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Review of Accounting Studies
https://doi.org/10.1007/s11142-025-09873-9
Innovation incentives and competition for corporate
resources
Sunil Dutta1·Qintao Fan2
Accepted: 21 January 2025
© The Author(s) 2025
Abstract
This paper investigates how competition for scarce corporate resources impacts
innovation incentives within multidivisional firms and, consequently, shapes firms’
preferences for fostering or restricting intra-firm competition. In our model, divisions
become privately informed about the potential value of new investment opportunities
generated through their innovation initiatives. We demonstrate that intra-firm com-
petition unambiguously reduces divisions’ ex ante innovation incentives. However, it
benefits ex post resource allocation by enabling the firm to (i) select the most promising
project and (ii) limit the rents divisions earn from their private information. Conse-
quently, a firm’s preference to limit or encourage interdivisional competition hinges on
balancing ex post allocative efficiency, which favors increased intra-firm competition,
against ex ante innovation incentives, which favor reduced competition. Our analysis
identifies plausible conditions under which each organizational design—competitive
or exclusive innovation—emerges as the optimal choice.
Keywords Innovation ·Incentives ·Scarce resources ·Competition ·Asymmetric
information
JEL Classification D82 ·G34 ·G31 ·G32 ·L22
1 Introduction
Innovation is widely regarded as a crucial driver of value creation, especially as the
rapid pace of technological advancement has heightened its significance for modern
firms. The growth and survival of companies in various sectors now depend heavily
BSunil Dutta
dutta4@berkeley.edu
Qintao Fan
qfan@uoregon.edu
1Haas School of Business, University of California at Berkeley, Berkeley, CA, USA
2Lundquist College of Business, University of Oregon, Eugene, OR, USA
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S. Dutta, Q. Fan
on their ability to effectively foster and harness innovation. However, providing incen-
tives for innovation within companies presents significant challenges. Unlike routine
tasks, innovation often involves intangible ideas and know-how, the value of which is
difficult to assess until these ideas are transformed into concrete products or services.
In addition, innovators, such as employees, managers, or business units, often possess
private knowledge about the potential value of their innovations. Agency problems
arising from asymmetric information and the lack of verifiable measures for innova-
tion become further exacerbated when multiple agents compete for limited corporate
resources, such as investment budgets or human capital. In this context, agents must
be motivated to pursue innovation, even when the value of their efforts may never
materialize because the firm is constrained in the number of projects it can undertake
due to the scarcity of necessary resources.
This paper theoretically examines the interaction between intrafirm competition for
resources and innovation and its implications for managerial incentives and organiza-
tional design. To explore this, we develop a model that captures key characteristics of
innovation within firms: ex ante innovation efforts, asymmetric information, and ex
post competition for limited resources. We focus on a multi-divisional firm consisting
of a headquarters and two divisions. Each division is led by a risk-neutral manager
who can undertake personally costly and unobservable efforts to develop new invest-
ment opportunities. These innovation efforts stochastically increase the value of the
managers’ respective projects. To model intra-firm competition, we assume the firm
can undertake at most one of the two divisional projects. During the innovation phase,
divisional managers become privately informed about the potential payoffs of their
projects.
We explore how competition for scarce corporate resources influences managers’
innovation incentives and how this relationship affects the firm’s preference for orga-
nizational designs that either promote or limit competition. To limit competition, the
firm could designate one division as the “innovative” type. Under this organizational
design, which we term exclusive innovation, the chosen division has exclusive rights
to develop and seek approval for new investment projects. This contrasts with the
competitive innovation design, where both divisions are allowed to innovate. Under
the competitive innovation design, divisional managers engage in upfront innovation
efforts to enhance their investment opportunities, knowing that the firm will imple-
ment at most one of the two potential projects. Our analysis reveals that the choice
between competitive and exclusive innovation comes down to a trade-off between ex
post allocative efficiency and ex ante innovation incentives.
At the project implementation stage, the firm takes the managers’ innovation choices
as given and makes sequentially rational investment decisions to maximize its ex post
expected profit.1This opportunism on the part of the firm creates a holdup problem,
undermining divisional managers’ incentives to engage in costly innovation upfront.
From the ex post perspective of resource allocation, we find that competition offers two
1Though innovation activities often span long horizons, managerial incentive plans and investment policies
are frequently subject to revisions in light of new information. To capture this tendency for flexible invest-
ment and compensation rules in our reduced-form single-period setting, we follow Arya et al. (2000)and
others in assuming that the firm commits to investment decision rules after the managers have contributed
their innovation efforts.
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Innovation incentives and...
benefits. First, competition creates a sampling or “winner-picking” benefit, allowing
the firm to select the most valuable project from a pool of options. Second, competition
enables the firm to capture a larger share of the investment surplus. Specifically, com-
petition reduces the amount of rent the firm must leave to managers in order to elicit
their private information. This rent extraction benefit is akin to a well-known result
from auction theory: a seller can capture a greater fraction of the trade surplus as the
number of privately informed bidders increases. In addition, competition decreases
the likelihood that any given project will be implemented. Since a division can only
earn rent if its project is selected, these two effects of competition work together to
reduce the expected rents of each manager.
Information rents are typically seen as private benefits that accrue to asymmetrically
informed managers at the expense of shareholders. However, in our model, these rents
also play a crucial role in providing incentives for managers to engage in innovation.
In our model, the output of innovation is the respective manager’s private information
and cannot be directly used to reward the managers. Instead, a manager is motivated
to invest in costly innovation efforts because she expects these efforts to yield higher
future rents. In a competitive innovation setting, where multiple divisions compete
for scarce resources, competition reduces the managers’ expected information rents,
thereby lowering the marginal returns from innovation. As a result, the equilibrium
level of innovation is always lower in a competitive environment compared to an
exclusive innovation setting. This result readily generalizes to firms with more than two
divisions, with innovation levels decreasing monotonically as the number of divisions
competing for scarce resources increases.
The optimal organizational design for managing interdivisional competition, there-
fore, requires balancing the firm’s dual objectives: providing innovation incentives,
which favors less competition, and improving ex post allocative efficiency, which
favors more competition. Are there conditions under which competition’s negative
impact on innovation incentives outweighs its rent limiting and winner-picking bene-
fits? We find that limiting competition through the exclusive innovation organization
design is indeed optimal if innovation efforts are sufficiently productive; i.e., if project
quality is sufficiently sensitive to innovation efforts. In such environments, innovation
not only enhances the expected value of projects more effectively but also plays a
more significant role in determining managerial rents, thereby increasing its impact
on firm value and making it easier to incentivize. Therefore, the provision of inno-
vation incentives is more critical and exclusive innovation outperforms competitive
innovation. Conversely, when innovation efforts are relatively less productive, the
firm benefits more from encouraging competition, making the competitive innovation
design the optimal choice.
We also find that competitive innovation is more likely to be optimal when there is
greater uncertainty about the returns from investment projects. This is because both
beneficial effects of competition—sampling and rent limitation—become amplified
as project returns become more volatile. The sampling benefit of the option to select
the best project from a draw of two projects becomes more valuable when project
outcomes are more uncertain. Additionally, higher project value volatility generates
greater expected rents for managers, and hence competition has a greater impact on
curtailing these rents. At the same time, when project values are more uncertain, the
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S. Dutta, Q. Fan
expected payoff of the firm, as well as managerial rents, becomes less sensitive to
the managers’innovation efforts. These three factors combine to make competitive
innovation more attractive to firms under conditions of high uncertainty in investment
returns.
Our finding that intra-firm competition weakens innovation incentives is consistent
with the empirical findings of Seru (2014), who documents that single-divisional firms
tend to be more innovative than multi-divisional firms. It also lends theoretical support
to the practice of firms spinning off parts of their businesses, based on the argument that
such spinoffs foster corporate entrepreneurship and innovation (see, e.g., de Cleyn and
Braet 2010). Our analysis further predicts that firms that rely heavily on innovation are
more likely to create spinoffs and that such spinoffs will enhance value. This prediction
is consistent with empirical evidence showing that the market reaction to spinoffs is
generally positive (see, e.g., Daley et al. 1997; Desai and Jain 1999; Krishnaswami
and Subramaniam 1999).
Our paper relates to the work of Piccione and Tan (1996), who examine a model in
which agents invest to stochastically reduce their production costs before competing
for a procurement contract from a buyer. Unlike our model, their initial focus is on a
setting where the buyer can precommit to a procurement contract before the invest-
ment stage (that is, before agents learn their types). They demonstrate that in this
post-contract private information setting, standard auction mechanisms can be used
to extract all surplus from the agents, resulting in a first-best outcome for the buyer.
They then explore an alternative commitment scenario in which the buyer commits to
a procurement mechanism after the investment stage, which parallels the commitment
scenario in our study. However, in contrast to our findings, they conclude that com-
petition is unequivocally beneficial to the buyer. As discussed in Section 4, Piccione
and Tan (1996) reach this result under restrictive assumptions about innovation tech-
nology, which are not applicable in our model. Instead, we develop a tractable model
of innovation that not only identifies the specific conditions under which either of the
two intra-firm competition scenarios (i.e., competitive or exclusive) emerges as the
optimal choice, but also links these conditions to observable firm characteristics, such
as innovation uncertainty and sensitivity to innovation efforts.
Gershkov et al. (2021) also explore an auction setting in which agents can exert
efforts to influence their private valuations. However, their model differs from ours
in two key ways. First, in our model, the principal commits to a contract only after
the agents have already exerted their innovation efforts, leading to a holdup problem
concerning these initial efforts. In contrast, Gershkov et al. (2021) assume that the
principal can precommit to an auction mechanism before the agents exert their efforts,
thereby avoiding the holdup problem that is central to our analysis. Secondly, in our
paper, as well as in Piccione and Tan (1996) and Jeitschko and Wolfstetter (2000),
innovation is modeled as a stochastic process, where innovation efforts shift the dis-
tributions of types and are made before the agents possess any private information.
In contrast, following the standard model of Laffont and Tirole (1993) and Gershkov
et al. (2021) assume that agents exert efforts after learning their types, and these efforts
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Innovation incentives and...
affect their private valuations deterministically.2Their analysis departs from the stan-
dard model and becomes more complex due to the assumption that the principal can
only elicit the agents’ private information after they have exerted their efforts.
