Overinvestment and the Fear of Missing Out∗

Gregory Buchak†

University of Chicago

January 20, 2016

Abstract

I show that investors skip due diligence to quickly fund uncertain projectsat high valuations because they rationally fear missing out. Investorsmay either fund projects immediately or engage in cost-justified buttime-consuming learning to evaluate project quality. Time spent learn-ing puts the investor at risk of losing his investment opportunity, leadingto an inefficiently low level of learning in equilibrium. Inefficient learningleads to overinvestment in unprofitable projects and often underinvest-ment in profitable projects. This offers a rational explanation of boom-like overinvestment and bust-like underinvestment for venture capital,mortgage lending, and mergers along price, quantity, and funding speeddimensions.

∗Thanks to Professors Zhiguo He, Stavros Panageas, William Lin Con, Harald Uhlig,and Casey Mulligan for guidance, and to Yiyao Wang for help with the idea that led tothis paper. Thanks also to Johnathan Loudis, Michael Barnett, Klakow Akepanidtaworn,students from Econ 33603, Winter 2015, and Sarah Kang.†gbuchak@chicagobooth.edu

1 Introduction

“Valuations...” the New York Times reported in 2015, “are inflating, leadingsome people to worry that investment decisions are being guided by somethingventure capitalists call FOMO—the fear of missing out.”1 2015 saw at least73 private technology companies valued above $1 billion, compared to 41 theprevious year. One venture capitalist noted in the Wall Street Journal thatprivate technology firms were receiving “sums of money usually reserved forIPO offerings... with the kind of confidence usually associated with investorswho’ve perused regulatory filings for detailed financial information.”2 Thesestories of high-valuation, low-information venture fundings resemble those ofthe late 1990s tech bubble. Pets.com, an infamous case study of that era,received roughly $130 million in funding despite its investors conducting almostno due diligence regarding the firm’s business viability; the firm was liquidatedless than a year after its IPO.3 In 2015, investors wondered if it was déjà vuall over again. Yet despite the lessons of the late 1990s and the apparentfoolishness of investment without due diligence, investors may not always havea choice: Facing gobs of rival capital looking for investment opportunities, canan investor afford the time spent distinguishing a future Facebook or Amazonfrom a future Pets.com, or will a fear of missing out entirely make uncertainventures attractive even at sky-high valuations?

This paper asks if investors will skip due diligence to quickly fund uncer-tain projects at high valuations because they rationally fear missing out. Here,due diligence is cost-justified but time-consuming, and careful evaluation putsthe investor at risk of losing his investment opportunity altogether. I showthat when there are many investors relative to investment opportunities, and

1http://www.nytimes.com/2015/05/23/technology/overvalued-in-silicon-valley-but-not-the-word-that-must-not-be-uttered.html

2http://blogs.wsj.com/digits/2015/02/20/bill-gurley-fomo-in-the-private-ipo-market-fuels-valuations/

3http://en.wikipedia.org/wiki/Pets.com Facts that rudimentary due diligencewould have revealed include: Pets.com was selling merchandise for one-third of its cost,spent 1,900% of its revenues on advertising, and tried to absorb the costs of shipping heavybags of cat litter, which elsewhere sold at razor-thin margins.

2

investors have sufficient bargaining power, the equilibrium amount of due dili-gence will be inefficiently low. This gives rise to bubble-like high-quantity,high-valuation investment behavior with many funded projects failing. Fore-going due diligence is not irrational exuberance but rather a calculated invest-ment rush motivated by the fear of losing investment opportunities. Moreover,the same mechanism that drives overinvestment in unprofitable projects maylead investors to underfund profitable projects: Insufficient bargaining powermay discourage them from funding enough projects of any type, or in a rush toinvest without due diligence, they may deploy too much capital on unprofitableprojects and have none remaining for the profitable projects.

The fear of missing out arises in settings with matching frictions betweeninvestors and borrowers such as venture capital financing, mortgage lending,and mergers and acquisitions. These settings share four important features.First, investors do not have ex-ante exclusive rights to fund particular projects.Second, both investors and potential projects are in limited supply and projectsare not scalable. Third, investors choose investment timing. Fourth, projectschoose when to accept funding. These features will generate investment inef-ficiencies driven by the fear of missing out and lead to a price, quantity, andspeed theory of rational bubbles.

The economic mechanism behind the fear of missing out is as follows: Be-cause investors face limited investment opportunities, they will try to preempteach other by sacrificing due diligence and investing earlier than they other-wise would. Projects can choose whether to accept offers, and therefore candemand high valuations for early investment. The resulting price mechanismdoes not lead to an efficient amount of due diligence, however, because thematching market forces price competition to occur only across time. Rushinginvestors can offer higher valuations to projects than what patient investorscould commit to offering the following period, even though patient due dili-gence matches are more productive in expectation. Valuations, rather thaninducing an efficient allocation, merely serve as a way for early investors andprojects to appropriate value from patient investors.

The paper first gives conditions under which too little due diligence occurs

3

in the simple case of perfect matching—that is, when the technology matchinginvestors and projects forms all possible matches. In this case, there may bean inefficient rush to fund projects that results in overinvestment in projectsthat fail ex-post. That the inefficiency occurs in the case of perfect matchingdemonstrates that the externality is distinct from well-known congestion ex-ternalities in the employment search literature. Instead, the inefficiency arisesbecause the rushing investor and project jointly appropriate value from thehypothetical non-rushing investor.

The paper’s main result is to characterize conditions under which thereis too little or too much due diligence in the case of imperfect matching. Inthis case, congestion externalities in search make some early investment—investment without due diligence—socially desirable, but the private equilib-rium will not achieve the efficient outcome in general. A wedge arises betweenthe private equilibrium and social optimum for two reasons: First, as in thecase of perfect matching, early investment removes investment opportunitiesfrom late investors, and marginal early investors do not internalize this costwhen making decisions. This wedge leads to too much early investment. Sec-ond, in order to compensate projects for their outside option of waiting forlate funding, early investors must offer projects high valuations. While thesetransfers impact the investors’ decision making, they do not impact aggregatewelfare and consequently do not impact the social planner’s allocation. Thiswedge leads to too little early investment. Which wedge dominates depends onthe relative bargaining strength of investors and projects, with high investorbargaining power leading to excess early investment.

The paper offers a rational explanation of investment bubbles along price,quantity, and funding speed dimensions consistent with recent data and makesthe following predictions. First, when investors have strong bargaining positions—for instance, following good performance—there will be more high-speed, high-valuation, and low-information investment leading to overinvestment into badprojects due to the fear of missing out. Second, large inflows of savings intoa FOMO-prone investment sector exacerbate the high-speed, high-valuation,and low-information associated with the fear of missing out. Third, tech-

4

nological improvements in matching investors to savers exacerbates the fearof missing out problem. Fourth, improvements in the unconditional pool ofprojects—for example, advancements in nearly instant, high-speed screening—exacerbates the problem and can reduce social welfare. Consistent with thesepredictions, data show that as flows into the venture capital sector have in-creased, new companies seeking funding have received it more quickly. Figure1 compares hazard rates for founded companies receiving funding by companyage between 2004-2009 to 2010-2014. The figure shows that both the abso-lute probability of funding as well as the speed at which funding occurs hasincreased. According to Pitchbook Data, this increase has been accompaniedwith an increase in median valuation.4 As the funding speed and valuation hasincreased, however, the probability of success as measured by IPOs or acquisi-tion has decreased. This paper argues that this increased speed and valuationand lower quality of funded project is driven partly by VC investors’ rationalfear of missing out.

4http://files.pitchbook.com/pdf/PitchBook_1H_2015_VC_Valuations_and_Trends_Report.pdf

5

Figure 1: Dotted lines are for the period 2004-2009. Solid lines are for the period 2010-2014. The top panel shows the probability of a company receiving venture funding t yearsafter being founded conditional on not yet being funded (and founding + t date being inthe sample). The bottom panel shows the probability of a successful exit t years afterreceiving funding conditional on not having exited yet (and funding + t date being in thesample). The figure shows that (1) the probability of being funded is higher recently, (2) thespeed at which funding takes place is faster recently, and (3) the quality of firms receivingfunds, as measured by probability of a successful exit—IPO or acquisition—is lower recently.Founding/funding/exit data from crunchbase.com.

1.1 Literature

This paper proposes a theory of over and underinvestment centered on an ineffi-cient timing-of-investment decision. A large literature offers alternate theoriesof waves of over and underinvestment. For example, papers such as Moore andKiyotaki (1997), He and Krishnamurthy (2008), Brunnermeier and Sannikov(2014), and many others5 emphasize firm-level financial frictions leading toaggregate inefficiencies. In these models, individual firms fail to internalizethe fact that their borrowing in booms leads to fire-sale prices for other firmsduring busts. He and Kondor (2012) emphasize a firm-level liquidity manage-

5See He and Kondor (2012) for a complete discussion.

6

ment channel. Rather than focusing on within-firm financial frictions and howthey impact aggregates, this paper focuses on how one investor’s investmenttiming decision impacts the decisions of others.

The starting place for this paper’s theory of over and underinvestmentrelates most closely to the tragedy of the commons in natural resource extrac-tion. This large literature, starting with Hardin (1968) studies the harvestingof common-pool, often renewable resources.6 For example, Huang and Smith(2014), find evidence of dynamic overharvesting in fisheries. Over-extraction ofnon-renewable natural resources prior to learning about extraction-site qualityis a prediction of a special case of this paper. In particular, when investmentprojects in this model do not have the power to reject funding offers, they canbe interpreted as passive resources in the commons waiting to be claimed. Forexample, by calling investors “prospectors,” projects “potential gold miningsites,” and requiring the sites to accept any prospector’s digging, this papermodels a Gold Rush. If the quality of mining sites is discoverable only withtime-consuming mineral studies, prospectors may dig before evaluating a siteor risk being claim jumped by a rival.7 The Diamond and Dybvig (1983) stylebank run has a similar mechanism: The pressure to claim jump, interpretedhere as withdrawing deposits before illiquid projects are complete, can trig-ger society-wide claim jumping—a run. In the bank run, depositors do nothave particular property rights to withdrawal deposits; here, investors do nothave particular rights to fund specific projects. I extend the analysis past theclassical run by introducing a search friction and bargaining in the spirit ofMortensen and Pissarides (1994). Interestingly, I find that a less severe searchfriction leads to more severe claim jumping.

