Capturing direct and cross price
effects in a differentiated products
duopoly model
Michael Kopel§ , Luca Lambertini# and Anna Ressi§
§ University of Graz
Department of Organization and Economics of Institutions
Universitätsstraße 15/E4, 8010 Graz, Austria
[email protected]
[email protected]
# University of Bologna, Department of Economics
Strada Maggiore 45, 40125 Bologna, Italy
[email protected]
February 11, 2015
Abstract
We show that a frequently used direct demand system with product differentiation in a duopoly market generates unexpected effects
of increasing substitutability of firms’ products on prices, outputs,
and profits. Using the original demand system introduced by Bowley
(1924) as a reference, we argue that this alternative model does not
capture a consumer’s taste for variety. Moreover, we demonstrate that
positive values for the parameter which represents cross-price effects
in the alternative demand system corresponds to the regime of complementary products in the original Bowley model. As a consequence,
for increasing values of this parameter – meant to capture increasing
competition – prices do not converge towards marginal costs and profits do not vanish. Finally, we study two well-known models from the
economics and accounting literature and compare the outcomes and
payoffs for the two demand systems.
1
Keywords: Product substitutability; Demand systems; Price and
quantity competition
JEL Codes: L13
1
Introduction
In this paper, we point out that a commonly used direct demand system
with product differentiation generates unexpected consequences in terms of
comparative statics. In contrast to conventional wisdom, a decrease in the
degree of product differentiation, i.e. an increase in the substitutability of
firms’ products, should be expected to boost prices, outputs, and profits as
well. We demonstrate that this demand system is not equivalent to the wellknown demand system dating back to Bowley (1924) and revived by Spence
(1976), Dixit (1979) and Singh and Vives (1984). In particular, we show that
only the utility function which leads to the inverse demands in the Bowley
formulation captures a consumer’s taste for product variety. Furthermore, we
point out that the degree of differentiation in the Bowley case can be seen as a
measure for the level of competition in the usual sense with prices converging
towards marginal costs and profits vanishing as competition gets stronger. In
contrast, the parameter which represents cross-price effects in the alternative
demand system does not yield the same conclusions for positive values of
this parameter. In fact, we demonstrate that this case corresponds to the
regime of complementary products in the original Bowley model. In order
to demonstrate the practical implications, we discuss two well-known models
from the economics and accounting literature which employ the alternative
specification of demands and compare the equilibrium outcomes with the
results for the Bowley demand.
2
The model
Consider a price-setting differentiated duopoly where symmetric firms share
the same marginal and average production cost c. Direct demands are given
by
q1 = a − p1 + bp2
q2 = a − p2 + bp1
2
(1)
where p1 and p2 denote the prices of the firms and q1 and q2 are the quantities.
We have a > 0 and b < 1 since the effect of own prices on demand should be
stronger than the cross-price effect.
This model is frequently used in the literature to capture price competition in a differentiated duopoly market and regularly it is assumed that
b ∈ [0, 1). For example, Gibbons (1992) provides a textbook treatment of differentiated Bertrand duopoly. Sklivas (1987), Aggarwal and Samwick (1999),
Miller and Pazgal (2001, 2005), Jansen et al. (2007, 2009), and Manasakis et
al. (2010) demonstrate the impact of using (strategic) incentive contracts in
oligopoly markets. Hoernig (2012) and Chirco and Scrimitore (2013) study
the impact of network effects in a duopoly. Amir and Wooders (2000) and
Barcena-Ruiz and Olaizola (2006) investigate firms’ R&D investments under
imperfect competition. Wang and Wang (2010) study vertically related markets. Choi (1991, 1996) gives insights on optimal supply chain configurations.
In accounting, Dürr and Göx (2011) study the strategic use of a single transfer price for managerial and tax purposes (referred to as a one-set-of-books
policy); see also Narayanan and Smith (2000). Göx and Schöndube (2004)
demonstrate that in a setting with risk-averse agents, transfer prices can be
employed as a commitment device even if they are not observable since they
are also used to provide incentives to the agent. All in all, it seems justified
to say that the demand system (1) is standard in many subdisciplines of
economic research.
The parameter b is commonly considered as a measures for the degree
of substitutability between the firms’ products. A lower value of b indicates
lower substitutability and the firms are closer to being local monopolists. A
higher value of b indicates greater substitutability, ”... that is, an industry that is more competitive in the sense that price is closer to marginal
cost.” (Aggarwal and Samwick 1999, p. 2007). This interpretation is also
adopted, for example, by Gibbons (1992), Miller and Pazgal (2005), Jansen
et al. (2007), Dürr and Göx (2011), and Göx and Schöndube (2004). At
first glance, this seems reasonable, as the demand for firm i´s product for
b = 0 only depends on the price of firm i and for increasing values of b
the dependence of firm i´s demand on the price of the competitor increases.
As we will demonstrate below, however, viewing the parameter b as a measure of increasing competition results in unexpected conclusions about the
dependence of equilibrium outcomes and payoffs on b.