In our model, information rents incentivize managers to exert innovation efforts,
a concept similarly emphasized by Arya et al. (2000), Che et al. (2021), and Dutta
and Fan (2012). Arya et al. (2000) demonstrate that a firm may prefer to implement a
less detailed information system to mitigate the holdup problem related to the agent’s
project search efforts. Dutta and Fan (2012) argue that a firm could opt to delegate
investment decision rights to its divisions, thereby committing to avoid centralized
monitoring of divisional investment opportunities. Aghion and Tirole (1997) reach a
similar conclusion within an incomplete contracting framework. A common theme in
the works of Aghion and Tirole (1997)), Arya et al. (2000), and Dutta and Fan (2012)
is that excessive information can disadvantage a principal with limited commitment
capabilities. However, unlike our study, these papers do not explore the impact of
competition on incentives, as they all investigate single agent settings. In contrast to
these works and to our study where innovation outcomes are agents’ private informa-
tion, Che et al. (2021) assume that innovation outcomes are public and contactable
information.
The idea that competition fosters innovation is prevalent in practitioner literature
and is a key theme in the research contest literature, which examines how to best
structure competition to stimulate innovation.3Unlike our model, where innovators
have private information about the value of their projects, most studies in the research
contest literature assume that innovation outcomes are public (albeit unverifiable)
information. Although unlimited competition in the form of free and open entry of
suppliers is generally suboptimal (e.g., Taylor 1995; Fullerton and McAfee 1999;Che
and Gale 2003), some competitive pressure among suppliers generally benefits the
procurer. Without competition, a single supplier could win the contest by remain-
ing idle, thus having no incentive to exert any research effort. In contrast, our study
demonstrates that an exclusive organizational structure—where only one division is
permitted to innovate—always induces more innovation than a competitive structure
and, under many plausible conditions, results in higher expected profits for the firm.
Our paper is also related to the works of Baldenius et al. (2007), Inderst and
Laux (2005), and Stein (1997). Similar to our model, these studies assume that firms
are resource-constrained and therefore cannot undertake all positive npv projects.
Baldenius et al. (2007) characterize optimal capital budgeting mechanisms for such
environments in a dynamic setting. However, unlike our paper, Baldenius et al. (2007)
assume that project payoff distributions are exogenous and do not explore how inno-
vation influences investment opportunities. Although Inderst and Laux (2005) and
2Thus, agents’ effort choices depend on their types in Gershkov et al. (2021). Specifically, they show that
optimal mechanisms exclude types below a certain threshold from the auction: such types do not exert effort
or submit bids, since they cannot win. Moreover, the deterministic technology model of Gershkov et al.
(2021) seems more descriptive of routine production than of innovation.
3A frequently expressed viewpoint in the practitioner literature is that intra-firm competition fosters innova-
tion. For instance, Hughes et al. (2021) observe: “Vigorous competition between business units or technical
teams can be a valuable means of accelerating innovation and an excellent way to hedge bets.”
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S. Dutta, Q. Fan
Stein (1997) also model resource-constrained firms, their research focus and model-
ing choices differ significantly from ours. Specifically, these studies examine the role
of corporate headquarters in reallocating scarce capital across competing divisions
and raising external funds. In contrast to our model, in which innovation creates pri-
vate information for divisions, corporate headquarters and divisions are symmetrically
informed in their models.
The remainder of the paper proceeds as follows. Section 2describes the model.
Section 3characterizes the equilibrium investment and innovation decisions for
exclusive and competitive innovation settings. Section 4compares the performance of
competitive and exclusive organizational designs and identifies sufficient conditions
for each of these designs to emerge as optimal. Section 5investigates several exten-
sions of our base model. Section 6discusses the empirical implications. Section 7
concludes.
2Model
We consider a one-period model of a risk-neutral firm consisting of two divisions.4
Each division is managed by a risk-neutral manager and can engage in innovation
activities to create a potentially profitable investment project. We describe the specifics
of the relation between innovation and project profitability later in this section.
For simplicity, we focus on a setting in which the divisional investmentopportunities
are ex ante identical. For each i∈{1,2}, the investment project of division i,if
undertaken, requires an initial cash expenditure of kdollars at the beginning of the
period and generates a gross cash flow of k+vidollars at the end of the period. Project
i’s net cash flow is thus given by vi. Without loss of generality, we normalize the firm’s
cost of capital to zero, and hence vialso measures the project i’s npv.
Let sidenote the compensation of the manager i. The earnings of the division i(or
net cash flows) are then given by:
πi=vi·Ii+yi−si−bi,(1)
where Ii∈{0,1}is an indicator variable that denotes whether division i’s project
is undertaken and yidenotes earnings from the division’s existing assets. While the
project value viis the division’s private information, yiis a commonly known param-
eter which we normalize to zero for brevity.
To model intra-firm competition for scarce resources, we assume that the firm can
undertake at most one of the two divisional projects because it has limited financial or
human capital resources. Hence, a feasible investment policymust satisfy the constraint
that 2
i=1Ii≤1. One interpretation of this assumption is that the firm’s capital cost
would increase sufficiently with additional investments such that the additional project
payoffs could not cover the higher cost of capital. Another interpretation is that the firm
has only a limited supply of skilled human resources needed to implement investment
projects. Baldenius et al. (2007), Inderst and Laux (2005), and Stein (1997) examine
4We show in Section 5.1 that our findings readily generalize to settings with more than two divisions.
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Innovation incentives and...
similar models of resource-constrained firms that are unable to invest in all positive
npv projects.
The firm observes the aggregate earnings in Eq. (1), but not its individual com-
ponents. In Eq. (1), bi≥0 denotes the amount of divisional resources that the
manager diverts for private benefits (e.g., consumption of perquisites). If the project is
undertaken (i.e., Ii=1), managerial consumption of perquisites cannot be verifiably
separated from the regular operating expenditures because viis the manager’s private
information. In contrast, the manager’s diversion of corporate resources for personal
benefits can be directly detected when the project is not undertaken (i.e., Ii=0), since
the earnings from the division’s existing assets are commonly known. The inability to
distinguish between regular operating costs and managerial diversion of resources for
private benefits, combined with the divisional managers’ private information, creates
an agency problem that prevents the firm from achieving the first-best investment out-
comes.5In particular, the managers have incentives to consume perks and then ascribe
the resulting poor performance to low investment payoffs.
Next, we specify how innovation affects divisional investment opportunities. Each
divisional manager can undertake a personally costly and unobservable effort ei≥0
to develop her division’s investment opportunity. For each i∈{1,2}, manager i’s
personal cost of innovation effort eiis denoted by c(ei). The effort cost function c(·)
is strictly increasing and convex, and such that c(0)=c(0)=0 and c(ei)→∞as
ei→∞.
For each i∈{1,2}, the project value viis drawn from [v, ¯v]according to a cumula-
tive distribution, F(·|ei).Let f(·|ei)denote the corresponding density function, which
is strictly positive over the support [v,¯v]with v≤0 and ¯v>0. We allow the support
to extend over the entire real line (i.e. v→−∞and ¯v→∞). The values of the two
divisions’ projects, v1and v2, are assumed to be stochastically independent. To model
that an increase in eiresults in a more “favorable” distribution of vi, we assume that
the inverse hazard rate:
H(vi|ei)=1−F(vi|ei)
f(vi|ei)
increases in ei. That is, He(vi|ei)>0 for all vi∈[v, ¯v]and ei≥0. This monotone
hazard rate property implies that Fe(vi|ei)<0 for all vi∈[v, ¯v]and ei≥0; that
is, innovation effort eishifts the return distribution F(·|ei)in the first order stochastic
dominance sense.6We note that He(·|ei)>0 is a less restrictive condition than the
usual assumption of monotone likelihood ratio property.7We also assume that the
usual monotone inverse hazard rate condition in viholds (i.e., Hv(vi|ei)<0) and
innovation has diminishing marginal returns (i.e., Fee(·)≥0).
5The possibility of excessive perk consumption allows us to introduce a conflict of interests between the
firm and the divisions in a simple manner. Without affecting the main insights of our analyses, however,
such agency conflicts can also be modeled to arise for other reasons, such as divisional managers shirking
on productive efforts or spending excessively on their pet projects.
6Note that dln[1−F(vi|ei)]/dvi=−1/H(vi|ei), and hence 1−F(vi|ei)=exp −vi
vH(v|ei)−1dv.
Differentiating with respect to eiyields Fe(vi|ei)<0 because H(·|ei)is increasing in ei.
7See Poblete and Spubler (2012) for an application of this assumption in a moral hazard setting.
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S. Dutta, Q. Fan
The firm seeks to maximize its expected profits 2
i=1πiwith divisional net earnings
πias given by Eq. (1). For each i∈{1,2},let
Ui=si+wbi
denote manager i’s utility gross of the cost of innovation effort. Here, w∈(0,1)
denotes the compensation that the manager will accept in exchange for giving up a
dollar of perk consumption. The assumption w<1 reflects that each manager assigns
a higher value to direct compensation, which allows her unrestricted consumption
choices, than to an equivalent amount of divisional resources consumed as perquisites.
Each manager’s reservation utility is normalized to zero without loss of generality,
and hence the participation constraints at the contracting date require that Ui≥0.
Manager i’s net utility (i.e., utility net of the cost of effort) is given by Ui−c(ei)=
si+wbi−c(ei).
The sequence of events is illustrated in Fig. 1. Initially, the firm commits to an orga-
nization structure that determines the level of intra-firm competition. In the exclusive
innovation structure, the firm eliminates competition between divisions by designat-
ing one division as the sole innovator, granting it exclusive rights to develop and
seek approval for new investment projects. In contrast, in the competitive innovation
structure, both divisions are allowed to engage in innovation.
We argue that the choice of organizational design offers a credible way for the firm
to precommit to a level of intra-firm competition. Organization design tends to be a
long-term choice that is difficult to revise frequently. For instance, many technology
companies have established separate “innovative” units (e.g., Google X and Amazon
Lab126) with dedicated resources to shield them from the competitive pressures of the
core business. Additionally, firms often spin off parts of their business into separate
entities.
After observing the choice of the organization structure of the firm, each division
chooses a level of project innovation activity ei≥0 and privately learns the value of
its project vi. Subsequently, the firm contracts with the managers. For each manager,
the contract specifies a compensation rule si(·), as a function of the managers’ reports
about their projects and divisional earnings, and an investment decision rule Ii(·),as
a function of the managers’ reports. We discuss the contracting stage in more detail in
the next sections. Finally, investment decisions are implemented, earnings are realized,
and managers are paid according to the contracts in place.
