The Allocation of Socially Responsible Capital

Daniel Green & Benjamin N. Roth*

November 25, 2020

Abstract

A rapidly increasing share of asset allocation decisions incorporate social values in addition

to financial considerations. We argue that the most common strategies for socially motivated

investing, which only consider the social value of the firms in an investors’ portfolio, are mis-

guided. We develop a tractable framework in which commercial and social investors compete,

and identify alternative strategies for social investors that result in higher social welfare and de-

liver higher financial returns. We discuss several normative implications for socially-motivated

investors. From the enterprise perspective, we demonstrate that a focus on increasing profitability

can have a greater social impact than a focus on direct social value creation.

*broth@hbs.edu, dgreen@hbs.edu. We thank Malcolm Baker, Vivek Bhatacharya, Paul Brest, Robert Gertner (discus-sant), Oliver Hart, Dean Karlan, Scott Kominers, Ernest Liu, Ludwig Straub, Mark Wolfson and participants at HarvardBusiness School, Brandeis, and the 2020 Economics of Social Sector Organizations conference for helpful comments.

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1 Introduction

The last several decades have seen an invigoration of investing in companies that rank favorably onmetrics of social value, such as environmental stewardship, social responsibility, and good governancepractices (collectively referred to as ESG). For instance, one quarter of assets under management byprofessional investors–$30 trillion– are now allocated with such considerations (US SIF Foundation,2018). This large shift in investment strategies has the potential to dramatically alter the allocation ofcapital in the economy. In fact, many argue the entire purpose of this movement is to help reallocateresources to socially beneficial uses and away from socially harmful ones. Thus, it is centrally im-portant to understand whether and how this style of investing generates impact. This paper developsa general framework to explore how investing with social convictions results in the creation of socialvalue and which investment strategies generate social value most efficiently.

We focus on “passive” strategies based entirely on selecting which assets to hold or avoid. By farthe most common of such strategies in practice are constructed with attention to the financial returnsand the social value of the companies included in an investor’s portfolio. For example, ESG indexfunds attempt to track the returns of a benchmark index while maximizing some composite measureof the social good of the companies in the portfolio. Proponents of such “values-aligned” investingclaim that they increase the valuation of (or equivalently decreases the cost of capital for) economicendeavors that contribute the most positively to society. This in turn shifts the set of projects thatmarkets will finance towards those that create social value and away from those that destroy it.

We argue that the folk wisdom justifying “values-aligned” investing is misguided, and such in-vestment strategies are an inefficient way to use asset allocation decisions to influence social valuecreation. Our framework builds on the insight that an investor’s true contribution to social value isnot reflected in the social value of the companies in their portfolio, but rather by the additional socialvalue created relative to if the investor did not exist at all (e.g. Brest et al., 2016). The distinctionbetween these perspectives is driven by the fact that many companies that have high social valuecould attract investors with a purely financial objective. Therefore, socially motivated investors whofinance these companies may not be contributing to social value creation. In fact, their behavior couldeven result in social value destruction if it displaces investors unconstrained by social considerationsinto financing socially harmful projects. We formalize this critique in a general equilibrium modelof capital allocation and characterize its implications for the behavior of social investors. We furtheridentify an alternative to “values-aligned” investment strategies that can generate more impact andachieve higher financial returns.

To understand the basic logic of why social investors who attend only to the social value of theirportfolio companies might achieve sub-optimal outcomes, consider the following stylized example.

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Suppose there are two investors each of whom holds one unit of capital. One commercial investorcares only about financial returns and the other social investor cares about both financial returns andsocial value. Suppose further that there are three enterprises, each of whom needs one unit of capitalto survive:

• Firm A generates a 10% profit and 10 units of social value.

• Firm B generates a 8% profit and 5 units of social value.

• Firm C generates a 9% profit and 0 units of social value.

Investors finance companies at prices that ensure them a financial return no greater than the firm’stotal return on investment. A social investor who makes investment decisions based only on their ownreturns and the social value of the companies they finance would want to invest in Firm A. Such aninvestor would be willing to pay more for this opportunity than the commercial investor, leaving thecommercial investor to finance Firm C and earn a 9% return. As a result, the social investor financesFirm A, enjoys a financial return of 9%, and 10 units of social value are created. If instead the socialinvestor took a holistic view, they would appreciate that Firm A is profitable enough to attract thesupport of the commercial investor. The social investor might then want to invest in firm B. In thiscase the social investor would receive a financial return of 8% and 15 units of social value would becreated.

This example highlights that social investors narrowly focused on the social value attributable tocompanies in their own portfolio are not effective at generating social value through their investmentdecisions.1 We develop a framework to embed this logic in a competitive financial market, in whichmany commercial investors and social investors coexist and firms’ costs of capital are determinedin equilibrium. To highlight the nuances arising from the two approaches to social investing in theprevious example, we introduce two types of social investors. Values-aligned social investors formportfolios based on the financial returns and social welfare generated by the companies they support.Sophisticated social investors are similar, but consider implications of their investment decisions fortotal social welfare, not just the social value of firms in their portfolio.

Beyond admitting a tractable analysis of equilibrium behavior, our model yields several normativeimplications for social investors and entrepreneurs. First we demonstrate that capital held by differentclasses of investors has different social values, reflecting the investors’ contribution to social welfare.We define the social value of capital for a particular class of investors as the increase in total socialwelfare associated with marginally expanding the pool of capital held by those investors. Both typesof social investors have socially valuable capital. However, values-aligned investors have a lower so-

1In fact, notice in this example that the social investor’s choice of Firm A results in zero additional social value creationrelative to an economy in which neither investor had preferences for social value.

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cial value of their capital. Relative to sophisticated investors, values-aligned investors over-prioritizecompanies that have both high social impact and high profitability. These companies could attractcommercial investment independent of whether social investors support them. Values-aligned socialinvestors who finance these companies bid up their prices, crowding out commercial investors, whoinstead finance less profitable and socially impactful firms.

While values-aligned social investors do lower the cost of capital for socially valuable projects,their displacement of commercial investors also lowers the cost of capital for, and enables financing of,projects that the values-aligned social investors themselves would not want to support. Sophisticatedsocial investors, by definition, internalize this effect of their investment behavior and do not supportprojects that could have been financed by commercial investors in their absence.

An important implication of this result is that there are improvements to the investment strategy ofvalues-aligned social investors. Any capital that values-aligned investors deploy to fund profitable butalso impactful projects that could have been commercially financed should be redeployed to projectswith lower profitability (and potentially lower social value). Perhaps surprisingly, this can increaseboth the social impact and financial return to this capital. Why? As the result of values-alignedinvestors competing with each other to own shares in certain projects, they are both pushing commer-cial investors into lower impact projects and transferring excessive value to entrepreneurs (or a firm’sexisting owners). Any financial concession made to own a firm that could have attracted commercialinvestment does not serve to expand the set of socially valuable projects that are economically viable.This generates scope to put a financial concession to more effective use. Instead investing directly inprojects that are less profitable but more impactful than what a displaced commercial investor wouldhave chosen can thus result in higher social value creation and higher financial returns to the socialinvestor.

Our baseline model considers an environment where all firms have a fixed scale. In this case,socially conscious investors maximize by investing in firms that could not attract any commercialfinancing. This is not a realistic option for small socially conscious investors, who are typically limitedto making investments through established capital markets. We extend our model to investigate theimplications of socially responsible investing when firms have an intensive margin of scale and showthat in this case there is scope for creating impact. However, this depends on the ability of socialand commercial investors to provide capital at different terms. When there is just a single security forwhich all investors must pay the same price, social investors can only subsidize a firm’s new projects ifthey are the marginal investor in the security. Enabling new projects therefore requires social investorsnot only to finance the new projects, but also to displace existing commercial investors. The socialcapital used to displace commercial investors generates no marginal social impact. We argue thatthe creation of a second security, similar to existing “green bonds,” provides more scope for social

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investors to generate impact. This allows social investors to provide low-cost financing for sociallyvaluable projects without displacing a firm’s existing commercial investors. However, green bonds intheir current form do not necessarily serve this goal, because they do not restrict proceeds to be usedonly to fund investment that is not commercially viable.