This is straight-forward to see. Given the direct demands in (1) and
3
assuming that firms set prices to maximize profits, the (symmetric) Nash
equilibrium prices, p∗1 = p∗2 = p∗B , are given by
p∗B =
a+c
.
2−b
(2)
The corresponding quantities and profits in equilibrium are
a − (1 − b) c
,
2−b
(a − (1 − b) c)2
= (qB∗ )2 .
π1∗ = π2∗ = πB∗ =
(2 − b)2
q1∗ = q2∗ = qB∗ =
(3)
Several observations can be made here. First, note that for b ∈ [0, 1) the
Bertrand-Nash price p∗B is larger than the monopoly price (which is obtained
by setting b = 0). Likewise, the profit πB∗ is larger than the corresponding
monopoly profit. Even more revealing are the following results on comparative statics.
Consider
∂p∗B
a+c
=
>0
∂b
(2 − b)2
∂qB∗
a+c
=
> 0,
∂b
(2 − b)2
2(a + c)(a − (1 − b)c)
∂πB∗
=
> 0.
∂b
(2 − b)3
(4)
(5)
Consequently, interpreting b as the degree of competition would result in
the conclusion that equilibrium prices, outputs and profits are increasing for
increasing levels of competition between firms.
In the case of quantity competition, inverting (1) yields
a
1
b
−
q1 −
q2
2
1−b 1−b
1 − b2
1
b
a
−
q
−
q1 .
p2 =
2
1 − b 1 − b2
1 − b2
p1 =
(6)
It is easy to see that the quadratic utility function which gives rise to the
4
linear inverse demand system (6) is given by
V (q1 , q2 ) =
1
a
(q1 + q2 ) −
(q12 + 2bq1 q2 + q22 ).
2
1−b
2(1 − b )
(7)
In order to ensure that the utility function V (q1 , q2 ) is strictly concave, we
have to assume that b2 < 1 or equivalently b ∈ (−1, 1). This condition again
confirms that the own-price effect should dominate the cross-price effect.
Furthermore, total demand should not increase with an increase in prices.
Assuming that firms (non-cooperatively) choose their quantity to maximize
the firm’s profit, it is now straightforward to derive the (symmetric) CournotNash quantities, q1∗ = q2∗ = qC∗ , with
qC∗ =
[a − (1 − b) c] (1 + b)
.
2+b
(8)
The corresponding equilibrium prices and profits are
a + (1 − b2 )c
(2 + b)(1 − b)
[a − (1 − b) c]2 (1 + b)
∗
∗
∗
π1 = π2 = πC =
.
(2 + b)2 (1 − b)
p∗1 = p∗2 = p∗C =
(9)
Looking at the relationship between b ∈ (0, 1) and the equilibrium outcomes
above, we obtain
∂p∗C
∂πC∗
∂qC∗
> 0,
> 0,
> 0.
(10)
∂b
∂b
∂b
Therefore, with regard to prices and profits the same unexpected conclusions
for increasing values of the “competition parameter” b are derived.
In order to understand why the widely used model (1) yields comparative
statics which are at odds with intuition, we compare it to the original formulation introduced by Bowley (1924) and then revived by Spence (1976), Dixit
(1979) and Singh and Vives (1984), inter alia.1 A simplified2 and frequently
1
We employ the, in our view, more frequently used model introduced by Bowley
(1924) rather than the alternative (but qualitatively equivalent) formulation of Shubik
and Levithan (1980). For a recent review of various demand functions, see Huang et al.
(2013).
2
A more general form is U (q1 , q2 ) = α1 q1 +α2 q2 − 12 (β1 q12 +2γq1 q2 +β2 q22 ). We consider a
representative consumer and set β1 = β2 = 1 and α1 = α2 = α since we also study welfare
properties. This does not have any qualitative effect on the results of our paper.
5
used form of the utility function of the original Bowley formulation is
1
U (q1 , q2 ) = α(q1 + q2 ) − (q12 + 2γq1 q2 + q22 ),
2
(11)
which gives rise to the inverse demand system
p1 = α − q1 − γq2 ,
p2 = α − q2 − γq1 .
(12)
This leads to the direct demands
α
1
γ
−
p1 +
p2
2
1+γ 1−γ
1 − γ2
1
γ
α
−
p2 +
p1 .
q2 =
2
1+γ 1−γ
1 − γ2
q1 =
(13)
To guarantee (strict) concavity of the utility function U (q1 , q2 ), we assume
γ 2 < 1 or equivalently γ ∈ (−1, 1). The parameter γ 2 measures the degree of
differentiation between the firms’ products (e.g. Vives, 1999; Shy, 1995). For
γ = 0, products are independent. Products become increasingly homogenous
for γ 2 → 1. In other words, as γ 2 is increasing, competition gets stronger.
Note that products are substitutes (complements) for γ > 0 (γ < 0). For
γ 2 = 1 products are perfect substitutes (perfect complements).
