Theoretical Economics Letters, 2012, 2, 355-360
http://dx.doi.org/10.4236/tel.[phone] Published Online October 2012 (http://www.SciRP.org/journal/tel)
Strategic Delegation in Price Competition
Werner Güth1, Kerstin Pull2, Manfred Stadler2
1Max Planck Institute of Economics, Strategic Interaction Group, Jena, Germany
2Department of Business and Economics, University of Tübingen, Tübingen, Germany
Received July 6, 2012; revised August 1, 2012; accepted September 3, 2012
ABSTRACT
We study price competition in heterogeneous markets where price decisions are delegated to agents. Principals imple-
ment a revenue sharing scheme to which agents react by commonly charging a sales price. The results of our model
exemplify the importance of both intra- and interfirm interactions of principals and agents in competition. We show that
price delegation can increase or decrease the firms’ surplus depending on the heterogeneity of the market and the num-
ber of agents employed by the firms.
Keywords: Strategic Delegation; Agency Theory; Revenue Sharing
1. Introduction
Whereas principal-agent theory typically restricts itself to
the analysis of intrafirm interaction by neglecting inter-
firm competition, most models in the theory of Industrial
Organization (IO) focus purely on interfirm competition
by assuming a unitary decision maker for each of the
competing firms. While studying only one of these two
interaction types certainly answers many questions, in
some cases it may suggest questionable implications for
real-world behavior facing typically both types of in-
teraction. For instance, a standard principal-agent frame-
work (see e.g. Grossman and Hart [1]) neglects interfirm
competition and, hence, the influence of market con-
ditions on intrafirm compensation schemes. Similarly,
assuming a unitary decision maker in IO models of in-
terfirm competition ignores the principal-agent relation-
ships and thus the decisive reason why firms may abstain
from profit maximization.
Of course, principals may be the only ones determining
both, intrafirm and interfirm interactions. If principals,
for instance, confront their agents with piece rates, all
what agents have to do is match their efforts with the
given piece rates, i.e., agents neither interact strategically
with other agents of the same nor with those working in
other firms. Thus if only principals are involved in
intrafirm and interfirm interaction, the analysis is rather
simple and straightforward. Here we focus, however, on
situations where not only principals are “running the
mill”: in our model principals only determine the in-
centives for their agents who then choose their firm’s
sales policy. Hence, in this scenario, principals as well as
agents are engaged in interfirm competition.
Our analysis is related to the strategic delegation analysis
of Vickers [2], Fershtman [3], Fershtman and Judd [4],
Sklivas [5], Caillaud, Jullien, and Picard [6] and Schmidt
[7] who study intrafirm incentives for managers facing a
market with interfirm quantity or price competition. In
these delegation games, the profit-maximizing principals
(the owners) implement an incentive scheme for their
agents (the managers) based on a weighted difference of
revenue and cost. Fumas [8] and Miller and Pazgal [9]
consider an incentive scheme based on a weighted sum
of a firm’s own profit and its rival’s profit. Kräkel [10]
investigates tournament-like interfirm competition based
on a principal-(one) agent framework. These models
have in common that they restrict analysis to the dele-
gation to manager agents who decide on prices or quan-
tities but are not involved in production, i.e. they face no
cost of producing. In contrast we are interested in the
decisions of worker agents who will anticipate the con-
sequences of their price or quantity decisions on their ef-
fort.
Güth, Pull and Stadler [11] have studied intrafirm and
interfirm interaction between principals and agents in an
integrative model. They analyzed how revenue sharing
affects the behavior and payoffs on a homogeneous
oligopoly market with quantity competition1. On most
markets firms, however, compete in prices. Therefore,
we explore the case where principals implement a re-
venue sharing scheme and agents compete via prices.
The price decisions of all competing firms in the market
determine the quantities to be produced by firms and,
1Quantity competition can be justified by the necessity of firms to set
up capacities before engaging in price competition (see e.g. Kreps and
Scheinkman [12]).
C
opyright © 2012 SciRes. TEL

W. GÜTH ET AL. 356
hence, the effort costs of their agents.
The remainder of this paper is structured as follows:
Section 2 describes a benchmark case with price com-
petition between two monolithic firms. Section 3 introduces
delegation and price competition by workers allowing for
an integrative analysis of strategic intra- and interfirm in-
teraction based on the realistic assumptions of price com-
petition. Section 4 concludes.
2. The Benchmark: Price Competition
between Two Monolithic Firms
We consider two competing firms in a heteroge-
neous market with firm specific sales amounting to
=1,2i
,
=;= 1,2,= 1,...,,
iik
k
qei k
n
,
where ki denotes the effort level of agent em-
ployed in firm . Sales are assumed to serve demand for
differentiated products. To keep the model analytically
tractable, we rely on linear demand functions of the
standardized form
e,k
i
 
