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The double marginalization problem of transfer pricing: Theory and experiment
European Journal of Operational Research 196 (2009) 434–439
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Continuous Optimization
The double marginalization problem of transfer pricing: Theory and experiment Björn Lantz * School of Business, Economics and Law, University of Gothenburg, Box 610, 405 30 Gothenburg, Sweden
a r t i c l e
i n f o
Article history: Received 6 February 2007 Accepted 2 April 2008 Available online 11 April 2008 Keywords: Pricing Transfer pricing Experiment
a b s t r a c t In this paper, we find that the idea of using optional two-part tariffs as a basis for tariff renegotiations in a bilaterally monopoly setting is a solution to the double marginalization problem that theoretically (1) creates a stable equilibrium, (2) at the overall efficient level, (3) without the presence of a central management. Through experimental testing, we find that the efficiency of this mechanism is significantly higher than the efficiency of simple direct negotiation, both under symmetrically and asymmetrically distributed information. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The general double marginalization problem refers to the fact that the actors in a bilateral monopoly setting will reduce their combined profit by simultaneously exercising their market power against each other, compared to a situation where the actors are vertically integrated. This problem can be traced back to Cournot (1838), even though Edgeworth (1889) and Pareto (1896) probably were the first to provide formal analysis of the problem. The basis of the problem is that, compared to the perfect competition pricequantity solution (or the vertically integrated solution), a classical monopolistic seller wants to reduce the quantity transferred in order to increase the price, while a classical monopsonistic buyer wants to reduce the quantity transferred in order to decrease the price. But if no actor can dictate the price, this actually results in two distinct problems: (1) there is no stable equilibrium on the market, which means that the actors will have to negotiate price and quantity; (2) the negotiation is likely to result in an inefficient solution, since both actors were trying to reduce the quantity in the first place. The problem of double marginalization has been discussed in several settings, beside the pure economic bilateral monopoly context. For example, Jeuland and Shugan (1983) analyze how distributions channels can be coordinated and suggest, e.g. joint ownership, profit sharing or quantity discounts as possible ways to deal with the double marginalization. However, neither of these suggestions actually solved the two distinct problems. Another application is made by Economides (1999), who analyzes how quality is affected in a bilateral monopoly setting. He concludes that the double marginalization creates a situation
* Tel.: +46 317865245. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.04.007
where ‘‘. . .independent vertically-related (disintegrated) monopolists will provide products of lower quality level than a sole integrated monopolist” (Economides, 1999). But no solution to the problem was provided. Yet another setting where double marginalization has been a major issue is the transfer pricing context. For example, Ronen and McKinney (1970) acknowledged the Hirshleifer (1956) proposition that an optimal transfer price must reflect the seller’s marginal cost, but claimed that dual pricing with support from a central administration was necessary in order to make the marginal cost solution an equilibrium (i.e. in order to remove the problem of double marginalization). This model, however, requires the presence of a central management and is sensitive to asymmetric information since it does not remove the incentive for the actors to provide false information during the process of determining the pricing structure. The literature on the double marginalization problem and proposed solutions to it is rather extensive (see, e.g. Rey and Vergé, 2005; Tirole, 1988) but the only proposed solution that (1) creates a stable equilibrium (2) at the overall efficient level (3) without the presence of a central management at some point in the process seems to be the bilaterally optional two-part (BOT) tariff model suggested by Lantz (2000).1 The basic idea is to use the current two-part tariff as a ”threat tariff” à la Sibley (1989) in the negotiations (the model was presented in a transfer pricing setting) for a new two-part tariff. Either the buyer or the seller can propose a change in the present two-part tariff at any time and for any reason, but if the counterpart does not accept that proposition the present tariff remains unchanged. The logic is that the model by definition
1 A two-part tariff is defined as a pricing principle where the price of a product or service is composed of two parts – a lump-sum fee (‘‘the fixed part”) and a per-unit charge (‘‘the variable part”).
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prevents any actor from exploiting the other, since either both actors gain from a tariff change, or no one increases profit from keeping the old tariff. Exactly the same idea was presented later by Cheng (2002) as ‘‘a breakthrough in transfer pricing”, even though he denoted the fixed part of the tariff ‘‘an option”. However, neither Lantz (2000) nor Cheng (2002) provide a formal analysis of the model, and even though Lantz (2000) does provide some empirical evidence, it is not really conclusive. The purpose of this paper is to provide a formal analysis and the results of an experimental test of this mechanism. The remainder of the paper is organized as follows. Firstly, there is an analytic section where the properties of a direct negotiation (DN) in a standard bilateral monopoly setting under no regulation and under the BOT model, respectively, are derived. The main claim here is that BOT actually does provide incentives for the seller and the buyer to autonomously find the overall efficient solution in the negotiations. Then the experimental design is discussed, and the results from the experiments are presented. The main claim of this section is that the BOT model provides significantly better incentives than DN, both under symmetric and asymmetric information. Finally, there is a short discussion which concludes the paper. 2. The direct negotiation model and the bilaterally optional two-part tariff model
Analogously, the buyer’s decision problem is maximize rðqðpt ÞÞ pt qðpt Þ; which has the solution qðp Þ pt ¼ r 0 ðqðpt ÞÞ þ 0 t ; q ðpt Þ
We assume that the functions c(q(pt)) and r(q(pt)) are suitably smooth and twice continuously differentiable and that c00 (q(pt)) 6 0 and that r00 (q(pt)) P 0 for all q. Note that q(pt) will coincide with r0 (q(pt)) when pt P p* and with c0 (q(pt)) when pt 6 p*. Proposition 1. Profit maximizing behaviour leads to a situation where efficiency is reduced compared to the overall efficient solution.
ð5Þ
which is equivalent to reducing quantity from the overall efficient level by using a monopsony ‘‘mark-down” on net marginal revenue. The problem, of course, is that both actors in the above analysis assume that the opponent does not exercise his market power. Yet, it is easy to see that the dominant strategy for both actors is to try to use their market power no matter what the opponent does. In a game setting, there are four possible outcomes. There are two actors, and both can choose to exercise market power or not exercise market power. For each actor, we rank the outcomes from 1 to 4 in terms of profit. The best outcome for either actor is the monopoly/monopsony position, which is reached if one actor exercises market power while the other one does not. This is of course also the worst outcome for the other actor. The second best is the perfect competition solution, which is reached if neither actor exercises market power. The third best solution (which is indeterminate, but generally better than being subject to the other actor’s market power, but also worse than the perfect competition solution) is reached when both actors try to exercise their respective market power simultaneously.
Under DN, we assume that we have one seller and one buyer in a bilateral monopoly setting, where both want to maximize their individual profit are about to determine short term price and quantity for the good in question. Throughout the paper, we will use these definitions: ft is the fixed part of a two-part tariff (where applicable) in period t; pt is the variable part of a two-part tariff (where applicable – or else the standard linear price) in period t; q(pt) is the quantity in period t; c(q(pt)) is the seller’s total cost function in period t; r(q(pt)) is the buyer’s net total revenue function in period t; p* is the overall efficient level of p, i.e. where r0 (q(p)) = c0 (q(p)).
