Intrinsically Motivated Agents: Blessing or Curse
For-Profit Maximizing Firms?
Ester Manna∗
March 13, 2013
Abstract
I investigate whether the presence of motivated agents benefits firms in a competitive
environment. Without competition, the firm has a clear advantage to hire motivated agents
because the motivation impacts positively on its marginal profits. This is due to the fact
that the firm can pay a lower salary for motivated agents. I show that this result no longer
holds in a competitive environment. In a duopoly, the firms may obtain higher profits by
hiring self-interested agents than by hiring motivated agents. But I show that a prisoner’s
dilemma arises in which the strategy by hiring self-interested agents is strictly dominated by
hiring motivated agents. Hence, the presence of motivated agents can hurt firms.
1
Introduction
It is often argued that preferences and work motivations of employees differ depending on the
nature of the jobs. Many jobs involve helping people in need or contributing to society at large,
rendering these jobs attractive to people who have a strong willingness to serve the others (see
Buurman, Dur and Van den Bossche, 2009). An employee might be interested not only in his
wage, but also in how his work affects the well-being of the others. In this sense, individuals can
be intrinsically motivated.1 The literature has focused on the examples of bureaucracy, public
services provision (e.g. health care, education), or non-profits organizations (see, for instance,
∗
1
Ulb (ECARES); [email protected]
Some economists with theoretical model have assumed that individuals are not only motivated by monetary
incentives, but also by non-pecuniary incentives, i.e. intrinsic motivation (see for example Ma, 2004, Andreoni,
1990, Rabin, 1993). This assumption has been confirmed by empirical studies (see for example Ochs and Roth,
1989, Fehr, Kirchsteiger and Riedl, 1993).
1
Francois (2000), Dixit (2002) and Dewatripont, Jewitt and Tirole (1999)).2 Undoubtedly, it is
natural to think of motivated agents in these kinds of organizations. However, also maximizingprofit firms can attract motivated employees, offering jobs which have a socially valuable impact
or a socially-oriented mission. Suitable examples would be those of engineers working in developing countries or for firms offering environmental services (see for example Parsons, 1996). In
addition, all those jobs that have interaction with customers can be attractive for individuals
who are intrinsically motivated.3 What these examples have in common is that individuals
take satisfaction from rendering a good service to the customers. This explains why motivated
employees decide to work for-profit firms given the nature of the job (see for example Boltanski
and Chiapello, 2002). Ghatak and Mueller (2010) analyze the choice of for-profits vs. notfor-profits in a labor market setting where firms and workers match endogenously. They show
that motivated workers are better off working in a for-profit firm compared to a not-for-profit
firm. As a result, they explain the presence of motivated workers in not-profit firms with the
overabundance of them.
The aim of this article is to study the impact of intrinsic motivation on the firms’ profits when
there is competition between firms maximizing their profits.
Without competition, a firm has a clear advantage to hire motivated agents because the motivation impacts positively on its marginal profits. This is due to the fact that the firm can pay a
lower salary for motivated agents setting a monopolistic price. I show that this result no longer
holds in a competitive environment.
In a duopoly, the agents’ intrinsic motivation has a countervailing effect on the marginal profits obtained by the firms. On the one hand, motivation has a negative impact on each firm’s
marginal profits due to a reduction of the price of the product offered by the firms. The economic
intuition behind this result is the following: the agents’ motivation has a positive impact on the
quality offered by the firms. This reduces the product differentiation between firms thereby
increasing competition and reducing prices. With higher qualities, the product differentiation
becomes relatively less important leading to fiercer competition. On the other hand, motivation
has a positive impact on the marginal profits. This is due to a reduction of the wage. Motivated
employees provide a given level of quality for a lower wage. This result supports the previous
literature in which motivation is effective in stimulating work effort even in absence of monetary
2
Francois argues that the financial motivation for the manager in the for-profit sector crowds out intrinsic
motivation of the worker. Dixit (2002) discusses the agents’ motivations in the public sector. He argues that organizations with an idealistic or ethical purpose may be attractive to workers who share these goals. Dewatripont,
Jewitt and Tirole (1999) assume that bureaucrats are career concerned. They show that this motivation leads to
efficient outcomes if the goal they have to achieve is narrow and precisely defined. Hence, the provision of the
public services is often viewed as “mission oriented”. When the goals of the agent are aligned with the principal’s
mission, civil servants’ social concern can motivate their performance without the need of financial incentives.
3
As an example take an advertisement for a job at a beauty salon: “If you are motivated and passionate about providing high quality hair and beauty services, love being part of a team and understand the
value of fantastic client service, why not apply” (http://www.seek.com.au/Job/senior-hairdresser-and-seniorbeautician/in/brisbane-brisbane/23448759). Another suitable example would be the entertainment industry.
2
rewards (see for example Gneezy and Rustichini, 2000a,b and Benabou and Tirole, 2003, 2006).4
What effect dominates depends on the degree of substitutability of the services and on the degree
of intrinsic motivation. More specifically, a reduction of the price has a stronger effect than a
reduction of the wage on the firms’ profits in markets with more substitutable products and a
lower degree of motivation or in markets with less substitutable products and a higher degree
of motivation.
The firms may obtain higher profits by hiring self-interested agents than by hiring motivated
agents but a prisoner’s dilemma arises in which cooperation by hiring self-interested agents is
dominated by hiring motivated agents.