Our framework also has implications for evaluating the social impact of a firm, sometimes calledits enterprise impact (Brest et al., 2016). Enterprise impact depends not only on the amount of capitalused by the enterprise, but also on the type of capital used by the enterprise. All else equal, enterprisesthat attract the capital of socially minded investors have lower contribution to social welfare than thosethat attract the capital of purely commercial investors. Holding fixed the social value created by a firm,it can raise its enterprise impact by reducing its dependence on social capital, freeing social capital tofund another enterprise that is unable to obtain commercial financing. The more profitable is a firm,the less likely it is to rely on scarce, socially valuable capital. Our framework thus provides a newconnection between the profitability of an enterprise and its contribution to social welfare.

Finally we consider an extension in which social investors exhibit varying degrees of altruism.When social investors are values-aligned we identify a familiar result: social investors and entrepreneursexhibit positive assortative matching, in that investors who care more about social welfare match toentrepreneurs that create more social value. However when social investors are sophisticated, thisresult partially reverses. Holding fixed the level of an entrepreneur’s profitability, social investorswho value social welfare more highly match to entrepreneurs who create less social value. This resultarises from the fact that social investors have interdependent utility in that they internalize the socialvalue created by all firms that receive financing.

Before reviewing the relevant literature, we make one note about the interpretation of our modeland results. The investment strategies adopted by values-aligned investors in our model strongly mir-ror those of real-world socially conscious investors. There are at least two ways to understand this.First, investors may not care about social welfare creation, but instead derive utility from associa-tion with social value. They may not want to be associated with companies that are heavy pollutersand would be willing to forgo higher returns from investing in those companies, for example, evenif over-weighting low-polluting companies in their portfolio does little to reduce aggregate pollution.According to this interpretation of the motives of social investors, our analysis should be understoodas exploring the positive implications of values aligned versus sophisticated social investment strate-gies on aggregate resource allocation. Alternatively, real-world socially conscious investors could bemotivated by making the world a better place, and therefore would be making a mistake in adoptingvalues-aligned investment strategies. If they knew better, they might prefer to invest in a way thathad a bigger impact on social value and generated higher returns. If investors are indeed making amistake, our model delivers normative suggestions that can help investors improve asset allocation

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decisions. We return to this discussion in our conclusion.

This paper contributes to the literature on investing with social preferences. A number of papersstudy financing environments where social and commercial investors coexist, and ask how social in-vestors should behave to maximize their impact. Oehmke and Opp (2019) and Landier and Lovo(2020) both study activist social investors who aim to resolve a moral hazard problem amongst en-trepreneurs. The two papers investigate how the amount of social and commercial capital influencesthe bargaining power of social investors and their resulting social impact. Broccardo et al. (2020)highlights that if activist investors all vote as if they are pivotal, well-diversified investors with even asmall concern for social welfare will vote in line with the social planner. Like in our analysis, thesepapers consider social investors who care about social value independently of if it was created by thecompanies they support. In contrast to these papers, we study passive investors in a complete infor-mation environment, whose goal is to enable new projects by offering cheaper capital to firms withsocially valuable projects.

Pedersen et al. (2019) and Pastor et al. (2020) study social and commercial investors within aMarkowitz framework, derive the optimal portfolios for each class of investors and shows an ESG-adjusted CAPM emerges. The social investors in their framework maximize a combination of finan-cial return and the impact of firms in their portfolio, and hence correspond to our values-aligned socialinvestors.2 Heinkel et al. (2001) demonstrates that when socially motivated investors divest from so-cially unproductive firms their stock price declines, as the remaining investors hold more concentratedportfolios. In contrast to these papers, we study a model without uncertainty, and focus our analysison the behavior of sophisticated social investors who aim to maximize social welfare rather than theimpact of their own portfolio.

Several papers analyze the behavior of individual firms and their prosocial investors. Focusing onthe single investor case, Chowdhry et al. (2019) and Roth (2020) analyze when a socially mindedinvestor can have more impact through an investment in a social enterprise than they can througha grant. Hart and Zingales (2017) fleshes out several cases for a stakeholder view of the firm, inwhich management’s objective encompasses more than profits alone, and considers arrangements bywhich activist shareholders can induce the firm to take socially efficient actions. Morgan and Tum-linson (2019) argue that corporate social responsibility might be a vehicle to overcome collectiveaction concerns amongst a firm’s prosocial shareholders, and argues that corporate social responsi-bility might be an efficient channel for prosocial actions when the firm’s production imposes societal

2Pastor et al. (2020) considers an extension in which investors incorporate the social value of all firms into theirobjective function, but the analysis assumes that individual investors cannot meaningfully influence the economy andhence their preferences for financial return and the social value of firms within their own portfolio fully determine theirinvestment decisions. In contrast our sophisticated social investors value their own financial return on the same order astheir contribution to total social value, and hence the fact that they partially internalize the value created by all firms doesinfluence their decisions. We expand on this discussion in Section 2.1.

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externalities. This latter point is also highlighted in Nilsson and Robinson (2017). Dewatripont andTirole (2020) study how competition affects the degree to which firms’ behaviors reflect the ethicalconcerns of their stakeholders.

Our study also relates to the broader economic literature on altruistic motives. Andreoni (1990)highlights the distinction between “pure altruists,” who derive utility from social welfare, and “impurealtruists” who derive utility, or “warm glow” from having directly improved social welfare. In thislight, our sophisticated social investors can be understood as pure altruists, and our values-alignedsocial investors can be understood as impure altruists.

Our analysis bears a technical resemblance to assignment matching models, commonly employedin trade and labor economics (e.g. Roy, 1951, Becker, 1973, Sattinger, 1979, Costinot and Vogel,2010). We contribute to this literature by providing a model in which agents sort along two dimensionsof heterogeneity, as in Gola (2020), and by studying an environment where one side of the market hasinterdependent utility in the sense that they care not only about their own match but also the matchesof others. As discussed in Section 5, we show that this latter feature can partially reverse the classicresult of positive assortative matching.

The rest of the paper proceeds as follows. In Section 2 we outline the model for the case whereentrepreneurs have binary projects. In Section 3 we analyze the model separately for the case whereall social investors are values-aligned and for the case where all social investors are sophisticated. InSection 4 we study a market in which both types of social investors coexist. In Section 5 we considerinvestors with varying degrees of concern for social welfare. Section 6 concludes.

2 Model

Players, Technology, and Contracts

There is a mass E of entrepreneurs. Each one is endowed with a project that requires one unit ofcapital. If entrepreneur i receives the requisite capital, his project returns πi ∈ R+ profit and wi ∈ R“social value,” where πi and wi represent the private and social return of the project respectively.3 Weassume that πi and wi have atom-less and full-support distribution with CDF F , and that the featuresof each project are perfectly observable to all players.

There is a mass C of commercial investors and a mass S of social investors, each of whom controlsone unit of capital. We assume that C+S < E so that some projects go unfinanced.

3We assume that wi encompasses the full social return of the project, including the private return πi, as well as anyconsumer and employee surplus and externalities arising from the project.

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A contract between some investor and an entrepreneur i specifies the transfer of one unit of capitalfrom the investor to the entrepreneur at a price pi ≥ 1. The investor receives financial return pi ontheir invested capital, the entrepreneur receives πi− pi, and wi social value is created. Because weare studying a complete information environment without contracting frictions, this contract can beunderstood as either debt of equity.

Preferences

Index investors and entrepreneurs such that investor i matches with entrepreneur i. Each en-trepreneur’s utility is their share of the profit, (πi− pi)di and each commercial investor’s utility istheir return on capital pidi. We will separately examine two classes of social investors.

Values-aligned social investors make investment decisions based on financial returns they receiveand the social value created by the entrepreneur they’ve financed. That is, their utility is

(pi +θwi)di,

where θ represents the strength of investor i’s social preference.

Sophisticated social investors receive utility from their income and from the total social valuecreated by all entrepreneurs who receive financing. That is, their utility is

pidi+θ∫

j∈Ēw jd j = (pi +θwi)di+θ

∫j∈Ē\i

w jd j,

where Ē is the set of entrepreneurs who receive financing. We discuss the formalization of thesepreferences in Section 2.1. We can observe that the difference between the utility functions of values-aligned and sophisticated social investors is that values-aligned investors do not derive utility fromthe social output generated by entrepreneurs they do not finance, while sophisticated social investorsderive utility equally from all social output regardless of who financed it.