ð4Þ
Seller
Buyer
Does exercise
Does not exercise
market power
market power
Does exercise
S: 3rd
S: 4th
market power
B: 3rd
B: 1st (monopsony)
Does not exercise S: 1st (monopoly) S: 2nd market power
B: 4th
B: 2nd
From the above table, it is easy to see that the dominant strategy for both actors is to exercise market power no matter what the other actor does. Thus, the equilibrium (though indeterminate in terms of price and quantity) is that both actors try to exercise market power. h Under BOT, the seller and the buyer are both assumed to maximize their individual profit, to reject any suggested tariff change that will lead to lower individual profit compared to the present tariff, and only to suggest tariff changes that will lead to a higher individual profit compared to the present tariff. Thus, the sellers’s decision problem is
Proof. The seller’s decision problem is maximize pt qðpt Þ cðqðpt ÞÞ;
ð1Þ
maximize pt qðpt Þ þ ft cðqðpt ÞÞ subject to pt qðpt Þ þ ft cðqðpt ÞÞ P pt1 qðpt1 Þ þ ft1 cðqðpt1 ÞÞ rðqðpt ÞÞ pt qðpt Þ ft P rðqðpt1 ÞÞ pt1 qðpt1 Þ ft1 :
which has the solution o½pt qðpt Þ cðqðpt ÞÞ ¼ 0; opt
ð6Þ ð2Þ
which yields pt ¼ c0 ðqðpt ÞÞ
qðpt Þ ; q0 ðpt Þ
Under symmetric information, the seller is aware of the fact that the buyers’s decision problem is maximize
ð3Þ
which is equivalent to reducing quantity from the overall efficient level by using a monopoly mark-up on marginal cost (since q(pt)/ q0 (pt) < 0 by definition).
subject to
rðqðpt ÞÞ pt qðpt Þ ft rðqðpt ÞÞ pt qðpt Þ ft P rðqðpt1 ÞÞ pt1 qðpt1 Þ ft1 pt qðpt Þ þ ft cðqðpt ÞÞ P pt1 qðpt1 Þ þ ft1 cðqðpt1 ÞÞ: ð7Þ
The situation facing the buyer is of course simply the reversed one.
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Note that the restrictions in (6) are identical to the restrictions in (7). By adding the restrictions vertically and simplifying, we can express both actors’ decision problems with one restriction, i.e. for the seller maximize
pt qðpt Þ þ ft cðqðpt ÞÞ
subject to rðqðpt ÞÞ cðqðpt ÞÞ P rðqðpt1 ÞÞ cðqðpt1 ÞÞ
ð8Þ
and for the buyer maximize rðqðpt ÞÞ pt qðpt Þ ft subject to rðqðpt ÞÞ cðqðpt ÞÞ P rðqðpt1 ÞÞ cðqðpt1 ÞÞ:
Thus, the stationary solution is characterized by o½pqðpÞ þ f cðqðpÞÞ ¼ pq0 ðpÞ þ qðpÞ c0 ðqðpÞÞq0 ðpÞ ¼ 0 op and o½rðqðpÞÞ pqðpÞ ft ¼ r 0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ qðpÞ ¼ 0: op
ð9Þ
qðpÞ ¼ c0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ
Now, facing the decision problem (8), the seller can be in one of two possible situations. Either it is possible to suggest a new tariff that accomplishes the restrictions in (6) and (8) simultaneously, or it is not.2 Proposition 2. If it is impossible for either the seller or the buyer to propose a new tariff that both increases the individual profit and is accepted by the opponent, the present tariff is an overall efficient tariff. Proof. If it is impossible for either the seller or the buyer to propose a new tariff that both increases the individual profit and is accepted by the opponent, the decision problem for the seller is
ð11Þ
Substituting c(q(pt)) with c(q(pt1)) + r(q (pt)) r(q(pt1)) in the objective function in (6), we can express the decision problem unrestrictedly as maximize pt qðpt Þ þ ft cðqðpt1 ÞÞ rðqðpt ÞÞ þ rðqðpt1 ÞÞ
ð12Þ
The solution for this problem is characterized by o½pt qðpt Þ þ ft cðqðpt1 ÞÞ rðqðpt ÞÞ þ rðqðpt1 ÞÞ ¼ 0: opt
ð13Þ
But if no new tariff can be suggested, we have by definition p = pt = pt1 and f = ft = ft1 (i.e. a stationary solution), and the decision problem (12) then simplifies to maximize pqðpÞ þ f cðqðpÞÞ:
ð14Þ
Following the same logic, if the buyer cannot suggest a new tariff that satisfies the restrictions in (7) and (9) simultaneously, his decision problem simplifies to maximize rðqðpÞÞ pqðpÞ f :
ð19Þ
Substituting (18) in (19) gives r0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ ¼ c0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ
ð20Þ
which simplifies to r0 ðqðpÞÞ ¼ c0 ðqðpÞÞ:
ð21Þ
Thus, if the model does converge, it converges to the overall efficient solution where pt = p*. Hence, the next step in the analysis should be to see if the model converges or not. The condition for convergence is rather obvious: If it is possible to increase overall efficiency, that possibility will be exploited if, and only if, it is possible to find a new tariff that increases both actors’ profit. Proposition 3. When it is possible to increase overall efficiency, it is also possible to find a new tariff that increases both actors’ profit. Proof. A new tariff (pt,ft) dominates the present tariff (pt1, ft1) iff it increases both actors’ profit, i.e. iff both the conditions pt qðpt Þ þ ft cðqðpt ÞÞ > pt1 qðpt1 Þ þ ft1 cðqðpt1 ÞÞ
ð22Þ
and ð10Þ
which can be rearranged to subject to cðqðpt ÞÞ ¼ cðqðpt1 ÞÞ þ rðqðpt ÞÞ rðqðpt1 ÞÞ
ð18Þ
and r0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ ¼ qðpÞ:
A tariff change suggested by one actor, must improve not only that actor’s efficiency, but also the overall efficiency. The fixed part of the tariff cancels out from the restrictions, which means that neither the level of ft nor the difference ft ft1 in the tariff has anything to do with the overall efficiency. It only affects the distribution of profits between the seller and the buyer.
maximize pt qðpt Þ þ ft cðqðpt ÞÞ
ð17Þ
Rearranging (16) and (17) yields
These decision problems have several interesting implications, e.g.