When both principals hire self-interested agents, the principals have an advantage to deviate
by hiring intrinsically motivated agent. The principal that decides to deviate gains a quality
comparative advantage over its rival. This advantage permits to steal the market share to the
rival firm and to increase the price.5 These effects have a positive impact on the marginal profits
obtained by this firm. When a firm hires motivated agents, its competitor’s best response is
to follow suit. The unique Nash equilibrium is the one in which both principals hire motivated
agents.
This article contributes to the theoretical literature on the effects of competition on managerial incentives. While some articles argue that competition reduces managerial incentives (see,
for example, Hart, 19836 ), others show that the relationship is ambiguous (see, for example,
Scharfstein, 1988, and Hermalin, 19927 ). Schmidt (1997) explains that greater competition may
lead to stronger incentives for agents because greater effort is required to avert the threat of
bankruptcy. Raith (2003) examines how the degree of competition among firms in an industry
with free entry and exit affects the incentives for their managers. Then, the effect of competition on incentives and effort takes place through a change in the equilibrium number of firms
in the industry. His results suggest an unambiguous positive relationship between competition
and incentives. Baggs and De Bettignies (2007) also propose a duopoly model with two firms
that compete on prices and quality. My article shares with this paper the strategic interactions
between principals. However, these authors focus on the effect of transport costs on the degree
of substitutability between the products when both firms maximize their profits and do not
4
This literature shows that monetary incentives can influence negatively the individuals’ behavior in terms of
their levels of contribution. The reason is that monetary incentives give the agent a selfish motive to operate.
Explicit incentives from principals may change how tasks are perceived by agents (Gneezy and Rustichini, 2000a)
and they may also reduce the value of generous or civic minded acts as a signal of one’s moral character (Benabou
and Tirole, 2003, 2006). If extrinsic incentives are not large enough, this change in perception can even lead to
undesired effects on behavior (Gneezy and Rustichini, 2000b).
5
This comparative advantage is due to the higher effort elicited from the motivated agent. The principal pays
higher incentives to the motivated agent in the way to maintain this advantage and to improve the quality.
6
Hart (1983) is the first to formalize this idea by modeling the effect of competition on the agency problems
between a firm’s owner and a manager.
7
Scharfstein (1988) reconsiders Hart’s model while relaxing the assumption of infinitely risk-adverse managers.
Hermalin (1992) considers additional effects of competition on the agency problem, all of which are of potentially
ambiguous sign.
3
consider potential differences in the agents’ preferences.
My work is also related to the literature on psychological incentives in organizations (Benabou
and Tirole, 2003, 2006; Gneezy and Rustichini, 2000a, 2000b). Most articles in this literature are
principally focused on working relationship between the public and private sectors. Besley and
Ghatak (2005) emphasize the importance of the precise definition of the target of a public sector
institution in increasing organizational efficiency. They, moreover, show that public incentives
are likely to be more low-powered than in the private sector. In Benabou and Tirole (2003) and
Delfgaauw and Dur (2007), workers may enjoy to exert effort at work. In Prendergast (2007),
many individuals are motivated to exert effort because they care about their jobs, rather than
because there are monetary consequences to their actions. He assumes that workers differ in
altruism for clients and shows that government prefers to attract different workers types for
different agencies. Moreover, he shows that, when agents’ types are not observable, agencies are
likely to attract both the most preferred and the least preferred workers.
The previous articles focus on the matching of employees with different motivations in different
sectors. That is not the purpose of this work. By contrast, I am interested in the study of the
impact of intrinsic motivation on the firms’ outcome in terms of quality, price and wage when
both firms maximize profits.
2
The Set-Up of the Model
To model competition, I build a duopoly model where two profit-maximizing firms are positioned
at each end of a Hotelling line, with locations xi = 0 and xj = 1, respectively. There are a
continuum of costumers of mass 1 distributed uniformly along the line. Each firm consists of a
principal and an agent, both risk neutral. The principal-agent relationship can be interpreted
as the relationship between the owner of the firm that wants to delegate the decision about the
quality of the product q to an employee (the agent).
The agents are wealth constrained with zero initial wealth and have a reservation wage of zero.
The agents have quadratic effort costs, which are observable to the principal. The exerted effort
ǫ determines the quality of the services. For convenience, I assume that quality q depends
linearly on workers’ effort: q = ǫ in both firms. There is no asymmetric information between
the principal and the agent. Since quality is verifiable, the principals do not need to offer an
incentive to the agents because they have all the necessary information to implement the efficient
levels of quality.
In addition, the agents’ utilities might positively depend on the benefits of the customers. The
measure of this utility depends on the parameter θ that represents the intrinsic motivation of
the agent. It influences the optimal levels of quality and price. There are only two types of
agents: the self-interested agents with θ = 0 and the motivated agents with θ > 0. There is an
infinite number of agents of both types. The principals offer a wage that covers the cost of effort
4
paid by the agent minus his intrinsic motivation.
After the employment decision, the two firms offer imperfectly substitutable services, competing
against each other on quality q and prices p.
The timing of the model is as follows. At the initial stage 0, each principal decides whether
to hire a motivated agent or a non motivated agent; At stage 1, each principal makes an offer
(ω, q) to their agent. The agents accept any contract with an expected utility of at least their
reservation utility, which I normalize to 0; At stage 2, after agents have exerted effort determined
by the contract, the principals simultaneously choose prices; At stage 3, the customers choose
from which firm to buy the good.
2.1
The Objective Functions
A customer enjoys a utility
Ui = v(qi ) − pi − tx from the service offered by the firm i and
Uj = v(qj ) − pj − t(1 − x) from the service offered by the firm j.
