Sustainable Investing Under Delegated
Investment Management∗
Meg Adachi-Sato† and Hiroshi Osano‡March 30, 2024∗

The authors would like to thank Yuuichi Fukuta, Keiichi Hori, Shinsuke Ikeda, Jason Vladusic, andseminar participants at the Asset Pricing Workshop (Osaka University) for their helpful comments andsuggestions.
†
Faculty of Business Administration, Osaka University of Economics, 2-2-8, Osumi, Higashiyodogawaku, Osaka 533-8533, Japan and Research Institute for Economics and Business Administration, Kobe
University, Kobe 657-8501, Japan. E-mail: m-sato@osaka-ue.ac.jp; work phone number: +81-6-63282431.
‡
Corresponding author. Faculty of Economics, Konan University, 8-9-1, Okamoto, Higashinada-ku,
Kobe 658-8501, Japan. e-mail address: osano@kier.kyoto-u.ac.jp; work phone number: +81-78-431-3024.

1

Sustainable Investing Under Delegated Investment Management
Abstract
This paper examines how profit-motivated fund managers of sustainable funds withdivestment strategies and passive funds choose their level of engagement with portfolio
firms to enhance profitability and mitigate any negative externalities, particularly whenaiming to attract capital from socially responsible investors. Considering the fund ownership eﬀect resulting from investor capital allocation, we derive several novel implications
regarding the eﬀects of growing interest in environmental, social, and governance (ESG)
investments, the diminishing impact of shareholder ESG engagement on firms with highESG scores, and the increasing proportion of ESG-indexed funds within sustainable funds
on the ESG and financial performances generated by portfolio firms.
JEL Classification: D83, G23, G32, M14.
Keywords: delegated asset investment, ESG, passive fund, social impact, socially responsible investing, sustainable fund.

2

1. Introduction
In recent decades, there has been a growing preference for environmental, social, andgovernance (ESG) investments among investors, who seek nonpecuniary as well as pecuniary benefits from these investments. Consequently, there has been a rapid expansion insustainable investment.1 However, a significant portion of sustainable investments is madeby institutional investment funds, such as sustainable funds, which are often managed byasset managers who may prioritize their own pecuniary gains. Therefore, it is crucial tomotivate fund managers to actively engage in managing their portfolio firms, aiming not
only to enhance pecuniary returns but also to improve ESG performance. One dominant
investing strategy among these funds is to buy only the stocks of firms with high-ESG
performance. We refer to a fund that pursues this strategy as a D-fund, which comprises
active sustainable funds and ESG-indexed funds.2 However, some scholars (e.g., Hartzmark and Shue, 2022) argue that a sustainable investing strategy that directs capital awayfrom firms with low-ESG performance toward firms with high-ESG performance may be
counterproductive because this strategy may lead the former firms to lower their ESG
performance without incentivizing the high-ESG firms to raise theirs.
In recent years, fund markets have been characterized by the significant presence of
passive funds (P-funds).3 P-fund managers create market portfolios aligned with an index
and hence are considered to have little or no governance role in their portfolio firms. Many
practitioners and scholars suggest that P-funds present a considerable problem to ESG
activists because the eﬀorts of ESG activists to convince investors to avoid stocks with
low-ESG characteristics are undermined by the presence of P-funds.4 However, there ismixed empirical evidence on the governance role of P-funds.5 Furthermore, recently, P1

According to Alyssa Stankiewicz (Sustainable Fund Flows Reach New Heights in 2021’s First
Quarter, Morningstar, April 30, 2021, https://www.morningstar.com/articles/1035554/sustainable-fundflows-reach-new-hights-in-2021s-first-quarter), as of March 21, 2021, assets in US sustainable funds totaled
nearly $266 billion, which represents a 12% increase over the previous quarter and a 125% increase year
over the year.
2
Recently, BlackRock and other exchange traded fund providers have oﬀered many ESG-indexed funds.
3
The fraction of P-funds grew to just under 40% of the US fund market at the end of 2019 (Investment
Company Institute 2020 Factbook).
4
For example, see Billy Nauman (How Passive Investment Dulls the Green Wave, Financial Times,
February 7, 2022, https://www.ft.com/content/ abd2a946-48d5-11ea-aee2-9ddbdc86190d). This problem
is also related to the passive funds’ incentive problem of underinvestment in stewardship (see Bebchuk and
Hirst, 2019). In this context, note that ESG-indexed funds are excluded from the definition of P-funds.
5
See the literature cited in Corum, Malenko, and Malenko (pages 2—3, 2021). Indeed, Fichner,

3

funds have begun to exert their own ESG engagement eﬀorts (see Chu, 2021). Thus, thegovernance role of the P-fund managers is becoming increasingly important.
Given the divestment and engagement strategies of D-funds and the engagement strategies of P-funds, investors make capital allocation decisions among assets, thereby influencing the fund ownership stakes of each firm. These ownership stakes impact on theengagement strategies of fund managers within firms. Such engagement strategies encompass two aspects of fund manager engagement: first, managing their portfolio to improve
ESG performance (i.e., reducing any negative externalities), and second, improving pecuniary returns. In this study, we refer to eﬀorts to mitigate the negative externalitiesof portfolio firms and eﬀorts to increase their pecuniary returns as the fund manager’s
engagement eﬀorts. These eﬀorts may be cost complementary (substitutable), which indicates the degree to which increased engagement eﬀorts to enhance ESG performance
reduce (increase) the cost of engagement eﬀorts to improve profit. In particular, in Dfunds, the degree of cost complementarity may depend on the proportion of ESG-indexedfunds within D-funds, as discussed in Section 5.
In this context, this paper explores the governance role of profit-motivated managers of
D- and P-funds when they aim to attract capital from investors with ESG preferences. We
address the following questions. First, how does growing interest in ESG aﬀect the negative externalities and pecuniary returns of various types of firms when D- and P-funds arepresent in the fund market? Does growing interest in ESG truly have a counterproductive
eﬀect on negative externalities and pecuniary returns when D- and P-funds are present?
Second, how does the decreasing impact of fund managers’ engagement eﬀorts regarding
ESG in firms with high-ESG performance aﬀect the negative externalities and pecuniary
returns of various types of firms when D- and P-funds are present in the market? Finally,
we examine how the cost complementarity in fund managers’ eﬀorts between improving
profit and ESG performances aﬀects the negative externalities and pecuniary returns ofvarious types of firms when D- and P-funds exist.
To the best of our understanding, this paper is the first theoretical study to examine how
Heemskerk, and Garcia-Bernardo (2017) find that the top three passive index fund companies in theUS, which are large stakeholders in most of the Standard & Poor’s (S&P) 500 firms–BlackRock, Vanguard, and State Street–use coordinated voting strategies and influence the management of their invested
companies through private engagements. McCahery, Sautner, and Starks (2016) report that large institutional investors with a long time horizon and low concerns about stock liquidity intervene more intensively
with management through private engagements.

4

fund managers’ engagement eﬀorts toward their portfolio firms aﬀect negative externalities
and pecuniary returns across diﬀerent types of firms, while considering the divestment
and engagement strategies of D-funds and the engagement strategies of P-funds. This
analysis responds to growing interest in ESG, the decreasing impact of shareholder ESG
engagement on firms with high-ESG performance, and the cost complementarity of fundmanagers’ eﬀorts.
To address these questions, we build an asset management market model in which ESGconscious fund investors allocate their wealth between a D-fund, a P-fund, and outside
investment opportunities, such as public bonds. These fund investors can invest theirwealth in firms only via funds.6 However, these funds are managed by for-profit fundmanagers who may not favor ESG investments. If fund investors decide to invest in a Dfund, P-fund, or both funds, they must search for these funds, which incurs a search cost.
After matching, fund investors and the fund manager bargain over the asset management
fee. Then, the fund manager invests in firms that diﬀer in the sustainability of theiractivities; the D-fund buys only the stocks of firms with high-ESG performance and the
P-fund invests all of its money in a value-weighted market portfolio.7 After investing,each fund manager chooses his costly level of ownership engagement eﬀort to mitigate thenegative externalities and to increase the profits of his portfolio firms.
Our first main results are as follows. When investors’ ESG preferences are strengthening, the following results are obtained. (i) If the D-fund managers have a comparative costadvantage over the P-fund managers in improving firms’ profit performance over improving their ESG performance, the expected negative externality decreases for the high-ESG
firms (hereafter referred to as G-firms) and all the firms in the market portfolio; otherwise, under particular conditions, the expected negative externality may increase for thesefirms. (ii) Growing interest in ESG always reduces the expected negative externality but
raises the expected profit for the low-ESG firms (hereafter, B-firms).
The intuition behind these results is decomposed into two parts, namely the direct
eﬀect and the fund ownership eﬀect. As the eﬀort cost parameters are fixed, the fundmanagers’ incentives to exert ESG eﬀort are determined by their expected marginal return6

We also consider liquidity investors, who directly invest in firms for various reasons but do not engage
in the governance of the firms.
7
In Appendix B.2, we introduce a non-sustainable fund that buys only the stocks of firms with low-ESG
performance. Under certain conditions, our main results are unaﬀected.

5

from such ESG eﬀort. Thus, when investors’ ESG preferences are strong, this directly
raises the fund managers’ expected marginal return from ESG eﬀort. However, the fundownership eﬀect, which occurs through a change in the fund ownership stake, is more
complicated because it involves investors’ capital allocation among assets.
Indeed, if investors’ preferences for ESG strengthen, they may allocate less of theirwealth to the D-fund, even though they allocate more wealth to the P-fund. This possibility arises because the expected negative externality released by G-firms is smaller thanthat released by B-firms. Then, the fund ownership eﬀect arising from the reduction of
the D-fund’s ownership weakens the D-fund manager’s level of engagement eﬀort. However, if the D-fund managers have a comparative cost advantage in improving firms’ profit
performance over their ESG performance compared with the P-fund managers, the fundownership eﬀect from the reduction of ownership by the D-fund is dominated by that fromthe rise of ownership by the P-fund. Then, the expected negative externality decreases
for both G-firms and market portfolio firms.
By contrast, if, compared with the P-fund managers, the D-fund managers have acomparative cost advantage in improving firms’ ESG performance over their profit performance, the fund ownership eﬀect from the reduction of ownership by the D-fund may
dominate the remaining eﬀects, including the direct eﬀect, in a particular situation. Then,the expected negative externality may increase for G-firms and/or market portfolio firms.
However, for B-firms, the fund ownership eﬀect works only through a rise of ownershipby the P-fund and a decline of ownership by liquidity investors. Consequently, growinginterest in ESG always reduces the expected negative externality but raises the expected
profit because the liquidity investors do not make engagement eﬀorts.
Our second main result examines the result of a decrease in shareholder ESG engagement in G-firms relative to that in B-firms. The decreasing impact of shareholder ESG
engagement on G-firms increases both the expected negative externality and the expected
profit for G-firms and market portfolio firms, but does not aﬀect the expected negative
externality or the expected profit for B-firms.
Intuitively, such a decreasing impact directly weakens the incentives for ESG eﬀort
for both D- and P-fund managers in G-firms, but does not aﬀect this incentive for the
P-fund managers in B-firms. Again, we need to consider how the fund managers’ eﬀort
incentive is impacted by a fund ownership eﬀect occurring through a change in fund6

ownership stakes. However, although the fund ownership eﬀect operates in G-firms, thefund investors’ expected rate of return can be adjusted in equilibrium so that it does notchange in either fund in this case. Furthermore, the decreasing impact of shareholder
ESG engagement on G-firms has no eﬀect on the expected negative externality or the
expected profit for B-firms because it does not directly aﬀect the P-fund manager’s ESG
eﬀort in B-firms, and the fund ownership eﬀect does not operate in B-firms either.
Our third main result examines the eﬀect of greater cost complementarity in fundmanagers’ eﬀorts, which refers to the degree to which increased engagement eﬀorts toenhance ESG performance reduce the cost of engagement eﬀorts to improve profit. For
both G-firms and market portfolio firms, regardless of whether D- or P-fund managers
are considered, the greater cost complementarity in these eﬀorts reduces both the expected negative externality and the expected profit. In addition, for B-firms, the greater
cost complementarity in the P-fund managers’ eﬀorts reduces both the expected negative
externality and the expected profit.
Intuitively, when cost complementarity increases for one fund manager, this directly
strengthens his incentives for ESG eﬀorts, but it does not directly aﬀect these incentives
for other fund managers. Hence, the opposite arguments regarding the decreasing impact
of shareholder ESG engagement on G-firms apply to the greater cost complementarity casefor G-firms and market portfolio firms. Furthermore, because the P-fund buys B-firms andthe D-fund does not, the B-firms are aﬀected only by the greater cost complementarityof the P-fund manager.
The theoretical implications of our results are as follows. Our first results show that theeﬀect of growing interest in ESG on the expected negative externality in the presence of Dand P-funds depends on the comparative cost advantage between each fund in improving
ESG or generating profit. This is attributed to the influence of growing interest in ESG
on the fund ownership stakes. Thus, under certain conditions, growing interest in ESG
can reduce the expected negative externality across all types of firms in the presence ofD- and P-funds. However, if these conditions are not met, growing interest in ESG in thepresence of D- and P-funds may not necessarily reduce the expected negative externality
for all firms, but will do so only for B-firms. This discrepancy arises because of the fundownership eﬀect, which diﬀers from the argument of Hartzmark and Shue (2023), which
is primarily based on the limited ESG impact on G-firms.
7

