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Corporate Governance in the Presence of Active and Passive Delegated Investment

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We examine the governance role of delegated portfolio managers. In our model, investors decide how to allocate their wealth between passive funds, active funds, and private savings, and asset management fees are endogenously determined. Funds' ownership stakes and asset management fees determine their incentives to engage in governance. Whether passive fund growth improves aggregate governance depends on whether it crowds out private savings or active funds. In the former case, it improves governance even if accompanied by lower passive fund fees, whereas in the latter case, it improves governance only if it does not increase fund investors' returns too much. Regulations that decrease funds' costs of engaging in governance may decrease total welfare. Moreover, even when such regulations are welfare improving and increase firm valuations, they can be opposed by both fund investors and fund managers.
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Corporate governance in the presence of active and
passive delegated investment
Adrian Aycan CorumyAndrey MalenkozNadya Malenkox
August 2020
Abstract
We examine the governance role of delegated portfolio managers. In our model,
investors decide how to allocate their wealth between passive funds, active funds, and
private savings, and asset management fees are endogenously determined. Funds’own-
ership stakes and asset management fees determine their incentives to engage in gov-
ernance. Whether passive fund growth improves aggregate governance depends on
whether it crowds out private savings or active funds. In the former case, it improves
governance even if accompanied by lower passive fund fees, whereas in the latter case,
it improves governance only if it does not increase fund investors’returns too much.
Regulations that decrease funds’ costs of engaging in governance may decrease total
welfare. Moreover, even when such regulations are welfare improving and increase …rm
valuations, they can be opposed by both fund investors and fund managers.
Keywords: corporate governance, delegated asset management, passive funds,
index funds, competition, investment stewardship, engagement
JEL classi…cations: G11, G23, G34, K22
We are grateful to Alon Brav for helpful comments and suggestions.
yCornell University. Email: corum@cornell.edu.
zUniversity of Michigan and CEPR. Email: amalenko@umich.edu.
xUniversity of Michigan, CEPR, and ECGI. Email: nmalenko@umich.edu.
1
1 Introduction
Institutional ownership has grown tremendously over the last decades, rising to more than
70% of US public …rms. The composition of institutional ownership has also changed, with
a remarkable growth in index fund ownership. The fraction of equity mutual fund assets
held by passive funds is now greater than 30%, and the Big Three index fund managers
(BlackRock, Vanguard, and State Street) alone cast around 25% of votes in S&P 500 …rms
(Appel et al., 2016; Bebchuk and Hirst, 2019a). How active and passive asset managers
monitor and engage with their portfolio companies has thus become of utmost importance
for the governance and performance of public …rms. In 2018, the SEC chairman Jay Clayton
encouraged the SEC Investor Advisory Committee to examine “how passive funds should
approach engagement with companies,”and during the 2018 SEC Roundtable on the Proxy
Process, Senator Gramm noted that “what desperately needs to be discussed [in the context
of index fund growth] ... is corporate governance.”
1
There is considerable debate in the literature about the governance role of asset managers
and the di¤erent incentives faced by active vs. passive fund managers. Some argue that
index funds “have incentives to underinvest in stewardship”(Bebchuk and Hirst, 2019b) and
even propose that “lawmakers consider restricting passive funds from voting at shareholder
meetings”(Lund, 2018). Others disagree and counter that passive investors have “signi…cant
incentives ... to play their current roles in corporate governance responsibly” (Rock and
Kahan, 2019) and that “existing critiques of passive investors are unfounded”(Fisch et al.,
2019). The existing empirical evidence is also mixed: on the one hand, Appel, Gormley,
and Keim (2016, 2019) …nd that passive ownership is associated with more independent
directors, fewer antitakeover defenses, and greater success of activist investors. On the other
hand, Brav et al. (2019) and Heath et al. (2020) conclude that index funds vote against
management more rarely than active funds, and Schmidt and Fahlenbrach (2017) and Heath
et al. (2020) …nd that passive ownership is associated with more CEO power, less board
independence, and worse pay-performance sensitivity.
Motivated by these ongoing academic and policy discussions, the goal of our paper is to
1See the SEC chairman’s statement at https://www.sec.gov/news/public-statement/statement-clayton-
iac-091318 and the 2018 SEC roundtable transcript at https://www.sec.gov/…les/proxy-round-table-
transcript-111518.pdf.
2
provide a theoretical framework to analyze the governance role of active and passive asset
managers. We are particularly interested in the following questions. How does competition
among funds a¤ect their assets under management and fees and, in turn, fund managers’
incentives to engage in governance? What are the e¤ects of passive fund growth? What
is the relation between asset management fees and governance? And what are the expected
ects of policy proposals that have been put forward to improve the governance role of asset
managers?
In our model, fund investors decide how to allocate their capital by choosing between
three options: they can either save privately or invest with either an active or a passive (in-
dex) fund manager by incurring a search cost. If an investor decides to delegate his capital
to a fund manager, the two negotiate an asset management fee, which is a certain frac-
tion of the realized value of the fund’s assets under management (AUM) at the end of the
game. Next, trading takes place. Passive funds invest all their AUM in the value-weighted
market portfolio. Active funds invest strategically, exploiting trading opportunities due to
liquidity investors’demand: they buy stocks with low liquidity demand, i.e., those that are
undervalued,”and do not invest in “overvalued”stocks with high liquidity demand. After
investments are made, fund managers decide how much costly e¤ort to exert in order to
increase the value of their portfolio companies. ort captures multiple actions that a share-
holder can take to increase …rm value: interacting and engaging with the …rm’s management
and board, investing resources to make informed voting decisions, ongoing monitoring activ-
ities, and more confrontational tactics such as submitting shareholder proposals, nominating
directors, and aggressively questioning management at annual meetings and on conference
calls. All of these tactics are regularly employed by institutional investors, as evidenced by
the survey of McCahery, Sautner, and Starks (2016). We refer to these actions as engaging
in governance or monitoring.
The key determinants of a fund manager’s incentives to engage in governance are the
fund’s stake in the …rm and the fees it charges for assets under management: The higher is
the fund’s stake, the more its AUM increase in value due to monitoring; and the higher are
the fees, the more is captured by the fund manager from this increase in value.2(See Lewellen
2These properties are consistent with the observed empirical evidence. For example, Iliev and Lowry
(2015) and Iliev, Kalodimos, and Lowry (2020) show that funds with higher equity stakes are more likely to
3
and Lewellen (2018) for an empirical estimate of funds’incentives to engage in governance
based on the analysis of their portfolios and asset management fees.) The equilibrium stake
and fees, in turn, depend on the fund’s combined AUM, the fees and expected returns of other
funds in the market, and liquidity investors’demand since it determines funds’portfolios.
While the model captures all of these e¤ects, it is very tractable, allowing us to analyze
the e¤ect of important market characteristics on the equilibrium level of governance, …rm
valuations, investors’payo¤s, and total welfare.
Our analysis produces several implications. First, we emphasize that the relation between
passive funds’fees and the equilibrium level of governance is far from obvious and could be
negative. It is frequently argued that the growth in passive funds is detrimental to governance
because of the low fees they charge to investors which, in turn, lead to lower incentives to
be engaged shareholders. However, this argument does not take into account that fees do
not change in isolation, and a decrease in fees is typically accompanied by other changes
that are relevant for governance, such as the reallocation of investor funds from private
savings to asset managers, the reallocation of funds across di¤erent types of asset managers,
and changes in funds’investment portfolios. Our model analyzes the combination of these
general equilibrium e¤ects and shows that greater availability of passive fund managers could
simultaneously decrease passive funds’fees but improve the overall corporate governance.
Intuitively, when passive funds are more easily available (or formally, in the context of the
model, are easier for investors to search for) and charge lower fees, their aggregate AUM
increase, which, in turn, increases their stakes in the …rms and improves their incentives to
engage in governance. If investors’aggregate wealth is su¢ ciently large, the entry of passive
fund managers does not signi…cantly a¤ect active funds’AUM and fees, because passive fund
managers primarily crowd out fund investors’private savings. Hence, active fund managers
continue to engage in governance, and the combined e¤ect of passive fund growth is positive
despite the decrease in fund fees.
However, if investors’aggregate wealth is more limited, the growth in passive funds could
be detrimental to governance. In this case, their entry no longer crowds out private savings
conduct governance research and to vote “actively”instead of relying on proxy advisors’recommendations,
while Heath et al. (2020) document that index funds with high expense ratios are more likely to vote against
management than those with low expense ratios.
4
but instead crowds out investors’allocations to active fund managers.3The competition
for investor funds substantially reduces active asset management fees and their AUM and
this, in turn, decreases active funds’incentives to engage in governance. As a result, there
is a heterogeneous e¤ect of passive fund entry on the governance of di¤erent types of …rms.
Firms that are particularly “undervalued”(because of liquidity investors’low demand), and
hence are primarily held by portfolio-optimizing active fund managers, experience a decrease
in the overall level of investor monitoring due to active fund managers’ lower incentives.
In contrast, …rms that are not in active fund managers’portfolios see an improvement in
governance: without passive funds, they are primarily held by liquidity investors who do not
engage in governance, whereas in the economy with passive funds, they are held by passive
fund managers, who have incentives to be engaged.
Given these heterogeneous e¤ects, what is the e¤ect of passive fund growth on the ag-
gregate level of governance in the economy? We show that whether this e¤ect is positive
or negative depends on whether the growth of passive funds substantially increases fund
investors’returns on their investment. There is a trade-o¤ between the two: if passive fund
entry is su¢ ciently bene…cial for fund investors’ welfare, it is detrimental to governance,
and vice versa. Intuitively, passive fund entry increases fund investors’returns on their in-
vestment only if its presence increases competition among funds and substantially decreases
asset management fees. But lower fees decrease funds’incentives to invest in monitoring and
hence are detrimental to governance. Put di¤erently, e¤ective fund manager engagement
requires that funds earn su¢ cient rents from managing investors’assets.