(1)
where v(qi ) = v + qi and v(qj ) = v + qj represent the costumer’s benefit from the service offered
by the firm i and the firm j, respectively. Costumers have a common and positive v that is
common knowledge. Customers derive some positive utility from the good irrespective of its
quality. At location x, a customer i incurs a transport cost tx for traveling to firm i and a cost
t(1 − x) to firm j. For given pi , pj , qi , qj there is a cutoff x, such that all consumers with x < x
choose the service of firm i, and with x > x choose firm j. The parameter t represents the degree
of differentiation of the products/services offered by the two firms. When t is low the firms offer
similar products/services, implying fierce competition.
Profits are given by the difference between the revenue of the sales and the wage of the agent.
The two firms maximize the following profit functions:
πi = pi di − ωi
πj = pj dj − ωj
(2)
The key assumption of this model is that agents can be intrinsically motivated. The agents’
utility function consists of their own “egoistic” payoff, given by the difference between wage
and effort costs, and of their intrinsic motivation. There are two types of agents: self-interested
agents and motivated agents. θ = 0 represents self-interested agents and θ > 0 motivated agents.
The agents’ utility function from working in the firms can be written as:
1
Vi = ωi − qi2 + θi U i
2
1 2
Vj = ω j − q j + θ j U j
2
5
(3)
where U i and U j are the utilities of the average consumer buying a product from firm i and firm
j, respectively. The utilities of the average consumer buying a product from firm i and firm j
are, respectively, equal to: U i = v(qi ) − pi − t
x
2
and U j = v(qj ) − pj − t
(1−x)
2 .
Assumption 1. I make the following assumption to guarantee an interior solution.
• v∈
1
1
5, 3
;
• t ∈ 0.281,
2
9
+ 32 v ;
• and θ ∈ 0, θ max
with θ max =
8+45t+36v−
√
−8(−10−54t)+(−8+45t−36v)
.
2(−54t−10)
The role of this assumption will become clear in the next sections.
3
The Characterization of the Equilibrium
The equilibrium is determined by backward induction.
At stage 3, the customers choose which good to buy. A customer located at x is indifferent
between the two firms if and only if Ui = Uj , or equivalently qi − pi − tx = qj − pj − t(1 − x). x
represents the demand for the firm i and (1 − x) the demand for the firm j:
1 (qi − qj ) + (pj − pi )
+
2
2t
(4)
1 (qj − qi ) + (pi − pj )
(1 − x) = dj (qi , qj , pi , pj , t) = +
2
2t
At stage 2, the principals choose their prices to maximize their objective functions, taking
x = di (qi , qj , pi , pj , t) =
qualities and wages as given.
Both principals maximize their profits, respectively:
1 (qi − qj ) + (pj − pi )
max πi = pi
+
− ωi
pi
2
2t
1 (qj − qi ) + (pi − pj )
+
− ωj
max πj = pj
pj
2
2t
(5)
Taking the first order condition with respect to the prices, I obtain the equilibrium prices as
function of the levels of quality offered by the firms:
(qi − qj )
1
pi = t(1 − θi ) − tθj +
2
3
1
(qj − qi )
pj = t(1 − θj ) − tθi +
2
3
6
(6)
Substituting equilibrium prices into the equation (5), I obtain an expression for profits as a
function of the levels of quality and wages offered by the two firms:
(qi − qj ) 1 qi − qj
1
1
+
+ (θi − θj ) − ωi
πi = t(1 − θi ) − tθj +
2
3
2
6t
4
(7)
1
(qj − qi ) 1 qj − qi 1
πj = t(1 − θj ) − tθi +
+
+ (θj − θi ) − ωj
2
3
2
6t
4
At stage 1, these functions are maximized with respect to ωi , qi and ωj , qj , respectively, under
the following participation constraints:
1
ωi − qi2 + θi U i ≥ 0
2
1 2
ωj − qj + θj U j ≥ 0
2
(8)
The participation constraints guarantee that the agents do not choose their outside option.
In the following subsections, I will characterize the equilibrium for different degrees of intrinsic
motivation: when both agents are self-interested, i.e. θi = θj = θ = 0 (subsection 3.1); when
agents are homogeneous and motivated, i.e. θi = θj = θ > 0 (subsection 3.2); when only the
agent i is motivated, i.e. θi = θ > 0 and θj = θ = 0 (subsection 3.3).8 All the mathematical
computations for these next subsections are in the Appendix A.
3.1
When Agents are Self-Interested
I begin by characterizing the equilibrium when agents are self-interested, i.e. θi = θj = 0. I
obtain the standard Hotelling model.
At stage 1, the optimal levels of quality and wages are determined. The unique solution is the
symmetric first-best equilibrium in which both principals elicit quality’s levels
qi∗ = qj∗ =
with wages
ωi∗ = ωj∗ =
1
3
(9)
1
.
18
(10)
The chosen prices at stage 2 are:
p∗i = p∗j = t.
(11)
A reduction of t leads to a reduction of the prices of both firms. As t goes to 0 the firms offer
always more similar services. The two firms are more competitive and the prices go down.
And at stage 3, the demands are realized with
d∗i = d∗j =
8
1
2
Obviously, the case in which only the agent j is motivated is symmetric to this one.
7
(12)
and the principals obtain the following profits
1
1
πi∗ = πj∗ = t − .