Although we examine a reduced-form, one-period model, the incentive problems
modeled here are likely to span multiple time periods in most contexts. Innovation
activities, such as the introduction of a new product, often begin years in advance
of their eventual implementations. A complete long-term contract in such contexts
Fig. 1 Sequence of Events
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Innovation incentives and...
would require the firm to precommit to the precise ways in which investment projects
will be evaluated for all future periods. For example, we show later that a higher
hurdle rate for investment dampens ex ante innovation incentives. Consequently, if
the firm could precommit to an investment decision rule for the long term, it would
sometimes be optimal for the firm to commit to a hurdle rate lower than its cost of
capital. However, such an ex ante commitment to destroy value ex post seems neither
credible nor descriptive of practices in most firms.
Instead, investment policies are frequently revised in light of new information and
managerial compensation plans are often governed by short-term contracts. To capture
this limited commitment ability in our reduced-form, one-period model, we assume
that the firm commits to compensation and investment rules after the completion of
the innovation phase. Thus, the firm makes its compensation and investment choices
to maximize its ex post payoffs, taking the managers’ earlier innovation effort choices
as sunk and given. This opportunistic behavior by the firm creates a holdup problem
concerning the managers’ ex ante choices of innovation efforts.8
3 Equilibrium investment and innovation decisions
We first characterize the equilibrium for the exclusive innovation case in which the
firm eliminates interdivisional competition by preselecting one of the two divisions as
the “innovative” type. Such an asymmetric organization design not only serves as a
useful benchmark but also, as we demonstrate later, emerges as optimal under certain
circumstances.
3.1 Exclusive innovation
Since the two divisions are ex ante symmetric, it does not matter which division is
designated as the innovative type. For brevity, we therefore drop the subscript “i”
from the variables in this subsection. The division with exclusive innovation rights
chooses its innovation level anticipating the firm’s subsequent choice of investment and
compensation rules. At the contracting stage, the firm takes the division’s innovation
choice as given and chooses an incentive scheme that is sequentially optimal; that is,
it maximizes the firm’s ex post expected profit.
To characterize the equilibrium innovation effort and the investment decision rule,
we work backward and first solve for the firm’s optimization problem for a given
level of innovation effort e. By the revelation principle, the firm’s problem can be
formulated as that of choosing a mechanism that induces truthful reporting. The usual
methodology to solve this standard adverse selection problem yields the following
result:
Lemma 1 In the exclusive innovation setting, the project is undertaken if and only if
its npv, v, exceed some hurdle value h >0. For a given e, the optimal hurdle for
8Arya et al. (2000), Baiman and Rajan (1995), and Dutta and Fan (2012) examine similar holdup problems
arising from limited commitment ability in single-agent settings.
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S. Dutta, Q. Fan
investment h is given by:
h(e)=wH(h|e). (2)
If the project is undertaken, the manager earns information rent with type v manager’s
rent given by
U(v) =wmax{v−h,0}.(3)
Proof All proof are in the appendix.
The manager has a natural incentive to understate the project’s payoff prospects,
which allows her to consume perks and then ascribe the resulting poor earnings to low
investment returns. To counteract these incentives for inefficient perk consumption,
the firm must provide the manager with information rent if the project is implemented.
Equation (3) shows that the manager’s utility exceeds her reservation utility of zero
whenever the project is undertaken (that is, U(v) > 0ifv>h). Consistent with the
findings of Antle and Eppen (1985) and Harris et al. (1982), the optimal threshold
for investment hexceeds its first-best value of zero; that is, the firm forgoes some
marginally profitable projects in order to limit managerial rents.
Information rents, though necessary for eliciting managers’ private information,
are often viewed as private benefits that self-interested managers earn at the expense
of shareholders. In our model, however, these rents also serve an important incentive
role by motivating managers to exert innovation efforts. We now characterize the
equilibrium choice of innovation. The manager makes her ex ante innovation choice
anticipating the firm’s subsequent choice of incentive scheme and investment decision
rule, as characterized in Lemma 1.
Specifically, the manager chooses her innovation effort to maximize E[U(v)]−c(e),
where E[U(v)]is the manager’s expected information rent and c(e)is the cost of
innovation effort. The expected information rent, wE[max{v−h,0}], can be expressed
as follows:
E[U(v)]=w¯v
h
[1−F(v|e)]dv. (4)
The following first-order condition characterizes the manager’s optimal choice of e:
c(e)=w¯v
h
−Fe(v|e)dv. (5)
While the right-hand side of the above equation measures the marginal benefit of
innovation, the left-hand side reflects its marginal cost. At the optimum, the manager
equates the marginal benefit of innovation with its marginal cost. Equation (5) thus
defines the manager’s optimal response e(h)as a function of the firm’s choice of
investment hurdle value h.
A pure strategy equilibrium e0,h0is given by a point at which the manager’s
response function e(h), as defined by Eq. (5), intersects with the firm’s response
function h(e), as defined by Eq. (2). That is, e0=e(h0)and h0=h(e0).
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Innovation incentives and...
Proposition 1 In the exclusive innovation setting, the equilibrium levels of innovation
e0and investment hurdle value h0are given by the unique solution to the following
equations:
c(e0)=w¯v
h0
−Fe(v|e0)dv, (6)
h0=wH(h0|eo), (7)
Furthermore, e0>0and h0∈(0,¯v).
The equilibrium is unique because the reaction curve slopes upwards for the firm and
downwards for the manager. The manager’s reaction curve slopes downward because
the marginal information rent, as given by the right-hand side of Eq. (5), decreases in
the hurdle value h. To understand why the firm’s reaction curve h(e)slopes upward,
note that the type vmanager’s rent in Eq. (3) can be viewed as the manager receiving
w“call options” on the investment payoff, each with a strike price of h. Since the value
of a call option increases in the expected value of the underlying asset, the manager’s
information rent increases in the expected npv of the project. An increase in the value
of eincreases the project’s expected npv, and hence the firm optimally adjusts the
investment hurdle value hupwards to curtail managerial rent.
3.2 Competitive innovation
We now investigate the alternative organization design in which both divisions are
permitted to innovate, but the firm can undertake at most one of the resulting projects.
As before, each division chooses its innovation activity eirationally anticipating the
firm’s subsequent choice of incentive schemes and investment decision rules. Taking
the managers’ earlier choices of innovation efforts e1and e2as given, the firm chooses
sequentially optimal incentive schemes at the contracting stage.
We first characterize the sequentially optimal compensation and investment rules.
In the competitive setting, a direct revelation mechanism specifies each manager’s
compensation si(˜v1,˜v2), the investment decision rule Ii(˜v1,˜v2), and the performance
target ˜vi·Ii(˜v1,˜v2), all as functions of the two managers’ reports ˜v1and ˜v2.Given
truthful reporting by the other manager, let Ui(˜vi,v
i|vj)represent manager i’s utility
contingent on reporting ˜viwhen her true type is vi. That is,
Ui(˜vi,v
i|vj)=s(˜vi,vj)+w·b(˜vi,v
i|vj),
where b(˜vi,v
i|vj)=max{0,(v
i−˜vi)Ii(˜vi,vj)}is the amount of divisioinal resources
that type vimanager can divert for personal consumption and yet deliver perfor-
mance consistent with her report ˜vi.LetUi(vi,vj)≡U(vi,v
i|vj)and bivi,vj≡
bi(vi,v
i|vj)denote the manager’s utility and perk consumption, respectively, when
she reports her private information truthfully. The firm’s optimization problem can
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S. Dutta, Q. Fan
then be written as follows:
max
{Ii(·),bi(·),si(·)}E2
i=1vi·Ii(vi,vj)−si(vi,vj)−bi(vi,vj)
subject to:
Ej[Ui(vi,vj)]≥Ej[Ui˜vi,v
i|vj]for each iand all viand ˜vi,(IC)
Ej[Ui(vi,vj)]≥0 for each iand all vi,(PC)
2
i=1
Ii(vi,vj)≤1,(RC)
Here, Ej(·)denotes the expectation over vjand E(·)represents the expectation over
both viand vj. The incentive compatibility constraints in (IC) ensure that truthful
reporting constitutes a Bayesian Nash equilibrium. The participation constraints in
(PC) are required to hold in an interim sense; i.e., each manager must at least break
even in expectation over the other manager’s type. The resource constraint in (RC)
reflects that the firm can undertake at most one of the two projects.
As in the exclusive innovation setting, the firm faces a trade-off between curtailing
rents and maximizing investment efficiency. The proof of Lemma 2 shows that the
firm’s optimization problem simplifies to the following program:
max
{Ii(·)}2
i=1
E2
i=1
[vi−wH(vi|ei)]·Ii(vi,vj)
subject to the feasibility constraint that I1(·)+I2(·)≤1. Hence, the firm invests in
project iif and only if its virtual npv,vi−wH(vi|ei), exceeds both zero and the
virtual npv of the competing project. That is, Ii(vi,vj)=1 if and only if
vi−wH(vi|ei)>max 0,vj−wH(vj|ej).
Since the divisions are ex ante identical, we will focus on symmetric equilibria in
which the two managers exert the same level of innovation efforts.
Lemma 2 In the competitive innovation setting, project i is undertaken if and only
if vi>max{h,vj}, where h >0is given by Eq. (2). Type vimanager’s utility in
expectation over the other manager’s types is given by
Ej[Ui(vi,vj)]=wEj[max{vi−max{h,vj},0}] (8)
for each i ∈{1,2}.
For given levels of innovation efforts, Lemma 2 is essentially a one-period ver-
sion of the optimal revelation mechanism in Baldenius et al. (2007). Equation (8)
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Innovation incentives and...
shows that interdivisional competition lowers the investing division’s information rent.
Specifically, while the investing division earns w(vi−max{h,vj})under competitive
innovation, its rents would be w(vi−h)if it were the only division. Interdivisional
competition mitigates the divisions’ misreporting incentives because (i) a division can
earn rents only if its project is chosen for investment, and (ii) a project is chosen for
investment if and only if its reported value exceeds both hand the value of the compet-
ing project. This is why the investing division’s rent is proportional to the difference
between its true type viand the maximum of the investment hurdle value, h, and the
value of the competing project, vj. This rent extraction benefit of competition is anal-
ogous to a well-known result from auction theory: the seller of a good can extract a
greater fraction of trade surplus as the number of privately informed bidders increases.