Our model also has implications for policy proposals suggesting to reduce institutional
investors costs of engaging in governance. A common criticism, especially about passive
funds, is that they do not have su¢ cient resources to monitor the governance of their portfo-
lio …rms and engage with them. Based on this criticism, it is natural to suggest regulations
inducing passive funds to increase investments in their stewardship teams, which in the con-
text of the model can be interpreted as reducing ex-post costs of e¤ort. However, our model
shows that the relationship between such regulations and total welfare is generally subtle.
3Passive funds seem to be replacing active funds in recent years: according to Morningstar (2019), actively
managed U.S. stock funds have posted net out‡ows in 11 out of the last 12 years, while passive funds have
posted net in‡ows in all these years. See https://www.morningstar.com/insights/2019/06/12/asset-parity.
5
On the one hand, decreasing fund managerscosts of engaging in governance induces them
to monitor and engage more, which increases the value of their portfolio …rms. On the other
hand, this increase in …rm valuations can come at the expense of fund investors’well-being.
Intuitively, traders in …nancial markets rationally anticipate the e¤ects of increased engage-
ment on valuations and bid up the prices, so that the fund does not make trading pro…ts
on its monitoring. Moreover, increased prices imply a lower ability of the fund to realize
gains from trade, which can harm fund investors. Overall, lower realized gains from trade
can make such regulations welfare-decreasing despite their positive e¤ects on governance.
Indeed, we show that decreasing the costs of engagement beyond a certain threshold always
harms welfare.
Likewise, fund managers themselves do not always bene…t from decreasing their costs of
engagement, e.g., increasing the size of their stewardship teams, even if it is costless. Since
this induces the fund to monitor more, and more monitoring is, in turn, detrimental to fund
investors, the fund may experience out‡ows and thereby a reduction in its asset management
fees. We show that the active fund is more susceptible to such out‡ows since it has a higher
relative advantage in realizing gains from trade and hence is hurt more when prices increase.
Thus, while passive funds often …nd it optimal to decrease their costs of engagement, active
funds do not. Interestingly, this implies that regulations inducing funds to increase their
stewardship teams can be welfare improving but nevertheless be strongly opposed both by
fund managers and fund investors. More generally, our analysis suggests that to understand
the e¤ects of governance regulations, it is important to consider the potential e¤ects of
regulation on funds’assets under management.
Our paper contributes to the literature on shareholder activism, which emphasizes voice
and exit as the two key mechanisms through which shareholders can increase value. The focus
of our paper is on voice. Many papers examine the interaction between trading in …nancial
markets and shareholder activism through voice.4Our paper also studies the interaction
between shareholders’trading and activism decisions, but di¤erently from the literature, we
focus on shareholders who are delegated asset managers and examine how the competition
between funds and the simultaneous presence of active and passive funds a¤ect funds’fees,
4E.g., Admati, P‡eiderer, and Zechner (1994), Kahn and Winton (1998), and Maug (1998), among others.
See Edmans and Holderness (2016) for a survey.
6
AUM, investment decisions, and through this, their engagement in governance. Given our
interest in these questions, we abstract from more speci…c details of the activism process, such
as negotiations with management (Corum, 2020), the role of the board (Cohn and Rajan,
2013), communication (Levit, 2019), pushing for the sale of the …rm (Burkart and Lee,
2019; Corum and Levit, 2019), as well as the interaction between multiple shareholders (e.g.,
Edmans and Manso, 2011; Brav, Dasgupta, and Mathews, 2019). Dasgupta and Piacentino
(2015) and Cvijanovic, Dasgupta, and Zachariadis (2019) also study the governance role of
asset managers, but di¤erently from our paper, focus on how governance via exit is a¤ected
by their ‡ow-based incentives. Edmans, Levit, and Reilly (2019) and Levit, Malenko, and
Maug (2020) analyze index funds in extensions of their models but focus, respectively, on
the interaction between voice and exit, and on index funds’role in voting.
Our paper is also related to empirical studies of index reconstitutions, which examine
how the resulting changes in …rms’ownership structures a¤ect corporate governance.5In
the context of our model, if institutional investors replace liquidity investors (who can be
thought of as retail shareholders) in the …rm’s ownership structure, the …rm’s governance
is expected to improve. On the other hand, if index inclusion primarily a¤ects the mix
between active and passive funds (as, e.g., in Bennett, Stulz, and Wang, 2020, and Heath et
al., 2020), the e¤ects on governance are more subtle and depend on the active and passive
funds’ownership stakes, fees, and costs of engagement. This can potentially reconcile the
con‡icting …ndings on the e¤ects of index inclusion in the literature. More importantly, while
the index reconstitution papers focus on the cross-sectional di¤erences between individual
rms, our key implications concern the time-series e¤ects of passive fund growth on the
aggregate governance in the economy. As we emphasize, these time-series e¤ects crucially
depend on whether passive funds crowd out households’private savings or their investments
into active funds. Hence, the aggregate time-series implications of passive fund growth could
be quite di¤erent from the cross-sectional e¤ects of index reconstitutions.
Finally, our paper contributes to the literature on delegated asset management and the
role of passive investing. This literature examines investor learning about fund manager
skills (e.g., Berk and Green, 2004; Pastor and Stambaugh, 2012), endogenous formation of
5They include Appel, Gormley, and Keim (2016, 2019), Bennett, Stulz, and Wang (2020), Crane,
Michenaud, and Weston (2016), Heath et al. (2020), Schmidt and Fahlenbrach (2017), and others.
7
mutual funds by informed agents (e.g., Admati and P‡eiderer, 1990; Garcia and Vanden,
2009), and the asset pricing implications of benchmarking and asset management contracts
in general (e.g., Cuoco and Kaniel, 2011; Basak and Pavlova, 2013; Bu¤a, Vayanos, and
Woolley, 2019). Within this literature, our paper is most related to studies that examine the
equilibrium levels of active and passive investing and their implications for price e¢ ciency
and welfare (Brown and Davies, 2017; Bond and Garcia, 2019; Garleanu and Pedersen, 2020;
Malikov, 2019). Among these papers, the closest is Garleanu and Pedersen (2020), as we
build on Garleanu and Pedersen (2018, 2020) in modeling the asset management industry
with endogenously determined fees and investors’search costs. But di¤erently from all the
above papers, our focus is on the corporate governance role of delegated asset management.
In particular, while the asset payo¤s in the above papers are exogenous, the asset payo¤s in
our paper are determined endogenously by fund managers’decisions on monitoring. Like our
paper, Buss and Sundaresan (2020) and Kashyap et al. (2020) also study the e¤ects of asset
managers on corporate outcomes, but through very di¤erent channels: Buss and Sundaresan
(2020) show that passive ownership reduces …rms’cost of capital and induces them to take
more risk, while Kashyap et al. (2020) show that due to benchmarking in asset management
contracts, …rms inside the benchmark are more prone to invest and engage in mergers.
The remainder of the paper is organized as follows. Section 2 describes the setup of
the model. Section 3 derives the equilibrium allocation of capital by investors, fundsfees,
investment portfolios, and governance decisions. Section 4 analyzes the implications for
governance, fund investor returns, and total welfare. Finally, Section 5 concludes.
2 Model setup
Our model is motivated by Garleanu and Pedersen (2018, 2020): we follow their approach in
modeling investors’search for fund managers and their bargaining over asset management
fees. Our trading and governance stages are broadly based on Admati, P‡eiderer, and
Zechner (1994). We extend their model to a continuum of …rms (rather than one …rm in
Admati et al.), multiple shareholders that can take actions (rather than one shareholder
in Admati et al.), and we introduce active and passive delegated asset management. In
addition, di¤erently from Admati et al., in which agents are risk-averse, we assume that all
8
agents are risk-neutral, and trading occurs not due to risk-sharing motives but because of
heterogeneous private valuations.
There are three types of agents: (1) fund investors, who decide how to allocate their
capital; (2) fund managers, who make investment and governance decisions; and (3) liquidity
investors. All agents are risk-neutral.
Timeline
The timeline of the model is illustrated in Figure 1. At t= 1, fund investors decide whether
to search for a fund manager or invest their capital outside the …nancial market, which we
refer to as private savings. At t= 2, investors who meet a fund manager negotiate with
the fund manager over the asset management fees. At t= 3, fund managers decide how to
invest their assets under management and trading takes place. At t= 4, each fund manager
decides on e¤ort to exert for each …rm in his portfolio. Finally, at t= 5, all …rms pay o¤,
and the payo¤s are split between fund managers and their investors according to the asset
management fees decided upon at t= 2.
Figure 1. Timeline of the model.
We now describe the three types of agents and each of these stages in more detail.
Fund managers and fund investors
There are two types of risk-neutral fund managers, active and passive (index). The number of
active managers is NA; the number of passive managers is NP. For now, we focus on the case
of NA=NP= 1. While an active fund manager optimally chooses his investment portfolio,
a passive fund manager is restricted to hold a value-weighted index of stocks. Assets in
nancial markets can be accessed by fund investors only through fund managers. Each fund
9
manager o¤ers to invest the capital of fund investors in exchange for an asset management
fee. To focus on the e¤ects of contractual arrangements that are observed in the mutual fund
industry, we ignore the issues of optimal contracting and, following Pastor and Stambaugh
(2012), assume that the fee charged to fund investors is a fraction of the funds realized
value of AUM at date 5. In particular, let fAand fPdenote the fees as the percent of AUM
charged by the active and passive fund manager, respectively. These fees are determined by
bargaining between investors and fund managers, as described below. Then, if the realized
value of fund manager is portfolio at date 5 is ~
Yi, he keeps fi~
Yito himself and distributes
(1 fi)~
Yiamong fund investors in proportion to their original investments to the fund.
There is a mass of risk-neutral investors with capital, who have combined capital (wealth)
W. Each investor has an in…nitesimal amount of capital. At t= 1, each investor decides
whether to invest his capital in the …nancial market by delegating his capital to one of the
fund managers, or whether to invest outside the …nancial market (private savings). We
normalize the return of the outside asset to zero. It can be interpreted as saving at a bank
deposit or simply keeping the funds under the mattress.
