cooperative game theory to find the negotiated transfer prices assuring that the downstream members are better (by achieving more profits) in the cooperative setting than in the non-cooperative setting.

Therefore, for the multidivisional firm with a single upstream division and $n$ downstream divisions with $n\ge 2$, we construct an $(n+1)$-division cooperative game in characteristic function form, and prove that the characteristic value function is supermodular, that is, the game is convex and thus superadditive. We also show that our game has a non-empty core, and use the concept of Shapley value [12] to find a unique allocation scheme. Note that the Shapley value is a proper concept for our analysis because it is in the core due to the convexity of our game. We then calculate $n$ transfer prices for the $n$ downstream divisions, using the Shapley value-based allocation scheme. Our proofs for all theorems and corollaries are delegated to online Appendices C and D, respectively.

An important contribution of our paper to the literature is the application of the $n$-player cooperative game theory (with $n\ge 3$) to transfer pricing problems. To the best of our knowledge, very few transfer pricing-related publications (for example, [11]) applied this important methodology. Our model differs from [11] because of the following three facts. (i) As mentioned above, we consider the quantity-dependent production cost and the price-sensitive demand functions; Rosenthal [11] assumed a constant production cost and a deterministic demand. (ii) We investigate the two-echelon system involving a single upstream division and multiple downstream divisions, whereas Rosenthal [11] considered an $n$-echelon ($n\ge 3$) system in which there is a single division at each level. (iii) We compute the firm-wide profit surplus as the difference between the profit in the cooperative setting and that in the non-cooperative setting. In [11], each division’s profit in the non-cooperative setting was assumed to be zero; thus, each player’s profit surplus was equal to the player’s profit in the cooperative setting.