We now characterize the Nash equilibria of the innovation stage game between the
firm and the divisions. The proof of Proposition 2 shows that manager i’s ex ante
expected information rent can be expressed as:
E[Ui(vi,vj)]=w¯v
h
F(vi|ej)[1−F(vi|ei)]dvi.(9)
Compared to Eq. (4) for the exclusive innovation case, the integrand on the right-hand
side of Eq. (9) contains an extra multiplicative term: namely, F(vi|ej)<1.For any
given hurdle value h, therefore, each manager earns a lower amount of expected rent in
the competitive setting than in the exclusive setting. An intuition for this is as follows.
Conditional on vi>h, the project iis approved for investment if and only if the
competing project is less profitable (that is, vj<v
i), which happens with probability
F(vi|ej). In contrast, conditional on vi>h, division i’s project would be undertaken
with probability one if it were the only division.
Taking the hurdle value hand the other manager’s innovation effort ejas given,
the manager ichooses eito maximize her expected utility net of the cost of the inno-
vation effort, E[Ui(vi,vj)]−ci(ei). The following first-order condition characterizes
manager i’s optimal effort choice eias a function of hand ej:
c(ei)=w¯v
h
−F(vi|ej)Fe(vi|ei)dvi.
As before, the right-hand side of the above equation represents the marginal bene-
fit of innovation (i.e., marginal information rent), and the left-hand side represents
the marginal cost of innovation. In a symmetric equilibrium, the common response
function of the two managers e(h)is implicitly defined by the following equation:
c(e)=w¯v
h
−F(v|e)Fe(v|e)dv. (10)
The minimum threshold for investment is still determined by the condition that the
virtual net present value of the threshold project is zero (i.e., h−wH(h|e)=0), and
hence the firm’s response function h(e)is the same as defined by Eq. (2).
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S. Dutta, Q. Fan
An equilibrium is then given by a point (ec,hc)at which the managers’ reaction
curve e(h), as defined by Eq. (10), intersects with the firm’s reaction curve h(e)in
Eq. (2). That is, ec=e(hc)and hc=h(ec). The following result characterizes the
unique symmetric equilibrium.
Proposition 2 In equilibrium, the innovation effort, ec>0, and the hurdle value for
investment, hc∈(0,¯v), uniquely solve the following equations:
c(ec)=w¯v
hc
−F(v|ec)Fe(v|ec)dv, (11)
hc=wH(hc|ec). (12)
Compared to the single-division setting, the equilibrium levels of innovationand invest-
ment hurdle are lower in the competitive setting; that is,
ec<e0and hc<h0.
A comparison of the right-hand side of Eq. (10) with the right-hand side of Eq. (5)
makes clear that for any given investment hurdle value h, the marginal benefit of
innovation is smaller in the competitive setting than in the single-division setting.
This is intuitive because competition reduces the likelihood that a given project is
undertaken and allows the firm to extract a greater share of surplus when the project is
undertaken. These two effects of competition work in the same direction and together
reduce the managers’ expected rents, and hence their marginal returns from innovation.
Therefore, for any given h, managers choose a lower level of innovation effort in the
competitive setting than in the single division setting. Since the firm’s response curve
h(e)is the same in both settings, the equilibrium level of innovation is lower with
competition than without competition (i.e., ec<e0). This result that competition
undermines innovation is consistent with a finding of Piccione and Tan (1996), who
study an auction setting in which potential bidders invest to lower their production
costs before competing for a procurement contract from a buyer.9
Furthermore, ec<e0implies hc<h0because the optimal hurdle value for invest-
ment increases in the innovation effort. Figure 2provides a graphical illustration of
this intuition. The effort response curve e(h)for competitive innovation is everywhere
below that for exclusive innovation. The firm’s response curve h(e)is the same in
both cases, and thus the intersection point of the two reaction curves for competitive
innovation is below that for exclusive innovation along both axes.
4 Competitive versus exclusive innovation
The analysis in the previous section has shown that competition for corporate resources
has a negative effect on innovation incentives. An obvious next question is whether
the firm would ever find it optimal to restrict competition by instituting the exclusive
9Section 4reviews the key modeling differences between our paper and Piccione and Tan (1996).
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Innovation incentives and...
Fig. 2 Competitive versus Exclusive Innovation
innovation organization structure at the outset. It will be instructive to first examine
this organizational design problem when there are no moral-hazard problems with
regard to innovation. In particular, suppose that project values viare drawn from some
exogenous distribution F(vi), which cannot be influenced by the innovation efforts of
the divisions. We have the following result:
Proposition 3 If there are no opportunities to innovate, the competitive organization
structure outperforms the exclusive organization structure.
In our setting with asymmetric information and limited resources for investment,
the competitive form of organizational design improves efficiency for two different
reasons. First, competition generates a sampling or winner-picking benefit for the
firm by providing it with the option to choose the best project from two independent
draws. Hence, the winning project (i.e., the one chosen for investment) is on average
more profitable in the competitive setting than in the single division setting. Second,
competition lowers the expected rent that the investing division earns on account
of its private information. This rent extraction benefit of competition results in a
higher virtual npv under competition than under no competition. These two effects
of competition work together to improve investment efficiency, and hence the firm
would always prefer the competitive organization design if there were no innovation
incentive problems.
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S. Dutta, Q. Fan
When divisions can engage in innovation activities to improve their investment
opportunities, the previous section shows that competition has a detrimental effect on
innovation incentives. An obvious question is whether and when this negative effect
of competition will be sufficiently strong to offset its winner-picking and rent-limiting
benefits.
To investigate this, we analyze a more tractable specification of our model that
allows an explicit comparison of firm profits in the two organizational forms. In
particular, we assume that the stochastic relationship between project value viand
innovation effort eitakes the following additive form:
vi=mei+σ
i,(13)
where iis a standard normal random variable. As before, project values are drawn
from stochastically independent distributions; i.e., Cov(1,
2)=0. The parameter
m>0 measures the marginal product of the innovation effort.
Equation (13) implies that viis normally distributed with mean meiand vari-
ance σ2. For a given ei, the distribution and density functions of viare given by
F(vi|ei)=vi−mei
σand f(vi|ei)=1
σφvi−mei
σ, respectively, where (·)and
φ(·)are the corresponding functions for the standard normal distribution. Since the
inverse hazard rate of the standard normal distribution, 1−(·)
φ(·), is a decreasing func-
tion, the inverse hazard rate of visatisfies our earlier monotonicity conditions; i.e.,
H(vi|ei)decreases in viand increases in ei. To guarantee that the managers’ objective
functions are concave in their innovation effort choices, we impose the following
technical requirement:
wm2max φ(·)<σ. (14)
The above condition requires that the payoff uncertainty σ2is not too small. We
note that this is a considerably weaker requirement than our earlier assumption of
diminishing marginal returns (i.e., Fee(·)≥0). We also assume that each manager’s
personal cost of innovation effort is quadratic; i.e., c(ei)=e2
i
2.
We have shown that the innovation effort in the competitive equilibrium is always
lower than that in the exclusive innovation setting. The following result shows that as
innovation efforts become increasingly more productive, the gap between equilibrium
efforts in exclusive and competitive settings widens. For sufficiently large values of
m, this gap between the equilibrium innovation efforts, and hence in the average
project qualities, more than offsets the winner-picking and rent-limiting benefits of
competition.
Proposition 4 i. Exclusive innovation dominates competitive innovation when the
marginal product of innovation, m, is sufficiently large and w≤1
2.
ii. Competitive innovation dominates exclusive innovation when the marginal product
of innovation, m, is sufficiently small.
Exclusive innovation generates stronger incentives to exert innovation effort, but
is inefficient from the ex post resource allocation perspective because it eliminates
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Innovation incentives and...
the winner-picking and rent-limitation benefits of competition. The optimal choice
of organizational design thus comes down to a trade-off between ex post allocative
efficiency, which calls for competitive innovation, and ex ante innovation incentives,
which calls for exclusive innovation.
For small values of m, innovation efforts are relatively ineffective in improving the
expected project value, and hence the ex post winner-picking and rent limiting benefits
of competition are far more critical determinants of firm value. As a result, competitive
innovation outperforms exclusive innovation for small values of m. In contrast, when
the marginal product of innovation is sufficiently large, it is more critical to motivate
innovation, and the firm prefers exclusive innovation to competitive innovation. This
later result requires that the parameter wis not too large (i.e., w≤1
2). The reason is
that the managers earn higher rents for larger values of w, and hence the rent extraction
advantage of competition becomes far more critical for larger values of w.10
Although we cannot analytically characterize the optimal organization design for
intermediate values of m, numerical simulations suggest that competitive innovation
outperforms (underperforms) exclusive innovation for values of mbelow (above) a
unique threshold. Figure 3shows the expected firm profit as a function of the marginal
product of innovation, m, for the two organizational designs when w=0.1 and
σ2=1. This figure shows that the optimal organization design entails competitive
(exclusive) innovation when the marginal product of innovation is below (above) a
unique threshold.
We next characterize how the firm’s preference between exclusive and competitive
innovation varies as investment returns become more or less uncertain. The additive-
normal specification of our model in Eq. (13) is convenient to address this question,
since it allows us to vary the variance of viwithout simultaneously altering its mean.
Proposition 5 Competitive innovation dominates exclusive innovation if project pay-
offs are sufficiently uncertain (i.e., σ2is large).
An intuitive explanation for this result is as follows. For a given hurdle value h,
the expected information rent increases in σ2. The reason is that the optimal infor-
mation rent of type vimanager is a convex function of vi, and hence the expected
information rent increases as vibecomes more volatile.11 This implies that compe-
tition makes a greater impact on limiting managerial rents when investment returns
are more volatile. Furthermore, the sampling benefit of competition also increases as
investment returns become more uncertain because the winning project’s expected
npv increases in σ2. Intuitively, choosing the best project from a draw of two projects
10 Let E(vi)and E(v) denote the expected project values in the competitive and exclusive innovation
settings, respectively. For large values of m,E(vi)≈E(v)/2 because the equilibrium innovation level is
proportional to the probability of investment, which approaches one (one-half) in the exclusive (competitive)
setting. Furthermore, for large values of m, the firm can extract almost all the managerial rents in the
competitive innovation environment, and hence the expected firm profit approaches E(vi)≈E(v)/2. The
expected firm profit always exceeds (1−w)E(v) in the exclusive innovation setting. Thus, for a largeenough
m, a sufficient condition for exclusive innovation to dominate competitive innovation is that w≤1/2.