If the investor decides to invest his capital with a fund manager, he needs to incur a
search cost. This cost captures the time and resources that investors typically spend to …nd
an asset manager (see Appendix B in Garleanu and Pedersen (2018) for a detailed description
of investors’search process and associated costs). Speci…cally, to …nd a passive (active) fund
manager, an investor with wealth "needs to incur a cost P"( A").6We assume that
A P, i.e., it is more costly to …nd an active fund manager than a passive fund manager.
Intuitively, active fund managers in our model have skill in that they successfully exploit
trading opportunities and thus outperform passive fund managers, who simply invest in the
market portfolio. Hence, fund investors face a trade-o¤ between earning a higher rate of
return on their portfolio but at a higher cost (we can think of Aas the cost of searching for
skill) vs. a lower rate of return at a lower cost.
If an investor incurs a search cost i", he …nds fund manager of type i2 fA; P g, and
they negotiate the asset management fee ~
fithrough Nash bargaining, as in Garleanu and
Pedersen (2018). Suppose that fund managers have bargaining power , and fund investors
6The assumption that search costs are proportional to wealth "is just a normalization, which substantially
simpli…es the exposition.
10
have bargaining power 1. Each investor consumes the proceeds at t= 5.
Let WAand WPdenote the assets under management of the active and passive fund after
the investors make their capital allocation decisions.
Assets and trading
There is a continuum of measure one of …rms (stocks), indexed by j2[0;1]. Each stock is
in unit supply. The date-5 payo¤ of stock jis
Rj=R0+
Mj
X
i=1
eij ;(1)
where R0is publicly known, Mjis the number of shareholders of …rm j, and eij is the amount
of “ort”exerted by shareholder iin …rm jat date 4, as described below.
The initial owners of each …rm are assumed to have low enough valuations to be willing
to sell their shares at any positive price (for example, we can assume that their valuations
are zero), so that the supply of shares in the market is always one. In addition to the initial
owners, there are three types of traders who initially do not hold any stocks: active fund
managers, passive fund managers, and competitive liquidity investors.
The trading model is broadly based on Admati, P‡eiderer, and Zechner (1994), augmen-
ted by passive fund managers: The active fund is strategic in that it takes into account the
impact of its trading on the price, the passive fund buys the index portfolio, and the price
is set to clear the market (i.e., a Walrasian trading mechanism). It can be microfounded
by the following game: …rst, the active and passive fund each submits a market order, then
competitive liquidity investors submit their demand schedules as a function of the price, and
the equilibrium price is the one that clears the market. Short sales are not allowed.
More speci…cally, for each stock, there is a large mass of competitive risk-neutral liquidity
investors, who can each submit any demand of up to one unit. Liquidity investors value
an asset at its common valuation, given by (1), perturbed by an additional private value
component. In particular, liquidity investors valuation of stock jis RjZj, where Zj
captures the amount of liquidity demand driven by hedging needs or investor sentiment:
Stocks with large Zjhave relatively low demand from liquidity investors, while stocks with
11
small Zjhave relatively high demand. We assume that Zjare i.i.d. (across stocks) draws
from a binary distribution: Pr (Zj=ZL) = Pr (Zj=ZH) = 1
2, where ZL> ZH. We will refer
to these two types of stocks as L-stocks and H-stocks, i.e., stocks with low and high liquidity
demand, respectively. The realizations of Zjare publicly observed for all j. We assume that
ZL+ZH
2>0, i.e., the liquidity investors’private valuations of the market portfolio are negative,
which automatically also implies ZL>0. In other words, the market portfolio and, even
more so, the L-stocks, are undervalued by liquidity investors, which enables fund managers
to realize gains from trade by buying these stocks. The role of di¤erent realizations of Zjfor
di¤erent stocks (ZL> ZH) is to create potential gains from active portfolio management.
When trading: (1) liquidity investors have rational expectations in their assessment of
asset payo¤s and trade anticipating the equilibrium level of e¤ort exerted by fund managers;
(2) fund managers of active funds are not price takers: they are strategic in that they take
into account the price impact of their trades; and (3) fund managers of passive funds follow
the mechanical rule of investing all assets under management in a value-weighted portfolio
of all stocks. We denote xij the number of shares held by investor iin …rm j.
Governance stage
After establishing a position in …rm j, each fund manager decides on the amount of e¤ort
to exert in the …rm. If he exerts e¤ort eand is of type i2 fA; P g, he bears a private
cost of e¤ort ci(e). This cost is not shared with fund investors, capturing what happens in
practice (although the equilibrium fees charged to fund investors will be indirectly a¤ected
by these costs). We impose the standard assumptions that ci(0) = 0,c0
i(e)>0,c00
i(e)>0,
c0
i(0) = 0, and c0
i(1) = 1, which guarantee an interior solution to fund managers’decisions
on governance.
As discussed in the introduction, we think of the fund’s e¤ort as any action that a
shareholder can take to increase value: informed voting, monitoring, engagement with man-
agement, as well as more confrontational activism tactics. We refer to these actions broadly
as engagement in governance or monitoring. We allow for derent cost functions for active
and passive funds: for example, active funds’ trading in the …rm’s stock could give them
access to …rm-speci…c information, which could be helpful for their engagement e¤orts and
12
reduce their costs of monitoring.
3 Analysis
We solve the model by backward induction, starting with the fund managers’decisions about
monitoring.
3.1 Governance stage
If fund manager i2 fA; P gwith fee fiand xij shares in …rm jexerts e¤ort eij , his pay,
up to a constant that does not depend on eij , is fixij eij ci(eij ). The …rst-order condition
implies that the fund manager’s optimal e¤ort level satis…es
eij =c01
i(fixij ):(2)
Thus, the fund manager exerts more e¤ort if his fund owns a higher fraction of the …rm
(higher xij) or if he keeps a higher fraction of the payo¤ to himself rather than giving it out
to his investors (higher fee fi).
3.2 Trading stage
During the trading stage, all players rationally anticipate that the e¤ort decisions will be
made according to (2).
Liquidity investors. Each liquidity investor has rational expectations about the e¤ort
that the active and passive fund managers will undertake. Speci…cally, if he expects the
active fund to hold xAj shares and the passive fund to hold xP j shares of stock j, then his
assessment of the pay (1) of the stock is
Rj(xAj ; xP j ) = R0+c01
A(fAxAj ) + c01
P(fPxP j ):(3)
Thus, each liquidity investor …nds it optimal to buy stock jif and only if Rj(xAj; xP j )Zj
Pj, i.e., his valuation of this stock exceeds its price. We focus on the parameter range such
13
that liquidity investors are the marginal traders in each type of stock, Land H. This holds
when the combined AUM of active and passive funds, WA+WP, are not too high, so that
their combined demand for the stock is lower than its supply (a su¢ cient condition for this
to hold is speci…ed in Proposition 1 below). Thus, the price of stock jis given by:
Pj=RjZj:(4)
Equation (4) has intuitive properties. First, the price is decreasing in Zj: all else equal,
the price is lower if demand from liquidity investors is lower, for example, if there is lower
hedging demand or lower investor sentiment (i.e., higher Zj). Second, the price is higher if
Rj=Rj(xAj ; xP j )is higher, i.e., if either the active fund or the passive fund holds more
shares. This is because higher ownership by a fund manager implies higher value creation
given (2), and consequently, higher demand from liquidity investors, leading to a higher price.
We assume that R0> ZL, which ensures that the price of each stock is always positive.
Passive fund manager. The passive fund manager is restricted to investing his assets
under management WPinto the value-weighted portfolio of stocks. Denote this market
portfolio by index M, and note that its price, i.e., the total market capitalization, is PM
R1
0Pjdj =PL+PH
2. The passive fund manager would like to buy xP j units of stock jsuch that
the proportion of his AUM invested in this stock, xP j Pj
WP, equals the weight of this stock in
the market portfolio, i.e., Pj
PM. It follows that xPj is the same for all stocks and equals
xP=WP
PM
:(5)
Note that the passive fund manager’s demand for each stock does not depend on the stock’s
individual price and only depends on the price of the market portfolio.
Active fund manager. The active fund manager strategically chooses which assets to
invest in, choosing between stocks of type L, stocks of type H, and the outside asset with
return zero. We focus on the case when the active fund manager …nds it optimal to only
buy L-stocks, but not H-stocks or the outside asset, and to diversify across all L-stocks (a
su¢ cient condition for this to hold is speci…ed in Proposition 1). Intuitively, stocks with
14
higher liquidity demand are “overpriced”relative to stocks with lower liquidity demand, and
the active fund manager only …nds it optimal to buy the relatively cheaper stocks. Since the
total wealth of the active fund manager is WAand it is allocated evenly among mass 1
2of
L-stocks, the fund manager’s investment in each L-stock is
xAL =2WA
PL
:(6)
Summary of the equilibrium at the trading and governance stage. Combining the
above arguments, we can characterize the equilibrium in the …nancial market and the payo¤s
of all stocks as functions of funds’assets under management WAand WPand the fees fA
and fPthat are determined at stages 1 and 2. Denote the aggregate liquidity demand for
the market portfolio by ZMZL+ZH
2. Since active fund managers only invest in L-stocks,
which constitute half of all stocks, the equilibrium prices and payo¤s of L-stocks and of the
market portfolio are given by the following equations:
PL=RLZL;(7)
PM=RMZM;(8)
RL=R0+c01
A(fAxAL) + c01
P(fPxP);(9)
RM=R0+1
2c01
A(fAxAL) + c01
P(fPxP);(10)
where xPand xAL are given by (5) and (6), respectively. Note that there is a one-to-one
mapping between WAand xAL, and between WPand xP. Therefore, we can treat xAL and
xPas state variables at date 3, which will simplify the exposition.
We next consider fund investors’capital allocation decisions and their bargaining with
fund managers over fees.
