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\begin{document}

\title{A Theory of Socially Responsible Investment \thanks{For helpful comments and
suggestions, we thank Ulf Axelson, Markus Brunnermeier, Patrick Bolton, Peter DeMarzo, Diego Garcia, Anastasia
Kartasheva, Ailsa Ro\"ell, Joel Shapiro, Paul Woolley, and seminar
participants at the ASU Sonoran Conference, Bonn, Copenhagen Business School, CU Boulder, \'Ecole Polytechnique,
Erasmus University, EWFC, Frankfurt School of Finance, Humboldt Universit\"at zu Berlin, LSE, OxFIT, Princeton,
Stanford, Stockholm School of Economics, UCLA, USC, and Warwick Business
School. } }
\author{Martin Oehmke\thanks{London School of Economics and CEPR, e-mail:
m.oehmke@lse.ac.uk.}
\and Marcus Opp\thanks{Stockholm School of Economics, e-mail: marcus.opp@hhs.se.} }
\date{\today}
\maketitle

\begin{abstract}
We characterize necessary conditions for socially responsible investors to impact firm behavior. Impact requires a \textit{broad mandate} (socially responsible investors need to internalize social costs irrespective of whether they are investors in a firm) and is optimally achieved by enabling a scale increase for clean production. Socially responsible and financial investors are complementary: jointly they can achieve higher welfare than either investor type alone. Scarce socially responsible capital should be allocated according to a \textit{social profitability index} (SPI), which captures not only a firm's social status quo but also counterfactual social costs produced in the absence of socially responsible investors. 

\bigskip

\noindent \emph{Keywords}:\ Socially responsible investing, ESG, SPI, capital
allocation, sustainable investment, social ratings.

\bigskip

\noindent \emph{JEL\ Classification}:\ G31 (Capital Budgeting; Fixed Investment
and Inventory Studies; Capacity), G23 (Non-bank Financial Institutions;
Financial Instruments; Institutional Investors).

\end{abstract}

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%\end{center}


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%\section{Introduction}


In recent years, the question of the social responsibility of business,
famously raised by \cite{Friedman1970}, has re-emerged in the context of the
spectacular rise of socially responsible investment (SRI). Assets under
management in socially responsible funds have grown manifold,\footnote{For
example, the Global Sustainable Investment Alliance (2018) reports sustainable
investing assets of \$30.7tn at the beginning of 2018, an increase of 34\%
relative to two years prior.} and many traditional investors consider
augmenting their asset allocation with environmental, social, and governance
(ESG) scores \citep{PastorEtAl2019, PedersenEtAl2019}. From an asset
management perspective, this trend raises immediate questions about the
financial performance of such investments
\citep{HongKacperczyk2009, Chava2014, BarberMorseYasuda2018}. However, if
socially responsible investing is to generate real impact, it must affect
firms' production decisions. This raises additional, fundamental questions:
Under which conditions can socially responsible investors impact firm
behavior? And how should scarce socially responsible capital be allocated
across firms?

Answering these questions requires taking a corporate finance view of socially
responsible investment. To this end, we incorporate socially responsible
investors and the choice between clean and dirty production into an otherwise
standard model of corporate financing with agency frictions, building on
\cite{HolmstromTirole1997}. The model's main results are driven by the
interaction of financing constraints (leading to underinvestment in socially
desirable clean production) and negative production externalities (which can
lead to overinvestment in socially undesirable dirty production).

We find that socially responsible investors can indeed push firms to adopt
clean production. They optimally do so by raising a firm's financing capacity
under clean production beyond the amount that purely profit-motivated
investors would provide. The resulting increase in clean production raises
welfare, even compared to the scenario where all capital is held by socially
responsible investors. However, increasing clean production beyond the scale
that profit-motivated investors would fund is only possible if socially
responsible make financial losses. Therefore, a necessary condition for
socially responsible investors to break even, in social terms, on their impact
investments is that they follow a broad mandate, in the sense that they
internalize social costs generated by firms
%(e.g., carbon emissions)
regardless of whether they are investors in these firms. When faced with an
investment decision across many heterogeneous firms, scarce socially
responsible capital should be allocated according to a social profitability
index (SPI). One key feature of this micro-founded ESG metric is that avoided
social costs are relevant for ranking
investments. Hence, investments in \textquotedblleft sin\textquotedblright%
\ industries are not necessarily inconsistent with the mandate of being
socially responsible.

We develop these results in a parsimonious model, initially focusing on the
investment decision of a single firm. The firm is owned by an entrepreneur
with limited wealth, who has access to two constant-returns-to-scale
production technologies, dirty and clean. Dirty production has a higher
per-unit financial return, but entails significant social costs. Clean
production is financially less attractive but socially preferable, because it
generates lower (although not necessarily zero) social costs. Production under
either technology requires the entrepreneur to exert unobservable effort, so
that not all cash flows are pledgeable to outside investors. The firm can
raise funding from (up to) two types of outside investors. Financial investors
behave competitively and, as their name suggests, care solely about financial
returns. Socially responsible investors also care about financial returns,
but, in addition, they satisfy two conditions that, as our analysis reveals,
are necessary for impact. First, they care \textit{unconditionally} about
external social costs generated by the firm (i.e., irrespective of whether
they are investors in the firm). Second, they act in a coordinated fashion, so
that they internalize the effect of their investments on production decisions.
Socially responsible investors in our model are therefore most easily
interpreted as a large (e.g., sovereign wealth) fund.
%\footnote{This condition rules out a free-rider problem that arises when infinitesimal socially responsible investors take the firm's investment decisions as given.}


As a benchmark, we initially consider a setting in which only financial
investors are present. Because of the entrepreneur's moral hazard problem, the
amount of outside financing and, hence, the firm's scale of production are
limited. Since the dirty production technology is financially more attractive,
financial investors offer better financing terms for dirty production,
enabling a larger production scale than under the clean technology. As a
result, the entrepreneur may adopt the socially inefficient dirty production
technology, even if she partially internalizes the associated externalities
due to an intrinsic preference for clean production.

We then analyze whether and how socially responsible investors can address
this inefficiency. We show that the optimal way to achieve impact
(i.e., induce a change in the firm's production technology) is to relax
financing constraints for clean production, thereby enabling additional value
creation. One way the firm can implement the resulting financing agreement is by
issuing two bonds, a green bond purchased by socially responsible investors
and a regular bond purchased by financial investors. However, because
financial investors are not willing to provide this scale on their own, the
extra financing must involve a financial loss to socially responsible
investors. Therefore, the green bond is issued at a premium, consistent with
evidence in \cite{Bakeretal2018} and \cite{Zerbib2019}.

Our results highlight a complementarity between socially responsible and
financial investors. Because of this complementarity, welfare (which, in our
model, is equivalent to the total scale of clean production) is generally
higher when both types of investors are present. The complementarity between
socially responsible and financial investors arises from each investor's comparative advantage. Compared to socially responsible
investors, financial investors are more aggressive in providing financing as
they view a project (under either technology) as more profitable, thereby
alleviating the underinvestment problem that results from agency frictions.
However, because of their disregard for externalities, financial investors
facilitate the adoption of the socially inefficient dirty technology, creating
scope for socially responsible investors to guide the firm's choice of
technology. In doing so, the threat of dirty production relaxes the
participation constraint of socially responsible investors and, thereby, acts
like a \emph{quasi asset} to the entrepreneur that allows the financing of the
clean technology. Therefore, the presence of financial investors is
instrumental to unlocking the additional capital socially responsible
investors provide for the clean technology.

Our analysis identifies three necessary conditions for this complementarity to
arise. First, socially responsible investors need to follow a broad mandate.
This means that they must care unconditionally about external social costs of
production, whether or not they are investors in the firm that produces them.
If socially responsible investors follow a narrow mandate, so that they only
care about social costs generated by their own investment, their presence does not reduce the social costs generated by the firm, since dirty production will then simply be financed by
financial investors.\footnote{Our analysis is motivated by social costs such as
carbon emissions, so that socially responsible investors are driven by the mitigation of negative
production externalities. We discuss positive production externalities in Section \ref{sec: robustness production technology}. This extension reveals an interesting asymmetry: In the case of positive externalities, a narrow mandate (i.e., only accounting for the positive externalities generated by one's own investment) is more effective.} Second, the clean technology must be subject to financing constraints, so that
additional clean scale is socially valuable. Third, socially responsible
capital needs to be in sufficient supply to be able to discipline the threat
of dirty production. If this is not the case, dirty production is not merely
an off-equilibrium threat, but occurs in equilibrium.

While socially responsible capital has seen substantial growth over the last
few years, it is likely that such capital remains scarce relative to financial
capital that only chases financial returns. This raises the question of how
scarce socially responsible capital is invested most efficiently. Which firms
should impact investors target? A multi-firm extension of our model yields a
micro-founded investment criterion for scarce socially responsible capital,
the \emph{Social Profitability Index} (SPI).

Similar to the profitability index, the SPI measures \textquotedblleft bang
for buck\textquotedblright--- i.e., value created for socially responsible
investors per unit of socially responsible capital consumed. However, unlike
the conventional profitability index, the SPI not only reflects the (social)
return of the project that is being funded, but also the counterfactual social
costs that a firm would have generated in the absence of investment by
socially responsible investors. For example, investment metrics for socially
responsible investors should include estimates of carbon emissions that can be
avoided if the firm adopts a cleaner production technology. Because avoided
externalities matter, it can be efficient for socially responsible investors
to invest in firms that, in an absolute sense, generate a high level of social
costs even under clean production. Accordingly, investments in sin industries
(see \cite{HongKacperczyk2009}) can be consistent with socially responsible
investing. In contrast, it is efficient to not invest in firms that are
already committed to clean production (e.g., because, intrinsically, the
entrepreneur cares sufficiently about the environment), because clean
production will occur regardless of investment by socially responsible investors.

The SPI also rationalizes why environmental, social, and governance issues are
usually bundled into one ESG score. In our model, a connection between these
distinct aspects of corporate behavior arises naturally, because the severity
of the manager's agency problem (a proxy for governance) determines the
financing constraints the firm faces with respect to both financial and
socially responsible investors.  The
SPI reflects these financing constraints because they interact with the
(environmental and social) externalities generated by the firm. Intuitively, financing constraints reduce welfare if they limit the scale of clean
production, but can be welfare-enhancing if they limit dirty production.
%Therefore, the SPI incorporates all three ESG elements.

Throughout the paper, we abstract away from government intervention. One way
to interpret our results is therefore as characterizing the extent to which
the market can fix problems of social cost before the government imposes
regulation or Pigouvian taxes. Another interpretation is that our analysis
concerns those social costs that remain after the government has intervened.
For example, informational frictions and political economy constraints may
make it difficult for governments to apply Pigouvian taxes or ban dirty
production (see, e.g., \citealp{Tirole2012}).\footnote{Examples of social
costs for which government intervention is likely to be particularly difficult
include those where the relevant externality is global in nature, as is the
case for carbon emissions or systemic externalities caused by large financial
institutions.} However, even if government intervention is possible, our
analysis reveals that text-book regulation, in the form of Pigouvian taxes or
a ban on dirty production, may result in lower welfare than the allocation
achieved via co-investment by financial and socially responsible investors.
While Pigouvian taxes or bans on dirty production would certainly ensure the
adoption of the clean technology (even when financing is provided by financial
investors only), such regulation also eliminates the threat of dirty
production, which is necessary to unlock additional capital by socially
responsible investors. The broader point is that regulation that targets one
source of inefficiency (externalities) but does not address the other
(financing constraints) has limited effectiveness in our setting. Optimal regulation, which is beyond the scope of this paper, needs to account for both sources of inefficiencies.
%\footnote{Optimal regulation, which is beyond the scope of this paper, therefore, needs to account for both production externalities and financing constraints.}