The decreasing impact of shareholder ESG engagement on G-firms has a direct eﬀect on
the ESG eﬀorts of both the D- and P-fund managers in G-firms. Because of the possible
eﬀect of changes in fund ownership stakes, determining the eﬀects of this decreasing impact
is not straightforward. However, we can show that the fund ownership eﬀect works onlyin G-firms. Hence, as we only need to analyze the interaction between the direct eﬀect
and the fund ownership eﬀect in G-firms, we can derive our second result.
Our theoretical results have several empirical implications as a result of imposing restrictions on the model parameters that cause D- and P-funds to have diﬀerent types ofcosts and to specialize in diﬀerent types of engagement. To this end, we define expectedfinancial returns as [the expected profit of the firm] − [the price of the firm]. Next, sup-pose that improving ESG performance can be achieved to a certain degree by setting
broad, market-wide standards for ESG rather than focusing on firm-specific operational
improvements, or that most of the D-funds are made up of ESG-indexed funds. Then,as will be discussed in Section 5, the D-fund has a comparative cost advantage comparedwith the P-fund in improving profit performance over ESG performance.
Now, under this setup, we suggest the following. First, growing interest in ESG is likely
to improve expected ESG performance but decrease expected financial returns in any typeof firm if the cost complementarity in the D-fund manager’s eﬀorts is not suﬃciently large,
particularly if most of the D-funds comprise ESG-indexed funds. However, in firms with
low-ESG scores, growing interest in ESG always improves expected ESG performancebut decreases expected financial returns. In addition, it creates a positive association
between ESG scores and the earnings before interest, taxes, depreciation, and amortization(EBITDA) attained by firms with low-ESG scores.
Second, the decreasing impact of shareholder ESG engagement on firms with highESG scores reduces expected ESG performance but improves expected financial returns
in firms with high-ESG scores and all the firms in the market portfolio. Furthermore,it causes a negative association between ESG scores and the EBITDA attained by firms
with high-ESG scores and all the firms in the market portfolio.
Third, as sustainable active funds are more inclined to focus on firm-specific operational
engagements compared with ESG-indexed funds, there is greater cost complementarity
between the eﬀorts of fund managers in sustainable active funds than is the case for ESGindexed funds. Thus, the higher proportion of ESG-indexed funds in D-funds worsens
8expected ESG performance but improves expected financial returns in firms with highESG scores and all the firms in the market portfolio. It also causes a negative association
between ESG scores and the EBITDA attained by firms with high-ESG scores and by all
the firms in the market portfolio. Furthermore, the higher cost complementarity between
the engagement eﬀorts of firm-specific operational involvements and the establishment ofbroad, market-wide standards for ESG improves expected ESG performance but reduces
expected financial returns in any type of firm. It also causes a negative association between
ESG scores and the EBITDA attained by any type of firm.
The remainder of the paper is organized as follows. Section 2 reviews the related
literature, and Section 3 presents the basic model. Sections 4.1 and 4.2 examine fundmanagers’ eﬀorts and trading decisions, taking asset management fees and investment
decisions by fund investors as given. Section 4.3 derives the asset management fees and theinvestment allocation decisions by fund investors. Section 4.4 characterizes the equilibrium
and Section 4.5 discusses the comparative static results. Section 5 considers the empiricalimplications of our main results. The final section concludes the paper. Appendix A
provides the proofs of all propositions. Appendix B discusses the robustness of our main
results in the case of multiple D- and P-funds and in the presence of non-sustainable
funds, and provides additional comparative static results regarding P-fund growth.
2. Related Literature
The analysis in this paper is related to the delegated asset management literatureon the interaction between active and passive funds in general equilibrium (Gârleanu andPedersen, 2018, and Corum, Malenko, and Malenko, 2021). In particular, by extending themodel of Gârleanu and Pedersen (2018) with shareholder engagement, Corum, Malenko,
and Malenko (2021) mainly examine how the fund managers’ governance incentives toimprove the expected profit of their portfolio firms are aﬀected by passive fund growthevidenced by a decrease in fund investors’ search costs for the P-fund.8 By contrast, basedon the model of Corum, Malenko, and Malenko (2021), we consider multiple task-settingby D- and P-fund managers who make engagement decisions to improve not only the8

Kakhbod, Loginova, Malenko, and Malenko (2023) discuss the eﬀect of shareholder engagement;that is, shareholders communicating their views to management, under growing ownership by passively
managed funds. However, they do not consider the agency relationship between fund investors and fundmanagers.

9expected profit but also the social performance (i.e., impact) of their portfolio firms wheninvestors have an ESG preference.
Our model is related to theoretical studies of sustainable investing and provides implications for the existence of socially responsible investors. Pástor, Stambaugh, and Taylor(2021b) derive an ESG factor in an equilibrium asset-pricing model when investors havean ESG preference. They show that sustainable investing brings about a positive social
impact by making firms greener and by shifting real investment toward green firms. Goldstein, Kopytov, Shen, and Xiang (2022) develop a model with socially responsible andtraditional for-profit investors, and suggest that an increase in the fraction of socially responsible investors and an improvement in the ESG information quality can reduce priceinformativeness about the financial payoﬀ and raise the financial returns for investors.
Under the interaction between socially responsible investors and traditional for-profit
investors when firms are subject to financing constraints, several papers consider the effect of socially responsible investors providing more capital to sustainable investment.
Chowdhry, Davies, and Waters (2019) indicate that socially responsible investors must
hold financial claims to counterbalance project owners’ tendencies to overemphasize profits, if project owners cannot commit to social objectives when they finance capital fromprofit-motivated investors alone. Green and Roth (2021) examine the situation in whichsocially responsible and commercial investors compete to finance for-profit entrepreneurs,
and characterize alternative strategies for socially responsible investors that result inhigher social welfare and higher financial returns. Oehmke and Opp (2023) identify optimal investment choices for socially responsible investors who bargain with the entrepreneur with ESG preferences, and show that socially responsible and for-profit financial
investors are complementary. Using the model in which entrepreneurs search for capital,
Landier and Lovo (2022) indicate that a firm’s incentive to reduce negative externalities
increases with the size of socially responsible capital.
Heinkel, Kraus, and Zechner (2001) and Edmans, Levit, and Schneemeier (2022) derive conditions concerning whether holding stocks of a “brown” firm taking a corrective
action dominates divestment of stocks of “brown” firms. Broccardo, Hart, and Zingales(2022) find that engagement (i.e., through voting rights) is more eﬀective than divesting
stocks to make firms internalize negative externalities. Inderst and Opp (2022) investigate
a situation in which the social planner sets a minimum susceptibility standard that all
10

investment and production must satisfy, and ask whether such labeling is socially optimal. Hartzmark and Shue (2022) discuss that the current sustainable investment strategy
mainly rewards green firms for trivial reductions in their already low levels of emissions.
Our paper is the first in the literature to explore how the expected negative externality
and expected profits of firms vary in the presence of sustainable funds with divestment
strategies and passive funds, considering (i) investors’ ESG preferences, (ii) the relative
impact of shareholder ESG engagement in high-ESG firms compared with low-ESG firms,
and (iii) the cost complementarity between fund managers’ eﬀorts. In particular, we
examine these issues in the case where profit-motivated fund managers encounter a multitask agency scenario, wherein they apply engagement eﬀorts to improve ESG and profit
levels for socially responsible investors. Although the direct eﬀects of the key parametersin these problems are crucial, a significant aspect of our study is that, in contrast withthe literature, we examine the interaction between fund investors reallocating investments
across assets and fund managers reallocating engagement eﬀorts across tasks in the presence of D- and P-funds. This interaction may result in the failure of growing interest in
ESG to eﬀectively contribute to improving expected ESG performance.
3. The Basic Model
3.1. Firms, fund managers, and investors.–
The model considers a single period, from time 0 to time 1, in which there are three
types of agents: fund managers, who invest in firms on behalf of fund investors but are
purely interested in their monetary payoﬀs; fund investors, who indirectly invest in firms
through the fund managers and consider both firm profit and ESG performances; andliquidity investors, who directly invest in firms for various reasons, and consider both
firm profit and ESG performances. Fund investors can be interpreted as aggregationsof both socially responsible and non-socially responsible investors. In Appendix B.2, we
extend the basic model by incorporating non-socially responsible investors, who do notcare about ESG performance. All agents are risk neutral.
There is a mass one of firms with observable high-ESG performance (which we referto as G-firms), and a mass one of firms with observable low-ESG performance (B-firms).
G-firms generate lower negative externalities (e.g., they are cleaner and pollute the en-

11vironment less) than B-firms. We index G-firms (B-firms) by Gj ∈ [0, 1] (Bj ∈ [0, 1]).

Each firm’s stock is in unit supply because the initial owners of the firm have suﬃciently
low valuations such that they are willing to sell their shares regardless of the price.
The observable profit of firm Gj is represented byeGj ≡ RGj + ²RGj = R0 + P eiGj + ²RGj ,
R

(1)

i∈ΥGj

where R0 is the base profit without shareholder engagement, eiGj is the amount of unobservable eﬀort exerted by shareholder i to improve the profit of firm Gj , ΥGj is theset of shareholders of firm Gj , and ²RGj is unobservable noise that has a mean of zero.9
Similarly, the observable profit of firm Bj iseB ≡ RB + ²BG = R0 + P eiB + ²RB ,
R
j
j
j
j
j

(2)

i∈ΥBj

where eiBj is the amount of unobservable eﬀort exerted by shareholder i to improve theprofit of firm Bj , ΥBj is the set of shareholders of firm Bj , and ²RBj is the unobservablenoise that has mean zero. For simplicity, note that the base profit is the same for bothG- and B-firms.
The observable negative externality released by firm Gj is given byP
ZeGj ≡ ZGj + ²ZGj = ZG0 − λaiGj + ²ZGj ,

(3)

i∈ΥGj

where ZG0 is the negative externality of G-firms without shareholder engagement, λ ≥ 0

is a constant parameter, aiGj is the amount of unobservable eﬀort exerted by shareholder
i to improve the ESG performance of firm Gj , and ²ZGj is the unobservable noise thathas mean zero. Similarly, the observable negative externality released by firm Bj isP
aiBj + ²ZBj ,ZeBj ≡ ZBj + ²ZBj = ZB0 −

(4)

i∈ΥGj

where ZB0 is the negative externality of B-firms without shareholder engagement, aiBj is
9

The noise terms ²RGj and ²RBj (²ZGj and ²ZBj ) prevent fund investors from stipulating the eﬀorts
for their fund managers directly by observing RGj and RBj (ZGj and ZBj ).

12the amount of unobservable eﬀort exerted by shareholder i to improve the ESG performance of firm Bj , and ²ZBj is the unobservable noise that has mean zero. Comparing (3)and (4), note that the eﬀect of each shareholder on improving the ESG performance ismore (or less) eﬀective in B-firms than in G-firms if 0 ≤ λ < 1 (or 1 < λ).

It follows from (3) and (4) that in response to an increase in the eﬀort of each share-

holder, there is a marginal improvement in the ESG performance of G-firms that is smaller
than that of B-firms if 0 ≤ λ < 1. The scenario where 0 ≤ λ < 1 reflects the observationthat the shareholder eﬀort would be more valuable for firms that lag behind in terms of
ESG; that is, for B-firms (for empirical evidence, see Hartzmark and Shue, 2023).

We assume that ²RGj , ²RBj , ²ZGj , and ²ZBj are distributed independently. It follows
eGj , ReBj , ZeGj , andfrom (1)—(4) that RGj , RBj , ZGj , and ZBj are the expected values of R

ZeBj , respectively. As G-firms have better ESG performance than B-firms, we focus on
the case of ZGj < ZBj , which implicitly assumes that ZB0 is suﬃciently larger than ZG0 .

In the subsequent analysis, we focus on the case of (ZGj , ZBj ) > 0.

We assume that the stocks of firms can be accessed by fund investors only throughtwo fund types: namely, D-funds and P-funds. For simplicity, there is one fund managerof each type of fund, although we can easily extend to any number of D-funds and Pfunds, ND and NP .10 The D-fund is restricted to holding stocks of firms with high-ESG
performance, whereas the P-fund is restricted to holding a value-weighted portfolio of all
stocks according to a mechanical rule. In our framework, the D-fund can be taken as
either sustainable active funds or ESG-indexed funds. On the other hand, the P-fund
can be interpreted as index funds excluding ESG-indexed funds. In Appendix B.2, weconsider a non-sustainable fund (N-fund) that only invests in stocks of firms with low-ESG
performance. The fund manager of each fund oﬀers to invest the wealth of fund investors
in stocks of firms in exchange for an asset management fee.
The fund manager of type i ∈ (D, P ) chooses the amount of unobservable eﬀorts(eihj , aihj ) to improve the profit and ESG performance of his portfolio firms hj at time

0, where h ∈ (G, B). If he exerts an eﬀort regarding the firm profit eihj and an eﬀort
regarding ESG performance aihj , he incurs a private cost cRi (eihj ) + cCi (aihj ) for i ∈10

Although we extend our model to the case of multiple D- and P-funds, the only things that matterfor fund managers’ engagements with portfolio firms are the combined holdings of all D-funds and thoseof all P-funds. The individual fund’s ownership stakes do not matter; therefore, our results continue tohold. See Appendix B.1 for further details.