11 Specifically, type vimanager’s information rent in Eq. (9) can be thought of as a call option on the
investment payoff vi.
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S. Dutta, Q. Fan
Fig. 3 Firm Profit and Marginal Product of Innovation
is more advantageous when project values are more uncertain. Thus, both the rent-
limiting and sampling benefits of competition increase in σ2. At the same time, the firm
becomes less concerned about innovation incentives for larger values of σ2because its
expected gross profit becomes less sensitive to the mean-shifting innovation effort. As
a result, ex post allocative efficiency concerns dominate and competitive innovation
outperforms exclusive innovation for sufficiently large values of σ2.
Figure 4illustrates this result by plotting the expected firm profits in the two orga-
nizational forms for different levels of uncertainty when w=0.1 and m=6. This
numerical analysis shows that competitive innovation dominates (is dominated by)
exclusive innovation for values of σ2above (below) a unique threshold.
To conclude this section, we relate our study to the work of Piccione and Tan
(1996), who study an auction setting wherein potential suppliers invest to reduce
their production costs before competing in a procurement auction organized by a
buyer. Unlike our study, Piccione and Tan (1996) initially assume that the buyer can
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Innovation incentives and...
Fig. 4 Firm Profit and Project Value Variability
precommit to an auction mechanism before suppliers make their investments and
become privately informed. In this post-contract private information setting, Piccione
and Tan (1996) show that standard auction mechanisms can be readily adopted to
extract all the surplus from the sellers and generate the first-best outcome for the
buyer.
In a later section of their paper, Piccione and Tan (1996) analyze a commitment
scenario similar to ours in which the buyer commits to an auction mechanism after
the innovation stage. In contrast to our result in Proposition 4, however, they find that
competition is unamiguously beneficial to the buyer. This discrepancy arises because
of differences in the way the two papers model innovation. Translated into the context
of our model, a key assumption in Piccione and Tan (1996) is that Fe(vi|ei)
F(vi|ei)increases
in ei; that is,
Fee(vi|ei)> [Fe(vi|ei)]2
F(vi|ei).(15)
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S. Dutta, Q. Fan
This requirement is even more stringent than the already restrictive cdfc condition,
and hence is unlikely to be satisfied by most innovation technologies as represented
by F(vi|ei).12
In contrast, we develop a more tractable model of innovation that allows us to
provide a richer characterization of the potential costs and benefits of intra-firm com-
petition. Specifically, to derive our results in Propositions 4 and 5, we assume that
innovation technology takes the normal additive form in Eq. (13), which satisfies nei-
ther the condition in Eq. (15) nor the cdfc assumption. However, the mild condition
in Eq. (14) is sufficient to ensure that each division’s objective function is concave
in its innovation effort. The results in Propositions 4 and 7 highlight how our mod-
eling framework allows us to identify specific conditions under which each of the
two intra-firm competition scenarios (i.e., competitive and exclusive) emerges as the
optimal choice. Furthermore, we are able to relate these conditions to observable char-
acteristics of firms such as innovation uncertainty and innovation sensitivity. We draw
empirically testable implications of these comparative statics results in Section 6.
5 Extensions
In this section, we explore three extensions of our base model. The first extension
examines the implications of increased competition for corporate resources, param-
eterized by the number of competing divisions. The second extension investigates a
setting in which project payoffs are positively correlated across divisions. In the final
subsection, we demonstrate that the firm can replicate the performance of optimal
revelation mechanisms using linear compensation rules, which also provide dominant
strategy incentives for truthful reporting.
5.1 Innovation and competition intensity
We have thus far investigated a model with two divisions. In this base model, competi-
tion intensity is essentially a binary variable (that is, either competition exists or it does
not). In this subsection, we extend our base model to allow for an arbitrary number
of divisions, enabling us to vary competition intensity more finely and investigate its
impact on equilibrium innovation and investment outcomes. We show that innovation
incentives decline monotonically as the number of divisions competing for scarce firm
resources increases.
Suppose that the firm has nsymmetric divisions with stochastically independent
investment opportunities. As before, the firm is resource constrained and can undertake
at most one project. It can then be shown that the firm will optimally invest in the
highest npv project but the “winning” (i.e., investing) division’s information rent will
be determined by the second highest npv.Letv−i=(v1,...,v
i−1,v
i+1,...,v
n)
12 To prove that competition is beneficial, Piccione and Tan (1996) also need that the distribution function
F(vi|ei)is exponentially separable. That is, there exist increasing functions G(vi)and α(ei)such that
F(vi|ei)=G(vi)α(ei)with G(v)=0andG(¯v) =1.
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Innovation incentives and...
denote the net present values of all projects other than project i, and let vh=max{v−i}
denote the highest npv of these projects.
Using the same arguments as in the two division setting, it can be shown that project
iwill be undertaken if and only if
vi>max{h,v
h}
and an incentive compatible mechanism must provide the investing division with an
interim utility of wE−i[vi−max{h,v
h}], where E−i(·)denotes the expectation over
other divisions’ types v−i. In a symmetric equilibrium (that is, ej=efor all j), the
expected information rent of the division ican be expressed as follows:
E[U(vi,v
−i)]=w¯v
h
[F(vi|e)]n−1[1−F(vi|ei)]dvi,
where [F(·|e)]n−1is the distribution function of vh. Taking the firm’s choice of
investment threshold hand other managers’ choice of innovation effort eas given,
division ichooses its innovation effort eito maximize its ex ante expected utility,
E[Ui(vi,v
−i)]−c(ei). In a symmetric equilibrium, each division’s response function
e(h)is thus given by the following first order condition:
c(e(h)) =w¯v
h
−[F(v|e)]n−1Fe(v|e)dv. (16)
The firm’s response function h(e)is the same as before and is given by Eq. (2). The
following proposition characterizes the unique symmetric equilibrium:
Proposition 6 Suppose that there are n symmetric divisions and that the firm can
undertake at most one project.
i. In equilibrium, the innovation effort, ec>0, and the hurdle value for investment,
hc∈(0,¯v), uniquely solve the following equations:
c(ec)=w¯v
hc
−[F(v|ec)]n−1Fe(v|ec)dv,
hc=wH(hc|ec).
ii. The equilibrium levels of innovation ecand the investment hurdle hcare both
decreasing in the level of competition as parametrized by n.
Division iearns information rent only if its project is undertaken, which happens
if the npv of its project exceeds the highest npv of the remaining n−1 projects
(i.e., vi≥vh). The probability of viexceeding vhis given by [F(vi|e)]n−1, which
decreases in the number of divisions. Moreover, the investing division’s rent is a
decreasing function of the second highest npv,vh, which is increasing on average in
the number of divisions. Combined together, these two effects imply that the expected
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S. Dutta, Q. Fan
information rent as well as the marginal rent declines in the level of competition. For
any given h, therefore, managers choose a lower level of innovation as the number
of divisions nincreases. Since the firm’s response curve in Eq. (2) does not directly
depend on n, the equilibrium level of innovation ecdecreases in n. Furthermore, a
lower level of innovation ecimplies a lower threshold for investment, hc, because the
firm’s reaction curve h(e)slopes upward.
The following result establishes that Proposition 4 also readily generalizes to the
ndivision setting. For this result, we again assume that the stochastic relationship
between project value viand innovation effort eitakes the normal additive form in
Eq. (13) and the cost of the innovation effort is quadratic; i.e., c(e)=e2
2.
Proposition 7 i. Exclusive innovation dominates competitive innovation when the
marginal product of innovation, m, is sufficiently large and w≤n−1
n.
ii. Competitive innovation dominates exclusive innovation when the marginal product
of innovation, m, is sufficiently small.
To illustrate this result, we numerically compare the firm profits under different
competition scenarios for a firm with three divisions. Let ν≤3 denote the number
of divisions that are allowed to innovate. Figure 5plots the expected profits of the
firm as a function of the marginal product of innovation mfor three possible levels of
competition: (i) full competition (that is, ν=3), (ii) partial competition (i.e., ν=2),
and (iii) exclusive innovation (i.e., ν=1) for the same parameter values as used in
generating Fig. 3. Figure 5shows that partial competition is always dominated by
full competition or no competition in this example, and full competition (exclusive
innovation) is optimal when the marginal product of innovation is below (above) a
unique threshold.
5.2 Correlated projects
Our analysis thus far has assumed that the returns from innovations are stochastically
independent across divisions. In this subsection, we examine the incentive implications
of correlation in the returns of divisional projects.
As before, suppose that the two divisions are ex ante symmetric and each project’s
net present value viis drawn from F(vi|ei). To allow for correlation between invest-
ment returns, suppose that the firm and the divisions observe the realization of a random
state variable, S∈{SC,SI}, before contracting. At the initial innovation stage, the
parties share common beliefs that the realized state will be SCwith probability ρ>0
and SIwith the complementary probability of 1 −ρ. The project values v1and v2
are drawn from perfectly correlated distributions in state SC, and from independent
distributions in state SI.
In this information structure, the parameter ρmeasures the degree of stochastic
correlation between the values of the two projects. In state SC, the divisions earn no
information rents because the firm extracts all the surplus by exploiting the correlation
between the project values (Cremer and McLean 1988).13 In state SI, each manager’s
13 With perfectly correlated types, the firm can costlessly obtain the divisions’ private information by
paying each manager zero (reservation utility) if the two reports coincide (i.e., ˜v1=˜v2) and imposing a
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Fig. 5 Firm Profit and Number of Competing Divisions
expected information rent is given by Eq. (9), and hence the ex ante expected rent
takes the following form:
E[Ui(vi,vj)]=w(1−ρ)¯v
h
F(vi|ej)[1−F(vi|ei)]dvi.
Proposition 8 The equilibrium levels of innovation and the investment hurdle decrease
in the degree of correlation between the divisional investment returns ρ.
The correlation in project payoffs negatively impacts innovation incentives by
reducing the manager’s marginal information rent. As the equilibrium hurdle value
for investment increases with project quality, the resulting lower upfront innovation
efforts are also associated with a lower threshold value for investment. Our finding
penalty on each manager if the reported values differ (i.e., ˜v1=˜v2.) Truthful reporting is then a dominant
strategy equilibrium.