maximize pt qðpt Þ þ ft cðqðpt ÞÞ subject to rðqðpt ÞÞ cðqðpt ÞÞ ¼ rðqðpt1 ÞÞ cðqðpt1 ÞÞ
ð16Þ
ð15Þ
2 The buyer’s situation when facing the decision problem (9) is handled in symmetric manner.
rðqðpt ÞÞ pt qðpt Þ ft > rðqðpt1 ÞÞ pt1 qðpt1 Þ ft1
ð23Þ
are satisfied. By adding (22) and (23) vertically and simplifying we get rðqðpt ÞÞ cðqðpt ÞÞ > rðqðpt1 ÞÞ cðqðpt1 ÞÞ: ð24Þ Thus, if there is a tariff (pt,ft) that satisfies (24), there is also a tariff that satisfies (22) and (23). An inefficient tariff (pt1,ft1) is generally characterized by either pt1 < p* or pt1 > p* since a tariff where pt1 = p* by definition is an efficient tariff. Assume that pt1 > p*.3 Then the new tariff (pt, ft) must obviously be characterized by pt < pt1 and ft > ft1 in order to be able to satisfy both (22) and (23). Then c(q(pt)) > c(q(pt1)) and r(q(pt)) > r(q (pt1)) by definition. But since r0 (q(pt)) > c0 (q(pt)) when p1 – p*, r(q(pt)) r(q(pt1)) > c(q(pt)) c(q(pt1)) and thus also r(q(pt)) c(q(pt)) > r(q(pt1)) c(q(pt1)). Hence, the new tariff (pt,ft) provides a higher overall efficiency than (pt1,ft1). Since there is a new tariff (pt,ft) where pt < pt1 that satisfies (19) when pt1 > p*, it is also possible to choose ft > ft1 so that the new tariff also satisfies (22) and (23). Since this possibility exists, the actors will exploit it. Thus, it is optimal for any actor to refrain from market power no matter what the opponent does. This is easy to see by using the same kind of game setting as above. There are still four possible outcomes. We know that the result will be p11 = pt if any, or both, actors try to exercise market power. Hence, three of the four outcomes of the game are identical with respect to profit. Assuming p11 – p*, we also know that there is a potential to increase the overall efficiency. Since the actors can only increase 3
The case where pt1 < p* can easily be handled in a symmetric way.
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B. Lantz / European Journal of Operational Research 196 (2009) 434–439
their individual profit by not exercising market power, the equilibrium of the game is that both actors refrain from market power.
Table 1 Quantity data from the experiment Mechanism
Period 1
Seller
Buyer
S: 2nd
market power
B: 2nd
B: 2nd
Does not exercise S: 2nd
S: 1st
market power
B: 1st
B: 2nd
6
7
6 7 6 7 6 8 6 6 8 7 7
6 8 6 7 6 7 6 6 8 7 7
6 6 6 8 6 7 6 7 8 7 8
BOT-s
8 10 8 8 6 10 6 7 7 6 7
8 8 6 8 8 8 6 8 7 8 8
8 5 6 8 8 8 7 8 7 8 8
8 5 7 8 8 8 7 8 8 8 8
8 6 7 8 8 8 7 8 8 8 8
8 8 8 8 8 8 8 8 8 8 8
DN-a
5 1 0 9 9 10 0 4
6 2 0 7 10 6 5 4
6 2 5 4 7 10 6 4
7 8 9 4 8 5 6 4
8 8 9 5 7 10 6 8
8 8 9 4 7 7 7 8
8 8 9 4 8 10 7 8
BOT-a
10 8 6 10 4 8 6 4
10 4 7 9 6 8 6 4
10 10 8 8 6 9 8 4
10 5 8 8 6 9 8 4
10 8 8 8 9 8 10 8
10 8 8 8 9 8 8 8
10 10 8 8 9 8 10 8
3. Experimental design
4 This is the same kind of design as in Dejong et al. (1989) and Avila and Ronen (1999), where various transfer pricing models also were tested experimentally.
5
6 7 5 7 7 8 6 7 8 7 7
h
Economic experiments often give the subjects an opportunity to learn, by having them performing the task of interest repeatedly. Hence, this study was designed as a two-factor experiment (mechanism and trials) with repeated measures on one factor (trials).4 Upper level business students were used as subjects. The subjects were paired randomly, and the members of each pair were randomly designated as either a seller or a buyer. The pairs were informed that they could trade up to 10 units in each period. The periods were 8 min long. In each period, the buyer could resell units of an imaginary product to the experimenter, and the seller could purchase units of the same product from the experimenter. The sellers were given their true revenue schedule derived from the marginal revenue function MR = 1025-50Q and the buyers were informed that their true incremental cost was 625 for all units except for the first unit where the incremental cost was 725. Obviously, the pareto efficient (and also the monopsonistic) solution under these conditions is to trade 8 units, but the monopolistic solution is to trade only 4 units. In the experiments where information were symmetrically distributed, the sellers were also given the cost information and the buyers were given the revenue schedule, but in the experiments based on asymmetric information, the cost and revenue information was hidden from the counterpart. All subjects were rewarded based on their individual profits in the negotiations. The experiment was run in two sessions. In the first session, which was based on direct negotiation under symmetric information (DN-s) and bilaterally optional two-part tariffs under symmetric information (BOT-s), 44 subjects participated. They were divided into 22 pairs and the session was run for six periods. The second session was based on direct negotiation under asymmetric information (DN-a) and bilaterally optional two-part tariffs under asymmetric information (BOT-a). Thirty-two subjects participated there, and the session lasted 7 periods. All negotiations were carried out without any interference whatsoever of any external actor, e.g. central management. In total, 244 negotiations took place. The quantities of each negotiation are indicated in Table 1. Of course, the actual efficiency of an agreement is equal to the proportion of the maximum attainable efficiency that the agreement exploits. In Table 2, the actual efficiencies of the different quantities are indicated.
4
6 7 7 7 6 7 6 6 9 7 7
market power market power S: 2nd
3
6 6 6 6 6 7 6 6 5 10 8
Does exercise Does not exercise
Does exercise
2
DN-s
Table 2 Efficiencies for the different quantity alternatives Quantity
Total cost
Total revenue
Aggregate profit
Efficiency
0 1 2 3 4 5 6 7 8 9 10
0 725 1350 1975 2600 3225 3850 4475 5100 5725 6350
0 1000 1950 2850 3700 4500 5250 5950 6600 7200 7750
0 275 600 875 1100 1275 1400 1475 1500 1475 1400
0.0000 0.1833 0.4000 0.5833 0.7333 0.8500 0.9333 0.9833 1.0000 0.9833 0.9333
Price data are not reproduced here, since only the quantities are of interest in an efficiency perspective. The pairs negotiating under DN were not given any specific input to their negotiations beside the above cost and revenue functions. The pairs negotiating under BOT were in addition told that, unless they agreed on something else in some period, the fixed and variable price elements would remain the same as in the previous period. Since the BOT model relies on the idea that the actors in all periods will be subject to a known threat tariff unless they agree about some other tariff, the threat tariff in the very first period of negotiation has to be specified in some other way than ‘‘the tariff of the previous period”. So
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Table 3 Descriptive statistics Mechanism
n
Mean
Standard deviation
Median
DN-s DN-a BOT-s BOT-a
66 56 66 56
0.960 0.842 0.984 0.945
0.034 0.260 0.033 0.081
0.983 0.933 1.000 0.983
Average efficiency
0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 1
2
3
Table 4 Pairwise comparisons Alternative hypothesis
p value for t-test
DN-s > DN BOT-s > BOT DN-s < BOT DN-a < BOT DN-a < BOT
a a s s a
5
6
Fig. 1a. Average efficiency for DN-s.
<0.01 <0.01 <0.01 <0.01 <0.01
Average efficiency
1.2
Table 5 Average efficiency in the different periods
1 0.8 0.6 0.4 0.2
Period
DN-s DN-a BOT-s BOT-a
4
Period
0
1
2
3
4
5
6
0.936 0.583 0.965 0.900
0.965 0.721 0.986 0.904
0.965 0.813 0.977 0.940
0.964 0.902 0.983 0.929
0.964 0.960 0.991 0.981
0.965 0.958 1.000 0.990
1
7
2
3
4
5
6
Period
0.954
Fig. 1b. Average efficiency for DN-a.