13

(D, P ) and h ∈ (G, B) to improve profit performance, whereas he incurs a private costcZi (aihj ) for i ∈ (D, P ) and h ∈ (G, B) to improve ESG performance. We assume thatcki (0) = 0, c0ki (·) > 0, c00ki (·) > 0, c0ki (0) = 0, and c0ki (∞) = ∞ for k ∈ (R, Z) and i ∈ (D, P );and cCi (0) = 0, c00Ci (·) > 0, c0Ci (0) = 0, and c0Ci (∞) = ∞ for i ∈ (D, P ). These standard

assumptions ensure an interior solution to (eihj , aihj ) for i ∈ (D, P ) and h ∈ (G, B). In
addition, c0Ci (·) < 0 (c0Ci (·) > 0) implies that for each type of fund manager, improving

the profit performance of his portfolio firms hj is cost complementary (substitutable) toimproving the ESG performance of his portfolio firm hj .
The eﬀort eihj (aihj ) exerted by the fund manager i for i ∈ (D, P ) and h ∈ (G, B)

includes any actions, such as communicating with management, submitting shareholder
proposals, nominating directors, and voting on proxy contests. These engagement strategies are more extensively discussed in Section 5. Although large passive and index funds
charge substantially smaller management fees than actively managed funds, the large
amount of assets under their management and ownership stakes can compensate for their
low management fees and strongly incentivize their fund managers to exert managerial
eﬀort (see Brav, Malenko, and Malenko, 2022, for a numerical discussion; and Kahn andRock, 2020, Lewellen and Lewellen, 2022, and Brav, Jiang, and Pinnington, 2024, forempirical evidence regarding financial incentives of P-funds). In particular, recently, fundmanagers of large passive and index funds have been involved in engagement throughvoting and communications with management (see the literature review of Section 4.2 in
Brav, Malenko, and Malenko, 2022, for the empirical evidence). Hence, greater ownershipof passive and index funds has various eﬀects on governance. In the subsequent analysis,
we focus on the case of (eDGj , aDGj ) > 0 and (eP hj , aP hj ) > 0 for h ∈ (G, B) and any j.
Note that the D-fund does not hold stakes in B-firms.

There is a large mass of homogeneous fund investors, who have a certain amount ofwealth to invest, ε.11 We denote their aggregate wealth by W , which is given exogenously. Although fund investors want to receive more pecuniary investment returns, theyalso derive disutility from holding stocks of firms generating negative externalities. For
simplicity, we assume that the amount of disutility incurred by fund investors per unit of
their stock holdings is equal to η × (the negative externality generated by their holding
11

Fund investors typically include pension funds, sovereign wealth funds, and wealthy retail investors
who have invested in family oﬃces in which hedge funds manage their covert operations.

14

firms per unit of their stock holdings), where η (> 0) is a scalar measuring the degree ofinvestors’ “ESG” preference.12
At time 0, each fund investor chooses whether to invest in the D-fund and/or the P-fund
by delegating her wealth to the fund manager, and whether to invest in an alternative
investment opportunity such as public bonds that generates a fixed net return of 0. Wheneach fund investor with wealth ε seeks an D-fund (a P-fund) manager, she must search
for and vet fund managers by incurring a search cost ψD ε (ψP ε). Fund investors are
assumed to be homogeneous; therefore, they incur the same search cost ψD ε (ψP ε) when
they try to find an D-fund (a P-fund) manager. Thus, the proportional parameter εcan be interpreted as a normalization by adjusting the scale of ψD and ψP . These costs
indicate that most fund investors must spend significant resources to find a fund manager
whom they trust with their money, and to examine the funds’ investment strategies andfee structures.13
After a fund investor finds a fund manager i ∈ (D, P ), they negotiate the fee fi throughgeneralized Nash bargaining at time 0.14 The fund manager of the D-fund (P-fund) has

bargaining power ωD (ωP ) ∈ (0, 1), whereas the fund investor has bargaining power 1 −ωD (1− ωP ). We assume that the fee charged by the fund manager to the fund investor
is a fraction of the sum of the realized values of the profit and the disutility of the negative externality of their portfolio firms, which is assumed to be observable. The pecuniary
amount deducted from the fund manager’s compensation as a result of the negative externality arises directly from bargaining between the fund manager and the fund investor.15
12

The utility of fund investors depends on the pecuniary returns that they receive and the social value
created by firms they have financed through funds. Green and Roth (2021) refer to these investors asvalue-aligned social investors. The assumption that investors prefer socially responsible performance issupported by empirical studies in the mutual fund literature. For example, see Riedl and Smeets (2017)
and Hartzmark and Sussman (2019).
13
For more information, see Corum, Malenko, and Malenko (2021) and the references listed in footnote
9 in their text. Furthermore, in our model, the positive search costs, (ψD , ψP ) > 0, ensure that the fundinvestors’ indiﬀerence conditions, (18), can be compatible with the Nash bargaining outcomes, (19) and(20), under the equity pricing rule, (14) and (16).
14This assumption is also made in Gârleanu and Pedersen (2018) and Corum, Malenko, and Malenko(2021).
15
Note that the deduction amount, fi × (the disutility of the negative externality), in turn, increases
the pecuniary payoﬀ of fund investors, while decreasing the pecuniary payoﬀ of the fund manager. In
addition, the fund manager’s payoﬀ is positive even though this amount is deducted because we impose
the assumption, as discussed below, that the stock price is always positive. Adachi-Sato (2022) clarifies
that a profit-oriented agent pursues improvement in ESG performance under the bargaining between asocially and environmentally aware principal and the agent.

15

However, it may be viewed as the fund manager’s reward reduction being tied to ESG
criteria and/or the fund manager’s reputation loss from the nonaccomplishment of directcontract commitments on ESG goals or from public sentiment that the fund manager
may impair improvement in firms’ ESG performance. Regardless of which interpretationis chosen, the bargaining provides an incentive for the profit fund manager to improve
ESG performance.
Finally, there is a large mass of liquidity investors, who directly invest in firms at time
0 for various reasons, such as liquidity need, hedging demand, firms’ issuance, repurchase of shares, or investor sentiment, although they also incur a disutility of amount as
η × (the negative externality generated by their holding firms per unit of their holding

stocks).16 Liquidity investors apply rational expectations in predicting the value of eachstock. However, their valuation is perturbed by an additional factor that captures theamount of liquidity need, hedging demand, firms’ issuance, repurchase of shares, or investor sentiment. As a result, liquidity investors’ expectation of the value of the stockof each firm is equal to the sum of the expected values of the profit and the disutilityof the negative externality generated by the firm hj , Rhj − ηZhj , minus the additional

component, Lhj > 0; that is, Rhj − ηZhj − Lhj for h ∈ (G, B).17
3.2. Equilibrium.–

The equilibrium includes the search and investment allocation strategies of fund investors, management fees, each agent’s trading strategy, each fund manager’s engagement
eﬀort strategy, and the market clearing price, as follows.
At the beginning of time 0, fund investors make their search and investment allocation
decisions with the aim of maximizing the sum of their expected profit and their expected
disutility from negative externalities minus the search cost. After a fund investor finds afund manager, they negotiate the management fee through generalized Nash bargaining.
Then, each fund manager invests the delegated amount of fund investors’ wealth in stocks
according to the trading strategy of each fund, as described below. First, the D-fund isrestricted to holding stocks of firms with high-ESG performance. Specifically, we assume
that the D-fund manager invests only in G-firms. Second, the P-fund is restricted to
16
17

These traders include insurance companies and retail investors.
All of our results continue to hold even when LGj = LBj .

16

holding a value-weighted portfolio of all stocks according to a mechanical rule.
However, liquidity investors trade according to their predictions about the stock priceof each firm. Their predictions are made by anticipating the equilibrium eﬀort of fundmanagers under rational expectations.
After trading, each fund manager selects his eﬀort at time 0 to maximize his expectedcompensation minus his eﬀort cost by improving the profit and ESG performance of his
portfolio firms, given his management fee.
Finally, at time 1, the profit and negative externality generated by each firm is realized.
Then, the payoﬀs of the fund manager and fund investors in each fund are determinedaccording to the management fee.
The stock price of each firm is set to clear the market at time 0. Short sales are ruledout. We restrict our analysis to the case in which liquidity investors hold at least someshares of each type of stock.18 The assessment of both the D-fund and the P-fund of eachstock h ∈ (G, B) reflects the fund investors’ valuation of the stock h, which is higher thanthat of the liquidity investors; therefore, the market clearing price of stock h ∈ (G, B)

may be determined by the liquidity investors’ assessment of stock h.19
Figure 1 illustrates the model timing.
4. The Analysis

We solve this model by backward induction. First, we determine the eﬀort decisionsof fund managers and examine the trading decisions of liquidity investors and fund managers. Then, we clarify the investment allocation decisions of fund investors and thedetermination of asset management fees. Finally, the equilibrium is defined as a set ofthese decisions and market clearing conditions.
We drop the subscript j from each variable in the subsequent discussions because Gfirms (B-firms) are all homogeneous and because the D-fund finds it optimal to diversify
equally across all G-firms.20
4.1. Fund managers’ eﬀort decisions.–
18

Proposition 1 provides a suﬃcient condition for this to hold.
For an alternative justification, we can assume that the D- and P-funds submit a market order,
whereas liquidity investors submit a limit order reflecting their valuation of each stock.
20
We provide a suﬃcient condition for this in Proposition 1.
19

17

Suppose that the D-fund manager charges the management fee fD , holds xDG shares,
and exerts the eﬀorts (eDG , aDG ). Then, the D-fund manager’s expected payoﬀ is given
byfD xDG (RG − ηZG ) − cRD (eDG ) − cCD (aDG ) − cZD (aDG ).

(5)

Note that the D-fund holds only the stock of the G-firm, and that the sum of the expected
profit and the expected disutility from the negative externality of the G-firm is RG −

ηZG . As discussed in Section 3.1, note that the fund manager’s compensation includes
the pecuniary amount deducted as a result of the negative externality.
Next, suppose that the P-fund manager charges the management fee fP , holds xP hshares, and exerts the eﬀorts (eP h , aP h ) for h ∈ (G, B). Then, the P-fund manager’s
expected payoﬀ is given byX

h∈(G,B)[fP xP h (Rh − ηZh ) − cRP (eP h ) − cCP (aP h ) − cZP (aP h )] .

(6)

Note that the P-fund holds stocks of both the G-firm and the B-firm, and that the sumof the expected profit and the expected disutility from the negative externality of firm h
is Rh − ηZh for h ∈ (G, B).

For simplicity, we assume that cRi (eih ), cC i (aih ), and cZi (aih ) are quadratic; that is,cRi (e) =cRi 2
cC i 2cZi 2e , cC i (aih ) =a , and cZi (a) =
a,
2
2
2

for i ∈ (D, P ),(7)

where cRi > 0 and cZi > 0 for i ∈ (D, P ). However, for i ∈ (D, P ), either cC i > 0 or cC i <0 is possible, although we assume that cZi + cC i > 0. Then, it follows from (1)—(4) thatmaximizing (5) and (6) yields the following first-order conditions; that is, the optimal
eﬀort decisions of the D-fund manager, (eDG , aDG ), and the P-fund manager, (eP h , aP h )for h ∈ (G, B), satisfyeDG =eP h =fD xDG
ηλfD xDG
and aDG =,cRD
cZD + cCD

fP xP hηλh fP xP h
and aP h =,cRP
cZP + cCPfor h ∈ (G, B),

(8)

(9)

where λG = λ and λB = 1. Next, (8) and (9) imply the following: for i ∈ (D, P ) and18h ∈ (G, B), (i) each fund manager exerts more eﬀort regarding both firm profit and

ESG performance in all firms if he holds more shares of his portfolio firms (i.e., if xih is

higher) and/or if his management fee is higher (i.e., if fi is higher); (ii) each fund manager
exerts more eﬀort regarding ESG performance in all firms if the degree of investors’ ESG
preference is larger (i.e., if η is larger); (iii) each fund manager exerts more eﬀort regarding
ESG performance in G-firms if his eﬀort is more valuable in G-firms (i.e., if λ higher);and (iv) each fund manager exerts more eﬀort regarding ESG performance in all firms if
the cost complementarity between eﬀorts regarding firm profit and ESG performance islarger (i.e., if cC i is smaller).
4.2. Trading decisions.–
In making their trading decisions, liquidity investors and fund managers rationally
anticipate the fund managers’ eﬀort decisions given by (8) and (9).
Under rational expectations, if liquidity investors expect the D- and P-funds to holdxDG and xP G shares in G-firms, it follows from (1), (3), (8), and (9) that the liquidity
investors’ assessment of the stock of G-firms is¶µ
fD xDG fP xP G
ηλfD xDG
ηλfP xP G− LG .
+− η ZG0 − λ−λ
RG − ηZG − LG = R0 +cRD
cRP
cZD + cCD
cZP + cCP
(10)
Similarly, using (2), (4), and (9), liquidity investors’ assessment of the stock of B-firms is¶µfP xP BηfP xP B
RB − ηZB − LB = R0 +− LB .
− η ZB0 −cRP
cZP + cCP

(11)

Note that the D-fund holds only the stock of G-firms.
Each liquidity investor purchases stock h ∈ (G, B) if his valuation exceeds the price;that is, Rh − ηZh − Lh ≥ Ph for h ∈ (G, B). We focus on the case in which liquidity

investors hold at least some shares of stocks in each type of firm; that is, G- and B-firms.
As mentioned at the end of Section 3.2, this implies that the market clearing price ofstock h ∈ (G, B) is determined by the liquidity investors’ assessment of stock h:
Ph = Rh − ηZh − Lh ,19

for h ∈ (G, B),(12)where Rh − ηZh − Lh for h ∈ (G, B) is given by (10) and (11). Here, we assume that R0> max (ηZG0 + LG , ηZB0 + LB ), which is also provided in Proposition 1. Given (10) and(11), this assumption ensures that the price of each stock is always positive.
To characterize the trading decisions of the D- and P-funds, let WD and WP denote thesizes of the D- and P-funds, respectively, which are endogenously determined in equilibrium. The D- and P-fund managers use all of WD and WP delegated to them as long asliquidity investors hold at least positive shares of stocks of each firm.21
The D-fund invests only in G-firms. As mentioned, the D-fund finds it optimal todiversify equally across all G-firms. Given that the D-fund can use all WD to purchasexDG units of the stock of G-firms, we have
WD
.
PG

xDG =

(13)

Because of (12), note that
PG = RG − ηZG − LG .

(14)

The P-fund is restricted to holding a value-weighted portfolio of all stocks, with thismarket portfolio denoted by index M. As there is a mass one of G-firms and a mass
R1
R1
one of B-firms, the price of the market portfolio is PM = 0 PG dj + 0 PB dj = PG + PB .

