Operational Involvement of Angel Investors

Incentive Provision in Light of Expertise and
Operational Involvement of Angel Investors
Anil Arya*, Brian Mittendorf
Fisher College of Business, The Ohio State University, 2100 Neil Avenue, Columbus, Ohio 43210, USA, [email protected],
[email protected]
Thomas Pfeiffer
Faculty of Business, Economics and Statistics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, 1090, Austria,
[email protected]
A ngel investors provide a unique role in startup entities: in addition to providing financing, they also provide exper- tise and make critical operational and managerial decisions. As a result, a startup not only seeks funding from its
angel investor, but also strives to ensure that their expertise promotes entrepreneurial success. This study formalizes a
model of optimal equity sharing with angel investors in light of the facts that (i) they have private information about their
expertise, (ii) the expertise impacts the efficacy of subsequent operating decisions, and (iii) equity sharing alters control
rights linked to these decisions. We demonstrate that an entrepreneur will exhibit seemingly excessive optimism in decision-making when angel investors claim they bring high expertise (skills) to the table. This apparent optimism is not a
sign of naivete but rather an effective device to discipline boasting by angels. The results demonstrate the critical role of ´
expertise and decision rights in constructing ownership-sharing agreements with angel investors. They also demonstrate a
case in which the private information of investors (rather than that of insiders) is important, and how such adverse selection concerns entail aggressive operational choices, rather than the typical response of caution, coupled with limited
equity usage that accounts for not just the transfer of ownership, but also the transfer of decision rights.
Key words: adverse selection; angel investing; control rights; equity sharing; expertise; operational decisions
History: Received: August 2019; Accepted: February 2021 by Kalyan Singhal, after 1 revision.
*Corresponding author.
1. Introduction
Traditional views of entrepreneurial financing
revolve around a privately informed entrepreneur
convincing capital providers of the potential of their
venture. In many startups, however, the role of the
capital provider is more nuanced in that they too
bring their own skills and information to the table.
While established institutional investors are typically
sought primarily for capital, startups often rely more
on angel investors who bring not only financing but
also hands-on expertise and operational acumen in
launching new ventures. Popularized (often to the
extreme) in the hit television show “Shark Tank,” the
expertise brought by angels and their role in subsequent operational decision-making can be the true
key to a successful startup launch. This study seeks to
formally represent the role of angel expertise in entrepreneurial success, and examines how this influences
optimal contractual arrangements and the venture’s
core decision-making.
We demonstrate that when an entrepreneur contracts with an angel who will be active in project
implementation, she must use equity grants to
balance the benefit of ceding cash flow risks to the
skilled angel with the desire to retain control as a
means of restraining adverse selection. In particular,
in order to prevent an angel from securing overly
generous terms by overstating his skills, the entrepreneur opts for less ownership transfer in response
to reports of high expertise. Although ownership
transfer would, all else equal, force the angel investor to “put his money where his mouth is,” it also
cedes operational control to him. The entrepreneur
instead opts to retain ownership because it gives
her the ability to control subsequent operating
choices, and these choices can themselves be a critical device to restrain information rents—an overly
boastful angel is dissuaded since he knows his
boasting will result in the entrepreneur taking
greater risks which will sink the angel’s stake in the
business. Interestingly, the case of a privately
informed angel runs counter to the typical view of
a privately informed entrepreneur (e.g., Leland and
Pyle 1977, Myers and Majluf 1984). Although a privately informed entrepreneur seeking to state high
potential must retain ownership to keep “skin in
the game,” a privately informed angel must agree
2890
Vol. 30, No. 9, September 2021, pp. 2890–2909 DOI 10.1111/poms.13409
ISSN 1059-1478|EISSN 1937-5956|21|3009|2890 © 2021 Production and Operations Management Society
to less ownership (less control) if he wants to represent high potential.
To elaborate on the results, consider first a baseline
case in which a privately informed angel contracts
with an entrepreneur but neither party makes subsequent operational decisions relevant to the firm’s success. That is, the firm’s success is solely determined
by the firm’s potential and the angel’s ability to bring
it to fruition. In this case, a typical adverse-selection
tradeoff exists: an angel is tempted to overstate his
skills in order to secure better terms from the entrepreneur. In order to restrain the angel’s boasting, the
entrepreneur can force him to stand behind his assertion by accepting more of his compensation in the
form of equity. This points to the traditional view of
equity being used as a disciplining device and a prediction that more skilled angels will receive more
equity (and entrepreneurs will retain less) in
equilibrium.
The added wrinkle in this model is that subsequent
to agreeing on contract terms, both the entrepreneur
and angel can make consequential operational decisions that influence the firm’s success. If one presumes that the relative power to make such decisions
is independent of ownership, the notion that transferring equity can restrain boasting remains and is in fact
reinforced. However, in reality, the relative decisionmaking power of the parties naturally depends on
their relative ownership. The fact that ownership and
control go hand-in-hand creates an offsetting force:
the entrepreneur can use operating choices as an
additional screening instrument because she can
respond to excessive boasts with a decision to make
more aggressive operational choices and thereby
place the boastful angel’s capital at risk. Importantly,
to make this second instrument have power, the
entrepreneur must back away partially from the first
instrument. That is, the entrepreneur balances the
ability to control firm decisions (by retaining ownership) with the ability to impose some pain on the
angel (by ceding ownership to him).
The end result of this dual-instrument screening
mechanism is that providing equity to angel investors
is used to alleviate adverse selection, and it is done so
in a way with notable comparative statics: (i) higherskill angels are compensated via smaller equity
grants, (ii) entrepreneurs working with higher-skill
angels opt for more aggressive operational decisions,
and (iii) entrepreneurs working with higher-skill
angels are associated with apparent over-optimism—
they make decisions that are indicative of better economic conditions than are reflected in reality. Among
other things, these comparative statics suggest a different interpretation of a commonly discussed empirical phenomenon: higher ownership retention by
entrepreneurs may indeed indicate better prospects
but that may be due to an investor’s private information rather than the entrepreneur’s. After all, under
optimal contracting in our model, entrepreneurs
retain more ownership when funding is provided by
angels who bring more expertise to the engagement.
This is not to say that that entrepreneurs fail to bring
private information; rather, the point is to show that
when other parties do so too, the effects are subtle
and can be difficult to untangle empirically.
To underscore the economic forces and their
robustness, the paper also examines some modeling
variants. First, our model allows for the possibility of
the entrepreneur transferring shares that would cede
ownership to angels without ceding control. Such
dual-class shares are a particularly popular phenomenon when entrepreneurs seek external funding.
Consistent with their popularity among entrepreneurs, we demonstrate that in our setting a dual-class
split between ownership and control is the preferred
route for the entrepreneur. However, we also endogenize the inherent tie-in between ownership and decision rights by considering how an entity interested in
maximizing venture value (e.g., a state or federal
securities regulator) would view dual-class ownership. In this case, we demonstrate that the decision
maker strictly prefers stock ownership to come with
commensurate and proportional decision rights.
Second, we examine the case when the decisions of
the parties have a team coordination effect wherein a
mismatch between the underlying condition and the
parties’ joint actions also reduces firm value. In this
case, over-optimism exhibited by the entrepreneur is
tempered by pessimism in the actions taken by the
angel. That is, the need to align team choices actually
manifests in realized decisions being substitutes, as
the distortions brought by adverse selection lead to
offsetting postures in decisions by both parties. Third,
we also consider the possibility that the angel investor
gains from visibility of his choices and thus has incentives to make aggressive decisions even at the expense
of firm value. Here, the key forces remain in play with
the additional feature that decision-making is now
complementary in that each party is aggressive in
their choices.
The remainder of the paper proceeds as follows.
Section 2 reviews and connects to related literature.
Section 3 presents the baseline model. Section 4
details the results: 4.1 provides the benchmark of full
information wherein the angel’s expertise is known to
all; 4.2 layers asymmetric information and operational
decisions in a binary setting to provide intuition for
the economic forces in play; and 4.3 characterizes the
optimal contract in the continuous case and discusses
the ownership-control tie-in. Section 5 examines two
extensions: 5.1 considers the consequence of team
coordination and 5.2 addresses the effects of angel
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preferences for visibility in operational decision-making. Section 6 concludes the analysis.
2. Related Literature
This study builds on existing research that examines
contracting between an entrepreneur and a capital
provider. Starting with Leland and Pyle (1977) and
Myers and Majluf (1984), much of this literature