At first sight, the (properties of the) utility functions U and V seem
identical. However, one crucial difference which can be noticed is that the
parameter γ only affects the cross term which measures the utility of joint
consumption of products 1 and 2. In particular, given that quantities are
positive this leads to ∂U/∂γ = −q1 q2 < 0. In other words, since an increase
in γ (in absolute terms) reduces product differentiation, the representative
consumer’s utility is strictly increasing in the degree of product differentiation. This property of the utility U can be seen as capturing a consumer’s
taste for variety. In contrast, a change in the parameter b affects all coefficients of the utility function V . Hence, it changes the utility from consuming
each of the products individually as well. In particular, in equilibrium we
have ∂V /∂b > 0 for price and quantity competition. This would lead to the
conclusion that for increasing degrees of competition between the products,
6
consumer utility increases, quite in contrast to the original Bowley formulation.
In the original Bowley model, the parameter γ in the inverse demand
system (12) only affects the influence of firm j’s quantity on firm i’s price,
whereas in the alternative model (6) a change in the parameter b affects not
only the influence of firm j’s quantity on firm i’s price, but also the choke price
(given by a/(1 − b)) and the slope of the inverse demand with regard to firm
i’s own quantity (i.e., 1/(1 − b2 )). For example, if the parameter b ∈ [0, 1)
increases, then the maximum willingness to pay and the sensitivity with
respect to a firm’s own quantity variations increase simultaneously. Likewise,
the dependence of a firm’s demand on the price of the competitor increases
as well. As a result of these three effects, the equilibrium price and the
equilibrium quantity increase in b, which results in increasing profits.
Table 1 provides more details on the impact of the parameter b on (equilibrium) prices, profits, consumer surplus (CS) and welfare (W) for price
and quantity competition. Consumer surplus for the alternative model (the
Bowley model) is calculated as
1
(q12 + 2bq1 q2 + q22 ),
2
2(1 − b )
1 2
(CS = (q1 + 2γq1 q2 + q22 )).
2
CS =
Welfare is calculated as the sum of the firms’ profits plus consumer surplus.
For the alternative model, all partial derivatives with respect to b ∈ [0, 1)
are positive independent of the strategic variable (price or quantity). This is
in stark contrast to the “competition parameter” γ ∈ [0, 1), where the sign
of the partial derivatives is (except in one case) opposite with regard to the
alternative model.
To gain deeper insights into the relation between the two demand models (1) and (13) for price competition in a differentiated duopoly, we take a
closer look at the correspondence between the parameters b and γ and industry output and prices in equilibrium (see Figure 1).3 We consider three
situations.
3
We owe this illustration to an anonymous reviewer who suggested this helpful figure
in order to clarify the relation between the two models.
7
Bowley Model
γ ∈ [0, 1)
Price
Competition
∂p∗B
∂γ
<0
∂p∗B
∂b
>0
∂Π∗B
∂γ
<0
∂Π∗B
∂b
>0
∗
∂CSB
∂γ
∗
∂WB
∂γ
∗
∂qC
∂γ
∂Π∗C
∂γ
Quantity
Competition
Alternative Model
b ∈ [0, 1)
∗
∂CSC
∂γ
∗
∂WC
∂γ
>0
<0
<0
<0
<0
<0
∗
∂CSB
∂b
∗
∂WB
∂b
∗
∂qC
∂b
∂Π∗C
∂b
∗
∂CSC
∂b
∗
∂WC
∂b
>0
>0
>0
>0
>0
>0
Table 1: Comparative statics of prices, profits, consumer surplus, and welfare in equilibrium (superscript *). Subscript B indicates (Bertrand) price
competition, C indicates (Cournot) quantity competition.
8
1. We start with the case where competition is the lowest. For independent products, i.e. γ = 0 and b = 0, both models yield the monopoly
prices and quantities, i.e. pM = (α + c)/2 and q M = (α − c)/2 for the
Bowley model and pM = (a + c)/2 and q M = (a − c)/2 for the alternative model. This situation is indicated by the point in the middle of
Figure 1.
2. Consider next the situation where products are substitutes, γ > 0, and
best responses are positively sloped (light-grey region). For γ → 1,
i.e. increasing degrees of substitution between the firms’ products,
competition is increasing and – as expected – equilibrium prices decrease towards marginal cost, p∗B = ((1 − γ)α + c)/(2 − γ) → c. The
industry output converges towards Q∗ = q1∗ + q2∗ = α and profits converge towards zero. In contrast, for the alternative model the situation
where both firms set their prices equal to marginal costs is reached for
b = −(a − c)/c < 0, in which case the industry output and profits
are zero. In Figure 1 this situation is represented by the point in the
lower-left corner of the figure. In the light-grey region, b < 0, and best
responses are negatively sloped. An increase of b in absolute terms
decreases prices and quantities like in the Bowley model.