\textbf{Related Literature}. Despite the growing interest in socially
responsible investing \citep[see, e.g.,][]{LandierNair2009}, the theory
literature on this topic is still relatively small. In a pioneering
contribution, \cite{HeinkelKrausZechner2001} show that firms that are excluded
by socially responsible investors suffer a reduction in risk-sharing among
their investor base. The resulting increase in the cost of capital can induce
firms to clean up their activities.\footnote{However, as
\cite{DaviesVanWesep2018} point out, divestment can also have unintended
consequences, for example, by inducing firms to prioritize short-term profit
at the expense of long-term value.} \cite{HartZingales2017} characterize the
objective of a firm with prosocial investors, who dislike social costs if they
feel directly responsible for them. They argue that firms should maximize
shareholder welfare instead of shareholder value. Socially responsible
investors in our model are similar to prosocial investors, with the important
difference that they care about externalities regardless of whether they are
directly responsible for them.
%They show that, within the context of their framework, frequent corporate votes are a way to implement the correct objective.
%These papers take the firm's ownership structure as given. In contrast, we endogenize the assignment of socially responsible investors to firms. Another key difference is that our paper features a moral hazard problem, and, therefore, underinvestment, which is a key ingredient for the complementarity between financial and social capital.
\cite{ChowdhryDaviesWaters2018} study the financing of a firm that cannot
commit pursuing social goals.
%They show that issuing securities to socially-minded investors can blunt a firm's profit motives, thereby allowing the firm to commit to emphasize social goals.
A common theme with our paper is that the firm can monetize the
socially-minded investors' social preference. However, their analysis focuses
on how socially-minded investors can blunt a firm's profit motive
\citep[in the spirit of][]{GlaeserShleifer2001} thereby allowing the firm to
commit to emphasize social goals. In contrast, we focus on the ability of
socially responsible investors to impact firms by relaxing financial
constraints for clean production. \cite{LandierLovo2020} show how search frictions in financial markets can allow an ESG fund to affect production decisions and how the effect of impact investors can be transmitted along the supply chain.  \cite{Roth2019} contrasts impact investing
with grants or donations, highlighting the ability of investors to withdraw
capital as a potential advantage over grants. While in our model socially
responsible investors must necessarily make a financial loss,
\cite{GollierPouget2014} provide a model in which a large activist investor
can generate positive abnormal returns by reforming firms and then selling
them back to the market.
%Another important difference is that we develop an investment criterion, the SPI, to guide scarce socially responsible capital in a multi-firm setting.
Related to our result that, in order to have impact, socially
responsible investors need to act in a coordinated fashion, 
\cite{MorganTumlinson2018} provide a more detailed analysis of potential
free-rider problems among investors and how those can be overcome. Finally, socially responsible investors in our model are characterized by an intrinsic motivation to reduce social costs. Our analysis therefore provides a corporate finance perspective on the economics of motivated agents \citep[e.g.,][]{BesleyGhatak2005, BenabouTirole2006}.

\section{Model Setup}

Our modeling framework aims to uncover the role of socially responsible
investing in a setting in which \emph{production externalities} interact with
\emph{financing constraints}. It builds on the canonical model of corporate
financing in the presence of agency frictions laid out in
\cite{HolmstromTirole1997} and \cite{Tirole2006}. The main innovation is that
the firm has access to two different production technologies, one of them
\textquotedblleft clean\textquotedblright \ (i.e., associated with low social
costs) and the other \textquotedblleft dirty\textquotedblright \ (i.e.,
associated with larger social costs). To focus on the role that socially
responsible investors can play in alleviating distortions, we abstract away
from government intervention for most of the paper.\footnote{The two
technologies can therefore be interpreted as those available to the firm after
government intervention has taken place. Because government intervention is
usually subject to informational and political economy constraints, it seems
reasonable that the social costs of production cannot be dealt with by the
government alone, creating a potential role for socially responsible
investors. Alternatively, our analysis can be interpreted as establishing what
market forces (in the form of socially responsible investors) can achieve
before government intervention takes place.} In Section
\ref{sec: robustness regulation}, we discuss the effects of standard
regulatory policies, such as Pigouvian taxes or banning the dirty production technology.

\paragraph{The entrepreneur, production, and moral hazard.}

Our setting considers a risk-neutral entrepreneur who is protected by limited
liability and endowed with initial liquid assets of $A$. The entrepreneur has
access to two production technologies $\tau \in \left \{  C,D\right \}  $, each
with constant returns to scale.\footnote{In Section
\ref{sec: robustness production technology}, we discuss robustness of our
results to $N>2$ technologies and decreasing returns to scale.}
%\footnote{All of our results continue to hold if the production technology exhibits decreasing returns to scale.}
The technologies are identical in terms of revenue generation. Denoting firm
scale (capital) by $K$, the firm generates positive cash flow of $RK$ with
probability $p$ (conditional on effort by the entrepreneur, as discussed
below) and zero otherwise.

Where the technologies differ is with respect to the required ex-ante
investment and the social costs they generate. In particular, the dirty
technology $D$ generates a non-pecuniary negative externality of $\phi_{D}>0$
per unit of scale and requires a per-unit upfront investment of $k_{D}$. The
clean technology results in a lower per-unit social cost $0<\phi_{C}<\phi_{D}%
$, but requires a higher per-unit upfront investment of $k_{C}>k_{D}$.\footnote{The assumption that $0<\phi_{C}<\phi_{D}$ reflects that our analysis focuses on the mitigation of negative production externalities by socially responsible investors. We discuss the case of positive production externalities in Section \ref{sec: robustness production technology}.}  The
entrepreneur internalizes a fraction $\gamma^{E}\in \left[  0,1\right)  $ of
social costs, capturing potential intrinsic motives not to cause social harm.

To generate a meaningful trade-off in the choice of technologies, we assume
that the ranking of the two technologies differs depending on whether it is
based on financial or social value. Denoting the per-unit financial value by
$\pi_{\tau}:=pR-k_{\tau}$ and the per-unit social value (welfare) by $v_{\tau
}:=\pi_{\tau}-\phi_{\tau}$, we assume that the dirty technology has higher
financial value, $\pi_{D}>\pi_{C}$, but clean production generates higher
social value, $v_{C}>0>v_{D}$.\footnote{Once we allow for $N$ technologies
(see Section ~\ref{sec: robustness production technology}), we show how our
results readily extend to cases where the dirtiest technology may no longer be
the profit-maximizing technology. The case where the clean technology also
maximizes profits is uninteresting for our analysis of socially responsible
investment, since even purely profit-motivated capital would ensure clean
production in this case.} The final inequality implies that the social value
of the dirty production technologies is negative, meaning that the
externalities caused by dirty production outweigh its financial value. The
assumption that the dirty production technology has negative social value is
not necessary for our results, but it simplifies the exposition because it
implies that, from a social perspective, the dirty technology should never be
adopted.
%\footnote{The assumptions that we make regarding the production technology can be relaxed relatively straightforwardly. In Section \ref{sec: discussion}, we discuss extensions to $N$ production technologies, allowing for social goods, and decreasing returns to scale.}


As in \cite{HolmstromTirole1997}, the entrepreneur is subject to an agency
problem. Whereas the choice of production technology is assumed to be
observable (and, hence, contractible) effort is assumed to be unobservable
(and, therefore, not contractible). Under each technology, the investment pays
off with probability $p$ only if the entrepreneur exerts effort ($a=1$). The
payoff probability is reduced to $p-\Delta p$ when the entrepreneur shirks
$\left(  a=0\right)  $, where $p>\Delta p>0$. Shirking yields a per-unit
non-pecuniary benefit of $B$ to the entrepreneur, for a total private benefit
of $BK$. A standard result (which we will show below) is that this agency
friction reduces the firm's unit pledgeable income by $\xi:=p\frac{B}{\Delta
p}$, the per-unit agency cost. A high value of $\xi$ can be interpreted as an
indicator of poor governance, such as large private benefits or weak
performance measurement. We make the following assumption on the per-unit
agency cost:
%\footnote{The term $\frac{p}{\Delta p}$ is an inverse function of the likelihood ratio for the \textquotedblleft success\textquotedblright \ outcome, $\frac{p}{p-\Delta p}$, the relevant metric for performance measurement in our setting.}


\begin{assumption}
\label{ass: agency} For each technology $\tau,$ the agency cost per unit of
capital $\xi:=p\frac{B}{\Delta p}$ satisfies%
\begin{equation}
\pi_{\tau}<\xi<pR-\frac{p}{\Delta p}\pi_{\tau}.
\end{equation}

\end{assumption}

This assumption states that the moral hazard problem, as characterized by the
agency cost per unit of capital $\xi$, is neither too weak nor too severe. The
first inequality implies a finite production scale. The second inequality is a
sufficient condition that rules out equilibrium shirking and ensures
feasibility of outside financing. To streamline notation, $\pi$ and $v$ are
defined conditional on the relevant case, in which the entrepreneur exerts
effort (as usual, shirking is an off-equilibrium action).

\paragraph{Outside investors and securities.}

The entrepreneur can raise financing from (up to) two types of risk-neutral
outside investors $i\in \left \{  F,SR\right \}  ,$ \emph{financial investors}
and \emph{socially responsible investors}. Both investor types care about
expected cash flows, but only socially responsible investors internalize
social costs of production (i.e., $\gamma^{SR}>0$, whereas $\gamma^{F}%
=0$).\footnote{The assumption that at least some investors internalize social
costs is consistent with evidence in \cite{RiedlSmeets2017} and
\cite{BonnefonEtAl2019}.}$^{, }$\footnote{In our model, financial investors
literally do not care about social costs. However, an alternative setting, in
which financial investors do care about social costs but do not act on them
because of a free-rider problem, would yield equivalent results.} Regardless
of whether the entrepreneur raises financing from both investor types or just
one, it is without loss of generality to restrict attention to financing
arrangements in which the entrepreneur issues securities that pay a total
amount of $X:=X^{F}+X^{SR}$ upon project success and 0 otherwise, where
$X^{F}$ and $X^{SR}$ denote the payments promised to financial and socially
responsible investors, respectively.
%\footnote{The optimality of these arrangements follows directly from \cite{Innes1990}.}
Given that the firm has no resources in the low state, this security can be
interpreted as debt or equity. The entrepreneur's utility can then be written
as a function of the investment scale $K$, the total promised repayment $X$,
the effort decision $a$, upfront consumption by the entrepreneur $c$, and the
technology choice $\tau \in \left \{  C,D\right \}  $,
\begin{align}
U^{E}\left(  K,X,\tau,c,a\right)  =  &  p\left(  RK-X\right)  -\left(
A-c\right)  -\gamma^{E}\phi_{\tau}K\nonumber \\
&  +\mathbbm{1}_{a=0}\left[  BK-\Delta p\left(  RK-X\right)  \right]  .
\tag{$U^E$}\label{UE}%
\end{align}


The first two terms of this expression, $p\left(  RK-X\right)  -\left(
A-c\right)  $, represent the project's net financial payoff to the
entrepreneur under high effort, where $A-c$ can be interpreted as the upfront
co-investment made by the entrepreneur. The third term, $\gamma^{E}\phi_{\tau
}K$, measures the social cost internalized by the entrepreneur. The final
term, $BK-\Delta p\left(  RK-X\right)  $, captures the incremental payoff
conditional on shirking ($a=0$). Exerting effort is incentive compatible if
and only if $U^{E}\left(  K,X,\tau,c,1\right)  \geq U^{E}\left(
K,X,\tau,c,0\right)  $, which limits the total amount $X$ that the
entrepreneur can promise to repay to outside investors to
\begin{equation}
X\leq \left(  R-\frac{B}{\Delta p}\right)  K. \tag{IC}\label{IC}%
\end{equation}


Per unit of scale, the entrepreneur's pledgeable income is therefore given by
$pR-\xi$. The resource constraint at date $0$ implies that capital
expenditures, $Kk_{\tau}$, must equal the total investments made by the
entrepreneur and outside investors,
\begin{equation}
Kk_{\tau}=A-c+I^{F}+I^{SR}, \label{resource}%
\end{equation}
where $I^{F}$ and $I^{SR}$ represent the investments of financial and socially
responsible investors, respectively.