A small number of papers apply these common-pool resource models specif-

6The range of applications to natural resource extraction is voluminous and well-summarized in Huang and Smith (2014): Oil extraction (Libecap and Wiggins (1984)),ground water (Provencher and Burt (1993)), hunting (Smith (1975)), and forestry (Mendel-sohn (1994) and Linde-Rahr (2003)), to name only a few.

7Clay and Wright (2005) document this phenomenon and the legal context during theCalifornia Gold Rush: Potential extraction sites were limited, and the law encouraged minersto mine in rivals’ inactive sites. In this setting, a miner wanting to evaluate a mining sitebefore digging risked losing out to a less prudent miner.

7

ically to project funding. Jovanovic and Szentes (2013), Toxvaerd (2008), andXie (2015) study investment entry or exit decisions in settings without propertyrights over limited extractable resources. These resources are interpreted asprojects or ideas for Xie or merger targets for Toxvaerd and typically generateinefficiently early investment decisions and a “quantity” theory of investmentbubbles. This result is analogous to my results when projects lack the right toreject offers. An important difference between natural resources and projects,however, is that projects and merger targets possess the legal right to reject anearly investor’s offer in anticipation of a better offer from a patient investor.Models relating investment bubbles to natural resource extraction thereoforemiss an important aspect of the investor-investee relationship, which is thatinvestees have the agency to reject inefficient offers in order to capture somegains to waiting for themselves. My model contributes to this literature by giv-ing projects this agency and showing when and why inefficient investment stilltakes place. In doing so, the model generates “price” predictions in additionto the “quantity” and “speed” predictions of existing literature.

The fear of missing out drives claimants to forgo information-producingactivities that are efficient but time-consuming because they risk losing outon the claim altogether. The option to invest only after learning is a valuablereal option, but that option belongs in the commons. The strategic interactioncoming from claim jumping elaborates on the option value of waiting literaturethat started with McDonald and Siegel (1986) and includes a large literaturefrom Industrial Organization, such as Fudenberg et al. (1983), regarding patentraces. Jovanovic and Rousseau (2001) consider the optimal investment timeof a firm that acquires information about its production function. Jovanovic(2009) considers a model where new investment requires “seeds” and calculatesthe optimal policy for planting a potentially constrained measure of seeds.Garleanu et al. (2012) examines the role and exercise of real options in makingtechnological investments on the business cycle and technology booms. In thisliterature, the real options are privately owned, and their value is mediated—sometimes driven to zero—by a price mechanism in output. Unlike much ofthis literature, here the option value of waiting is positive, but this value is in

8

the commons. Models of option exercise and patent races, moreover, do notcapture an important aspect of project funding in that the project itself hascontrol over when “exercise” takes place: Venture capital or merger targets,for instance, can be strategic and calculating in which offers they accept or re-ject; buried treasure or a particular—potentially monopolized—market cannotmake this choice. Ownership of the intellectual property reduces or reversesthe claim jumping problem but rarely eliminates it. Many policy prescriptionsregarding patent or innovation races focus on intellectual property-based solu-tions. This paper shows that the financial structure of an innovating industryis also important even if intellectual property rights exist.

The model offers a rational explanation of speculative over-investment ob-served in venture capital booms and mortgage lending. Harris et al. (2010) andHarris et al. (2014) show that while both venture capital and private equityinflows spiked in the late 1990s, venture capital returns plummeted while pri-vate equity returns did not. Similarly, Gompers and Lerner (2000) show highvaluations accompanied high inflows into venture capital between 1987 and1995. In my model, early and late investing differ in the amount of informa-tion available when investment takes place. In venture capital, large quantitiesof information are produced about new firms near the beginning of their lives,which makes early and late investing meaningfully different. In private equity,take-private targets are already subject to heightened information disclosuresand therefore the information differences between early and late investing isless relevant. Thus, venture capital investment is prone to claim jumping be-havior while private equity investment is not. The fear of missing may explainthe difference in returns of these sectors following large capital inflows. Thefear of missing out may also be partly behind changing practices in mortgagelending. Rajan et al. (2010) and Keys et al. (2010) show that over 2002-2006,mortgage lenders engaged in more mechanical and faster screening techniquesat the expense of softer and often slower techniques. Iyer et al. (2015) showthat these soft techniques aid in screening, but my model predicts that lenderswill rationally forgo these slow-but-effective screening techniques because theyrisked being claim jumped by faster screeners.

9

Finally, my paper has implications for the financial technology literature.Much of this literature, e.g., Philippon (2013) and Philippon and Reshef(2013), treats financial technology as essentially a factor-augmenting tech-nology whose improvement is welfare-enhancing to society. My model demon-strates how the particular form that financial innovation takes is importantfor outcomes and welfare. In particular, financial innovations improving in-vestor and project matching—online social media, for instance—increase thetendency for rushed investment and can be welfare-reducing. Additionally,new instant screening technologies like automated mortgage screening can im-prove the unconditional pool of investment opportunities and make investingwithout additional screening activities profitable. This may cause investorsto forgo more careful screening activities, even if they are efficient, for fearof missing out. While the automated screening will avoid some of the worstprojects, the adoption of faster screening can overall lead to more bad projectsbeing funded and a decrease in social welfare. In these examples, financialinnovation takes on a darker quality and raises questions, like Zingales (2015),about what sorts of financial innovation benefit society, especially when thetechnologies are chiefly about rent allocation among various intermediaries.

The paper proceeds as follows. In Section 2, I set up and discuss the model.In Section 3, I characterize the equilibrium and social planner’s solution. InSection 4, I discuss the model and its implications, and in Section 5, I conclude.

2 Model Setup

The model combines (1) projects with potentially profitable ideas requiringfunding from investors, and (2) a search market structure where investorschoose the timing of funding.

2.1 Projects

There is a continuum of measure p of unscalable projects. Each project hasone speculative idea with unproven business potential and must receive one

10

unit of outside funding to execute the idea. Ideas are good with probability π;good ideas produce R gross returns with certainty; bad ideas produce nothing.Idea quality cannot be detected without investor due diligence. In periodone—the early period—good and bad ideas are indistinguishable. In periodtwo—the late period—participants’ due diligence activities allow good andbad ideas to be distinguished. After the second period, projects’ ideas becomewidely known and copyable, and neither project nor investor can profit frominvestment.

Projects may receive offers of funding in both early and late periods, butbecause ideas are not scalable, if funded early, a project will not require (norwill it accept) late funding. A project may choose to reject an early offer ifits outside option of waiting is better. In equilibrium, it will never reject alate offer because its outside option is obsolescence in the next period. Mo-tivated by the often one-off nature of these deals requiring bespoke sourcing,negotiating, and legal work, I assume that projects receive at most one offerper period.

A match between an investor and project in the early period produces πRin expected gross output; a match between an investor and a project in thelate period produces R with certainty. A riskless outside savings technologyproduces gross return r. I assume that projects are unconditionally positivenet present value investments, i.e., πR − r > 0. Denote by re and rl theexpected production from an early or late match, net of the outside savingstechnology. These quantities are given by

re ≡ πR− r (1)

rl ≡ R− r (2)

Note that rl > πrl > re and that rl − re = (1 − π)r, equal to the outsideproduction from bad projects avoided when investing late.

11

2.2 Matching Market

Investors and projects must match for funding to occur. With s atomistic in-vestors, each with enough capital to fund one project, and p available projectsmatching within a period, their efforts produce m(s, p) total matches. Fol-lowing the matching literature, I assume that m(s, p) has a constant returnsto scale Cobb-Douglas form, with the addition of a technical truncation sothat the measure of formed matches exceeds neither s nor p.8 Under thisassumption,

m(s, p) = min{s, p, µsαp1−α

}(3)

where µ and α are technological parameters. As µ becomes large, matchingbecomes perfect in the sense that all possible matches are formed, and m(s, p)

becomes the perfect matching function, m̃(s, p), where

m̃(s, p) = min {s, p} (4)

Formed matches are allocated among investors uniformly and among projectsuniformly. The probability of an investor or project receiving a match is qs(s, p)and qp(s, p), defined as follows:

qs(s, p) ≡m(s, p)

s= min

{1,p

s, µ[ps

]1−α}(5)

qp(s, p) ≡m(s, p)

p= min

{s

p, 1, µ

[ps

]−α}(6)

Note that when s > p, even perfect matching does not guarantee that aparticular investor will match. In this case, only p matches can occur, andthey must be rationed to the over-abundant investors. To avoid rationing,investors will be tempted to claim jump.

8The truncation is necessary in the discrete-time setup. Though the results hold forall values of m(s, p), including those in the “truncation region,” for clarity of expositionI present results under the assumption that the truncation does not bind. Proofs in theappendix treat the general case with truncation.

12

2.2.1 Claim Jumping in the Matching Market

Investors choose at the outset whether they will try to learn, invest late, andfund only projects with good ideas, or whether they will claim jump—forgolearning, invest early, and risk funding projects with bad ideas. Both activitiesare time-consuming, so investors must choose only one. Let s and p be theeconomy-wide measures of investors and projects, respectively. Let se denotethe endogenous measure of early investors to be determined in equilibrium.Early investors and projects match first and produce m(se, p) matches, whichare allocated among the se investors. In equilibrium investors will only attemptan early match if they can make offers that will be accepted, so the m(se, p)

matches are funded and removed from the potential project pool.In the late investing round, investors can distinguish good ideas and will

therefore only attempt to match with good-idea projects. After the earlyprojects have been funded and idea quality has been revealed, there are π(p−m(se, p)) projects with good ideas remaining to be funded. The s−se remain-ing investors attempt to match with the π(p−m(se, p)) projects, matches areallocated, and the good projects are funded. The measure of matches formedin each market given the measure of early matchers se is

Early Matches: m(se, p)

Late Matches: m(s− se, π(p−m(se, p)))

Successful matches are allocated among attempted matches pro-rata, so thatthe probabilities of matching are as follows:

Investor Probability Project ProbabilityEarly: m(se,p)

se

m(se,p)p

Late: m(s−se,π(p−m(se,p)))s−se

m(s−se,π(p−m(se,p)))π(p−m(se,p))

2.2.2 Bargaining in the Marching Market

A matched investor and project engage in Nash Bargaining, where the in-vestor’s weight in the negotiations is β. Having made his decision regarding

13

the timing of his investment, his outside option is the outside storage technol-ogy r. To calculate the project’s outside option, we work backwards from thelate period.