,=1 ;=1,2,
iijij i
qpppppii j   
for the two substitute goods with restrictions
where
>0,= 1,2,
i
qi


10,  
indicates the degree
of market heterogeneity. In the limit case =0

there
are two coexisting monopoly markets without interfirm
competition. In the other extreme case where


the market becomes homogeneous since, in the limit, any
price difference will not leave (positive) demand for the
more expensive seller. Costs of the agents’ sales efforts
are private and commonly known. All agents share the
same quadratic effort-cost function

2
,,
=2
ik ik
cee .
n
To provide a benchmark solution without intrafirm in-
teraction let us first assume that both firms maximize
their surplus for example by assuming a unitary decision
maker for each firm who dictates effort levels and mone-
tarily compensates his agents for their effort costs.
Due to the strictly convex cost function, each firm i will
impose the same effort level , for all workers
. Thus the surplus for each firm can be ex-
pressed by
n
=
ik i
ee
=1, ,k






2
,=11
112;=1,2,.
iij iij
ij
Spppp p
pp nii


 
 j
From the first-order conditions the equilibrium prices
are derived as

*1
=,
12
n
pn



 
leading to each firm’s output



*1
=12
n
qn




 
and surplus

 

*
2
112
=.
21 2
nn
Sn








Some numerical results are summarized in Table 1 for
a fixed number of agents per firm and in Table 2
for a fixed intermediate degree of heterogeneity,
=2n
=1

.
A decreasing degree of heterogeneity (an increasing

)
lowers equilibrium prices and the firms’ surplus even if
agents’ individual efforts and sales increase (see Table 1).
For a given degree of heterogeneity (=1

) all out-
come variables react monotonically to an increase of the
same number of employees in both firms. Prices de-
cline and agents’ individual efforts converge to 0 whereas
sales and surplus levels increase monotonically (see Table
2).
n
Rather than assuming that all members (principal and
agents) of each firm are interested in maximizing the
firm’s surplus or that only one type of actor (the principal)
essentially “runs the mill”, we now include vertical and
horizontal interaction by analyzing strategic delegation
of price decisions.
3. Strategic Delegation of Price Decisions
For the integrative analysis of intrafirm and interfirm
interaction, we assume that principals share revenues
with their agents. Let i denote the revenue share for
all the agents of firm as a whole. Agents are assumed
s
i
Table 1. Numerical solution of the price-competition game
with monolithic firms, n = 2.

0 1 ...  
*
p
0.600 0.500 ... 0.333
*
q 0.400 0.500 ... 0.666
*
e 0.200 0.250 ... 0.333
*
S 0.200 0.188 ... 0.111
Table 2. Numerical solution of the price-competition game
with monolithic firms,

=1.
n 1 2 3 ... 100 ...n
*
p
0.6000.5000.455 ... 0.338 ...0.333
*
q0.4000.5000.555 ... 0.662 ...0.667
*
e0.4000.2500.185 ... 0.007 ...0.000
*
S0.1600.1880.198 ... 0.221 ...0.222

W. GÜTH ET AL. 357
to be identical and to distribute their overall revenue
share iii
s
pq proportionally to each agent’s individual
contribution ,ik i eq . Agents can observe and control the
efforts of the team members. This means that each in-
dividual agent in firm chooses the same effort
, for all workers and realizes the net
utility
=1,2i
=1,k =
ik
ei e,
delegation game can be solved analogously to the quan-
tity-delegation game. Indeed, in the standard principal-
agent scenario where and =1n=0

the results for
our price-delegation game coincide with those of the
quantity-delegation game (Güth, Pull and Stadler [11, p.
372]). n

 
 

2
22
,, =2
=11
112,
iii jiiii
iii j
ij
Usppspe e
sppp n
pp n



 
 
3.1. The Delegation Game with a Variable
Degree of Market Heterogeneity
To study the influence of market heterogeneity on the
compensation scheme, we first restrict our analysis to the
case of agents in each firm and neglect fixed
compensation payments which would leave the strategic
decisions as well as the firms’ surplus unchanged. Maxi-
mization of agent utility
=2n
where is the price decision made by the agents of
firm , given the rival firm’s price
i
p
=1,2i,
j
pi j

. The
analysis of this delegated price competition complements
our former analysis of delegated quantity competition
(Güth, Pull and Stadler [5]). Both scenarios have in
common that agents anticipate the effects of their sales
choice (price or quantity) and principals anticipate these
decisions when implementing the revenue sharing
scheme. Thus, from a technical point of view, the price-





 
2
,, =112
118
iii jiiij
ij
Uspp spp p
pp


 
 
with respect to the price yields the equilibrium prices
choices i
p
   
  