3. Now consider the situation where products are complements, γ < 0,
(dark-grey region). It is well-know that price competition with complements is dual to quantity competition with substitutes (Singh and
Vives 1984). That is, the two models share similar strategic properties.
For example, reaction functions slope downwards and computations
done for one mode of competition can be used in the other mode by
exchanging prices and quantities. In this region, as γ 2 increases, products become increasingly homogenous until for γ 2 = 1 products are
perfect complements. In this sense, as γ → −1, competition is increasing. Note that, as γ → −1 prices (and quantities) will increase. In the
dark-grey region, b > 0 and best responses are positively sloped. An
increase of b increases prices, quantities, and profits like in the Bowley
model. For b → 1, prices converge towards a + c and industry output
towards 2a. In Figure 1 this situation is represented by the point in
the upper-right corner.
The surprising conclusion derived from the arguments above is that the
9
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ܾ ൌ െሺܽ െ ܿሻȀܿ
ܿ
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‫ݍ‬ଵ ൅ ‫ݍ‬ଶ ൌ Ͳ ‫݁ݒ݅ݐܽ݊ݎ݁ݐ݈ܣ‬
‫ݍ‬ଵ ൅ ‫ݍ‬ଶ ൌ ߙሺ‫ݕ݈݁ݓ݋ܤ‬ሻ
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Figure 1: Comparison of prices and industry outputs in equilibrium for representative values of b and γ.
alternative model for substitute products, b ∈ [0, 1), leads to similar qualitative conclusions as the original Bowley model for complementary products
where γ ∈ (−1, 0]. Consequently, modeling price competition using the demand system (1) and interpreting b as the degree of substitutability between
the firms’ products does not lead to the usual predictions that prices decrease
towards marginal costs and profits vanish for increasing levels of competition
between firms.
In order to avoid confusion, we believe that is preferable to start from the
optimization problem of a representative consumer using (e.g.) the general
utility function4
1
U (q1 , q2 ) = α1 q1 + α2 q2 − (β1 q12 + 2γq1 q2 + β2 q22 )
2
4
For more details and generalizations to n-firm oligopoly, see e.g. Vives (1999), Häckner
(2000), Hsu and Wang (2005).
10
and derive the indirect demand system pi = αi − βi qi − γqj , i, j = 1, 2, i 6= j
to analyze quantity competition in a differentiated duopoly. For studying
price competition, the direct demand system has to be derived from the
inverse demands by inverting, qi = (αi βj − αj γ)/(βi βj − γ 2 ) − βj pi /(βi βj −
γ 2 ) + γpj /(βi βj − γ 2 ). For simplicity, the (commonly used) normalization
β1 = β2 = 1 with α1 = α2 = α can be introduced. Obviously, the structure
of this demand system can then indeed be written as
qi = a − εpi + bpj
but with
a=
α
1
γ
;ε =
;b =
.
2
1+γ
1−γ
1 − γ2
(14)
(15)
As our analysis has demonstrated, the particular dependence of the parameters a, ε, and b on the competition parameter γ is important for obtaining
sensible results on comparative statics of the outcomes.
3
Examples
In order to demonstrate the importance of this issue, we look at two wellknown models taken from the literature.
3.1
Strategic incentive contracts based on relative performance
The first paper we consider is Aggarwal and Samwick (1999; henceforth AS),
who study the strategic effects of compensation contracts for managers based
on relative performance in a differentiated duopoly. In particular, they analyze how the degree of competition affects the relative weight put on own-firm
profits when managers are compensated according to a linear combination of
own and rival firm’s profit. We focus on the case where firms are price setters.
The AS model consists of two firms which sell differentiated products in a
market characterized by the direct demands (1).5 Recall that a higher value
5
Aggarwal and Samwick use the more general form (14), but neglect the dependence
on the parameter γ. For the sake of consistency, we consider the simplified form (1). This
does not affect the conclusions we derive from our analysis.
11
of b > 0 indicates a higher degree of substitutability and supposedly captures a higher degree of competition. Owners delegate the pricing decision
to managers who are compensated according to
wi = ki + λi Πi + µi Πj .
(16)
Here, ki represents a fixed salary (which is only used to fulfill the manager’s
participation constraint), while λi and µi are the weights put on own firm’s
profit (Πi ) and the rival firm’s profit (Πj ) respectively. The game sequence
is as follows. First, owners decide on the incentive parameters λi and µi that
maximize their expected profit
Πi = (pi − c)(a − pi + bpj ) + ξ,
(17)
where ξ is a normally distributed error term. Then, managers set prices pi
to maximize total compensation wi . Owners and managers are assumed to
be risk neutral, i.e. the focus is on the strategic use of contracts.6 The game
is easily solved by backwards induction. This yields the solutions depicted
in the first column of Table 2. For comparison, the second column shows the
equilibrium outcomes when using the original Bowley demand function (13).