In the late period, quality is observed so investors only match with goodprojects. A funded project produces R with certainty. A project rejectingfunding will not have another chance to get funding before the idea enters thepublic domain, so its outside option is zero. Total surplus from a late matchis therefore

Late Surplus: rl = R− r

Under the Nash Bargaining assumption, total payments to project and investorfrom a late match are

Late Investor Payment : ξls = βrl

Early Investor Payment : ξlp = (1− β)rl

In the early period, idea quality is unobserved. A project receiving an offer offunding has the option to reject it, determine its idea’s quality, and wait forfunding in the late period. The rejecting project will match in the late periodwith probability qp(s − se, π(p − m(se, p))), and conditional on funding, willreceive (1− β)rl. Expected surplus from an early match is therefore

Early Surplus: re − πqp(s− se, π(p−m(se, p)))ξlp

=re − πqp(s− se, π(p−m(se, p)))(1− β)rl

Importantly, the early surplus is not simply the difference between what isproduced early, πR, and what is produced late, R. Rather, the surplus relevantto an early match is the difference between what is produce early and whatthe project would receive in a late match.

Finally, given aggregate behavior se, expected surplus from the early matchis divided among investor and project according to their bargaining strength,

14

so the surplus going to each party net of the outside option is

ξes(se) = βmax{(re − πqp(s− se, π(p−m(se, p)))ξ

lp

), 0}

(7)

ξep(se) = (1− β) max{(re − πqp(s− se, π(p−m(se, p)))ξ

lp

), 0}

(8)

2.3 Savings Timing Decision

Returns to early and late investment, net of the investor’s outside option onfailing to match, are

re(se) = qs(se, p)ξse(se) (9)

rl(se) = qs(s− se, π(p−m(se, p)))ξsl (10)

Investors are risk-neutral and given aggregate behavior se, pick the greater of(9) and (10).

Before characterizing the equilibrium, notice that while conditional onfunding a project, late investing offers a higher return than early investing,an investor’s probability of receiving a late match may be so low that it isworth accepting a lower-quality early match with higher match probability.

3 Equilibrium

I look for the equilibrium quantity of early investors, se, who choose to enterthe early search market. The remaining s−se search in the late market amongthe remaining π(p−m(se, P )) projects with good ideas. I use spe to denote aprivate equilibrium early investment level, and sse to denote the social planner’soptimal early investment level. Before moving forward, I formally define theequilibrium condition, which is simply a Nash equilibrium. There are threevarieties of equilibria:

Definition 1. spe is an all-early equilibrium if for spe = s, re(spe) ≥ rl(spe).

Definition 2. spe is an all-late equilibrium if for spe = 0, re(spe) ≤ rl(spe).

15

Definition 3. spe is a mixed equilibrium if for spe ∈ (0, s), re(spe) = rl(spe).

If, when all others choose early investment, an individual prefers to investearly, an all-early equilibrium occurs. If, instead all others choose late in-vestment, an individual preferes to invest late, an all-late equilibrium occurs.Finally, if at a given interior early investment level, returns to early and lateinvestment are equal, a mixed equilibrium occurs.

Of particular interest will be stable mixed equilibria, defined below.

Definition 4. A mixed equilibrium is stable if given the equilibrium quantityof early investment spe ∈ (0, s), r′e(spe) < r′l(s

pe).

A stable mixed equilibrium occurs if adding a marginal amount of earlyinvestors makes early investment strictly worse than late investment: A smalldeviation encourages correction towards the equilibrium. Graphically, a mixedstable equilibrium occurs when as a function of Se, returns to late investmentcross returns to late investment from below.

Finally, in the case of perfect matching technology, I consider a modificationof the all early equilibrium, the strong all-early equilibrium:

Definition 5. spe is a strong all-early equilibrium if for spe = s, re(spe) >rl(s

pe) and projects receiving early offers accept them, even if the project could

surely match with a late investor.

A strong all-early equilibrium has the interpretation that an ε measureof trembling early investors will instead match late. In an ordinary all-earlyequilibrium, projects’ outside options in the early period are zero because theywill have no investors with whom to match late. Lacking a non-zero outsideoption, they will accept any non-negative offer from investors. The strongall-early equilibrium considers the case where even if projects had a betteroutside option—a sure-thing late match—early investors would still be able tooffer acceptable valuations to projects. The strong all-early equilibrium willillustrate the important role that the appropriation of late matching investorshas in the inefficiency.

16

Finally, I define the social planner’s problem. The social planner faces thesame matching friction but can allocate early and late savings as she pleases.Social welfare is equal to the total production from all matches plus outsideproduction of unmatched investors (r per investor) and outside production ofunmatched projects (0 per project):

Definition 6. Given aggregate savings s and early savings se, social welfareis given by:

w(se) = m(se, p)re +m(s− se, π(p−m(se, p)))rl + sr (11)

The social planner’s solution sse satisfies

sse = arg maxse∈[0,s]

w(se) (12)

For purposes of interpretation, notice the following:

Lemma 1. The Social Planner’s Problem, given in (12), is equivalent to max-imizing the total ex-ante expected welfare of investors and projects with equalPareto weights.

Proof. See Appendix, Section 6.1.1.

I first characterize the social planner’s solution and private equilibriumwith perfect matching, as it illustrates simply and starkly the frictions arisingfrom the opportunity to claim jump. I will then characterize private equilibriaand the social planner’s problem with imperfect matching technology, whichprovides meaningfun notions of valuations, overinvestment, and underinvest-ment.

3.1 Perfect Matching Technology

The following proposition characterizes the social planner’s solution with per-fect matching from (4):

Proposition 1. The social planner’s solution is sse = 0.

17

Proof. See Appendix 6.1.2.

The following proposition characterizes private equilibria with perfect match-ing:

Proposition 2. With perfect matching technology, the existence and unique-ness of private equilibria are as follows:

1. All-late: An all-late equilibrium exists if

rerl− π(1− β) ≤ πp

s

2. All-early: An all-early equilibrium exists if p ≤ s and

re ≥ 0

3. Strong all-early: A strong all-early equilibrium exists if p ≤ s and

.re − π(1− β)rl ≥ 0

4. Mixed: A mixed equilibrium exists if all-early and all-late equilibria ex-ist.

Proof. See Appendix 6.1.3.

18

Figure 2: Perfect matching. The solid and dashed lines are the expected returns to earlyand late investors, respectively, as functions of aggregate early savings se. The dotted lineis social welfare. The first panel shows a unique all-late equilibrium because late investmentdominates for all se. The second panel shows a unique all-early equilibrium because earlyinvestment dominates for all se. The last panel shows all three types: At se = 0, latedominates—an all-late equilibrium; at se = s, early dominates—an all-early equilibrium.There is also a mixed equilibrium at the se where early and late returns are equal. Note inall cases, the socially optimal level is se = 0.

Broadly, the existence of all-late depends primarily on project scarcity, andthe existence of all-early depends on project returns and bargaining power.Figure 2 shows the possibilities graphically. First, absent a savings glut, in-vestors can learn and thus invest late without fear of losing opportunities torival late matchers or claim jumpers. Because they will surely match, theychoose the larger return. On the other hand, a sufficiently severe savings glutcan rule out the late equilibrium altogether. For p much smaller than s, anlate investor may be so unlikely to match with a project that he would preferthe lower conditional early payment in exchange for a sure match.

Once s ≥ p, an all-early equilibrium can exist so long as early investingis NPV-justified. This is because if all invest early, all projects are consumed

19

so the returns to late investment is zero. More interestingly, the condition fora strong all-early equilibrium illustrates both why investors choose to investearly and why it is socially inefficient. The strong early equilibrium conditionis equivalent to (1 − π)r ≤ πβrl. The left-hand side is what society savesby avoiding bad ideas. The right-hand side is what the project would payto a late investor if it rejected an early offer and matched with one of theε trembling investors. This is a division of late surplus between project andinvestor. The early investor and project have the opportunity to appropriatethis transfer from the absent third party and will do so if it is larger than theexpected savings from avoiding bad ideas. Their gain is merely a transfer, buttheir loss is a net loss to society. All-early equilibria are self-defeating from theinvestors’ perspectives. The motivation for deviation is to avoid crowdednessand expropriation from others, but when all claim jump, the market is equallycrowded but projects have lower expected returns.

Finally, early, late, and mixed equilibria are possible for a moderate savingsglut. This has a similar intuition to a bank run in Diamond and Dybvig(1983), where claim jumping—running—can become a self-fulfilling prophesy.An individual who fears a claim jumping “run” will himself claim jump or loseout on the investment opportunities altogether. This triggers a rush to earlyinvestment.

For the social planner, all-late investment is always optimal. The socialplanner wants (1) if there are not enough projects, to make sure that eachproject gets matched late, and (2) if there are not enough investors, to makesure that each investor matches late. Claim jumping occurs in equilibriumbecause investors and projects can appropriate future payments from late in-vestors and because investors fear missing out on investment opportunitiesaltogether. These fears concern, from the social planner’s perspective, meretransfers. There is scope for divergence between the social planner’s solutionand the private equilibrium in a moderate savings glut, and there is necessarilydivergence in a severe savings glut.