0.973
Average efficiency
1.01
in the first period of negotiation, they were told to assume that the two-part tariff in ‘‘the previous period” was characterized by a fixed part equal to 200 and a variable part equal to 700.5 The negotiating pairs got an audio warning when there was only 1 min left, unless they already had reached an agreement. Since all negotiations took place face-to-face, the actors were allowed to use all kind of strategic manipulation and other tactics possible, except threats. All agreements were written down on paper and submitted to the experimenter between each trading period.
1 0.99 0.98 0.97 0.96 0.95 0.94 1
2
3
4
5
6
Period Fig. 1c. Average efficiency for BOT-s.
4. Experimental results
5
These numbers were not entirely arbitrarily chosen. Since marginal cost was a constant, the variable part of this initial tariff should be slightly above marginal cost for the situation to be realistic to the actors. And the level of the fixed part was chosen in order to make the profit distribution between the actors roughly equal, given the level of the variable part. It might of course be interesting to test the model using other initial values. 6 Since the normality assumption of the t-tests and other tests used here may be violated, corresponding non-parametric tests (e.g. Mann-Whitney’s test) have been done. The results were qualitatively the same as for the parametric tests in all cases.
Average efficiency
1
Table 3 describes the data from the experiment on an aggregate level. Initially, all four mechanisms were found to produce an efficiency below 1000 (one-tailed t-tests, p-value <0.01). Of higher interest was of course whether BOT generally performed better than DN. Pairwise t-tests between the four mechanisms revealed a number of significant differences indicated by Table 4.6 DN-s performed significantly better that DN-a, like BOT-s outperformed BOT-a, which was expected. Ceteris paribus, symmetric information should be of some value in the negotiations since it reduces the negotiating power of both actors. Of higher interest is that BOT did significantly better than DN both under symmetric and asymmetric information. Thus, the presence of BOT in the negotiation situation increased the efficiency.
0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 1
2
3
4
5
6
Period Fig. 1d. Average efficiency for BOT-a.
Table 6 Regression results Mechanism
Coefficient of determination
p value for significant slope
DN-s DN-a BOT-s BOT-a
0.04 0.23 0.08 0.14
>0.1 <0.01 <0.05 <0.01
The next step was to analyze the trial effect. Did the subjects learn something from accumulating experience that increased the efficiency in the negotiations? In Table 5, the mean values from
B. Lantz / European Journal of Operational Research 196 (2009) 434–439 Table 7 ANOVA on DN-s and BOT-s over all periods Source of variation
SS
df
MS
F
p-value
F crit
ANOVA Sample Columns Interaction Within
0.018992 0.013369 0.001753 0.129849
1 5 5 120
0.018992 0.002674 0.000351 0.001082
17.55154 2.471066 0.323999
5.37E05 0.036114 0.897702
3.920124 2.289851 2.289851
Total
0.163963
131
Table 8 ANOVA on DN-a and BOT-a over all periods Source of variation
SS
df
ANOVA Sample Columns Interaction Within
0.300357 0.764297 0.309782 3.006319
1 6 6 98
Total
4.380755
111
MS
F
p-value
F crit
0.300357 0.127383 0.05163 0.030677
9.791042 4.152425 1.683044
0.00231 0.000926 0.133134
3.938111 2.192518 2.192518
the different periods for the different mechanisms are reproduced. There seem to be an increasing tendency in the data. A perhaps more illustrative way to indicate this phenomenon is to plot the mean values for each mechanism against the OLS regression line (see Figs. 1a–1d). It is obvious that there may be a ‘‘trial effect” for at least some of the mechanisms. In Table 6, the coefficient of determinations for the regression lines are displayed, along with the p-values for t-tests of whether the true slope is significantly different from 0. For all mechanisms except DN-s, there was a significant ‘‘trial effect”. On the other hand, the correlation can obviously not be linear for further trials, so these figures and tests should only be seen as descriptive. Instead, the two factors ‘‘mechanism” and ‘‘trials” were tested through two-way analyses of variance. The ANOVA table from the comparison of DN-s and BOT-s over all periods is reproduced in Table 7. Both the mechanism effect (p-value <0.01) and the trial effect (p-value <0.05) are significant. The ANOVA table from the comparison of DN-a and BOT-a over all periods is reproduced in Table 8. As in the case of symmetric information, both the mechanism effect (p-value <0.01) and the trial effect (p-value <0.01) is significant. Thus, BOT is significantly more efficient than DN under symmetric as well as under asymmetric information, and the differences between periods are significant. On the other hand, there is no significant interaction between the factors. 5. Concluding remarks In this paper, we have seen that the idea of using optional twopart tariffs as a basis for tariff renegotiations in a bilaterally monopoly setting does have the nice properties claimed, but not formally proven, by Lantz (2000) and Cheng (2002). Thus, this idea is a solution to the problem of double marginalization that (1) creates a stable equilibrium, (2) at the overall efficient level, (3) without the presence of a central management. Even though several theoretical models have been suggested as solutions to the double marginalization problem (Rey and Vergé, 2005), there does not seem to exist some other model with the same nice properties in the literature at this point. One should note that the incentive compatibility of the BOT model only covers the bargaining for pt. The process of determining ft once pt is set is generally a zero-sum game, and may thus consti-
439
tute a problem. Under symmetric information, this could be solved by an ex ante agreement to split any increase in overall efficiency due to a change in pt according to some specific rule (e.g. equally).7 In fact, this model might work better under asymmetric information. A tough negotiator could, under symmetric information, be able to keep close to the entire increase of total surplus due to a tariff change. But under asymmetric information, the weaker part has an information advantage that he may use to prevent the opponent from exploiting him when ft is to be set. No matter how the information is distributed, both actors have, ceteris paribus, an interest in finding a tariff (pt,ft) characterized by pt = p*. Both actors know that it is useless to suggest a new tariff that the opponent does not gain on, so it is easy to see that the model should result in the overall efficient solution under asymmetric information too. This is also the main results of the experiments performed in this study. Under BOT, two actors in a bilateral monopoly setting perform better in an efficiency perspective than in an unregulated direct negotiation. It does not matter whether the actors have private information or not. It is interesting to note that Dejong et al. (1989) found similar results in another experimental study of theoretically optimal transfer pricing mechanisms – the mechanisms significantly outperformed direct negotiation. The theoretically easiest way to solve the problem of double marginalization between two actors is of course to integrate the actors vertically. However, in practice, it is for other reasons often necessary in large organizations to disintegrate, divisionalize and decentralize. This means that control mechanisms can be important management tools in transfer pricing settings, channel coordination situations and other contexts where two actors with some market power have to agree about price and quantity. References Avila, M., Ronen, J., 1999. Transfer-pricing mechanisms: An experimental investigation. International Journal of Industrial Organization 17, 689–715. Cheng, J.M.A., 2002. Breakthrough in transfer pricing. Management Accounting Quarterly 3 (2)
7 One might argue that double marginalization is not a problem on the aggregate level under symmetric information and two-part tariffs, since a monopolistic and fully informed seller will choose a tariff characterized by pt = p* anyway in order to maximize profit through ft, see, e.g. Oi (1971).