examines the importance of entrepreneur private
information on contractual relationships. Each of the
seminal works considers different contractual forms
but reaches a unified theme that a privately informed
entrepreneur must retain an equity stake when seeking capital in order to convey the quality of the venture. Among other things, this view has played a
critical role in expanding our understanding of capital
structure and the “pecking order” theory.
Despite the frequent focus on entrepreneurial private information and incentives, it has also been recognized that investors too can bring their own
information and expertise to the relationship. The role
of investor information and decision-making is particularly notable in the cases of venture capital and angel
investing. Empirical evidence consistently suggests
that such investors bring substantial expertise that
changes the prospects of startup ventures (Gompers
et al. 2016, Gompers et al. 2020, Hellmann and Puri
2002, Hellmann and Thiele 2019, Madill et al. 2007,
Wiltbank et al. 2009). The critical nature of the expertise and decision-making of each party—investors
and entrepreneurs—in angel markets specifically has
also been documented (Bammens and Collewaert
2014, Murnieks et al. 2016, Wiltbank 2005). Not only
are these features prominent in angel financing, but
they also represent a vital and expanding share of
capital markets (Prowse 1988, Shane 2012).
Given the importance of investor information in
venture capital and angel investing, theoretical work
has also examined their effect on prevailing views of
financing arrangements. Sahlman (1990) notes that
the skills brought by venture capitalists can lead to
them taking on more ownership risks when working
with passive investors. Admati and Pfleiderer (1994)
demonstrate that the information advantage of such
investors makes them an important part of early
financing rounds, while Ueda (2004) also demonstrates that this feature can ultimately influence the
decision to seek venture capital vs. bank financing.
Booth et al. (2004) examine how this adverse selection
can influence venture-backed debt financing. Following this theme, Garmaise (2007) then revisits pecking
order theory in light of venture capital investor expertise and demonstrates that this can point toward
junior equity provisions for investors, a contrast from
conventional views.
Research focused on the upside potential of entrepreneurial ventures also demonstrates the benefits
and consequences of staged investments (Erzurumlu
et al. 2019; Gompers 1995), the role of venture capitalists in promoting growth (Keuschnigg 2004), the
advantage of equity contracts in conjunction with royalty payments (Savva and Taneri 2015), and the
importance of bargaining power in ultimate arrangements (Koskinen et al. 2014). Other studies recognize
and incorporate the feature that choices of the capital
provider can increase the value of the venture even
when such investments are potentially subject to
moral hazard (e.g., de Bettignies 2008, Elitzur and
Gavious 2003, and see, for an overview, Edelman
et al. 2017). Importantly, Casamatta (2003) shows how
the presence of moral hazard may justify relying on a
capital provider also as an advisor, thereby endogenizing the decision-making role of angels.
With these empirical and theoretical results emphasizing investor expertise in place, the current paper
too incorporates the key feature of angel investing
that the capital provider possesses critical private
information and expertise. The unique aspects that
we introduce are that (i) private expertise can influence subsequent operational decisions and (ii) ownership influences the degree to which each party can
influence such decisions. We demonstrate that when
these features are considered in tandem, novel consequences for both equity provision to capital providers
and the firm’s operational decisions arise. And, as
noted, these consequences provide a different interpretation to known empirical phenomena.
By highlighting the role of angel investors in startups, this study is also in line with recent calls in the
literature to advance “theory in entrepreneurship from
the lens of operations management” (Joglekar and
Levesque 2013, Phan and Chambers 2013). More ´
widely, this study also builds on and adds to the vast
literature on the use of stock-based pay to alleviate
incentive issues with parties who add expertise and
private information (e.g., employees, funders, suppliers). The use of stock-based compensation has largely
been credited to the notion that participants need “skin
in the game” to ensure appropriate effort incentives
(e.g., Benmelech et al. 2010, Garen 1994, Lewellen et al.
1987). A complement to this view is that privately
informed parties need to “put their money where their
mouth is” when seeking to claim high expertise, a
view that suggests higher skilled individuals should
receive a greater share of their compensation via stock
(e.g., Arya and Mittendorf 2005, Lazear 2005). These
extant views on stock-based pay abstract from implications of stock grants on control and decision rights of
the parties. Accounting for such considerations, this
study’s results suggests that the optimal instrument
reduces reliance on stock-based compensation and
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
2892 Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society
increases emphasis on subsequent operational decisions to discipline boasting of expertise.
3. Model
An entrepreneur contracts with an angel investor to
provide capital K, K > 0, and business expertise θ,
θ∈½0, θ, to establish and run a startup company. In
exchange, the entrepreneur commits to providing
equity ownership to the angel, with the angel’s ownership share denoted by α, α∈½0, 1. The arrangement
can also entail a subsequent fixed monetary exchange
between the two parties, with t denoting the transfer
amount (i.e., in addition to receiving equity ownership,
the angel can also receive a fixed-income commitment
from the entrepreneur). The ultimate value of the venture depends on (i) the angel’s expertise (i.e., his type
θ) and (ii) the impact of the decisions made in running
the firm as chosen by the angel and the entrepreneur,
denoted dA, dA ≥ 0, and dE, dE ≥ 0, respectively. In particular, the value of the venture is given by:
Vðθ, dA, dE, αÞ ¼ Kθ λα θ ½ dA 2 ½1 λα½ θ dE 2 þΩ:
(1)
The infusion of capital and expertise from the
angel combine in complementary fashion to generate
venture value of Kθ in Equation (1). To the degree
there is a mismatch between operational decisions
and the true underlying circumstances, the venture’s
value is undercut. That is, the efficacy of operations
depends on how well decisions match economic conditions. This framework permits the decisions to be
interpreted broadly, covering key aspects of running
a firm such as product choice and placement, capacity utilization, inventory management, selecting
among growth options, etc. (see, e.g., Chen et al.
2010).
The parameter λ, λ∈½0,1, in Equation (1) represents the degree to which the angel’s ownership
translates into influence over decision-making in the
firm. The value of λ ¼ 0 implies that the angel’s role
is confined to bringing financing and business expertise—all subsequent decisions are solely at the discretion of the entrepreneur even if ownership is
transferred. The value of λ ¼ 1 implies that management and operational decisions of both participants
can be critical, and their importance is fully in proportion to their stock holding (ownership).1 Notice
that the λα formulation captures the natural feature
that a greater transfer of stock to the angel is accompanied by an increased transfer of control rights that
boosts the influence of the angel’s decision-making
vision.
Finally, Ω in VðÞ captures all other components of
firm value, and it is assumed to be large enough that
it is always worthwhile to operate the company. The
generated value is shared by the two parties, with
payoffs as follows: (i) the angel’s payoff is
αVðÞþtK reflecting his compensation αVðÞþt net
of capital supply K and (ii) the entrepreneur’s payoff
is ½1 αVðÞ t reflecting her residual claimant
status.2
At the contracting stage, the angel knows his expertise θ, but the entrepreneur is uncertain of it. Her
beliefs over θ are represented by a differentiable and
positive probability density function fðθÞ and the
associated cumulative distribution function FðθÞ. As
is standard (e.g., Bagnoli and Bergstrom 2005, Laffont
and Tirole 1994), the hazard rate, HðθÞ ¼ FðθÞ=fðθÞ, is
assumed to be increasing, that is, H0ðθÞ>0. This hazard rate property is satisfied by several common distributions defined over positive support including
uniform, exponential, power, half-normal, lognormal,
Rayleigh, and chi-square.
In contracting with the privately informed angel,
the entrepreneur can solicit a report of the angel’s
expertise, denoted ^ θ, upon which she can pre-commit
to contractual terms, αð^ θÞ and tð^ θÞ, and her impending
operational decision dEð^ θÞ. The angel’s decision is
denoted dAð^ θ, θÞ, reflecting the fact that the angel
alone is privy to θ. The angel’s and entrepreneur’s
respective utilities are thus given by:
UAð^ θ, θÞ ¼ αð^ θÞVð^θ, θÞ þtð^ θÞ K
and UEð^ θ, θÞ ¼ ½1 αð^ θÞVð^ θ, θÞtð^ θÞ,
(2)
where, for sake of simplicity, we denote with
Vð^ θ, θÞ ¼ Vðθ, dAð^ θ, θÞ,dEð^ θÞ, αð^ θÞÞ the company value
given the angel’s reported expertise and his actual
expertise.
The entrepreneur’s contracting problem (P) is presented in Equation (3) below. In particular, she
designs a take-it-or-leave-it menu of contracts
tð^ θÞ, αð^θÞ, dEð^ θÞ , ^ θ∈½0, θ, to maximize her expected
utility subject to the following constraints.
First, the individual rationality constraints (IR)
ensure that in equilibrium the θ-type angel, if he
selects the contract ð Þ tðθÞ, αðθÞ, dEðθÞ , receives at least
his reservation utility of δθ. This reservation utility
naturally reflects that the angel must be reimbursed
for the opportunity cost of capital, where his next-best
opportunity depends on his skill. Here, δ > 0 is
assumed sufficiently large that multiple binding (IR)
constraints are a nonissue (Rochet and Chone 1998). ´
Second, invoking the Revelation Principle
(Myerson 1979), the incentive compatibility (IC) constraints guarantee that the θ-type angel reports truthfully. Third, for any given contract, the constraints
ðDÞ recognize that the angel’s decisions are noncontractible and he undertakes operational decisions
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society 2893
in self-interest for all values of his report and private
information.
Program (P).