Perfect matching starkly illustrates the early investment option’s dangers—it can lead to a rush to fund projects before their payoffs are known due to the

20

fear of missing out, which in turn leads to overinvestment. I next consider themore general case with imperfect matching. The results will illustrate clearlyhow the wedge between the private equilibrium and social planner’s solutionarises and show when overinvestment and underinvestment occur.

3.2 Imperfect Matching Technology

A key difference with imperfect matching technology is that the social planner,in order to fund more good ideas, will want some speculative early investmentalong with safe late investment. As before, there will be scope for overinvest-ment in bad ideas in the private equilibrium, but there will also be scope forunderinvestment in good ideas, because private investors may (1) fail to en-gage in enough speculation or (2) use too much of their capital funding badideas early, both of which will lead to unfunded good ideas after period two.Generally, an inefficient amount of early investing will occur in the privateequilibrium, and whether there is too much or too little will depend on theinvestors’ bargaining power.

3.2.1 Social Planner’s Solution

I begin by characterizing the social optimum. The following proposition char-acterizes the social planner’s solution and gives a simple criterion for deter-mining whether a given allocation with se has more early investment than thesocial planner would choose.

Proposition 3. With imperfect matching technology, the social planner’s so-lution (12) is characterized as follows:

1. All-late: If w′(0) = re − [ms(s, πp) + πmp(s, πp)] rl < 0 then sse = 0 issocially optimal.

2. FOC: If ∃se ∈ [0, s] such that w′(se) = 0, then such an se is unique andsse = se is socially optimal.

3. Corner: If (1) and (2) are not satisfied, then the unique sce satisfyingµ(sce)

αp1−α = sce is socially optimal.

21

Moreover, if w′−(se) is the left-side derivative of w, if for any se ∈ (0, s],w′−(se) < 0, then sse < se.

Proof. See Appendix, 6.1.4.

Figure 3: The possible solutions of the social planner’s problem. Each panel shows thesocial welfare function for different values of µ in the matching function. The upper leftshows that for high µ, the social planner’s optimum is to set se = 0 and match all late. Theupper right shows for a moderate µ, some early investment is optimal, and the optimal pointoccurs when µsαe p1−α = se. The lower left shows that for a low µ, the optimum occurs at alocation where w′(se) = 0. Finally, the lower right panel shows the social welfare functionsfrom each other panel over the entire range of se, to give a sense of the shape of the function.

Though there are several cases, the intuition is simple: The social plannerwants to fund as many of the good projects as possible while avoiding badprojects. All-late funding, which will avoid bad projects, may fail to fundall good projects due to the overconcentration of investors in the matchingmarket. The social planner therefore trades off better conditional returnsagainst less crowded matching and picks the optimal. If the matching frictionmild relative to the differences in returns, she chooses all late, and otherwisechooses an interior quantity of early investment. An intuitive corollary of

22

Proposition 3 is that if m(s, p) = p, then sse = 0 is optimal. If the socialplanner can match all projects in a single round, he would prefer to matchthem all late. Figure 3 illustrates the cases.

The last line of the proposition is helpful in comparing the private equilibriato the social planner’s solution. In particular, to determine whether spe ≶ sse,it will be sufficient to compute the sign of the derivative of the social welfarefunction at spe: If social welfare is increasing at spe, then there is insufficientearly investment; if social welfare is decreasing at spe, then there is too muchearly investment; if social welfare is flat at spe, then spe and sse are equal.

3.2.2 Private Equilibrium

I first give necessary and sufficient conditions for the existence of corner equi-libria where spe = 0 or spe = s:

Proposition 4. Given imperfect matching technology m(s, p),

1. An all early equilibrium exists if and only if m(s, p) = p.

2. An all late equilibrium exists if and only if qs(s, πp) >πR−r−π(1−β)qp(s,πp)

R−r .

Proof. See Appendix 6.1.5

The intuition for this proposition is similar to that of perfect matching. If itis possible for all projects to match in a single matching market—m(s, p) = p—then all-early is an equilibrium because no projects will remain for late match-ing. If any projects remain, some investors will invest late because they canmatch with certainty and get the larger late payment. The all-late equilibriumcan exist so long as the crowding effects are not worse than the difference inpayments. Notice that when an all-early equilibrium is possible correspondsexactly to when the social planner would choose all-late investment: When itis possible to match all projects, an early equilibrium is possible because itsends late returns to zero; at the same time, this is precisely when the socialplanner would only like to invest late. This illustrates how too much savingscan make the inefficiency worse.

23

For the remainder of the paper, I will rule out pure private equilibria.These equilibria are analogous to the perfect matching technology case and sothe analysis is similar. To that end, I assume the following:

Assumption 1. Ruling out corner equilibria:

1. No all-early equilibrium: m(s, p) < p.

2. No all-late equilibrium: qs(s, πp) >πR−r−π(1−β)qp(s,πp)

R−r .

Having ruled out corner equilibria, the following proposition establishes theexistence of stable mixed equilibria.

Proposition 5. Under Assumption 1, there is a stable mixed equilibrium quan-tity of early investment spe satisfying

re(spe) = rl(s

pe)

Proof. See Appendix 6.1.6.

24

Figure 4: Imperfect matching. Along the horizontal axis is the (not necessarily equi-librium) amount of early investment. The solid and dashed lines are expected returns toearly and late investors, respectively. The first and second panels show unique all-late andall-early equilibria. The third panel shows an interior, stable, mixed equilibrium.

Figure 4 shows the possibilities. Assumption 1 rules out the first two pan-els, and Proposition 5 guarantees the crossing point shown in the last panel. Infocusing on stable mixed equilibria, we are considering points at which early in-vesting is a strategic substitute. Marginal early investment has two effects: (1)early investment becomes more crowded relative to late investment, pushingtowards strategic substitutability, and (2) future late investment opportuni-ties are consumed, pushing towards strategic complementarity. In the perfectmatching case, latter effect dominates and the only mixed equilibrium is un-stable. In the imperfect case, the former dominates. Part of the force pushingtowards inefficiently high early investment in the case of perfect matching isthis strategic complementarity that creates the Diamond and Dybvig (1983)run-like equilibrium. Imperfect matching dulls this force: When bank runnersmay fail to reach the bank, other depositors are less worried about a run.When claim jumpers may fail to find the claim, investors who want to learn

25

are less worried about having their claims disturbed while they learn. Thispreviews a result to come: improvements in the matching technology generatemore claim jumping.

Having characterized the social planner’s solution and the private equilib-ria, we come to the main proposition of this section:

Proposition 6. Let spe denote the private equilibrium, and sse denote the socialplanner’s level of early investment. The efficiency of equilibrium early invest-ment depends on the investors’ borrowing strength β relative to their share inthe matching function α. In particular, under the maintained Assumptions 1,

1. Efficient: spe = sse if β = α.

2. Too much early: spe > sse if β > α.

3. Too little early: spe < sse if β < α.

Proof. See Appendix 6.1.7

26

Figure 5: Comparison of the private equilibrium and the social planner’s solution. Solidand dashed curves show early and late expected returns, respectively. The dotted curveshows the social surplus. The solid vertical line marks the equilibrium spe. The dottedvertical line marks the socially optimal level sse. When β = α, spe = sse. When β > α, theinvestor’s high bargaining power leads to spe > sse. When β < α, the investor’s low bargaiingpower leads to spe < sse.

Figure 5 shows a graphical comparison of the private equilibrium and thesocial planner’s solution. The efficiency of the private outcome depends on howthe investors’ bargaining power β compares to their importance in matchingwith projects α. To see why, the following is instructive. Define

h(se) ≡ (1− qp(se, p))1−α

Interpret h(se) as a transformation of the probability that a project is notfunded early. Then, the β ≶ α condition is equivalent to the following before

27

simplification:

β > α ⇐⇒

−h′(se)m(s− spe, πp)rl︸ ︷︷ ︸marginal interference externality

>∂m(spe, p)

∂speπqlp(s

pe)(1− β)rl︸ ︷︷ ︸

marginal transfer to project

The left-hand side measures the marginal interference of early investment onlate investment. The right-hand side the transfer that an early investor mustpay to a project in order to compensate it for its outside option of matchinglate. These generate opposing wedges in determining whether there will betoo much or too little early investment.

The interference term arises because early investment prevents some latematches from forming at a marginal rate of h′(se) < 0. Marginal early in-vestors impart this external cost on society. The social planner takes this intoaccount but the private participants do not. The interference term is a wedgebetween the social planner and private investors that pushes the equilibriumtowards too much early investment. The magnitude of the interference termis decreasing in α, because 1−α measures projects’ share in matching. When1− α is large, the removal of the early-matched projects prevents a relativelylarge amount of late matches from forming.

The transfer term occurs because early investors must pay projects abovetheir outside option of matching late. A project could refuse an early offer andobtain πqlP (spe)(1− β)rl in expectation from matching late. The early investormust therefore compensate the project for this amount. Multiplying the per-investor transfer payment by the marginal number of early matches formedyields the total transfer payments that the additional early investors must payto projects.9 Because the social planner cares about aggregate welfare, thistransfer payment does not impact her optimal allocation to early investment.Nonetheless, private investors must make the transfer, and they take this into

9Notice that neither marginal early investor nor social planner care about the pecuniaryexternality coming from the fact that increased early investing changes the valuation otherearly investors must pay to projects.

28

account when deciding whether to match early. This makes early investingless desirable and reduces the amount of early investing. The transfer termis a wedge between the social planner and private investor and pushes theequilibrium towards too little early investment. The magnitude of the transferterm is decreasing in β: When 1− β is small, the project will receive little ina late match, which makes the necessary transfer payment in an early matchsmall.