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Continuous Optimization
The double marginalization problem of transfer pricing: Theory and experiment Björn Lantz * School of Business, Economics and Law, University of Gothenburg, Box 610, 405 30 Gothenburg, Sweden
a r t i c l e
i n f o
Article history: Received 6 February 2007 Accepted 2 April 2008 Available online 11 April 2008 Keywords: Pricing Transfer pricing Experiment
a b s t r a c t In this paper, we find that the idea of using optional two-part tariffs as a basis for tariff renegotiations in a bilaterally monopoly setting is a solution to the double marginalization problem that theoretically (1) creates a stable equilibrium, (2) at the overall efficient level, (3) without the presence of a central management. Through experimental testing, we find that the efficiency of this mechanism is significantly higher than the efficiency of simple direct negotiation, both under symmetrically and asymmetrically distributed information. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The general double marginalization problem refers to the fact that the actors in a bilateral monopoly setting will reduce their combined profit by simultaneously exercising their market power against each other, compared to a situation where the actors are vertically integrated. This problem can be traced back to Cournot (1838), even though Edgeworth (1889) and Pareto (1896) probably were the first to provide formal analysis of the problem. The basis of the problem is that, compared to the perfect competition pricequantity solution (or the vertically integrated solution), a classical monopolistic seller wants to reduce the quantity transferred in order to increase the price, while a classical monopsonistic buyer wants to reduce the quantity transferred in order to decrease the price. But if no actor can dictate the price, this actually results in two distinct problems: (1) there is no stable equilibrium on the market, which means that the actors will have to negotiate price and quantity; (2) the negotiation is likely to result in an inefficient solution, since both actors were trying to reduce the quantity in the first place. The problem of double marginalization has been discussed in several settings, beside the pure economic bilateral monopoly context. For example, Jeuland and Shugan (1983) analyze how distributions channels can be coordinated and suggest, e.g. joint ownership, profit sharing or quantity discounts as possible ways to deal with the double marginalization. However, neither of these suggestions actually solved the two distinct problems. Another application is made by Economides (1999), who analyzes how quality is affected in a bilateral monopoly setting. He concludes that the double marginalization creates a situation
* Tel.: +46 317865245. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.04.007
where ‘‘. . .independent vertically-related (disintegrated) monopolists will provide products of lower quality level than a sole integrated monopolist” (Economides, 1999). But no solution to the problem was provided. Yet another setting where double marginalization has been a major issue is the transfer pricing context. For example, Ronen and McKinney (1970) acknowledged the Hirshleifer (1956) proposition that an optimal transfer price must reflect the seller’s marginal cost, but claimed that dual pricing with support from a central administration was necessary in order to make the marginal cost solution an equilibrium (i.e. in order to remove the problem of double marginalization). This model, however, requires the presence of a central management and is sensitive to asymmetric information since it does not remove the incentive for the actors to provide false information during the process of determining the pricing structure. The literature on the double marginalization problem and proposed solutions to it is rather extensive (see, e.g. Rey and Vergé, 2005; Tirole, 1988) but the only proposed solution that (1) creates a stable equilibrium (2) at the overall efficient level (3) without the presence of a central management at some point in the process seems to be the bilaterally optional two-part (BOT) tariff model suggested by Lantz (2000).1 The basic idea is to use the current two-part tariff as a ”threat tariff” à la Sibley (1989) in the negotiations (the model was presented in a transfer pricing setting) for a new two-part tariff. Either the buyer or the seller can propose a change in the present two-part tariff at any time and for any reason, but if the counterpart does not accept that proposition the present tariff remains unchanged. The logic is that the model by definition
1 A two-part tariff is defined as a pricing principle where the price of a product or service is composed of two parts – a lump-sum fee (‘‘the fixed part”) and a per-unit charge (‘‘the variable part”).
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prevents any actor from exploiting the other, since either both actors gain from a tariff change, or no one increases profit from keeping the old tariff. Exactly the same idea was presented later by Cheng (2002) as ‘‘a breakthrough in transfer pricing”, even though he denoted the fixed part of the tariff ‘‘an option”. However, neither Lantz (2000) nor Cheng (2002) provide a formal analysis of the model, and even though Lantz (2000) does provide some empirical evidence, it is not really conclusive. The purpose of this paper is to provide a formal analysis and the results of an experimental test of this mechanism. The remainder of the paper is organized as follows. Firstly, there is an analytic section where the properties of a direct negotiation (DN) in a standard bilateral monopoly setting under no regulation and under the BOT model, respectively, are derived. The main claim here is that BOT actually does provide incentives for the seller and the buyer to autonomously find the overall efficient solution in the negotiations. Then the experimental design is discussed, and the results from the experiments are presented. The main claim of this section is that the BOT model provides significantly better incentives than DN, both under symmetric and asymmetric information. Finally, there is a short discussion which concludes the paper. 2. The direct negotiation model and the bilaterally optional two-part tariff model
Analogously, the buyer’s decision problem is maximize rðqðpt ÞÞ pt qðpt Þ; which has the solution qðp Þ pt ¼ r 0 ðqðpt ÞÞ þ 0 t ; q ðpt Þ
We assume that the functions c(q(pt)) and r(q(pt)) are suitably smooth and twice continuously differentiable and that c00 (q(pt)) 6 0 and that r00 (q(pt)) P 0 for all q. Note that q(pt) will coincide with r0 (q(pt)) when pt P p* and with c0 (q(pt)) when pt 6 p*. Proposition 1. Profit maximizing behaviour leads to a situation where efficiency is reduced compared to the overall efficient solution.
ð5Þ
which is equivalent to reducing quantity from the overall efficient level by using a monopsony ‘‘mark-down” on net marginal revenue. The problem, of course, is that both actors in the above analysis assume that the opponent does not exercise his market power. Yet, it is easy to see that the dominant strategy for both actors is to try to use their market power no matter what the opponent does. In a game setting, there are four possible outcomes. There are two actors, and both can choose to exercise market power or not exercise market power. For each actor, we rank the outcomes from 1 to 4 in terms of profit. The best outcome for either actor is the monopoly/monopsony position, which is reached if one actor exercises market power while the other one does not. This is of course also the worst outcome for the other actor. The second best is the perfect competition solution, which is reached if neither actor exercises market power. The third best solution (which is indeterminate, but generally better than being subject to the other actor’s market power, but also worse than the perfect competition solution) is reached when both actors try to exercise their respective market power simultaneously.
Under DN, we assume that we have one seller and one buyer in a bilateral monopoly setting, where both want to maximize their individual profit are about to determine short term price and quantity for the good in question. Throughout the paper, we will use these definitions: ft is the fixed part of a two-part tariff (where applicable) in period t; pt is the variable part of a two-part tariff (where applicable – or else the standard linear price) in period t; q(pt) is the quantity in period t; c(q(pt)) is the seller’s total cost function in period t; r(q(pt)) is the buyer’s net total revenue function in period t; p* is the overall efficient level of p, i.e. where r0 (q(p)) = c0 (q(p)).