Values-aligned investors do not fully internalize the implications of their investment decision onsocial welfare. This does not imply that the preferences of values-aligned investors are incorrect. Asdiscussed in the introduction, values-aligned investors may derive intrinsic utility from owning firmsthat create social value, similar to the conception of warm-glow altruists in Andreoni (1990). In sucha case, the analysis to follow should be understood as exploring the positive implications of these twomodes of investment behavior. Alternatively values-aligned preferences may represent the behaviorof socially conscious investors, while sophisticated social preferences may more faithfully representthe intentions of socially-conscious investors to affect social change. Under this interpretation our

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analysis of the behavior of sophisticated social investors offers normative guidance to real-worldinvestors with social preferences. We return to this discussion in our conclusion.

Timing of Actions

The central focus of our analysis is to understand how social investors make decisions given thatsome entrepreneurs can receive commercial financing and some cannot. Therefore we study a modelin which entrepreneurs’ “outside option” is to seek capital on the commercial market if they cannotattract capital from social investors. This is relevant not only for determining the price of equity thatsocial investors must offer, but also for the determination of which projects could attract financingwithout the intervention of social investors. We follow Oehmke and Opp (2019) and Broccardo et al.(2020) in assuming that commercial investors make their decisions after observing those of socialinvestors.

Specifically, we study a two-period model. In period 1 each social investor offers a contract toan entrepreneur. In period 2 each commercial investor observes the period 1 actions and then offersa contract to an entrepreneur. Entrepreneurs who have received at least one contract then choose atmost one contract to accept, and payoffs are realized.

Equilibrium

The solution concept is Subgame Perfect Equilibrium. All investors choose contracts that aremutual best responses. Among other things this implies that no entrepreneur ever receives offersfrom more than one investor. Therefore we maintain the convention that entrepreneur i matches withinvestor i. Further, we adopt the convention that an entrepreneur i who receives no offers for financinghas a price of equity pi = πi.

Social Welfare

Our measure of social welfare is W =∫

i∈Ē wi , where Ē is the set of entrepreneurs that receivefinancing. Our interpretation is that wi is the total social value created by firm i if it receives financing,including the value to the firm’s owners.4 Sophisticated social investors can therefore be understoodto be maximizing a modified variant of social welfare that increases the weight placed on their ownconsumption.

4Under this interpretation, the value accruing to the firm’s owners is determined independently of how ownership isdivided, i.e. the welfare weights placed on entrepreneurs and investors are the same.

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Note that we do not adopt a social welfare function that sums over the utility of entrepreneursand investors. Doing so would induce a standard analysis of public good provision, wherein none ofthe investors we consider invest sufficiently in businesses that produce high wi because they do notinternalize the benefit accruing to other investors. Instead, we adopt the convention that the socialplanner cares about value creation wi, but does not care about the intrinsic “altruistic” utility thatsocial investors derive from the creation of wi. In this sense we follow Hart and Zingales (2017) andBroccardo et al. (2020).

2.1 Discussion of the Preferences of Sophisticated Social Investors

Sophisticated social investors differ from values-aligned social investors in that they value total so-cial value rather than privileging the social value of the firms that they support. Because we studya model with a continuum of projects, social investors cannot influence aggregate social welfare∫

Ē w jd j. An intuitive formulation of the utility of sophisticated social preference – which we donot employ – would be

pi +θ∫

Ēw jd j (1)

Because a single investor cannot influence aggregate social welfare, social investors with the abovepreferences would single-mindedly optimize their financial return.5 To ensure that sophisticated socialinvestors do not behave this way, we employ a different modeling approach.

Formally, we model the preferences of sophisticated social investors as arising from the limit of asequence of discrete models, each of which has a finite but increasing number of projects n that canbe financed. In each of these models we assume that sophisticated social investors have utility

1n

(pi +θ ∑

j∈Ēw j

)=

1n

(pi +θwi)+θ ∑j∈Ē\i

w j

. (2)As the number of projects financed increases, the contribution of investor i to social welfare as afraction of total social welfare vanishes, yet so does the amount that she values her own financialreturn. These preferences might be understood to represent the fact that a sophisticated social investorplaces the same relative value on her own financial return and the welfare of a fixed set of others nomatter how large is the set of total financed projects.

In contrast, the social preferences represented by Equation 1 can be understood as arising from the

5Indeed, this aligns with Pastor et al. (2020), which briefly considers social investors who have preferences for aggre-gate social value and concludes that this preference does not influence investor behavior.

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continuous limit of the same set of discrete models, in which social investor preferences are

pi +1n

θ ∑j∈Ē

w j.

This would correspond to the idea that relative to her own financial return, a social investor places lessvalue on the welfare of a fixed set of others, as the set of all financed projects grows.

For the remainder of the analysis, we adopt the preferences in Equation 2. In the continuous limit,we represent these preferences as

pidi+θ∫

Ēw jd j.

3 Equilibrium Structure with Just One Type of Social Investor

To understand the behavior of values-aligned and sophisticated social investors we first characterizethe equilibrium the model in which all investors are either values-aligned or sophisticated. In Section4 we present our main results in the model in which both types of social investors coexist.

3.1 Values-Aligned Social Investors

We begin by characterizing the equilibrium of the model where all social investors are values-aligned.

In the period 2 commercial market, investors’ returns must be equalized across all entrepreneurswho receive financing. That is, prices offered to any two entrepreneurs i and j who are both supportedby a commercial investor satisfy pi = p j ≡ π̄ . Further, for any entrepreneur k who is not, it must beeither that πk ≤ π̄ , or that they have already received an offer from a social investor with price pk ≤ π̄ .Entrepreneurs who do not receive support in period 2 are those who cannot offer sufficiently highfinancial return to commercial investors.

In period 1 prices offered to any two entrepreneurs i and j who are both supported by a socialinvestor satisfy pi +θwi = p j +θw j.6 And for an entrepreneur i supported by a social investor andan entrepreneur k who is not, prices must satisfy pi +θwi ≥ pk +θwk.

6The preceding equality holds so long as prices are finite. In equilibrium a social investor may provide funding inexchange for zero share of the proceeds (p = 0) if the project has sufficiently high social impact (akin to philanthropy).In such a case, the above equality need not hold.

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𝜋

𝑤ഥ𝑤

ത𝜋

𝐶

𝑆

Figure 1: Equilibrium investment with values-aligned social investors

With the above pricing equations we can now characterize the set of entrepreneurs financed byeach type of investor in equilibrium. The equilibrium investment allocation is depicted in Figure 1.

Importantly, commercial investors only invest in projects with social value below some w̄ and prof-its above some π̄ determined in equilibrium. Marginally profitable projects remain unfinanced, andbecause commercial investors could finance them in exchange for 100% of their proceeds, commercialinvestors earn returns of π̄ .

Because commercial investors support all entrepreneurs with profit higher than π̄ and with socialvalue lower than w̄, π̄+θ w̄ is the social investor’s effective outside option utility. Prices in equilibriumare set such that social investors achieve this outside option utility. Social investors receive financialreturn of π̄ − θ (wi− w̄). Thus, they are willing to pay (in terms of reduced financial return) forprojects that generate high social value. Commercial investors do not find it attractive to invest incompanies with πi > π̄ and wi > w̄ precisely because social investors are willing to invest in thesecompanies at higher prices.

A formal characterization of equilibrium investment allocations is relegated to the Appendix Sec-tion B.1.

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𝜋

𝑤𝑤′

𝜋 𝐶

𝐶

𝑆

Figure 2: Equilibrium investment with sophisticated social investors

3.2 Sophisticated Social Investors

Next we analyze the equilibrium allocation of capital when social investors are all sophisticated.The equilibrium allocation is depicted in Figure 2.

Similar to the case with values-aligned social investors, there is an equilibrium level of profit π̃below which commercial investors do not fund projects, and all commercial investors receive returnsof π̃ .