The P-fund purchases xP h units of stock h according to the market portfolio rule, underwhich the proportion of the amount invested in stock h in the fund, xPWhPPh , equals theweight of this stock in the market portfolio, PPMh . This implies that xP h is the same forany h ∈ (G, B) and is equal to

xP =

WP
.
PM

(15)

Furthermore, let RM = RG + RB , ZM = ZG + ZB , and LM = LG + LB . Then, it follows
from (10)—(12) with xP h = xP for any h ∈ (G, B) thatPM = RM − ηZM − LM ,
21

(16)

Note that these fund managers evaluate each stock more highly than do liquidity investors or submit
a market order.

20

where¶µfD xDG fP xP
ηλfD xDG
ηλfP xP
RM − ηZM − LM = R0 ++− η ZG0 − λ
−λ
cRD
cRP
cZD + cCD
cZP + cCP¶µ
fP xPηfP xP− LM .
− η ZB0 −(17)+ R0 +cRP
cZP + cCP
Note that from (9) and (15), eP h is the same for any h ∈ (G, B). Thus, (9) is rewritten
so that

eP =

fP xP
ηλh fP xP
and aP h =,cRP
cZP + cCPfor h ∈ (G, B),(90 )where λG = λ and λB = 1.
4.3. Investment allocation decisions and asset management fees.–
Now, we discuss the investment allocation decision by infinitesimal fund investors, whochoose between investing in the D-fund and/or the P-fund and investing in an alternative
investment opportunity, such as public bonds, that generates the fixed return 0.
Consider a fund investor with wealth ε. If the fund investor invests εD in the D-fund, thefund manager buys PεDG stocks. Then, the fund investor’s expected payoﬀ is (RG − ηZG ) PεDG

− fD (RG − ηZG ) PεDG − ψD εD = (1 − fD )(RG − ηZG ) PεDG − ψD εD because she incurs asearch cost ψD ε and pays the fee fD . Similarly, if the fund investor invests εP in theP− ψP εP . Hence,P-fund, her expected payoﬀ with the P-fund is (1 − fP )(RM − ηZM ) PεMtime 0, the objective function of the fund investor is represented by(1 − fD )(RG − ηZG )εDεP− ψD εD + (1 − fP )(RM − ηZM )− ψP εP + ε − εD − εP .
PG
PMNote that the net return of the alternative investment opportunities apart from the Dand P-funds is equal to zero.
To characterize the investment allocation decisions of fund investors, we begin by examining their indiﬀerence conditions regarding the investment allocation. Under the condition of Proposition 1 derived below, we can ensure that WD + WP < W if the aggregate
wealth of fund investors such as pension funds is suﬃciently large that they purchasepublic bonds.22 Then, fund investors make a positive investment in public bonds with the22

For each investor level, this implies that εD + εP < ε.

21

fixed net return of 0. This implies that fund investors earn the same rate of expected net
return regardless of whether they invest with D- or P-funds, and this expected net returnis equal to 1.23 Specifically, the fund investors’ indiﬀerence conditions are(1 − fD )RG − ηZG
RM − ηZM− ψD = (1 − fP )− ψP = 1.
PG
PM

(18)

Next, we deal with bargaining in the D-fund. After a fund investor investing εD in
the D-fund incurs the cost ψD εD and finds an appropriate D-fund, she bargains with the
D-fund manager over the fee fbD . The outcome of the bargaining depends on each player’s

expected payoﬀ in the event of agreement and no agreement. If the fund investor and thefund manager agree on the fee fbD , the fund investor’s expected payoﬀ is (1 − fbD )(RG

− ηZG ) PεDG . If no agreement is reached, the fund investor can either find the P-fund byincurring the cost ψP εD and invest with the P-fund or invest in the alternative investment

opportunity. Under (18), the P-fund yields the same rate of net return 1 for the fundinvestor as the alternative investment opportunity so that her expected payoﬀ is εD whenno agreement is reached.
To specify the fund manager’s expected payoﬀ, we must provide his additional expected
payoﬀ from agreeing on the fee fbD and obtaining the additional funds εD . The additional

expected payoﬀ is represented by fbD (RG − ηZG ) PεDG .24 However, the fund manager’s gainfrom no agreement is zero.

Given the fund manager’s (investor’s) bargaining power ωD (1 − ωD ), the bargaining

outcome maximizes the product of the expected payoﬀ gains from agreement with respect
to fbD :∙¸1−ωD ∙
¸ω
εD
εD D
b
b
fD (RG − ηZG )max (1 − fD )(RG − ηZG )
− εD
.
PG
PG
fD
23

This assumption forces us to neglect the case where the rates of net return from investing with theD- and P-funds are larger than 1, which is analyzed by Corum, Malenko, and Malenko (2021). However,
the assumption enables us to focus on the eﬀects that occur through the eﬀort decisions of fund managers
on multiple tasks under the negative externality released by their portfolio firms.
24
By adding εD to the fund with xDG , it follows from (1), (3), and (90 ) that the expected payoﬀ ofxP
P xP
the D-fund manager is max {fD [RG0 + e + fcPRP− η(ZG0 − λa − λ cηλf)]xDG + fbD [RG0 + e +ZP +cCP
e,a

fP xP
cRP

P xP
− η(ZG0 − λa − λ cηλf)] PεDG − cRD (e) − cCD (a) − cZS (a)}. Using the envelope theorem, theZP +cCP
derivative with respect to εD at εD = 0 yields fbD [RG0 + e + fP xP − η(ZG0 − λa − λ ηλfP xP )] 1 , thatcRP

G)
is, fbD (RGP−ηZ.
G

22

cZP +cCP

PG

The solution must satisfy
∙
¸
εD
εD
b
fD (RG − ηZG )= ωD (RG − ηZG )− εD .
PG
PG

As the D-fund fee is the same for all fund investors, we have fbD = fD . Thus,µfD = ωD 1 −PG
RG − ηZG

¶

.

(19)

Similarly, the P-fund fee is the same for all investors, and is given byµfP = ωP 1 −PM
RM − ηZM

¶

.

(20)

4.4. Characterization of equilibrium.–
The equilibrium is defined as a solution to the following system of equations: (i) thefund managers’ eﬀort decisions (8) and (90 ); market clearing conditions (10) and (13)—(17);fund investors’ capital allocation conditions (18); and fee bargaining conditions (19) and(20). These equations determine the following endogenous variables: the fund managers’
eﬀort decisions, (eDG , aDG , eP , aP G , aP B ); the asset management fees, (fD , fP ); the trading
decisions and investment asset allocations, (xDG , xP , WD , WP ); the total expected payoﬀs,(RG − ηZG , RM − ηZM ); and the asset prices, (PG , PM ).

Now, we obtain the following proposition that characterizes the equilibrium.

Proposition 1: Suppose that R0 > max (ηZB0 + LB , ηZG0 + LG ) and R0 − ηZG0 − LG

> W ≥ W , where W is given by (A9) in Appendix A. Then, we have the following.

D ψD
P ψP
and fP = ψPω+1−ω, and fD ≥ fP if ωD(i) The asset management fees are fD = ψDω+1−ω
DP≥ ωP .

D
LG + ηZG(ii) The expected profit of G-stocks and the market portfolio are RG = ψD +1−ωψDP
and RM = ψP +1−ωLM + ηZM .
ψP(iii) The expected negative externalities generated by B-firms, G-firms, and the firms included in the market portfolio satisfy the following equations:
ZB + ZG = ZM ,

23

(21)

ZG = ZG0 − ηλ2cRDcRP(2RG − RM ) − ηλ2(RM − RG − R0 ),cZD + cCD
cZP + cCP

ZM = ZG0 + ZB0 − ηλ2
−η

(22)

cRD
cRP
(2RG − RM ) − ηλ2(RM − RG − R0 )cZD + cCD
cZP + cCP

cRP
(RM − RG − R0 ),cZP + cCP

(23)

where RG and RM are given above.
DP
LG and PM = 1−ω
LM .
(iv) The prices of G-stocks and the market portfolio are PG = 1−ωψDψPThe restrictions R0 > max (ηZB0 + LB , ηZG0 + LG ), R0 − ηZG0 − LG > W , and W ≥W ensure that the liquidity investors’ holding ratio in each stock is positive, that is, xDG+ xP < 1. The restriction R0 > ηZG0 + LG guarantees that the D-fund finds it optimal
to diversify equally across all G-firms. Finally, the restriction W ≥ W further ensuresthat fund investors make a positive investment in an alternative investment opportunity
such as public bonds; that is, WD + WP < W .
As WD + WP < W , fund investors’ aggregate wealth is relatively large. Then, theiroutside options in negotiations are eventually limited by an alternative investment opportunity such as public bonds with the fixed net return 0.25 In addition, when ψD ≥ ψP ,the fee charged by the D-fund is higher than that charged by the P-fund if the bargaining
power of the D-fund manager is equal to or exceeds that of the P-fund manager.
Some recent studies indicate the limited or even counterproductive impact of D-funds on
ESG because high-ESG firms have little scope for further improvement in their impact,
whereas low-ESG firms, which face an increase in financing costs, must produce large
negative externalities (e.g., see Hartzmark and Shue, 2023). In the present model, thelimited impact of the ESG performance in G-firms in response to each fund manager’s
eﬀort can be captured by supposing λ < 1.
4.5. Comparative statics.–
We examine the eﬀects of the key parameters of the model on the expected negative
externalities, the expected profits, and the asset management fees. The key parameters
are the intensity of investors’ ESG preferences, η, the diminishing impact of shareholder
25

To be consistent with the assumption of R0 − ηZG0 − LG > W that ensures xDG + xP < 1, we needto suppose the situation where R0 is also suﬃciently large. Hence, our analysis may be less applicable tothe case where the economy faces a financial crisis such that R0 is not suﬃciently large.

24

ESG engagement on high-ESG firms, λ, and the eﬀort cost complementarity of each Dand P-fund manager, cCD and cCP , respectively.
First, we consider how growing interest in ESG aﬀects the ESG and pecuniary performances of firms. Then, we obtain the following proposition.
Proposition 2: Suppose that investors’ ESG preferences are strengthened (i.e., η increases).
RP
ZP +cCP≥ ccZD.
(i) The expected negative externality released by G-firms, ZG , decreases if ccRD
+cCD
RP
ZP +cCP< ccZDand if ZZBG is suﬃciently large.
However, it may increase if ccRD
+cCD(ii) The expected negative externality released by B-firms, ZB , decreases, whereas the expected profit of B-firms, RB , increases.
(iii) The expected negative externality released by all the firms in the market portfolio,
RP
ZP +cCPRP
ZP +cCP≥ ccZD. However, it may increase if ccRD
< ccZDand if ZZBG
ZM , decreases if ccRD
+cCD
+cCD

is suﬃciently large.

(iv) The asset management fees of both funds are unaﬀected.
cRP
cRD

is equal to the eﬀort cost ratio of the P-fund manager to the D-fund manager for

ZP +cCPis the eﬀort cost ratio for improving the negative
improving firms’ profits, whereas ccZD
+cCD
RP
ZP +cCP(or ccZD) implies that increasing profits (or decreasing theexternality. A larger ccRD
+cCD

negative externalities) of the portfolio firms is relatively less costly for the D-fund manager
RP
ZP +cCP(or ccZD) can be interpreted suchthan for the P-fund manager. Thus, the larger ccRD
+cCD

that the D-fund has a greater cost advantage in improving profit (ESG performance)compared with the P-fund.
Considering this interpretation, Propositions 2(i) and 2(iii) show that when η increases,
ZG and ZM decrease if the D-fund has a comparative cost advantage in improving profit
RP
ZP +cCP≥ ccZD). However,
performance over ESG performance to the P-fund (i.e., if ccRD
+cCD

Propositions 2(i) and 2(iii) indicate that when η increases, ZG and ZM may increase if the

D-fund has a comparative cost advantage over the P-fund in improving ESG performance
RP
ZP +cCP< ccZD) and if the ratio of the negative externalities
over profit performance (i.e., if ccRD
+cCD

of B-firms to those of G-firms is suﬃciently large. Proposition 2(ii) also suggests that anincrease in η always decreases ZB and increases RB . Proposition 2(iv) states that anincrease in η does not aﬀect the fees of either the P-fund or the D-fund.
The intuition behind Proposition 2 is as follows. For convenience, we begin by con-25

sidering the eﬀect of an increase in η on the expected negative externality by taking themanagement fee of each fund as given. Later, we clarify the eﬀect of an increase in η byconsidering its eﬀect on the asset management fees.
Now, from the perspective of the fund manager’s optimizing behavior, the eﬀort incentive for each fund manager to reduce the negative externality in a firm depends not only
directly on η but also on the fund ownership stakes in the firm, (xDG , xP ), as indicatedin (8) and (90 ). The direct eﬀect of an increase in η on the ESG eﬀort incentive increases
the engagement eﬀorts for the D- and P-fund managers, (aDG , aP G , aP B ), to reduce thenegative externality, thus decreasing ZG , ZB , and ZM . However, an increase in η changes
the fund ownership stakes in each firm, (xDG , xP ), by aﬀecting the investment allocationof the fund investors in each fund. We refer to this as the fund ownership eﬀect.
Suppose that (ZG , ZM ) is initially given. Then, an increase in η increases the expected disutilities of the fund investors, ηZG and ηZB . The larger values of ηZG andηZM increase the fund investors’ rates of expected gross return from each fund (note)/∂(ηZ) > 0, where P = R − ηZ − L). However, in equilibrium, fund inthat ∂( R−ηZ
P

vestors must be indiﬀerent between investing in each fund and the alternative investment

opportunity (see (18)).
Given that ZG is assumed to be smaller than ZB and that η directly aﬀects fundinvestors’ indiﬀerence conditions, we can show that xDG may increase or decrease, butthat xP must increase to restore (18). This implies that fund investors may increase or
decrease their investment in the D-fund, enabling the D-fund manager to take larger orsmaller stakes xDG in G-firms, whereas the fund investors must increase their investmentin the P-fund, inducing the P-fund manager to take larger stakes xP in all the firms inthe market portfolio. Thus, the fund ownership eﬀect in response to an increase in η
increases or decreases aDG , while increasing aP G and aP B . Furthermore, such changes inxDG and xP do not necessarily reduce the ownership stakes held by liquidity investors in
G-firms, 1 − xDG − xP , but they reduce the ownership stakes held by liquidity investors

in the market portfolio, 1 − xP . However, even in this situation, if the D-fund has acomparative cost advantage in improving profit performance over ESG performance toRP
ZP +cCP≥ ccZD), the fund ownership eﬀect as a whole in response to
the P-fund (i.e., if ccRD
+cCD

an increase in η decreases ZG and ZM because the fund ownership eﬀects on aP G and aP B
are dominant and liquidity investors do not make any ESG engagement eﬀorts.
26