3.3 Capital allocation by investors
In…nitesimal investors decide whether to invest their capital into an outside asset and get a
return of zero, or whether to search for an active or passive fund manager and invest with
them. Our baseline analysis focuses on the case where the equilibrium AUM of each fund are
15
positive (a su¢ cient condition for this to hold is speci…ed in Proposition 1).7Then, there are
two possible cases, depending on, as we show below, the aggregate wealth of investors W.
First, if Wis su¢ ciently large, then in equilibrium, investors earn a low rate of return and
are indi¤erent between all the three options: investing in the outside asset (private savings),
investing with the active fund, and investing with the passive fund. Second, if Wis small,
then investors are indi¤erent between investing with the active fund and the passive fund,
and both options dominate investing in the outside asset, i.e., they earn a su¢ ciently high
rate of return. Consider each of these cases separately.
3.3.1 Case 1: Low investor returns
Suppose …rst that private savings occur in equilibrium, i.e., investors earn a low rate of return
from investing in the …nancial market.
Negotiations over fees. We start by …nding the active fund manager’s fees. Consider
an investor with wealth ", and suppose this investor has already incurred the cost to …nd
an active fund manager. To determine the Nash bargaining solution, we …nd each party’s
pay upon agreeing and upon negotiations failing.
First, consider the fund investor. The investor’s payo¤ from agreeing on fee ~
fAis
(1 ~
fA)"
PLR0+c01
A(fAxAL) + c01
P(fPxP):(11)
This is because the fund manager will invest all the investor’s wealth into L-stocks, which
have price PL, and the pay of each of these stocks is given by (9). The investor’s payo¤
if negotiations fail is "because the net return of private savings is zero. The investor also
has an option to search for the passive fund manager, but given the assumption that private
savings occur in equilibrium, the investors are indi¤erent between all three options, so it is
su¢ cient to consider her private savings as the outside option.
Consider the active fund manager. Note that by the envelope theorem, the e¤ect of a
marginal additional investment on the fund manager’s utility via a change in e¤ort is second-
7Lemma 1 in the appendix analyzes equilibria where only one of the funds raises positive AUM, and we
examine these equilibria in some of the implications.
16
order.8Hence, the fund managers additional utility from agreeing on fee ~
fAand getting
additional assets under management "is ~
fARL"
PL, where RLis given by (9). Given the fund
manager’s bargaining power , fee ~
fAis determined via the Nash bargaining solution:
max
~
fA(1 ~
fA)RL
"
PL"1~
fARL
"
PL
:(12)
Since the total surplus created from bargaining is RL"
PL", the fee must be such that the
fund manager gets fraction of this surplus:
~
fARL
"
PL
=RL
"
PL";(13)
or ~
fA=1PL
RL. This implies that the active management fees for all investors are the
same, ~
fA=fA, and satisfy the following …xed point equation:
fA=1PL
RL:(14)
Second, consider the passive fund manager. By exactly the same arguments, the Nash
bargaining solution ~
fPsatis…es:
~
fPRM
"
PM
=RM
"
PM";(15)
or ~
fP=1PM
RM. This implies that the passive management fees for all investors are the
same, ~
fP=fP, and satisfy the following …xed point equation:
fP=1PM
RM:(16)
Asset allocation. Finally, we need to determine the assets under management. In equilib-
rium, investors must be indi¤erent between searching for the active fund manager, searching
8To see this, note that the active manager’s payo¤ is 1
2[fAxAL R0+e+c01
P(fPxP)+
~
fAR0+e+c01
P(fPxP)2"
PLcA(e)], and by the envelope theorem, the derivative with respect to "is
~
fAR0+e+c01
P(fPxP)1
PL, where e=c01
A(fAxAL):
17
for the passive fund manager, and investing in the outside asset, which gives:
(1 fA)RL
"
PL A"= (1 fP)RM
"
PM P"=": (17)
Dividing by ", we get the following conditions for investor inderence
1 + A= (1 fA)RL
PL
;(18)
1 + P= (1 fP)RM
PM
:(19)
Combining these arguments, the equilibrium (fA; fP; xAL; xP; PL; PM; RL; RM)is given by
the solution to the following system of equations: market clearing and optimal monitoring
decisions (7)-(10); fee negotiation conditions (14) and (16); and investor capital allocation
conditions (18) and (19). We characterize this equilibrium in Proposition 1 below.
3.3.2 Case 2: High investor returns
Next, suppose that investors earn a high rate of return from investing in the …nancial market
and thus private savings do not occur in equilibrium. The solution follows the same steps as
those in Section 3.3.1, but with two di¤erences. First, the investor inderence conditions at
the capital allocation stage, (17)-(19), are replaced by: (a) the indi¤erence condition between
investing with active and passive funds,
(1 fA)RL
PL A= (1 fP)RM
PM P;(20)
and (b) the condition that the combined AUM of the funds are equal to W:
WA+WP=W: (21)
The second di¤erence is that during bargaining, the fund investor’s outside option is now
to invest with the other fund manager, which is no longer equivalent to using private savings.
The fund managers’outside options remain unchanged. First, consider negotiations with the
active fund manager. Since the investor’s outside option is to search for the passive fund
18
manager and get (1 fP)RM"
PM P", the total surplus created from bargaining is now
RL"
PL(1 fP)RM"
PM+ P". Hence, the fee must be such that the fund manager gets
fraction of this surplus:
~
fARL
"
PL
=RL
"
PL(1 fP)RM
"
PM
+ P";(22)
which yields ~
fA=fAthat satises the following …xed point equation:
fA=PL
RL
RL
PL(1 fP)RM
PM
+ P:(23)
Similarly, in negotiations with the passive fund manager, the investor’s outside option is
to search for the active fund manager and get (1 fA)RL"
PL A". Therefore, fee ~
fPis
determined from:
~
fPRM
"
PM
=RM
"
PM(1 fA)RL
"
PL
+ A";(24)
which yields ~
fP=fPthat satises the following …xed point equation:
fP=PM
RM
RM
PM(1 fP)RL
PL
+ A:(25)
Combining these arguments, the equilibrium (fA; fP; xAL; xP; PL; PM; RL; RM)is given by
the solution to the following system of equations: market clearing and optimal monitoring
decisions (7)-(10); fee negotiation conditions (23) and (25); and investor capital allocation
conditions (20) and (21). We characterize this equilibrium in Proposition 1 below.
3.4 Equilibrium
We derive the equilibrium in each of the above cases by combining the market clearing and
optimal monitoring conditions, fee negotiation conditions, and investor capital allocation
conditions derived above. From this point on, we assume that fund managers’costs of e¤ort
are quadratic, i.e.,
ci(e) = ci
2e2:
19
While the assumption of quadratic costs is not necessary to characterize the equilibrium
and is not important for many equilibrium properties discussed after Proposition 1 and in
Section 4,9assuming quadratic costs allows us to formulate in closed form the su¢ cient
conditions for the existence of this equilibrium and simpli…es the exposition. In particular,
funds’equilibrium e¤ort levels are then given by eP=fPxP
cPand eAL =fAxAL
cA.
Denote
(1 fA)RL
PL A(26)
the equilibrium gross rate of return that fund investors earn on their investment. In Case
1 above, = 1 since investors are indi¤erent between investing in the outside asset (that
earns a net return of zero) and investing with the fund managers, while in Case 2,  > 1.
Moreover, and intuitively, we show in the appendix that investors’equilibrium rate of return
is decreasing in aggregate investor wealth W. Hence, there exists a cuto¤ on aggregate
investor wealth,
W, such that Case 1 with = 1 obtains for W
W, while Case 2 with
 > 1obtains for W >
W. Together, this allows us to fully characterize the equilibrium.
Proposition 1 (equilibrium).Suppose cP P
AcA,r1<ZM
ZL< r2and W1< W < W2,
where ri; Wiare given by (40)-(41) in the appendix. Then the equilibrium is as follows.
(i)The asset management fees are fA=A
A+(1)and fP=P
P+(1).
(ii)The payo¤s of the L-asset and the market asset are RL= (1 + 1
A+(1)(1))ZLand
RM= (1 + 1
P+(1)(1))ZM.
(iii)The prices of the L-asset and the market asset are PL=1
A+(1)(1)ZLand PM=
1
P+(1)(1)ZM.
(iv)There exists
Wsuch that if W
W, the investors’gross rate of return satis…es = 1,
whereas if W
W,decreases in Wand satis…es the …xed point equation
W=cA
fA
(RLRM)PL+cP
fP
(2RMRLR0)PM:(27)
9For example, for general costs of e¤ort, the equilibrium characterized by Proposition 1 takes exactly the
same form, except that equation (27) becomes W=PL
2fAc0
A(2 (RLRM)) + PM
fPc0
P(2RMRLR0). The
proof of Proposition 1 in the appendix is presented for this more general case.
20
The restrictions on parameters in the statement of the proposition ensure that we consider
the interesting case, i.e., one in which both the active and the passive fund raise positive
AUM, liquidity investors are marginal in both types of stocks, and the active fund …nds it
optimal to invest in L-stocks, and not in H-stocks or the outside asset. As a result, the
active fund holds a less diversi…ed portfolio than the passive fund, which is consistent with
the observed evidence. For the remainder of the paper, we assume that these assumptions
hold with a few exceptions that we explicitly point out.
The assumption cP P
AcAis intuitive: assuming that passive and active funds have
relatively similar monitoring technologies (cPcA), it automatically follows from the as-
sumption that active funds are harder to search for, A P. In addition, Lund (2018)
notes that “governance interventions are especially costly for passive funds, which do not
generate …rm-speci…c information as a byproduct of investing.”
The properties of the equilibrium are as follows. If aggregate investor wealth is large,
investors’outside options in negotiations are limited, which makes the fees charged by asset
managers relatively high and investorsrate of return equal to the rate of investing in the
outside asset, = 1. If, in contrast, aggregate investor wealth is limited, asset managers
compete for investor funds and have to o¤er relative low asset management fees, allowing
investors to earn a rate of return  > 1(fees fi=i
i+(1)decrease in and increase in
aggregate investor wealth W).