In equilibrium, social investors and entrepreneurs in period 1 expect that those entrepreneurs withprofits above π̃ can receive financing and earn πi− π̃ from a commercial offer in period 2.7 There-fore, sophisticated social investors recognize that by supporting an entrepreneur i with profits πi ≥ π̃ ,their marginal contribution to social welfare is not wi, but instead the social value corresponding tomarginally expanding the set of entrepreneurs supported by commercial investors.

We next determine the price offered to any entrepreneur supported by a social investor.Lemma 1. For any entrepreneur i supported by a social investor, pi = π̃ if πi ≥ π̃ and pi = πi else.

Lemma 1 dictates that if a social investor were to support an entrepreneur with profit πi ≥ π̃ ,they must offer contracts that allow the entrepreneur to retain πi− π̃ of their profit, as this is theprofit the entrepreneur could achieve from a commercial investment. In contrast, social investors whosupport entrepreneurs with πi < π̃ can extract the entrepreneur’s full profits. These entrepreneursexpect not to be able to attract commercial financing in period 2. Further, because sophisticated

7Note that this holds even if in period 1 there are more than a mass C of entrepreneurs who meet this criterion, sinceall players expect that in period 2 a mass of exactly C will meet this criterion.

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social investors recognize that they cannot contribute to social welfare by undercutting another socialinvestor, and because in equilibrium all social investors prefer their investment to earning π̃ andmarginally expanding the pool of commercial investments, no social investor will undercut anotherwho is supporting an entrepreneur with profits πi < π̃ .

In equilibrium, sophisticated social investors do not invest in any projects that generate at least π̃profit, as they anticipate that all of these projects will receive commercial financing. Instead, theyinvest in the mass S of entrepreneurs i who have the highest πi + θwi, amongst those that generateprofits πi < π̃ (so that they would not receive commercial financing). This is depicted in Figure 2.In Appendix Section B.2, we formally characterize the allocation of sophisticated social investor’scapital.

3.3 Comparison of the Two Equilibria

Before analyzing the model where sophisticated and values-aligned social investors coexist, wecompare social welfare across the regimes with only one type of social investor. Recall, our measure ofsocial welfare is W =

∫i∈Ē wi , where Ē is the set of entrepreneurs that receive financing in equilibrium.

Proposition 1. Social welfare is higher when all social investors are sophisticated than when allsocial investors are values-aligned.

Though all social investors place weight θ on social value, sophisticated social investors createmore social value in equilibrium. This is because values-aligned social investors prioritize investmentin entrepreneurs that have high social value wi even if they could also attract investment in the com-mercial market. The marginal contribution of these investments to social welfare is not wi but ratherthe social value created by the marginal project financed by commercial investors. In other words, bysupporting an entrepreneur i that could have attracted commercial financing, the investor’s marginalcontribution to social welfare is not wi but rather the social value created by firm that the displacedcommercial investor goes on to support. As commercial investors focus single mindedly on financialreturns, the marginal commercial investment has relatively low social value.8

In contrast, sophisticated social investors focus on the set of entrepreneurs who could not attractfinancing on the commercial market. Relative to values-aligned social investors, sophisticated socialinvestors may support entrepreneurs with lower contributions to social welfare. But the sophisticatedinvestor’s contribution to social welfare is higher, as they are not displacing a commercial investorwho would have supported the firm in their absence.

8Note, this does not require any assumption about the joint distribution of profits and social value amongst projects inthe population. Instead we are observing that in equilibrium, the marginal projects that commercial investors support havelower social value than the projects that social investors support, as social investors place positive weight on social valuewhen making their investment decision.

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𝜋

𝑤ഥ𝑤

ത𝜋

𝐶 𝑆𝑁

𝑆𝑆

Figure 3: Equilibrium investment with both types of social investors

4 Main Results

In this section we discuss a number of normative results about social investing, in a market in whichboth sophisticated and values-aligned social investors coexist. In particular, we demonstrate that thereare investment strategies that deliver higher financial returns and create more social value than thoseemployed by values-aligned social investors. And we draw a new link between the profitability of afirm and the firm’s social value.

First we characterize the equilibrium in the market with both types of social investors.

4.1 Equilibrium Structure with Both Types of Social Investors

Let there be a mass SN of values-aligned social investors, and SS sophisticated social investors, sothat S≡ SN +SS.

There is no longer a unique equilibrium in this market. Commercial investors’ behavior is stilluniquely pinned down as a function of period 1 actions. But social investors may follow multipleinvestment profiles in equilibrium leading to different allocations of capital. In Appendix SectionB.3 we characterize the full set of equilibria. In this section we focus on the unique, welfare-optimalequilibrium, depicted in Figure 3.

As in Section 3.1 there is some w̄ such that values-aligned social investors only support en-trepreneurs with social value greater than w̄ and commercial investors only support entrepreneurs

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with social value less than w̄. Moreover, as in Section 3.1, there exists a π̄ such that entrepreneurswith profits higher than π̄ and social value less than w̄ are supported by commercial investors.

To characterize the behavior of sophisticated social investors, the following analogue of Lemma 1holds.Lemma 2. For any entrepreneur i supported by a sophisticated social investor,

if πi ≥ π̄ and wi ≤ w̄, pi = π̄ ,

if min{πi, π̄}+θwi ≥ π̄ +θ w̄ then pi solves p+θwi = π̄ +θ w̄ if such a p exists and pi = 0 if nosuch p exists

and pi = πi else.

The price that a sophisticated social investor pays is disciplined by the commercial market if the en-trepreneur could have attracted commercial financing, and by the values-aligned social investors if theentrepreneur could have attracted financing from values-aligned social investors. Else, a sophisticatedsocial investor can demand an entrepreneur’s entire profit, as in Section 3.2.

The above lemmas apply across all equilibria. In the welfare-optimal equilibrium sophisticatedsocial investors do not compete with either commercial investors or values-aligned social investors,and instead support the set of entrepreneurs i who maximize πi + θwi amongst those who couldnot attract financing from other investors. We defer formal characterization of this equilibrium toAppendix B.3. For the remainder of our analysis we focus on the welfare-optimal equilibrium toilluminate the model’s comparative statics.

4.2 The Social Value of Capital

Next we discuss the social value of capital held by various investors, defined to be the socialvalue created by marginally expanding a given pool of capital. Let W (C,SN ,SS) be the total socialvalue created in the investor-optimal equilibrium given masses of investors C, SN , and SS. DefineνC ≡ dW (C,SN ,SS)dC to be the social value of commercial capital, νSN ≡

dW (C,SN ,SS)dSN

to be the social value

of values-aligned social capital, and νSS ≡dW (C,SN ,SS)

dSSto be the social value of sophisticated social

capital. We have the following result.Lemma 3. The social value of commercial capital is less than the social value of values-alignedsocial capital, which is less than the social value of sophisticated social capital: νC ≤ νSN ≤ νSS .

Figure 4 depicts how the equilibrium asset allocation changes as each pool of capital is expanded.The grey lines define the equilibrium prior to adding new capital. Each type of marginal capital has a

16

Expand Naïve Social Investors

Expand Sophisticated Social Investors

Expand Commercial Investors

Figure 4: Expanding the pool of Commercial, Values-aligned Social, and Sophisticated Social Capital

different impact on the equilibrium set of funded projects and which investors finance them. The bluelines denote the shift in the equilibrium induced by expanding a particular set of capital.

Increasing the mass of commercial investors C results in the least social value creation, as commer-cial investors disregard social welfare entirely. New commercial investors create social value in twoways. First, they fund previously unfinanced marginally profitable projects that have incidental socialvalue. Second, they bid up the price of claims on profits and displace social investors, who substituteinto higher social value projects.

Increasing the mass of values-aligned social investors SN causes a displacement of both commer-cial and sophisticated social entrepreneurs, effectively expanding the pool of each. This results in alarger increase in social welfare than does directly increasing the mass of commercial investors, asit effectively expands the pool of commercial capital by less and the pool of sophisticated capital bymore.

17

Increasing the mass of sophisticated social investors SS provides more social value than increas-ing the mass of values-aligned social investors SN , as some of the firms that values-aligned socialinvestors support could have attracted commercial capital and therefore they merely serve to expandthe pool of commercial capital. In contrast, expanding sophisticated social capital does not result inthe displacement of either of the other two types of investors.