The equilibrium values of the incentive parameters λi coincide if the differentiation parameters b and γ are exchanged (see row 2 of Table 2). There
is a continuum of optimal contracts for both models. The crucial point of AS
is that as competition, i.e. b > 0, increases (i.e. the products become more
substitutable), more weight is put on rival firm’s profit relative to own-firm
profits. That is, λi /µi = (2−b)/b is decreasing for increasing b. Interestingly,
this holds for both models. However, all the other equilibrium outcomes and
comparative statics with respect to the differentiation parameters differ. In
particular, for the Bowley model comparative statics are as expected. For
example, as γ increases, profits are decreasing and the price comes closer to
marginal costs c. In contrast, in the AS model equilibrium profits and prices
are increasing with b. Consequently, a higher value of b does not represent
“... an industry that is more competitive in the sense that price is closer to
marginal cost.” (Aggarwal and Samwick 1999, p. 2007).
6
Interestingly, Aggarwal and Samwick (1999) show that observability of the contract is
not needed in order to obtain strategic effects (see Appendix B in their paper for a model
of credibly signaled contracts).
12
Π∗i
λ∗i
Alternative Model
Bowley Model
(4−b2 )(c(1−b)−a)2
(4−γ 2 )(α−c)2
16(1−b)
16(1+γ)
µi
2−b
b
µi
2−γ
γ
p∗i
2(a+c)−b2 c−b(a+c)
4(1−b)
2(α+c)−γ(α−c)
4
qi∗
(2+b)(a−c(1−b))
4
(2+γ)(α−c)
4(1+γ)
Table 2: Comparison of equilibrium outcomes for the model of Aggarwal and
Samwick (1999).
3.2
Keeping one set of books in international transfer
pricing
The second paper we consider is Dürr and Göx (2011; henceforth DG), who
study the impact of different accounting policies in a multinational duopoly
with price competition. In particular, DG analyze the difference between
keeping one set of books (use the same transfer price for internal and for tax
accounting) versus two sets of books (use two potentially different transfer
prices). The benefit of keeping two sets of books is that a firm is better
able to disentangle managerial objectives (coordinating multiple independent
divisions) and tax objectives (shifting profits to low-tax jurisdictions). The
cost of using two sets of books is that since the internal transfer price is
hardly observable it cannot be used to strategically influence the behavior
of the competitor in order to dampen the intensity of price competition in
the final product market. In contrast, with one set of books there may
be a conflict between tax and managerial objectives, but the transfer price
used for tax purposes is publicly observable and can therefore be used as a
strategic commitment device. The major insight of DG is that the optimal
accounting policy crucially depends on the degree of substitutability between
the firms’ products. In particular, keeping one set of books is the dominant
strategy if the intensity of competition is high since the strategic benefit of
price coordination is important.
The model is set up as follows. There are two multinational companies M1
13
and M2 . Both companies consist of headquarters HQi and one selling division
Si , both located in the domestic country, and a buying division Bi , located in
a foreign country. The selling division Si produces an intermediate product
and sells it to the buying division Bi . The buying division then produces the
final product and sells it on the market in the foreign country. Firms compete
in prices on the final product market which is characterized by the inverse
demand function (1), where a > 0 measures the market size and b ∈ [0, 1)
denotes the degree of substitutability between products. According to DG
(p. 290), “...b can can be taken as a measure for the intensity of competition.
The higher b, the higher is the reaction of a price change of firm j on the
demand of product i.” Furthermore, they state that “...if b approaches zero,
firm i becomes a monopolist in its market, and as b goes to one, the products
become perfect substitutes.”
The two divisions Bi and Si are taxed at flat tax rates. In the domestic
country there is a tax rate of τ ∈ [0, 1]. The foreign tax rate is τ +∆ ∈ [0, 1].
Hence, ∆ ∈ [−τ, 1−τ ] measures the tax difference between the two countries,
so that ∆ > 0 indicates that the home country has a tax advantage (and
vice versa). The internal transfer price is denoted by Ti and the (publicly
observable) tax transfer price by ti . Denoting the marginal production costs
of the selling divisions by c and assuming zero costs of finalizing the product,
the payoffs of the parties under two sets of books (superscript t) are
ΠSt
i = (Ti − c)qi − τ (ti − c)qi ,
ΠBt
i = (pi − Ti )qi − (τ + ∆)(pi − ti )qi ,
for the selling and the buying division i respectively. Under one set of books
(superscript o), we have
ΠSo
i = (1 − τ )(ti − c)qi ,
ΠBo
i = (1 − τ − ∆)(pi − ti )qi .
In both cases, a firm’s total profit is
Bt
So
Bo
Πi = ΠSt
i + Πi = Π i + Πi =
= (1 − τ )(pi − c)qi − ∆(pi − ti )qi .