We impose two conditions on the behavior of socially responsible investors. As
we will show later, both of these conditions are \emph{necessary} for socially
responsible investors to have impact.

\begin{condition}
[Broad Mandate]\label{C1} Socially responsible investors are affected by
externalities $\gamma^{SR}\phi_{\tau}K$ regardless of whether they invest in
the firm.
\end{condition}

\begin{condition}
[Coordination]\label{C2} Socially responsible investors allocate their capital
in a coordinated fashion.
\end{condition}

Effectively, these conditions imply that socially responsible investors care
unconditionally about externalities (Condition~\ref{C1}) and are
willing to act on them (Condition \ref{C2}). Because the benefits of socially
responsible investment in the form of lower externalities are non-rival and
non-excludable, coordination is necessary to ensure that socially responsible
investors take into account that their investment affects the social cost
generated by the firm (see literature on public goods following
\cite{Samuelson1954}). This condition is naturally satisfied if
socially responsible capital is directed by one large fund, such as the
Norwegian sovereign wealth fund. Another interpretation is that socially
responsible investors are dispersed, but have found a way to overcome the
free-rider problem that would usually arise.\footnote{One such example is the
establishment of the Poseidon Principles, an initiative by eleven major to
promote green shipping, see \cite{Nauman2019}. \cite{DimsonEtAl2019}
empirically document coordinated engagements by large investors.
%\cite{MorganTumlinson2018} provide a model in which shareholders of a company value public good production but are subject to free-rider problems.
}

Under Conditions~\ref{C1} and \ref{C2} the respective utility functions of
outside investors, given an incentive-compatible financing arrangement, can be
written as%
\begin{align}
U^{F}  &  =pX^{F}-I^{F},\tag{$U^F$}\label{UF}\\
U^{SR}  &  =pX^{SR}-I^{SR}-\gamma^{SR}\phi_{\tau}K, \tag{$U^{SR}$}\label{USR}%
\end{align}
where $\gamma^{SR}$ captures the degree to which socially
responsible investors internalize the social costs generated by the firm. The
sum $\gamma^{E}+\gamma^{SR}\in \left(  0,1\right]$  represents the
fraction of total externalities that is taken into account by investors and
the entrepreneur. When $\gamma^{E}+\gamma^{SR}=1$, agents in the model jointly internalize all externalities. In the case %
$\gamma^{E}+\gamma^{SR}<1$ some externalities are not internalized. This can be interpreted in two ways. First, it
may represent a situation where some externalities (e.g., those imposed on future
generations) are simply unaccounted for by decision makers. Second, it may
capture, in reduced form, the effect of imperfect coordination among socially
responsible investors. In the extreme case $\gamma^{SR}=0,$ the
free rider problem among socially responsible investors is so severe that they
effectively act like financial investors.

\section{The Effect of Socially Responsible Investment\label{sec:SR}}

In this section, we investigate whether and how socially responsible investors
can impact a single firm's investment choice, assuming that socially
responsible capital is abundant relative to the funding needs of the firm. Our
subsequent multi-firm setting in Section \ref{sec:SPI} analyzes how scarce
socially responsible capital should be optimally allocated across firms. In
Section \ref{sec:benchmark}, we first solve a benchmark case of firm financing
without socially responsible investors. This benchmark shows that, when
investors care exclusively about financial returns, the dirty technology may
be chosen even when the entrepreneur has some concern for the higher social
cost generated by dirty production (i.e., when $\gamma^{E}>0$). In Section
\ref{sec:impact}, we add socially responsible investors to the model and
characterize conditions under which their presence has impact, in the sense
that it changes the firm's production decision.

\subsection{Benchmark: Financing from Financial Investors Only}

\label{sec:benchmark}

The setting in which the entrepreneur can borrow exclusively from competitive
financial investors corresponds to the special case $I^{SR}=X^{SR}=0$. Then,
the entrepreneur's objective is to choose a financing arrangement (consisting
of scale $K\geq0$, repayment $X^{F}\in \left[  0,R\right]  $, upfront
consumption by the entrepreneur $c\geq0$, and technology choice $\tau
\in \left \{  C,D\right \}  $) that maximizes the entrepreneur's utility \ref{UE}
subject to the entrepreneur's \ref{IC} constraint and financial investors' IR
constraint, \ref{UF}$\geq0$.

As a preliminary step, it is useful analyze the financing arrangement that
maximizes scale for a given technology $\tau$.\footnote{As discussed below,
given competitive investors, maximizing scale is indeed optimal if the
technology generates positive surplus from the entrepreneur's perspective.}
Following standard arguments \citep[see][]{Tirole2006}, this agreement
requires the entrepreneur to co-invest all her wealth (i.e., $c=0$) and that
the entrepreneur's \ref{IC} constraint as well as the financial investors' IR
constraint bind. The binding \ref{IC} constraint ensures that the firm
optimally leverages its initial resources $A$, whereas the binding IR
constraint is a consequence of competition among financial investors.

When all outside financing is raised from financial investors, the maximum
firm scale under production technology $\tau$ is then given by
\begin{equation}
K_{\tau}^{F}=\frac{A}{\xi-\pi_{\tau}}. \label{eq:maxscale}%
\end{equation}
This expression shows that the entrepreneur can scale her initial assets $A$
by a factor that depends on the \emph{agency cost per unit of investment},
$\xi:=p\frac{B}{\Delta p}$, and the financial return under technology $\tau$,
$\pi_{\tau}$. As $\xi>\pi_{D}$ (see Assumption~\ref{ass: agency}), the maximum
investment scale is finite under either technology. The key observation from
Equation \eqref{eq:maxscale} is that the maximum scale that the entrepreneur
can finance from financial investors is larger under dirty than under clean
production, $K_{D}^{F}>K_{C}^{F}$, since dirty production generates larger
financial value, $\pi_{D}>\pi_{C}$, and financial investors only care about
financial returns.

The following lemma highlights that the entrepreneur's technology choice
$\bar{\tau}$ is then driven by a trade-off between achieving scale and her
concern for externalities. Of course, if the entrepreneur completely
disregards externalities $\left(  \gamma^{E}=0\right)  $, no trade-off arises
and the entrepreneur always chooses dirty production to maximize scale.
%, as $K_{D}^{F}>K_{C}^{F}$.


\begin{lemma}
[Benchmark: Financial Investors Only]\label{lemma: benchmark}When only
financial investors are present, the entrepreneur chooses
\begin{equation}
\label{eq:TauBar}\bar{\tau}=\argmax_{\tau}(\xi-\gamma^{E}\phi_{\tau})K_{\tau
}^{F}\text{.}%
\end{equation}
The firm operates at the maximum scale that allows financial investors to
break even, $K_{\bar{\tau}}^{F}$. The entrepreneur's utility is given by%
\begin{equation}
\label{eq:UEfin}\bar{U}^{E}=(\xi-\gamma^{E}\phi_{\bar{\tau}})K_{\bar{\tau}%
}^{F}-A.
\end{equation}

\end{lemma}

According to Lemma \ref{lemma: benchmark}, the entrepreneur chooses the
technology that maximizes her payoff, which is given by the product of the
per-unit payoff to the entrepreneur (agency rent net of internalized social
cost) and the maximum scale under technology $\tau$ (given by Equation
\eqref{eq:TauBar}). Maximum scale is optimal because, under the equilibrium
technology $\bar{\tau}$, the project generates positive surplus for the
entrepreneur and financial investors. It follows that the entrepreneur adopts
the dirty technology whenever
\begin{equation}
(\xi-\gamma^{E}\phi_{D})K_{D}^{F}>(\xi-\gamma^{E}\phi_{C})K_{C}^{F}.
\end{equation}
Given that the maximum scale is larger under the dirty technology, $K_{D}%
^{F}>K_{C}^{F}$, this condition is satisfied whenever the entrepreneur's
concern for externalities $\gamma^{E}$ lies below a strictly positive cutoff
$\bar{\gamma}^{E}$.

\begin{corollary}
[Financial Investors Can Induce Dirty Production]\label{cor: benchmark}When
only financial investors are present, the entrepreneur adopts the dirty
production technology when $\gamma^{E} < \bar{\gamma}^{E}:= \frac{\xi(\pi
_{D}-\pi_{C})}{\phi_{D}(\xi-\pi_{C})-\phi_{C}(\xi-\pi_{D})}$.

%chooses the dirty
%technology whenever $\gamma^{E}<\bar{\gamma}^{E}$, where $\bar{\gamma}^{E}:=
%\frac{\xi(\pi_{D}-\pi_{C})}{\phi_{D}(\xi-\pi_{C})-\phi_{C}(\xi-\pi_{D})}>0$.

\end{corollary}

Note that the entrepreneur can be induced to choose the dirty
technology when financing from financial investors is available even if she
were to choose the clean technology under autarky (i.e., self-financing).\footnote{The entrepreneur prefers the clean
technology under self-financing if and only if $\frac{A}{k_{C}}\left(
\pi_{C}-\gamma^{E}\phi_{C}\right)  > \frac{A}{k_{D}}\left(  \pi_{D}-\gamma
^{E}\phi_{D}\right).$ Hence, the entrepreneur is \textquotedblleft
corrupted\textquotedblright \ by financial markets when $\gamma^{E}\in \left(
\widetilde{\gamma}^{E},\bar{\gamma}^{E}\right)  $ where %
$\widetilde{\gamma}^{E}:= \frac{k_C \pi_D - k_D \pi_C}{k_C \phi_D - k_D \phi_C}
%\left (\frac{\pi_{D}}{k_{D}}-\frac{\pi_{C}}{k_{C}}\right ) \big / \left ( \frac{\phi_{D}}{k_{D}}-\frac{\phi_{C}}{k_{C}} \right )
$.} Given that
financing from financial investors is always available, this benchmark case
shows that there is a potential role for socially responsible investors to
steer the entrepreneur towards the clean production technology as long as
$\gamma^{E}<\bar{\gamma}^{E}$.

\subsection{Equilibrium with Socially Responsible Investors\label{sec:impact}}

We now analyze whether and how the financing arrangement and the resulting
technology choice are altered when socially responsible investors are present.
Because the entrepreneur could still raise financing exclusively from
financial investors, the utility she receives under the financing arrangement
with financial investors only, $\bar{U}^{E}$, now takes the role of an outside
option to the entrepreneur.

\subsubsection{Optimal Financial Contract with Socially Responsible Investors}

Due to the broad mandate Condition~\ref{C1}, socially responsible investors
are affected by the social costs of production regardless of whether they have
a financial stake in the firm. In particular, if socially responsible
investors remain passive, their (reservation) utility is given by
\begin{equation}
\bar{U}^{SR}=-\gamma^{SR}\phi_{\bar{\tau}}K_{\bar{\tau}}^{F}<0,
\label{USR outside}%
\end{equation}
which reflects the social costs generated when the entrepreneur chooses the
optimal production technology $\bar{\tau}$ and scale $K_{\bar{\tau}}^{F}$ (see
Lemma \ref{lemma: benchmark}) when raising financing from financial investors
only.\footnote{If the entrepreneur could not raise financing from
financial investors, the outside option for socially responsible investors
would be determined by the entrepreneur's technology choice under
self-financing.}

To improve their payoff relative to this outside option, socially
responsible investors can engage with the entrepreneur. Because socially
responsible investors act in a coordinated fashion (see Condition \ref{C2}),
they make a take-it-or-leave-it contract offer that specifies the technology
$\tau$, scale $K$ as well as the required financial
investments and payoffs for all investors and the
entrepreneur.\footnote{The assumption that socially responsible investors make a take-it-or-leave-it offer implies that they extract the entire joint surplus. This assumption on surplus sharing is not crucial.}$^{,}$\footnote{One may wonder what happens if multiple socially responsible investors could compete. Interestingly, unlike financial investors, they would not have an incentive to compete with each other. This is because the reduction in social cost is non-rival and non-excludable, so that all socially responsible investors profit from it. Moreover, as we will show below, the (excludable) financial part of their overall return is negative, so that no socially responsible investors has an incentive to undercut on this dimension.} This contract solves the following maximization problem:

\begin{problem}
[Maximization Problem Faced by Socially Responsible Investors]%
\label{Problem: SR}%
\begin{equation}
\max_{I^{F},I^{SR},X^{SR},X^{F},K,c,\tau}pX^{SR}-I^{SR}-\gamma^{SR}\phi_{\tau
}K\text{ } \label{obj}%
\end{equation}
subject to the entrepreneur's IR constraint:%
\begin{equation}
U^{E}\left(  K,X^{SR}+X^{F},\tau,c,1\right)  \geq \bar{U}^{E}, \tag{$%
IR^E$}\label{IRE}%
\end{equation}
as well as the entrepreneur's \ref{IC} constraint, the resource constraint
(\ref{resource}), the financial investors' IR constraint \ref{UF}$\geq0,$ and
the non-negativity constraints $K\geq0,c\geq0$.
\end{problem}

The key difference relative to the benchmark case is that financing and
technology are now chosen to maximize socially responsible investors' utility
subject to the constraint that the entrepreneur is weakly better off than
under the outside option of raising financing exclusively from financial
investors, $\bar{U}^{E}$. Note that this formulation permits the possibility
of compensating the entrepreneur with sufficiently high upfront consumption
$(c>0)$ in return for smaller scale $K$, possibly even shutting down
production completely
\citep[as in the typical Coasian solution, see][]{Coase1960}. However, because
(at a minimum) the clean production technology generates positive joint
surplus for the entrepreneur and socially responsible investors, the optimal
financing arrangement rewards the entrepreneur with larger scale (relative to
what could be funded by financial investors alone) rather than upfront consumption.