The relative sizes of the marginal interference externality and the marginaltransfer payment determine whether the amount of early matching is efficient.Too many projects match early if the private costs of matching early are lessthan the public externalized costs of matching early. Conversely, too fewprojects match early when the private costs of matching early are greater thanthe public costs of matching early. In the knife-edge case where they are equal,the amount of early investing is efficient.10

The number of good and bad projects funded are:

Good Fundings: πm(se, p) +m(s− se, π(p−m(se, p)))

Bad Fundings: (1− π)m(se, p)

While the social planner will fund some ex-post bad projects to insure match-ing of good projects, overinvestment occurs when private investors fund morebad projects than the social planner would. Underinvestment occurs when pri-vate investors fail to fund as many good projects as the social planner would.This can happen for two reasons: First if α > β, there will be insufficientearly investment, and crowding in the late period will leave good projects un-funded. In this case, there is underinvestment in both good and bad projects.Second, if α < β, there will be too much early investment, and if the amountof early investment is so large, investors will use most of their capital on spec-ulative early projects, leaving too little capital to fund known good projectsin the late period. In this case, there is overinvestment in bad projects and

10This relationship between bargaining β and share in the match production function αis similar to the result in Hosios (1990) regarding labor markets.

29

underinvestment in good projects.

4 Discussion and Comparative Statics

The fear of missing out can lead to a rush to claim investment opportunitiesbefore conducting due diligence. This rush leads to overinvestment in badprojects and often underinvestment in good projects. The problem is due tothe separation of ownership between patent holder and financier: Hence, it isa problem of financial property rights rather than intellectual property rights.Models with projects-in-the-commons setups often prescribe intellectual prop-erty rights as the solution, but as shown here, such property rights do notentirely address the problem. Here, projects have the intellectual propertyrights over their ideas and therefore the power to accept or reject early financ-ing. Nevertheless, an inefficient rush to investment will often occur. If theintellectual property belonged to the financier, or if the IP-owner did not re-quire external finance, the problem would be solved. However, as Arora et al.(2015) show, investment in new technology is increasingly taking place outsidelarge corporations rather than through internal capital markets. This suggestswe are likely to see these problems continue or even grow going forward.

I now discuss three comparative statics shaping the size and direction ofthe inefficiency: (1) the investor’s bargaining power, (2) aggregate savings inthe sector, and (3) technology. These comparative statics will help illuminatethe model and offer predictions.

4.1 Investor’s Bargaining Power

Whether there is too much or too little early investment depends on the bar-gaining strength of investors, β. Interpret β as representing the parties’ ed-ucation levels, bargaining abilities, or un-modeled institutional backing. Re-call from the discussion of Proposition 6 that β impacts investment decisionsthrough how much an early investor must compensate a project to accept hisoffer. In particular, β > α determines when there is too much or too little early

30

investment. This in turn determines whether there is over or underinvestment.Figure 6 shows equilibrium early investment, and number of projects of

each type funded as a function of bargaining power. Increases in β lead to moreearly investment. More early investment leads one-to-one to more bad projectsbeing funded, with overinvestment in bad projects when β > α. Interestingly,the number of good projects funded is not monotonic in β: For low β andbelow-efficient early investment, too many investors match late and due tomatching frictions, many good projects go unfunded. For moderately-high β,more good projects are funded than the social planner would fund, but thisis because there is higher-than-efficient early investment and more projects ofall types are funded. For very-high β, there is again underinvestment in goodprojects because so much early investment leaves little savings to fund goodprojects in period two.

Figure 6: The equilibrium effects of increases investor bargaining power β. The top panelshows the total quantity of early investment. The middle panel shows the number of goodprojects funded. The bottom panel shows the number of bad projects funded.

These results show the importance of the bargaining process between in-

31

vestors and projects, especially if this bargaining power is time-varying. Forinstance, if following a VC bust investors have an institutionally weak bargain-ing position due to demands from worried clients and hence a low β, there willbe underinvestment and the market will recover even more slowly. If, on theother hand, VC firms have been producing good returns and can bargain confi-dently, the high β will exacerbate subsequent overinvestment in bad projects.This is the model’s first prediction: There will be slow funding and under-investment in good projects when investors have a weak bargaining positionand fast funding and overinvestment in bad projects when they have a strongbargaining position.

4.2 Aggregate Savings

With intellectual-property-owner and financier separate, investors compete forproject funding opportunities. Projects enable and benefit from this compe-tition, which leads to inefficient quantities of early investment. When totalsavings in the sector increases, there is more between-investor competition,leading to greater inefficiencies. Data suggests that increased inflows have cor-responded with faster and lower quality fundings, and the comparative staticspresented here generate the same result. Figure 7 shows how equilibrium earlyinvestment, returns, and project valuations—the amount paid to a project inearly matching—vary with increases in society-wide savings, s, under the as-sumption that β > α.

32

Figure 7: The equilibrium effects of increases in total savings s, with β > α. For all s,spe > sse, and differences grow for increasing s. The top panel shows social planner andprivate allocation to early investment. The middle panel shows return to investors. Thebottom panel shows the valuation of early projects—how much early projects receive whenfunded.

The key result is that the wedge between equilibrium early investment andthe social planner’s early investment is increasing in total savings. Valuationsincrease with allocation to early investment: More savings mean better outsideoptions for waiting projects, and hence investors must pay more to invest early.Increasing prices, however, are not sufficient to push early investment back tothe efficient level. This is the model’s second prediction: Greater savingsin a rush-prone investment sector leads to increased speed and valuations,and decreased quality. The model stops short of explaining entry into thesector into the first place, but increased savings in the sector is consistent, forexample, with an economy-wide increase in the supply of savings or poorerinvestment opportunities in other sectors.

33

4.3 Technological Improvements

Perversely, technological innovations can make worsen inefficiencies and reducewelfare. I consider two types of technology improvements: First, technologi-cal improvements in matching technology, and second, technological improve-ments in rudimentary screening technology.

4.3.1 Improvments in Matching Technology

Recent innovations in social media and other online communications tools fa-cilitate matching, which I interpret as increases in µ, the scale parameter inthe matching technology (3). Interestingly, these matching tools exacerbatethe inefficiencies in the model and can even decrease equilibrium welfare. Fig-ure 8 shows how private equilibria and the social planner’s solution change asmatching technology improves and how unconditional returns change in bothcases.

34

Figure 8: The equilibrium effects of improvements in matching technology. The top panelshows equilibrium early investment and social planner’s early investment as µ increases.The bottom panel shows equilibrium returns to savings and the returns to an unconditionalinvestor’s returns in the social planner’s solution. Note that equilibrium returns can decreasewhen µ increases.

With bad matching technology, social planner and private individual allo-cate similarly to early investment because the preemption effect is weak. Astechnology improves, however, the allocations rapidly diverge. Eventually, theprivate equilibrium has all early investment, and the social planner has all lateinvestment. The bottom panel shows returns to these allocations. Note thatthe social planner is always able to weakly increase unconditional returns astechnology improves. Private returns initially increase with the matching tech-nology, but when marginally better technology prompts large switches to earlyinvesting, technological improvements reduce returns to private individuals.

We saw from Proposition 6 that the social planner and private equilibriumdiverge because (1) early matching interferes with late matching, leading to toomuch early matching, and (2) early investors must make transfer payments toprojects, leading to too little early matching. Improved matching technology

35

worsens the interference effect directly because more projects match early andare hence interfered with. Improved matching technology similarly worsens thebargaining power effect because if projects have a higher chance of matchinglate, they demand greater compensation early. Which dominates depends on αrelative to β. Interestingly, improvements in matching technology can decreasewelfare by triggering a rush to early investment. This gives the model’s thirdprediction: Improved matching rates between investors and projects increasesthe speed of financing and decreases the quality of funded projects.

4.3.2 Improvements in Project Screening

Next, I consider small improvements in technology that pre-screens projects.That is, I assume that the unconditional pool of projects have good ideas withprobability π̃ > π. Interpret increases π as being improvements in baselinemechanical screening—a computer or robo-signer that does fast, cheap, andrudimentary screening to screen out obviously bad ideas or scammers from theproject pool. Figure 9 shows how private equilibria and the social planner’ssolution change as π increases, and how unconditional returns change in bothcases.

In the top panel, with a relatively bad unconditional project pool, neitherprivate investors nor the social planner choose to invest early. However, as theproject pool gets better, private investors start to rush to invest early. Thisreduces welfare because investors are now funding both good ideas and badideas, even though there more good ideas overall. Here, technological advancesin screening make early investing feasible in equilibrium, but the effect of thetechnology is only to enable—and necessitate—a rush to invest. The technol-ogy does not create a new investment opportunity; it merely provides a chanceto take on an existing project in a less efficient, more opportunistic way. Thisis the model’s fourth prediction: Higher pre-screened project quality leads tofaster funding and can decrease unconditional project returns. This is largelyconsistent with the rise of computerized screening and robo-sign programs inthe run-up to the 2008 financial crisis.

36

Figure 9: The equilibrium effects of improvements in preliminary screening. The top panelshows equilibrium early investment and social planner’s early investment as π increases.The bottom panel shows equilibrium returns to savings and the returns to an unconditionalinvestor in the social planner’s solution. Note that equilibrium returns can decrease whenthe unconditional project pool improves.

These situations are particularly likely to arise in many financial contextswhere matching frictions result in investors earning rents. In a setting rifewith rents and opportunities to wastefully cut in line, as Zingales (2015) warnsus, we should be cautious before concluding that the widespread adoption ofa new financial technology is efficient and beneficial simply because it is anequilibrium outcome. Often times, the technologies may simply allow investorsto obtain rents faster—and less prudently—than competitors, which makeseveryone worse off.

5 Conclusion

Investors will skip due diligence to quickly fund uncertain projects becausethey rationally fear missing out. Early investors and projects can appropriate

37

value from learning investors, and in order to protect themselves, investors ra-tionally choose to forgo learning. The model applies naturally to the contextof venture capital funding: Projects must fund unproven ideas, matching be-tween investors and projects is difficult, and there are no property rights overthe financing opportunities. Unlike settings concerning resource extraction orfirm entry, projects have the opportunity to reject early offers in anticipationof better ones. Nevertheless, early investors can offer high valuations to induceacceptance, and a glut of savings relative to projects makes paying the highvaluation worthwhile for investors in order to avoid congestion later on. Thisphenomenon generates rational boom-like overinvestment with bad projectsfunded in large quantities and at high valuations, and often underinvestmentin good projects. The model’s predictions are consistent with past episodes intech financing, mortgage lending, and recent venture funding data.