Note that the internal transfer price Ti does not influence a firm’s total profit
14
(Ti is just used to coordinate the two divisions). Also recall that the internal
transfer price Ti is not observable by the competitor, so it cannot be used
strategically to soften competition in the final product market. Consequently,
headquarters choose Ti to align the objective of the selling division (maximize ΠBt
i ) with the firm’s objective (maximize Πi ). Equating the payoffs,
Bt
Πi = Πi , shows that perfect goal congruence is obtained with an internal
transfer price of Ti∗ = (1 − τ )c + τ ti . As a consequence, headquarters can be
sure that Bi sets the market price which maximizes total profits Πi and the
selling division just charges a transfer price which corresponds to its “virtual”
production costs.
We follow DG and assume, for simplicity, that marginal production costs
are zero, c = 0. We also assume that the tax transfer price must not be
lower than marginal production costs and must not be higher than the final
product price. This leads to the condition ti ∈ [t, t], where t ≥ c = 0 and
t ≤ pi . The timeline of the model is as follows. In stage 1, headquarters Hi
choose between the two accounting policies of keeping two sets of books (t)
or one set of books (o). Then, Hi determine the transfer prices to maximize
total profit. In the last stage, the buying divisions Bi set the final product
prices to maximize after tax division profit. In our analysis we will focus on
the two symmetric cases where both firms either choose to keep two sets of
books or choose to keep one set of books.
In the case where both firms choose to keep two sets of books, division Bi
chooses the price pi such that division profit ΠBt
i is maximized. Simultaneously solving the first-order conditions of both buying divisions and taking
into account that headquarters will select Ti∗ = τ ti to achieve goal congruence
yields (i, j = 1, 2, i 6= j)
pti =
a(2 + b)(1 − ∆ − τ ) + ∆(2ti + btj )
.
(4 − b2 ) (1 − ∆ − τ )
(18)
Denoting the resulting market quantities as qit , total profit at this stage
can simply be written as Πti = (1 − ∆ − τ )(qit )2 . Differentiating with respect
to the tax transfer price ti and assuming symmetry (ti = tj = t∗t ), we obtain
2(2 − b2 )
∂Πti
=∆
qi (t∗t , t∗t ).
∂ti
4 − b2
Hence, whenever the domestic country has a tax advantage (∆ > 0), the
tax transfer price is set as high as possible in order to shift profits from the
15
foreign to the domestic country. On the other hand, if the foreign country
has a tax advantage (∆ < 0), the tax transfer price is set as low as possible.
Consequently, the optimal tax transfer price is simply determined by
(
t if ∆ > 0,
(19)
t∗t =
t if ∆ < 0.
The symmetric equilibrium outcomes and profit under a two sets of books
policy are summarized in the first column in table 3. For the sake of brevity,
we set µ = 1 − ∆ − τ . The second column shows the equilibrium outcomes
when using the original Bowley demand function (13).
Alternative Model
Bowley Model
Πti
(aµ+(1−b)∆t∗t )2
µ(2−b)2
(1−γ)(αµ+∆t∗t )2
µ(2−γ)2 (1+γ)
pti
aµ−∆t∗t
µ(2−b)
αµ(1−γ)−∆t∗t
µ(2−γ)
qit
aµ+(1−b)∆t∗t
µ(2−b)
αµ+∆t∗t
µ(2−γ)(1+γ)
tti
t∗t
t∗t
Table 3: Comparison of equilibrium outcomes with two sets of books for the
model of Dürr and Göx (2011) with µ := 1 − ∆ − τ .
In the case where both firms choose to keep one set of books, division Bi
chooses the price pi such that division profit ΠBo
is maximized. This gives
i
the price reaction functions
a(2 + b) + btj + 2ti
.
(20)
4 − b2
Substituting the price reaction functions into total profits Πi , differentiating and solving with respect to ti leads to the symmetric transfer price
poi =
t∗ =
a(4∆ + b2 (1 − 2∆ − τ ))
.
4(1 + ∆ − τ ) − 2b(1 + 2∆ − τ ) − b2 (1 + 2(1 − b)∆ − τ )
16
(21)
Consequently, the optimal transfer


t if
∗o
t = t∗ if


t if
price is determined by
t∗ > t,
t∗ ∈ [t, t]
t∗ < t.
The equilibrium values for the case where both firms keep one set of books
are again shown in the first column of table 4. The values for the Bowley
model can be found in the second column.
Alternative Model
Bowley Model
Πoi
(a−(1−b)t∗o )(aµ+t∗o (µ+∆(2−b)))
(2−b)2
(α−t∗o )(α(1−γ)µ+t∗o (µ+∆(2−γ)))
(2−γ)2 (1+γ)
poi
a+t∗o
2−b
α(1−γ)+t∗o
2−γ
qio
a−(1−b)t∗o
2−b
α−t∗o
(2−γ)(1+γ)
toi
t∗o
t∗o
Table 4: Comparison of equilibrium outcomes with one set of books for the
model of Dürr and Göx (2011) with µ := 1 − ∆ − τ .