4.3 Reallocating Values-Aligned Social Capital to Improve Social Welfare andFinancial Returns

In this section we consider two thought exercises. First, holding fixed the equilibrium behavior ofall other investors, we consider the possibility of reallocating the investment of a single values-alignedsocial investor. We demonstrate that generically, any values-aligned social investor who supportsa firm with πi ≥ π̄ could reallocate their capital to increase total social welfare and increase theirfinancial return. In this sense values-aligned investors leave money on the table. We then considerthe possibility of converting a values-aligned social investor into a sophisticated social investor, andshow that this always leads to an increase in social welfare and sometimes (but not always) leads toan increase in the investor’s financial return.Proposition 2. Consider any values-aligned social investor i that supports a firm with πi > π̄ inequilibrium. Generically there exists an unfinanced firm j with profits π j > pi, such that if the values-aligned social investor i were to deviate and offer firm j financing at price π j, total social welfarewould increase as would investor i’s financial return.

The logic of Proposition 2 can be understood with reference to Figure 5. Fix any values-alignedsocial investor i that supports a firm i with profits πi ≥ π̄ and who earns financial return pi < π̄(generically this holds for all values-aligned investors who support firms with πi≥ π̄). These investorssupport the firms highlighted in blue. And consider among the set of unfinanced firms some firm jwith profits π j > pi and with social value w j greater than the social value of the marginal firm receivingcommercial support. This firm is guaranteed to exist by the assumption that the distribution of firmshas full support, and one such firm is highlighted in green.

The contribution to social welfare of the equilibrium investment for investor i is low, regardless ofthe social value wi of firm i, as investor i is merely displacing commercial investment. Firm j createsless social value than any firm in the blue region of the diagram, but by reallocating investor i’s capitalto firm j social welfare increases, as firm j creates more social value than the marginal firm no longersupported by a commercial investor.

18

!

"#"

$!

Figure 5: Reallocating Values-aligned Social Capital Out of Equilibrium

Further, by offering firm j a cost of capital p j = π j, investor i can earn higher financial returnas well. As with social value, firm j earns lower profits than any firm in the blue region of thediagram. But social investor i can demand the full profit of firm j in exchange for financing, whereasthe price that values-aligned investor i offered to firm i was disciplined by competition with othervalues-aligned social investors. Values-aligned social investors compete up the prices of firms withlarge contribution to social value even when these firms could have attracted commercial financing.The financial compromise made by values-aligned investors to support such firms results in a transferof wealth to the entrepreneur rather than expanding the pool of socially valuable firms. In contrast, thefinancial compromise made to support a firm that could not attract commercial financing goes entirelytoward expanding the pool of socially valuable firms rather than transferring rents to entrepreneurswhose projects would anyway have been feasible.

Proposition 2 demonstrates that values-aligned social investors leave money on the table in thesense that, relative to the firms these investors support, there are unfinanced firms that could deliverhigher financial returns and increase social welfare. However, while it is straightforward to showthat converting values-aligned social investors to sophisticated social investors would result in highertotal social welfare, in general we cannot guarantee that this conversion would lead investors to earnhigher financial returns. The simple reason is that once values-aligned investors have been convertedto sophisticated social investors, while they would prefer to finance firms in the green region of Figure5 relative to any firm in the blue region, there may be yet another firm j′ which contributes more tosocial welfare but has lower financial return than firms in the green region. Nevertheless, we candemonstrate the following result.Proposition 3. In the welfare-optimal equilibrium, there exists a set of values-aligned social investors

19

𝜋

𝑤ഥ𝑤

ത𝜋

𝐶 𝑆𝑁

𝑆𝑆

Figure 6: Converting Values-aligned to Sophisticated Investors

such that were they to be converted to sophisticated social investors they would earn higher returns.

Moreover total social welfare would increase.

The logic of Proposition 3 is depicted in Figure 6. We identify a mass of values-aligned socialinvestors, shaded in blue, who are supporting entrepreneurs that satisfy two properties.

1. These entrepreneurs could attract commercial capital at market rates, i.e. πi ≥ π̄ ,

2. These entrepreneurs have very high contribution to social value wi, so that values-aligned socialinvestors make a large financial compromise to support them.

By converting these values-aligned social investors to sophisticated social investors, they instead sup-port the firms shaded in yellow. As above, this results in an increase in social welfare, as the firmsin yellow could not have attracted commercial capital, and have contribution to social value higherthan rC. Moreover, by identifying values-aligned social investors who were making a sufficientlylarge financial compromise to support firms i with high wi, we can guarantee that the newly convertedsophisticated social investors earn higher profits, as these sophisticated social investors can demandthe full profits of the firms they support.

Finally, we note that while converting these values-aligned social investors to sophisticated socialinvestors increases their profits and total social welfare, it does not increase their utility judged ac-cording to the utility function of values-aligned social investors. Nevertheless, Proposition 3 offersencouraging news about the prospect of converting values-aligned social investors to sophisticatedsocial investors in practice. A substantial amount of effort has gone into investigating the hypothesisthat ESG investing can increase impact and financial returns (e.g. Eccles et al., 2014), suggesting

20

investors are sensitive to the financial implications of values investing. Our model demonstrates thatrelative to conventional ESG strategies there is room for improvement in this dimension.

4.4 Enterprise Impact

How should one judge the contribution to social welfare of a particular entrepreneur, sometimesreferred to as enterprise impact (e.g. Brest et al., 2016)? On first pass it might seem natural for wi tobe the measure of enterprise impact. However, in Section 4.2 we demonstrated that different socialvalue is attributable to different classes of capital, suggesting that the impact of an enterprise shouldalso reflect the social value of the capital it employs. Along these lines, we define the enterpriseimpact of firm i to be ei ≡ wi− νi where νi is the social value of capital attributable to the investorwho supports entrepreneur i in the welfare-optimal equilibrium. We define the enterprise impact to be0 for firms that do not receive financing.

This definition of enterprise impact might have practical value for socially motivated investorsaiming to quantify the social value of a particular enterprise. Frontier efforts in the impact investingindustry often attempt to account for the social value created by the enterprise and the amount ofcapital employed by the enterprise, such as in the impact multiple of money method (Addy et al.,2019). Our analysis highlights that it is also critical to account for the composition of social capitalversus commercial capital raised by an enterprise in judging its impact.

This definition of enterprise impact also highlights an alignment between the enterprise impactand profitability of a firm. Firms can increase their enterprise impact by increasing their profitabilityeven holding fixed their social value wi. Increasing the profitability of the firm makes it more likelyto attract commercial capital, freeing up capital that is willing to accept lower returns to fund highersocial value endeavors. In particular, we have the following result.Proposition 4. Suppose firm i attracts financing from some investor in equilibrium. Increasing itsprofits πi while holding fixed its social value wi weakly increases its enterprise impact ei and totalsocial value created in equilibrium.

Proposition 4 states that making a firm more profitable increases its enterprise impact even holdingits social value wi fixed. Importantly, this result is not driven by an assumption that a firm’s profitabil-ity and its social value are correlated. Instead, this result is driven by the observation that the moreprofitable is a company, the more likely it is to attract a class of investor whose capital has low socialvalue. Every firm that is successful in attracting financing is effectively crowding out another firmthat could have attracted capital in its absence. The more profitable is a firm, the less socially valuableis the firm that it displaces from being funded.

21

4.5 Impact on the Intensive Margin

So far we have assumed that every firm has a single project, which is completed if and only if itraises a unit of capital. Within this setting, we demonstrated that the sophisticated investing strategycan outperform the values-aligned investment strategy on both total social value creation and financialreturns. However, the sophisticated approach required that investors allocate their capital to firms thatare not commercially viable. In reality, this would effectively relegate sophisticated social investorsto private capital markets, which is likely infeasible for small investors. In this section we consideran extension of the model where firms have a non-trivial intensive margin of scale and demonstratethat sophisticated social investors can have impact by inducing commercially viable firms to changetheir scale of operation. Therefore, there may be room for sophisticated social investors to inducechange in public markets. We present this case as a formal extension to the model in the Appendixand discuss the results briefly here.