ð4Þ
Seller
Buyer
Does exercise
Does not exercise
market power
market power
Does exercise
S: 3rd
S: 4th
market power
B: 3rd
B: 1st (monopsony)
Does not exercise S: 1st (monopoly) S: 2nd market power
B: 4th
B: 2nd
From the above table, it is easy to see that the dominant strategy for both actors is to exercise market power no matter what the other actor does. Thus, the equilibrium (though indeterminate in terms of price and quantity) is that both actors try to exercise market power. h Under BOT, the seller and the buyer are both assumed to maximize their individual profit, to reject any suggested tariff change that will lead to lower individual profit compared to the present tariff, and only to suggest tariff changes that will lead to a higher individual profit compared to the present tariff. Thus, the sellers’s decision problem is
Proof. The seller’s decision problem is maximize pt qðpt Þ cðqðpt ÞÞ;
ð1Þ
maximize pt qðpt Þ þ ft cðqðpt ÞÞ subject to pt qðpt Þ þ ft cðqðpt ÞÞ P pt1 qðpt1 Þ þ ft1 cðqðpt1 ÞÞ rðqðpt ÞÞ pt qðpt Þ ft P rðqðpt1 ÞÞ pt1 qðpt1 Þ ft1 :
which has the solution o½pt qðpt Þ cðqðpt ÞÞ ¼ 0; opt
ð6Þ ð2Þ
which yields pt ¼ c0 ðqðpt ÞÞ
qðpt Þ ; q0 ðpt Þ
Under symmetric information, the seller is aware of the fact that the buyers’s decision problem is maximize
ð3Þ
which is equivalent to reducing quantity from the overall efficient level by using a monopoly mark-up on marginal cost (since q(pt)/ q0 (pt) < 0 by definition).
subject to
rðqðpt ÞÞ pt qðpt Þ ft rðqðpt ÞÞ pt qðpt Þ ft P rðqðpt1 ÞÞ pt1 qðpt1 Þ ft1 pt qðpt Þ þ ft cðqðpt ÞÞ P pt1 qðpt1 Þ þ ft1 cðqðpt1 ÞÞ: ð7Þ
The situation facing the buyer is of course simply the reversed one.
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Note that the restrictions in (6) are identical to the restrictions in (7). By adding the restrictions vertically and simplifying, we can express both actors’ decision problems with one restriction, i.e. for the seller maximize
pt qðpt Þ þ ft cðqðpt ÞÞ
subject to rðqðpt ÞÞ cðqðpt ÞÞ P rðqðpt1 ÞÞ cðqðpt1 ÞÞ
ð8Þ
and for the buyer maximize rðqðpt ÞÞ pt qðpt Þ ft subject to rðqðpt ÞÞ cðqðpt ÞÞ P rðqðpt1 ÞÞ cðqðpt1 ÞÞ:
Thus, the stationary solution is characterized by o½pqðpÞ þ f cðqðpÞÞ ¼ pq0 ðpÞ þ qðpÞ c0 ðqðpÞÞq0 ðpÞ ¼ 0 op and o½rðqðpÞÞ pqðpÞ ft ¼ r 0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ qðpÞ ¼ 0: op
ð9Þ
qðpÞ ¼ c0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ
Now, facing the decision problem (8), the seller can be in one of two possible situations. Either it is possible to suggest a new tariff that accomplishes the restrictions in (6) and (8) simultaneously, or it is not.2 Proposition 2. If it is impossible for either the seller or the buyer to propose a new tariff that both increases the individual profit and is accepted by the opponent, the present tariff is an overall efficient tariff. Proof. If it is impossible for either the seller or the buyer to propose a new tariff that both increases the individual profit and is accepted by the opponent, the decision problem for the seller is
ð11Þ
Substituting c(q(pt)) with c(q(pt1)) + r(q (pt)) r(q(pt1)) in the objective function in (6), we can express the decision problem unrestrictedly as maximize pt qðpt Þ þ ft cðqðpt1 ÞÞ rðqðpt ÞÞ þ rðqðpt1 ÞÞ
ð12Þ
The solution for this problem is characterized by o½pt qðpt Þ þ ft cðqðpt1 ÞÞ rðqðpt ÞÞ þ rðqðpt1 ÞÞ ¼ 0: opt
ð13Þ
But if no new tariff can be suggested, we have by definition p = pt = pt1 and f = ft = ft1 (i.e. a stationary solution), and the decision problem (12) then simplifies to maximize pqðpÞ þ f cðqðpÞÞ:
ð14Þ
Following the same logic, if the buyer cannot suggest a new tariff that satisfies the restrictions in (7) and (9) simultaneously, his decision problem simplifies to maximize rðqðpÞÞ pqðpÞ f :
ð19Þ
Substituting (18) in (19) gives r0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ ¼ c0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ
ð20Þ
which simplifies to r0 ðqðpÞÞ ¼ c0 ðqðpÞÞ:
ð21Þ
Thus, if the model does converge, it converges to the overall efficient solution where pt = p*. Hence, the next step in the analysis should be to see if the model converges or not. The condition for convergence is rather obvious: If it is possible to increase overall efficiency, that possibility will be exploited if, and only if, it is possible to find a new tariff that increases both actors’ profit. Proposition 3. When it is possible to increase overall efficiency, it is also possible to find a new tariff that increases both actors’ profit. Proof. A new tariff (pt,ft) dominates the present tariff (pt1, ft1) iff it increases both actors’ profit, i.e. iff both the conditions pt qðpt Þ þ ft cðqðpt ÞÞ > pt1 qðpt1 Þ þ ft1 cðqðpt1 ÞÞ
ð22Þ
and ð10Þ
which can be rearranged to subject to cðqðpt ÞÞ ¼ cðqðpt1 ÞÞ þ rðqðpt ÞÞ rðqðpt1 ÞÞ
ð18Þ
and r0 ðqðpÞÞq0 ðpÞ pq0 ðpÞ ¼ qðpÞ:
A tariff change suggested by one actor, must improve not only that actor’s efficiency, but also the overall efficiency. The fixed part of the tariff cancels out from the restrictions, which means that neither the level of ft nor the difference ft ft1 in the tariff has anything to do with the overall efficiency. It only affects the distribution of profits between the seller and the buyer.
maximize pt qðpt Þ þ ft cðqðpt ÞÞ
ð17Þ
Rearranging (16) and (17) yields
These decision problems have several interesting implications, e.g.