2
222
12 121122123423
(, )=12 121(24)4483
ij
iij
ij ij
ij
s
ss
pss ss ss
 
 

 
s
j
(1)
for , as functions depending on the com-
pensation schemes, i.e. on the strategic variables
=1,2,ii


,
ij
s
s

chosen by the two principals. These prices imply the out-
put levels



  

22
222
21 214253
(, )=,
12 121(24)4483
iij
ii j
ij ij
sss
qss
s
sss

 

 
 
the individual effort levels =
ii
eq2, and the principals’
profits


π,=1= ,=1,2,
iiji ii,
s
sspqNDii





22
2
2
=1212124
4483.
ij
ij
Ds
ss
 

 


 
s
j
where
2
22
=2(1) (12)(1)2(1)(12 )
2(1)(23)4(23)
[(12)(1)2(253)]
ii
j
ij
ii
Ns
j
s
s
ss
ss
The first-order conditions for maximizing


π,
iij
s
s
with respect to , and the obvious symmetry of
the solution lead to a sixth-order polynomial equation
whose solution is
,=1,2
i
si

 
=;=20,n  **
s
** **
ss
s


 
 





 

1.
The equilibrium revenue shares imply the iden-
tical prices
and
2****
**
22**
(12)(1 )2(1 )(35)4(23)
=,
(12)(1 )4(1 )(24)4(483)
ss
pss
 
 

 
2
2**2
output levels





2**2 **2
**
22** 2**2
21(132)4253
=,
12 141244483
ss
qss
 
 
 
 
and effort levels




 
 
2** 2**2
**
22**2*2
11322253
=.
12 141244483
ss
e
s
s

 
 
 

W. GÜTH ET AL. 358
Finally, principals' profits are

****** **
π=1
s
pq.
Due to the nonlinearity of the reaction functions the
game can in general be solved only by using numerical
techniques. An exception is the case of monopoly, i.e. the
market structure characterized by two independent markets
due to =0

2. In this case agents’ price decisions (1) sim-
plify to


12 4812
=14
14 16
ijiji
ii i
ij ij
ssss
=
s
ps
s
ss ss
 


and imply the output levels

2
=.
14
i
ii i
s
qs
s

Hence, the principals’ profits are



2
2
21 2
π=.
14
iii
i
i
s
ss
ss


The symmetric first-order conditions from maximizing
the profits with respect to lead to the cubic equation
i
s
32
8621=0
iii
sss ,
which has the unique real solution

** =0;=2 =14sn
 . Given this revenue-sharing rule
the agents charge the prices **=34p implying output
levels ** =1 4q and effort levels ** =18e such that
the principals realize the profits ** π=9 64, agents the
utility ** =164U, and the firm as a whole the surplus
** =11 64S.
Numerical solutions (****** ******** ,,,,π,,
s
pqe US

=0 
) for
a varying degree of heterogeneity are presented in Table
3. As can be seen, a decreasing degree of heterogeneity
induces principals to offer higher revenue shares to their
agents. Higher revenue shares imply higher sales efforts
by the agents corresponding to lower prices. In the case
of two separate markets agents charge the
highest prices. Starting from that benchmark case
declining prices result from declining heterogeneity
Table 3. Numerical solution of the price-delegation game, n =
2.

0 1 ...


**
s
0.250 0.298 ... 0.366
**
p
0.750 0.685 ... 0.577
**
q 0.250 0.315 ... 0.423
**
e 0.125 0.157 ... 0.211
**
π 0.141 0.151 ... 0.155
**
U 0.016 0.020 ... 0.022
**
S 0.172 0.190 ... 0.200
(larger

) thereby increasing the revenues to be shared
between principals and agents. The revenue effect domi-
nates leading to (slightly) higher profits as a result of a
lower market heterogeneity. Agents’ utility and, hence,
the firms’ surplus also increase.
Compared to the benchmark solution (Table 1) of Sec-
tion 2, which neglects intrafirm conflicts, delegation leads
to higher prices and lower efforts of agents throughout.
The surplus in the delegation game is higher if markets
are homogeneous but is lower in case of very heteroge-
neous markets. This interesting result suggests that it de-
pends on the basic conditions of the market under con-
sideration whether a firm as a whole gains from delega-
tion.
3.2. The Delegation Game with a Variable
Number of Agents
To study how the number of agents hired by both prin-
cipals affects the market outcome, we set the parameter

equal to an intermediate degree of heterogeneity,
=1

, and vary the number of agents symmetrically
across firms3. Maximization of agent utility
n






22
,, =12122
iiijiii ji j
Usppspp p nppn 
with respect to the prices leads to the equilibrium price
choices
i
p