We assume that each firm i operates at scale k ∈ N, producing πi (k) profit and wi (k) social value,and we assume that πi (·) and w(·) feature decreasing returns to scale. Though the model can beused to study the behavior of either values-aligned or sophisticated social investors, in this sectionwe restrict attention to sophisticated social investors. We first consider the case where all of a firm’sinvestors must offer capital at the same terms.

For sophisticated social investors to have an impact they must expand (or contract) the scale of afirm relative to if it were financed exclusively by commercial investors. Simply put, to induce firmsto adopt marginal high social value projects, social investors need to offer the firm a lower cost ofcapital. But because all investors provide capital at the same terms, previously marginal commercialinvestors will not find these terms attractive. Social investors would thus need to replace the capitalprovided by these marginal commercial investors, which we have argued in the binary-project caseis a low-impact use of social capital. We further demonstrate that sophisticated social investors arenot only prioritizing firms that have a high marginal social value project, but also firms for which theelasticity of demand from commercial investors is low, so that this displacement effect we highlightis small.9

If sophisticated social investors can provide financing at different terms than commercial investorsthen they can achieve higher social value creation. In this case, social investors can effectively providesubsidized capital to support high social-value marginal projects in a way that does not dilute theclaims of commercial investors. This allows social investors to exclusively finance socially valuableinvestment opportunities that are not profitable enough to be financed by commercial investors.

9In our model, investors are homogeneous and in equilibrium every investor is marginal, so sophisticated investorswould need to displace all commercial investors to impact the firm. Thus, in our model inducing a firm to undertake agiven marginal socially valuable project requires the displacement of less commercial capital at smaller firms.

22

The model outlined in the Appendix shows that the economics of having impact is similar on theintensive and extensive margins. In both cases the issue is adopting an investment strategy that avoidsdisplacing commercial investors. On the extensive margin, this is achieved by supplying capital tocompanies that could not raise it in commercial markets. On the intensive margin, sophisticatedsocial investors can have impact by inducing a change in the scale of commercially viable firms, butonly if they are able to provide subsidized financing.

This highlights the potential for companies to issue a new class of security in order to facilitatesocial impact creation. Importantly, this new security is only impactful if it is used to finance marginalinvestment opportunities that are not feasible to finance with commercial capital. This differs from theexisting green bond market, which raises capital to finance socially valuable investment, regardlessof their commercial viability.10 Thus, the efficacy of green bonds could be improved if a greenbond certification required verification that the project being financed would not create value at itscommercial cost of capital.

5 Extension: Heterogeneous Investor Altruism θ

In this section we consider an extension of the model in which we allow the altruism parameter θto vary across investors. Our aim is to explore the model’s implications with regards to assortativematching. A classic exercise in the assignment matching literature is to identify conditions underwhich agents exhibit positive assortative matching (e.g. Roy, 1951, Becker, 1973, Sattinger, 1979,Costinot and Vogel, 2010, Gola, 2020) -– i.e. when do agents with higher “types” match with oneanother? We demonstrate when social investors in our model are values-aligned, investors with higheraltruism match with entrepreneurs with higher social value for familiar reasons. In contrast, wheninvestors are sophisticated, they exhibit a variant of negative assortative matching. This latter resultarises from the fact that sophisticated social investors have interdependent utility; their utility dependsnot only on the terms of their own match but also on the matches of other investors.

5.1 Model

The model is the same as in Section 2 with the exception that for the set of social investors we nowindex their altruism parameter θi by i, and let it vary across investors. Specifically we assume that thereis a finite set Θ≡

{θ 1, . . . ,θ n

}of potential levels of altruism, with θ j < θ k for 1≤ j < k≤ n. We make

10See VanEck (2020) for background on the green bond market. There is mixed evidence on whether green bondsare valued at a premium to conventional bonds (Baker et al. (2018), Larcker and Watts (2020), and Partridge and Medda(2020).)

23

𝜋

𝑤ഥ𝑤1

ത𝜋

ഥ𝑤2 ഥ𝑤3

𝐶 𝑆1 𝑆2 𝑆3

Figure 7: Equilibrium sorting with values-aligned social investors

no assumption about the distribution of θi. As before we let C be the mass of commercial investorsand we now let Sl be the mass of social investors with altruism parameter θ l . In both the modelwith values-aligned social investors and the model with sophisticated social investors, commercialinvestors can be understood as having an altruism parameter of 0. We maintain all other assumptionsof the model in Section 2.

5.2 Values-aligned Social Investors

We now characterize the equilibrium of the model where all social investors are values-aligned anddemonstrate that social investors and entrepreneurs exhibit positive assortative matching on θ and w.

The period 2 commercial market operates in exactly the same manner as in Section 3.1; there isa level π̄ such that firms that generate profits πi ≥ π̄ and who do not already have investment offerswith pi ≤ π̄ receive commercial financing at price pi = π̄ .

In period 1 prices offered to any two entrepreneurs i and j who are both supported by a socialinvestor with type θ l satisfy pi+θ lwi = p j +θ lw j.11 And for an entrepreneur i supported by a socialinvestor of type θ l and an entrepreneur k who is not, prices must satisfy pi +θ lwi ≥ pk +θ lwk.

With the above pricing equations we can now characterize the set of entrepreneurs financed byeach type of investor in equilibrium. The equilibrium investment allocation is depicted in Figure 7.

11The preceding equality holds so long as prices are finite. In equilibrium a social investor may provide funding inexchange for zero share of the proceeds (p = 0) if the project has sufficiently high social impact (akin to philanthropy).In such a case, the above equality need not hold.

24

𝜋

𝑤

𝜋

𝐶

𝑆1

𝑆2

𝑆3

Figure 8: Equilibrium sorting with sophisticated social investors

Relative to Section 3.1 the principle novelty is that we can now establish assortative matchingin equilibrium. Namely, investors partition the set of entrepreneurs who receive financing such thatinvestors with higher θi match with entrepreneurs who have higher wi. This stems from the fact thatthe utility of investor i is supermodular in θi and wi, and hence social investors with higher altruismhave a higher willingness to pay for projects with high social value. This positive assortative matchingechoes many results in the assignment matching literature cited above. As we will see in the followingsection, this result breaks down, and partially reverses when social investors are sophisticated.

5.3 Sophisticated Social Investors

When social investors are sophisticated there is a multiplicity of equilibria; Figure 8 depicts theinvestment allocation in the welfare-optimal equilibrium. Appendix Section B.5 offers a formal char-acterization of the welfare-optimal equilibrium.

Relative to when social investors are values-aligned, the equilibrium allocation features two impor-tant differences. First, as in Section 3.2, so long as social capital is sufficiently scarce, sophisticatedsocial investors exclusively support firms that could not attract commercial financing. Second, andnovel to this section, positive assortative matching breaks down, even among the set of firms supportedby social investors. In fact, holding fixed a level of profits π ′, social investors exhibit negative assor-tative matching; the higher is the social investor’s altruism parameter θi, the lower is the social valuewi of the firm they support. This negative assortative matching holds despite the fact that the utility ofsophisticated social investors is still supermodular in their altruism parameter θi and the social value

25

wi of the firm they support.12

In equilibrium, in order for a sophisticated social investor i not to deviate and support a firm thatcould have attracted commercial investment it must be that

πi +θiwi ≥ π̃ +θiw′

where π̃ is the equilibrium level of commercial returns and w′ is the social value created by the firmfinanced by the displaced commercial investor. This incentive compatibility condition is easier tosatisfy for social investors with higher altruism parameters. Therefore in the welfare optimal equi-librium, it is the sophisticated social investors who care the least about social welfare that match tothe most impactful entrepreneurs for a given level of profitability, as these are the entrepreneurs whoare most able to entice social investors away from commercial markets. In contrast, sophisticatedsocial investors with higher altruism parameters are willing to forgo commercial returns to supportentrepreneurs with lower contribution to social welfare for a given profit level. And because sophisti-cated social investors derive utility from the social value created by all firms supported in equilibrium,social investors with high altruism parameters do not compete with social investors with low altruismparameters, as they recognize that doing so would not expand social value. Therefore, that positive as-sortative matching breaks down, and partially reverses when social investors are sophisticated arisesfrom the fact that sophisticated social investors have interdependent utilities in the sense that theirutility depends not only on the firm they match to and the price they reach but also on the matchingof other investors and entrepreneurs.