\begin{proposition}
[Financing in the Presence of Socially Responsible Investors]%
\label{Theorem: Scale}Let $\hat{v}_{\tau}:=\pi_{\tau}-\left(  \gamma
^{E}+\gamma^{SR}\right)  \phi_{\tau}\geq v_{\tau}:=\pi_{\tau}-\phi_{\tau}$
denote the joint surplus, per unit of scale, accruing to all investors and the
entrepreneur. Then, in any optimal financing arrangement, production is
characterized by%
\begin{align}
\hat{\tau}  &  =\argmax_{\tau}\frac{\hat{v}_{\tau}}{\xi-\gamma^{E}\phi_{\tau}%
},\label{tau_hat}\\
\hat{K}  &  =\frac{\xi-\gamma^{E}\phi_{\bar{\tau}}}{\xi-\gamma^{E}\phi
_{\hat{\tau}}}K_{\bar{\tau}}^{F}. \label{K_hat}%
\end{align}
The entrepreneur consumes no resources upfront, $\hat{c}=0$. The total
date-$0$ investment (by both investors) is $\hat{I}=\hat{K}k_{\hat{\tau}}-A$
and the total payout (to both investors) satisfies $\hat{X}=\left(  R-\frac
{B}{\Delta p}\right)  \hat{K}$. The set of optimal co-investment arrangements
can be obtained by tracing out the cash-flow share accruing to socially
responsible investors $\lambda \in \left[  0,1\right]  $ and setting $\hat
{X}^{SR}=\lambda \hat{X}$, $\hat{X}^{F}=\left(  1-\lambda \right)  \hat{X}$,
$\hat{I}^{F}=p\hat{X}^{F}$ and $\hat{I}^{SR}=\hat{I}-\hat{I}^{F}$. The utility
of socially responsible investors is given by:%
\begin{equation}
\hat{U}^{SR}=\left(  \pi_{\hat{\tau}}-\xi \right)  \hat{K}+A-\gamma^{SR}%
\phi_{\hat{\tau}}\hat{K}. \label{USRhat}%
\end{equation}

\end{proposition}

Proposition \ref{Theorem: Scale} contains the main theoretical results of the
paper. It shows that when both types of investors are present, the optimal
choice of technology maximizes total joint surplus, which is governed by the
joint surplus that is created per unit of capital, $\hat{v}_{\tau}$, and a
term, $\frac{1}{\xi-\gamma^{E}\phi_{\hat{\tau}}}$, that reflects the optimal
scale $\hat{K}$ (see Equation (\ref{K_hat})).
%\footnote{The key difference relative to \cite{BerkGreen2004} is that the scale is not limited by decreasing-returns-to-scale investment opportunities, but rather a moral hazard problem.}
An immediate implication is that if the entrepreneur and socially responsible
investors jointly internalize all externalities, $\gamma^{E}+\gamma^{SR}=1$,
production will always be clean, since, in this case, joint surplus $\hat
{v}_{\tau}$ coincides with social welfare $v_{\tau}$.\footnote{Since $v_{D}%
<0$, the dirty technology would never be chosen. Note that, once we allow for
multiple technologies with $v_{\tau}\geq0$ (see
Section~\ref{sec: robustness production technology}), technologies are not
ranked exclusively according to the per-unit surplus because, in this case,
maximum scale matters as well.} Moreover, Proposition \ref{Theorem: Scale}
shows that the optimal financing arrangement rewards the entrepreneur entirely
with scale, in the sense that the optimal capital stock $\hat{K}$ is chosen so
that the entrepreneur obtains the same utility as under her outside option
$\bar{U}^{E}$. Intuitively, any upfront consumption by the entrepreneur is
suboptimal in the presence of a moral hazard problem that gives rise to
capital rationing and, consequently, underinvestment.

\paragraph{Implementation.}

While the optimal financing arrangement uniquely pins down the production side
(i.e., technology choice and scale), there exists a continuum of co-investment
arrangements between financial and socially responsible investors that solve
Problem~\ref{Problem: SR}. Intuitively, any increase in the cash flow share
accruing to financial investors translates at competitive terms into higher
upfront investment by financial investors, $\hat{I}^{F}$. Because also the
entrepreneur remains at her reservation utility $\bar{U}^{E}$, the payoff to
socially responsible investors as well as aggregate surplus remain unchanged.

There are two particularly intuitive ways in which the optimal financing
arrangement characterized in Proposition~\ref{Theorem: Scale} can be
implemented.\footnote{Under both implementations, the security targeted at
socially responsible investors is issued at a premium in the primary market
(see Corollary~\ref{Cor: Financial loss} below), ensuring that only socially
responsible investors have an incentive to purchase this security.}

\begin{corollary} [Implementation]
\label{Cor: implementation}The following securities implement the optimal
financing agreement:\newline \textbf{1. Green bond and regular bond:} The
entrepreneur issues two bonds with respective face values $\hat{X}^{F}$ and
$\hat{X}^{SR}$ at respective prices $\hat{I}^{F}$ and $\hat{I}^{SR}$. The
green bond contains a technology-choice covenant specifying technology
$\hat{\tau}$. \newline \textbf{2. Dual-class share structure: }The entrepreneur
issues voting and non-voting shares, where shares with voting rights yield an
issuance amount of $\hat{I}^{SR}$ in return for control rights and a fraction
$\lambda$ of dividends. The remaining proceeds $\hat{I}^{F}$ are obtained in
return for non-voting shares with a claim on a fraction $1-\lambda$ of dividends.
\end{corollary}

It is worthwhile pointing out that it is not necessary for the optimal financing agreement to restrict the entrepreneur
from seeking financing for additional dirty production. The financing agreement described in Proposition \ref{Theorem: Scale} and Corollary \ref{Cor: implementation} exhausts all pledgeable assets, so that financial investors would not provide any additional financing for the dirty technology.




\subsubsection{Impact\label{sec: impact}}

To highlight the economic mechanism behind Proposition \ref{Theorem: Scale},
this section provides a more detailed investigation of the
%most interesting
case in which socially responsible investors have impact, where we define
impact as an induced change in the firm's production decision, through a
switch in technology from $\bar{\tau}=D$ to $\hat{\tau}=C$ and/or a change in
scale.\footnote{If investment by socially responsible investors does not
result in a change in production technology compared to the benchmark case
(i.e., $\hat{\tau}=\bar{\tau}$), there is no impact and we obtain the same
level of investment and utility for all agents in the economy as in the
benchmark case. This (less interesting) situation occurs either if the
entrepreneur adopts the clean production technology even in the absence of
investment by socially responsible investors, or if the entrepreneur adopts
the dirty technology irrespective of whether socially responsible investors
provide funding.}

\begin{corollary} [Impact]
Socially responsible investors have impact if and only if $\gamma^{E}%
<\bar{\gamma}^{E}$ and $\gamma^{SR}>\bar{\gamma}^{SR}$, where the threshold
$\bar{\gamma}^{SR}$ is a decreasing function of $\gamma^{E}$.
\end{corollary}

\paragraph{Complementarity between Financial and Social Capital.}

The following proposition highlights that when this condition for impact is
satisfied, the equilibrium features a complementarity between financial and
socially responsible investors.

\begin{proposition}
[Financial and Social Capital Are Complementary]\label{prop: complementarity}
Suppose that $\gamma^{E}<\bar{\gamma}^{E}$ and $\gamma^{SR}>\bar{\gamma}^{SR}%
$, then financial capital and socially responsible capital act as complements
in that the equilibrium clean scale with both investor types, $\hat{K},$ is
larger than the maximum clean scale that can be financed in an economy with
only one of the two investor types,
\begin{equation}
\hat{K}>K_{C}^{F}>K_{C}^{SR}, 
\end{equation}
where $K_{C}^{F}=A / (\xi-\pi_{C})$ and $K_{C}^{SR}=A / (\xi-\pi_{C}+\gamma^{SR}\phi_{C})$.

\end{proposition}

Proposition \ref{prop: complementarity} is a consequence of the interaction of
financing constraints and externalities. Financing constraints imply that,
conditional on choosing the clean technology, there is underinvestment (recall
that $v_{C}>0$ and there are constant returns to scale, so that the socially
optimal scale of clean production is infinite). If only financial investors
are present or, equivalently, if they hold all the capital in the economy (as
in the benchmark case presented in Lemma~\ref{lemma: benchmark}), the maximum
clean scale is given by $K_{C}^{F}$, which exceeds the
corresponding break-even scale in the opposite case when only socially
responsible investors are present, $K_{C}^{SR}$. Hence, financial investors in isolation are better
positioned to alleviate the financing constraints with respect to the clean
technology, precisely because they disregard externalities and, therefore,
perceive each scale unit of the clean technology as more valuable (by
$\gamma^{SR}\phi_{C}$).\footnote{Only in the case in which the clean
technology generates no externalities are the break-even scales offered by
each investor type in isolation identical.} In this context, it is important
to note that the objective of socially responsible investors (see \ref{USR})
differs from the maximization of social surplus $v_{C}K$ even if $\gamma
^{SR}=1$, because socially responsible investors do not internalize rents that
accrue to the entrepreneur.

The key feature of Proposition~\ref{prop: complementarity} is that the
equilibrium scale in the presence of both investor types, $\hat{K}$, strictly
exceeds the financing that is available with only one of the investor types.
This complementarity arises because financial investors are willing to finance
an even larger scale under the dirty production technology, $K_{D}^{F}%
>K_{C}^{F}$, thereby providing the entrepreneur with the outside option of
adopting dirty production.
%(whenever $\frac{\pi_{D}-\gamma^{E}\phi_{D}}{\xi-\pi_{D}}>\frac{\pi_{D}-\gamma^{E}\phi_{C}}{\xi -\pi_{C}}$).
The resulting pollution threat relaxes the participation constraint of
socially responsible investors, through its effect on their reservation
utility, which is given by $\bar{U}^{SR}=-\gamma^{SR}\phi_{D}K_{D}^{F}$. This
reduction in reservation utility unlocks additional financing from socially
responsible investors, relative to the clean scale that is possible when
raising financing from financial investors, $\hat{K}>K_{C}^{F}$. The optimum
scale $\hat{K}$ is chosen by socially responsible investors to just induce the
entrepreneur to switch to the clean production technology.\footnote{When the
entrepreneur does not internalize any of the social costs of production
($\gamma^{E}=0$), this switch from dirty to clean requires that the production
scale under the clean technology is the same as the scale that financial
investors would fund under the dirty technology (i.e., $\hat{K}=K_{D}%
^{F}>K_{C}^{F}$). If $\gamma^{E}>0$, it is sufficient for socially responsible
investors to partially make up for lost scale, because the entrepreneur
internalizes part of the reduction in social costs that results from the
switch to clean production (i.e., $K_{D}^{F}>\hat{K}>K_{C}^{F}$).}
%TAKEN OUT, WE SAID THIS BEFORE Intuitively, it is cheapest to facilitate a scale increase of the clean technology because of the underinvestment problem (rather than rewarding the entrepreneur with upfront consumption).