2
18
3.2
(13)
When Agents are Homogeneous and Motivated
In this subsection, I determine the equilibrium when both agents are homogeneous and intrinsically motivated, i.e. θi = θj = θ > 0.
The optimal levels of quality are the following:
1
(1 + θ)
3
q ∗i = q ∗j =
(14)
The quality of the services offered in the firms will be higher or equal to the standard case
studied in the previous subsection, i.e. q ∗i = q ∗j ≥ 31 .
The wages are given by:
ω ∗i
=
ω∗j
1
=
2
1+θ
3
2
1+θ
1
3
−θ v+
−t 1− θ − t
3
2
4
(15)
The agents’ intrinsic motivation has a negative impact on wages. This is because motivated
employees provide a given level of quality for a lower wage.9
Prices are given by:
3
p∗i = p∗j = t 1 − θ
2
(16)
When the agents are intrinsically motivated, the prices of the services offered by the firms will
be lower or equal to the standard case, p∗i = p∗j ≤ t. The effect of θ on the prices is negative.
Motivation has a positive impact on the quality offered by the firms. It implicitly reduces the
product differentiation between firms stiffening competition and reducing prices. With higher
qualities, the location becomes relatively less important, leading to fiercer competition. This
fall in prices has a positive impact on the customer’s utility. The value of θ has to be at most
equal to 32 .10 For higher values of θ, the prices are negative.
The firms share the demand in the market such as in the standard Hotelling model
∗
∗
di = dj =
1
2
(17)
and the profits are realized:
π ∗i
9
=
π ∗j
1
2
1
5 2
3
+ θ + θ − tθ 2 − θ + vθ
= t−
2
18 9
18
2
The value of θ has to be at most equal to
8+45t+36v−
wages are negative.
10
This condition holds due to assumption 1.
8
√
−8(−10−54t)+(−8+45t−36v)
.
2(−54t−10)
(18)
For higher values of θ, the
3.3
When Only the Agent i is Motivated
Now, suppose that only the agent i is intrinsically motivated, i.e. θi = θ and θj = 0.
At stage 1, agents exert effort and the optimal levels of quality are determined:
qbi∗ =
q ∗j
1 θ(27t − 2)
+
3
6(9t − 2)
1 θ(9t + 2)
= −
3 6(9t − 2)
(19)
In the standard Hotelling model, the unique solution is the symmetric first-best equilibrium in
which principal i and j elicit effort level qi∗ = qj∗ = 13 . When agent i is intrinsically motivated,
the quality of the products/services offered in the firm i will be higher or equal to the standard
case, qbi∗ > 13 , while the quality in the firm j will be lower or equal to the standard case, q∗j < 13 .
To guarantee that both firms are active, θ ≤
2(9t−2) 11
(9t+2) .
Otherwise, firm i pushes the rival firm
out the market and obtains monopolistic profits. My results will be confirmed.12
The principals pay the following wages to their agents:
2
1 1 θ(27t − 2)
ω
bi =
− θ Ui
+
2 3
6(9t − 2)
2
1 1 θ(9t + 2)
ωj =
−
2 3 6(9t − 2)
(20)
By increasing the level of quality offered by the firm i, the principal i has to provide more
incentives and elicit higher agent effort in order to improve quality. This firm produces a larger
output and provides stronger incentives, i.e. ω
bi > ωj .13
The prices are given by:
(4 − 9t)
=t 1+
θ
(9t − 2)
(2 + 9t)
∗
pj = t 1 −
θ
2(9t − 2)
pb∗i
(21)
A high agent i’s degree of motivation has a positive effect on the price of the firm i. A high θ
produces an improvement of the quality offered by the firm i, which increases its price. This
effect has a positive impact on its marginal profits.
In contrast, a high θ has a negative effect on the price of the product offered by the firm j.
On the one hand, the reduction in the quality of the products/services lead a reduction of the
price; on the other hand, the fact that firm i has a comparative advantage, due to the agent’s
motivation, reduces the price of the rival firm.
The demands will be equal to:
1 θ(2 + 9t)
db∗i = +
2 4(9t − 2)
d∗j
11
1 θ(2 + 9t)
= −
2 4(9t − 2)
This condition is guaranteed by assumption 1.
The proof of this result is available under request.
13
The computations to show this result are in the appendix A.
12
9
(22)
The demand of the firm i is higher or equal than the demand of the firm j. Compared with
the result obtained in the standard Hotelling Model, db∗i > 12 and d∗j < 12 . This is due to the
heterogeneity in intrinsic motivation of the workers working in the firm i and j. The firm i gains
a quality comparative advantage over its rival, and obtains the “business stealing effect”: an
increase of quality permits to steal the market share to the rival firm. This effect has a positive
impact on the marginal profits obtained by the firm i.
At stage 3, the profits are realized:
1 θ(2 + 9t)
(4 − 9t)
π
bi =
+
t 1+
θ
−ω
bi
2 4(9t − 2)
(9t − 2)
1 θ(2 + 9t)
(2 + 9t)
θ
− ωj
πj =
−
t 1−
2 4(9t − 2)
2(9t − 2)
(23)
(24)
The principal i, by hiring a motivated agent, obtains a comparative advantage respect to its rival,
i.e. π
bi > π j .14 This comparative advantage is due to the higher effort elicited by the motivated
agent. It increases the demand and the price offered by the firm i having a positive impact on
its marginal profits. In addition, the principal i pays higher incentives to the motivated agent in
the way to maintain this advantage and to improve quality. This firm produces a larger output
and provides stronger incentives.