We close this section with one final remark. We view this result as an interesting contribution tothe literature on assignment matching models but are cautious in interpreting it as a normative resultfor social investors, as it relies on a feature of the model that may be our biggest departure from thereal world. Namely, while sophisticated social investors recognize that firms they do not finance canstill search for financing on the commercial market, sophisticated social investors who support firmsthat are not eligible for commercial financing are all pivotal. Consider an equilibrium in which socialinvestor i with low altruism parameter is assigned to support a high social value firm j that cannotattract commercial investment. Because sophisticated social investors all choose their investmentdecisions at the same time, if social investor i deviated to invest in a higher profit firm, then firmj would go unfinanced. However, in a richer model in which social investors make their decisionsdynamically, social investor i might expect that another social investor with higher altruism parameterwould replace them if they were to deviate and leave firm j unfinanced. In such a case, this negative

12All equilibria with the same investment frontier depicted in Figure 8 are welfare-optimal. Therefore, formally, thereexists a welfare-optimal equilibrium such that holding the level of entrepreneur profit fixed, social investors engage innegative assortative matching. But there may be other equilibria with equivalent allocations that do not feature negativeassortative matching.

26

assortative matching result would break down.

The above not withstanding, we conjecture that some variant of negative assortative matchingmight survive other extensions of the model that more closely align with reality. Perhaps most plau-sibly, if social investors’ utility was some combination of values-aligned and sophisticated — i.e.social investors derive utility from both total social value and also especially from the social value ofthe firm they support — then social investors with low altruism who were assigned to support the mostimpactful firms might not be tempted to deviate even if they anticipated that another social investorwould willingly replace them. And social investors with higher altruism assigned to support lowersocial value firms would now face an increased temptation to undercut their competitors financinghigher social value firms, but the knowledge that low altruism investors will only support high socialvalue firms might still deter this kind of competition.

6 Conclusion

This paper provides a new framework understand to how values-based investing generates impact.We consider a model in which investors influence social outcomes only through their asset allocation.Investors’ choices affect the set of companies or projects that are financed in equilibrium. Preciselyhow socially conscious investors think about social value when making investment decisions matters.If investors act as if they only care about the social value generated by companies in which they invest– what we term values-aligned investing –they have limited impact on total social value creation.

Investors following values-aligned investment strategies, which closely resemble the constructionof conventional ESG and emissions reduction portfolios, have limited impact because they displacecommercial investors who do not care about social value creation but would have supported somesocially valuable companies anyway. The impact of socially conscious investors is therefore in partdetermined by the preferences of the investors displaced by the arrival of socially conscious capital.The mechanism for this displacement further constrains their impact. By competing with each otherto invest in high social value companies, values-aligned investors are wasting a financial concessionthat could have been better spent investing in a socially valuable project that could not have attractedcommercial capital. Because of this, we show it is possible for a values-based investor to alter theirinvestment strategy in a way that generates more impact and delivers higher financial returns.

The idea of displacement also has efficiency considerations from a firm’s perspective. By investingin their own profitability, companies are able to reduce their reliance on social capital, freeing it toinvest in other endeavors and thus contributing to social value creation. This represents a new linkbetween a firm’s profitability and its social value.

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An important question not addressed by our analysis regards the true preferences of socially awareinvestors. There has been an explosive growth in investment strategies that resemble the behaviorvalues-aligned preferences in our model, and are thus consistent with preferences for association withsocial value rather than creation of social value. If these are indeed the true preferences of investors,our analysis delivers a positive prediction that preferences for social value association result in onlylimited social value creation.

On the other hand, it is not unreasonable to believe that investors have preferences for social valuecreation, yet, perhaps mistakenly, pursue strategies that are inefficient in generating social value.Investment advisers and mutual fund managers advertising ESG portfolios routinely allude to thepositive impact of their investment strategies. For a few examples of many, Nuveen, an investmentmanagement firm with one trillion dollars in assets under management asserts that ESG investing isthe approach “that is most likely to produce optimal financial and societal outcomes.”13 Candriam, asocially responsible investment fund with C130 billion under management asserts on its webpage thatthey “invest in the future, channeling capital for the common good. ”14 While these quotes indicatethat ESG investment funds aim to attract investors with a direct preference for the creation of socialvalue, Calvert, another provider of ESG mutual funds with $23 billion in assets under management,goes one step further; their website allows investors to calculate the impact of their investment in aCalvert mutual fund across a variety of outcomes.15

To the extent that ESG funds are marketed to investors that have a preference for the creation ofimpact rather than merely being the owner of impactful firms, our analysis offers normative guidance.Namely, we offer investment strategies that dominate typical ESG approaches on both financial returnand the creation of social impact.

13See: https://www.tiaa.org/public/pdf/how_nuveen_uses_responsible_investing_across_asset_classes.pdf Last Ac-cessed: November 27, 2020

14See: https://www.candriam.com/en/private/about-us/ Last Accessed: November 27, 202015In fact, Calvert’s calculation of the impact of an investment conflates measures of social value of portfolio companies

with the impact of an investor in these companies. For example, it reports that a $10,000 investment in Calvert USLarge-Cap Core Responsible Index Fund results in an annual reduction in emissions equivalent to burning 147 gallons ofgasoline. This figure is based on the difference between the value-weighted emissions of constituents in the Calvert fundand the Russell 1000 Index. See: https://www.calvert.com/what-is-your-impact.php Last Accessed: November 27, 2020

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A Model Extension to Firms with Variable Scale

A.1 Model

There is a mass E of entrepreneurs. Each entrepreneur i is endowed with a production function thattakes as an input k ∈ N capital. If entrepreneur i invests k capital, they produce πi (k) ∈ R+ profit andwi (k) ∈ R “social value,” where πi (k) and wi (k) represent the private and social return of the projectrespectively. We assume that πi (·) and wi (·) are perfectly observable to all players. For expositionalclarity we assume that πi (k+1)−πi (k)> πi (k+2)−πi (k+1) for all k > 0 and i, and similarly forwi. Importantly, we do not assume that wi is positive or strictly increasing, so social investors maydesire to reduce the scale of a firm’s operations, for example a heavy polluter.

There is a mass C of commercial investors and a mass S of social investors, each of whom controlsone unit of capital.

A contract between some investor and an entrepreneur i specifies the transfer of one unit of capitalfrom the investor to the entrepreneur at a price pi. Therefore, letting k̄i be the amount of capitalinvested in entrepreneur i’s project, each of entrepreneur i’s investors receive return pi on their investedcapital, the entrepreneur receives πi

(k̄i)− pik̄i financial return, and wi

(k̄i)

social value is created.

Market Clearing Mechanism

We assume that the market is cleared via uniform price auction. Each entrepreneur i reports anintention to raise ki units of capital. Each investor then submits a bid pi ∈

[0, πi(ki)ki

]to some en-

trepreneur i. Entrepreneur i then accepts his ki highest bids.16 Each investor receives a return p̄iwhere p̄i is the ki’th lowest bid that entrepreneur i received.

Preferences

Each entrepreneur i’s utility is their share of the profit,(πi(k̄i)− p̄ik̄i

)di. Each commercial investor

who supports entrepreneur i has utility equal to her return on capital p̄idi. We will separately examinetwo classes of social investors.

Values-aligned social investors receive utility from their financial return and from the averagesocial value per unit of capital created by the entrepreneur they’ve financed. That is, a values-aligned

16Entrepreneurs that fail to raise their goal of ki capital instead raise nothing.

31

social investor who supports entrepreneur i receives utility(p̄i +θ

wi(k̄i)

k̄i

)di,

where θ represents the strength of the investor’s social preference.

Sophisticated social investors receive utility from their income and from the total social value cre-ated by all entrepreneurs who receive financing. That is, a sophisticated social investor who supportsentrepreneur i receives utility

p̄idi+θ∫

jw j(k̄ j)

d j.