Given that B-firms are bought only by the P-fund and liquidity investors, such changes
in the fund ownership stakes reduce ZB . Because the direct eﬀect of an increase in η on
the ESG eﬀort derived at the beginning also decreases ZG , ZB , and ZM , the total eﬀect,consisting of the direct eﬀect and the fund ownership eﬀect in response to an increase inη, subsequently decreases ZG , ZB , and ZM .
By contrast, if the D-fund has a comparative cost advantage in improving ESG perRP
ZP +cCP< ccZD), theformance over profit performance compared with the P-fund (i.e., if ccRD
+cCD

fund ownership eﬀect of decreasing aDG may dominate that of increasing aP G and aP B if
ZB
ZG

is suﬃciently large. Then, the fund ownership eﬀect as a whole may increase ZG and

ZM and dominate the direct eﬀect of an increase in η on the ESG eﬀort if ZZBG is suﬃciently
large. Hence, the total eﬀect, consisting of the direct eﬀect and the fund ownership eﬀect
in response to an increase in η, may increase ZG and ZM .
In addition, even in this case, for B-firms, the fund ownership eﬀect works only throughchanges in the ownership of the P-fund and liquidity investors. Consequently, combining
the direct eﬀect, we show that an increase in η always decreases ZB and increases RB .
Finally, we return to examining the eﬀect of an increase in η on the asset management
fee of each fund. An increase in η changes the expected gross return earned by fundinvestors. The reason is that an increase in η changes the expected disutilities of fundinvestors, ηZ, as discussed above. However, given (14), (16), (19), and (20), note that, depends on the expected
the fund investors’ expected gross return rate, (1 − f ) R−ηZ
P

profit of the portfolio firms of the fund, R, minus the expected disutility of the negative

externality released by the firms, ηZ. When η increases, the eﬀect of the increase in η on
the expected disutility is canceled out by its eﬀect on the expected profit, restoring thefund investors’ capital allocation conditions (18). This implies that the change in η has
no eﬀect on the fund investors’ expected gross return rate, and thus has no eﬀect on the
management fee of either fund when fund investors’ bargaining power is fixed.26
Now, we discuss the comparative statics of λ, which represents the eﬀectiveness of eachfund manager’s ESG eﬀort in G-firms relative to that in B-firms. We have the following
proposition.
26

Under fund investors’ capital allocation conditions (18), the D-fund (P-fund) investors’ expected grossGM
return rate (1 − fD ) RGP−ηZ((1 − fP ) RMP−ηZ) must remain constant for a fixed ψD (ψP ). CombiningGM
this with (19) ((20)), we can show that both the fund fee fD (fP ) and the expected gross return rate
RG −ηZG RM −ηZM
( PM ) must remain unaﬀected by the change in η.
PG

27

Proposition 3: Suppose that the improvement of the ESG performance of G-firms relative to that of B-firms in response to each fund manager’s ESG eﬀort decreases (i.e., λdecreases).
(i) The expected negative externality and the expected profit of G-firms, ZG and RG , increase.
(ii) The expected negative externality and the expected profit of B-firms, ZB and RB , areunaﬀected.
(iii) The expected negative externality and the expected profit of all the firms in the market
portfolio, ZM and RM , increase.
(iv) The asset management fees of both funds are unaﬀected.
Proposition 3 indicates that when λ decreases, the expected ESG performances of Gfirms and of all the firms in the market portfolio are negatively aﬀected, whereas the
expected profit performances of these firms are improved. However, in the case of Bfirms, neither expected ESG nor expected profit performances are aﬀected. In addition,
the asset management fees of both funds are unaﬀected.
The intuition for Proposition 3 is as follows. Again, we start by assuming that the
management fee is fixed. Then, the lower λ directly decreases both the D-fund manager’s
D xDG) and the P-fund manager’s ESG eﬀort in
ESG eﬀort in G-firms (see aDG = cηλf
ZD +cCD
P xP), thereby directly increasing both ZG and ZM .27 However,G-firms (see aP G = cηλf
ZP +cCP

as discussed in the case of an increase in η, we need to consider the eﬀect of a change
in the fund ownership stakes. In fact, the crucial diﬀerences between the comparative
static mechanisms of η and λ are that η aﬀects the fund investors’ indiﬀerence conditions,(18), not only directly but also through changes in the fund ownership stakes, whereas λaﬀects (18) only through changes in the fund ownership stakes. Given this, in the caseof a decrease in λ, increases in ZG and ZM in response to the direct eﬀect of λ result inincreases in RG and RM to restore (18) because a larger ZG and ZM (or a larger RG andRM ) increase (or decrease) the fund investors’ rate of expected gross returns from each]/∂Z > 0 and ∂[(1 − f ) R−ηZ]/∂R < 0, where P = R − ηZ −fund (see ∂[(1 − f ) R−ηZ
P
PL). The increases in RG and RM are achieved by inducing fund investors to increase theirinvestment in the D-fund. Such an increase in investment compels the D-fund manager topurchase a larger number of G-firm shares, thus increasing xDG . The eﬀect of the larger
27

However, the lower λ does not aﬀect the P-fund manager’s ESG eﬀort in B-firms.

28

xDG
xDG raises the fund managers’ eﬀorts regarding firm profit (see eDG = fDcRD). Hence, RG

and RM increase. However, the eﬀect of the larger xDG on aDG does not oﬀset the initial
direct eﬀect of decreasing aDG and aP G . Thus, ZG and ZM continue to increase.
In fact, under fund investors’ capital allocation conditions (18), the eﬀect of the lower
λ increasing the expected disutility, ηZ, cancels out its eﬀect of decreasing the expected
profit, R. This implies that the change in λ does not aﬀect fund investors’ expected gross
return rate, although the initial direct eﬀect of decreasing aDG and aP G is retained, as is
the fund ownership eﬀect of an increase in eDG through an increase in xDG . Thus, thechange in λ does not influence the management fee of either fund,28 and both ZG and ZM
continue to increase, along with RG and RM . In addition, as these eﬀects do not apply to
B-firms, neither ZB nor RB is aﬀected as long as the management fee remains unaﬀected.
The cost complementarity in each fund manager’s eﬀort between improving profit and
ESG performances has significant eﬀects on the expected negative externality and the
expected profit of each firm. In the context of our model, an increase in complementarity
can be understood as a decrease in cCD and cCP .
Proposition 4: Suppose that the cost complementarity in the D-fund manager’s eﬀort
between improving profit and ESG performances is higher (i.e., cCD is lower).
(i) The expected negative externality and the expected profit of G-firms, ZG and RG , bothdecrease.
(ii) The expected negative externality and the expected profit of B-firms, ZB and RB , respectively, are unaﬀected.
(iii) The expected negative externality and the expected profit from all the firms in themarket portfolio, ZM and RM , respectively, decrease.
(iv) The asset management fees of both funds are unaﬀected.
Proposition 5: Suppose that the cost complementarity in the P-fund manager’s eﬀort
between improving profit and ESG performances is higher (i.e., cCP is lower). Then, statements (i), (iii), and (iv) in Proposition 4 hold, whereas the expected negative externality
and the expected profit of B-firms both decrease.
Propositions 4 and 5 show that the lower cCD and the lower cCP both improve expected
ESG performance in G-firms and in all the firms in the market portfolio, but reduce28

A remark similar to that of footnote 26 holds in this case.

29the expected profit performance of G-firms and of all the firms in the market portfolio.
However, these propositions indicate that only the lower cCP improves B-firms’ expected
ESG performance but harms their expected profit performance. Neither the lower cCD
nor the lower cCP aﬀects the asset management fees of either type of funds.
Intuitively, the mechanism by which the lower cCD operates is similar to that of thehigher λ, except that the lower cCD directly increases only the D-fund manager’s ESG
eﬀort. Regardless of this diﬀerence, similar to the case of the higher λ (i.e., opposite to
the case of lower λ), the lower cCD decreases ZG , RG , ZM , and RM , but does not aﬀect
ZB , RB , or the asset management fees of either fund. For the eﬀect of the lower cCP ,we can apply a similar logic, except that the lower cCP raises the P-fund manager’s ESG
eﬀort in the market portfolio firms. Consequently, the lower cCP reduces ZB and RB .
Several remarks regarding Propositions 2—5 are in order. First, even though growinginterest in ESG may reduce fund investors’ investment in the D-fund, it decreases the
expected negative externality released by G-firms and in all the firms in the market
portfolio if the D-fund has a comparative cost advantage over the P-fund in improving
profit performance relative to ESG performance. Furthermore, growing interest in ESG
always reduces the expected negative externality released by B-firms. However, if the
D-fund has a comparative cost advantage over the P-fund in improving ESG performance
relative to profit performance, and if the negative externalities of B-firms are suﬃciently
larger than those of G-firms, growing interest in ESG may have a counterproductive eﬀect
on ESG: the dominant eﬀect is the decrease in the fund investors’ allocation to the D-fund
and the expected negative externality released by G-firms and all the firms in the market
portfolio also increases. As some recent studies, such as Hartzmark and Shue (2022),
suggest–albeit for reasons other than the limited or counterproductive impact of the Dfund on ESG–the latter result theoretically suggests the possibility of strengthening ESG
preferences preventing the reduction of the expected negative externality under particularconditions.
Second, Goldstein, Kopytov, Shen, and Xiang (2022) suggest that an increase in thegreen investor share leads to an increase (a decrease) in the cost of capital and the expected asset returns when most investors are traditional (green) investors. Their results
depend on changes in the composition of the investor base and in the price informativeness. Pástor, Stambaugh, and Taylor (2021b) indicate that a higher ESG appetite leads
30

green firms to become greener but reduces the expected returns of green firms. Theirresults depend on the shift of real investment from brown to green firms. By contrast,
our results regarding the growing interest in ESG are primarily derived from changes in
the fund managers’ governance eﬀorts, which are caused not only by the direct eﬀectof strengthened ESG preferences but also by the change in the fund ownership stakes.
Accordingly, our results depend on the comparative cost advantage of each fund manager
in improving ESG performance relative to firm profit. Furthermore, our results suggestthat growing interest in ESG may not necessarily improve expected ESG performance in
G-firms or in all the firms in the market portfolio in a particular situation, where theD-fund has a comparative cost advantage over the P-fund in improving ESG performance
relative to firm profit and the negative externalities of B-firms are suﬃciently larger thanthose of G-firms. This result diﬀers from the findings of Pástor, Stambaugh, and Taylor(2021b) because it is driven by the change in the fund ownership stakes in each firm rather
than by investment shifting from brown to green firms.
Third, Proposition 3 shows that when shareholders’ ESG engagement becomes lessvaluable for high-ESG firms, the expected ESG performance of the high-ESG firms andall the firms in the market portfolio worsens, whereas their expected profit performances
improve. Although this change aﬀects fund managers’ ESG eﬀorts toward G-firms in boththe D- and P-funds, the expected ESG and profit performances in B-firms remain unaffected because neither the direct eﬀect nor the fund ownership eﬀect operates in B-firms.
The fund ownership eﬀect operates only through reducing fund investors’ investment in
the D-fund in this case.
Fourth, there is empirical support for first-order eﬀects arising from the change in thefund ownership structure that underlies the mechanism in Propositions 2—5. Although
D-funds are considered ESG-indexed funds, there is growing evidence that passive fundgrowth may aﬀect information production and the information content of asset prices (seeIsraeli, Lee, Sridharam, 2017; Glosten, Nallareddy, and Zou, 2021; and Coles, Health,and Ringgenberg, 2022). These changes may have first-order eﬀects on shareholders’willingness to make costly engagements in their portfolio firms.
Fifth, Corum, Malenko, and Malenko (2021) report that passive fund growth improves
firm governance and increases firm returns if it replaces liquidity investors with institutional investors, whereas its eﬀects on governance are subtler and depend on the active
31

and passive funds’ ownership stakes if the passive fund growth primarily aﬀects the composition of active versus passive funds. Our results regarding growing interest in ESG
depend on the ownership stakes of D- and P-funds. However, our model incorporates
investors’ ESG preferences, the costs associated with engagement and complementarity
for each fund manager in reducing the negative externality, and the costs associated withimproving firm profit. Hence, our findings are contingent upon the ratio of the negative
externality released by each type of firm and the comparative cost advantage of improving
ESG performance relative to firm profit for each fund manager.
Sixth, Propositions 4 and 5 show that greater cost complementarity in the eﬀorts of
D-fund managers enhances expected ESG performance in both G-firms and all firms
within the market portfolio, but does not aﬀect expected ESG performance in B-firms.
Conversely, greater cost complementarity in the eﬀorts of P-fund managers enhances
expected ESG performance even in B-firms. Hence, the comparative static results forsuch complementarities are not straightforward because we need to examine an indirectchannel that operates through fund investors’ capital allocations that could impede thereduction of expected negative externalities.
5. Discussions and Empirical Implications
Propositions 2—5 oﬀer several empirical predictions regarding the eﬀects of shareholder
engagement resulting from growing interest in ESG, the relatively reduced impact of suchengagement on ESG performance in G-firms, and the heightened cost complementarity ofeﬀort costs for each fund manager that impact on both the expected negative externality
and the expected financial return. Although ESG performance includes various aspects,
one could rely on various proxies proposed in the empirical literature that capture diﬀerent
ESG aspects (e.g., see Pedersen, Fitzgibbons, and Pomorski, 2021) to test the predictionsof our model empirically.
To discuss the empirical implications, using (12) and (16), we observe that the expectedfinancial returns of G-stocks, B-stocks, and the market portfolio for investors, RG − PG ,RB − PB , and RM − PM , are defined by

Rh − Ph = Lh + ηZh ,32

for h ∈ (G, B, M).