4
A Prisoner’s Dilemma
In stage zero, both firms choose simultaneously which type of agent to hire. Given prices,
quantities and wages, the type choice reduces to the following game:
θj
θj
θi
(πi∗ , πj∗ )
θi
(b
πi∗ , π ∗j )
(π ∗i , π
bj∗ )
(π ∗i , π ∗j )
Figure 1: The Type-Choice Game
I start comparing the profits obtained by hiring self-interested agents (equation 13) with those
obtained by hiring only a motivated agent (equation 23).
Lemma 1. πi∗ < π
bi∗ for any value of t, v and θ.
Proof. See Appendix B.
14
The computations are shown in the appendix A.
10
When both principals hire self-interested agents, a principal finds it profitable to deviate by
hiring a motivated agent. By doing so, he obtains a comparative advantage in terms of demand
and price. Compared to the case in which both principals hire self-interested agents, the revenues
are considerably higher. Indeed, the demand was higher than the one obtained in the standard
model, i.e. db∗ > 1 , and the effect of the agent’s motivation on the price was positive. Regarding
i
2
the wage, the principal has to pay a higher compensation to the motivated agent to maintain
the comparative advantage. This wage will be higher than the wage obtained when both agents
are self-interested. However, the profits obtained when only a principal hires a motivated agent
are higher than the case in which both agents are self-interested.
Furthermore, when I compare the profits obtained by hiring motivated agents (equation 18)
with those obtained by hiring a non-motivated agent when the rival firm hires a motivated
agent (equation 24), I obtain that:
Lemma 2. π ∗i > π ∗i for any value of t and v and θ.
Proof. It is shown in the Appendix B.
This leads to the following proposition.
Proposition 1. The unique Nash equilibrium is the one in which both firms hire motivated
agents.
Proof. See lemma 1 and 2.
The best response of the principal i is always to hire a motivated agent. Regardless of whether
principal j hires a motivated agent or not, principal i is always better-off hiring a motivated
agent. This is because when both principals hire self-interested agents, the principal i has an
advantage to deviate. By hiring an intrinsically motivated agent, he will obtain higher profits,
i.e. π
bi∗ > πi∗ . When a firm hires a motivated agent, its competitor’s best response is to follow suit,
i.e. π ∗ > π ∗ . The unique Nash equilibrium is the one in which both principals hire motivated
agents: (π ∗i , π ∗j ).
However, both firms may obtain higher profits by hiring self-interested agents than by hiring
motivated agents.
1
5 2
3
1
1
2
1
5θ + 18v + 4
π = t−
+ θ + θ − tθ 2 − θ + vθ < π ∗ = t −
if t >
2
18 9
18
2
2
18
36 − 27θ
∗
(25)
When v takes his minimum value, i.e. v = 51 , the equation (25) is satisfied for any value of θ
if t ≥ 0.305. The higher value of v, the higher the value of t that holds true the inequality.
11
Indeed, when v = 14 , π ∗i ≤ πi∗ if t ≥ 0.324 for any value of θ. And, finally, when v = 13 , π ∗i ≤ πi∗
if t ≥ 0.337 for any value of θ > 0.
In this situation, a prisoner’s dilemma arises in which collaboration by hiring self-interested
agents is dominated by hiring motivated agents.
To understand the reason why we are in presence of a prisoner’s dilemma, I study more in detail
the effect of intrinsic motivation on the outcome of the firms when both principals hire motivated
agents. To analyze the effect of the agent’s intrinsic motivation on the profits, I differentiate
equation (18) with respect to θ and I obtain the following:
∂π
1
2
1
1
∂p ∗ ∂ω
3
1
d −
=
= − t − θ − + v + θ + − t + 3tθ − t
4
9
9
3
3
4
∂θ
∂θ
∂θ
A higher motivation influences the profits in different ways.
First, it has a negative impact on the prices of the product offered by the firms. When both firms
∂pi
∂θ
∂q i
1
=
3
∂θ
hire intrinsically motivated agents, the effect of θ on the prices is negative, i.e.
Motivation has a positive impact on the quality offered by the firms, i.e.
= − 23 t < 0.
> 0. With
higher qualities, the product differentiation becomes relatively less important, leading to an
increase of the competition and a reduction of the price. This negative effect of the agent’s
intrinsic motivation has a negative impact on the profits of the firm.
Second, the agents’ intrinsic motivation has a negative impact on the wages. Motivated employees provide a given level of quality for a lower wage. It impacts positively on the profits.
The overall effect of the agents’ intrinsic motivation on the profits depends on the degree of
substitutability of the services t and on the degree of intrinsic motivation θ. More specifically,
∂π
∂θ
< 0 if t >
2+5θ+9v
.
9(2−3θ)
In this case, a reduction of the price has a stronger effect than a reduction
of the wage and it has a negative impact on the profits obtained by the firms.
5
Future Research and Conclusions
In this article, I have shown that the firms could have an incentive to cooperate by hiring selfinterested agents. This was explained by the effect of the agents’ intrinsic motivation on the
prices and on the wages in a competitive environment. More specifically, a higher motivation has
a negative effect on the price of the product offered by the firms. The economic intuition behind
this result is the following: the agents’ motivation has a positive impact on the quality offered
by the firms, and this reduces the relative importance of product differentiation between firms,
leading to stiffened competition and reduced prices. This effect has a negative impact on the
profits. In contrast, the overall effect of the intrinsic motivation on the wages is negative and it
impacts positively on the profits. What effect dominates depends on the degree of differentiation
of the products offered by the firms and on the degree of intrinsic motivation. The firms obtain
higher profits by hiring self-interested agents in markets with more substitutable products and
a lower degree of motivation or in markets with less substitutable products and higher degree
12
of motivation.