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S. Dutta, Q. Fan
that innovation incentives decline with increased correlation contrasts sharply with a
well-known result from the agency theory literature on relative performance evalu-
ation, which states that incentive efficiency increases with the degree of correlation.
This discrepancy arises because, unlike our model where innovation outcomes are
the divisions’ private information, outputs are commonly observed and verifiable in
standard moral-hazard settings. In such settings, incentive efficiency improves with
higher correlation, as it enables the principal to base each agent’s rewards on a less
noisy performance measure.
An obvious question is how the firm’s preference between competitive and exclusive
structures is affected by correlation. Numerical simulation shows that the impact of
the correlation is generally ambiguous. Intuitively, the benefit of choosing a winner
from a draw of two projects is highest when the project values viand vjare distributed
independently. This winner-picking benefit decreases in the degree of correlation and
vanishes entirely when the project values become perfectly correlated (i.e., ρ=1).
In the absence of innovation concerns, therefore, correlation can either strengthen
or weaken the firm’s preference for competition depending on which one of the two
benefits of competition, winner-picking or rents exaction, is more important. This
makes it difficult to draw general conclusions on how correlation affects the firm’s
preference between the two organizational designs.
5.3 Delegated incentive schemes
The optimal revelation mechanism in Lemma 2 is derived subject to the constraints
that each manager’s (i) truthful reporting incentives constitute a Bayesian Nash equi-
librium, and (ii) participation constraints hold on an interim basis (i.e., in expectation
over the other manager’s types). In this subsection, we show that the performance
of these optimal revelation mechanisms can be attained by linear compensation con-
tracts that provide dominant strategy incentives for truthful reporting and ensure ex
post satisfaction of the agents’ participation constraints.
Suppose investment decisions are delegated to the managers and each manager is
compensated on the basis of a linear contract of the following form:
si=αi+βi·RI
i,(17)
where αiis fixed salary, βi>0 is a bonus parameter, and RI
iis division i’s residual
income, which is defined as operating income less a charge for capital. Since operating
income equals viIi−bi, residual income is given by
RI
i=viIi−bi−rikI
i,
where ri≥0 denotes the capital charge rate applied to the initial investment amount of
kdollars. For setting divisional capital charge rates, the firm can rely on the divisions’
reports about the values of their projects.
Suppose that the capital charge rate for the division iis set equal to 1
kmax{h,vj}
where vjis the value reported by the other division. Type vimanager will then derive
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Innovation incentives and...
the following amount of utility from the compensation rule in Eq. (17):
Ui(vi,v
−i)=αi+βi·[(vi−max{h,vj})·Ii−bi]+w·bi.(18)
Under this delegated incentive scheme, each manager will find it optimal to choose
Ii=1 if and only if vi>max{h,vj}. Furthermore, the managers will have no
incentives to consume excess perks if each bonus coefficient βiis set equal to w.
Setting αiequal to the manager’s reservation utility then ensures that each manager’s
interim utility is the same as in the optimal revelation scheme. Hence, the linear
compensation rule in Eq. (17) replicates the performance of the optimal revelation
mechanism in Lemma 2. Furthermore, since the utility in Eq. (18) does not depend
on a manager’s own report, it is a dominant strategy for each manager to report her
private information truthfully. We summarize the preceding discussion as follows:
Proposition 9 The linear compensation scheme in Eq. (17) generates optimal incen-
tives (i) the capital charge rate riis set equal to 1
kmax{h,vj}, and (ii) the bonus rate
βiis set equal to w, for each i. Furthermore, Eq. (17) provides dominant strategy
incentives for truthful reporting.
For exogenously given innovation efforts e1and e2, the above delegated incentive
scheme is essentially a one-period version of the optimal incentive contract based on
residual income as derived by Baldenius et al. (2007) in their dynamic setting.14 This
linear compensation scheme based on residual income with the capital charge rate of
1
kmax{h,vj}ensures that Ui(vi,vj)≥0 for each viand vj. Thus, each manager’s
participation constraints are satisfied on an ex post basis; i.e., manager i’s participation
constraints hold point-wise for each value of vj(rather than in expectation over vj).
6 Empirical implications
Taken together, our analysis generates several interesting empirical predictions. First,
Propositions 2 and 6 show that intra-firm competition is unambiguously detrimental to
innovation incentives (i.e., ec<e0). This prediction aligns with the empirical findings
of Seru (2014), which indicate that single-divisional firms are more innovative than
multi-divisional firms. Spin-offs, where parent companies separate parts of their busi-
nesses into new entities, are often considered effective means of fostering corporate
entrepreneurship and innovation (e.g., de Cleyn and Braet 2010). Many firms have
adopted this strategy of creating separate entities for innovation. Our results in Propo-
sitions 2 and 6 offer theoretical support for such spin-off practices and the argument
that they improve innovation incentives.
Second, Propositions 4 and 7 predict that firms whose values depend more critically
on innovation are more likely to find it optimal to insulate their innovation activities
from intra-firm competition. Instead of altering their internal organizational structures
to limit intra-firm competition, firms sometimes also shield their innovation activities
by spinning off their innovations into separate entities. Our analysis predicts that
14 Dutta and Reichelstein (2002) derive a similar result in a single division setting.
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S. Dutta, Q. Fan
such spin-offs are more likely to be undertaken by firms that depend more critically
on innovation and that these spin-offs will enhance their value. This prediction is
consistent with empirical evidence showing that the market reaction to spin-offs is
positive (e.g., Daley et al. 1997; Desai and Jain 1999; Krishnaswami and Subramaniam
1999).
Finally, Proposition 5 establishes that when innovation outcomes are more uncer-
tain, competitive innovation is more likely to outperform exclusive innovation.
Although we are not aware of any studies directly investigating how innovation uncer-
tainty affects the relationship between firm value and intra-firm competition, some
work has been done in the research contest literature. Many firms and public sector
organizations use contests to procure innovative solutions to their specific problems
from external agents (i.e., contestants). In an empirical study, Boudreau et al. (2011)
investigated how the performance of innovation contests varies with the number of
agents participating in them. Consistent with our result in Proposition 5, they find that
more contestants lead to better outcomes for the contest-sponsoring firm, but only
when the problems are highly uncertain.
7 Conclusion
We have examined how competition for scarce corporate resources affects innovation
incentives in a multi-divisional firm, and, in turn, how this relationship affects the firm’s
preference for either encouraging or limiting inter-divisional competition. Although
intra-firm competition is beneficial to the firm from an ex post resource allocation
perspective, it has a detrimental effect on the divisions’ ex ante innovation incentives.
From a resource allocation perspective, competition is beneficial for two reasons. First,
competition generates multiple investment opportunities, allowing the firm to select
the most promising ones for implementation. This winner-picking option is especially
advantageous when the firm can undertake only a limited number of projects. Second,
competition allows the firm to curtail the amount of rent that the divisions earn due to
their private information.
However, this rent-extraction benefit of competition becomes a cost from the per-
spective of ex ante innovation incentives. The reason is that innovation outputs are
the divisions’ private information and hence cannot be used to motivate innovation.
Instead, innovation incentives are entirely dependent on future information rents. Since
competition lowers the divisions’ marginal rents, and hence their marginal returns from
innovation, the equilibrium level of innovation declines with the degree of intra-firm
competition. The firm’s preference for limiting or encouraging interdivisional compe-
tition thus comes down to a fundamental tradeoff between ex post allocative efficiency,
which calls for competitive innovation, and ex ante innovation incentives, which call
for exclusive innovation. Our analysis identifies plausible conditions under which
each of these two organizational designs (i.e., competitive and exclusive innovation)
emerges as the optimal choice.
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Appendix
Proof of Lemma 1A direct mechanism specifies the manager’s compensation s(˜v),
investment decision rule I(˜v), and performance target ˜vI(˜v), all as functions of the
the manager’s report ˜v.LetU(˜v, v) =s(˜v) +w·b(˜v, v) denote the utility of type v
manager when she reports ˜v, where b(˜v, v) =max{0,(v −˜v)I(˜v)}is the maximum
amount of corporate resources that type vmanager can divert for personal consumption
and yet deliver performance consistent with her report ˜v.
Let U(v) ≡U(v, v) and b(v) ≡b(v , v) denote the manager’s utility and perks,
respectively, when she reports her private information truthfully. The firm’s optimiza-
tion problem can then be expressed as follows:
max
{I(·),b(·),s(·)}E[v·I(v) −s(v) −b(v)]
subject to:
U(v) ≥U(˜v, v) for all vand ˜v, (IC)
U(v) ≥0 for all v. (PC)
The incentive compatibility constraints in (IC) ensure that the manager finds it in
her self-interest to report truthfully. The participation constraints in (PC) guarantee
that the manager will earn at least her reservation utility, which we normalize to zero
without loss of generality.
An incentive compatible mechanism must satisfy the “local” first-order condition
that dU(v)
dv=∂
∂vU(˜v, v)˜v=v.Substituting b(˜v, v) =[v−π(˜v)]I(˜v) in the expres-
sion for U(˜v, v) and differentiating with respect to vgives ∂
∂vU(˜v, v)|˜v=v=wI(v).
Combined with the fact that the participation constraint U(v) ≥0 will optimally hold
with equality for the lowest type v, this local first-order condition implies
U(v) =wv
v
I(u)du,
Integrating by parts yield the following expression for the manager’s expected utility:
E[U(v)]=w¯v
v
[1−F(v|e)]·I(v)dv. (19)
Note that U(v) =s(v) +w·b(v). The firm will optimally induce b(v) =0 for all
vbecause the manager derives a lower utility from consuming perks bthan the same
amount of unrestricted compensation s(i.e., w<1). Substituting E[s(v)]=E[U(v)]
into the firm’s objective function and using Eq. (19) yield the following unconstrained
optimization problem:
max
I(v) ¯v
v
[v−w·H(v|e)]I(v) f(v|e)dv.
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S. Dutta, Q. Fan
By point-wise maximization, it follows that the firm will choose I(v) =1 if and only
if v≥h, where the threshold has a function of eis given by the unique solution to
the following equation:
h(e)=w·H(h|e).
It remains to be verified that the above incentive scheme is globally incentive com-
patible, which requires that ∂
∂vU(˜v, v) is (weakly) increasing in ˜v. For the above
mechanism, we note that ∂
∂vU(˜v, v) =wI(˜v), which is increasing in ˜vsince the
optimal I(·)is an upper-tail investment policy.