In sum, Proposition~\ref{prop: complementarity} implies that social surplus is
higher when both financial and socially responsible investors deploy capital,
relative to the case in which all capital is allocated by either financial or
socially responsible investors. Intuitively, the counterfactual social cost
$\gamma^{SR}\phi_{D}K_{D}^{F}$, which is enabled by the presence of financial
investors, acts as a \textit{quasi asset} to the firm, thereby generating
additional financing capacity from socially responsible
investors.\footnote{This interpretation can be formalized as follows. The
break-even scale when financing is raised form both financial and socially
responsible investors can be written as $\frac
{A+\textcolor{red}{\widetilde{A}}}{\xi-\pi_{C}+\gamma^{SR}\phi_{C}}$. The
quasi asset $\tilde{A}:=\gamma^{SR}\phi_{D}K_{D}^{F}>0$ enters the maximum
scale in exactly the same way as the entrepreneur's financial assets $A$.}

\paragraph{The cost of impact.}

Even though socially responsible investors only invest if doing so increases
their utility relative to the case in which they remain passive,%
\begin{equation}
\Delta U^{SR}:=\hat{U}^{SR}-\bar{U}^{SR}=\hat{v}_{C}\hat{K}-\hat{v}_{D}%
K_{D}^{F}\geq0, \label{Delta_U}%
\end{equation}
they do not break even in financial terms on their impact investment.

\begin{corollary}
[Socially Responsible Investors Make a Financial Loss]%
\label{Cor: Financial loss} Impact (a switch from $\bar{\tau}=D$ to $\hat
{\tau}=C$) requires that socially responsible investors make a financial loss.
That is, in any optimal financing arrangement as characterized in
Proposition~\ref{Theorem: Scale},
\begin{equation}
p\hat{X}^{SR}<\hat{I}^{SR}\text{.}%
\end{equation}

\end{corollary}

Intuitively, to induce a change from dirty to clean production, socially
responsible investors need to enable a scale for the clean technology greater
than the clean scale offered by competitive financial investors in isolation.
Because financial investors just break even at that scale, socially
responsible investors must make a financial loss on any additional scale they
finance.\footnote{Socially responsible investors are nevertheless willing to
provide financing because their financial loss, $p\hat{X}^{SR}-\hat{I}^{SR}$,
is outweighed by the utility gain resulting from reduced social costs,
$\gamma^{SR}\left(  \phi_{D}K_{D}^{F}-\phi_{C}\hat{K}\right)  $.} Empirically,
Corollary \ref{Cor: Financial loss} therefore predicts that impact funds must
have a negative alpha or, equivalently, that funds generating weakly positive
alpha cannot generate (real) impact.

Our model also predicts that the financial loss for socially responsible
investors, $p\hat{X}^{SR}-\hat{I}^{SR}$, occurs at the time when the firm
seeks to finance investment in the primary market, consistent with evidence on
the at-issue pricing of green bonds in \cite{Bakeretal2018} and
\cite{Zerbib2019}. However, if socially responsible investors were to sell
their cash flow stake $\hat{X}^{SR}$ to financial investors after the firm has
financed the clean technology, our model does not predict a price premium for
the \textquotedblleft green\textquotedblright \ security in the secondary
market (i.e., in the secondary market, the security would be fairly priced at
$p\hat{X}^{SR}$).\footnote{Intuitively, in our static model, control (or a
technology covenant) only matters once, at the time of initial investment. In
a more general, dynamic setting, control could matter multiple times (e.g.,
whenever investment technologies are chosen).}

\paragraph{Necessary conditions for impact.}

The analysis above reveals why Conditions \ref{C1} and \ref{C2} are both
necessary for socially responsible investors to have impact. To see the
necessity of the broad mandate, suppose first that Condition \ref{C1} is
violated and that socially responsible investors follow a narrow mandate, in
that they only care about social costs that are a direct consequence their own
investments. Because, under the narrow mandate, socially responsible investors
ignore the social costs of firms that are financed by financial investors, the
threat of dirty production does not relax their participation constraint.
Hence, the key force that generates the additional financing capacity for
clean production (see Proposition~\ref{prop: complementarity}) is absent and,
therefore, no impact can be achieved.\footnote{In
Section~\ref{sec: robustness production technology}, we revisit the
necessity of the broad mandate in the context of social goods (i.e., technologies with positive
production externalities). We discuss risk aversion and
the risk-premia effects of exclusion under a narrow mandate in Section
\ref{sec: risk-premia}.}

Next suppose that socially responsible investors are infinitesimal and
uncoordinated, so that Condition \ref{C2} is violated. Then, due to the
resulting free-rider problem, each individual investor takes social costs
generated by the firm as given and, therefore, behaves as if $\gamma^{SR}=0$.
No impact can be achieved because socially responsible investors behave like
financial investors.

Finally, a third necessary condition for impact is that socially responsible
capital is available in sufficient amounts to ensure adoption of the clean
production technology. When this is not the case, the presence of financial
capital can induce firms to adopt the dirty production technology, leading to
a social loss. We discuss this case in Section \ref{sec:SPI}, where we
consider an economy with multiple firms and limited socially responsible
capital. This analysis will shed further light on how the composition of
investor capital (and not just the aggregate amount) matters for welfare.

\section{The Social Profitability Index\label{sec:SPI}}

Based on the framework presented above, we now derive a micro-founded
investment criterion to guide scarce socially responsible capital. To do so,
we extend the single-firm analysis presented in Section \ref{sec:SR} to a
multi-firm setting with limited socially responsible capital.

Let $\kappa$ be the aggregate amount of socially responsible capital (we
continue to assume that financial capital is abundant) and consider an economy
with a continuum of infinitesimal firms grouped into distinct firm
types.\footnote{The assumption that firms are infinitesimally small is made
only to rule out well-known difficulties that can arise when ranking
investment opportunities of discrete size.} Firms that belong to the same firm
type $j$ are identical in terms of all relevant parameters of the model,
%$A_{j},B_{j},R_{j}% ,p_{j},\Delta p_{j}$,$\phi_{C,j},\phi_{D,j},k_{C,j},k_{D,j},\gamma_{j}^{E}$
whereas firms belonging to distinct types differ according to at least one
dimension (with Assumption~\ref{ass: agency} satisfied for all types). Let
$\mu(j)$ denote the distribution function of firm types, then the aggregate
social cost in the absence of socially responsible investors is given by
\begin{equation}
\int_{\gamma_{j}^{E}<\bar{\gamma}_{j}^{E}}\phi_{D,j}K_{D,j}^{F}d\mu
(j)+\int_{\gamma_{j}^{E}\geq \bar{\gamma}_{j}^{E}}\phi_{C,j}K_{C,j}^{F}d\mu(j).
\label{Agg externality}%
\end{equation}
The first term of this expression captures the social cost generated by firms
that, in the absence of socially responsible investors, choose the dirty
technology ($\gamma_{j}^{E}<\bar{\gamma}_{j}^{E}$), whereas the second term
captures firm types run by entrepreneurs that have enough concern for external
social costs that they choose the clean technology even in absence of socially
responsible investors ($\gamma_{j}^{E}\geq \bar{\gamma}_{j}^{E}$).

Given this aggregate social cost, how should socially responsible investors
allocate their limited capital? One direct implication of
Proposition~\ref{Theorem: Scale} is that any investment in firm types with
$\gamma_{j}^{E}\geq \bar{\gamma}_{j}^{E}$ cannot be optimal as these firms
adopt the clean technology even when raising financing from financial
investors only. For the remaining firm types, the payoff to socially
responsible investors from \textquotedblleft reforming\textquotedblright \ a
firm of type $j$ is given by:
\begin{equation}
\Delta U_{j}^{SR}=\left(  \pi_{C,j}-\xi_{j}\right)  \hat{K}_{j}+A_{j}%
+\gamma^{SR}\left[  \phi_{D,j}K_{D,j}^{F}-\phi_{C,j}\hat{K}_{j}\right]  .
\end{equation}
The first two terms of this expression capture the total financial payoff to
socially responsible investors, net of the agency cost that is necessary to
incentivize the entrepreneur. The third term captures the (internalized)
change in social cost that results from inducing a firm of type $j$ to adopt
the clean production technology.

Given limited capital, socially responsible investors are generally not able
to reform all firms. They should therefore prioritize investments in firm
types that maximize the \textit{impact per dollar invested}. This is achieved
by ranking firms according to a variation on the classic profitability index,
the \textit{social profitability index} (SPI). The SPI divides the change in
payoffs to socially responsible investors, $\Delta U_{j}^{SR}$, by the amount
socially responsible investors need to invest to impact the firm's behavior,
$I^{SR}$.\footnote{The change in the payoff to socially responsible investors,
$\Delta U_{j}^{SR}$, is the same across all financing agreements characterized
in Proposition~\ref{Theorem: Scale}. Absent other constraints, it is therefore
optimal for socially responsible investors to choose the minimum co-investment
that implements clean production.}
\begin{equation}
\text{SPI}_{j}=\mathbbm{1}_{\gamma_{j}^{E}<\bar{\gamma}_{j}^{E}}\frac{\Delta
U_{j}^{SR}}{I_{j}^{SR}}. \label{SPI}%
\end{equation}


\begin{proposition}
[The Social Profitability Index (SPI)]\label{prop: SPI} Socially responsible
investors should rank firms according to the social profitability index,
SPI$_{j}$. There exists a threshold $SPI^{*}\left(  \kappa \right)  \geq0$ such
that socially responsible investors with scarce capital $\kappa$ should invest
in all firms for which $SPI_{j}\geq SPI^{\ast}\left(  \kappa \right)  $.
\end{proposition}

According to Proposition \ref{prop: SPI}, it is optimal to invest in firms
ranked by the SPI until no funds are left, which happens at the cutoff
$SPI^{\ast}\left(  \kappa \right)  $. Social capital is scarce if and only if
the amount $\kappa$ is not sufficient to fund all firm types with $SPI_{j}>0$.
The SPI links the attractiveness of an investment by socially responsible
investors to underlying parameters of the model, thereby shedding light on the
types of investments that socially responsible investors should prioritize.

\begin{proposition}
[SPI Comparative Statics]\label{Prop: SPI comparative}As long as $\gamma
_{j}^{E}<\bar{\gamma}_{j}^{E}$, the SPI is increasing in the avoided social
cost, $\Delta \phi:=\phi_{D}-\phi_{C}$, and the entrepreneur's concern for
social cost, $\gamma^{E}$, and decreasing in the financial cost associated
with switching to the clean technology, $\Delta \pi:=k_{C}-k_{D}$.
\end{proposition}

According to Proposition \ref{Prop: SPI comparative}, socially responsible
investors should prioritize in firms for which avoided social cost is
high, as reflected in the difference in social costs under the clean and the
dirty technology, $\Delta \phi$. Because the SPI reflects the relative social
cost, it can be optimal for socially responsible investors to invest in firms
that generate significant social costs provided that these firms would have
caused even larger social costs in the absence of engagement by socially
responsible investors. Of course, the potential to avoid social costs, as
summarized by the $\Delta \phi_{j}$, has to be traded off against the
associated financial costs, as measured by the resulting reduction in profits
$\Delta \pi_{j}$.

Another implication is that, as long as $\gamma_{j}^{E}<\bar{\gamma}_{j}^{E}$,
firms with more socially minded entrepreneurs should be prioritized, because
they require a smaller investment from socially responsible investors to be
convinced to reform. However, as soon as the entrepreneur internalizes enough
of the externalities, so that she chooses the clean technology even if
financed by financial investors (i.e., $\gamma_{j}^{E}\geq \bar{\gamma}_{j}%
^{E})$, the $SPI$ drops discontinuously to zero. Socially responsible
investors should not invest in these firms.

To obtain a closed-form expression for the SPI, it is useful to consider the
special case of $\gamma^{E}=0$ and $\gamma^{SR}=1$. Moreover, while strictly
speaking it is optimal to minimize socially responsible investors' investment
by selling all cash flow rights to financial investors, suppose that socially
responsible investors need to receive a fraction $\lambda_{j}$ of a firm's
cash flow rights. This minimum cash-flow stake pins down $I_{j}^{SR}$. The
assumption of a required cash-flow stake for socially responsible investors
can be justified on two grounds. First, it seems natural that socially
responsible investors cannot rely purely on the \textquotedblleft
warm-glow\textquotedblright \ utility that results from reducing social costs,
but require a certain amount of financial payoffs alongside non-pecuniary
payoffs.
%\footnote{it may be interpreted as the simplest possible way to account for complementarities of financial wealth and concern for social cost in the utility function. That is,In the baseline specification, cash flows and externalities are perfect substitutes.}
Second, the minimum cash flow share $\lambda_{j}$ can be interpreted as a
reduced form representation of the control rights that are necessary to
implement ensure that firm $j$ implements the clean technology.\footnote{This
could be the case because the entrepreneur cannot commit to the adoption of
the clean technology. In this case, a cash-flow stake for socially responsible
investors and blunt the entrepreneur's profit motive
\citep[see][]{ChowdhryDaviesWaters2018} or may allow socially responsible
investors to enforce appropriate technology adoption, for example, via voting
rights.} Given these assumptions, the SPI takes the following simple
expression,%
\begin{equation}
\text{SPI}_{j}=\frac{\Delta \phi_{j}-\Delta \pi_{j}}{\Delta \pi_{j}+\lambda
_{j}\left(  p_{j}R_{j}-\xi_{j}\right)  }. \label{eq:SPIspecial}%
\end{equation}


This expression not only transparently reveals the comparative statics
described in Proposition~\ref{Prop: SPI comparative}, but also highlights the
subtle role of agency costs. Agency costs, through the resultant financing constraints, have two
opposing effects on the SPI. On the hand, an increase in agency costs means that a larger fraction of cash flows needs to go to the entrepreneur under clean production, making it more expensive for socially
responsible investors to finance a scale increase in clean production. On the
other hand, an increase in agency costs also implies that a larger
fraction of the cash flows needs to go to the entrepreneur under dirty
production, which lowers the financeable scale under the dirty technology and,
hence, makes it cheaper for socially responsible investors to induce the
entrepreneur to switch to the clean technology. In our baseline setup, this second effect dominates,
which explains why the SPI given in Equation (\ref{eq:SPIspecial}) is increasing in
$\xi$. In an extension presented in Section~\ref{sec: robustness production technology}, we allow for agency
costs that are technology-specific. In this case, the SPI  decreases in the agency cost under the clean technology and increases in the agency cost under the dirty technology.