I show that a prisoner’s dilemma arises in which cooperation by hiring self-interested agents
is dominated by hiring motivated agents. Each firm is willing to deviate by the equilibrium
in which both firms hire self-interested agents in the way to obtain a comparative advantage
respect to its rival increasing the profits and pushing the other firm out of the market. If one
firm decides to deviate the best response of the other firm is to deviate too. Hence, the presence
of motivated agents can hurt firms.
For future research, it might be interesting to analyze the optimal position of the firms on the
Hotelling line and to study different models of competition. Moreover, a possible extension of
the model is the one in which the quality is unobservable.
13
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14
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15
A
A.1
Appendix
The Characterization of the Equilibrium when Both Agents are SelfInterested
Both principals maximize their profits, respectively:
1 (qi − qj ) + (pj − pi )
+
−
max πi = pi
pi
2
2t
1 (qj − qi ) + (pi − pj )
max πj = pj
+
−
pj
2
2t
1 2
q
2 i
1 2
q
2 j
(26)
Taking the first order condition with respect to the prices, I obtain the equilibrium prices as
function of the levels of quality offered by the firms:
(qi − qj )
3
(qj − qi )
pj = t +
3
pi = t +
(27)
Substituting equilibrium prices into equation (27), I obtain an expression for profits as a function
of the levels of quality offered by the two firms:
(qi − qj ) 1 qi − qj
max t +
+
−
qi
3
2
6t
(qj − qi ) 1 qj − qi
max t +
+
−
qj
3
2
6t
1 2
q
2 i
1 2
q
2 j
(28)
Deriving with respect to qi and qj , I obtain the optimal levels of quality:
qi∗ = qj∗ =
1
3
(29)
1
18
(30)
Then, the wage paid to the agents will be equal to:
ωi∗ = ωj∗ =
The prices are given by:
p∗i = p∗j = t
(31)
1
2
(32)
1
1
t−
2
18
(33)
The demands are realized:
d∗i = d∗j =
and the principals obtain the following profits:
πi∗ = πj∗ =
16
A.2
Characterization of the Equilibrium when Both Agents are Motivated
The principal i maximizes his profits:
1 (q i − qj ) + (pj − pi )
1
(34)
+
− q 2i +
max π i = pi
pi
2
2t
2
t 1 (q i − q j ) + (pj − pi )
+
+θ v + q i − pi −
2 2
2t
The principal j, similarly, maximizes:
1 (q j − q i ) + (pi − pj )
1
max π j = pj
(35)
+
− q 2j +
pj
2
2t
2
t 1 (q j − qi ) + (pi − pj )
+θ v + q j − pj −
+
2 2
2t
Taking the first order condition with respect to the prices, I obtain the equilibrium prices as
function of the levels of quality offered by the firms:
(q i − q j )
3
pi = t 1 − θ +
2
3
(q j − q i )
3
pj = t 1 − θ +
2
3
(36)
Substituting equilibrium prices into equation (34) and (35), respectively, I obtain an expression
for profits as a function of the levels of quality offered by the two firms:
(q i − qj ) 1 q i − q j
1
3
+
− q 2i +
max π i = t 1 − θ +
qi
2
3
2
6t
2
(q i − q j )
3
t 1 (q i − q j )
+
+θ v + q i − t 1 − θ +
−
2
3
2 2
6t
(q j − q i ) 1 q j − q i
3
1
max π j = t 1 − θ +
+
− q 2j +
qj
2
3
2
6t
2
(q j − qi )
3
t 1 (q j − qi )
+
+θ v + q j − t 1 − θ +
−
2
3
2 2
6t
Deriving with respect to qi and qj , I obtain the optimal levels of quality:
1
(1 + θ)
3
q ∗i = q ∗j =
The prices are equal to:
3
p∗i = p∗j = t 1 − θ
2
The demands are realized:
∗
∗
di = dj =
1
2
(37)
(38)
(39)
(40)
(41)
and the wage paid to the agents is equal to:
2
2
1
5θ
tθ
− θ−
−v θ+
5 − 6θ
18 9
18
4
The principals obtain the following profits:
ω∗i = ω ∗j =
(42)
2
π ∗i
=
π ∗j
1
2
tθ
1
5θ
+ θ+
−
4 − 3θ + v θ
= t−
2
18 9
18
2
17
(43)
A.3
Characterization of the Equilibrium when only Agent i is Motivated
Assume the only the agent i is intrinsically motivated. He maximizes an utility function that
consists of his own “egoistic” payoff, given by a difference between wage and exerted effort, and
of his intrinsic motivation, θ U i . Then, the agent i utility function from working in the firm i
can be written as:
1
Vbi = ω
bi − qbi2 + θ U i
(44)
2
In contrast, I assume that the agent working in the firm j is only interested in his own monetary
payoff, i.e. θj = 0. The agent j’s utility from working in the firm j is described by:
1
V j = ω j − q 2j
2
The principal i maximizes his profits:
"
#
qi − q j ) + (pj − pbi )
1
1 (b
− qbi2 +
+
max π
bi = pbi
2
2t
2
pbi
!#
"
qi − q j ) + (pj − pbi )
t 1 (b
+θ v + qbi − pbi −
+
2 2
2t
In contrast, the principal j maximizes the following profits function:
"
#
pi − pj )
1 (q j − qbi ) + (b
1
max π j = pj
− q 2j
+
pj
2
2t
2
(45)
(46)
(47)
Taking the first order condition with respect to the prices, I obtain the equilibrium prices as
function of the levels of quality offered by the firms:
(b
qi − q j )
pbi = t(1 − θ) +
3
(q − qb )
i
1
j
pj = t 1 − θ +
2
3
18
(48)
Substituting equilibrium prices into equation (46) and (47), respectively, I obtain an expression
for profits as a function of the levels of quality offered by the two firms:
#"
#
"
(b
qi − q j )
1 qbi − q j
1
1
+
+ θ − qbi2 +
max t(1 − θ) +
3
2
6t
4
2
qbi
!