Proof of Proposition 1Asafunctionofh, the manager’s optimal response e(h)is given
by the necessary first-order condition in Eq. (5), which is also sufficient given the
assumption that Fee ≥0.
To prove this result, we show that there exists a unique point (e0,h0)at which
the firm’s response function h(e)in Eq. (2) intersects with the manager’s response
function e(h)in Eq. (5). That is, e0=e(h0)and h0=h(e0).
Implicitly differentiating the firm’s optimal response function in Eq. (2) yields
dh
de =wHe(h|e)
1−Hv(h|e).
Since Hv(v|e)<0 and He(v|e)>0, the firm’s optimal response function is upward
slopping; i.e., dh
de >0. Implicitly differentiating the manager’s optimal response func-
tioninEq.(5)gives
de
dh =wFe(h|e)
c(e)+w¯v
hFee(v|e)dv
.
The numerator of the above fraction is strictly negative because Fe(·)<0 for all
h∈(v,¯v) and all e≥0. The denominator is strictly positive because c(·)>0 and
Fee(·)≥0. It thus follows that the manager’s optimal response function is downward
sloping; i.e., de
dh <0.
An equilibrium is the point e0,h0at which the manager’s response function e(h)
intersects with the firm’s response function h(e). There exists a unique h0∈(v,¯v)
and e0>0 because (i) the response function e(·)is strictly decreasing and h(·)is
strictly increasing, (ii) e(v)>0 and e(¯v) =0, and (iii) h(0)>0.
Proof of Lemma 2The participation constraint Ej[Ui(vi,vj)]≥0 will hold with
equality for the lowest type vi=v. This boundary condition combined with the
fact that bi(˜vi,vj|vi)=(vi−˜vi)·I(˜vi,vj)and the “local” incentive compatibility
condition implies that manager iwill earn the following interim utility:
Ej[Ui(vi,vj)]=w¯v
vvi
v
Ii(u,vj)duf(vj|ej)dvj.(20)
Taking expectation over vi, we get the following expression for the expected utility:
E[Ui(vi,vj)]=w¯v
v¯v
v
(vi,vj)f(vi|ei)dvif(vj|ej)dvj,
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Innovation incentives and...
where (vi,vj)≡vi
vIi(u,vj)du.
Evaluating the inner integral by parts and using the fact that ∂(vi,v j)
∂vi=Ii(vi,vj)
yield the following expression for manager i’s ex ante expected utility:
E[Ui(vi,vj)]=w¯v
v¯v
v
H(vi|ei)Ii(vi,vj)f(vi|ei)f(vj|ej)dvidvj
=EH(vi|ei)Ii(vi,vj),(21)
where H(·)=1−F(·)
f(·)denotes the inverse hazard rate. Since w<1, the firm will
optimally set bi(vi,vj)=0, and hence E[si(vi,vj)]=E[Ui(vi,vj)]for all viand
vj. Substituting this into the firm’s objective function and using Eq. (21) yield the
following optimization problem:
max
{Ii(·)}2
i=1
E2
i=1
[vi−w·H(vi|ei)]·Ii(vi,vj)
subject to the feasibility constraint that I1(·)+I2(·)≤1. Pointwise maximization
yields that Ii(vi,vj)=1 if and only if
vi−wH(vi|ei)>max 0,vj−wH(vj|ej).
Given that the divisions are ex ante identical, we focus on a symmetric equilibrium
in which the two divisions choose the same level of innovation efforts; i.e., ei=ej.
Hence, the above investment decision rule simplifies to the rule that Ii(vi,vj)=1if
and only if
vi>max{h,vj},
where his the optimal threshold for investment in the single division setting and
given by Eq. (2). Substituting the above investment decision rule in Eq. (20) yields
the interim expected utility Eq. (8). The above incentive scheme also meets the global
incentive compatibility requirement because
∂Ui(˜vi,vj|vi)
∂vi
=wIi(˜vi,vj),
which is weakly increasing in ˜vi, since the optimal Ii(·,vj)is an upper-tailed invest-
ment policy.
Proof of Proposition 2We first derive Eq. (9)fortheex ante expected utility. Given the
optimal investment decision rule that Ii(vi,vj)=1 if and only if vi>max{h,vj},
manager i’s interim utility in Eq. (8) simplifies to:
Ej[Ui(vi,vj)]=wh
v
(vi−h)f(v j|ej)dvj+wvi
h
(vi−vj)]f(v j|ej)dvj.
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S. Dutta, Q. Fan
Integrating the second term by parts and simplifying yield
Ej[Ui(vi,vj)]=wvi
h
F(v j|ej)dvj.
Taking expectation over vigives the following expression:
E[Ui(vi,vj)]=w¯v
hvi
h
F(v j|ej)dvjf(vi|ei)dvi.
Equation (9) follows by integrating the right hand side of the above equation by parts
as shown below.
E[Ui(vi,vj)]=wF(vi|ei)·vi
h
F(v j|ej)dvj
vi=¯v
vi=h
−w¯v
h
F(vi|ej)F(vi|ei)dvi
=w¯v
h
F(vi|ej)[1−F(vi|ei)]dvi.
Taking hand ejas given, manager ichooses eito maximize her expected utility
net of the cost of innovation, E[Ui(vi,vj)]−c(ei). Since the two divisions choose the
same level of innovation (i.e., ei=ej=e) in a symmetric equilibrium, their response
functions are identical and given by the first order condition in Eq. (10). The second
order condition holds given the assumption that Fee(·)≥0.Implicitly differentiating
Eq. (10) with respect to hyields
de
dh =wFe(h|e)F(h|e)
c(e)+w¯v
hF(vi|e)Fee(vi|e)dvi
,
which is strictly negative because Fe(·)<0, Fee(·)≥0, and c (·)>0.It thus follows
that the managers’ reaction curve slopes downward; i.e., de
dh <0. The firm’s response
function, again given by Eq. (2), slopes upward.
An equilibrium is the point (ec,hc)at which the managers’s response function
e(h)in Eq. (10) intersects with the firm’s response function h(e)in Eq. (2). That is,
hc=h(ec)and ec=e(hc). There exists a unique hc∈(v,¯v) and ec>0 because
(i) the response function e(·)is strictly decreasing and h(·)is strictly increasing, (ii)
e(v)>0 and e(¯v) =0, and (iii) h(0)>0.
Proof of Proposition 3In the absence of innovation, the optimal threshold for invest-
ment is identical in the exclusive and competitive settings (i.e., h0=hc=h). The
expected profit of the firm in the exclusive setting is given by
E(π0)=¯v
h
vdF(v) −w¯v
h
(1−F(v)]dv,
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where the first term represents the expected gross payoffs from the investment project
and the second term is the expected payoff of the manager. Integrating the first term
by parts gives
E(π0)=h[1−F(h)]+¯v
h
(1−w)[1−F(v)]dv. (22)
To calculate the expected firm profit under competition, define vH≡max{v1,v
2}.
Let G(·)=F(·)2denote the distribution function of vH. The expected firm profit in
the competitive setting can then be written as
E(πc)=¯v
h
vHdG(vH)−2E[Ui(vi,vj)],
where E[Ui(vi,vj)]is each manager’s expected rent as given by Eq. (9) and the first
term represents the expected gross payoff from the investment project. Integrating the
first term by parts and substituting for E[Ui(vi,vj)]from Eq. (9) yield
E(πc)=h1−F(h)2+¯v
h
[1+F(v) −2wF(v)]·[1−F(v)]dv. (23)
The first term on the right hand side of Eq. (23) is larger than the corresponding term
of Eq. (22) because 1 −F(h)2>1−F(h)for all h<¯v. The second term of Eq. (23)
is also larger than the second term of Eq. (22) because 1 +F(v) −2wF(v) > 1−w
for all v≤¯vand w≤1. Thus, it follows that E(π c)>E(π 0).
Proof of Proposition 4Under the additive normal information structure in Eq. (13),
note that F(vi|ei)=((vi−mei)/σ )and f(vi|ei)=1
σφ((vi−mei)/σ).Thus,it
follows that Fe(v|e)=−mf(v|e).
In the exclusive innovation setting, the optimal response of the manager satisfies
the following first-order condition:
e=wm[1−(z)],
where z≡(h−me)/σ . The second-order condition requires that
wm2
σφ(z)<1(24)
which is satisfied given Eq. (14). The unique equilibrium outcomes in the exclusive
setting are thus given by
e0=wm1−(z0)
h0=wσ(z0),
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S. Dutta, Q. Fan
where
z0≡h0−me0
σ=w(z0)−wm2
σ[1−(z0)],(25)
and (·)≡1−(·)
φ(·)denotes the invese hazard rate for the standard normal distribution,
which is decreasing.
In the competitive innovation setting, manager i’s optimal effort choice satisfies
the following first-order condition:
ei=wm∞
h
F(vi|ej)f(vi|ei)dvi.
In a symmetric equilibrium, the two managers exert the same level of innovation
effort (that is, ei=ej=e.) Integrating by parts yields e=wm[1−F(h|e)2]−
wm∞
hF(v|e)f(v|e)dv=wm1−(z)2−e, and hence
e=wm
21−(z)2,
where, as before, z≡(h−me)/σ. The second-order condition requires that
wm2
2σ(z)φ(z)<1,(26)
which is satisfied given Eq. (14). Hence, the unique symmetric competitive equilibrium
effort is given by:
ec=wm
21−(zc)2,
and
hc=wσ(zc),
where
zc≡hc−mec
σ=w zc−wm2
2σ(1−zc2). (27)
The expected profit of the firm is given by
E(π0)=h0Pr(v ≥h0)+(1−w)E[max{v−h0,0}] (28)
in the exclusive innovation setting, and by
E(πc)=hcPr(h≥zc)+(1−w)σ E[max{h−zc,0}] + wσ E[max{l−zc,0}].
(29)
in the competitive innovation setting, where h=max{1,
2}and l=min{1,
2}.
To show that exclusive innovation can outperform competitive innovation, we eval-
uate E(π0)and E(πc)for large values of m.Asmbecomes arbitrarily large, it can be
verified from Eq. (25) that z0→−∞and wm2
σφ(z0)→w. Similarly, Eq. (27) can
be used to verify that zc→−∞and wm2
2σφ(zc)→was mbecomes arbitrarily large.
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Innovation incentives and...
Hence, the second-order conditions in Eqs. (24) and (26) hold for sufficiently large
values of m.