Comparing the active and the passive fund, we note that the active fund manager out-
performs the passive fund manager before fees. Indeed, the active fund manager earns a
return of RL
PL= A
1+on his investments, which is greater than P
1+=RM
PM, the return
of the passive fund manager. Accordingly, and consistent with practice, the fee charged
by the active fund manager is higher than the fee charged by the passive fund manager:
fA=A
A+(1)P
P+(1)=fP.
Note also that the payo¤s of both the L-asset and the market portfolio increase with
aggregate investor wealth (RLand RMdecrease in and hence increase in W): higher
investor wealth and thereby higher funds’ AUM imply larger fund managers ownership
stakes, which, in turn, lead to more monitoring and hence higher asset payo¤s. As a result,
the prices of both assets are also higher as funds’AUM increase.
21
Because we are interested in the role of passive funds for corporate governance, it is useful
to understand how the search cost Pects the equilibrium. The growing availability of
passive funds over time can be interpreted as a decrease in P.
Proposition 2.As passive funds become more readily available ( Pdecreases): (1) funds’
fees, fAand fP, decrease; (2) funds’AUM, WA+WP, increase; and (3) fund investors’rate of
return, , increases. In particular, there exists a cuto¤
Psuch that = 1 for P
Pand
 > 1for P<
P.
Intuitively, greater availability of passive funds is generally bene…cial for fund investors: it
decreases fund fees and increases investors’returns on their investment. As a result, investors
allocate more funds from private savings to fund managers, so funds’combined AUM grow.
Proposition 2 is broadly consistent with the observed empirical evidence if we think of
the recent trends in the asset management industry as stemming from the greater availability
of passive funds over time, i.e., a decrease in investors’search costs P. The assets held by
passive funds have increased substantially over the last decades, both in absolute value and
as a fraction of all fund assets. For example, the total AUM of passive funds have grown
from less than $1 trillion in the early 2000s to more than $5 trillion in recent years. These
trends have been accompanied by a decrease in both active and passive funds’expense ratios
(captured by fAand fPin the model), from around 1% (0.23%) for active (passive) funds in
the 2000s, to less than 0.7% (0.15%) in recent years.10
4 Policy implications
In this section, we examine the properties of the equilibrium and derive the implications of
delegated asset management for corporate governance, investor returns, and total welfare.
10 These stylized facts are based on the data on funds’AUM and expense ratios from the CRSP Mutual
Fund database. We thank Davidson Heath, Daniele Macciocchi, Roni Michaely, and Matthew Ringgenberg
for generously sharing these data with us.
22
4.1 The governance role of passive funds
It is often argued that the growth in passive funds is detrimental to corporate governance
due to lower fees that passive fund managers charge and, thereby, their lower incentives to
stay engaged. This argument implicitly assumes that fees fPdecrease, while other factors
that a¤ect fund managers’monitoring orts do not change. However, in reality, fees do not
change exogenously and in isolation: a change in fees is likely to be accompanied by other
changes, such as changes in AUM of di¤erent types of funds, changes in funds’ownership
stakes, the substitution between delegated asset management and private savings, and others.
In this section, we use our model to analyze the governance role of passive funds, while
formally accounting for these other e¤ects.
To study the e¤ect of passive funds on governance, we consider the comparative statics
of parameter P, i.e., the cost of searching for a passive fund. The growth of passive funds
over the last decades can be interpreted as a decrease in P. As we show next, if aggregate
investor wealth is large, so that the entry of passive funds primarily crowds out investors’
private savings, then passive fund growth improves aggregate governance; moreover, this
happens even though passive fund fees decrease. In contrast, if aggregate investor wealth is
small, so that the entry of passive funds crowds out investors’allocations to active funds,
then passive fund growth can have a detrimental e¤ect on aggregate governance. Moreover,
our key conclusion is that there is generally a trade-o¤ between the e¤ect of passive funds
on corporate governance and their e¤ect on the well-being of fund investors.
The case of small investor returns. First, consider the case of large aggregate investor
wealth (W >
W), such that investors’rate of return is = 1. Since fP=P
P+1, a decrease
in Pdecreases the passive fund fee fP. At the same time, since RM= (1 + 1
P)ZM, a
decrease in Pincreases the return RMon the market portfolio and hence, the average stock
price. Intuitively, when Pdecreases, the passive fund’s AUM increase, which increases the
passive fund’s equity stake in each …rm and thereby strengthens its incentives to engage in
governance. Since active funds do not own the “relatively more expensive H-…rms, the
governance and payo¤s of H-…rms improve, while the payo¤s of relatively “cheap” rms,
RL= (1 + 1
A)ZL, are not a¤ected.
23
Together, these two e¤ects imply that as passive funds become more readily available,
the aggregate investments in governance and the payo¤ of the market portfolio increase, even
though passive fund fees decline. This suggests that the link between asset management fees
and governance is not immediate.
Corollary 1. If W >
W, then greater availability of passive funds (lower P) improves
aggregate corporate governance, even though it decreases passive fund fees fP.
The case of large investor returns. Second, consider the case of small aggregate investor
wealth (W <
W), so that  > 1. Denote Rpassive
L,Rpassive
H, and Rpassive
Mthe equilibrium payo¤s
from Proposition 1 under some baseline value of P.
To understand the governance e¤ect of passive funds, consider a second scenario in which
Pis so large that investing with the passive fund becomes unpro…table, as if it did not exist
( P=1), and compare this scenario with the one under the baseline P. Assume also that
while Wis small enough to crowd out private savings for the baseline P, it is not so small
as to crowd out private savings when P=1, i.e., that = 1 without the passive fund.
Lemma 1 in the appendix presents su¢ cient conditions for such a “corner equilibrium to
exist and for investors’rate of return in this equilibrium to be = 1.
In this equilibrium, since there is no monitoring by the passive fund, the pay of L-…rms
is Rno passive
L= (1 + 1
A)ZL, and the payo¤ of the market portfolio is
Rno passive
M=R0+1
2
fAxAL
cA
=R0+1
2(RLR0) = R0
2+1
2(1 + 1
A
)ZL:(28)
This leads to several observations. First, note that under the baseline P,Rpassive
L=
(1 + 1
A+(1)(1))ZLdecreases in . Therefore, Rno passive
L> Rpassive
L, i.e., the presence of
the passive fund makes governance of the relatively “cheap”rms worse. Intuitively, this is
because the presence of the passive fund and the resulting competition pushes the fee and
AUM of the active fund down, which decreases its incentives to engage in governance of its
portfolio …rms.
On the other hand, the presence of the passive fund improves the governance of the
relatively “expensive” H-…rms: Rno passive
H=R0< R0+fPxP
cP=Rpassive
H. This is because
24
in the economy without the passive fund, these H-…rms are entirely owned by liquidity
investors, who do not engage in governance at all. In contrast, when the passive fund is
present, these …rms are partly owned by the passive fund, which has incentives to engage.
Finally, consider the e¤ect on the aggregate market portfolio. Note that
Rno passive
M> Rpassive
M,R0
2+1
21 + 1
AZL>(1 + 1
P+ (1) (1 ))ZM;(29)
which is equivalent to  >
for some cuto¤
. Since captures the equilibrium rate of
return that fund investors earn on their investment, this leads to the following result:
Proposition 3.If W <
W, the presence of passive funds always worsens governance at
L-…rms; always improves governance at H-…rms; and improves aggregate governance if and
only if it does not increase fund investors’returns too much.
In other words, if the entry of passive funds improves the well-being of fund investors by
enough (by substantially reducing the fees and making investing in the stock market very
attractive), its e¤ect on the overall governance is negative. Intuitively, e¤ective governance
requires su¢ cient incentives of fund managers to stay engaged, and this, in turn, requires that
fund managers earn enough rents from managing investors’portfolios and do not leave too
much money to the investors. Hence, passive fund growth is only bene…cial for governance
if it does not improve fund investor well-being by a lot.
Figure 2 presents a numerical example illustrating this result. The parameters in this
example satisfy the conditions of Proposition 1 for the case with passive funds, P= 0:09,
and satisfy the conditions of Lemma 1 in the appendix for the case without passive funds,
P=1. To illustrate how the presence of passive funds a¤ects governance as a function of
investor returns, we vary the aggregate investor wealth W: as shown in Proposition 1 and
illustrated in panel (a) of the …gure, investors’equilibrium rate of return in the presence
of passive funds decreases in Wup to the point W=
W2:1, where it stabilizes at the
level = 1. Panels (b) and (c) compare average …rm value, RM, and total welfare for
the case with a passive fund (solid line) and without a passive fund (dashed line). As the
gure demonstrates, and consistent with Proposition 3, the presence of passive funds is only
25
bene…cial for …rm value if Wis large enough (W > 1:216), i.e., in the presence of passive
funds is not too high. Moreover, when Wis very small (W < 1:198), the presence of passive
funds is even detrimental to total welfare: the negative e¤ect on …rm value (and hence the
welfare of initial owners of the …rm) dominates the positive e¤ect of passive funds on fund
investor well-being.
Figure 2. The …gure plots fund investors’gross rate of return, average …rm value, and total welfare
as a function of aggregate investor wealth W. The solid line corresponds to the case where the
passive fund is present, P= 0:09. The dashed line corresponds to the case without a passive fund,
P=1. The parameters are = 0:01,cA= 0:001,cP= 0:002, A= 0:1,ZL= 1,ZH= 0:81,
R0= 10:75.