Max
ð Þ tðθÞ,αðθÞ,dEðθÞ ,θ∈½0, θ

Eθf g UEðθ,θÞ

(3)

subject to:

UAðθ,θÞ ≥ δθ 8θ
UAðθ,θÞ ≥ UAð^θ,θÞ 8ð^θ,θÞ
dAð^ θ,θÞ∈argmaxdA UAð^ θ,θÞ 8ð^ θ,θÞ

(IR)
(IC)
(D)

The solution to program (P) is denoted by
ðt∗ðθÞ, α∗ðθÞ, d∗ EðθÞÞ. The Figure summarizes the timeline of events.
4. Results
4.1. Full-Information Benchmark
As a reference point, we first present the solution
in the absence of private information, that is, when
the entrepreneur also knows the angel’s business
expertise. Formally, this corresponds to solving
Equation (3) without the (IC) constraints. In this
case, the angel receives precisely his reservation
utility, and it is immaterial whether this is accomplished via a fixed transfer only, equity, or a mix
of instruments. In other words, given the risk neutrality of both parties and symmetric information,
the precise ownership rule is not critical as long as
the angel is sufficiently compensated for her investment. Moreover, each participant simply undertakes their operational decision to eliminate any
value loss due to mismatch between decision and
state. The following observation summarizes this
solution, denoted by “**” (All proofs are detailed
in the appendix.)
OBSERVATION. When the entrepreneur has perfect information about the angel’s expertise,
(i) any tðθÞ and αðθÞ choice under which the angel
receives his reservation utility is optimal, that is, any
t∗∗ðθÞ and α∗∗ðθÞ that satisfies t∗∗ðθÞ þ α∗∗ðθÞ
½ Kθ þΩ K ¼ δθ is optimal; and
(ii) the operational decisions of the angel and the
entrepreneur are each efficient, that is, d∗∗ A ðθ;θÞ ¼
d∗∗
E ðθÞ ¼ θ:
The Observation demonstrates that irrespective of
the degree of the angel’s control, λ, providing company ownership to the angel is inconsequential absent
information differences. This feature permits us to
focus on implications for compensation and operational decisions stemming solely due to information
asymmetry.
4.2. Intuition: The Binary Setting
This subsection demonstrates how information asymmetry creates a nontrivial demand for equity compensation and alters operational decisions, starting with a
simple binary setting. In particular, suppose
λ∈f g 0,1 and θ∈ 0, θ , with p denoting the probability
of the high type. The contracting problem in the binary setting has a structure that mirrors that in the continuous case of program (P) in Equation (3). That is,
the entrepreneur again maximizes her expected utility
subject to the angel’s individual rationality, the incentive compatibility, and the constraints for the angel’s
operational decision choice for each of his two type
values. We refer to the entrepreneur’s problem as
(PB), with “B” denoting binary.
Program (PB).
Max
ð Þ tðθÞ,αðθÞ,dEðθÞ ,θ∈f g 0, θ
½1p½ ð Þ 1 αð0Þ Vð0,0Þ tð0Þ
þp½ ð Þ 1 αð θÞ Vð θ, θÞ tðθÞ (4)
subject to:
Figure. Timeline
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
2894 Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society
As is standard, we solve (PB) in the following steps.
First, we conjecture which of the constraints are binding. Next, we drop the presumed nonbinding constraints, and solve the relaxed problem. Finally, we
verify that the solution to the relaxed problem indeed
satisfies the deleted constraints. In line with intuition,
we conjecture that the incentive problem entails the
angel overstating his true expertise, that is, the lowtype angel has incentives to report his type as θ, so
ðIC0Þ binds. To deter such over-reporting, the entrepreneur is compelled to let the low-type angel earn
information rents (i.e., earn more than his reservation
utility). In contrast, there is no “room” for the hightype angel to overstate his type. Absent misreporting
incentives, the high-type angel is held precisely to his
reservation utility, so ðIR θÞ binds.
Simultaneously solving the binding ðIC0Þ and the
ðIR θÞ constraints allows us to characterize the transfer
payments for the low-type and the high-type angel as
follows:
tð0Þ ¼ K þ δ θ αð θÞ½ Vð θ, θÞ Vð θ,0Þ αð0ÞVð0,0Þ and
tð θÞ ¼ K þ δ θ αð θÞVð θ, θÞ:
(5)
The transfer payments in Equation (5) ensure that
the high-type angel receives exactly his outside
option, whereas the low-type angel receives rents.
For any contract, the angel selects his operational
decision to maximize the firm value, yielding
dAð^ θ,0Þ ¼ 0 and dAð^ θ, θÞ ¼ θ as the solution to the ðD0Þ
and ðD θÞ constraints in Equation (4). That is, the
angel’s decisions depend only on his private information (his report is moot), and are always efficient.
Given the angel’s operational decisions, and the transfer payments in Equation (5), the entrepreneur’s
problem (PB) can be restated as the following unconstrained maximization problem:
Max
ð Þ αðθÞ,dEðθÞ ,θ∈f g 0, θ
½1pVð0,0Þ þpVð θ, θÞ K
½1p½ δ θ αð θÞð Þ Vð θ,θÞ Vð θ,0Þ p½δ θ:
(6)
The objective function consists of the expected firm
value that is generated net of capital provision (i.e.,
the first three terms in Equation (6)) less the expected
utility of the angel (i.e., the last two terms). In particular, the fourth term depicts the utility (including information rents) of the low-type angel that we derived
from the binding incentive compatibility constraint,
ðIC0Þ. The last term depicts the utility of δ θ for the
high-type angel that we derived from the individual
rationality constraint, ðIR θÞ.
An inspection of Equation (6) reveals that the ownership share and the entrepreneur’s decision choice
for a low-type report, that is, αð0Þ and dEð0Þ, do not
impact the angel’s rents. These values are thus determined to maximize firm value Vð0,0Þ, implying
α∗ð0Þ ¼ 1 and d∗ Eð0Þ ¼ 0 without loss of generality. That
is, the familiar “no distortion” result for the boundary
type applies.
Next we turn our attention to the contract design
for the high-type angel. Given dAð^ θ, θÞ ¼ θ from ðD θÞ,
and the firm value in Equation (1), the objective function in Equation (6) can be stated as:
Max
ð Þ αð θÞ,dEð θÞ
p K h i θ ½1 λαð θÞ½ θ dEð θÞ 2 þΩ δθ
þ½1p Ω δ θ αð θÞ θ½ K þ ½1 λαð θÞ½ 2dEð θÞ θ K:
(7)
The first term represents the firm value net of angel
utility generated by the high-type angel. The second
term depicts the same when the angel employed is of
low type; the term in curly brackets denotes the information rents earned by the low-type angel. If the
entrepreneur does not grant the high-type angel
shares, that is, αð θÞ ¼ 0, then the low-type angel
receives rents of δ θ. Ceteris paribus, the entrepreneur
can reduce the low-type angel’s rents by increasing
the number of shares used to compensate the hightype angel. This is because relative to the high-type
angel, the low-type angel has a stronger preference for
cash payment vs. a payoff conditioned on firm value.
In addition, from Equation (7), the entrepreneur can
also reduce the low-type angel’s rents with his operational decision. In particular, the entrepreneur’s choice
preference dEð θÞ is weighted by the entrepreneur’s
level of ownership and the degree of ceded control,
that is, 1 λαðθÞ. The resulting tradeoff necessitates
that the entrepreneur has to balance the use of αð θÞ
and dEð θÞ as instruments to limit information rents.
Maximizing the entrepreneur’s payoff in Equation
(7) shows that she uses a mix of instruments to optimally achieve the “production vs. rents” tradeoff. For
the high-type angel, the entrepreneur transfers ownership and undertakes operational decisions above
the efficient level irrespective of the degree of ownership-decision rights alignment, that is, the entrepreneur commits to decisions that appear optimistic in
that they would match circumstance only if the
angel’s type was higher than what is truly is. To elaborate, the high-type angel’s ownership share depends
on the degree of control: he receives full ownership
for λ ¼ 0 whereas the tradeoff described above can
dictate he receives less than full ownership for λ ¼ 1,
allowing the entrepreneur to use her operational decisions as an instrument to mitigate information rents.
In all cases, the entrepreneur engages in operational
decisions above the efficient level when dealing with
the high-type angel. Proposition 1 presents the complete solution to the binary agency problem.
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
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PROPOSITION 1. When the angel’s expertise is private, in
the binary setting,
(i) for λ ¼ 0 the angel receives ownership share of
α∗ð0Þ ¼ α∗ð θÞ ¼ 1 whereas for λ ¼ 1 the angel
receives ownership share of:
α∗ð0Þ ¼ 1 and α∗ðθÞ ¼
min 1
3 11p pþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ3p½1pKþp θ
( ) ! s ½1p2 θ ,1 ;
(ii) the operational decision of the angel is matched
and efficient, whereas the operational decision
of the entrepreneur is mismatched and more
than the efficient level when dealing with the
high-expertise angel:
d∗ Að0,0Þ ¼ 0, d∗ Að θ, θÞ ¼ θ,
d∗
Eð0Þ ¼ 0, d∗ EðθÞ ¼ θþα∗ð θÞ θ½1p
p
; and
(iii) the transfer payments, tð θÞ and tð0Þ, are determined such that the angel’s utility equals:
UAð θ, θÞ ¼ δ θ and
UAð0,0Þ ¼ δ θα∗ð θÞ θ K þ½1λα∗ð θÞ½2d∗ Eð θÞ θ :
4.3. Solution in the Continuous-Type Setting
In this subsection, we characterize the solution in the
general continuous-type setting. This solution reinforces and builds on the intuition developed in the
binary case. The optimal compensation arrangement
and operational decisions are obtained by solving
problem (P). To do so, we note the following simplifying features in Equation (3). First, for any αð^θÞ, tð^ θÞ,
and dEð^ θÞ, the angel selects dA to maximize the firm
value, yielding dAð^ θ,θÞ ¼ θ as the solution to ðDÞ-constraints in Equation (3); thus, again, the level of the
angel’s operational decisions is efficient and conditioned only on his type (more on this in extensions).
We also replace the global (IC) constraints in Equation
(3) by their local counterpart. The appendix proves
that this substitution ensures that the angel has incentives to truthfully report not just in the local neighborhood of θ but for all θ-values. The local incentive
constraints, presented in Equation (8), reflect the fact
that the compelling incentive problem is the angel
overstating his type to inflate the value he can generate by “going it alone,” that is,
UAðθ,θÞ ¼ UAð θ, θÞZ
θ
θ
αð~ θÞ Kþ2½1λαð~ θÞ dEð~ θÞ~ θ d~θ:
(8)
From Equation (8), if no equity is transferred, that
is, αð~ θÞ ¼ 0, then the entrepreneur is forced to pay
each type of angel as though he were of the highest
type, that is, UAðθ,θÞ ¼ δ θ. Transferring ownership to
the angel helps to limit the angel’s utility (or, equivalently, information rents, UAðθ,θÞδθ). By itself, this
force suggests maximizing the use of equity transfer.
However, again from Equation (8), note that the entrepreneur can also curtail the angel’s rents by choosing
an optimistic “higher-than-efficient” operational decision. But this ability to use dE as an instrument to control rents is limited to the degree the entrepreneur
retains decision rights (formally, dE is weighted by
the entrepreneur’s retained control based on ceded
control, i.e., 1λα). For λ≠0, these forces suggest a
tradeoff that results in both a sharing of ownership
and the entrepreneur choosing a mismatched operational decision.
From Equation (8), dAð^ θ,θÞ ¼ θ, and the fact that the
individual rationality constraint for the highest type
angel binds, the angel’s expected utility equals:
Eθf g UAðθ,θÞ ¼ δ θEθf g HðθÞαðθÞ½ Kþ2½1λαðθÞ½dEðθÞθ :
(9)
From Equation (2), EθfUEðθ,θÞg ¼ EθfVðθ,θÞg
EθfUAðθ,θÞgK. Using Equation (9) and dAð^ θ,θÞ ¼ θ,
the entrepreneur’s program in Equation (3) can be written as the following unconstrained problem:

