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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

e¤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. E¤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 managers’costs 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, funds’fees,

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 fund’s 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 i’s 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 “e¤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 di¤erent 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 payo¤,

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 =c01

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 payo¤ (1) of the stock is

Rj(xAj ; xP j ) = R0+c01

A(fAxAj ) + c01

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+c01

A(fAxAL) + c01

P(fPxP);(9)

RM=R0+1

2c01

A(fAxAL) + c01

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

payo¤ upon agreeing and upon negotiations failing.

First, consider the fund investor. The investor’s payo¤ from agreeing on fee ~

fAis

(1 ~

fA)"

PLR0+c01

A(fAxAL) + c01

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 payo¤ 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 manager’s 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+c01

P(fPxP)+

~

fAR0+e+c01

P(fPxP)2"

PLcA(e)], and by the envelope theorem, the derivative with respect to "is

~

fAR0+e+c01

P(fPxP)1

PL, where e=c01

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 indi¤erence

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 indi¤erence 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 satis…es 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 satis…es 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 investors’rate 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 Pa¤ects 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 e¤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 payo¤ 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 bene…ts from investing in governance?

It is frequently noted that asset managers may not have su¢ cient resources to engage in

e¤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 i’s 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 i’s 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’payo¤s (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’payo¤s, 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 in‡ows 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

e¤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 payo¤ is zero. Hence, the

e¤ect of such policies on total welfare depends on the trade-o¤ between their positive e¤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.

e¤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 indi¤erent 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

e¤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

e¤ort would raise …rms’payo¤s (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 bene…cial 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,

e¤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 di¤erent 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:

1ZL

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

2c01

A(fAxAL),c0

A(2 (RLRM)) = fAxAL;(34)

2RMRL=R0+c01

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,PM–are 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 una¤ected 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 manager’s 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 payo¤s 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 payo¤ 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

AZL1 + 1

AZL1 + 1

PZM

+cP

P1 + 1

PZM21 + 1

PZM1 + 1

AZLR01

A;(51)

W=1

0

@

cA

A1 + 1

AZL1 + 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

AZL1 + 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

AZL1 + 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

PZM1 + 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

AZL1 + 1

AZLR0:(58)

Then, the equilibrium where = 1 and only the active fund raises AUM exists if and

only if 1 + 1