4.2 Who benets from investing in governance?
It is frequently noted that asset managers may not have su¢ cient resources to engage in
ective monitoring of their portfolio companies. For example, Bebchuk and Hirst (2019b)
point out that for each of the Big Three passive fund families, the size of its stewardship
team is between 12 and 45 people, even though it manages more than 11,000 portfolio …rms,
and that its stewardship budget is less than 0.2% of the fees it charges for managing equity
assets. Based on this criticism, some observers propose regulations inducing asset managers,
and especially passive funds, to invest more resources into their stewardship teams. In the
context of our model, we can think of these regulations as reducing the ex-post costs of
engaging in governance (cAand cP) at the expense of some unmodeled ex-ante cost. In
this section, we study the e¤ects of such proposals on governance, fund investors’and fund
26
managers’payo¤s, and total welfare. The next result shows that while they generally have a
positive e¤ect on governance and …rm valuations, they can be detrimental to fund investors
and fund managers themselves.
Proposition 4. Suppose fund manager is cost of monitoring cidecreases. Then:
(i)rms’payo¤s and prices always weakly increase, and strictly increase if W <
W;
(ii)fund investors’rate of return always weakly decreases, and strictly decreases if W <
W;
(iii)fund manager is payo¤ strictly decreases if W
W.
This result emphasizes that policy proposals that decrease investors’costs of engagement
for example, by inducing funds to invest more resources into their stewardship teams –
are not universally bene…cial. While a decrease in ciincreases the fund’s engagement and
thus …rms’pays (RLand RM), it can make fund investors and, potentially, fund managers
worse o¤. Intuitively, because liquidity investors have rational expectations about the e¤ect
of cion the fund’s equilibrium e¤ort and …rms’pays, a decrease in citranslates into higher
prices. In particular, even though Rjincreases as cidecreases, the price Pj=Rj+Zj
increases by the same amount, so the fund can only make money on gains from trade, Zj,
and neither fund investors nor fund managers can bene…t from the fund’s monitoring. In
fact, they can be made worse o¤: higher prices imply that funds can buy a lower number of
shares and hence realize lower gains from trade. More precisely, as part (ii)of Proposition
4 shows, fund investors do not bene…t from increased monitoring when W
W(when their
rate of return is = 1) and are harmed by the fund’s increased monitoring when W <
W.
Thus, while initial owners of the …rm are better o¤ as they can now sell their shares for a
higher price, the new owners of the …rm, i.e., fund investors, are weakly worse o¤.
Whether decreasing the costs of monitoring is bene…cial for the fund itself depends on
the interaction of several forces. The positive e¤ect is that for a given level of e¤ort, the
fund’s costs of engagement decrease. However, there can also be a negative e¤ect: given
that greater monitoring decreases fund investors’return, the fund may experience out‡ows,
leading to lower management fees. This is exactly what happens when aggregate investor
wealth is large, W
W: because fund investors can invest in the outside asset that earns a
27
gross return of one, the fund ends up with lower AUM when cidecreases, and as part (iii)
of Proposition 4 shows, this e¤ect dominates the decrease in the costs of e¤ort.
In contrast, when aggregate investor wealth is small, W <
W, so that investors only
choose between the active and passive fund (and earn a return higher than that of the outside
asset,  > 1), the fund manager may …nd it optimal to decrease its costs of monitoring.
Moreover, the passive fund manager’s incentives to decrease cPare generally stronger than
the active fund manager’s incentives to decrease cA. The reason is that the passive fund can
actually experience in‡ows as a result of such a policy change. Intuitively, more monitoring
by the funds increases stock prices and decreases funds’ability to realize gains from trade.
Since the ability to realize gains from trade is relatively more important for the active fund,
this hurts the active fund more than the passive fund, resulting in out‡ows from the active
fund and inows into the passive fund. Note that this e¤ect arises due to the interaction
between the active and passive fund and would not arise with a single fund.11 Hence, while
the active fund manager is often hurt when funds’costs of monitoring decrease, the passive
fund manager can bene…t from such a change.
Figure 3 illustrates this logic. It considers the same set of parameters as in Figure 2,
but varies parameters cAand cP. Panels (a) and (b) show the trade-o¤ between the positive
ect of lower monitoring costs on …rm valuations and its potential negative e¤ect on fund
investors (parts (i)and (ii)of Proposition 4). Panels (d) and (e) show the di¤erence between
active and passive funds: while the active fund would prefer to keep its costs of monitoring
high, the passive fund bene…ts from decreasing its costs of monitoring.
4.2.1 Implications for total welfare
Whether decreasing funds’ costs of monitoring is bene…cial for total welfare depends on
its combined e¤ect on …rms’initial owners, fund investors, fund managers, and liquidity
investors. Since liquidity investors are marginal traders, their pay is zero. Hence, the
ect of such policies on total welfare depends on the trade-o¤ between their positive ect
on governance and hence initial owners’payo¤ on the one hand, and their potential negative
11 To show this formally, we analyze the setting in which A( P) is so large that only the passive (only the
active) fund manager raises positive AUM, as in Lemma 1 in the appendix. In this equilibrium, as Lemma
9 in the online appendix demonstrates, both the active and the passive fund manager are always worse o¤ if
their cost of monitoring decreases, similar to result (iii)of Proposition 4.
28
Figure 3. The …gure plots average …rm value, investors’rate of return, active and passive fund
managers’payo¤s, and total welfare as a function of fund managers’costs of monitoring cAand cP.
The parameters are cA= 0:001 (when cPvaries), cP= 0:002 (when cAvaries), = 0:01, A= 0:1,
P= 0:09,ZL= 1,ZH= 0:81,R0= 10:75,W= 2.
ect on fund investors and fund managers on the other hand.
In the example above, total welfare increases when either of the fund’s costs of monitoring
decrease (panels (c) and (f) of Figure 3), i.e., regulations that induce funds to increase the
size of their governance teams are welfare improving. Interestingly, however, both the active
fund manager and fund investors would push against such welfare improving regulations
because it would make them worse o¤.
However, as we point out next, such regulations are not always welfare improving. In
particular, decreasing funds’costs of engagement beyond a certain threshold is always det-
rimental to total welfare:
Proposition 5 (welfare e¤ects of decreasing the costs of monitoring). De…ne cias
the in…mum of cifor which  > 1. If ci<ci, then decreasing ciharms total welfare.12
12 If this in…mum does not exist, i.e., = 1 for all cisatisfying the conditions of Proposition 1, then
decreasing ciharms total welfare for all cisatisfying these conditions.
29
The logic is the following. According to Proposition 4, as a fund’s cost of engagement
decreases, fund investors’rate of return decreases as well, until it reaches the point (at ci= ci)
where investors are inderent between investing with the fund managers and their private
savings, i.e., = 1. At this point, a further decrease in the fund’s cost of engagement has
no additional marginal bene…t because, as follows from Proposition 1, the fund’s monitoring
levels and hence …rm valuations stay constant in ciwhen = 1. Therefore, the only welfare
ect of further decreasing ciis the decline in fund managers’pro…ts (condition W
Win
part (iii)of Proposition 4 corresponds to the case of = 1).
The reason why funds’monitoring and thus …rm value do not change with ciwhen = 1
is as follows. Suppose, for example, that the passive fund’s e¤ort increased as cPdecreased
(assuming for a moment that the fund’s ownership stakes xPwould not change). Higher
ort would raise …rms’pays (RM) and hence market prices (PM). Since, as discussed
above, the fund does not gain from increased monitoring, the only e¤ect of higher valuations
would be the fund’s lower ability to realize gains from trade. This would make investing
in the fund less attractive for investors relative to investing in the outside asset, leading
to out‡ows into private savings and decreasing the fund’s AUM. These out‡ows, in turn,
would lead the fund to take smaller positions in the underlying stocks, and these smaller
positions would have a counteracting e¤ect of decreasing the fund’s incentives to monitor.
In equilibrium, the fund’s AUM and, accordingly, its ownership stakes xPdecrease in a way
that the combined e¤ects of lower cPand lower xPon the fund’s e¤ort cancel out, so that
the equilibrium e¤ort and hence …rm valuations remain unchanged.
Overall and more generally, this logic emphasizes that to understand the e¤ects of gov-
ernance regulations, it is important to consider their potential e¤ects on funds’assets under
management, since those e¤ects can potentially counteract the desired e¤ects of regulations.
Note also that when passive funds are more readily available ( Pis lower), funds’AUM
are larger and investors are likely to strictly prefer investing with the fund managers over
their private savings (Proposition 2), which makes the counteracting e¤ect described above
less likely. Accordingly, as we show in the proof of Proposition 5, the threshold ciincreases
with P, which leads to the following policy implication: Regulations that reduce funds’
costs of engagement are more likely to be welfare improving if (1) passive funds are more
30
readily available, and (2) funds’assets under management are su¢ ciently large.
Proposals that restrict passive funds from voting. Lund (2018) suggests that law-
makers consider restricting passive funds from voting at shareholder meetings. In the con-
text of our model, this would be equivalent to substantially increasing passive funds’costs
of monitoring cP, and Proposition 5 implies that such a proposal could indeed be potentially
bene…cial for total welfare. However, the reasoning emphasized in our paper is very di¤erent
from the reasoning put forward by Lund (2018). In particular, Lund (2018) points out that
if a passive fund chooses to intervene, “it will rationally adhere to a low-cost, one-size-…ts-all
approach to governance that is unlikely to be in the company’s best interest,”or, in other
words, that passive fund monitoring decreases …rm value. In contrast, we emphasize that
even if passive fund monitoring has the potential to increase …rm value, restricting it could be
welfare-improving because too much monitoring may have a negative e¤ect on fund investors
and, potentially, fund managers.
5 Conclusion
The governance role of delegated portfolio managers, and the e¤ect of passive funds in
particular, is the subject of an ongoing debate among academics and policymakers. This
paper develops a theoretical framework to study the governance role of active and passive
asset managers and to evaluate the policy proposals put forward to a¤ect their engagement
with companies. In our model, investors decide how to allocate their funds between active and
passive funds, or whether to invest privately, and asset management fees are endogenously
determined. Passive funds invest their AUM in the market portfolio, while active funds
trade strategically to exploit mispricing. Funds’ownership stakes and asset management fees
determine their incentives to exert e¤ort to increase the value of their portfolio companies.