Max
ð Þ αðθÞ,dEðθÞ ,θ∈½0, θ
Kθ½1λαðθÞ½ θdEðθÞ 2 þΩ

θR 0
δ θHðθÞαðθÞ½ Kþ2½1λαðθÞ½dEðθÞθ fðθÞdθ:
(10)
The objective function in Equation (10) equals the
expected firm value net of capital infusion (represented in the first line) less the angel’s expected utility
in Equation (9) that we derived from the local (IC)
constraints (represented in the second line). To curtail
the angel’s information rents, the entrepreneur can
grant the angel an equity stake and adjust the level of
her operational decisions. Pointwise optimization
yields the solution of Equation (10), and this solution
is characterized next.
PROPOSITION 2. When the angel’s expertise is private,
(i) the angel receives an ownership share of:
α∗ðθÞ ¼
1 if θ ≤ θ∗
1 3λ
1þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 3Kλ
! s HðθÞ if θ>θ∗,
8><>:
where θ∗ is the unique θ-value that solves
½3λ2HðθÞK ¼ 0;
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
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(ii) the angel’s operational decision is matched and
efficient, whereas the entrepreneur’s operational decision is mismatched and more than
the efficient level, that is, d∗ Aðθ,θÞ ¼ θ and
d∗
EðθÞ ¼ θ þ α∗ðθÞHðθÞ; and
(iii) the cash transfer payment, t∗ðθÞ, is determined
such that the angel’s utility equals:
UAðθ,θÞ ¼ δ θ Z
θ
θ
α∗ð~ θÞ K þ2½1 λα∗ð~θÞ½d∗ Eð~ θÞ ~ θ d~ θ:
In contrast to the case of perfect information presented in the Observation, the degree of firm ownership is material when the angel has private
information. When the angel’s degree of potential
control is low, then the angel serves as the sole owner;
formally, λ<2=3 implies ½3λ 2HðθÞ K<0 and thus
α∗ðθÞ ¼ 1 for all θ∈½0, θ. For λ>2=3 and low levels of
expertise, θ ≤ θ∗, the angel serves as the sole owner
and, thus, the only consequential operational decision
is made by the angel. However, for higher levels of
expertise, θ>θ∗, the angel and the entrepreneur share
ownership. In this case, the entrepreneur’s operational decision choice also comes into play as it
impacts both firm value and curtails the angel’s rents.
It is this “dual role” of dE that spurs the entrepreneur
to intentional choose a mismatched optimistic decision. Crucial for this tradeoff is the angel’s degree of
potential control. If ceding ownership also means ceding decision-control (high λ), the entrepreneur cedes
less ownership. Ceding less ownership, in turn,
reduces the relative impact of an aggressive decision.
As a consequence, the entrepreneur’s operational
decisions, d∗ EðθÞ, as well as its deviation from the fullinformation level, d∗ EðθÞ d∗∗ E ðθÞ, are decreasing in the
degree of the ownership-decision rights alignment.
Further comparative static insights reveal that the
entrepreneur’s decision, d∗ EðθÞ, as well as its deviation
from the full-information level, d∗ EðθÞ d∗ Aðθ,θÞ ¼
α∗ðθÞHðθÞ, is increasing in the angel’s expertise; in
contrast, the ownership share transferred to the angel,
α∗ðθÞ, is decreasing in the angel’s expertise. Intuitively, from Proposition 2(iii), when the entrepreneur
increases her level of the operational decision for a
given type of angel, say θ ¼ ~ θ, it provides rent reduction not just for the type ~ θ but also for all lower θ<~ θ
types. It is this cumulative lower-tail effect that results
in d∗
EðθÞ and d∗ EðθÞ d∗∗ E ðθÞ each increasing in θ. Of
course, to the extent the entrepreneur can utilize dE as
an instrument to limit rents depends on its impact on
firm value and, hence, on angel rents—rents are
reduced by ½1 λα∗ð~ θÞd∗ Eð~ θÞ. This implies that the
more the entrepreneur intends to rely on dE as an
instrument to limit rents, the more ownership she
needs to retain for it to have an impact, so α∗ðθÞ is
decreasing in θ.
Put slightly differently, for the extreme case of the
lowest expertise angel, θ ¼ 0, the entrepreneur
chooses the level of her operational decisions efficiently, that is, d∗ Eð0Þ ¼ 0. Since this is also the angel’s
preferred decision choice, transferring complete ownership to the angel is costless. In fact, for small θ values (θ ≤ θ∗), the entrepreneur prefers to avoid angel
misreporting concerns by continuing to transfer full
ownership to the angel rather than by introducing
any distortions in the operational decisions. However,
for higher θ values (θ>θ∗), information rents earned
by the now large cohort of lower types become of sufficient concern to the entrepreneur. As a consequence,
the wedge between the entrepreneur’s and the angel’s
preferred operational decisions, d∗ EðθÞ d∗ Aðθ,θÞ, is
also sufficiently wide, and the entrepreneur shifts to
relying on both α and dE, rather than α alone, to curb
angel misreporting. Proposition 3(i) and (ii) summarize this discussion, and Proposition 3(iii) takes the
next step of showing how the severity of the agency
conflict influences the entrepreneur’s decision.
PROPOSITION 3. When the angel’s expertise is private,
(i) the angel’s ownership share as well as the
absolute and the relative level of the entrepreneur’s operational decisions decrease in the
degree of ownership-decision rights alignment,
that is, ∂α∗ðθÞ=∂λ ≤ 0, ∂d∗ EðθÞ=∂λ ≤ 0, and
∂½d∗ EðθÞ d∗∗ E ðθÞ=∂λ ≤ 0; and
(ii) the angel’s ownership share decreases in his
expertise, whereas the absolute and the relative
level of the entrepreneur’s operational decisions increase in the angel’s expertise, that is,
∂α∗ðθÞ=∂θ ≤ 0, ∂d∗ EðθÞ=∂θ ≥ 0, and ∂½d∗ EðθÞ d∗∗ E ðθÞ=
∂θ ≥ 0; and
(iii) the angel’s ownership share decreases in the
severity of the agency conflict (measured in
terms of hazard rate dominance), whereas the
absolute and the relative level of the entrepreneur’s operational decisions increase with the
severity of the agency conflict, that is,
∂α∗ðθÞ=∂H ≤ 0, ∂d∗ EðθÞ=∂H ≥ 0, and ∂½d∗ EðθÞ
d∗∗
E ðθÞ=∂H ≥ 0.
As identified, the degree of ownership-decision
rights alignment and the extent of information asymmetry between the entrepreneur and the angel are the
driving forces behind the entrepreneur’s choice of her
operational decisions and ownership transfer to the
angel. Given the comparative statics in parts (i) and
(ii), one might also conjecture that the entrepreneur’s
strategy of relying more on d∗ EðθÞ and less on α∗ðθÞ
gets more pronounced as the agency conflict becomes
more severe. Proposition 3(iii) confirms this intuition,
where the degree of information asymmetry is
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
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formalized via the familiar hazard rate dominance
condition (e.g., Laffont and Tirole 1994). In terms of
how the agency conflict manifests in empirically
observable outcomes, it is also useful to consider how
it translates into the impact on the equilibrium value
of the venture, that is, Vðθ,d∗ A,d∗ E,α∗Þ. In line with the
above logic, a greater agency conflict also serves to
undermine the venture’s ultimate value, that is,
∂Vðθ,d∗ A,d∗ E,α∗Þ=∂H ≤ 0.
We conclude this section by examining the feature
that the financing instrument which creates an ownership split of α and 1 α between the angel and the
entrepreneur translates into an λα and 1 λα sharing
of decision rights. The results in this study are robust
in that they are derived irrespective of the ownershipdecision rights tie-in, that is, the results hold for all
λ-values. A related question is to ask what λ-value
would be chosen as part of a regulatory process identifying the rights of the shareholders.
From a value-maximizing regulator’s perspective,
what is critical is the firm value generated, not how
this value is shared between the entrepreneur and the
angel. Given this focus, and recognizing the inefficiently aggressive decision that the entrepreneur is
sure to undertake (as noted in Proposition 2), the regulator prefers to grant maximal decision rights to the
angel to curtail the entrepreneur’s use of aggressive
operational decisions that undermine firm value. That
is, the equilibrium venture value, Vðθ, d∗ A, d∗ E, α∗Þ,
increases in the degree of ownership-decision rights
alignment λ. This occurs because the mismatch
between the entrepreneur’s operational decisions and
the true underlying circumstances decreases in λ, and
so does the weight assigned to the mismatch, that is,
∂½θ d∗ E2=∂λ ≤ 0 and ∂½1 λα∗=∂λ ≤ 0. As such, firm
value is maximized by the choice of λ ¼ 1.
Of course, the entrepreneur is not concerned with
firm value per se, but rather her share of the firm
value net of the angel’s information rents. The focus
on curtailing angel rents, in fact, proves to be the compelling driver for the entrepreneur in that her preference is λ ¼ 0 if provided free reign. This preference
reflects its ability maximize the use of aggressive decisions to curtail the angel’s rents without unduly
affecting an ability to transfer the risks of those decisions via ownership. As such, the opposing preferences of the regulator and the entrepreneur imply
that if both parties have influence on the design of the
financial instrument, an intermediate λ-value can
arise in equilibrium. Corollary 1 summarizes this discussion.
COROLLARY 1. A value-maximizing regulator prefers an
equity instrument that grants ownership with commensurate decision rights, that is, λ ¼ 1. In contrast, a utilitymaximizing entrepreneur prefers an instrument that
grants ownership but no decision rights, that is, λ ¼ 0.
5. Extensions
In this section, we extend our model in two ways.
First, we expand the firm value expression to incorporate the importance of coordinated team decisions by
the angel and the entrepreneur. Second, we append
the angel’s preferences to include private gains from
operational visibility and reputation. Both extensions
point to the robustness of our main findings: equity
sharing and seemingly inefficient entrepreneur decisions are used in tandem as control instruments.
Moreover, the extensions also demonstrate that the
equilibrium outcome can lead not only to the entrepreneur introducing inefficiencies in decision-making
but also the angel. The angel’s inefficient decision can
either partially offset the inefficiency in the entrepreneur’s decision (i.e., the parties’ decisions are substitutes under team coordination) or be aligned in that
his action too is excessively high (i.e., the parties’ decisions are complements when the angel has preferences for visibility).
5.1. Team Coordination
The mismatch terms in Equation (1) presume that the
entrepreneur’s and the angel’s operational decisions
individually and separately impact venture value,
that is, the decisions made by the two parties do not
interact. An advantage of this formulation is that the
entrepreneur’s and the angel’s operational decisions
differ solely due to information reasons and not on
any inability to coordinate their choices. This is not to
say that incorporating team coordination derails our
main findings. To elaborate, in this subsection, we
extend Equation (1) to include a team coordination
term as follows:
Vðθ,dA,dE,α;τÞ ¼ Kθ λα θ ½ dA 2 ½1 λα½ θ dE 2
τ θ ½ ½λαdA þ ð1 λαÞdE 2 þΩ: (11)
In Equation (11), the parameter τ, 0 ≤ τ ≤ 1, reflects a
role for team coordination. In particular, τ ¼ 0 reduces
Equation (11) to the base scenario in Equation (1)
where the parties’ decisions have separate impacts on
firm value; higher values of τ value, however, reflect
the added need for the two parties’ decisions to jointly
be in alignment with underlying circumstances. The
team coordination term is again in line with each
party’s control rights in that decisions are weighted
by their relative influence and equity holding.
The team coordination effect induces an interdependence in the decisions of the two parties. In particular, the angel’s operational decision as a best
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
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response to the entrepreneur’s decision dE is as
follows:
d∗
Að^ θ,θ;τÞ ¼ θ τ 1 λαð^ θÞ
1 þ τλαð^ θÞð Þ dE θ : (12)
Intuitively, Equation (12) follows from the angel
selecting dA to maximize his expected utility,
αð^ θÞVð^θ,θÞ þ tð^ θÞ, for any αð^ θÞ>0. With the firm value
as in Equation (11), the angel recognizes that his decision impacts not just the separable mismatch cost
term, but also the mismatch cost due to the non-separable team coordination term. Balancing both considerations then implies that if the entrepreneur’s choice
deviates from θ, the angel has incentives to offset the
entrepreneur-induced mismatch by making a decision that mismatches in the reverse direction. In other
words, the angel’s and entrepreneur’s operational
decisions exhibit a negative complementarity as the
angel is pessimistic in his decision in order to offset
when the entrepreneur is optimistic (and vice versa).
The angel only undertakes the efficient decision when
the entrepreneur commits to the efficient decision
herself.
Given the angel’s decision in Equation (12), the firm
value in Equation (11) can be written as:
Vðθ,d∗ Að^θ,θ;τÞ,dEð^ θÞ,α;τÞ ¼
Kθ ½1 λαð^ θÞ½1 þ τ
½1 þ τλαð^ θÞ ½θ dEð^ θÞ2 þΩ: (13)
The firm value in Equation (13) is reduced when
the entrepreneur deviates from the efficient decision,
that is, dEð^ θÞ≠θ. However, undertaking an aggressively high operational choice allows the entrepreneur to reduce the angel’s rents which, in line with
Equation (9), are given by:
Eθf g UAðθ,θÞ
¼ δ θ Eθ HðθÞαðθÞ K þ2½1 λαðθÞ½1 þ τ
½1 þ τλαðθÞ ½ dEðθÞ θ :
(14)
Formally, greater need for team coordination exaggerates the firm value vs. rents dichotomy: as τ
increases, firm value in Equation (13) decreases but so
do the angel’s rents in Equation (14).
Using Equation (2), EθfUEðθ,θÞg ¼ EθfVðθ,θÞg
EθfUAðθ,θÞg K, the entrepreneur’s goal is to
maximize the expected firm value in Equation
(13) net of the provided capital (the first line in
Equation (15)) less the angel’s expected utility in
Equation (14) (the second line in Equation (15)).
That is,
Max
ð Þ αðθÞ,dEðθÞ ,θ∈½0, θ
θR 0
Kθ ½1 λαðθÞ½1þ τ
½1þ τλαðθÞ ½θ dEðθÞ2 þΩ K
δ θ HðθÞαðθÞ K þ 2½1 λαðθÞ½1 þ τ
½1þ τλαðθÞ ½ dEðθÞ θ fðθÞdθ:
(15)
As noted earlier, the firm value and the angel’s
rents both decrease when team coordination gets
more important. The “net effect” of these two terms is
scaled by ½1 þ τ=½1 þ τλαðθÞ and equals ½1 þ τ=½1þ
τλαðθÞ ½1 λαðθÞh i 2HðθÞαðθÞ½ ½ dEðθÞ θ θ dEðθÞ2 .
Accordingly, the entrepreneur determines her operational decisions as before above the efficient level, that
is, d∗ Eðθ;τÞ ¼ θ þ αðθÞHðθÞ. Given the entrepreneur’s
decision, the only issue left is the angel’s equity share
and that is determined via pointwise optimization of
Equation (15). The complete solution is characterized
in Proposition 4.