38

ReferencesArora, A., Belenzon, S., and Patacconi, A. (2015). Killing the golden goose?

the decline of science in corporate r&d. Technical report, National Bu-reau of Economic Research.

Brunnermeier, M. K. and Sannikov, Y. (2014). International credit flows,pecuniary externalities, and capital controls.

Clay, K. and Wright, G. (2005). Order without law? property rights during thecalifornia gold rush. Explorations in Economic History, 42(2):155–183.

Diamond, D. W. and Dybvig, P. H. (1983). Bank runs, deposit insurance, andliquidity. Journal of Political Economy, 91(3):401–419.

Fudenberg, D., Gilbert, R., Stiglitz, J., and Tirole, J. (1983). Preemption,leapfrogging and competition in patent races. European Economic Re-view, 22(1):3–31.

Garleanu, N., Panageas, S., and Yu, J. (2012). Technological growth and assetpricing. The Journal of Finance, 67(4):1265–1292.

Gompers, P. and Lerner, J. (2000). Money chasing deals? the impact offund inflows on private equity valuation. Journal of financial economics,55(2):281–325.

Hardin, G. (1968). The tragedy of the commons. science, 162(3859):1243–1248.

Harris, R., Jenkinson, T., and Kaplan, S. N. (2014). Private equity perfor-mance: What do we know? Journal of Finance, 69(5):1851–1882.

Harris, R., Jenkinson, T., and Stucke, R. (2010). A white paper on privateequity data and research. UAI Foundation Consortium.

He, Z. and Kondor, P. (2012). Inefficient investment waves. Technical report,National Bureau of Economic Research.

He, Z. and Krishnamurthy, A. (2008). Intermediary asset pricing. Technicalreport, National Bureau of Economic Research.

Hosios, A. J. (1990). On the efficiency of matching and related models of searchand unemployment. The Review of Economic Studies, 57(2):279–298.

Huang, L. and Smith, M. D. (2014). The dynamic efficiency costs of common-pool resource exploitation. American Economic Review, 104(12):4071–4103.

Iyer, R., Khwaja, A. I., Luttmer, E. F. P., and Shue, K. (2015). Screening peerssoftly: Inferring the quality of small borrowers. Management Science.

Jovanovic, B. (2009). Investment options and the business cycle. Journal ofEconomic Theory, 144(6):2247–2265.

Jovanovic, B. and Rousseau, P. L. (2001). Why wait? a century of life beforeipo.

39

Jovanovic, B. and Szentes, B. (2013). On the market for venture capital.Journal of Political Economy, 121(3):493–527.

Keys, B. J., Mukherjee, T., Seru, A., and Vig, V. (2010). Did securitizationlead to lax screening? evidence from subprime loans. The QuarterlyJournal of Economics, 125(1):307–362.

Libecap, G. D. and Wiggins, S. N. (1984). Contractual responses to the com-mon pool: prorationing of crude oil production. The American EconomicReview, pages 87–98.

Linde-Rahr, M. (2003). Property rights and deforestation: the choice of fuel-wood source in rural viet nam. Land Economics, 79(2):217–234.

McDonald, R. and Siegel, D. (1986). The value of waiting to invest. QuarterlyJournal of Economics, 101(4):707–727.

Mendelsohn, R. (1994). Property rights and tropical deforestation. Oxfordeconomic papers, pages 750–756.

Moore, J. and Kiyotaki, N. (1997). Credit cycles. Journal of Political Economy,105(2).

Mortensen, D. T. and Pissarides, C. A. (1994). Job creation and job destruc-tion in the theory of unemployment. The review of economic studies,61(3):397–415.

Philippon, T. (2013). Has the us finance industry become less efficient? onthe theory and measurement of financial intermediation.

Philippon, T. and Reshef, A. (2013). An international look at the growth ofmodern finance. The Journal of Economic Perspectives, pages 73–96.

Provencher, B. and Burt, O. (1993). The externalities associated with thecommon property exploitation of groundwater. Journal of Environmen-tal Economics and Management, 24(2):139–158.

Rajan, U., Seru, A., and Vig, V. (2010). The failure of models that predictfailure: distance, incentives and defaults. Chicago GSB Research Paper,(08-19).

Smith, V. L. (1975). The primitive hunter culture, pleistocene extinction, andthe rise of agriculture. The Journal of Political Economy, pages 727–755.

Toxvaerd, F. (2008). Strategic merger waves: A theory of musical chairs.Journal of Economic Theory, 140(1):1–26.

Xie, D. (2015). A rush theory of financing and optimal patent design underuncertainty. Working Paper.

Zingales, L. (2015). Presidential address: Does finance benefit society? TheJournal of Finance, 70(4):1327–1363.

40

6 Appendix for Online Publication

6.1 Proofs of Propositions

6.1.1 Proof of Lemma 1

Begin with the social welfare function

w(se = me(se)πR +ml(se)R + (s−me(se)−ml(se))r

Adding and subtracting

me(se)πR = me(se)[πR− r + r − πqlp(se)(1− β)(R− r) + πqlp(se)(1− β)(R− r)

]= me(se)

[β(πR− r − πqlp(se)(1− β)(R− r)) + r

]+me(se)

[(1− β)(πR− r − πqlp(se(1− β)(R− r)) + πqlp(se(1− β)(R− r))

]= seq

es(se)(ξ

es(se)) + pqep(se)(ξ

ep(se))

Then,

me(se)(πR− r) = seqes(se)(ξ

es(se)− r) + pqep(se)(ξ

ep(se))

Next, using similar algebra,

ml(se)(R− r) = (s− se)qls(se)(ξls − r) + π(p−m(se, p))qlp(se)ξ

lp

Replace these in the social welfare function,

me(se)(πR− r) +ml(se)(R− r) + sr = seqes(se)ξ

es(se) + (s− se)qls(se)ξls

+ pqep(se)ξep(se) + π(p−m(se, p))q

lp(se)ξ

lp

Rearrange the top and bottom lines to

seqes(se)ξ

es(se) + (s− se)qls(se)ξls = s

[sesqes(se)ξ

es(se) +

s− ses

qls(se)ξls

]pqep(se)ξ

ep(se) + π(p−m(se, p))q

lp(se)ξ

lp = p

[qep(se)ξ

ep(se) + (1− qsp(se))πqlp(se)ξlp

]41

The top line is the total ex-ante expected welfare of investors. The bottom lineis the total ex-ante expected welfare of projects: the first term is the probabilityof matching early times the payments of matching early; the second term is theprobability of matching late times the payments of matching late—1− qep(se)is the probability the project does not match early; π is the probability thatthe project is good, and qlp(se) is the probability of matching conditional onnot matching early and being good.

6.1.2 Proof of Proposition 1

The social planner’s welfare function is

min(se, p)(πR− r) + min(s− se,max(p− se, 0))(R− r) + sr

Because πR− r < R− r, this is maximized at se = 0, and social welfare is

min(s, p)(R− r) + sr

6.1.3 Proof of Proposition 2

First, suppose that p > s. For all se, an investor choosing early matchingwill surely form a match and earn at most β(πR − r). For all se, an investorchoosing late matching will match with a project with probability π(p−se)

s−se ≥ π

and a match will pay β(R − r). Expected returns to early matching are atmost β(πR − r). Expected returns to late matching are at least βπ(R − r).Hence, late matching strictly dominates for p > s.

Before characterizing the equilibria for p < s, some preliminary helpful facts.Returns to early and late investing are

re(se) = min

{1,p

se

}β

(re − πmin

{1,

s− seπmax{0, p− se}

}(1− β)rl

)rl(se) = min

{1,πmax{0, p− se}

s− se

}βrl

42

Note that when p ≤ s, there are two cases for matching probabilities given se.

Case 1: se ≤ p:

min

{1,p

se

}= 1 Early investor matching probability

min

{1,πmax{0, p− se}

s− se

}=π(p− se)s− se

Late investor matching probabililty

min

{1,

s− seπmax{0, p− se}

}= 1 Late project matching probability

Case 2: s > se > p:

min

{1,p

se

}=

p

seEarly investor matching probability

min

{1,πmax{0, p− se}

s− se

}= 0 Late investor matching probabililty

min

{1,

s− seπmax{0, p− se}

}= 1 Late project matching probability

Case 2: s = se > p:

min

{1,p

se

}=

p

seEarly investor matching probability

min

{1,πmax{0, p− se}

s− se

}= 0 Late investor matching probabililty

min

{1,

s− seπmax{0, p− se}

}= 0

Late project matching probability (uncoordinated)

ql∗p = 1

Late project matching probability (coordinated)

Importantly, when se = s, mechanically the probability of a project match-ing late is zero because there is no potential funding to match with, but in

43

evaluating the strong all early equilibrium, we consider the possibility of aproject/investor coordinating to wait, in which case the matching probabilityis one.

To characterize when equilibria exist, I go through each case in turn.

All late: For an all late equilibrium to exist, it must be that for aggregatebehavior se = 0, investing late is weakly preferred. Using the above facts,when se = 0, returns to early and late are:

re(0) = β(re − π(1− β)rl)

rl(0) =πp

sβrl

All late exists when

πp

sβrl ≥ β(re − π(1− β)rl)

⇐⇒ reπrl− (1− β)

p

s

All early: For an all early equilibrium to exist, it must be that for aggre-gate behavior se = s, investing early is weakly preferred. When se = s, therequirement is

πp

sβre ≥ 0

⇐⇒ re ≥ 0

The above is true because when p < s, so long as early investing is NPV jus-tified, there is no incentive for an investor to deviate because there will be noprojects for the investor to invest in.