As in Sections 3 and 4, the difference between the utility functions of values-aligned and sophisticatedsocial investors is that values-aligned investors do not derive utility from the social value generatedby entrepreneurs they do not finance.

Timing of Actions

As before, we study a two period model. In period 1 each entrepreneur i proposes to raise ki capitalafter which each social investor submits a bid to an entrepreneur. In period 2 each entrepreneur canrevise their target ki, but in doing so relinquishes all bids placed in period 1.17 After observing all prioractions, each commercial investor submits a bid to an entrepreneur. Then the market clears accordingto the process described above.

Equilibrium

Our solution concept is Subgame Perfect Equilibrium. All investors choose contracts that aremutual best responses. When an entrepreneur is indifferent between two capital targets, they choosethe one that maximizes social value.

A.2 Results

For simplicity we focus separately on the case when all social investors are values-aligned andwhen all social investors are sophisticated.

17This is specified for completeness, but in equilibrium no entrepreneur who receives bids from social investors everrevises his target ki.

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Values-aligned Social Investors

The model with values-aligned social investors works much the same way as in the single projectanalysis. There exists a cutoff π̄ such that all projects receive capital at least to the point where themarginal unit of capital generates profits less than π̄ . Values-aligned social investors attend to a firmi’s average social value per unit of capital,

wi(k̄i)k̄i

, in equilibrium. Therefore in equilibrium there

is some cutoff w̄ such that firms i for which p̄i + θwi(k̄i)

k̄i> π̄ + θ w̄ are entirely financed by social

investors, firms that do not clear this hurdle but that yield average profitπi(k̄i)

k̄i≥ π̄ are financed by

commercial investors, and firms that cannot clear either hurdle remain unfinanced. Firms i supportedby social investors raise the smallest amount of capital k̄i such that the next unit would yield(

πi(k̄i +1

)+θwi

(k̄i +1

))−(πi(k̄i)+θwi

(k̄i))

< π̄ +θ w̄

and firms i supported by commercial investors raise the smallest amount of capital k̄i such that thenext unit would yield

πi(k̄i +1

)−πi

(k̄i)< π̄

Therefore, the equilibrium size of a firm is determined by not only the intrinsic properties of thefirm but also by the identity of its marginal investor. Firms that attract social capital have scalepinned down by the equilibrium opportunity cost of a social investor, which is different from that of acommercial investor.

There are two important takeaways from the exposition of values-aligned social investors whenprojects have variable scale. First, the investment strategy of these investors closely mirrors that ofESG investors, for example in public equities. Such investors purchase shares of companies that theybelieve will have high financial returns and whose existing operations are measured in some way tobe socially desirable. In the model, values-aligned social investors make investments to achieve high

financial returns and be associated with companies generating high average impactwk(k̄i)

k̄i, regardless

of whether this impact would have occurred without the social investors’ capital.

Second, an immediate feature of this model is that values-aligned social investors never coinvestwith commercial investors. This stark phenomenon arises because social and commercial investorsdisagree on the relative value of companies with the same profits but different contributions to so-cial value, so there is no price at which both sets of investors would be happy to finance the sameinvestment. Any company financed by social investors is fully financed by social investors. Whilethis extreme separation would not arise in a model with, for example, diversification motives, it illus-trates an important point. Disagreement about the value of a company among investors implies that

33

to change the scale of the company requires displacing some of its existing investors.

This idea is closely related to the observations of Heinkel et al. (2001) and Broccardo et al. (2020),that commercial investors will partially “undo” the actions of social investors, insofar as social in-vestors may partially crowd-out commercial investors in the firms they support. This notion of dis-placement is critical for understanding the behavior of sophisticated social investors in the next sectionand implications for security design in Section A.2.

Sophisticated Social Investors

In this section we examine the behavior of sophisticated social investors and draw out severalnormative implications. The analysis in this section differs qualitatively from the case where projectsare binary in Section 3.2, in that now all projects could potentially attract some level of commercialcapital. Thus, sophisticated social investors may no longer want to avoid investing in projects thatcould have attracted a positive amount of commercial capital. We demonstrate that, unlike values-aligned social investors, sophisticated social investors have a preference to support small firms.

In the welfare-optimal equilibrium, sophisticated social investors allocate their capital to maxi-mize18

max{ki}

∫i∈Ē

{(πi (ki)−qCi +θwi (ki)

)I(ki > 0)+θwi

(kCi)I(ki = 0)

}di

such that ∫kidi = S.

Here, kCi is the amount of commercial capital that firm i could attract in period 2, and letting pC be

the cost of capital in the commercial market, qCi ≡ π(kCi)− pCkCi is the entrepreneur’s anticipated pay-

off from commercial investment. As in the binary project case, Ē represents the set of entrepreneursthat receive a positive level of financing in equilibrium. 19

The term πi (ki)− qCi represents the financial return that sophisticated social investors draw frominvesting k capital in firm i. Namely, they can demand the full return from the project minus whatentrepreneur i could earn on the commercial market. The term θwi (ki) represents the social valuecreated by firm i supported by sophisticated social investors, and for all firms i not supported bysophisticated social investors, θwi

(kCi)

social value is created. Recall from the previous section thatany firm supported by social investors must be wholly supported by social investors, as social and

18The welfare-optimal equilibrium can be found by solving a maximization problem over the joint behavior of allsophisticated social investors. Nevertheless one can readily verify that their behavior is individually optimal in the sensethat it can be supported in subgame perfect equilibrium.

19Note that kCi , pC, qCi , and Ē are all implicitly functions of the allocation of social capital in period 1.

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commercial investors disagree about the value of each firm. Thus, the constraint that∫

kidi = S

represents the sophisticated social investors’ resource constraint; the sum of capital allocated to allfirms that receive social investors’ support must equal the sum of all social investors’ capital.

Let{

kSi}

denote the solution to this maximization problem. Note that in general, kSi may be lessthan kCi . In such a case, social investors are effectively paying entrepreneurs to reduce their scalerelative to their optimum if financed by commercial investors.

Implied by the above maximization problem, in the welfare-optimal equilibrium, for each firm i,sophisticated social investors maximize

vi (k)≡(

πi (k)−qCi +θ(

wi (k)−wi(

kCi)+ γkCi

))−λk (3)

where λ is the Lagrange multiplier on the resource constraint, and γ is the social value of the marginalproject supported by commercial investors in equilibrium.20 Sophisticated social investors support thefirms i for whom maxk vi (k)≥ 0.

This maximization problem captures two ideas. First, relative to values-aligned social investors, so-phisticated social investors care about their marginal contribution to social value, wi

(kSi)−wi

(kCi)+

γkCi , rather than the firm’s total social value wi(kSi), as they recognize that their contribution to social

value corresponds ot the additional social value of firm i relative to if it had received commercialfinancing, plus the social value created by the displaced commercial investors. This parallels thedifference in concerns of values-aligned and sophisticated investors discussed in Section 3.

Second, even though the marginal social value created by sophisticated social investors corre-sponds only to the output resulting from the final kSi − kCi invested capital plus the social value of thedisplaced commercial capital, to achieve this change they have to provide the full amount of capitalkSi , at cost λk

Si . This has important implications for social investors’ choice of which firms to support.

Consider investments in two firms i and j that would achieve the same marginal contribution tosocial welfare and the same marginal profits (for convenience, assume kSi > k

Ci and k

Sj > k

Cj ); i.e.

wi(

kSi)−wi

(kCi)= w j

(kSj)−w j

(kCj)

πi(

kSi)−πi

(kCi)= π j

(kSj)−π j

(kCj)

andkSi − kCi = kSj − kCj .

20Here we are taking kCi ,qCi and γ to be fixed at their levels implied by

{kSi}

. Taking as given the level of socialinvestment for all firms but j, v j (k) can be evaluated at any level of k without moving other equilibrium objects as theinvestment in firm j has mass 0.

35

Then sophisticated social investors prefer to support the firm that requires displacing less commercialcapital, i.e. vi

(kSi)> v j

(kSj)

if and only if kCi < kCj .

21 In this sense sophisticated social investorsprefer to support smaller firms. We codify this logic in the following observation.Observation 1. When there is just one security for which all investors must pay the same price (i.e.the model s