To conclude this section, we briefly revisit the complementarity result given
in Proposition \ref{prop: complementarity} in a setting with limited socially
responsible capital. The welfare change relative to the case without socially
responsible investors, $\Delta \Omega,$ results purely from the set of reformed
firms: firms for which $\gamma_{j}^{E}<\bar{\gamma}_{j}^{E}$ and $SPI_{j}\geq
SPI^{\ast}\left(  \kappa \right)  $. We can therefore write the change in
welfare as
\begin{equation}
\Delta \Omega=\int_{j : \gamma_{j}^{E}<\bar{\gamma}_{j}^{E}\And SPI_{j}\geq
SPI^{\ast}\left(  \kappa \right)  }\left(  v_{C,j}\hat{K}_{j}-v_{D,j}%
K_{D,j}^{F}\right)  d\mu(j).
\end{equation}
Clearly, if socially responsible capital is abundant, the results of
Proposition~\ref{prop: complementarity} still apply: Welfare is strictly
higher in an economy with both types of investors than in an economy where all
capital is held exclusively by either financial or socially responsible
investors. In contrast, when socially responsible capital is scarce there is a
trade-off. On the one hand, the set of reformed firms contributes towards
higher welfare, as before. On the other hand, the set of unreformed
\textquotedblleft dirty\textquotedblright \ firms exhibit overinvestment in the
dirty technology due to the presence of competitive financial capital without
regard for externalities. This trade-off implies that the right\emph{ balance}
between socially responsible and financial capital is important for a
complementarity between the two types of capital to arise.

\section{Discussion\label{sec: discussion}}

\subsection{Generalizing the production
technology\label{sec: robustness production technology}}

In our baseline model, we considered the choice between two
constant-returns-to-scale production technologies with identical cash flows
and agency rents. Moreover, we focused on the case, in which externalities of
production are negative for all production technologies. As we show in this
section, these assumptions can be relaxed relatively straightforwardly. In
particular, Proposition \ref{Theorem: Scale} generalizes to multiple
technologies and social goods. Moreover, even when the production technology
exhibits decreasing returns to scale, it remains optimal to reward the
entrepreneur with additional scale as long as financing frictions for the
clean technology are significant.

\paragraph{Many (heterogeneous) technologies and social goods.}

Let us first retain the assumption of constant returns to scale, but
generalize all other dimensions of the available production technologies. In
particular, suppose that the entrepreneur has access to $N$ production
technologies characterized by technology-specific cash flow, cost, and moral
hazard parameters $R_{\tau}$, $k_{\tau}$, $p_{\tau}$, $\Delta p_{\tau}$, and
$B_{\tau}$. The differences in parameters could reflect features such as increased willingness to pay for goods produced by firms with  clean  production technologies, implying $R_{C}>R_{D}$ \citep[see, e.g.,][]{AghionEtAl2019}, or the effects of
market structure (e.g. Bertrand vs. Cournot competition) on the pass-through of increased
 costs to consumers.  In addition, in contrast to the baseline model,
we now allow for the technology-specific social cost parameter $\phi_{\tau}$ to be negative,
in which case the technology generates a positive externality (a social good).

In analogy to the baseline model, we can then define, for each technology
$\tau \in \left \{  1,...,N\right \}  $, the financial value $\pi_{\tau}$, the
agency rent $\xi_{\tau}$, and the maximum scale available from financial
investors $K_{\tau}^{F}$,
%
%\begin{align*}
%\pi_{\tau}  &  :=p_{\tau}R_{\tau}-k_{\tau},\\
%\xi_{\tau}  &  :=p_{\tau}\frac{B_{\tau}}{\Delta p_{\tau}},\\
%K_{\tau}^{F}  &  :=\frac{A}{\xi_{\tau}-\pi_{\tau}},
%\end{align*}
maintaining the assumption that $\xi_{\tau}>\pi_{\tau}$ for all $\tau$, so
that the maximum scale of production is finite. A straightforward extension of
Lemma~\ref{lemma: benchmark} then implies that, in the absence of investment
by socially responsible investors, the entrepreneur chooses the technology
$\bar{\tau}$ such that:%
\begin{equation}
\bar{\tau}=\argmax_{\tau}\frac{\pi_{\tau}-\gamma^{E}\phi_{\tau}}{\xi_{\tau
}-\pi_{\tau}}. \label{tau_bar_general}%
\end{equation}


Equation \eqref{tau_bar_general} shows that even with $N$ general
technologies, the entrepreneur's choice of technology is essentially the same
as in Lemma \ref{lemma: benchmark}, with the exception that the agency cost
$\xi_{\tau}$ is now project specific. Moreover, Equation
\eqref{tau_bar_general} clarifies the entrepreneur's relevant outside option
in the presence of $N$ technologies: In particular, adopting any production
technology dirtier than $\bar{\tau}$ is not a credible threat.

The induced technology choice in the presence of socially responsible
investors $\hat{\tau}$ and the associated capital stock $\hat{K}$ are given
by
\begin{align}
\hat{\tau}  &  =\argmax_{\tau}\frac{\hat{v}_{\tau}}{\xi_{\tau}-\gamma^{E}%
\phi_{\tau}},\label{tau_hat_general}\\
\hat{K}  &  =\left \{
\begin{tabular}
[c]{l}%
$\frac{\xi_{\bar{\tau}}-\gamma^{E}\phi_{\bar{\tau}}}{\xi_{\hat{\tau}}%
-\gamma^{E}\phi_{\hat{\tau}}}K_{\bar{\tau}}^{F}$\\
$0$%
\end{tabular}
\  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \right.
%
\begin{array}
[c]{c}%
\hat{v}_{\hat{\tau}}>0\\
\hat{v}_{\hat{\tau}}\leq0
\end{array}
. \label{K_hat_general}%
\end{align}
These expressions mirror Proposition~\ref{Theorem: Scale}, except that
technology choice and scale in the presence of socially responsible investors
now also depend on the technology-specific severity of the agency problem
$\xi_{\tau}$. Ceteris paribus, a smaller agency problem makes it more likely
that a technology is adopted, both in the presence of financial investors only
 and when there is co-investment by socially
responsible investors (Equations (\ref{tau_bar_general}) and (\ref{tau_hat_general}), respectively).

Whereas the formal expressions are unaffected by whether the externality is negative or positive, there is one
important difference between these two cases. When externalities are negative, a broad mandate (Condition
\ref{C1}) is necessary to ensure that socially responsible investors have impact. A broad mandate reduces the outside option for socially responsible investors (see Equation (\ref{USR outside})), thereby unlocking the required additional financing capacity. In contrast, when the externalities under technology $D$ are positive, $\phi_{D}<0$, the outside option for socially responsible investors is higher under a broad mandate than under a narrow mandate (the outside option is positive under a broad mandate, whereas it is zero under a narrow mandate). Therefore, in the presence of positive externalities, impact is possible and, in fact, more likely to occur under a narrow mandate,  revealing an interesting asymmetry between preventing social costs and encouraging social goods.


The more general technology specification yields some additional insights
about cases that we previously excluded. First, it is possible that for some
industries the cleanest technology also maximizes financial value (e.g.,
because of demand by socially responsible consumers). In this case, there is
no trade-off between doing good and doing well and, hence, socially
responsible investors play no role. Second, the dirty technology may be the
socially optimal technology when cleaner technologies are too expensive. Also
in this case, there is no role for socially responsible investors. Finally, it
is possible that, for some industries, any feasible technology $\tau$ yields
negative social value (i.e., $\hat{v}_{\hat{\tau}}<0$). In this case, the
socially optimal scale is zero and the entrepreneur is optimally rewarded with
a transfer $\hat{c}>0$ to shut down production.

\paragraph{Decreasing returns to scale.}

We now consider the case in which the two production technologies $\tau
\in \left \{  C,D\right \}  $ exhibit decreasing returns to scale. In particular,
suppose that the \emph{marginal financial value} $\pi_{\tau}\left(  K\right)
$ is strictly decreasing in $K$. Then, the first-best scale $K_{C}^{FB}$ under
the (socially efficient) clean technology is characterized by the first-order
condition%
\begin{equation}
\pi_{C}\left(  K_{C}^{FB}\right)  =\phi_{C}.
\end{equation}
Now consider the scenario in which technology $D$ is chosen in the absence of
socially responsible investors, with an associated scale of $K_{D}^{F}$.
Moreover, for ease of exposition, focus on the case $\gamma_{E}+\gamma_{SR}%
=1$, so that socially responsible investors have incentives to implement
first-best scale. The optimal financing agreement that socially responsible
investors offer to induce the entrepreneur to switch to the clean technology
then comprises three cases.

\begin{enumerate}
\item If the financing constraints generated by the agency problem are severe,
so that the maximum clean scale under the benchmark financing agreement with
financial investors lies below a cutoff $\bar{K}$, i.e., $K_{C}^{F} \leq
\bar{K}<K_{C}^{FB}$, the optimal agreement offered by socially responsible
investors rewards the entrepreneur exclusively through an increase in scale
(rather than upfront consumption), as in Proposition~\ref{Theorem: Scale}.
Even with socially responsible investors, the resulting clean scale, $\hat{K}%
$, is smaller than first-best scale (i.e., $\hat{K}\leq K_{C}^{FB}$), with
equality when $K_{C}^{F}=\bar{K}$.

\item If the financing constraints generated by the agency problem are
intermediate, i.e., $\bar{K} < K_{C}^{F} \leq K_{C}^{FB}$,
%$K_{C}^{F}\in \left[  \bar{K},K_{C}^{FB}\right]  $,
the optimal agreement specifies the first-best scale, $\hat{K}=K_{C}^{FB}$. In
this case, it is efficient to increase clean scale up to the first-best level
but no further, since scale above and beyond $K_{C}^{FB}$ would reduce joint
surplus. Inducing the entrepreneur to switch technologies solely through an
increase in scale would require a production scale exceeds the first-best
level $K_{C}^{FB}$. It is therefore optimal to partially compensate the
entrepreneur through an upfront consumption transfer.

\item If financing constraints are mild, so that $K_{C}^{F}>K_{C}^{FB}$, then
financial investors alone would provide funding above and beyond the
first-best scale of the clean production technology (note that this case can
only occur if $\phi_{C}>0$). In this case, the optimal financing agreement
with socially responsible investors ensures that the clean production
technology is run at the first-best scale, $\hat{K}=K<K_{C}^{F}$. To induce
the entrepreneur to switch to the clean technology at a lower scale than
financial investors would finance, the agreement includes an upfront
consumption transfer to the entrepreneur.
\end{enumerate}

The above reasoning shows that, as long as financing constraints are
significant (as in cases 1 and 2), the main insights of the baseline analysis
continue to hold even under decreasing returns to scale: Socially responsible
investors optimally achieve impact by relaxing financing constraints and
increasing scale for the clean technology. Only when financing constraints are
mild (or absent), socially responsible investors optimally achieve impact by
reducing firm investment. Note that this latter case resembles a Coasian
solution in a setting without financing constraints. For example, a downstream
fishery might pay an upstream factory to reduce production
\citep[see][]{Coase1960}.\footnote{Note that in cases 2 and 3, the agreement
would need to explicitly limit the amount of firm investment (and not just
specify the technology). Otherwise, the entrepreneur would find it privately
optimal to convert the upfront consumption into additional firm investment.}

\subsection{Regulation\label{sec: robustness regulation}}

Our analysis so far focused on what socially responsible investors can achieve
in the absence of regulation (or when regulation is not optimally set). While
the design of optimal regulation is beyond the scope of this paper, our
analysis reveals that standard policy interventions (e.g., an outright ban of
dirty production or a Pigouvian tax) generally cannot restore efficiency in
the presence of financing constraints.\footnote{See \cite{HoffmannEtAl2017}
for an analysis of optimal regulation in a related setting with financial
constraints.} In fact, in some situations such interventions can lead to worse
outcomes than no intervention at all.