PROPOSITION 4.	Under team coordination,
(i) the	angel	receives	an	ownership	share

α∗ðθ;τÞ ¼ minf g αðθ;τÞ,1 , where αðθ;τÞ>0 is the
unique, positive αðθÞ-value that solves:
HðθÞ ¼ K½ 1 þ τλαðθÞ 2
½1 þ ταðθÞ½ λαðθÞ½3 þ 2τλαðθÞ τ 2 ,
with the angel’s share decreasing in his expertise, that is, ∂α∗ðθ;τÞ=∂θ ≤ 0.
(ii) the angel’s operational decisions appear pessimistic (less than efficient), while the entrepreneur’s operational decisions appear optimistic
(more than efficient), that is,

d∗
Eðθ;τÞ ¼ θ þ α

∗ðθ;τÞHðθÞ and

d∗
Aðθ,θ;τÞ ¼ θ τ 1 λα∗ðθ;τÞ
1 þ τλα∗ðθ;τÞα∗ðθ;τÞHðθÞ;
(iii) the cash transfer payment, t∗ðθ;τÞ, is determined such that the θ-type receives a utility of:
UAðθ,θÞ ¼
δθ Z
θ
θ
α∗ð~θ;τÞ K þ 2½1 λα∗ð~ θ;τÞ½1 þ τ
” # ½1 þ τλα∗ð~ θ;τÞ α∗ð~ θ;τÞHð~ θÞ d~ θ:
Proposition 4 shows that the solution exhibits similar characteristics as outlined in Proposition 2 in that,
to optimally curtail the angel’s rents and mismatching
costs, (i) the angel’s equity share decreases with his
expertise and (ii) the entrepreneur selects an aggressively high level for her operational decisions. There
is, however, an interesting added negative
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
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complementarity effect manifested by the parties’
decisions: accounting for the joint mismatch cost tied
to team coordination, both the entrepreneur and the
angel now deviate from the efficient decision level in
partially offsetting fashion.
Related to the above, as expected, when team coordination does not matter (τ ¼ 0), the solution in Proposition 4 equals our baseline results in Proposition 2. As
outlined, when team coordination increases in importance, the optimal arrangement prescribes a lower
equity share for the angel and thereby lower operational decision for the entrepreneur. In effect, the
greater the criticality of team coordination, the less
aggressive is the entrepreneur in her use of control
instruments. Intuitively, higher equity retained by the
entrepreneur incentivizes her to care more about the
loss in venture value attributed to mismatched team
action relative to the need to curtail agent rents, while
the use of such an action has less bite in curbing rents
since the agent takes actions to offset its effect. This
shift toward productive efficiency rather than
information rents leads the entrepreneur to choose
decisions to deviate less from θ, the underlying
condition. These comparative statics are confirmed in
Corollary 2.
COROLLARY 2. An increase in the importance of team
coordination results in (i) a decrease in the equity share
for the angel, (ii) a decrease in the entrepreneur’s operational decision, and (iii) a decrease in the angel’s operational decision. That is, ∂α∗ðθ;τÞ=∂τ ≤ 0, ∂d∗ Eðθ;τÞ=∂τ ≤ 0,
and ∂d∗
Aðθ,θ;τÞ=∂τ ≤ 0.
5.2. Angel Reputation and Visibility
While monetary return from successful financing of a
venture is an immediate source of wealth for an angel,
there are also potential long-term reputation gains to
the angel associated with the visibility and publicity
linked to his operational role in the scale and success
of the venture. As example, look no further than the
case of Sean Parker, who played a role in Facebook’s
initial funding but an even more prominent one in its
operations thereafter. This operational role, in turn,
resulted in enhanced reputation and visibility that
have far outlasted his official involvement in Facebook itself.
To reflect the possibility that the angel may derive
personal or professional benefits from a more visible
operational role, we add to the angel’s utility function
as follows:
UAð^ θ,θÞ ¼ αð^ θÞVð^ θ,θÞþtð^ θÞKþνλαð^ θÞdA: (16)
Here ν, ν ≥ 0, measures the angel’s “sensitivity-to-visibility” with the visibility term, in turn, contingent on
the angel’s operational decision dA, weighted by his
control rights, λα. Furthermore, presume the hazard
rate, HðθÞ ¼ FðθÞ=fðθÞ, is convex, that is, H00ðθÞ>0.
For any given contract, maximizing Equation (16),
the angel sets his operational decision above the efficient level. Intuitively, the angel’s decision is no
longer singularly focused on reducing mismatching
costs, but rather also on self-promotion by boosting
his visible actions. Formally, then, dAð^θ,θ;νÞ ¼
θþν=½2αð^ θÞ. The agent’s preferences for visibility
allow the entrepreneur to reduce the required payment to the agent which has a positive impact on her
utility, that is, UEðθ,θÞ ¼ Vðθ,θÞþνλαðθÞdAðθ,θ;νÞ
UAðθ,θÞK from Equations (2) and (16). Given the
angel’s decision, the unconstrained problem of the
entrepreneur equals:
Since the visibility-seeking angel sets his operational decisions aggressively high, mismatching costs
arise and reduce the venture value by ν2λ=½4αðθÞ,
as noted in Equation (17). On the plus side, the
entrepreneur takes advantage of the fact that the
angel derives personal benefits from visibility to
reduce both the angel’s required payments and rents.
The former is reflected in Equation (17) as
νλ½ vþ2αðθÞθ =2—when an angel is able to make visible decisions, the transfer required to entice the
angel to join (i.e., to satisfy the (IR)) is lower. The latter is reflected in the rent term (νHðθÞλαðθÞ) in Equation (17)—since a high-type angel is able to credibly
choose more visible decisions, the desire for visibility
is used as an additional tool to dissuade overrepresentation. Given these additional terms do not entail
dEðθÞ, it follows that the entrepreneur’s optimal decision takes the same form as before, albeit with the
angel’s visibility impacting this choice due to its
influence on the equilibrium equity sharing rule.
Max
ð Þ αðθÞ,dEðθÞ ,θ∈½0, θ
θR 0
Kθ ν2λ
4αðθÞ½1λαðθÞ½ θdEðθÞ 2
þΩKþνλ½ vþ2αðθÞθ
2
ð ÞÞ δ θ HðθÞαðθÞ½ Kþνλþ2½1λαðθÞ½dEðθÞθ fðθÞdθ:
(17)
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
2900 Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society
Pointwise optimization of Equation (17) yields the
solution that is detailed in Proposition 5.

PROPOSITION 5.
bility,

When the angel has a preference for visi

(i)	the	angel	receives	an	ownership	share
α∗ðθ;νÞ ¼ minfαðθ;νÞ,1g, where αðθ;νÞ ≥ 0 is the
unique	positive	α-value	that	solves

ν2λ ¼ 4α2f g ½ αHðθÞ½3λα 2 Kνλ HðθÞ νλθ ,
with the angel’s share decreasing in his expertise, that is, ∂α∗ðθ;νÞ=∂θ ≤ 0;

(ii)	the entrepreneur’s and the angel’s operational decisions are both optimistic (more than effi

cient) and are as follows:
d∗
Aðθ,θ;νÞ ¼ θ þ ν
2α∗ðθ;νÞ and
d∗
Eðθ;νÞ ¼ θ þα∗ðθ;νÞHðθÞ; and
(iii) the cash transfer payment, t∗ðθ;νÞ, is determined
such that the angel’s utility equals:
UAðθ,θÞ ¼ δ θ Z
θ
θ
α∗ð~ θ;νÞ½Kþ νλ
þ2 α∗ð~ θ;νÞ½1λα∗ð~ θ;νÞ Hð~ θÞd~ θ:
In line with the base results, Proposition 5 again
demonstrates that the entrepreneur optimally controls
the angel’s rents by committing to optimistic operational decisions and curtailing the ownership share
for the angel. The added effect is that the angel’s preferences for visibility increases his operational decision
above the efficient level as well and also helps the
entrepreneur reduce the required payments and rents
to the agent. This, in turn, leads the entrepreneur to
increase the angel’s ownership share: ownership is
used not only as a “stick” to discipline the angel’s
desire to overstate his expertise but also as a “carrot”
to provide the agent his desired visibility. The fact
that α∗ðθ;νÞ is increasing in visibility then implies the
same is true for the entrepreneur’s own decision.
Roughly stated, with increasing visibility gains there
is more of a disconnect between the angel and the
entrepreneur’s preferences leading to an increasing
reliance by the entrepreneur on both control instruments, ownership, and his decision. Corollary 3 formalizes this intuition.
COROLLARY 3. The angel’s ownership share as well as
the level of the entrepreneur’s and the angel’s decisions
increase in the angel’s degree for visibility, that is,
∂α∗ðθ;νÞ=∂ν ≥ 0, ∂d∗ Eðθ;νÞ=∂ν ≥ 0, and ∂d∗ Aðθ,θ;νÞ=∂ν ≥ 0.
6. Conclusion
The informal capital market introduced by “angel”
investors is often characterized by innovation, high
risk, and high return. These characteristics arise in
part due to the feature that the angel market matches
entrepreneurs with individual investors who bring
not only capital but also business expertise to startup
enterprises. This study models unique aspects of
angel investing—that angels bring expertise, that
angels and entrepreneurs both can make key strategic
operational decisions, that ownership influences the
degree to which the parties can influence these decisions, and that angel investors’ have incentives to
overstate what they bring to the table—to derive the
optimal contracting arrangement between entrepreneurs and angels.
We demonstrate that the distinguishing features of
angel investing lead to a circumstance where an entrepreneur may actually opt to retain more ownership
for herself precisely when an angel seeks to claim
high expertise. Rather than forcing the angel to “put
his money where his mouth is” by transferring ownership, the entrepreneur opts to take more risk herself
and instead offer more fixed income to the angel. This
happens because keeping ownership also naturally
permits the entrepreneur to maintain control, and the
entrepreneur retaining control (coupled with ceding
some ownership cash flow risks) proves an effective
technique to screen angel expertise. By threatening to
take aggressive operational decisions in response to
angel boasting and retaining ownership that permits
her to follow through on the threat, the entrepreneur
can better restrain the adverse selection problem of
the privately informed angel. In line with the frequent
use of different classes of ownership shares in venture
financing (e.g., Class A and B shares), our model also
permits for the possibility of different ownership
classes that allow the entrepreneur to maintain disproportionate control compared to the angel investor.
The entrepreneur undertakes more aggressive operational decisions and assigns more ownership shares
to the angel investor when the entrepreneur can retain
more control rights (i.e., when the degree of ownership-decision rights is less aligned). Among other
implications, our results provide a different perspective on entrepreneurs retaining ownership: it may
reflect a desire to restrain investor private information
rather than signal her own private knowledge. This
novel force is demonstrated in a model in which the
angel is the party with private information. Besides
providing a crisp characterization of the optimal
arrangement, this presumption helps isolate and
highlight the unique aspects of angel investing.
Future work can expand this view to solve for optimal
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society 2901
mechanisms when two-sided private information
(tied to expertise) and two-sided moral hazard (tied
to operational decisions) underlie angel–entrepreneur
relationships.
Acknowledgments
We thank Tyler Atanasov, Ammon Butcher, Hui Chen, John
Fellingham, Ralf Ewert, Hans Frimor, Robert Goex, Klaus
Haider, Michael Kopel, Ulf Schiller, Doug Schroeder, Alfred
Wagenhofer, workshop participants at Chapman University, the DAR & DART Accounting Theory Seminar, and
The University of Texas at Dallas, and especially Kalyan
Singhal (Department Editor), the anonymous Senior Editor,
and two anonymous referees for helpful comments. Anil
Arya and Brian Mittendorf gratefully acknowledge support
from the John J. Gerlach Chair and Fisher Designated Professorship in Accounting, respectively.
Appendix
Proof of the Observation
The proof follows immediately from solving the entrepreneur’s problem in Equation (3) without the (IC) constraints. In particular, the optimal operational decisions of the entrepreneur and the angel are both chosen to
eliminate mismatching costs, and equity sharing and fixed transfers are chosen to hold the angel of each type to
his reservation utility.
Proof of Proposition 1
From the ðDÞ constraints in the entrepreneur’s program (PB), it follows dAð^ θ,0Þ ¼ 0 and dAð^ θ, θÞ ¼ θ. Substituting
this in Equation (4), (PB) is solved presuming that ðIR0Þ and ðIC θÞ do not bind; subsequently, we show that the
solution to the relaxed program satisfies these constraints. We solve for tð0Þ and tð θÞ given the two remaining
constraints ðIR θÞ and ðIC0Þ hold as equalities at the optimal solution. This yields:

tð0Þ ¼ K þ δ θ αðθÞ θ½ K þ ½ 1 λαðθÞ ½ 2dEð θÞ θ αð0Þh i Ω ½ 1 λαð0Þ d2 Eð0Þ and
tð θÞ ¼ K þ δ θ αðθÞh i K θ ½1 λαð θÞ½ θ dEð θÞ 2 þΩ

(A1)

: Using Equation (A1), α∗ð0Þ ¼ 1, and d∗ Eð0Þ ¼ 0 in Equation (6) yields the unconstrained problem in Equation (7).
The first-order condition ∂Eθf g UE =∂dEð θÞ ¼ 2½1 λαð θÞ½ þ½ pdEð θÞ θ½p 1 pαð θÞ ¼ 0 of Equation (7) yields
d∗
Eð θÞ in Proposition 1. Since ∂2Eθf g UE =∂2dEðθÞ ¼ 2p½1 λαð θÞ<0 at d∗ EðθÞ, the second-order condition is satisfied. Finally, substituting for the entrepreneur’s optimal decision, the objective function in Equation (7) is a cubic
function in αðθÞ, with first-order condition as follows:
Eθf g UE ¼ K p ½ þ θ 1 Ω δ θ þ ½1 p½K θ þ θ2αð θÞ
þ
½1 p θ2½ ½1 p pλ
p
αð θÞ2 ½1 p2 θ2λ
p
αð θÞ3
∂Eθf g UE
∂αð θÞ ¼ ½1 p½K θ þ θ2 þ2½1 p θ2½ ½p1 p pλ αð θÞ 3½1pp2 θ2λαð θÞ2:
(A2)
For λ ¼ 0, Equation (A2) reduces to a linear function with a positive intercept and a positive slope, i.e.,
∂Eθf g UE =∂αð θÞ>0, implying α∗ð θÞ ¼ 1. For λ ¼ 1, solving ∂Eθf g UE =∂αð θÞ ¼ 0 yields:
α∗ð θÞ ¼ 1
3 11p p þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ3p½1 pK þ p θ
! s ½1 p2 θ ≥ 0: (A3)
It follows that if the α-value in Equation (A3) is less than 1, it is the optimal solution; else the optimal solution
is the boundary value α∗ð θÞ ¼ 1. The second-order condition at α∗ð θÞ is satisfied, ∂2Eθf g UE =∂αð θÞ2 ¼ 2½1
p θ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r θn o 3p½1 pK þ θh i p2 þ p½1 p þ ½1 p2 =p<0. Given the solution in Proposition 1, ðIR0Þ is satisfied for a
sufficiently large δ-value, namely, δ ≥ α∗ð θÞ K þ ½1 λα∗ð θÞ½2d∗ Eð θÞ θ . For λ ¼ 0, the left-hand side less the rightArya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
2902 Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society
hand side of ðIC θÞ equals: 2d∗ Eð θÞ θ>0. Finally, for λ ¼ 1, the left-hand side less the right-hand side of ðIC θÞ
equals:
½ 1 α∗ð θÞ θ α∗ð θÞ θ 1þ 2α∗ð θÞ½1 p
p K
¼ ½ 1 α∗ð θÞ θ2 p½1 α∗ð θÞ þ ½1 p½2 α∗ð θÞα∗ð θÞ
p
≥ 0:
(A4)
The first expression in Equation (A4) shows that ðIC θÞ is satisfied for the boundary value α∗ð θÞ ¼ 1. To show
that ðIC θÞ is also satisfied for the interior value, we restate the first-order condition in Equation (A2) as
K ¼ θn o p½ þ ½ 2α∗ð θÞ 1 1 ph i 3α∗ð θÞ2 2α∗ð θÞ =p and obtain the equivalent second expression in Equation (A4).
With ðIC θÞ satisfied, the proof of Proposition 1 is complete.
Proof of Proposition 2
From the ðDÞ constraints in Equation (3), it follows that dAð^ θ,θÞ ¼ θ. Making this substitution, we next derive the
local (IC) constraints. Since ^ θ ¼ θ is the maximizer of UAð^ θ,θÞ, from the Envelope Theorem it follows that:
dUAðθ,θÞ
dθ ¼
∂UAð^ θ,θÞ
∂θ

^ θ¼θ
¼
∂ ∂θ
αð^ θÞh i Kθ 1 λαð^ θÞ θ dEð^ θÞ 2 þΩ þ tð^ θÞ K ^ θ¼θ ¼
αð^ θÞ K þ 2 1 λαð^ θÞ dEð^ θÞ θ ^ θ¼θ:
(A5)
Integrating both sides yields:
UAðθ,θÞ ¼ UAð θ, θÞ Z
θ
θ
dUAð~ θ,~θÞ
d~ θ d~ θ ¼ UAð θ, θÞ Z
θ
θ
αð~ θÞ K þ 2 1 λαð~ θÞ dEð~ θÞ ~ θ d~ θ: (A6)
The ðIRÞ constraint for the highest type in Equation (3) binds, that is, UAð θ,θÞ ¼ δ θ. This, coupled with sufficiently large δ ensures the solution that satisfies Equation (A5) also satisfies the ðIRÞ constraints. Using the local
instead of the global (IC) constraints in Equation (3), the entrepreneur’s problem is as in Equation (10). Pointwise
optimization of Equation (10) with respect to dEðθÞ yields the following:
2½1 λαðθÞ½ ¼ dEðθÞ θ HðθÞαðθÞ 0: (A7)
Since ∂2Eθf g UE =∂2dEðθÞ ¼ 2½1 λαðθÞ<0, the second-order condition is satisfied for λαðθÞ<1. For λ ¼ 1 and
αðθÞ ¼ 1, the entrepreneur’s decision is inconsequential since Equation (10) is unaffected by dEðθÞ. Hence, without
loss of generality, dEðθÞ ¼ θ þ αðθÞHðθÞ. Substituting this in Equation (10), the entrepreneur’s problem is:
Max
ðαðθÞÞ,θ∈½0, θ
Eθf g UEðθ,θÞ
≡ Max
ðαðθÞÞ,θ∈½0, θ
θR 0
½Kθ þ KHðθÞαðθÞ þ HðθÞ2αðθÞ2½1 λαðθÞfðθÞdθ þΩ K δ θ:
(A8)
Taking the derivative of Equation (A8) with respect to αðθÞ yields:
∂Eθf g UEðθ,θÞ
∂αðθÞ ¼ KHðθÞ þ HðθÞ2αðθÞ½ 2 3λαðθÞ : (A9)
Note ∂Eθf g UEðθ,θÞ =∂αðθÞjαðθÞ¼1 ¼ HðθÞ ½ K þ ½2 3λHðθÞ is nonnegative for λ ≤ 2=3 or θ ≤ θ∗ and negative for
θ>θ∗, where θ∗ is the θ-value that solves ½3λ 2HðθÞ ¼ K: Hence, in the former case, the maximizer of
EθfUEðθ,θÞg is the boundary point α∗ðθÞ ¼ 1 while, in the latter case, the maximizer is interior and solves the
quadratic equation: KHðθÞ þ HðθÞ2½2αðθÞ 3λαðθÞ2 ¼ 0. This solution is α∗ðθÞ in Proposition 2. Finally, for the
interior α∗ðθÞ value, the second-order condition is satisfied, that is, ∂2EθfUEðθ,θÞg=∂αðθÞ2 ¼
2HðθÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½3Kλ þ HðθÞ<0, proving it is the unique interior solution.
To complete the proof, we now need to show that α∗ðθÞ, in conjunction with Equations (A5) and (A6), satisfies
the global incentive compatibility constraints. First, note that (the interior) α∗ðθÞ is decreasing in θ (recall H0ðθÞ > 0):
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society 2903
∂α∗ðθÞ
∂θ
¼
KH0ðθÞ
2HðθÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½3Kλþ HðθÞ ≤ 0: (A10)
Second, truth-telling is optimal if the condition in Equation (A11) holds:
UAð^θ,θÞ UAðθ,θÞ ¼ UAð^θ,^ θÞ UAðθ,θÞ þ UAð^ θ,θÞ UAð^ θ,^θÞ
¼ R
^θ θ
∂UAð~ θ,~ θÞ
∂~ θ
∂UAð^ θ,~ θÞ
! ∂~ θ d~ θ ≤ 0: (A11)
For ^ θ>θ, the condition in Equation (A11) is satisfied if the integrand is non-positive. Together with Equation