maximize pt qðpt Þ þ ft cðqðpt ÞÞ subject to rðqðpt ÞÞ cðqðpt ÞÞ ¼ rðqðpt1 ÞÞ cðqðpt1 ÞÞ
ð16Þ
ð15Þ
2 The buyer’s situation when facing the decision problem (9) is handled in symmetric manner.
rðqðpt ÞÞ pt qðpt Þ ft > rðqðpt1 ÞÞ pt1 qðpt1 Þ ft1
ð23Þ
are satisfied. By adding (22) and (23) vertically and simplifying we get rðqðpt ÞÞ cðqðpt ÞÞ > rðqðpt1 ÞÞ cðqðpt1 ÞÞ: ð24Þ Thus, if there is a tariff (pt,ft) that satisfies (24), there is also a tariff that satisfies (22) and (23). An inefficient tariff (pt1,ft1) is generally characterized by either pt1 < p* or pt1 > p* since a tariff where pt1 = p* by definition is an efficient tariff. Assume that pt1 > p*.3 Then the new tariff (pt, ft) must obviously be characterized by pt < pt1 and ft > ft1 in order to be able to satisfy both (22) and (23). Then c(q(pt)) > c(q(pt1)) and r(q(pt)) > r(q (pt1)) by definition. But since r0 (q(pt)) > c0 (q(pt)) when p1 – p*, r(q(pt)) r(q(pt1)) > c(q(pt)) c(q(pt1)) and thus also r(q(pt)) c(q(pt)) > r(q(pt1)) c(q(pt1)). Hence, the new tariff (pt,ft) provides a higher overall efficiency than (pt1,ft1). Since there is a new tariff (pt,ft) where pt < pt1 that satisfies (19) when pt1 > p*, it is also possible to choose ft > ft1 so that the new tariff also satisfies (22) and (23). Since this possibility exists, the actors will exploit it. Thus, it is optimal for any actor to refrain from market power no matter what the opponent does. This is easy to see by using the same kind of game setting as above. There are still four possible outcomes. We know that the result will be p11 = pt if any, or both, actors try to exercise market power. Hence, three of the four outcomes of the game are identical with respect to profit. Assuming p11 – p*, we also know that there is a potential to increase the overall efficiency. Since the actors can only increase 3
The case where pt1 < p* can easily be handled in a symmetric way.
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their individual profit by not exercising market power, the equilibrium of the game is that both actors refrain from market power.
Table 1 Quantity data from the experiment Mechanism
Period 1
Seller
Buyer
S: 2nd
market power
B: 2nd
B: 2nd
Does not exercise S: 2nd
S: 1st
market power
B: 1st
B: 2nd
6
7
6 7 6 7 6 8 6 6 8 7 7
6 8 6 7 6 7 6 6 8 7 7
6 6 6 8 6 7 6 7 8 7 8
BOT-s
8 10 8 8 6 10 6 7 7 6 7
8 8 6 8 8 8 6 8 7 8 8
8 5 6 8 8 8 7 8 7 8 8
8 5 7 8 8 8 7 8 8 8 8
8 6 7 8 8 8 7 8 8 8 8
8 8 8 8 8 8 8 8 8 8 8
DN-a
5 1 0 9 9 10 0 4
6 2 0 7 10 6 5 4
6 2 5 4 7 10 6 4
7 8 9 4 8 5 6 4
8 8 9 5 7 10 6 8
8 8 9 4 7 7 7 8
8 8 9 4 8 10 7 8
BOT-a
10 8 6 10 4 8 6 4
10 4 7 9 6 8 6 4
10 10 8 8 6 9 8 4
10 5 8 8 6 9 8 4
10 8 8 8 9 8 10 8
10 8 8 8 9 8 8 8
10 10 8 8 9 8 10 8
3. Experimental design
4 This is the same kind of design as in Dejong et al. (1989) and Avila and Ronen (1999), where various transfer pricing models also were tested experimentally.
5
6 7 5 7 7 8 6 7 8 7 7
h
Economic experiments often give the subjects an opportunity to learn, by having them performing the task of interest repeatedly. Hence, this study was designed as a two-factor experiment (mechanism and trials) with repeated measures on one factor (trials).4 Upper level business students were used as subjects. The subjects were paired randomly, and the members of each pair were randomly designated as either a seller or a buyer. The pairs were informed that they could trade up to 10 units in each period. The periods were 8 min long. In each period, the buyer could resell units of an imaginary product to the experimenter, and the seller could purchase units of the same product from the experimenter. The sellers were given their true revenue schedule derived from the marginal revenue function MR = 1025-50Q and the buyers were informed that their true incremental cost was 625 for all units except for the first unit where the incremental cost was 725. Obviously, the pareto efficient (and also the monopsonistic) solution under these conditions is to trade 8 units, but the monopolistic solution is to trade only 4 units. In the experiments where information were symmetrically distributed, the sellers were also given the cost information and the buyers were given the revenue schedule, but in the experiments based on asymmetric information, the cost and revenue information was hidden from the counterpart. All subjects were rewarded based on their individual profits in the negotiations. The experiment was run in two sessions. In the first session, which was based on direct negotiation under symmetric information (DN-s) and bilaterally optional two-part tariffs under symmetric information (BOT-s), 44 subjects participated. They were divided into 22 pairs and the session was run for six periods. The second session was based on direct negotiation under asymmetric information (DN-a) and bilaterally optional two-part tariffs under asymmetric information (BOT-a). Thirty-two subjects participated there, and the session lasted 7 periods. All negotiations were carried out without any interference whatsoever of any external actor, e.g. central management. In total, 244 negotiations took place. The quantities of each negotiation are indicated in Table 1. Of course, the actual efficiency of an agreement is equal to the proportion of the maximum attainable efficiency that the agreement exploits. In Table 2, the actual efficiencies of the different quantities are indicated.
4
6 7 7 7 6 7 6 6 9 7 7
market power market power S: 2nd
3
6 6 6 6 6 7 6 6 5 10 8
Does exercise Does not exercise
Does exercise
2
DN-s
Table 2 Efficiencies for the different quantity alternatives Quantity
Total cost
Total revenue
Aggregate profit
Efficiency
0 1 2 3 4 5 6 7 8 9 10
0 725 1350 1975 2600 3225 3850 4475 5100 5725 6350
0 1000 1950 2850 3700 4500 5250 5950 6600 7200 7750
0 275 600 875 1100 1275 1400 1475 1500 1475 1400
0.0000 0.1833 0.4000 0.5833 0.7333 0.8500 0.9333 0.9833 1.0000 0.9833 0.9333
Price data are not reproduced here, since only the quantities are of interest in an efficiency perspective. The pairs negotiating under DN were not given any specific input to their negotiations beside the above cost and revenue functions. The pairs negotiating under BOT were in addition told that, unless they agreed on something else in some period, the fixed and variable price elements would remain the same as in the previous period. Since the BOT model relies on the idea that the actors in all periods will be subject to a known threat tariff unless they agree about some other tariff, the threat tariff in the very first period of negotiation has to be specified in some other way than ‘‘the tariff of the previous period”. So
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Table 3 Descriptive statistics Mechanism
n
Mean
Standard deviation
Median
DN-s DN-a BOT-s BOT-a
66 56 66 56
0.960 0.842 0.984 0.945
0.034 0.260 0.033 0.081
0.983 0.933 1.000 0.983
Average efficiency
0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 1
2
3
Table 4 Pairwise comparisons Alternative hypothesis
p value for t-test
DN-s > DN BOT-s > BOT DN-s < BOT DN-a < BOT DN-a < BOT
a a s s a
5
6
Fig. 1a. Average efficiency for DN-s.
<0.01 <0.01 <0.01 <0.01 <0.01
Average efficiency
1.2
Table 5 Average efficiency in the different periods
1 0.8 0.6 0.4 0.2
Period
DN-s DN-a BOT-s BOT-a
4
Period
0
1
2
3
4
5
6
0.936 0.583 0.965 0.900
0.965 0.721 0.986 0.904
0.965 0.813 0.977 0.940
0.964 0.902 0.983 0.929
0.964 0.960 0.991 0.981
0.965 0.958 1.000 0.990
1
7
2
3
4
5
6
Period
0.954
Fig. 1b. Average efficiency for DN-a.