(24)

We start by deriving the empirical implication for the expected financial returns bycomparing RG − PG and RB − PB . In the previous sections, we assumed that ZB0 is

suﬃciently larger than ZG0 and that the D-fund holds only G-firms. Hence, it follows from(3), (4), and (24) that if LG is not so diﬀerent from LB and if the ESG engagement eﬀortof the P-fund manager in B-firms is not very large, the expected financial returns of stocks
with low-ESG scores outperform those of stocks with high-ESG scores. Many empiricalstudies provide predictions about the relationship between ESG aspects and the financial
returns of firms’ operations, but they document mixed results. For example, Hong and
Kacperczyk (2009), El Ghoul, Guedhami, Kwok, and Mishra (2011), Chava (2014), Zerbib(2019), Barber, Morse, and Yasuda (2021), and Bolton and Kacperczyk (2021) report anegative relationship between ESG performance and financial returns. However, Derwall,
Guenster, Bauer, and Koedijk (2005), Kempf and Osthoﬀ (2007), and Pastor, Stambaugh,
and Taylor (2021a) report that stocks with better environmental prospects have higher
financial returns. Green and Roth (2021) suggest that measurement issues are a significant

obstacle to resolving the problem of whether firms with good ESG performance face lower
financial returns. In addition, the opposite findings can be explained by the weak return
predictability of the overall ESG rating (Pedersen, Fitzgibbons, and Pomorski, 2021) andthe presence of uncertainty about the ESG profile (Avramov, Cheng, Lioui, and Tarelli,2022). Indeed, until recently, the construction of ESG ratings has not been regulated or
unified. Thus, the methodology of ESG ratings is opaque and proprietary.29
Next, we examine the empirical implication of the eﬀect of growing interest in ESG
because many practitioners and researchers question how growing interest in ESG reallyaﬀects ESG performance and financial returns. As mentioned in Section 3.1, we notethat the engagement eﬀort eih (aih ) exerted by the fund manager i for i ∈ (D, P ) andh ∈ (G, B) includes any actions such as communicating with management, submitting
shareholder proposals, nominating directors, and voting on proxy contests. Given this,we consider two cases, in which the D-fund is (i) a sustainable active fund; and (ii) an
ESG-indexed fund.
In the first case, in their review of the literature, Brav, Malenko, and Malenko (2022)
conclude that because of the diﬀering types of costs associated with actively managed
29

Avramov, Cheng, Lioui, and Tarelli (2022) report that there are substantial variations across diﬀerent
rating providers with the average rating correlation being 0.48.

33

funds and passive funds, and thus their probable specialization in diﬀerent types of engagement, passive funds may be better positioned to exert influence by setting broad,market-wide governance standards rather than focusing on firm-specific operational improvements.30 Given the disparity in the engagement strategies, the P-fund holds a comparative cost advantage over the D-fund in enhancing ESG performance relative to profit
performance if enhancing ESG performance can be achieved to some extent by establishing broad, market-wide standards for ESG, whereas improving the profit performance ofeach firm necessitates more firm-specific operational engagements. In addition, the engagement eﬀort concerning ESG may prove more valuable for firms that lag behind others
on ESG. As the D-fund invests solely in G-firms, this characteristic further strengthens
the comparative cost advantage of the P-fund over the D-fund in enhancing ESG performance relative to profit performance, although the parameter λ may primarily capture
this feature in our model.
In contrast, the cost complementarity in the D-fund manager’s eﬀorts between improving profit and ESG performances is greater than that in the P-fund manager’s eﬀorts
when the D-fund operates as a sustainable active fund. This diﬀerence arises because
both eﬀorts of the D-fund involve more firm-specific operational engagements than thoseof the P-fund. However, if the cost complementarity in the D-fund manager’s eﬀorts is
not significantly large, such that cCD is not notably smaller than cCP , then the P-fundretains its comparative cost advantage over the D-fund in improving ESG performanceRP
relative to profit performance, as argued above. Accordingly, we can assume that ccRD
ZP +cCP≥ ccZD
if improving ESG performance can be achieved to a certain extent by setting+cCD

broad, market-wide standards for ESG, and if the cost complementarity in the D-fundmanager’s eﬀorts is not significantly large.

For the second case, where the D-fund is an ESG-indexed fund, the engagement activityof the D-fund is similar to that of the P-fund, except that the D-fund invests only in Gfirms. If the engagement eﬀort regarding ESG is more valuable and less costly for firms
with low- vs high-ESG scores, the ESG engagement cost of the D-fund manager may be
30Kahn and Rock (2020) and Fish, Hamdani, and Solomon (2019) indicate that actively managed
funds may have an advantage over index funds in identifying firm-specific operational or financial issues
because they can specialize in collecting or acquiring such information as a by-product of their investment
activities. They also argue that large passive funds are well good positioned to reap economics of scalein collecting information on broad, market-wide issues and setting market-wide standards.

34

larger than that of the P-fund manager (i.e., cZD > cZP ) because the D-fund invests only
ZP +cCPRP
is not suﬃciently large and can be smaller than ccRD
. Hence,in G-firms. Then, ccZD
+cCD
RP
ZP +cCP≥ ccZD
without any further conditions.
in this case, we may assume that ccRD
+cCD

Now, considering that the strengthened ESG preference aﬀects both η and Zi (i = G,B, M) in (24), combining (24) with the above arguments and the results of Propositions
2(i)—2(iii) provides the following predictions (for the proof, see Appendix A):
Prediction 1A: If improving ESG performance can be achieved to a certain degree by
setting broad, market-wide standards for ESG, and if the cost complementarity in theeﬀorts of the D-fund manager is not significantly large, growing interest in ESG is likely
to reduce the expected negative externalities released by firms with high-ESG scores, aswell as by all the firms in the market portfolio, but also to reduce the expected financial
returns of these firms. However, growing interest in ESG always reduces the expected
negative externality by firms with low-ESG scores and their expected financial returns.
Prediction 1B: If most D-funds consist of ESG-indexed funds, growing interest in ESG
is likely to reduce the expected negative externalities and financial returns of any typeof firm.
Suppose that improving ESG performance cannot be achieved to a certain degree by
setting broad, market-wide standards for ESG, and/or that the cost complementarity
eﬀect in the D-fund manager’s eﬀorts is suﬃciently large. Then, Prediction 1A does
not necessarily suggest that growing interest in ESG is likely to improve expected ESG
performance in firms with high-ESG scores and in all the firms in the market portfolio.
However, Prediction 1A also shows that growing interest in ESG always improves expected
ESG performance in firms with low-ESG scores. By contrast, Prediction 1B suggests that
if most D-funds comprise ESG-indexed funds, then growing interest in ESG is likely to
improve expected ESG performance of any type of firm.
Now, we proceed to examine the empirical implication of the variation eﬀect in theeﬀectiveness of shareholder ESG engagement in G-firms relative to that in B-firms. Because such a diminishing impact is captured by a decrease in λ, it readily follows from
Propositions 3(i)—3(iii), along with (24), that the following predictions are obtained.
Prediction 2: When the impact of shareholder ESG engagement in G-firms relative to
B-firms is smaller, both the expected negative externalities and the expected financial
35

returns are increased for firms with high-ESG scores and all the firms in the market
portfolio, but remain unaﬀected for firms with low-ESG scores.
Prediction 2 particularly suggests that when the impact of shareholder ESG engagementin G-firms relative to that in B-firms decreases, both the expected ESG performance andfinancial returns in firms with high-ESG scores and market portfolio firms increase, but
those in firms with low-ESG scores remain unaﬀected.
Now, we discuss the empirical implication regarding the eﬀect of the cost complementarity in each fund manager’s eﬀorts between improving profit and ESG performances.
As argued above, sustainable active funds are more likely to focus on firm-specific operational involvements, whereas ESG-indexed funds and P-funds are more likely to exert
influence by setting broad, market-wide governance standards. Because improving the
profit performance of each firm necessitates more firm-specific operational engagements,cCD is smaller (larger) when the proportion of sustainable active (ESG-indexed) funds inD-funds is higher, whereas cCD and cCP are smaller when the eﬀort of firm-specific operational engagements is more complementary to the eﬀort of setting broad, market-wide
ESG governance standards. Hence, Propositions 4 and 5 along with (24) immediately
yield the following predictions:
Prediction 3A: Suppose that the proportion of sustainable active (ESG-indexed) funds
in D-funds increases. Then, the expected negative externalities released by firms withhigh-ESG scores and by all the firms in the market portfolio are reduced (increased).
However, the expected financial returns of firms with high-ESG scores and those of themarket portfolio also decline (increase).
Prediction 3B: Suppose that the eﬀort of firm-specific operational involvements is morecost complementary with the eﬀort of setting broad, market-wide ESG governance standards. Then, the expected negative externalities released by any type of firm decline.
However, the expected financial returns of any type of firm also decline.
Combining Predictions 1A and 3A, we can suggest that if interest in ESG is not growing but if the proportion of ESG-indexed funds in D-funds increases, the expected ESG
performance of firms with high-ESG scores and of all the firms in the market portfolio
does not improve, whereas the expected financial returns of these firms improve.
Now, the profit of each firm in Propositions 2—5 can be interpreted as the operating36

profits–that is, the EBITDA–of each firm. As a lower expected negative externality
implies higher ESG scores, Propositions 2—5 provide empirical implications regarding theassociation between ESG scores and the EBITDA achieved by each firm.
First, Propositions 2 and 3 imply the following predictions.
Prediction 4: Growing interest in ESG leads to a positive association between ESG
scores and the EBITDA attained by firms with low-ESG scores.
Prediction 5: The decreasing impact of shareholder ESG engagement on firms withhigh-ESG scores causes a negative association between ESG scores and the EBITDA
attained in firms with high-ESG scores and in all the firms in the market portfolio.
Prediction 4 does not necessarily suggest that growing interest in ESG creates a positive
association between ESG scores and the EBITDA attained by any type of firm.
Second, Propositions 4 and 5, combined with the discussion above for Predictions 3A
and 3B, imply the following predictions.
Prediction 6: A higher proportion of sustainable active (ESG-indexed) funds in D-funds
leads to a negative association between ESG scores and the EBITDA attained by firms
with high-ESG scores and by all the firms in the market portfolio.
Prediction 7: A higher cost complementarity regarding the eﬀort of firm-specific operational involvements with the eﬀort of setting broad, market-wide ESG governance standards in each fund causes a negative association between ESG scores and the EBITDA
attained by any type of firm.
To the best of our knowledge, Predictions 1—7 have not yet been tested. Regarding
Predictions 1A, 1B, and 4, although many empirical studies report the expected financial
returns of assets, they focus on cross-sectional analyses. In contrast, Predictions 1A, 1B,
and 4 provide time-series predictions created by the eﬀect of growing interest in ESG.
Testing of the other predictions, using cross-sectional and panel data analyses continues
to be required. To test Predictions 2 and 5, we need to identify the impact of shareholder
ESG engagement on each firm, which may be estimated using the method of Hartzmark
and Shue (2022). To test Predictions 3A, 3B, 6A, and 6B, diﬀerences in the accessibility
and plausibility of ESG data in diﬀerent industries and/or the growing accessibility andvariety of ESG proxies may be used to identify the cross-sectional diﬀerences and thetime-series changes in ESG engagement costs.
37

6. Conclusion
This paper explores how profit-motivated managers of various types of sustainable fundswith divestment strategies and passive funds govern their portfolio firms when these funds
aim to attract capital from socially responsible investors. We analyze a multitask situationin which the manager of each fund must determine the level of costly engagement eﬀort
to mitigate negative externalities and increase pecuniary returns in his portfolio firms,
considering the cost complementarity in these eﬀorts. We allow for variations in theimpact of shareholder ESG engagement in high-ESG firms.
Considering the fund ownership eﬀect resulting from investor capital allocation, we derive the following implications:(i) Growing interest in ESG is likely to improve the expected ESG performance of firms
with high-ESG scores and all the firms in the market portfolio, but to aggravate their
expected financial returns if the D-fund has a comparative cost advantage in improving
firms’ profit performance over their ESG performance compared with the P-fund. However, the growing interest in ESG may not contribute to improving the expected ESG
performance or to aggravating expected financial returns in these firms in a particularsituation. Conversely, for firms with low-ESG scores, growing interest in ESG always
improves expected ESG performance but aggravates expected financial returns. Indeed,
if most of the D-funds consist of ESG-indexed funds, growing interest in ESG is likely to
reduce the expected negative externalities and financial returns of any type of firm.
(ii) The decreasing impact of shareholder ESG engagement on firms with high-ESG scores
reduces the expected ESG performance but improves the expected financial returns offirms with high-ESG scores and all the firms in the market portfolio. However, it does
not aﬀect the expected ESG performance or expected financial returns of firms with lowESG scores.
(iii) The greater cost complementarity in each fund manager’s eﬀorts between improving
the profit and ESG performances improves the expected ESG performance as a whole,
but reduces the expected financial returns of any type of firm. Furthermore, the higher
proportion of ESG-indexed funds in D-funds reduces the expected ESG performance but
improves the expected financial returns of firms with high-ESG scores and all the firms
in the market portfolio.