Strong all early: For a strong all early equilibrium to exist, it must bethat for aggregate behavior se = s, even if a project and investor could coor-dinate to meet in the late period, one of the parties to the agreement would

44

deviate—either the project would accept and early offer, or the investor wouldmake an early offer. The investor would not deviate because he could get aguaranteed late match which is by assumption the highest return. The project,on the other hand, could receive an early offer that it would accept so long as

re︸︷︷︸surplus available to early investor

≥ π(1− β)rl︸ ︷︷ ︸expected value of late match

⇐⇒ re ≥ π(1− β)rl

Note that the above being true implies re > 0, that is, existence of a strongall early equilibrium implies existence of an all early equilibrium.

Mixed: re(se) and rl(se) are continuous in se, and therefore so is re(se) −rl(se). If an all-early equilibrium exists and an all-late equilibrium exists, thenre(s)− rl(s) ≥ 0 and re(0)− rl(0) ≤ 0. In this case, by the intermediate valuetheorem, there is a value s∗e ∈ (0, s) satisfying re(s∗e) = rl(s

∗e), whence s∗e is a

mixed equilibrium.

6.1.4 Proof of Proposition 3

I first rule out see = s as a possible equilibrium so that candidate optima are (1)se = 0, (2) points where w′(se) = 0, and (3) points where w′(se) does not exist.

Note thatπm(s, p) ≤ m(s, πp)

To see, case one: m(s, p) = s, then s ≤ p and s ≤ µsαp1−α. Note then thatπs ≤ πp and πs < µsα(πp)1−α, so πm(s, p) ≤ m(s, πp).

Case two: m(s, p) = p. Then p ≤ s and p ≤ µsαp1−α. Note then thatπp ≤ s and πp < πµsα(πp)1−α, so πm(s, p) ≤ m(s, πp).

Case three: m(s, p) = µsαp1−α. Then µsαp1−α ≤ s and µsαp1−α ≤ p. Note

45

then that πµsαp1−α ≤ πs, πµsαp1−α ≤ πp, and πµsαp1−α ≤ µsα(πp)1−α, sothat πm(s, p) ≤ m(s, πp).

Next, if m(s, p) = p, then sse = 0 is the socially optimal level of early in-vestment. Moreover, sse 6= s in general, because a solution with sse = s can betrivially improved upon by setting Sse = 0, and observing that

me(s)re ≡ m(s, p)re

< m(s, p)πrl

≤ m(s, πp)rl

= ml(s)rl

Hence, candidate optima are (1) se = 0, (2) points where w′(se) = 0 and (3)points where w′(se) does not exist.

The outline is as follows. We need to consider separately two connected inter-vals for se. Define,

sce ≡ se such that se = µsαe p1−α

s̄e ≡ se such that µ(s− s̄e)α(π(P −m(s̄e, p)))1−α = π(p−m(s̄e, p))

sce is the amount of early investment such that early investment becomes un-stuck from being at the upper bound of m(se, p) = se. It is the first potentialkink in the objective function. s̄e is the point at which late matching begins toconsume all projects available for late matching, i.e., when the late matchingfunction becomes stuck at the number of projects limitation.

46

We consider two intervals:11

I1 ≡ {se|se ∈ (0, sce)}

I2 ≡ {se|se ∈ (sce, s̄e)}

I show (1) that in the region se ∈ (s̄e, s], f ′(se) < 0 and therefore that theoptimum cannot occur in [s̄e, s]. I show (2) that inside each of I1 or I2, f(se)

has at most one point satisfying w′(se) = 0 and that it is a local maximum.I show (3) that if an se satisfying f ′(se) = 0 occurs in one of the intervals,it cannot occur in the other. I show (4) that if such an se occurs, it is theunique maximum. I show (5) that if w′(0) ≤ 0 then se = 0 is the maximizer,and finally I show (6) that if f ′(0) > 0 and no se exists with f ′(se) = 0, thense = sce is the maximizer. In making these arguments, I will show along theway that whenever f ′−(se) < 0 then sse < se.

For (1). I want to show that w′(se) < 0 for se ∈ (s̄e, s] where it exists,and if not that the derivative on both sides is negative. In this region, theobjective function and right derivative is always

w(se) = m(se, p)re + π(p−m(se, p))rl

w′−(se) =∂m(se, p)

∂se[re − πrl]︸ ︷︷ ︸

<0

< 0

Hence, the social planner can always reduce early investment to increase socialwelfare. Note that this shows that the objective function is decreasing overthis region.

For (2), we want to show there is at most one se ∈ I1 satisfying w′(se) = 0 andone se ∈ I2 satisfying w(se) = 0 and that in either case, these points are localmaxima. Take se ∈ I1. The goal is to show that if w′(se) = 0 then w′′(se) < 0,

11If one interval is empty the result goes through and we simply rule out an interior pointin that interval.

47

whence se is a local maximizer, and moreover, only one se satisfies w′(Se) = 0

in this region. In this region,

w(se) = sere +m(s− se, π(p− se))rl

If m(s−se, π(p−se)) = s−se then decreasing se then reducing se increases so-cial welfare by rl−re > 0, whence it is not a maximum. Ifm(s−se, π(p−se)) =

π(p−se) 6= µ(s−se)α(π(p−se))1−α, then decreasing se increases social welfareby πrl−re > 0. Hence, the only possibility for a local maximizer in this regionis that m(s− se, π(p− se)) = µ(s− se)α(π(p− se))1−α.12

It is sufficient to show then that for m(se, p) = se and m(se − s, π(p −m(se, p))) = µ(s − se)

α(π(p − se))1−α, that w′(se) = 0 =⇒ w′′(se) < 0.

Note that here, m(s− se, π(p− se)) = π1−αm(s− se, p− se) so we first factorthe π1−α. This leaves

w(se) = sere +m(s− se, p− se)π1−αrl

w′(se) = re −ms(s− se, p− se)π1−αrl −mp(s− se, p− se)π1−αrl

Take derivatives to obtain:

w′′(se) = π1−αrl [mss(s− se, p− se) + 2msp(s− se, p− se) +mpp(s− se, p− se)] < 0

⇐⇒ −α(1− α)µ(s− se)α−2(p− se)1−α

+ 2α(1− α)µ(s− pe)α−1(p− se)−α − α(1− α)µ(s− se)α(p− se)−1−α < 0

= −µα(1− α)(s− se)α−2(p− se)−1−α[(p− se)2 − 2(s− se)(p− se) + (s− se)2

]= −µα(1− α)(s− se)α−2(p− se)−1−α [p− s]2 < 0

And the last line is negative because by assumptino, p − se < 0. This showsthat w(se) is weakly concave inside the region and hence has at most one lo-cation where the derivative is equal to zero, and whenever it is equal to zero,

12If this occurs at the corner of the late matching function, the derivative must be chang-ing sign here in which case this is still the relevant critical point.

48

it is a maximum.

Similarly, if se ∈ [sce, s̄e], then m(s − se, p − m(se, p)) = µ(s − se)α(π(p −

m(se, p)))1−α = µ(s− se)α(p−m(se, p))

1−απ1−α the objective function is

w(se) = m(se, p)re +m(s− se, p−m(se, p))π1−αrl

w′(se) = ms(se, p)[re −mp(s− se, p−m(se, p))π

1−αrl]−ms(s− se, p−m(se, p))π

1−απ1−αrl

If w′(se) = 0, because ms > 0, this means that

re −mp(s− se, p−m(se, p))π1−αrl > 0

The second derivative is

w′′(se) = mss(se, p)[re −mp(s− se, p−m(se, p))π

1−αrl]

+

[mss(se − s, p−m(se, p)) + 2mspms(se, p)

+mpp(se − s, p−m(se, p))ms(se, p)2

]π1−αrl

The first term is negative when f ′(we) = 0. Next, performing the same fac-toring trick as above the second term is non-positive iff:

0 ≥ +mss(se − s, p−m(se, p)) + 2mspms(se, p) +mpp(se − s, p−m(se, p))ms(se, p)2

⇐⇒ 0 ≥ −[ms(se, p)

2(p−m(se, p)2)− 2ms(se, p)(p−m(se, p))(s− se) + (s− se)2

]⇐⇒ 0 ≥ − [ms(se, p)(p−m(se, p))− (s− se)]2

And the last line is true because it is the negative of a number squared. Thisshows that when w′(se) = 0 for se ∈ [sce, s̄e], it is a maximum, and because thederivative is continuous in this region, it happens at most once. Moreover, amaximum can occur in this region only if w′(sce) > 0.

(3) Now I show that if there is an se ∈ Ii satisfying w′(se) = 0 then there

49

is no se in I−i satisfying w′(se) = 0. Consider the derivative immediately tothe left and right of se = sce. We have,

w′−(se) = 1[re −mp(s− se, (p− se))π1−αrl

]−ms(s− se, (p− se))π1−αrl

w′+(se) = ms(se, p)[(re −mp(s− se, (p− se))π1−αrl

]−ms(s− se, (p− se))π1−αrl

Suppose there is an se ∈ I2 where w′(se) = 0. Then it must be that at thebeginning of the interval, w′+(sce) > 0. If w′+(sce) > 0 then it must be thatre −mp(s − se, π(p − se))π1−αrl > 0. Since ms(se, p) ≤ 1, we therefore musthave that w′−(se) ≤ w′+(se) > 0. Inside I1, w′ is weakly decreasing as shownin step (2), which means that there could not have been an se ∈ I1 satisfyingw′(se) = 0.

Hence, if there is a zero of the derivative in I2 then there is no zero of thederivative in I1, so a zero can occur in at most one of these regions.

Taking stock, we have so far showed that se satisfying w′(se) can only oc-cur in I1 or I2 and if it does occur in one it cannot occur in the other. Thistells us there is at most one unique w′(se) = 0 for all se ∈ [0, s].

(4) Now I show that if ∃se such that w′(se) = 0, it is the unique maximizer.Other candidate points are se = sce and se = 0. The former cannot occur:Suppose ∃w′(se) = 0 but the maximizer is sce = 0. sce can only be a maximumif w−(sce) > 0 and w+(sce) < 0. The derivative cannot go from negative to posi-tive on the left side of sce so no w′(se) = 0 exists on the left side. The derivativecannot go from negative to positive on the right-hand side in se ∈ [sce, s̄e], so now′(se) = 0 exists on the right side. Hence, there is no se such that w′(se) = 0

if sce is a maximizer, which is a contradiction.