0.973
Average efficiency
1.01
in the first period of negotiation, they were told to assume that the two-part tariff in ‘‘the previous period” was characterized by a fixed part equal to 200 and a variable part equal to 700.5 The negotiating pairs got an audio warning when there was only 1 min left, unless they already had reached an agreement. Since all negotiations took place face-to-face, the actors were allowed to use all kind of strategic manipulation and other tactics possible, except threats. All agreements were written down on paper and submitted to the experimenter between each trading period.
1 0.99 0.98 0.97 0.96 0.95 0.94 1
2
3
4
5
6
Period Fig. 1c. Average efficiency for BOT-s.
4. Experimental results
5
These numbers were not entirely arbitrarily chosen. Since marginal cost was a constant, the variable part of this initial tariff should be slightly above marginal cost for the situation to be realistic to the actors. And the level of the fixed part was chosen in order to make the profit distribution between the actors roughly equal, given the level of the variable part. It might of course be interesting to test the model using other initial values. 6 Since the normality assumption of the t-tests and other tests used here may be violated, corresponding non-parametric tests (e.g. Mann-Whitney’s test) have been done. The results were qualitatively the same as for the parametric tests in all cases.
Average efficiency
1
Table 3 describes the data from the experiment on an aggregate level. Initially, all four mechanisms were found to produce an efficiency below 1000 (one-tailed t-tests, p-value <0.01). Of higher interest was of course whether BOT generally performed better than DN. Pairwise t-tests between the four mechanisms revealed a number of significant differences indicated by Table 4.6 DN-s performed significantly better that DN-a, like BOT-s outperformed BOT-a, which was expected. Ceteris paribus, symmetric information should be of some value in the negotiations since it reduces the negotiating power of both actors. Of higher interest is that BOT did significantly better than DN both under symmetric and asymmetric information. Thus, the presence of BOT in the negotiation situation increased the efficiency.
0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 1
2
3
4
5
6
Period Fig. 1d. Average efficiency for BOT-a.
Table 6 Regression results Mechanism
Coefficient of determination
p value for significant slope
DN-s DN-a BOT-s BOT-a
0.04 0.23 0.08 0.14
>0.1 <0.01 <0.05 <0.01
The next step was to analyze the trial effect. Did the subjects learn something from accumulating experience that increased the efficiency in the negotiations? In Table 5, the mean values from
B. Lantz / European Journal of Operational Research 196 (2009) 434–439 Table 7 ANOVA on DN-s and BOT-s over all periods Source of variation
SS
df
MS
F
p-value
F crit
ANOVA Sample Columns Interaction Within
0.018992 0.013369 0.001753 0.129849
1 5 5 120
0.018992 0.002674 0.000351 0.001082
17.55154 2.471066 0.323999
5.37E05 0.036114 0.897702
3.920124 2.289851 2.289851
Total
0.163963
131
Table 8 ANOVA on DN-a and BOT-a over all periods Source of variation
SS
df
ANOVA Sample Columns Interaction Within
0.300357 0.764297 0.309782 3.006319
1 6 6 98
Total
4.380755
111
MS
F
p-value
F crit
0.300357 0.127383 0.05163 0.030677
9.791042 4.152425 1.683044
0.00231 0.000926 0.133134
3.938111 2.192518 2.192518
the different periods for the different mechanisms are reproduced. There seem to be an increasing tendency in the data. A perhaps more illustrative way to indicate this phenomenon is to plot the mean values for each mechanism against the OLS regression line (see Figs. 1a–1d). It is obvious that there may be a ‘‘trial effect” for at least some of the mechanisms. In Table 6, the coefficient of determinations for the regression lines are displayed, along with the p-values for t-tests of whether the true slope is significantly different from 0. For all mechanisms except DN-s, there was a significant ‘‘trial effect”. On the other hand, the correlation can obviously not be linear for further trials, so these figures and tests should only be seen as descriptive. Instead, the two factors ‘‘mechanism” and ‘‘trials” were tested through two-way analyses of variance. The ANOVA table from the comparison of DN-s and BOT-s over all periods is reproduced in Table 7. Both the mechanism effect (p-value <0.01) and the trial effect (p-value <0.05) are significant. The ANOVA table from the comparison of DN-a and BOT-a over all periods is reproduced in Table 8. As in the case of symmetric information, both the mechanism effect (p-value <0.01) and the trial effect (p-value <0.01) is significant. Thus, BOT is significantly more efficient than DN under symmetric as well as under asymmetric information, and the differences between periods are significant. On the other hand, there is no significant interaction between the factors. 5. Concluding remarks In this paper, we have seen that the idea of using optional twopart tariffs as a basis for tariff renegotiations in a bilaterally monopoly setting does have the nice properties claimed, but not formally proven, by Lantz (2000) and Cheng (2002). Thus, this idea is a solution to the problem of double marginalization that (1) creates a stable equilibrium, (2) at the overall efficient level, (3) without the presence of a central management. Even though several theoretical models have been suggested as solutions to the double marginalization problem (Rey and Vergé, 2005), there does not seem to exist some other model with the same nice properties in the literature at this point. One should note that the incentive compatibility of the BOT model only covers the bargaining for pt. The process of determining ft once pt is set is generally a zero-sum game, and may thus consti-
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tute a problem. Under symmetric information, this could be solved by an ex ante agreement to split any increase in overall efficiency due to a change in pt according to some specific rule (e.g. equally).7 In fact, this model might work better under asymmetric information. A tough negotiator could, under symmetric information, be able to keep close to the entire increase of total surplus due to a tariff change. But under asymmetric information, the weaker part has an information advantage that he may use to prevent the opponent from exploiting him when ft is to be set. No matter how the information is distributed, both actors have, ceteris paribus, an interest in finding a tariff (pt,ft) characterized by pt = p*. Both actors know that it is useless to suggest a new tariff that the opponent does not gain on, so it is easy to see that the model should result in the overall efficient solution under asymmetric information too. This is also the main results of the experiments performed in this study. Under BOT, two actors in a bilateral monopoly setting perform better in an efficiency perspective than in an unregulated direct negotiation. It does not matter whether the actors have private information or not. It is interesting to note that Dejong et al. (1989) found similar results in another experimental study of theoretically optimal transfer pricing mechanisms – the mechanisms significantly outperformed direct negotiation. The theoretically easiest way to solve the problem of double marginalization between two actors is of course to integrate the actors vertically. However, in practice, it is for other reasons often necessary in large organizations to disintegrate, divisionalize and decentralize. This means that control mechanisms can be important management tools in transfer pricing settings, channel coordination situations and other contexts where two actors with some market power have to agree about price and quantity. References Avila, M., Ronen, J., 1999. Transfer-pricing mechanisms: An experimental investigation. International Journal of Industrial Organization 17, 689–715. Cheng, J.M.A., 2002. Breakthrough in transfer pricing. Management Accounting Quarterly 3 (2)
7 One might argue that double marginalization is not a problem on the aggregate level under symmetric information and two-part tariffs, since a monopolistic and fully informed seller will choose a tariff characterized by pt = p* anyway in order to maximize profit through ft, see, e.g. Oi (1971).