38

In this paper, we focus on the fund manager’s multitask incentive problem in engaging in
the management of his portfolio firms, and fund investors’ investment allocation problems.
To elucidate these problems, we abstract from the tax and interest payments of theportfolio firms. Thus, in our model, the EBITDA and net income are indistinguishable.
However, in conducting empirical research, net income may be a more adequate measureof the profit of the portfolio firms for the fund manager. Hence, the tax and interest
payment considerations would be an interesting extension of the empirical analysis in
future research.

39

Appendix A
Proof of Proposition 1: We first derive statements (i), (ii), and (iv). Substituting PG
from (14) and fD from (19) into (18), we obtain(1 + ψD − ωD )LG = ψD (RG − ηZG ),D
D
LG + ηZG . Then, (14) yields PG = 1−ω
LG . Thus, it follows
which means RG = ψD +1−ω
ψD
ψD
D ψD
. Similarly, using (16), (18), and (20), we can derive thefrom (19) that fD = ψDω+1−ω
D
P ψP
P
P
, RM = ψP +1−ωLM + ηZM , and PM = 1−ω
LM . In addition, if
solution: fP = ψPω+1−ωψP
ψPPωD ≥ ωP , then ψD ≥ ψP implies that fD ≥ fP .

Next, we verify statement (iii). As the D-fund holds only the stock of G-firms, it follows
eG = RG , and E R
eB = RB
from (1) and (2) with RM = RG + RB , eP G = eP B = eP , E R

that

RM − RG = R0 + eP ,(A1)

2RG − RM = eDG .

(A2)

and

It is also found from (3) and (4) with ZM = ZG + ZB , E ZeG = ZG , and E ZeB = ZB that
ZG = ZG0 − λaDG − λaP G ,(A3)

ZM = ZG0 + ZB0 − λaDG − λaP G − aP B .

(A4)

and

Substituting aDG and eDG from (8) and aP G , aP B , and eP from (90 ) into (A3) and (A4)and rearranging them with (8), (90 ), (A1) and (A2), we obtain (22) and (23).
In the remaining part, we show that under the conditions of this proposition, (a) liquidity investors hold at least some shares in each type of stock, that is, xDG + xP < 1,(b) fund investors make a positive investment in an alternative investment opportunity
such as public bonds, that is, WD + WP < W , and (c) the D-fund finds it optimal todiversify equally across all G-firms.
We first prove (a). Given that the D-fund holds only the stock of G-firms, it follows
40

from (1)—(4) and (13)—(16) with RM = RG + RB and eP G = eP B = eP thatxDG + xP =

WD WP
+
.
PG
PM

(A5)

Note that
PG = R0 + eDG + eP − ηZG − LG ≥ R0 − ηZG0 − LG > 0,
PM = 2R0 + eDG + 2eP − ηZM − LM ≥ R0 − ηZG0 − LG > 0,
because we focus on the cases of (eDG , aDG ) > 0 and (eP , aP G , aP B ) > 0 and because R0> max (ηZB0 + LB , ηZG0 + LG ) > 0. Given WD + WP < W derived below, (A5) leads
toxDG + xP ≤WD + WP
W<.
R0 − ηZG0 − LG
R0 − ηZG0 − LG

It follows from the condition W < R0 − ηZG0 − LG that xDG + xP < 1.

We next proceed to prove (b). Rearranging (8) and (90 ) with (A1) and (A2), we havecRD
(2RG − RM ),
fD

(A6)

cRP
(RM − RG − R0 ).
fP

(A7)

xDG =

xP =

D
It is found from (13), (15), (A6), and (A7) with RG = ψD +1−ωLG + ηZG , RM =ψDψP +1−ωP
LM + ηZM , (A3), and (A4) thatψP

WD + WPPG cRD
PM cRP= PG xDG + PM xP =(2RG − RM ) +(RM − RG − R0 )
fD
fP
µ
¶
ψD + 1 − ωD
ψP + 1 − ωP
PG cRD2
LG −LM + ηZG0 − ηλaDG − ηλaP G − ηZB0 + ηaP B
=
fD
ψD
ψP
µ
¶
ψD + 1 − ωD
PM cRP ψP + 1 − ωP
(A8)
LM −
LG − R0 + ηZB0 − ηaP B .
+
fP
ψP
ψD
Define∙¸
ψD + 1 − ωD
1 − ωD ψD + 1 − ωD
ψP + 1 − ωP
LG cRD 2
LG −LM + ηZG0
W ≡
ψD
ωD ψD
ψD
ψP
41

1 − ωP ψP + 1 − ωP+
LM cRP
ψP
ωP ψP

µ

¶
ψP + 1 − ωP
ψD + 1 − ωDLM −LG − R0 + ηZB0 .
ψP
ψD
(A9)

D ψD
P ψP
D
P
LG , PM = 1−ω
LM , fD = ψDω+1−ω, and fP = ψPω+1−ωwith theThen, given PG = 1−ω
ψD
ψP
D
P

assumption of ZB0 > aP B , comparing (A8) with (A9) verifies that WD + WP < W if W
≥ W.

Finally, we prove (c). Indeed, applying a procedure similar to the proof of Lemma 2

in Online Appendix of Corum, Malenko, and Malenko (2021) under the condition R0 >
ηZG0 + LG and the assumption of a quadratic cost function, we can show that the D-fundfinds it optimal to diversify equally across all G-firms. k
DP
LG + ηZG and RM = ψP +1−ωLM
Proof of Propositions 2—5: Substituting RG = ψD +1−ωψDψP+ ηZM into (22) and (23) of Proposition 1(iii), we show that ZG , ZB , and ZM are determined by solving the following simultaneous equations
ZB + ZG = ZM ,(A10)

ZG = ZG0 − Γ1 − Γ2 ,(A11)

ZM = ZG0 + ZB0 − Γ1 − Γ2 − Γ3 ,

(A12)

where
cRD
Γ1 ≡ ηλcZD + cCD2µ¶
ψD + 1 − ωD
ψP + 1 − ωP
2
LG + 2ηZG −LM − ηZM ,
ψD
ψP

µ
¶
cRP
ψP + 1 − ωP
ψD + 1 − ωD
Γ2 ≡ ηλLM + ηZM −LG − ηZG − R0 ,cZP + cCPψP
ψDµ¶
ψP + 1 − ωP
cRP
ψD + 1 − ωD
LM + ηZM −LG − ηZG − R0 .
Γ3 ≡ η
cZP + cCP
ψP
ψD
2

Totally diﬀerentiating (A10)—(A12) with respect to ZB , ZG , ZM , η, λ, cCD , cCP , and ψP

42

yields

⎡

1
⎢
⎢
⎢ 0
⎣
0
⎡

1
∂Γ1
∂Γ2
1 + ∂Z
+ ∂Z
G
G
∂Γ1
∂Γ2
∂Γ3
+ ∂Z
+ ∂Z
∂ZG
G
G

⎤

0

⎤⎡

−1

⎤

dZB
⎥⎢
⎥
⎥
⎥
⎢
∂Γ1
∂Γ2
⎥
⎢
+
dZ
G ⎥
∂ZM
∂ZM
⎦⎣
⎦
∂Γ1
∂Γ2
∂Γ3
1 + ∂ZM + ∂ZM + ∂ZM
dZM
⎡

0

⎤

⎡

0

⎤

⎢
⎢
⎥
⎥
⎥
⎢ 2Γ1 2Γ2 ⎥
⎢
⎥
⎥
Γ1
dη
+
dλ
+
⎢ − λ − λ ⎥⎢ c +c⎥⎥ dcCD⎣⎣ ZD CD ⎦
⎦
⎦
Γ1
1
2
3
− ∂Γ
− ∂Γ
− ∂Γ
− 2Γλ1 − 2Γλ2∂η∂η∂ηcZD +cCD
⎡
⎡
⎤
⎤
0
0
⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
Γ
∂Γ
∂Γ
2
1
2
+ ⎢ c +c ⎥ dcCP + ⎢⎥ dψP .
− ∂ψP − ∂ψP
ZP
CP
⎣
⎣
⎦
⎦
Γ2 +Γ3
∂Γ1
∂Γ2
∂Γ3
−
−
−
cZP +cCP
∂ψP
∂ψP
∂ψP
⎢
⎢
=⎢
⎣

1
2
− ∂Γ
− ∂Γ
∂η
∂η

(A13)

D
P
Given (A6) and (A7) with RG = ψD +1−ωLG + ηZG and RM = ψP +1−ωLM + ηZM ,ψDψP2

fD xDG(2RG − RM ) = ηλ, Γ2 = ηλ2 cZPcRP(RM − RG − R0 )note that Γ1 = ηλ2 cZDcRD+cCD
cZD +cCD
+cCP
2

fP xP
ηfP xP, and Γ3 = η cZPcRP(RM − RG − R0 ) = cZP. It follows from (8) and (90 )
= cηλ
+cCP
+cCP
ZP +cCPthat the assumption (eDG , aDG ) > 0 and (eP , aP G , aP B ) > 0 means that Γ1 > 0, Γ2 > 0,and Γ3 > 0.
Now, solving (A13), we show
¶
µ
¶µ
cRP
cRD
Γ3
dZB
1
2 2
< 0,
+
=−
ηZB
1+η λ
dη
∆
cZP + cCPη
cZD + cCD

(A14)

µ
¶
µ
¶
dZG
cRP
ηλ2 ZBcRP
ηλ2 ZG
cRD
cRD
cZP + cCP
2
1+η
−
=−
−
dη
∆ cZD + cCD
cZP + cCP
∆ cZP + cCP cRD cZD + cCD
cRP
cRD
Γ1
Γ2
Γ1 2
Γ2 2
−
−
−
η
−
η
η∆ η∆ cZP + cCPη∆ η∆ cZD + cCD
cRP
cZP + cCP
< 0,
if
−
> 0,
(A15)
cRD cZD + cCD
dZM
ηλ2 ZGcRD=−
dη∆ cZD + cCD

µ
η2

∙
¶
cRP
cRP
cRD
ηZB 2 2η λ
+1 −cZP + cCP∆
cZP + cCP cZD + cCD

43

¶
¸
cRP
Γ1 2
cRP
cRP
Γ1
cRD
−
η
+
−
+λ
−
cZP + cCP
cZD + cCD
cZP + cCPη∆ η∆ cZP + cCP
¶
µ
cRD
cRP
Γ3
cZP + cCP
Γ3
if−
+ λ2 < 0,>,−2η2 λ2η∆ η∆
cZD + cCD
cRD
cZD + cCD
2

µ

(A16)

dZB
= 0,
dλ

¶
µ
cRP
dZh
2(Γ1 + Γ2 )
2
< 0,
=−
1+η
dλ
λ∆
cZP + cCPh = G, M,

(A17)

dZB
= 0;
dcCD

cRP
dZh
Γ1
(1 + η2=) > 0,
dcCD(cZD + cCD )∆cZP + cCPh = G, M,

(A18)

dZB
cRD
Γ3
(1 + η2 λ2=) > 0,dcCP(cZP + cCP ) ∆cZD + cCD
dZG
Γ2
cRD
(1 + η2=) > 0,dcCP(cZP + cCP ) ∆cZD + cCD
dZM
Γ3
cRD
(1 + λ2 + 2η 2 λ2=
) > 0,dcCP(cZP + cCP ) ∆cZD + cCD

(A19)

¶
cRP
cRD
cRP
2 2
> 0,
(A20)
+η λ
cZP + cCP
cZP + cCP cZD + cCD
µ
¶
cRD
dZG
cRP
cRP
(1 − ωP )ηλ2 LM
cZP + cCP
cZP + cCP
R 0, if and only if
=
−
R
,
2
dψP
(ψP ) ∆cZP + cCP cRD cZD + cCD
cRD
cZD + cCD∙
¸ (A21)cRD(1 − ωP )ηLM
cRP
cRP
cZP + cCP
dZM
(1 + λ2 )> 0,=+ η2 λ2− λ22dψP(ψP ) ∆ cZP + cCP
cRD
cZD + cCD
cZD + cCDdZB
(1 − ωP )ηLM=dψP(ψP )2 ∆µcZP + cCP
λ2
cRP
>
,
(A22)
if and only ifccRD
1 + λ2 + η2 λ2 cZDRD
c + cCD+cCD ZD
³
´
cRD
cRP
cRP
2 2
where ∆ = 1 + η2 λ2 cZDcRD> 0. It follows from
+
+
η
λ
+cCD
cZP +cCP
cZD +cCD cZP +cCP

(A14)—(A19) that the results of the expected negative externalities in Propositions 2—5

Γ1
Γ2
Γ3
, η∆
, and η∆express the direct eﬀects ofare obtained. In (A15) and (A16), note that η∆η on the ESG eﬀort incentives of the D- and P-fund managers, whereas the other terms
represent the fund ownership eﬀects of η on these ESG eﬀort incentives.
Given Proposition 1(ii), we havedZh
dRh
= Zh + η
,
dη
dη

44

h = G, M,

(A23)

dRh
dZh
=η
,
dλ
dλ

h = G, M,(A24)

h = G, M, and i = D, P,

(A25)

dRM
(1 − ωP )LM
dZM
=−
+η
.
2
dψP
(ψP )
dψP

(A26)

dRh
dZh
=η
,
dcCi
dcCi
dRG
dZG
=η
;
dψP
dψP
In addition,

dRB
dRM
dRG
=
−
,
dχ
dχ
dχ

χ = η, λ, cCD , cCP , ψP .