\paragraph{Banning the dirty technology.}

Suppose regulators could simply ban the dirty production technology. With
clean production as the only feasible technology, this intervention certainly
ensures that the entrepreneur adopts the socially efficient production
technology. However, in the context of our baseline model, the scale of
production under the clean technology would be strictly lower, equal to
$K_{C}^{F}$, compared to the case when dirty production is allowed but
socially responsible investors ensure that the entrepreneur adopts the clean
production technology with scale $\hat{K}>K_{C}^{F}$. Intuitively, the key
ingredient for the additional financing capacity from socially responsible
investors is the (credible) threat of dirty production (see
Proposition~\ref{prop: complementarity}). Banning dirty production eliminates
this threat, so that socially responsible investors are unwilling to extend
financing above and beyond the scale offered by financial investors,
$K_{C}^{F}$. Of course, we do not argue that a production ban is socially
harmful in all scenarios. In particular, if socially responsible investors do
not have enough capital or if their investment mandate does not satisfy
Conditions \ref{C1} and \ref{C2}, they are unable to steer the entrepreneur
towards the socially efficient choice. In this case, production bans can
increase welfare as they may prevent the adoption of the dirty technology by
firms that are not disciplined by socially responsible investors.

\paragraph{Pigouvian taxes.}

Now suppose that the regulator imposes a tax on the social cost generated by
the firm's production (e.g., a tax assessed on the firm's carbon emissions),
resulting in a total tax of $\phi_{\tau}K$ for a firm producing with
technology $\tau$ at scale $K$. Such a tax makes dirty production financially
nonviable. While this prevents dirty production, it reduces or eliminates the
threat of dirty production, resulting in similar effects to a production ban.
However, welfare can be even lower than under a production ban, because the
firm is taxed also on the clean technology, by an amount $\phi_{C}K$, lowering
the maximum feasible scale of clean production below $K_{C}^{F}$.

\subsection{Impact through Risk Premia\label{sec: risk-premia}}

Our framework features universal risk neutrality. As shown, under this
assumption a narrow mandate (i.e., the exclusion of polluting stocks by
socially responsible investors) has no effect in the presence of competitive
financial capital. If financial investors were risk averse, then such
exclusion would have an effect that works \emph{indirectly} through the price
of risk. In particular, equilibrium prices would feature differential risk
premia for clean and polluting stocks
\citep[see, e.g.,][]{HeinkelKrausZechner2001, PastorEtAl2019, PedersenEtAl2019}.
The mechanism for the differential pricing of risk is that the overweighting
of clean stocks by socially responsible requires underweighting of clean
stocks by financial investors, and, hence, generates imperfect risk sharing.
However, \cite{HeinkelKrausZechner2001} conclude that repricing due to
imperfect risk sharing may quantitatively be too small to generate
impact.\footnote{The size of the effect depends on the wealth share of
socially responsible investors and the correlation structure of stocks.
Intuitively, if all stocks were perfectly correlated, then exclusion has no
effects even in a setting with risk aversion since overweighting polluting
stocks does not affect diversification.}

Moreover, even if narrow exclusion generates a modest amount of impact under
risk aversion, our analysis shows that this is typically not the optimal
mechanism to achieve impact. For example, if a polluting firm produces safe
cash flows, then exclusion will not affect the firm's cost of funding and,
hence cannot generate impact. In contrast, an increase in clean scale (in the
presence of financing constraints) or a direct transfer could achieve impact.

\section{Conclusion}

%We have developed a simple model to show under which circumstances socially
%responsible investment can make a difference. Our analysis has shown that
%co-investment by socially responsible investors can have real impact, in the
%sense that it can induce firms to adopt cleaner production technologies, even
%when financial capital is abundant. In addition, our analysis has highlighted
%a complementarity between financial and socially responsible capital, in the
%sense that welfare in the presence of both types of capital can be higher than
%in an economy with only financial or only socially responsible capital. Our
%model delivers a micro-founded investment criterion for scarce socially
%responsible capital, the social profitability index (SPI). In contrast to many
%existing ESG metrics, a key ingredient to the SPI is the counterfactual
%pollution that would have occurred in the absence of engagement by socially
%responsible investors. For example, this can justify investment in companies
%that generate significant social costs (e.g., oil companies or gun
%manufacturers) provided that in the absence of engagement by socially
%responsible investors these companies would have done significantly more
%social harm.
%Q statement:


A key question in today's investment environment is to understand conditions
under which socially responsible investors can achieve impact. For example,
can investors with social concerns influence firms to tilt their production
technologies towards lower carbon emissions? To shed light on this question,
this paper develops a parsimonious theoretical framework, based on the
interaction of production externalities and corporate financing constraints.

Our analysis uncovers the necessity of a \textit{broad mandate} for socially
responsible investors. Given the abundance of competitive financial capital
chasing (financially) profitable investment opportunities, it is not enough
for socially responsible investors to internalize the social costs generated
by the firms they have invested in. Rather, their concern for social costs
must be unconditional---independent of their own investment. This condition
generates both normative and positive implications. From a positive
perspective, our model implies that if current ESG funds lack such a broad
mandate, they cannot have impact. From a normative perspective, it states
that, if, as a society, we want responsible investors to have impact, their
mandate needs to be broad. Moreover, because a broad mandate entails the
sacrifice of financial returns, socially responsible funds need to be
evaluated according to broader measures, explicitly accounting for real impact
rather than focusing solely on financial metrics.

To achieve impact in the most efficient way, it is optimal for socially
responsible investors to relax firm financing constraints for clean
production, thereby enabling a scale increase of clean technology relative to
what financial capital is willing to offer. Doing so generates a
complementarity between financial and socially responsible capital, in that
welfare is generally highest in an economy in which there is a balance between
financial and socially responsible capital.
%The presence of profit-motivated financial capital alleviates underinvestment for a given production technology, whereas socially responsible capital guides firms' towards the socially optimal production technology via co-investment.


From a practical investment perspective, our model implies a micro-founded
investment criterion for scarce socially responsible capital, the
\textit{social profitability index} (SPI), which summarizes the interaction of
environmental, social and governance (ESG) aspects. Importantly, in line with
the broad mandate, the SPI accounts for social costs that would have occurred
in the absence of engagement by socially responsible investors. Accordingly,
it can be optimal to invest in firms that generate relatively low social
returns (e.g., a firm with significant carbon emissions), provided that the
potential increase in social costs, if only financially-driven investors were
to invest, is sufficiently large. This contrasts with many common ESG metrics
that focus on firms' social status quo.
%(i.e., on how green the company is at the moment).


To highlight these ideas in a transparent fashion, our model abstracts from a number of
realistic features which could be analyzed in future work. First, our
model considers a static framework, where investment is best interpreted as
new  (greenfield)  investment. In a dynamic
setting, a number of additional interesting questions would arise: How to account for
dirty legacy assets? How to ensure the timely adoption of novel (and
cleaner) production technologies as they arrive over time? Because the
adoption of future green technologies may be hard to contract ex ante, a dynamic theory might yield interesting implications on the issue of control.

Second, our model considers the natural benchmark case where socially
responsible investors are homogeneous. These results can be extended in a
straightforward way if socially responsible investors have the same directional preferences (e.g., to lower carbon emissions), albeit with different intensity. More challenging is the case in which socially
responsible investors' objectives conflict or are multi-dimensional (e.g.,
there is a agreement on the goal of lowering carbon emissions, but disagreement on the social costs
imposed by nuclear energy).

Third, while our analysis is motivated by the large rise in the demand for ESG investments, the regulatory landscape is changing simultaneously.  Regulatory measures that have been discussed include the
taxation of carbon emissions as well as subsidies for investments in clean technology (e.g., subsidized loans for the purchases of electric cars or lower capital requirements for bank loans to clean firms). It would therefore be interesting to understand the conditions under which regulation and impact by socially responsible investors
are substitutes or complements.

Finally, we excluded the possibility that firms may interact (e.g., as part of a supply chain or as competitors). Yet it is
plausible that the financing of a green technology by one firm may impact
other firms (e.g., through cross-firm externalities related to production
technologies or by alleviating or worsening financing constraints). While such
spillovers are beyond the scope of this paper, they would be interesting to
study in follow-up work.

\bigskip

\bigskip

%\newpage

\bigskip

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\section{Proofs}

\noindent \textbf{Proof of Lemma~ \ref{lemma: benchmark}. }The Proof of
Lemma~\textbf{\ref{lemma: benchmark}} follows immediately from the proof of
Proposition \ref{Theorem: Scale} given below. First, set $\gamma^{SR}=0$ (so
that socially responsible investors have the same preferences as financial
investors). Second, to obtain the competitive financing arrangement (i.e., the
agreement that maximizes the utility of the entrepreneur subject to the
investors' participation constraint) one needs to choose the utility level of
the entrepreneur $u$ in (\ref{utility_SR}) such that $\hat{v}_{\tau}K_{\tau
}\left(  u\right)  -u=0$.\footnote{Note that $\hat{v}_{\tau}=\pi_{\tau}%
-\gamma^{E}\phi_{\tau}$ in the special case when $\gamma^{SR}=0$.} \bigskip

%\noindent \textbf{Proof of Corollary \ref{cor: benchmark}.} The entrepreneur
%payoff-prefers the clean technology if $\pi_{C} - \gamma^{E} \phi_{C} >
%\pi_{D} - \gamma^{E} \phi_{D}$, which is the case when $\gamma^{E} >
%\widetilde{\gamma}^{E} := \frac{\pi_{D} - \pi_{C}}{\phi_{D} - \phi_{C}}$. The
%entrepreneur adopts the dirty technology whenever $(\xi-\gamma^{E}\phi
%_{D})K_{D}^{F} > (\xi-\gamma^{E}\phi_{C})K_{C}^{F}$, which is the case when
%$\gamma^{E} < \bar{\gamma}^{E} := \frac{\xi(\pi_{D}-\pi_{C})}{\phi_{D}(\xi
%-\pi_{C})-\phi_{C}(\xi-\pi_{D})}$. To show that $\widetilde{\gamma}^{E}<
%\bar{\gamma}^{E}$, rewrite $\bar{\gamma}^{E} = \frac{\pi_{D} - \pi_{C}}%
%{\phi_{D} - \phi_{C} + \frac{\phi_{C} \pi_{D} - \phi_{D} \pi_{C}}{\xi}}$ and
%note that $\pi_{C} - \phi_{C} > \pi_{D} - \phi_{D}$ and $\xi>0$ imply that
%$\frac{\phi_{C} \pi_{D} - \phi_{D} \pi_{C}}{\xi}<0$, so that $\widetilde
%{\gamma}^{E} < \bar{\gamma}^{E}$. \bigskip


\noindent \textbf{Proof of Proposition \ref{Theorem: Scale}. }The Proof of
Proposition~\ref{Theorem: Scale} will make use of Lemmas \ref{Lemma: IRF} to
\ref{Lemma: Technology}.

\begin{lemma}
\label{Lemma: IRF}In any solution to Problem~\ref{Problem: SR}, the IR
constraint of financial investors, $pX^{F}-I^{F}\geq0$ must bind,
\begin{equation}
pX^{F}-I^{F}=0. \label{IRF}%
\end{equation}