"
!#
(b
qi − q j )
t 1 qbi − qj
1
+θ v + qbi − t(1 − θ) +
−
+
+ θ
3
2 2
6t
4
#
" (q − qb ) # "
i
1
1
1 q j − qbi 1
j
+
− θ − q 2j
max t 1 − θ +
qj
2
3
2
6t
4
2
(49)
(50)
Deriving with respect to qbi and qj , I obtain the optimal levels of quality:
qbi∗ =
q ∗j
The prices are equal to:
(51)
1 θ(9t + 2)
= −
3 6(9t − 2)
θ(4 − 9t)
=t 1+
(9t − 2)
θ(2 + 9t)
p∗j = t 1 −
2(9t − 2)
pb∗i
The demands are realized:
1 θ(27t − 2)
+
3
6(9t − 2)
(52)
1 θ(2 + 9t)
db∗i = +
2 4(9t − 2)
(53)
1 θ(2 + 9t)
= −
2 4(9t − 2)
The wage paid to the agents will be equal to:
2 1 θ(27t − 2)
t 1 θ(2 + 9t)
1 1 θ(27t − 2)
θ(4 − 9t)
∗
+
−θ v +
+
+
ω
bi =
− t+t
−
2 3
6(9t − 2)
3
6(9t − 2)
(9t − 2)
2 2 4(9t − 2)
(54)
2
1 1 θ(9t + 2)
ω ∗j =
−
2 3 6(9t − 2)
d∗j
The hwage paid toi the motivated
agent is higher than the wage
agent
i paidh to the unmotivated
i2
h
2
tθ(2+9t)
tθ(4−9t)
θ(27t−2)
θ(9t+2)
t
1 1
1
1 1
− θ 3 (1 + θ) − t − 9t−2 − 4 − 8(9t−2) ≥ 2 3 − 6(9t−2) . After some
if 2 3 + 6(9t−2)
computations, I obtain the following:
2
tθ
(9t−2)
2
+
tθ (34−63t)
8(9t−2)
+
36tθ
18(9t−2)
+ 54 tθ − 31 θ(1 + θ) ≥ 0. This
inequality always holds true due to the assumption 1. The motivated agent always receives a
higher wage.
And the principals obtain the following profits:
1 θ(2 + 9t)
∗
+
π
bi =
t 1+
2 4(9t − 2)
1 θ(2 + 9t)
∗
=
−
t 1−
πj
2 4(9t − 2)
19
θ(4 − 9t)
(9t − 2)
θ(2 + 9t)
2(9t − 2)
−ω
bi∗
−
ω ∗j
(55)
i
h
The profits obtained by firm i are higher than the profits obtained by firm j if 12 + θ(2+9t)
4(9t−2)
ih h i h
i
(2+9t)
(4−9t)
θ(2+9t)
1
t 1 + (9t−2) θ − 2 − 4(9t−2) t 1 − 2(9t−2) θ − (b
ωi − ω j ) ≥ 0 After some computations,
I obtain that
9tθ(1−2t)
2(9t−2)
+ 13 θ(1 + θ) +
2
tθ (572+4293t2 −2528t)
8(9t−2)2
inequality always holds true.
20
≥ 0. Given the assumption 1, this
B
B.1
Appendix
Proof of the lemma 1
I compare the profits when both principals hire self-interested agents with the one in which only
the principal i hires a motivated agent. And I want to show that πi∗ < π
bi∗ for any value of t, v
and θ. The explicit expression is the following:
1
1
θ(4 − 9t)
1 θ(2 + 9t)
∗
∗
∗
+
t 1+
−ω
bi − t −
>0
π
bi − πi > 0 if
2 4(9t − 2)
(9t − 2)
2
18
(56)
It is not immediate to see that the inequality holds. The proof unfolds in two steps. First, I
study the marginal effect of the agent’s degree of motivation θ, of the degree of differentiation
of the products t and of the customer’s interest v on the difference between profits. Secondly, I
show that the above inequality holds also in a limit case and this completes the proof.
I start by studying the marginal effect of the agent’s intrinsic motivation θ on the profits.
Deriving equation (56) with respect to θ, I obtain the following:
t[(4212t − 936) + θ(792 + 648t)] + 28[18t − 4 + θ(27t − 2)]
∂(b
πi∗ − πi∗ )
=v+
+
72(9t − 2)2
∂θ
243t2 [36t − 8 + θ(10 − 9t)]
6t + 189t2 − 8
>0
−
+θ
24(9t − 2)
72(9t − 2)2
(57)
Except for the last term of the equation (57), all the others are positive. The overall effect of the
agent’s degree of motivation on the differential profits is positive. A higher motivation influences
positively the profits obtained by the firm i when it is the only one to hire the motivated agent.