The two tables reveal that the equilibrium values of the two demand
models differ. Comparative statics of prices and profits are in line with our
general conclusions above. That is, for the alternative demand model prices
and profits are monotonically increasing whereas for the Bowley demand
specification prices and profits are monotonically decreasing towards zero
(recall that marginal costs are assumed to be zero). In order to check if “...the
practice of one set of books should be the prevalent accounting method in
markets with a small number of competitors and similar products”, we take a
closer look and compare the outcomes for the two accounting policies. As an
illustration we consider a numerical example studied in DG. Let a = α = 100
and τ = 0.5, and assume that the home country has a tax advantage, ∆ = 0.2.
Furthermore, assume that t = p∗i .7 Figure 2 depicts the (symmetric) profits
7
In Dürr and Göx (2011) the upper limit of the tax transfer price is set exogenously
17
of the two firms if they both keep one set of books (Πo ) and if they both
keep two sets of books (Πt ) for b ∈ [0, 1). Figure 3 depicts the corresponding
profits for the Bowley specification for γ ∈ [0, 1). Our analysis demonstrates
that the main qualitative conclusions drawn from the two models are identical
since the intersection points of the profit curves for Πo and Πt are the same
(γ ∗ = b∗ ≈ 0.391157). In other words, for higher substitutability of products
the policy of one set of books yields higher profits. On the other hand, the
dependence of profits for increasing values of substitutability of the products
is in line with intuition only for the original Bowley model.
Figure 2: Equilibrium profits for the alternative model of Dürr and Göx
(2011) with b ∈ [0, 1).
to t = 80. However, in this case the transfer price (in the case of two sets of books) is
higher than the product price violating an assumption of the model. To avoid such an
inconsistency, we set the upper bound t equal to the market price p∗i . In other words,
firms fully explore the possibilities for shifting profits to the lower-tax regime.
18
Figure 3: Equilibrium profits and intensity of competition for the Bowley
model for γ ∈ [0, 1).
4
Conclusions
This paper has shown that although some of the equilibrium outcomes and
qualitative insights are unaffected by adopting the alternative demand model,
the devil is in the details. Not only do some equilibrium outcomes differ.
Even more confusing, comparative statics lead to completely different conclusions. We believe that this insight is sufficiently important to promote
the use of a correctly specified version of the demand system derived from
microeconomic principles and to discourage the use of a seemingly simpler
alternative version.
As for future research, we are wondering if the same unexpected comparative statics may occur with demand system in multi-product settings. For
example, consider two upstream manufacturers, Mi , i = 1, 2, each producing a differentiated product. Each product can be sold on the final product
market by two retailers, Rh , h = a, b. Hence, consumers can choose among
19
four substitutable products (1a, 1b, 2a, 2b). The inverse demand for product
i sold by retailer h is (see e.g. Dobson and Waterson 1996, 2007)
pih = 1 − (qih + βqik ) − γ(qjh + βqjk ),
where i, j = 1, 2, i 6= j and h, k = a, b, h 6= k. Here γ ∈ (−1, 1) measures
the interproduct rivalry, i.e. how similar the products i, j are perceived to
be when sold by the same retailer h, k. The parameter β ∈ [0, 1) measures
the degree of intraproduct rivalry, i.e. how similar the retailers are perceived
to be when selling the same product. Dobson and Waterson (1996) show
that this demand system can be derived from a representative consumer’s
optimization problem. Inverting the inverse demand system above leads to
the direct demands and enables the study of price competition. Instead, one
could start analyzing price competition directly by employing the demand
system (see e.g. Rey and Vergé 2010, Anderson and Bao 2010, Choi 1996)
qih = 1 − pih + bpik + a(pjh + bpjk ),
where a > 0 measures the degree of interproduct rivalry and b > 0 represents
intraproduct rivalry. To make sure that demand decreases when all prices
increase, assume additionally that a + b + ab < 1. It might be interesting to compare the equilibrium outcomes for the two approaches and study
the impact of increasing inter- and intraproduct substitutability on prices,
quantities and profits.
Acknowledgements: We are grateful to Yossi Spiegel who provided the
idea for this paper and further remarks. We are further grateful to Paolo
Ramezzana, two anonymous reviewers and the editor, Antonio Nicolo, for
very helpful comments and suggestions. Financial support for this research
has been provided by EU COST Action IS1104, “The EU in the new economic
complex geography: models, tools and policy evaluation”.
20
References
[1] Aggarwal, R.K. and A.A. Samwick (1999), “Executive Compensation,
Strategic Competition, and Relative Performance Evaluation: Theory
and Evidence”, Journal of Finance, 54, 1999-2043.
[2] Amir, R. and J. Wooders (2000), “One-Way Spillovers, Endogenous Innovator/Imitator Roles, and Research Joint Ventures”, Games and Economic Behavior, 31, 1-25.
[3] Anderson, E.J. and Y. Bao (2010), “Price Competition With Integrated
and Decentralized Supply Chains”, European Journal of Operational
Research, 200, 227-234.