\end{lemma}

\begin{proof}
The proof is by contradiction. Suppose there was an optimal contract for which
$pX^{F}-I^{F}>0$. Then, one could increase $X^{SR}$ while lowering $X^{F}$ by
the same amount (until (\ref{IRF}) holds). This perturbation strictly
increases the objective function of socially responsible investors in
(\ref{obj}), satisfies by construction the IR constraint of financial
investors, whereas all other constraints are unaffected since $X=X^{SR}+X^{F}$
is unchanged. Hence, we found a feasible contract that increases the utility
of socially responsible investors, which contradicts that the original
contract was optimal.
\end{proof}

\begin{lemma}
\label{Lemma: IF0}There exists an optimal financing arrangement with
$I^{F}=X^{F}=0$.
\end{lemma}

\begin{proof}
Take an optimal contract $\left(  I^{F},I^{SR},X^{SR},X^{F},K,c,\tau \right)  $
with $I^{F}\neq0$. Now consider the following \textquotedblleft
tilde\textquotedblright \ perturbation of the contract (leaving $K,c$ and
$\tau$ unchanged). Set $\tilde{X}^{F}$ and $\tilde{I}^{F}$ to $0$ and set
$\tilde{I}^{SR}=I^{SR}+I^{F}$ and $\tilde{X}^{SR}=X^{SR}+X^{F}$. The objective
of socially responsible investors in (\ref{obj}) is unaffected since
\begin{align}
p\tilde{X}^{SR}-\tilde{I}^{SR}-\gamma^{SR}\phi_{\tau}K  &  =pX^{SR}%
-I^{SR}+\underset{0}{\underbrace{pX^{F}-I^{F}}}-\gamma^{SR}\phi_{\tau}K\\
&  =pX^{SR}-I^{SR}-\gamma^{SR}\phi_{\tau}K,
\end{align}
where the second line follows from Lemma \ref{Lemma: IRF}. All other
constraints are unaffected since $\tilde{X}^{F}+\tilde{X}^{SR}=X^{F}+X^{SR}$
and $\tilde{I}^{F}+\tilde{I}^{SR}=I^{F}+I^{SR}$
\end{proof}

Lemma~\ref{Lemma: IF0} implies that we can phrase Problem~\ref{Problem: SR} in
terms of total investment $I$ and total repayment to investors $X$ in order to
determine the optimal consumption $c$, technology $\tau$, and scale $K$.
However, to make the proof most instructive, it is useful to replace $X$ and
$I$ as control variables and instead use the expected repayment to investors
$\Xi$ and expected utility provided to the entrepreneur $u$, which satisfy
\begin{align}
\Xi &  :=pX,\\
u  &  :=\left(  pR-k_{\tau}-\gamma^{E}\phi_{\tau}\right)  K+I-pX.
\end{align}
Then, using the definition $\hat{v}_{\tau}:=\pi_{\tau}-\left(  \gamma
^{E}+\gamma^{SR}\right)  \phi_{\tau}\geq v_{\tau}$, we can write
Problem~\ref{Problem: SR} as:

\begin{problem}
\label{Problem: modified}%
\begin{equation}
\max_{\tau}\max_{u\geq \bar{U}^{E}}\max_{K,\Xi}\hat{v}_{\tau}K-u \label{obj2}%
\end{equation}
subject to
\begin{align}
K  &  \geq0\label{Kconstraint}\\
\Xi &  \leq \left(  pR-\xi \right)  K\tag{$IC$}\label{X1}\\
\Xi &  \geq-\left(  A+u\right)  +\left(  pR-\gamma^{E}\phi_{\tau}\right)  K
\tag{$LL$}\label{X2}%
\end{align}

\end{problem}

%\end{customthm}


Here, the last constraint (\ref{X2}) can be interpreted as a limited liability
constraint, since it refers to the constraint that upfront consumption is
weakly greater than zero (using the aggregate resource constraint in
(\ref{resource})). As the problem formulation suggests, it is useful to
sequentially solve the optimization in $3$ steps to exploit the fact that
$\Xi$ only enters the linear program via the constraints (\ref{X1}) and
(\ref{X2}), but not the objective (\ref{obj2}).

As is obvious from Problem~\ref{Problem: modified}, only a technology that
delivers positive surplus to investors and the entrepreneur (i.e., $\hat
{v}_{\tau}>0$) is a relevant candidate for the equilibrium
technology.\footnote{Note that $\hat{v}_{C}$ is unambiguously positive whereas
$\hat{v}_{D}$ could be negative or positive depending on whether the sum
$\gamma^{E}+\gamma^{SR}$ is sufficiently close to $1$.} Now consider the inner
problem, i.e., for a fixed technology $\tau$ with $\hat{v}_{\tau}>0$ and a
fixed utility $u\geq \bar{U}^{E}$ we solve for the optimal vector $\left(
K,\Xi \right)  $ as a function of $\tau$ and $u$.

\begin{lemma}
\label{Lemma: largest scale} For any $\tau$ with $\hat{v}_{\tau}>0\ $and
$u\geq \bar{U}^{E}$, the solution to the inner problem, i.e., $\max_{K,\Xi}%
\hat{v}_{\tau}K-u$ subject to (\ref{Kconstraint}), (\ref{X1}) and (\ref{X2}),
implies maximal scale, i.e.,
\begin{equation}
K_{\tau}\left(  u\right)  =\frac{A+u}{\xi-\gamma^{E}\phi_{\tau}}>0\text{.}
\label{K(u)}%
\end{equation}
The expected payment to investors is:
\begin{equation}
\Xi_{\tau}\left(  u\right)  =\left(  pR-\xi \right)  K_{\tau}\left(  u\right)
\text{.}%
\end{equation}

\end{lemma}

\begin{proof}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{Fig_Polygon.pdf}
\end{center}
\caption{\textbf{Feasible set of the inner problem: } The set of feasible
solutions is depicted in orange and forms a polygon. The objective function is
represented by the red line and the arrow: The red line is a level set of the
objective function of socially responsible investors, and the arrow indicates
the direction in which we are optimizing.}%
\label{Fig: LinProg}%
\end{figure}The feasible set for $\left(  K,\Xi \right)  $ as implied by the
three constraints (\ref{Kconstraint}), (\ref{X1}) and (\ref{X2}) forms a
polygon (see orange region in Figure~\ref{Fig: LinProg}). The upper bound on
$\Xi$ in (\ref{X1}) is an affine function of $K$ through the origin (i.e.,
linear in $K$) whereas the lower bound in Equation (\ref{X2}) is an affine
function of $K$ (with negative intercept $-\left(  A+u\right)  $). The slope
of the lower bound in Equation (\ref{X2}) is strictly greater than the slope
of the upper bound in Equation (\ref{X1}) since
\begin{align*}
\left(  pR-\gamma^{E}\phi_{\tau}\right)  -\left(  pR-\xi \right)   &
=\xi-\gamma^{E}\phi_{\tau}\\
&  >\pi_{\tau}-\gamma^{E}\phi_{\tau}\\
&  >\pi_{\tau}-\left(  \gamma^{E}+\gamma^{SR}\right)  \phi_{\tau}=\hat
{v}_{\tau}>0,
\end{align*}
where the second line follows from the finite scale that is implied by
Assumption~\ref{ass: agency} (i.e., $\xi>\pi_{\tau}$). Therefore, the
intersection of the upper bound (\ref{X1}) and the lower bound in (\ref{X2})
defines the maximal feasible scale of $K$. Choosing the maximal scale
$K_{\tau}\left(  u\right)  $ is optimal, since for any given $\tau$ with
$\hat{v}_{\tau}>0$ and any fixed $u\geq \bar{U}^{E}$, the objective function
$\hat{v}_{\tau}K-u$ is strictly increasing in $K$ and independent of $\Xi$.
The expression for $K_{\tau}\left(  u\right)  $ in Equation (\ref{K(u)}) is
obtained from $\left(  pR-\xi \right)  K=-\left(  A+u\right)  +\left(
pR-\gamma^{E}\phi_{\tau}\right)  K$.
\end{proof}

Given the solution to the inner problem, $\left(  K_{\tau}\left(  u\right)
,\Xi_{\tau}\left(  u\right)  \right)  $, we now turn to the optimal choice of
$u$ which maximizes $\hat{v}_{\tau}K_{\tau}\left(  u\right)  -u$ subject to
$u\geq \bar{U}^{E}$.

\begin{lemma}
\label{Lemma: Pareto}In any solution to Problem~\ref{Problem: modified}, the
entrepreneur obtains her reservation utility $u=\bar{U}^{E}$.
\end{lemma}

\begin{proof}
It suffices to show that the objective is strictly decreasing in $u$. Using
$K_{\tau}\left(  u\right)  =\frac{A+u}{\xi-\gamma^{E}\phi_{\tau}}$ and
$\hat{v}_{\tau}=\pi_{\tau}-\left(  \gamma^{E}+\gamma^{SR}\right)  \phi_{\tau}%
$, we obtain that:
\begin{equation}
\hat{v}_{\tau}K_{\tau}\left(  u\right)  -u=\frac{\hat{v}_{\tau}}{\xi
-\gamma^{E}\phi_{\tau}}A-\frac{\xi+\gamma^{SR}\phi_{\tau}-\pi_{\tau}}%
{\xi-\gamma^{E}\phi_{\tau}}u \label{utility_SR}%
\end{equation}
Since $\xi>\pi_{\tau}$ and $\xi>\gamma^{E}\phi_{\tau}$ (both by
Assumption~\ref{ass: agency}), both the numerator and the denominator of
$\frac{\xi+\gamma^{SR}\phi_{\tau}-\pi_{\tau}}{\xi-\gamma^{E}\phi_{\tau}}$ are
positive, so that Equation (\ref{utility_SR}) is strictly decreasing in $u$.
\end{proof}

Given that $u=\bar{U}^{E}$ the optimal payoff to socially responsible
investors \emph{for a given} $\tau$ is given by:%
\begin{equation}
U^{SR}=\hat{v}_{\tau}K_{\tau}\left(  \bar{U}^{E}\right)  -\bar{U}^{E}.
\label{payoff}%
\end{equation}
We now turn to the final step, i.e., the optimal technology choice.

\begin{lemma}
\label{Lemma: Technology}The optimal technology choice is given by:%
\begin{equation}
\hat{\tau}=\argmax_{\tau}\frac{\hat{v}_{\tau}}{\xi-\gamma^{E}\phi_{\tau}}.
\label{technology}%
\end{equation}

\end{lemma}

\begin{proof}
In the relevant case where $\hat{v}_{D}>0$, we need to compare payoffs in
(\ref{payoff}). The clean technology is chosen if and only if $\hat{v}%
_{C}K_{C}\left(  \bar{U}^{E}\right)  >\hat{v}_{D}K_{D}\left(  \bar{U}%
^{E}\right)  $, which simplifies to (\ref{technology}). If $\hat{v}_{D}\leq0$,
then \ref{technology} trivially holds as only $\hat{v}_{C}>0$.
\end{proof}

Lemmas \ref{Lemma: largest scale} to \ref{Lemma: Technology}, thus, jointly
characterize the solution to Problem~\ref{Problem: modified}, which, in turn,
allows us to retrieve the solution to the original Problem~\ref{Problem: SR}.
That is, substituting the expression for $\bar{U}^{E}$ in Equation
(\ref{eq:UEfin}) into $\hat{K}=K_{\hat{\tau}}\left(  \bar{U}^{E}\right)  $
yields Equation (\ref{K_hat}). Moreover, since (\ref{X2}) binds, we obtain
that $\hat{c}=0$. The aggregate resource constraint in (\ref{resource}) then
implies that total investment by both investors must satisfy $\hat{I}=\hat
{K}k_{\hat{\tau}}-A,$ whereas (\ref{IC}) implies that $\hat{X}=\left(
R-\frac{B}{\Delta p}\right)  \hat{K}.$ Since any agreement must satisfy
$X^{F}+X^{SR}=\hat{X}$ and $I^{F}+I^{SR}=\hat{I}$, we can trace out all
possible agreements using the fact that financial investors break even
(Lemma~\ref{Lemma: IRF}), meaning that $pX^{F}-I^{F}=0$ and $X^{F}\in \left[
0,R\right]  $.\bigskip

\noindent \textbf{Proof of Proposition \ref{prop: complementarity}. }See
discussion in main text.\bigskip

\noindent \textbf{Proof of Proposition \ref{prop: SPI}. }See discussion in main
text.\bigskip

\noindent \textbf{Proof of Proposition \ref{Prop: SPI comparative}. }The social
profitability index is defined as:%
\begin{equation}
\text{SPI}=\frac{\Delta U^{SR}}{I^{SR}}%
\end{equation}
Using Proposition~\ref{Theorem: Scale}, we obtain that the minimum investment
that is sufficient to induce a change in production technology is given by%
\begin{equation}
I_{\min}^{SR}=\left(  \xi-\pi_{C}\right)  \hat{K}-A.
\end{equation}
The corresponding (maximal) SPI is, hence, given by%
\begin{equation}
\text{SPI}_{\max}=\gamma^{SR}\frac{\Delta \phi}{\Delta \pi-\frac{\gamma^{E}}%
{\xi}\left(  \Delta \phi \left(  \xi-\pi_{C}\right)  +\Delta \pi \phi_{C}\right)
}-1
\end{equation}
which is increasing in $\Delta \phi$, $\xi$, and $\gamma^{E}$ and decreasing in
$\Delta \pi$.

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