2
2
12 6105
,=
12 141415
iji
iij
ij i
nsnsns s
pss nsnsns s
 
 
j
j
j for =1,2,ii

as functions of the strategic variables


,
ij
s
s, chosen by the two principals. The resulting pro-
fits are



12
π,=1 12=
=1,2, ,
iiiij
,
s
sspppN
iij


D
where


2
222 2322
23 2 23
222 332
= 27236120
605025
723612060
5025
ii ij
ij ij ij
iiij i
ij ij
Nns nsnss
nss nssnss
j
s
nsnssn s s
nss nss


 

and

2
2
=12 141415
ij i
Dnsnsns j s.
The first-order conditions for maximizing


π,
iij
s
s
2Note, however, that unlike in Section 2 it maintains intrafirm interac-
tion and thus non-monolithic firms.
3Endogenizing the number of agents in both firms would require to
study cases with different numbers of agents in the two firms. For such
a case, analytic results are very difficult to obtain. We thus restrict
ourselves to studying how a symmetric change in the number of agents
employed by each firm influences results via comparative statics.

W. GÜTH ET AL. 359
with respect to , and the obvious symmetry of
the solution again lead to a sixth-order polynomial equa-
tion for each number n of a agents hired by both prin-
cipals with the solution . Equi-
librium prices are
,=1,2
i
si
** **
=(;=1)(0,1)ssn  
**2 **2
**2 **2
5
=
15
nsn s
nsn s


** 12 16
12 28,p
output levels
**2 **2
**2 **2
10
=
15
n s
nsn s


** 12
12 28
ns ,q
and effort levels
** **2
**2 **2
10
=
15
sns
nsn s


** 12
12 28
**** **
,π,,
. e
Table 4 illustrates how the solution
(****** **
,,,
s
pqeUS) depends on the number
of agents employed by each seller firm. As can be seen,
an increasing number of agents inquires principals to
offer lower revenue shares to their agents. Declining
marginal effort costs of agents imply higher quantities
and lower prices. The principals’ profits first increase
and later on decrease with more and more agents. A
similar inverted-U shaped relationship holds for the
firms’ surplus whereas agents’ utility is monotonically
decreasing.
n
A comparison of the results to those of the benchmark
case (Table 2) in Section 2 shows that price delegation
results in higher prices and lower effort levels. The sur-
plus in the delegation game is lower in case of a small
number of agents but higher in case of a large number of
agents. Of course, in the limit case effort costs
go to zero and the solutions coincide. Therefore it de-
pends on intrafirm organization (treated as exogenous in
our analysis) whether the firm as a whole gains from
delegation or not.
n
4. Summary and Conclusion
Price delegation to sales managers is usual. But managers
Table 4. Numerical solution of the price-delegation game,

=1.
n 1 2 3 ... 100 ... n
**
s
0.363 0.298 0.259 ... 0.040 ... 0.000
**
p
0.765 0.685 0.641 ... 0.429 ... 0.333
**
q 0.235 0.315 0.359 ... 0.571 ... 0.667
**
e 0.235 0.157 0.120 ... 0.006 ... 0.000
**
π 0.115 0.151 0.171 ... 0.235 ... 0.222
**
U 0.038 0.020 0.013 ... 0.000 ... 0.000
**
S 0.153 0.190 0.209 ... 0.235 ... 0.222
do not suffer from the effort of producing what they sell.
In our analysis this effect is taken into account by as-
suming that the agents who set the sales prices are the
same who suffer from exerting effort. Our price-dele-
gation model assumes that both, principals and agents,
compete with each other. Principals implement a revenue
sharing scheme to which agents react by choosing a sales
price and by producing what is demanded. We thus com-
plement our former investigation of homogeneous markets
with quantity competition by an analysis of more or less
heterogeneous markets with price competition. Both types
of delegation can be observed in markets for specific
goods or services.
Our study demonstrates how implementing revenue
sharing affects intra- and interfirm interaction between
principals (the owners) and agents (the workers) who
suffer the cost of producing more. Thus low effort cost in
case of low output provides an incentive for choosing
high prices, an effect which is absent when sales manager
neglect producing efforts. Therefore, more intensive com-
petition due to a decreasing market heterogeneity or due
to an increasing number of agents, hired by both firms,
leads to lower prices and higher revenues. Accordingly
we derive an inverted U-shaped relationship between the
degree of market heterogeneity and the number of agents
on the one hand and the firms’ surplus on the other.
Whether price delegation increases or decreases the sur-
plus compared to the benchmark case of monolithic firms
depends decisively on the intrafirm organization and the
interfirm (market) structure.
The derived results pose quite a challenge for our in-
tuition of how complex markets operate. In our view, this
alone justifies the attempt to complement our former
analysis of homogeneous markets with quantity compe-
tition by one of more or less heterogeneous markets with
price competition. Both studies together will hopefully
help to understand more thoroughly what has to be ex-
pected from an integrative analysis of intrafirm delega-
tion and interfirm sales competition.
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