In contrast, it has no effect on the profits of the firm when both principals hire self-interested
agents. Then, the sign of this derivative is positive.
I also study the effect of t on the difference in the profits. Differentiating equation (56) with
respect to t, I obtain the following:
∂(b
πi∗ − πi∗ )
1 (18t − 4 + 27tθ − 2θ) + 18(18t − 4 + 2θ + 9tθ)
=− −
+
∂t
2
4(9t − 2)3
2
−
(58)
2
θ(428 − 558t) + θ (382 − 459t) + t(468 − 1170t2 θ) t2 (468 − 2349θ ) + 81t3 (1 − θ) + 80
+
<0
72(9t − 2)2
72(9t − 2)2
The last term in equation (58) is the unique one to be positive. The overall effect of the degree
of differentiation of the product on the differential profits is negative. This is because t has
a positive impact on the profits of the firm when both principals hire self-interested agents
but a negative impact when only a principal hires a motivated agent. When t increases the
comparative advantage by hiring a motivated agent is reduced.
Finally, I analyze the effect of v on the profits and I obtain that:
∂(πi∗ − π
bi∗ )
=θ>0
∂v
21
(59)
v has a positive impact on the profits of the firm that hires a motivated agent (because it has a
negative impact on the wages) but null on πi∗ .
To consider a limit case, I take the minimum value for v and θ, since their impact on the differential profits is positive, and the maximum value for t since it affects negatively the differential
profits. If inequality (56) holds in this limit case, it will be always satisfied for other values of
these parameters. When v takes its minimum value, i.e. v = 51 , t takes its maximum value to
0.356. I set t = 0.356, θ = 0.0005, and v =
B.2
1
5
and I obtain that πi∗ = 0.1224 < 0.1226 = π
bi∗ .
Proof of the lemma 2
In this subsection, I show that the profits obtained by hiring motivated agents (equation 18)
are higher than the profits obtained by hiring a non-motivated agent when the rival firm hires
a motivated agent (equation 24), for any value of t, v and θ.
2
1 θ(2 + 9t)
1
1 2 5θ tθ
θ(2 + 9t)
π ∗i −π∗i > 0 if
t− + θ+
−
4 − 3θ +v θ− −
t 1−
+ω ∗i > 0
2 18 9
18 2
2 4(9t − 2)
2(9t − 2)
(60)
It is not immediate to see that the inequality holds. Again, the proof unfolds in two steps. First,
I study the marginal effect of the agent’s degree of motivation θ, of the degree of differentiation
t and of the customer’s interest v on the differential profits. Secondly, I show that the above
inequality holds also in a limit case and this completes the proof.
I begin by studying the marginal effect of the agent’s intrinsic motivation on the profits. Deriving
equation (60) with respect to θ, I obtain the following:
∂(π ∗i − π ∗i )
1
=
(4 − 36t + 18v + 10θ + 54tθ)+
18
∂θ
+
(61)
(16 + 36t(81t2 − 9t − 4) + 2θ(4 − 243t2 − 729t3 )
>0
72(9t − 2)2
The difference in the profits is positively correlated with the degree of motivation θ. This is
mostly due to the fact that a higher degree of motivation of the agent hiring by the rival firm
when the principal i hires a selfish agent reduces its marginal profits, i.e.
∂π ∗i
∂θ
< 0, and it has a
positive impact on the differential profits.
When I study the marginal effect of t on the profits, I obtain the following:
2
2
∂(π ∗i − π ∗i )
1
(288 − 3240t + 8748t2 + 144θ + 648tθ − 8748t2 θ + 486tθ + 2187t2 θ )
2
= (9−36θ+27θ )−
+
∂t
18
72(9t − 2)2
(62)
2
2
2
2
3
2
3
2
3
(−16 + 288t − 1620t + 2916t − 16θ + 144tθ + 324t θ − 2916t θ − 4θ + 243t θ + 729t θ )
+
<0
4(9t − 2)3
After some computations, I obtain that
∂(π ∗i −π∗i )
∂t
< 0 if θ <
4(−32+360t−1458t2 +2187t3 )
.
−104+1404t−5346t2 +8018t3
This
statement is always guaranteed by assumption 1. t has a positive impact on the marginal profits
22
in both cases, i.e.
∂π ∗i
∂t
> 0 and
∂π ∗i
∂t
> 0 but
∂π ∗i
∂t
<
∂π ∗i
∂t .
Finally, I analyze the effect of v on the profits and I obtain that:
∂(π ∗i − π ∗i )
=θ>0
∂v
(63)
v has a positive impact on the profits of the firms that hire motivated agents but null on π ∗i .
To consider a limit case, I take the minimum value for v and θ, since their impact on the
differential profits is positive, and the maximum value for t since it affects negatively the profits.
If the inequality holds in this limit case, it will be always satisfied for other values of these
parameters. When v takes its minimum value, i.e. v = 51 , t takes its maximum value to 0.356.
I set t = 0.356, θ = 0.0005, and v =
1
5
and I obtain that π ∗i = 0.1221 < 0.1223 = π ∗i .
It is also possible to note that in this limit case the prisoner’s dilemma arises. The cooperation
by hiring self-interested agents is dominated by hiring motivated agents even if the profits in
this latter situation are lower, i.e. πi∗ = 0.1224 > 0.1223 = π ∗i .
23