(A27)

Inspecting (A23)—(A25) and (A27) with (A14)—(A19), we prove the results of the expected
profits in Propositions 2—5.
Finally, it follows from Proposition 1(i) that
dfi
dfi
dfi
= 0,
=
=
dη
dλ
dcZj
dfD
= 0;
dψP

i = D, P ; j = D, P,

dfP
ωP (1 − ωP )
=
> 0.
dψP
(ψP + 1 − ωP )2

(A28)

(A29)

It follows from (A28) that the results of the asset management fees in Propositions 2—5
are verified. k
Proof of Predictions 1A and 1B: Comparing Proposition 1(ii) with (24), we showi −Pi )i= dR
for i = G, B, M. Hence, it follows from Propositions 2(i)—2(iii) thatthat d(Rdηdη

the statements of Predictions 1A and 1B are obtained. k
Appendix B
B.1. Multiple D- and P-funds.–
Our basic model can be extended to the case of multiple funds in which there are ND
D-funds and NP P-funds. All ND D-funds only invest in and diversify the stocks of Gfirms, whereas all NP P-funds invest in the market portfolio. As we focus on symmetricequilibria, the same type funds choose the same eﬀort and bargaining strategies and thesame fund size. Then, under a quadratic cost function of eﬀorts, we can show that allof our propositions continue to hold by applying the discussion of Corum, Malenko, andMalenko (2021).

45

B.2. Non-sustainable funds and non-socially responsible investors.–
We can consider a third type of fund as a non-sustainable fund (N-fund), which invests
only in B-firms. The fund manager of the N-fund invests in B-firms on behalf of investors
and is interested purely in his monetary payoﬀs. There is also a large mass of non-socially
responsible investors, who have no ESG preference and are purely interested in theirmonetary payoﬀs. Then, we can discuss whether our results are robust to these changes.
For convenience, fund investors and liquidity investors in the basic model are denoted associally responsible investors.
For simplicity, we assume that socially responsible investors, due to their ESG preferences, avoid investing in the N-fund. Conversely, non-socially responsible investors basetheir investment decisions on whether to invest in the N-fund or alternative opportunities such as public bonds.31 We denote their aggregate wealth as W + , which is given
exogenously. Each investor of this category must search for and vet the N-fund manager,
incurring a search cost ψN εN , where εN represents the investment amount in the N-fundand is normalized to adjust the scale of ψN . Subsequently, upon finding the N-fundmanager, negotiation of the fee fN occurs through generalized Nash bargaining.
Under these assumptions, we begin with the scenario where the fund manager of the
N-fund has no governance role in his portfolio firms. This setup reflects the N-fund’s
primary objective of seeking higher financial returns primarily through arbitrage trading.
Then, the basic model is modified as follows. As shown in Section 4.4, in the basic model,
the endogenous variables consisting of the eﬀort decisions, (eDG , aDG , eP , aP G , aP B ), theasset management fees, (fD , fP ), the trading decisions and investment asset allocations,(xDG , xP , WD , WP ), the total expected payoﬀs of the D- and P-funds, (RG − ηZG , RM −

ηZM ), and the asset prices, (PG , PM ), are determined by (8), (90 ), (10), and (13)—(20). In
this extended model, we additionally need to determine the asset management fee of the
N-fund, fN , the holding shares of the N-fund in B-firms, xNB , the investment amount of
the N-fund in B-firms, WN , and the price of B-firms, PB .
31

Non-socially responsible investors, driven solely by monetary payoﬀs, seek assets with the highest
rate of expected net return among the D-fund, the N-fund, the P-fund, and alternatives like public funds.
However, we can show that the N-fund oﬀers the highest rate of expected net return if non-socially
responsible investors allocate a positive portion of their wealth to the N-fund.

46

First, PB is given by (12). Second, for WN and fixed W + , xN B satisfy
xNB =

WN
W+
≤
.
PB
PB

(B1)

Third, the N-fund fee fN is determined by generalized Nash bargaining between the Nfund manager and non-socially responsible investors. Because both agents are purelyinterested in their pecuniary returns, their concern is to distribute RB PεB in this case.
Hence, it follows from the discussion at the end of Section 4.3 thatµ¶PB,
fN = ωN 1 −
RB

(B2)

where ωN is the bargaining power of the N-fund manager and RB is given by (2) for eP
determined from the above-mentioned equation system of the basic model.
The remaining problem is to show how WN is chosen by non-socially responsible investors. As non-socially responsible investors decide whether they invest in the N-fundor the alternative investment opportunity such as public bonds, we need to specify the
expected payoﬀ of non-socially responsible investors attained by investing in the N-fund.
This expected payoﬀ is represented by(1 − fN )RB− ψN .
PB(B3)

Because RB and PB are given by (2), (4), and (12) for eP and aP B determined from theabove-mentioned equation system of the basic model, it follows from (B2) that the valueof (B3) may not be generically equal to 1; thus, it may be smaller than 1 or larger than
1. Hence, if (B3) is larger than 1, non-socially responsible investors invest their entire
wealth in the N-fund, that is, WN = W + . Otherwise, they do not invest in the N-fund,that is, WN = 0.
In either case, even in this extended model, the endogenous variables in the basic model
are still determined in the exactly same way as in the main text. Consequently, none ofour main results are aﬀected because Proposition 1 still holds in this extended model.
We next examine the case in which the fund manager of the N-fund exerts governance eﬀort eN B to increase the profit of B-firms by incurring a private engagement cost

47

cRN (eNB ), where cRN (e) = cRN
e2 . Then, as the negative externality released by each firm
2
is aﬀected by a fund ownership eﬀect because of a change in eNB , we cannot derive ourmain results generally. However, if socially responsible investors’ ESG preference, η, is notlarge, RB and PB are almost taken as exogenous because RB and PB are almost unaﬀected
by ZB . Thus, the eﬀect of a change in eNB is not large. Similarly, if the bargaining powerof the N-fund manager, ωN , is not large, and/or if the wealth of non-socially responsible
investors, W + , is not large, the eﬀect of a change in eNB is not large because eNB is notlarge in this case. Then, our main results are almost certainly true in these cases.
B.3. Eﬀect of P-fund growth.–
It has been discussed frequently that ESG investing is harmed by P-fund growth because
P-funds automatically invest their money in firms with lower ESG scores unlike D-funds.
However, as has been argued in this paper, P-funds have an ability to monitor theirportfolio firms and force the management of the firms to improve their ESG performance.
Hence, to investigate the above problem, we must analyze P-fund growth by incorporating
the P-fund manager’s engagement with their portfolio firms. A decrease in ψP can bethought of as indicating easy access to the P-fund over time and bring about P-fund growth
because it reflects more investor awareness about the fund and improved disclosure about
the investment strategy and the fee structure of the fund.32
We then have the following proposition.33
Proposition 6: Suppose that access to the P-fund becomes easier (i.e., ψP is lower).
RP
ZP +cCP> ccZD(i) The expected negative externality released by G-firms, ZG , decreases if ccRD
+cCD
RP
ZP +cCPbut increases otherwise. The expected profit of G-firms, RG , decreases if ccRD
> ccZD+cCD

but increases otherwise.
(ii) The expected negative externality released by B-firms, ZB , decreases, whereas the
expected profit of B-firms, RB , increases.
(iii) The expected negative externality released by all the firms in the market portfolio,
RP
ZP +cCP> (1 + λ2 + η 2 λ2 cZDcRD)−1 ccZD
but increases otherwise.
ZM , decreases if ccRD
+cCD
+cCD

(iv) The asset management fee of the D-fund, fD , is unaﬀected, whereas that of the P-fund,32

Corum, Malenko, and Malenko (2021) provide the same interpretation of a decrease in the search
cost.
33
The result of the expected negative externality of each firm is obtained from (A20)—(A22). The
results of the expected profits of G-firms and B-firms are derived from (A26) and (A27).

48

fP , decreases.
The intuition behind Proposition 6 is easily understandable by dividing the eﬀect of ψP
into two parts: the fund fee eﬀect and the fund ownership eﬀect. We begin by discussing
the eﬀect of the lower ψP on fD and fP . The eﬀect of the lower ψP on fP follows froman eﬀect of the decrease in the rate of expected gross return earned by the P-fund. The
reason is that the lower ψP increases fund investors’ rate of expected net return of the
P-fund from which ψP is deducted. In equilibrium, however, fund investors are indiﬀerent
between investing in the P-fund and the alternative investment opportunity (see (18)). To
restore (18), fund investors increase their investment in the P-fund until fund investors’rate of expected gross return from the P-fund decreases so that their rate of expected net
return returns to 1. A decrease in fund investors’ rate of expected gross return from the
P-fund then leads to a lower fP because the P-fund manager’s bargaining power ωP isassumed to be fixed. Indeed, in the expression for the fund fee in Proposition 1(i), thiseﬀect is featured as a dependence of fP on ψP . However, the lower ψP has no eﬀect on the
D-fund fee because it does not aﬀect fund investors’ expected net return from the D-fundexclusive of the search cost ψD .
Indeed, to examine the eﬀect of the lower ψP on the expected negative externality, weneed to consider both the fund fee eﬀect and the fund ownership eﬀect. Note that theeﬀort incentive for each fund manager to reduce the negative externality depends on thefund fee and the fund ownership stakes for fixed η, λ, cZD , cCD , cZP , and cCP , as indicatedin (8) and (90 ). For the eﬀect through the fund fee, a decrease in ψP reduces the P-fund
fee fP , but does not aﬀect the D-fund fee fD , as discussed above. This weakens the eﬀort
incentive for the P-fund manager to reduce the negative externality in all the firms in themarket portfolio and thus increases ZG and ZM . However, it has no eﬀect on the eﬀort
incentive for the D-fund manager to reduce the negative externality in G-firms.
A decrease in ψP also changes the fund ownership stakes in firms. If ψP decreases, fundinvestors increase their investment in the P-fund, WP , because their expected net returnfrom the P-fund increases, as argued above. This enables the P-fund manager to takeincreasingly large stakes xP in all the firms in the market portfolio. The increase in xP
reduces the stakes held by the D-fund, xDG , and those held by liquidity investors, 1 − xDG− xP , in G-firms, while it also reduces the stakes held by liquidity investors, 1 − xP , in themarket portfolio. Note that the D-fund does not buy the market portfolio. The eﬀect of49

the decrease in xDG of the D-fund in G-firms reduces the engagement eﬀort of the D-fundmanager in G-firms and thus increases ZG and ZM . However, the eﬀect of the decrease in
1 − xDG − xP (1 − xP ) of liquidity investors in G-firms (in the market portfolio) reduces

ZG (both ZG and ZM ). The reason is that if the P-fund replaces liquidity investors in
G-firms’ ownership (in the market portfolio ownership), this eﬀect decreases ZG (both ZG
and ZM ) because liquidity investors do not make any engagement eﬀorts.
In evaluating the eﬀects of the lower ψP on ZG (ZM ), note that the total eﬀects throughRPchanges in the fund fees and the fund ownership stakes depend on the diﬀerence in ccRD
ZP +cCPZP +cCPand ccZD((1 + λ2 + η2 λ2 cZDcRD)−1 ccZD). Accordingly, the lower ψP reduces ZG
+cCD
+cCD
+cCD
RP
ZP +cCPZP +cCP(ZM ) as long as ccRD
is larger than ccZD((1 + λ2 + η2 λ2 cZDcRD)−1 ccZD). However,
+cCD
+cCD
+cCD

for ZB , the total eﬀects of the lower ψP do not include any eﬀect through a change in thestakes held by the D-fund. As the eﬀect of replacing liquidity investors is dominant, thelower ψP always reduces ZB .
To investigate the eﬀect of the lower ψP on the expected profit, note that the expected
profit is positively associated with the expected negative externality in G-firms and in
market portfolio firms (see Proposition 1(ii)). Then, for RG , we show that the lower ψP
RP
ZP +cCPdecreases RG as long as ccRD
is larger than ccZD
. However, for RM , there exists an+cCD

additional direct eﬀect of the lower ψP on the eﬀort incentive for the P-fund manager
through the P-fund fee to increase the profit, which increases RM . Hence, the eﬀect of
the lower ψP on RM is ambiguous. For RB , as the eﬀect of replacing liquidity investors isdominant, the lower ψP always increases RB .
The theoretical implication of Proposition 6 is that because P-fund growth aﬀects fundinvestors’ capital allocation and then changes the fund ownership stakes, the eﬀect of
P-fund growth on the expected negative externality strongly depends on the comparative
cost advantage between each fund in improving ESG or profit performance, like the eﬀectof growing interests in ESG. Accordingly, despite the recent argument about the P-fund
not contributing to ESG, P-fund growth does not necessarily avoid the reduction in the
expected negative externality.
Given the argument in Section 5, it easily follows from Propositions 6(i)—6(iii) alongwith (24) that the following predictions are obtained:34
Prediction 7A: If improving ESG performance can be achieved to a certain degree by
34

cRD
cRP
2
2 2
−1 cZP +cCP
RP
ZP +cCP
Note that ccRD
> ccZD+cCD implies cRD > (1 + λ + η λ cZD +cCD )cZD +cCD .

50

setting broad, market-wide standards of ESG, and if the cost complementarity in the Dfund manager’s eﬀorts is not suﬃciently large, the growth in P-funds is likely to reduce
the expected negative externalities released by firms with high-ESG scores and by all thefirms in the market portfolio. However, it is also likely to reduce the expected financial
returns of these firms. Nonetheless, the growth in P-funds always reduces the expected
negative externalities released by firms with low-ESG scores and their expected financial
returns.
Prediction 7B: If most D-funds consist of ESG-indexed funds, the growth of P-funds islikely to reduce the expected negative externalities and expected financial returns of any
type of firm.
These predictions suggest that P-fund growth does not necessarily hinder the improvement in ESG performance, contrary to the argument of environmental activists.
Given Proposition 6, the argument in Section 5 also leads to the following prediction:
Prediction 8: P-fund growth causes a negative association between ESG scores and
EBITDA attained in firms with high-ESG scores. However, it brings about a positive
association between ESG scores and EBITDA attained in firms with low-ESG scores.

51

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