To place a lower bound on the value of E(π 0),letψ(h0)denote the right-hand side
of Eq. (28)asafunctionofh0. Since h0>0 is the unique maximizer of ψ(h),we
have ψ(h0)>ψ(0), and therefore
E(π0)>(1−w)E[max{v, 0}] >(1−w)E(v) =(1−w)me0
As m→∞,e0=wm[1−(z0)]→wm, which implies that
E(π0)
m2>(1−w)w. (30)
To evaluate the limiting value of E(πc), note that E[max i−zc,0]→E(i)−
zcfor each i∈{l,h}and Pr[h≥zc]→1asmbecomes arbitrarily large. Substituting
these into the expression for E(π c)in Eq. (29) yields
E(πc)
m2→hc−σzc
m2−σw[E(h)−E(l)]
m2.
Note that hc−σzc
m2=w
21−(zc)2→w
2and E(h)−E(l)
m2→0, and therefore
E(πc)
m2→w
2(31)
as mbecomes arbitrarily large.
It then follows from Eqs. (30) and (31) that for sufficiently large values of m,
E(π0)
m2−E(πc)
m2>(1−w)w −w
2=w1
2−w,
which is positive for w≤1
2.Hence, exclusive innovation dominates competitive
innovation when mis sufficiently large and w≤1
2.
When m=0, competition obviously dominates no competition due to its winner-
picking and rent extraction benefits (see Proposition 3). By continuity, competitive
innovation also dominates exclusive innovation (that is, E(π0)<E(πc)) when mis
small.
Proof of Proposition 5As σbecomes arbitrarily large, the terms with σin the denom-
inators of the right hand sides of both Eqs. (25) and (27) vanish so that lim
σ→∞ z0=
lim
σ→∞ zc=z, where z>0solves
z=w (z).
Therefore, e0→wm(1−(z)) and ec→wm
21−(z)2as σbecomes arbitrarily
large.
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S. Dutta, Q. Fan
Under exclusive innovation, the expected firm profit E(π 0)is given by Eq. (28),
which can be expressed as follows:
E(π0)=h0[1−(z0)]+(1−w)σ Emax{−z0,0},
Substituting h0=σz+me0and noting that limσ→∞ z0=zand limσ→∞ me0
σ=0,
we get
lim
σ→∞ E(π0)
σ=z[1−(z)]+(1−w)E[max{−z,0}].(32)
As before, let h=max {1,
2}and l=min {1,
2}. The expected profit of the
firm in the competitive setting can then be written as:
E(πc)=hc1−G(zc)+(1−w)Emax{h−zc,0}+wEmax{l−zc,0},
where G(·)=(·)2is the distribution function of h. Since lim
σ→∞ zc=z, and
lim
σ→∞
mec
σ=0, the optimal hurdle rate hc≡σzc+mecapproaches σzas σbecomes
arbitrarily large. This implies that
lim
σ→∞ E(πc)
σ=z1−(z)2+(1−w)Emax{h−z,0}+wEmax{l−z,0}.
(33)
The first term on the right hand side of Eq. (33) is larger than the corresponding term
of Eq. (32) because 1 −(z)2>1−(z). Moreover, we note that the distribution
of hdominates the distribution of in the first-order stochastic dominance sense
because G(u)=(u)2<(u)for all u∈(−∞,∞). This implies that
Emax{h−z,0}>E[max{−z,0}],
and hence the second term of Eq. (33) is also larger than the second term of Eq. (32).
Since the third term on the right-hand side of Eq. (33) is positive, it follows that:
lim
σ→∞ E(πc)
σ>lim
σ→∞ E(π0)
σ
for all w<1. Hence, competitive innovation dominates exclusive innovation when
the payoff uncertainty σis sufficiently large.
Proof of Proposition 6Using the same arguments as in Lemma 2, it can be shown that
the “local” incentive compatibility condition combined with the boundary condition
that the participation constraint E−i[Ui(vi,v
−i)]≥0 will hold with equality for the
lowest type imply that the manager iwill earn the following amount of interim utility:
E−i[Ui(vi,v
−i)]=wE−ivi
v
Ii(u,v
−i)du.
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Taking expectation over v−iyields the following expression for the expected utility:
E[Ui(vi,v
−i)]=wEH(vi|ei)·Ii(vi,v
−i).
Using the same arguments as used in the proof of Lemma 2, it thus follows that the
division i’s project is undertaken if and only if its virtual npv exceeds both zero and
the highest virtual npv of the remaining n−1 projects
vi−wH(vi|ei)>max{0,v
h−wH(vh|eh)},
where vh=max{v−i}denotes the highest npv of the remaining n−1 projects and ehis
the innovation effort choice of the division with the highest npv. Under a conjectured
symmetric equilibrium (that is, ei=efor each i), the above investment decision rule
simplifies to the rule that Ii(vi,vj)=1 if and only if
vi>max{h,v
h},
where his the optimal threshold for investment in the single division setting and given
by Eq. (2).
Since each of the n−1 components of the random vector v−iis iid with the
distribution function F(·), the distribution function of vh, the maximum value of v−i,
is given by F(·)n−1. Following the steps in the proof of Proposition 2, it can then be
verified that each manager i’s expected information rent takes the following form:
E[Ui(vi,v
−i)]=w¯v
h
[F(vi|e)]n−1·[1−F(vi|ei)]dvi.
Taking the firm’s choice of hand the other manager’s innovation choice eas given, the
manager ichooses eito maximize E[Ui(vi,v
−i)]−c(ei). In a symmetric equilibrium,
the first-order condition yields the response function in Eq. (16). The second order
condition is satisfied since Fee(·)≥0.
For any given n, an equilibrium is the point (ec,hc)at which the managers’s
response function e(h)in Eq. (16) intersects with the firm’s response function h(e)in
Eq. (2). That is, hc=h(ec)and ec=e(hc). There exists a unique hc∈(v,¯v) and
ec>0 because (i) the response function e(·)is strictly decreasing and h(·)is strictly
increasing, (ii) e(v)>0 and e(¯v) =0, and (iii) h(0)>0. This proves the first part
of the proposition.
To prove the second part, note that [F(vh|e)]n−1decreases in n, since F(vh|e)∈
(0,1)for all vh∈(v,¯v) and e. Thus, from Eq. (16) it follows that the effort response
curve e(h)shifts downward as nincreases. That is, e(h)decreases in nfor all h>0.
Since the firm’s response curve h(e)in Eq. (2) does not directly depend on n,ec
decreases in n. Furthermore, a lower level of ecimplies a lower level of hcbecause the
firm’s response function h(e)slopes upward. Therefore, ecand hcare both decreasing
in n.
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S. Dutta, Q. Fan
Proof of Proposition 7Based on the same arguments as used in the proof of Proposition
4, it can be shown that the equilibrium innovation effort ecin the competitive setting
is given by
ec=wm
n1−(zc)n
where hc=wσ(zc)is the equilibrium hurdle level for investment and
zc≡hc−mec
σ=w(zc)−wm2
nσ1−(zc)n.
As before, it can be verified that zcdecreases in mand approaches −∞ as mbecomes
large.
Let 1and 2denote the highest and second highest values of {1,
2,··· ,
n},
respectively. The expected firm profit in the competitive setting can then be expressed
as:
E(πc)=hcPr(1≥zc)+(1−w)σ E[max{1−zc,0}] + wσ E[max{2−zc,0}].
Next, we evaluate the limiting value of E(π c)as mbecomes arbitrarily large. As
before, it can be verified that wm2
nσφ(zc)→was mbecomes arbitrarily large, and
hence the second-order condition for the manager’s optimization problem holds for a
sufficiently large value of m.
Note that Emax i−zc,0 →E( i)−zcfor each i∈{1,2}and
Pr 1≥zc=(1−(zc)n)→1asmbecomes large. Hence,
E(πc)
m2→(hc−σzc)
m2−wσ[E(1)−E(2)]
m2.
Since hc−σzc
m2=w
n(1−(zc)n)→w
nand E(1)−E(2)
m2→0asmbecomes large, we
get
E(πc)
m2→w
n.
Let π0denote the expected firm profit in the exclusive innovation setting. The Eq.
(30) in the proof of Proposition 4 shows that E(π0)
m2>w(1−w) as m→∞. Thus, it
follows that for large enough m,
E(π0)
m2−E(πc)
m2>w(1−w) −w
n=wn−1
n−w≥0
for w≤n−1
n. Hence, exclusive innovation dominates competitive innovation when m
is sufficiently large and w≤n−1
n.
When m=0, competition obviously dominates no competition due to its winner-
picking and rent extraction benefits (see Proposition 3). By continuity, competitive
innovation also dominates exclusive innovation when mis small.
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Innovation incentives and...
Proof of Proposition 8As argued in the main text, each manager’s ex ante expected
utility is given by
E[Ui(vi,vj)]=w(1−ρ)¯v
h
F(vi|ej)[1−F(vi|ei)]dvi.
Hence, the equilibrium outcomes ecand hcfor the correlated competitive setting are
given by:
c(ec)=w(1−ρ)¯v
hc
−F(v|ec)Fe(v|ec)dv(34)
hc=wH(hc|ec). (35)
Implicitly differentiating Eq. (35) with respect to ρgives
dhc
dρ=wHe(hc|ec)
1−wHv(hc|ec)·dec
dρ.
Since He(·)>0 and Hv(·)<0, the above implies that dhc
dρand dec
dρare of the same
sign. That is, ecand hcboth increase or decrease in ρ.
Suppose that, contrary to the claim of the proposition, dhc
dρand dec
dρare both nonneg-
ative. This implies that the right-hand side of Eq. (34) strictly decreases in ρbecause
(i) the integrand of the right-hand side of Eq. (34) decreases in ρ, since Fee(·)≥0 and
dec
dρ≥0 by supposition, and (ii) the lower limit of the integral increases in ρbecause
of the supposition that dhc
dρ≥0. Hence, the left-hand side of Eq. (34), c(ec)must
also strictly decrease in ρ, which implies that ecstrictly decreases in ρbecause c(·)is
convex. However, this contradicts the supposition that dec
dρ≥0. Hence ecand hcmust
both decrease in ρ.
Acknowledgements We thank the Editor, two anonymous referees, and seminar participants at Columbia
University, Duke University, University of Minnesota, University of Vienna, University of Hawaii, and the
AES Webinar Series for helpful comments and suggestions.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
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