If se = 0 is a maximizer, then w′(0) ≤ 0. Again, because the derivativecannot cross from negative to positive inside either interval, it must be strictlynegative on the interior of I1 and therefore strictly negative on I2. Moreover

50

it must decrease going across sce, there can be no se satisfying f ′(se) = 0 ifse = 0 is a maximizer.

(5) Next I show that if w′(0) ≤ 0 then se = 0 is the maximizer. I showedthis in the above argument. If w′(0) ≤ 0 the derivative can only be negativethereafter.

Finally, (6), if w′(0) > 0 and there is no se satisfying w′(s0) = 0, then se = sce

is the maximizer. This is true simply because at this point we have ruled outall other candidates for maxima.

Notice that in making these arguments, whenever we found a maximizer,we saw that the objective function must be always increasing to the left ofthe point and decreasing to the right thereafter: If se = 0 is the maximizer,w′(0) ≤ 0 and arguments above showed that the derivative can never becomepositive thereafter. Hence, if se = 0 is the maximizer, any se with a negativederivative (aside from se = 0) satisfies se > sse. If se = sce is the maximizer, weargued that on the left side f(se) must be always increasing and on the rightside w(se) must be always decreasing. Hence if we find any se with w′(se) < 0,we know se > sse. If the maximizer is when w′(sse) = 0, again, to the right thefunction must be always decreasing and to the left it must be always increas-ing. Whence, w′(se) > 0 =⇒ se > sse.

This completes the proof.

6.1.5 Proof of Proposition 4

All early : If m(s, p) = p, then when se = s, the probability of matching late iszero, and hence, a private investor prefers to match early. Hence, m(s, p) = p

implies the existence of an early equilibrium. Next, suppose that se = s isa private equilibrium. This implies that qes(s)(πR − r) ≥ qls(0)(R − r). Butqls(s) = 1 unlessm(s, p) = p, and if qls(s) = 1, then qes(s)(πR−r) < qls(s)(R−r)because R− r > πR− r. Hence, it must be that that m(s, p) = p.

51

All late: This follows directly from the definition of the all late equilibrium.

6.1.6 Proof of Proposition 5

Consider the function f(se) ≡ qes(se)(πR−r)−qls(se)(R−r). Under the main-tained assumption that all late investment is not an equilibrium, f(0) > 0.Under the maintained assumption that all early investment is not an equilib-rium, f(s) < 0. f(·) is continuous so by the intermediate value theorem thereexists at least one spe where f(spe) = 0. This shows a mixed equilibrium exists.There must be a stable mixed equilibrium because f(0) must cross zero fromabove, i.e., returns to late investing cross returns to early investing from below,at at least one point.

6.1.7 Proof of Proposition 6

The maintained assumptions rule out spe = 0 and spe = s, and hence I onlyconsider interior equilibria. From Proposition 3, it is sufficient to check thederivative of the social welfare function at the candidate private equilibria spe.If it is negative, there is too much early investing; if it is positive, there is toolittle. Due to the technical truncation of m(s, p), I go through the functionin a piecewise manner. The “main” case is Case 5, but the others are shownhere as well for completeness.

Case 1: m(spe, p) = spe; m(s − se, π(p − spe)) = s − spe. Because the matchingprobability for early and late is one in this case, rl > re implies this is only anequilibrium if se = 0, which is ruled out by the maintained assumption.

Case 2: m(spe, p) = spe; m(s − spe, π(p − spe)) = µ(s − ppe)α(π(p − se))

1−α.Consider the social planner’s first-order condition evaluated at this point. I

52

will show that the FOC is negative:

FOC = re −[µα(s− se)α−1(p− se)1−α − µ(1− α)(s− se)α(p− se)−α

]π1−αrl

= re − µ(s− se)α(p− se)−α[αp− ses− se

+ (1− α)

]π1−αrl

Using the equilibrium condition that

re = µ

[p− ses− se

]1−απ1−αrl + π1−α(1− β)µ

[p− ses− se

]−αrl

= π1−αrlµ

[p− ses− se

]−α [p− ses− se

− (1− β)

]inject this into the FOC:

w′(se) = π1−αrlµ

[p− ses− se

]−α [p− ses− se

− (1− β)

]−µ(s− se)α(p− se)−α

[αp− ses− se

+ (1− α)

]π1−αrl

= µπ1−αrl

[p− ses− se

]−α [(1− α)

p− ses− se

− (1− β)− (1− α)

]= µ(1− α)π1−αrl

[p− ses− se

]−α [p− ses− se

− 1− β1− α

− 1

]Efficiency depends on whether the following is positive:

p− ses− se

− 1− β1− α

− 1 ≶ 0

⇐⇒ p− ses− se

≶(1− α) + (1− β)

1− α

In the case of β = 1, the LHS is greater than the RHS and the social welfarefunction is decreasing at se. As β becomes larger, this may go the other way.

Case 3: m(spe, p) = spe; m(s − se, π(p − spe)) = π(p − se). The social wel-

53

fare function issere + π(p− se)rl

which because re < πrl has a corner solution of se = 0. Hence, the privateequilibrium is spe > 0 = sse.

Case 4: m(se, p) = µsαe p1−α and m(s − se, π(p − m(se, p))) = s − se. No

equilibrium with se > 0 exists because it would require

qes(se)(re − πqlp(se)(1− β)rl) ≥ rl

and since qes(se) ≤ 1, this cannot happen. No corner equilibrium exists by themaintained assumptions.

Case 5: m(se, p) = µsαe p1−α and m(s−se, π(p−m(se, p))) = µ(s−se)α(π(p−

µsαe p(1−α)))1−α. Consider the social planner’s first-order condition evaluated

at the equilibrium condition:

In this case we can write

m(s− spe, π(p−m(spe, p))) = (1− qep(sce))1−α︸ ︷︷ ︸≡h(spe)

m(s− spe, πp)

= h(spe)m(s− spe, πp)

Interpret h(spe): 1−qep(sce) is the probability that a project is not matched withan early investor. This is a measure of interference of early matching with latematching—a higher h(spe) means less interference. Note that h′(spe) < 0, sothat more early matching causes more interference.

Next, the social planner’s problem as:

me(spe)re + h(spe)m(s− spe, πp)rl

54

Which we can rewrite as

speqs(spe, p)re + (s− spe)h(spe)qs(s− spe, πp)rl

The derivative of this is

qs(spe, p)re − h(spe)qs(s− spe, πp)rl

+spe∂qs(s

pe, πp)

∂sre + (s− spe)

[h′(spe)qs(s− spe, πp)− h(spe)

∂qs(s− spe, πp)∂s

]rl

Next, we use two facts. First, given the value of m in this region, we havese

∂qs(spe ,p)

∂s= −(1 − α)qs(s

pe, p) and (s − spe)

∂qs(s−spe ,πp)∂s

= (1 − α)q(s − spe, πp).Plug this in to simplify the above derivative:

α [qs(spe, p)re − h(se)qs(s− spe, πp)rl] + (s− spe)h′(se)qs(s− spe, πp)rl (13)

Next, spe being a private equilibrium implies that

qes(spe)(re − πqlp(spe)(1− β)rl) = qls(s

pe)rl

Now, add and subtract the project’s outside option term into (13):

α [qs(spe, p)re − h(se)qs(s− spe, πp)rl] + (s− spe)h′(se)q(s− spe, πp)rl

=α[qs(s

pe, p)(re − πqlp(spe)(1− β)rl)− h(se)qs(s− spe, πp)rl

]+ (s− spe)h′(se)q(s− spe, πp)rl + απqs(s

pe)q

lp(s

pe)(1− β)rl

Now, the first line gets canceled out with the equilibrium condition, and theimportant question is whether the second line is negative or positive. Cancel

55

the rl on each term and expand all the expressions:

(s− spe)h′(se)q(s− spe, πp) =∂

∂se

[p−m(se, p)

p

]1−αm(s− se, πp)

= −(1− α)ms(se, p)

p

[p−m(se, p)

p

]−αm(s− se, πp)

= −α(1− α)µsα−1e p−αm(s− se, πp)[p−m(se, p)

p

]−αAnd

παqs(spe)q

lp(s

pe)(1− β) = πα(1− β)

m(se, p)

se

m(s− se, π(p−m(se, p)))

π(p−m(se, p))

= πα(1− β)m(se, p)

se

m(s− se, π(p−m(se, p)))

π(p−m(se, p))

= πα(1− β)µsα−1e p1−αµ(s− se)1−α(π(p−m(se, p)))−α

= πα(1− β)µsα−1e p1−αµ(s− se)1−α(πp)1−α(πp)−1[

(p−m(se, p))

p

]−α= πα(1− β)µsα−1e p1−α(πp)−1m(s− se, πp)

[(p−m(se, p))

p

]−α= α(1− β)µsα−1e p−αm(s− se, πp)

[(p−m(se, p))

p

]−αAfter canceling stuff out, we’re left with,

(1− β)− (1− α) ≶ 0

That is, the derivative is negative if

β > α

and positive if

β < α

56

Case 6: m(se, p) = µsαe p1−α and m(s− se, π(p−µsαe p1−α)) = π(p−µsαe p1−α).

If the equilibrium is interior, notice that the social welfare function is

µsαe p1−αre + π(p− µsαe p1−α)rl

FOC : αµsα−1e p1−α(re − πrl)

This is strictly negative, whence, the social planner strictly prefers less earlyinvestment.

Cases 7, 8, 9: If m(se, p) = p, then the private equilibrium must be se = s

because there are no projects remaining for late investment. Next, if it isthe case that m(s, p) = p, then the social planner strictly prefers all late in-vestment because he can surely match all good projects in the late matchingmarket.

Finally, if at any time the social welfare function evaluated at the privateequilibrium is not differentiable, i.e., it occurs on a corner of the piecewisematching function, because the derivative is negative in all of these cases, theleft derivative will be negative which is sufficient to show that the social opti-mal occurs at a lower value sse < spe.

57