We show that whether the growth in passive funds is benecial for governance depends
on whether it crowds out investors’private savings or their allocation to active funds. In
the former case, passive fund growth improves governance because liquidity investors (who
play no governance role) are replaced by passive funds as …rms’shareholders, and passive
funds have incentives to engage given their large holdings in the …rms. Moreover, passive
31
fund growth can improve governance even if it is accompanied by a decrease in passive
fund fees. However, if passive fund growth crowds out investors’allocation to active funds,
it has a more subtle e¤ect. On the one hand, …rms primarily held by liquidity investors
experience improved governance. On the other hand, the increased competition between
funds decreases active funds’AUM and fees, which decreases their incentives to monitor and
worsens governance in …rms primarily held by active funds. We show that the combined e¤ect
of passive fund growth on aggregate governance is positive only if it does not substantially
improve the well-being of fund investors, i.e., there is a trade-o¤between the two. Intuitively,
ective engagement requires fund managers to earn su¢ cient rents from managing investors’
assets, and hence what is good for fund investors is bad for governance, and vice versa.
We also study the e¤ect of regulations that decrease funds’costs of engaging in gov-
ernance, e.g., by mandating larger stewardship teams. While such regulations increase funds’
monitoring and thus …rm valuations, they can be detrimental to fund investors and, poten-
tially, fund managers themselves. As a result, fund managers and fund investors may oppose
such regulations even when they are welfare improving. Moreover, if such regulations reduce
funds’costs of engagement beyond a certain threshold, they can harm total welfare.
To focus on the role of passive funds, asset management fees, and the competition between
funds, we abstract from several important features of the engagement process, such as the
interaction between derent shareholders in their engagement e¤orts or the role of fund
managers’ private information about the …rms. An in-depth look at these questions and
their interaction with the mechanisms we study in the paper provides interesting avenues for
future research.
32
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35
Appendix
Proof of Proposition 1. We consider each case separately.
(1) Equilibrium in Case 1: low investor returns, = 1.
Consider the three equations for active fund managers and L-assets, i.e., (7), (14), and
(18), which we can rewrite as:
fA=ZL
RL
(fee bargaining) (30)
(1 fA)RL
PL
= 1 + A(investor indi¤erence) (31)
RLPL=ZL(market clearing) (32)
Plugging fAfrom (30) and PLfrom (32) into (31) gives:
1ZL
RLRL
RLZL
= 1 + A,(1 + A)ZL= ARL:
Hence, RL=1 + 1
AZL. Then, (32) implies PL=RLZL=1
AZL, and (30) implies
fA=ZL
1+ A
AZL
=A
1 + A:
Similarly, we can rewrite the three equations for passive fund managers and the market asset,
i.e., (8), (16), and (19), as
fP=ZM
RM
(fee bargaining)
(1 fP)RM
PM
= 1 + P(investor indi¤erence)
RMPM=ZM(market clearing)
Since this system looks exactly the same as the corresponding system for active fund man-
agers and the L-asset, the solution looks the same: RM=1 + 1
PZM,PM=1
PZM, and
fP=P
1+ P, which completes the derivation of Case 1.
(2) Equilibrium in Case 2: high investor returns,  > 1.
We start by deriving (27). Using (5) and (6) and plugging them into (21), we get
W=1
2xALPL+xPPM:(33)
36
Next, using (9) and (10),
RLRM=1
2c01
A(fAxAL),c0
A(2 (RLRM)) = fAxAL;(34)
2RMRL=R0+c01
P(fPxP),c0
P(2RMRLR0) = fPxP:(35)
Plugging these into (33) gives (27).
We next characterize the equilibrium as a function of , using (7)-(10); (23), (25); and
(20), (27).
First, consider asset Land the active fund manager and use (23), (20), and (7):
fA
RL
PL
=RL
PL(fee bargaining) (36)
(1 fA)RL
PL
= A+(investor indi¤erence) (37)
PL=RLZL(market clearing) (38)
From (36), RL
PL=
fA, and plugging this into (37) gives
(1 fA)
fA
= A+,fA=A
A+(1 ):
Plugging this into (36) gives
RL
PL
1 A
A+(1 )= ,( A+(1 )) PL= (1 )RL;
and using (38) gives
( A+(1 )) ZL= ( A+(1 )) RL(1 )RL,
RL=1 + 1
A+ (1) (1 )ZL:(39)
Finally, using (38) and (39),
PL=RLZL=1
A+ (1) (1 )ZL:
Second, consider asset M(the market portfolio) and the passive fund manager. Since the
system of equations (8), (25), and (20) looks exactly the same as the corresponding system
for active fund managers and the L-asset (36)-(38), the solution looks the same as well, which
gives the expressions for fP,RM, and PMin the statement of the proposition.
Thus, all equilibrium outcomes –fA,fP,RL,RM,PL,PMare expressed as a function
of and the exogenous parameters of the model. The equilibrium is then determined from
37
the equilibrium condition that investors invest all of their capital either with the active or
with the passive fund manager, i.e., the …xed point solution to (27). This completes the
derivation of Case 2.
(3) Combining the two cases together.
According to Lemma 2 in the online appendix, if cP P
AcA, then is decreasing in W.
Hence, there exists
Wsuch that  > 1for W <
Wand = 1 for W
W. It remains
to verify that in the conjectured equilibrium: (1) the active fund indeed …nds it optimal to
only invest in L-stocks (and not H-stocks or the outside asset) and to diversify across all
L-stocks; (2) both the active and passive fund raise positive AUM; and (3) liquidity investors
are marginal in each stock. Lemma 3 in the online appendix shows that under the quadratic
cost function, the active fund will indeed diversify across L-stocks. Part (ii)of Lemma 4
and Part (ii)of Lemma 5 in the online appendix impose conditions that are su¢ cient for
the active fund to not deviate to investing in either H-stocks or the outside asset. Lemma
6 in the online appendix imposes su¢ cient conditions for both funds’AUM to be positive,
and Lemma 7 in the online appendix imposes su¢ cient conditions for liquidity investors to
be marginal. Combining these conditions together yields the following two conditions:
max 8
<
:0:64;
R0
ZL+h1 + 1
Ai
2h1 + 1
Pi;AP+AP
2
P
;
1
2+1
A
1 + 1
P
9
=
;<ZM
ZL
<1 + 1
A
1 + 1
P
;(40)
^
WW < R0ZL
2;(41)
where Aand Pare given by (81)-(82) and ^
W <
Wis de…ned in Lemma 6 in the online
appendix. The numerical example in Figure 2 satis…es this set of parameters, i.e., it is a
non-empty set.
Proof of Proposition 2. (1) We start by deriving the expressions for active and passive
funds’AUM. Using Proposition 1 and (70),
WP=xPPM=cPeP
fP
RM
P
1+=cP(2RMRLR0) P+(1 )
P
RM(1 )
P+(1 )
(42)
=1
cP
P
RM(2RMRLR0):
Similarly, using Proposition 1 and (69),
WA=1
2xALPL=1
2
cAeAL
fA
RL
A
1+ 1 =1
22cA(RLRM) A+(1 )
A
RL(1 )
A+(1 )
(43)
=1
cA
A
RL(RLRM):
38
Note, as an auxiliary result, that these expressions imply that in Case 1, AUM of fund
iare decreasing in i. Indeed, if = 1, then RLdoes not depend on P, and WPstrictly
decreases in Pif and only if
cP
2
P
RM(2RMRLR0) + cP
P
(4RMRLR0)dRM
d P
<0;
which holds since 2RMRLR0>0and dRM
d P<0. Similarly, if = 1, then RMdoes not
depend on A, and WAstrictly decreases in Aif and only if
cA
2
A
RL(RLRM) + cA
A
(2RLRM)dRL
d A
<0;
which holds since RLRM>0and dRL
d A<0. Note also that the same arguments hold for
the equilibria of Lemma 1, in which only one fund raises AUM –this is because the above
expressions for WA(WP) are still valid in the equilibrium where only the active (passive)
fund raises AUM.
(2) Next, we show that the combined AUM of active and passive fund managers, WA+WP,
strictly decrease in Pin Case = 1. This automatically implies that WA+WPalways
weakly decrease in P(because when  > 1,WA+WP=W). To show that total AUM
decrease in P, note, using (43)-(42), that
WA+WP=1
cA
A
RL(RLRM) + cP
P
RM(2RMRLR0):(44)
Since, in Case 1, RLdoes not depend on P, total AUM strictly decrease in Pif and only
if
cA
A
RL
dRM
d PcP
2
P
RM(2RMRLR0) + cP
P
(4RMRLR0)dRM
d P
<0,
cA
A
RL+cP
P
(4RMRLR0)dRM
d PcP
2
P
RM(2RMRLR0)<0:
Since 2RMRLR0>0and @RM
@ P<0, it is su¢ cient to show that
cA
A
RL+cP
P
(4RMRLR0)0:(45)
Note that eP= 2RMRLR00and hence 2RMRL>0, and summing up these two
inequalities gives 4RMRLR0> RL. This, together with the assumption of Proposition
1 that cP
PcA
A, implies (45), as required. The same result with respect to Palso applies
in the equilibrium of Lemma 1, in which only the passive fund raises AUM.
The fact that WA+WPdecrease in Pimplies the last statement of the lemma, i.e.,
that Case 1 of low investor returns (= 1) only applies when Pis large enough. Indeed,
39
in Case 1, fund investors invest their funds both with the fund managers and in the outside
asset, and hence WA+WP< W , while in Case 2, all investor funds are allocated to the fund
managers, i.e., WA+WP=W. Hence, Case 1 applies if and only if WA+WP< W , or if
and only if Pis large enough.