[4] Barcena-Ruiz, J.C. and N. Olaizola (2006), “Cost-Saving Production
Technologies and Strategic Delegation”, Australian Economic Papers,
45, 141-157.
[5] Bowley, A.L. (1924), “The Mathematical Groundwork of Economics”,
Oxford, Oxford University Press.
[6] Chirco, A. and M. Scrimitore (2013), “Choosing Price or Quantity? The
Role of Delegation and Network Externalities”, Economics Letters, 121,
482-486.
[7] Choi, S. C. (1991), “Price Competition in a Channel Structure With a
Common Retailer”, Marketing Science, 10, 271-296.
[8] Choi, S. C. (1996), “Price Competition in a Duopoly Common Retailer
Channel”, Journal of Retailing, 72, 117-134.
[9] Dixit, A. (1979), “A Model of Duopoly Suggesting the Theory of Entry
Barriers”, Bell Journal of Economics, 10, 20-32.
[10] Dobson, P.W., Waterson, M. (1996), “Product Range and Interfirm
Competition”, Journal of Economics and Management Strategy, 5, 317341.
[11] Dobson, P.W., Waterson, M. (2007), “The Competition Effects of
Industry-wide Vertical Price Fixing in Bilateral Oligopoly”,International
Journal of Industrial Organization, 25, 935-962.
21
[12] Dürr, O.M. and R.F. Göx (2011), “Strategic Incentives for Keeping One
Set of Books in International Transfer Pricing”, Journal of Economics
& Management Strategy, 20, 269-298.
[13] Gibbons, R. (1992), “Game Theory for Applied Economists, Princeton”,
NJ, Princeton University Press.
[14] Göx, R.F. and J.R. Schöndube (2004), “Strategic Transfer Pricing with
Risk-Averse Agents”, Schmalenbach Business Review, 56, 98-118.
[15] Häckner, J. (2000), “A Note on Price and Quantity Competition in
Differentiated Oligopolies”, Journal of Economic Theory, 93, 233-239.
[16] Hoernig, S. (2012), “Strategic Delegation Under Price Competition and
Network Effects”, Economics Letters, 117, 487-489.
[17] Hsu, J. and X.H. Wang (2005), “On Welfare under Cournot and
Bertrand Competition in Differentiated Oligopolies”, Review of Industrial Organization, 27, 185-191.
[18] Huang, J., L. Leng, and M. Parlar (2013), “Demand Functions in Decision Modeling: A Comprehensive Survey and Research Directions”,
Decision Sciences, 44, 557-608.
[19] Jansen, T., A. van Lier and A. van Witteloostuijn (2007), “A Note on
Strategic Delegation: The Market Share Case”, International Journal of
Industrial Organization, 25, 531-39.
[20] Jansen, T., A. van Lier and A. van Witteloostuijn (2009), “On the
Impact of Managerial Bonus Systems on Firm Profit and Market Competition: The Cases of Pure Profit, Sales, Market Share and Relative
Profits Compared”, Managerial and Decision Economics, 30, 141-53.
[21] Manasakis, C., E. Mitrokostas, and E. Petrakis (2010), “Endogenous
Managerial Incentive Contracts in a Differentiated Duopoly, with and
without Commitment”, Managerial and Decision Economics, 31, 53143.
[22] Miller, N.H. and A.I. Pazgal (2001), “The Equivalence of Price and
Quantity Competition with Delegation”, RAND Journal of Economics,
32, 284-301.
22
[23] Miller, N.H. and A.I. Pazgal (2005), “Strategic Trade and Delegated
Competition”, Journal of International Economics, 66, 215-31.
[24] Narayanan, V.G. and M. Smith (2000), “Impact of Competition on
Taxes on Responsibility Center Organization and Transfer Prices”, Contemporary Accounting Research, 17, 497-529.
[25] Rey, P. and T. Vergé (2010), “Resale Price Maintenance and Interlocking
Relationships”, Journal of Industrial Economics, 58, 928-961.
[26] Shubik, M. and R. Levitan (1980), “Market Structure and Behavior”,
Cambridge, MA., Harvard University Press.
[27] Shy, O. (1995), “Industrial Organization: Theory and Applications”,
Cambridge, MA., MIT Press.
[28] Singh, N. and X. Vives (1984), “Price and Quantity Competition in a
Differentiated Duopoly”, RAND Journal of Economics, 15, 546-54.
[29] Sklivas, S.D. (1987), “The Strategic Choice of Managerial Incentives”,
RAND Journal of Economics, 18, 452-58.
[30] Spence, M. (1976), “Product Differentiation and Welfare”, American
Economic Review, 66, 407-14.
[31] Vives, X. (1999), “Oligopoly Pricing - Old ideas and new tools”, MIT
Press, Cambridge, Massachusetts.
[32] Wang, L.F.S. and Y.-C. Wang (2010), “Input Pricing and Market Share
Delegation in a Vertically Related Market: Is the Timing Order Relevant?” International Journal of the Economics of Business, 17, 207-21.
23