(A5), d∗ EðθÞ ¼ θ þα∗ðθÞHðθÞ, and d∗ Eð^θÞ ¼ ^θ þα

∗ð^ θÞHð^ θÞ, this implies:

∂UAðθ,θÞ
∂θ

∂UAð^ θ,θÞ
∂θ
¼ α∗ðθÞ½ K þ 2½1 λα∗ðθÞα∗ðθÞHðθÞ
α∗ð^ θÞ K þ 2½1 λα∗ð^ θÞ½α∗ð^ θÞHð^ θÞ þ ^ θ θ ≤ 0:
(A12)
The following cases emerge for ^θ>θ.
Case (i) α∗ðθÞ ¼ α∗ð^ θÞ ¼ 1. In this case, the condition in Equation (A12) equals 2½^ θ þ Hð^ θÞ HðθÞ θ½1 λ ≤ 0
since the hazard rate is increasing, that is, Hð^ θÞ>HðθÞ.
Case (ii) 1>α∗ðθÞ>α∗ð^ θÞ. Rewriting the α∗ð^ θÞ and α∗ðθÞ expressions in terms of the hazard rate,
Hð^ θÞ ¼ K=½α∗ð^ θÞ½3λα∗ð^ θÞ 2 and HðθÞ ¼ K=½α∗ðθÞ½3λα∗ðθÞ 2, and noting that H>0 implies 3λα∗ð^ θÞ 2>0 and
3λα∗ðθÞ 2>0, the condition in Equation (A12) can be rewritten as:
∂UAðθ,θÞ
∂θ

∂UAð^ θ,θÞ
∂θ

¼ 2Kλ α∗ðθÞ
”

α∗ð^ θÞ½3λα∗ðθÞ 2
# 2

3λα∗ð½α^ θ∗Þ ðθÞ 2 ½ 3αλα ∗ð^ θ∗ðÞ θÞ 2
2α∗ð^θÞ 1 λα∗ð^ θÞ ½^ θ θ ≤ 0:
(A13)
Since 1>α∗ðθÞ>α∗ð^θÞ , 1>3λα∗ð^ θÞ 2>0, and 3λα∗ðθÞ 2>0, it follows that the term in Equation (A13) is negative, ensuring truth-telling.
Case (iii) 1 ¼ α∗ðθÞ>α∗ð^ θÞ. Rewriting the α∗ð^ θÞ expression as Hð^ θÞ ¼ K=½α∗ð^ θÞ½3λα∗ð^ θÞ 2 and noting that H ≥ 0
implies 3λα∗ð^ θÞ 2 ≥ 0, Equation (A12) yields:
∂UAðθ,θÞ
∂θ

∂UAð^θ,θÞ
∂θ
¼ K þ 2HðθÞ½1 λ λα∗ð^ θÞ2K
3λα∗ð^ θÞ 2 2α∗ð^ θÞ 1 λα∗ð^ θÞ ^ θ θ
≤
K 1 α∗ð^ θÞ h i n o 2 þ 2α∗ð^ θÞ α∗ð^ θÞ2 λ 2
α∗ð^ θÞ½3λα∗ð^ θÞ 2 2α∗ð^ θÞ 1 λα∗ð^ θÞ ^θ θ ≤ 0:
(A14)
The last line follows from HðθÞ ≤ Hð^ θÞ and setting HðθÞ ¼ Hð^ θÞ ¼ 1=½α∗ð^ θÞ½3λα∗ð^ θÞ 2. The term,
½2 þ 2α∗ð^ θÞ α∗ð^ θÞ2λ 2, increases in λ and is 2½1 α∗ð^ θÞ½2 þα∗ð^ θÞ=½3α∗ð^ θÞ>0 at the lower bound, λ ¼ 2=½3α∗ð^ θÞ.
If the angel instead understates his expertise, that is, ^ θ<θ, the condition in Equation (A11) is satisfied if the integrand is nonnegative. The expressions for the two cases (i) and (ii) are equivalent to the ones before. Noting that
^ θ<θ and thereby Hð^ θÞ<HðθÞ and αð^ θÞ>αðθÞ, it can be verified that both expressions are nonnegative. Case (iii)
α∗ð^ θÞ ¼ 1>α∗ðθÞ follows from applying the same logic as in Equation (A14). That is, rewriting the α∗ðθÞ expression as HðθÞ ¼ K=½α∗ðθÞ½3λα∗ðθÞ 2 and noting that H ≥ 0 implies 3λα∗ðθÞ 2 ≥ 0. Then using Equation (A12)
yields:
∂UAðθ,θÞ
∂θ

∂UAð^ θ,θÞ
∂θ
¼ K þ λα∗ðθÞ2K
3λα∗ðθÞ 2 þ 2½1 λ½θ ^ θ Hð^ θÞ
≥
K½ 1 α∗ðθÞ h i n o 2 þ 2α∗ðθÞ α∗ðθÞ2 λ 2
α∗ðθÞ½3λα∗ðθÞ 2 þ 2½1 λ½θ ^ θ ≥ 0:
(A15)
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
2904 Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society
The last line follows from HðθÞ ≥ Hð^ θÞ and setting Hð^ θÞ ¼ HðθÞ ¼ 1=½α∗ðθÞ½3α∗ðθÞλ 2. The term,
½2þ2α∗ðθÞ α∗ðθÞ2λ 2, increases in λ and is positive at the lower bound, λ ¼ 2=½3α∗ðθÞ. This completes the proof
of Proposition 2.
Proof of Proposition 3
Part (i) follows from two facts. First, from Proposition 2(i), the cut-off value, θ∗, (weakly) decreases in λ. Second,
for the interior α∗ðθÞ-value.
∂α∗ðθÞ
∂λ
¼
3Kλ þ2HðθÞ þ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½3Kλ þHðθÞ
6λ2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HðθÞ½3Kλ þHðθÞ ≤ 0: (A16)
Since d∗∗
E ðθÞ ¼ θ and d∗ EðθÞ ¼ θ þ α∗ðθÞHðθÞ, it follows that ∂d∗∗ E ðθÞ=∂λ ¼ 0 and ∂d∗ EðθÞ=∂λ ≤ 0, proving part (i). Part
(ii) follows from Equation (A10), that is, ∂α∗ðθÞ=∂θ ≤ 0 and the following (for interior values):

∂½d∗ EðθÞ d

¼
3Kλ þ2HðθÞ þ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½ 3Kλ þHðθÞ H0ðθÞ
≥ 0:

(A17)

∗∗ E ðθÞ
∂θ
6λ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½ 3Kλ þHðθÞ Since ∂d∗∗
E ðθÞ=∂θ ¼ 1, Equation (A17) implies ∂d∗ EðθÞ=∂θ ≥ 0. This proves part (ii). Part (iii) follows from θ∗
decreasing in H and the following (for interior values):
∂α∗ðθÞ
∂H ¼
K
2HðθÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½ 3Kλ þHðθÞ ≤ 0
∂d∗
EðθÞ
∂H ¼
∂½d∗ EðθÞ d∗∗ E ðθÞ
∂H ¼
3Kλ þ2HðθÞ þ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½ 3Kλ þHðθÞ
6λ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½ 3Kλ þHðθÞ ≥ 0:
(A18)
Proof of Corollary 1
The value-maximizing regulator’s goal is to choose λ to minimize the mismatching cost of the entrepreneur’s
decision, ½1 λα∗ðθÞ½θ d∗ EðθÞ2 ¼½1 λα∗ðθÞα∗ðθÞ2HðθÞ2. From Proposition 3(i), ∂α∗ðθÞ=∂λ ≤ 0. Given this, the
proof follows from the fact that 1 λα∗ðθÞ too is decreasing in λ as shown in Equation (A19) for the interior solution; thus, the entrepreneur selects the maximal λ-value of 1.
∂½1 λα∗ðθÞ
∂λ
¼
K
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pHðθÞ½3Kλ þHðθÞ ≤ 0: (A19)
From the entrepreneur’s perspective, for any λ ≥ 0, the expected utility is as in Equation (A8). Since Equation
(A8) presents the entrepreneur’s problem as an unconstrained optimization, using the envelope theorem with
Equation (A8) reveals that:
dEθf g UEð Þ θ,θ
dλ ¼ Z
θ
0
½HðθÞ2α∗ðθÞ3fðθÞdθ<0: (A20)
The entrepreneur’s preference for λ ¼ 0 follows from Equation (A20).
Proof of Proposition 4
Substituting d∗ Að^ θ,θ;τÞ from Equation (12) in UAð^ θ,θÞ, and using steps analogous to those in Proposition 2, the
local (IC) constraints are as follows:
UAðθ,θÞ ¼ UAð θ, θÞ Z
θ
θ
αð~ θÞ Kþ2½1 λαð~ θÞ½1þ τ
” # ½1þ τλαð~ θÞ ½dEð~ θÞ ~ θ d~ θ: (A21)
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society 2905
Again, the ðIRÞ constraint for the highest type binds, that is, UAð θ, θÞ ¼ δ θ, and a sufficiently large δ ensures that
the solution also satisfies the remaining ðIRÞ constraints. Using the local instead of the global incentive compatibility constraints, the entrepreneur’s problem is as in Equation (15). Pointwise optimization of Equation (15) with
respect to dEðθÞ yields the solution in part (ii). Substituting this in the entrepreneur’s objective function, and taking the derivative yields:
∂Eθf g UEðθ,θÞ
∂αðθÞ ¼ HðθÞK þ2½ Hð½1θÞð þ1τα þðτθÞ þ Þλ2τKλ αðθÞ
þHðθÞλ HðθÞð Þ τ 3 ð Þ þ 1þ τ τ2λK αðθÞ2 2HðθÞτð1þ τÞλ2αðθÞ3
½1þ τλαðθÞ2 :
(A22)
For λ ¼ 0, Equation (A22) reduces to ∂Eθf g UEðθ,θÞ =∂αðθÞ ¼ HðθÞ½ Kþ2HðθÞð1þ τÞαðθÞ , yielding the standard
solution α∗ðθ;τÞ ¼ 1. For λ>0, Descartes’ rule of signs shows that at most one positive root exists and the discriminant of the cubic polynomial of αðθÞ in Equation (A22) being positive, the equation ∂EθfUEðθ,θÞg=∂αðθÞ ¼ 0 has
exactly one positive root, and the root solves:
K HðθÞ½1þ ταðθÞ½ λαðθÞ½ 3þ2ταðθÞλ τ 2
½1þ τλαðθÞ2 ¼ 0: (A23)
From Equation (A23), αðθ;τÞ solves the equation HðθÞ ¼ H∗ðαðθ;τÞ;τÞ, where H∗ðα;τÞ is:
H∗ðα;τÞ ¼ K½ 1þ τλα 2
½ 1þ τ α λα ½ ½ 3þ2τλα τ 2 : (A24)
Since H∗ðα;τÞ is strictly decreasing in α, there exists a cut-off value θ∗ which satisfies Hðθ∗Þ ¼ H∗ð1;τÞ. The maximizer of EθfUEðθ,θÞg is thus the boundary point α ¼ 1 if θ ≤ θ∗ and the interior solution αðθ;τÞ if θ ≥ θ∗. This solution is denoted by α∗ðθ;τÞ in part (i). Note, ∂2Eθf g UE =∂α2 ¼ 2H2½1þ τ λα∗ 3þ3τλα∗ þ τ2λ2α∗2 1 =
½ 1þ τλα∗ 3<0 as a positive hazard rate implies λα∗½3þ2τλα∗ τ>2 which, in turn, implies
λα∗½3þ3τλα∗ þ τ2λ2α∗2>1. Hence, the second-order condition for the interior solution is satisfied. Furthermore,
αðθ;τÞ is (weakly) decreasing in θ:

∂αðθ;τÞ
∂θ
¼

½ 1þ τ αðθ;τÞ2½ λαðθ;τÞ½ 3þ2τλαðθ;τÞ τ 2 2H0ðθÞ
2K½ 1þ τλαðθ;τÞ h i λαðθ;τÞh i 3þ3τλαðθ;τÞ þ τ2λ2αðθ;τÞ2 1

≤ 0:

(A25)

Equation (A25) follows from the fact that a positive hazard rate implies λα∗½3þ2τλα∗ τ>2 and
λα∗½3þ3τλα∗ þ τ2λ2α∗2>1. The proof is complete by the identification of the local (IC) in part (iii), and using the
steps in the proof of Proposition 2 to prove that the global incentive compatibility constraints are also satisfied.
Proof of Corollary 2
The cut-off θ-value is given by Hðθ∗Þ H∗ð1;τÞ ¼ 0 where, using Equation (A24),
H∗ð1;τÞ ¼ K½1þ τλ2= ½1þ τ½3λ þ2τλ2 τλ 2 . From the implicit function theorem:
∂θ∗
∂τ
¼
2K½ 1 λ ½ 1þ τλ 2λ þ τλ2 1
½ 1þ τ 2 3λ þ2τλ2 τλ 2 2H0ðθ∗Þ ≤ 0: (A26)
The inequality in Equation (A26) follows from the fact that Hðθ∗Þ ¼ H∗ð1;τÞ ≥ 0 implies 3λ þ2τλ2 τλ 2 ≥ 0
which, in turn, implies 2λ þ τλ2 1 ≥ 0.
The angel’s optimal (interior) share is determined by HðθÞ ¼ H∗ðαðθ;τÞ;τÞ, and the implicit function theorem
yields:
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
2906 Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society
∂αðθ;τÞ
∂τ
¼
αðθ;τÞ½ 1 λαðθ;τÞ h i 2λαðθ;τÞ þτλ2αðθ;τÞ2 1
½ 1 þ τ h i 3λαðθ;τÞ þ 3τλ2αðθ;τÞ2 þτ2λ3αðθ;τÞ3 1 ≤ 0,
∂d∗
Eðθ;τÞ
∂τ
¼ HðθÞ∂α∗ðθ;τÞ
∂τ
≤ 0 and
∂d∗
Aðθ;τÞ
∂τ
¼ HðθÞ
α∗ðθ;τÞ½ 1 λα∗ðθ;τÞ τh i 2λα∗ðθ;τÞ þ τλ2α∗ðθ;τÞ2 1 ∂α∗∂ðτθ;τÞ
½ 1 þτλα∗ðθ;τÞ 2 ≤ 0:
(A27)
In assigning the sign of the expressions in Equation (A27), we use that the fact that the first-order condition for
αðθ;τÞ together with HðθÞ ≥ 0 implies λα∗½3 þ 2τλα∗ τ ≥ 2, which implies 2λα∗ þτλ2α∗2 ≥ 1.
Proof of Proposition 5
Substituting dAð^ θ,θ;νÞ ¼ θ þν=½2αð^ θÞ in UAð^ θ,θÞ and using steps analogous to those in Proposition 2, the local
(IC) constraints are as follows:
UAðθ,θÞ ¼ UAð θ, θÞ Z
θ
θ
αð~ θÞ K þ νλþ 2 1 λαð~ θÞ dEð~ θÞ ~ θ d~ θ: (A28)
Again, the (IR) constraint for the highest type binds, that is, UAð θ, θÞ ¼ δθ, and a sufficiently large δ ensures that
the solution also satisfies the remaining ðIRÞ constraints. From Equations (2) and (16), we get:
UEðθ,θÞ ¼ Vðθ,θÞ þνλαðθÞdAðθ,θ;vÞ UAðθ,θÞ K. Using dAðθ,θ;νÞ ¼ θ þ ν=½2αðθÞ and the local instead of the global incentive compatibility constraints, the entrepreneur’s problem is as in Equation (17). Pointwise optimization
of Equation (17) with respect to dEðθÞ yields the solution in part (ii). Substituting this in the entrepreneur’s objective function, and taking the derivative yields:
∂Eθf g UE
∂αðθÞ ¼
ν2λ 4αðθÞ2f g ½ αðθÞHðθÞ½ 3λαðθÞ 2 K νλ HðθÞ νλθ
4αðθÞ2 : (A29)
For λ ≤ 2=3, Equation (A29) is nonnegative for all θ, implying α∗ðθ;νÞ ¼ 1 for all θ. For the boundary value,
θ ¼ 0, Equation (A29) equals v2λ=½4α2ðθÞ>0 as Hð0Þ ¼ 0, implying α∗ð0;νÞ ¼ 1. For λ>2=3 and θ>0, Descartes’
rule of signs shows that at most one positive root exists. Absent a positive root, α∗ðθ;νÞ ¼ 1. In finding the positive
root, first note that the numerator of Equation (A29) equals λv2 ≥ 0 when α ¼ 0, thus, the positive root represents
a maximum. Denoting this positive root by αðθ;νÞ confirms the derivation of α∗ðθ;νÞ in part (i). Next, we show
that α∗ðθ;νÞ is decreasing in θ. Applying the implicit function theorem to the first-order condition in the numerator of Equation (A29) yields (for an interior α∗ðθ;νÞ):
∂α∗ðθ;νÞ
∂θ
¼
α∗ðθ;νÞ
2
h i 6λα∗ðθ;νÞ2HðθÞ 4α∗ðθ;νÞHðθÞ K νλ H0ðθÞ νλ
h i 6λα∗ðθ;νÞ2HðθÞ 3α∗ðθ;νÞHðθÞ K νλ HðθÞ νλθ
≤
α∗ðθ;νÞ
2θ
h i 6λα∗ðθ;νÞ2HðθÞ 4α∗ðθ;νÞHðθÞ K νλ HðθÞ νλθ
h i 6λα∗ðθ;νÞ2HðθÞ 3α∗ðθ;νÞHðθÞ K νλ HðθÞ νλθ ≤ 0:
(A30)
The first inequality in Equation (A30) is established by noting that a convex hazard rate implies
HðθÞ=θ<H0ðθÞ. This follows from d½H0ðθÞθ HðθÞ=dθ ¼ H00ðθÞ>0 and H0ð0Þ0 Hð0Þ ¼ 0: The second inequality
follows from the first-order condition in Equation (A29) and ν2λ ≥ 0 which imply 3Hλα∗2 2Hα∗ ≥ K þνλ and
½3Hλα∗2 2Hα∗ K νλH ≥ νλθ, and thus both the numerator and denominator are nonnegative. The proof is
complete by the identification of the local (IC) in part (iii), and following the steps in the proof of Proposition 2 to
prove that the global incentive compatibility constraints are also satisfied.
Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society 2907
Proof of Corollary 3
The cut-off θ-value at which α∗ðθ;νÞ ¼ 1 is obtained by solving ν2λ 4½ ¼ f g ½3λ 2Hðθ∗Þ Kνλ Hðθ∗Þ νλθ∗ 0,
and it is increasing in ν:
∂θ∗
∂ν
¼
λ ν ½ þ2θ∗ þ2Hðθ∗Þ
2f g ½ ½6λ4Hðθ∗Þ K νλ H0ðθ∗Þ vλ ≥ 0: (A31)
To show that the denominator is positive, note that the definition of the cut-off θ-value and ν2λ ≥ 0 together

imply	½3λ2H ≥ Kþ νλ	and	½ ½3λ 2H Kνλ H ≥ νλθ∗,	which rate	imply implies	½6λ4H>Kþ νλ H0ðθ∗Þ ≥ Hðθ∗Þ=θ∗	and and
½ ½6λ 4H Kνλ H>νλθ∗.	Finally,	a	convex	hazard

½ ½6λ 4H Kνλ H0 νλ ≥ f g ½ ½6λ4H Kνλ H νλθ∗ =θ∗ >0.
For the interior α∗ðθ;νÞ solution, the implicit function theorem yields:
∂α∗ðθ;νÞ
∂ν
¼
λ ν h i þ2α∗ðθ;νÞ2ðHðθÞ þθÞ
4α∗ðθ;νÞn o h i 6λα∗ðθ;νÞ2HðθÞ 3α∗ðθ;νÞHðθÞ Kνλ HðθÞ νλθ ≥ 0: (A32)
As with Equation (A30), the first-order condition in Equation (A29) implies that the denominator is nonnega

tive. From Proposition 5(ii), ∂d∗ Eðθ;νÞ=∂ν ¼ θ∂α

∗ðθ;νÞ=∂v ≥ 0; the inequality follows from Equation (A32). Finally,

from Proposition 5(ii),
∂d∗
Aðθ;νÞ
∂ν
¼
α∗ðθ;νÞ ν∂α∗∂ðνθ;νÞ
2α∗ðθ;νÞ2 : (A33)
For, α∗ðθ;νÞ at the boundary value of 1, Equation (A33) is clearly positive. For interior α∗ðθ;νÞ, using the expression in Equation (A32) and the first-order condition from Equation (A29) in Equation (A33) yields:

∂d∗ Aðθ;νÞ ∂ν	h i 6HðθÞλα∗ðθ;νÞ2 2HðθÞα∗ðθ;νÞ νλ HðθÞ νλθ
4α∗ðθ;νÞn o h i 6λα∗ðθ;νÞ2HðθÞ 3α∗ðθ;νÞHðθÞ Kνλ HðθÞ νλθ

¼
: (A34)
Again, Equation (A29) and ν2λ ≥ 0 guarantee that the denominator and the numerator are positive.
Notes
1In effect, while ownership and control typically go handin-hand (i.e., λ ¼ 1), we permit the possibility of different
ownership classes that permit the entrepreneur to maintain disproportionate control.
2While the fixed transfer is modeled as a payment made
directly from the entrepreneur to the angel for simplicity,
it can also equivalently be framed as a debt commitment
paid from the generated firm value. In this case, the transfer would be scaled by the ownership sharing arrangement, that is, the debt payment made by the firm to the
angel in the alternate formulation would equal t=ð1αÞ in
the current model.
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Arya, Mittendorf, and Pfeiffer: Incentive Provision for Angel Investors
Production and Operations Management 30(9), pp. 2890–2909, © 2021 Production and Operations Management Society 2909
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