(3) Next, we prove that decreases in Punder the conditions of Proposition 1. This is
weakly satis…ed for Case 1 because = 1. To see this for Case 2, note that the combined
AUM of the two funds, WA+WP, satisfy (44). In addition, for a …xed ,RLdoes not depend
on Pand RMdecreases in P, so repeating the steps subsequent to (44), implies that for a
xed ,WA+WPdecreases in P. Moreover, for Case 2, WA+WP=W. On the other hand,
as follows from the proof of Lemma 2 in the online appendix, equality (46) holds, where the
right-hand side decreases in . Combined, we have
WA(; P) + WP(; P) = W;
and hence,
@(WA+WP)
@
d
d P
+@(WA+WP)
@ P
= 0;
where @(WA+WP)
@ <0and @(WA+WP)
@ P<0. Thus, d
d P<0, as required.
(4) Finally, we prove the result for fund fees, i.e., that both fAand fPincrease in P.
Since fA=A
A+(1), it weakly increases in P(it does not depend on Pin Case 1 and
strictly increases in Case 2 given d
d P<0). And, since fP=P
P+(1), it always strictly
increases in P: In Case 1, this is because fP=P
P+1, while in Case 2, this is because
dfP
d P=@fP
@
d
d P+@fP
@ P>0, which follows from @ fP
@ <0,d
d P<0, and @fP
@ P>0. This completes
the proof.
Proof of Proposition 4.
We start by proving (ii). Fund investors’payo¤is characterized by their equilibrium rate
of return . When W
W, their rate of return is = 1 and is unected by ci. When
W
W,increases with ci. To see this, recall that is the solution to
W=cA
fA()(RL()RM()) PL() + cP
fP()(2RM()RL()R0)PM();(46)
where fA(),fP(),RL(),RM(),PL(), and PM()are given by the expressions in
Proposition 1. According to Lemma 2 in the online appendix, the right-hand side decreases
with whenever A> Pand cP P
AcA. Since the right-hand side increases in ci, it
follows that increases in ci(otherwise, if ciincreased, the right-hand side would increase
both through the e¤ect of ciand through the e¤ect of , while the left-hand side would not).
We next prove (i). Consider RLand RM. If W
W, they do not depend on ci. If
W
W, then RL= (1 + 1
A+(1)(1))ZLand RM= (1 + 1
P+(1)(1))ZM. Since
increases with cias shown above, then both RLand RMdecrease with ci, and thus PLand
PMdecrease with cias well.
40
Finally, we prove (iii). Let eP(eAL) denote the passive (active) fund manager’s equilib-
rium e¤ort. Then, the passive fund manager’s payo¤ is given by
VP=fPxPRMcP
2e2
P=cPePRM1
2eP=cP(2RMRLR0)RM1
2(2RMRLR0)
=cP
2(2RMRLR0) (RL+R0);
(47)
and the active fund managers payo¤ is given by
VA=1
2fAxALRLcA
2e2
AL=1
2cAeAL RL1
2eAL
=cA(RLRM)RL1
22 (RLRM)=cA(RLRM)RM:(48)
If W
W, then by Proposition 1, RLand RMdo not change with cPand cA, which implies
that VPstrictly increases with cPand VAstrictly increases with cA.
Proof of Proposition 5. Welfare equals the sum of the payo¤s of the initial shareholders,
the payo¤s of liquidity investors, the pays of fund managers, and the payo¤s of fund
investors:
W elf are =PM+0+1
2fAxALRL+fPxPRM1
2
cA
2e2
AL cP
2e2
P+ (1) W(49)
The …rst term is the payo¤ of the initial owners of the …rms, which is PL+PH
2up to a constant
(initial owners’ valuations). The second term equals zero because liquidity investors are
marginal traders. The third term, in the square brackets, captures the combined payo¤ of
the active and passive fund manager, which is their share of the fund’s pay minus their
costs of engaging in governance. The last term captures the payo¤ of the fund investors:
since their initial wealth is Wand they earn equilibrium rate of return on it, their …nal
payo¤ is W . Note that in the expression above, Whas a multiplier of (1), rather than
just . This has an e¤ect on the comparative statics of welfare only with respect to W,
and not any other parameters. The rationale behind this choice is that if Wincreases, the
increase in Wmust be …nanced from another source in the economy that is not explicitly
modeled in our framework. For example, if Wincreases by W, it must be that Wless is
invested in the rest of the overall economy, and to capture that, we subtract Wfrom our
welfare calculation, resulting in the term (1) W.
Using fAxAL =cAeAL,fPxP=cPeP,eAL = 2 (RLRM)0, and eP= 2RMRLR0
0, we can rewrite (49) as
W elf are =PM+1
2cAeALRL+cPePRM1
2
cA
2e2
AL cP
2e2
P+ (1) W
=PM+1
2cAeAL RL1
2eAL+cPePRM1
2eP+ (1) W
=PM+cA(RLRM)RM+cP
2(2RMRLR0) (RL+R0)+(1) W: (50)
41
Below, we show that ciis given by (51)-(52) and prove that  > 1for ci>ciand = 1 for
cici. Now, consider any ci<ci, so that = 1. Then, according to Proposition 1, PM,
RM, and RLdo not change with cPand cA. Note that RLRM=1
2eAL =1
2
fAxAL
cA>0and
2RMRLR0=eP=fPxP
cP>0, because fAand fPare positive by Proposition 1, and
both xAL and xPare positive by the proof of Proposition 1. Hence, (50) implies that welfare
strictly increases with cPand cA, as required.
We next show that cPand cAare given by
W=1
0
@
cA
A1 + 1
AZL1 + 1
AZL1 + 1
PZM
+cP
P1 + 1
PZM21 + 1
PZM1 + 1
AZLR01
A;(51)
W=1
0
@
cA
A1 + 1
AZL1 + 1
AZL1 + 1
PZM
+cP
P1 + 1
PZM21 + 1
PZM1 + 1
AZLR01
A;(52)
respectively. Indeed, recall that in equilibrium described by Proposition 1, WA+WPis given
by the right-hand side of (27). Consider any i2 fA; P g. We show that  > 1for ci>ciand
= 1 for cici. First, consider cici. Then, it must be = 1. This is because then, (51),
(52), and Proposition 1 imply WWA+WP, which is consistent with = 1. This also
implies that it cannot be  > 1, because if we had  > 1, then (51), (52), Proposition 1, and
Lemma 2 in the online appendix would imply that W > WA+WP, yielding a contradiction
since no investor would invest in the outside asset given  > 1. Second, consider ci>ci.
Then it must be  > 1. Indeed, if we had = 1, then (51), (52), and Proposition 1 would
imply W < WA+WP, yielding a contradiction since then the total investor endowment
would not be su¢ cient for funds to raise total AUM of WA+WP.
As an auxiliary result, we next also show that cPand cAstrictly increase with P. This
follows from (51) and (52), because the right-hand side in both of them is strictly decreasing
in P. To see this, take any i2 fA; P g, and let ci= ci. If i=P, consider (51), and if i=A,
consider (52). Then = 1, and using the expressions for RLand RMfrom Proposition 1,
the partial derivative of the right-hand side w.r.t. Pis negative if and only if
0>cA
A
RL+cP
P
(4RMRLR0)@RM
@ P
;
which always holds since @RM
@ P<0,cP P
AcA, and 4RMRLR0>2RM> RL, where
the last set of inequalities follow from 2RMRLR0=eP>0as argued after expression
(50) above.
While this completes the proof, in what follows, we also provide the su¢ cient conditions
that ensure that (1) cP> P
AcA>0and (2) cP> P
AcA. Together, these two inequalities
in turn ensure that the set of values of cithat satisfy both the conditions of Proposition 1
(cP P
AcA) and the condition ci<ci, is non-empty for each i2 fA; P g. We show that
42
these su¢ cient conditions are given by WL< W < WH, where
WL1
max 8
>
>
>
<
>
>
>
:
cA
A1 + 1
AZL1 + 1
AZL1 + 1
PZM
+cA
A1 + 1
PZM21 + 1
PZM1 + 1
AZLR0;
cP
P1 + 1
PZM21 + 1
PZM1 + 1
AZLR0
9
>
>
>
=
>
>
>
;
WH1
0
@
cP
P1 + 1
AZL1 + 1
AZL1 + 1
PZM
+cP
P1 + 1
PZM21 + 1
PZM1 + 1
AZLR01
A:
Note that WLWHis satis…ed since cP
PcA
Aas assumed in Proposition 1, and that
WL< WHwhenever cP
P>cA
A. The reason why WL< W < WHis a su¢ cient condition is
that from (51)-(52), it follows that WL< W implies that cP> P
AcAand cP> P
AcA>0,
and W < WHimplies cP> P
AcA, as required.
Finally, we point out that the set fW:WL< W < WHgoverlaps with the other
parameter assumptions made in Proposition 1. In other words, it does not result in an
empty set of parameters. To see this, consider the example provided in Figure 2 (that is,
= 0:01,cA= 0:001,cP= 0:002, A= 0:1, P= 0:09,ZL= 1,ZH= 0:81,R0= 10:75).
Then, WL= max f1:184 2;1:672 4g<2:631 6 = WH, and (WL; WH)is a subset of (W1; W2)
imposed by Proposition 1, since W1= 0:503 and W2= 4:875.
Lemma 1 (equilibria with one type of fund) Suppose
ZL<R0
1
A+1 + 1
AZH
R0
(53)
and R0ZL
2> W: (54)
(i)Suppose
W > WP(1) 1
cP
P1 + 1
PZM1 + 1
PZMR0:(55)
Then, the equilibrium where = 1 and only the passive fund raises AUM exists if and
only if 1 + 1
PZM1 + 1
AZL(56)
and 1 + 1
PZM> R0:(57)
43
If this equilibrium exists, then WP=WP(1),fP; RM, and PMare as described in
Proposition 1, RL=RM, and PL=RLZL. Moreover, if A>ZL
R0ZL, then this
equilibrium is unique.
(ii)Suppose
W > WA(1) 1
2
1
cA
A1 + 1
AZL1 + 1
AZLR0:(58)
Then, the equilibrium where = 1 and only the active fund raises AUM exists if and
only if 1 + 1