Reputation, Innovation, and Externalities inVenture Capital

Farzad Pourbabaee

Department of EconomicsUC Berkeley

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Outline

1. Motivations and contributions

2. Model

3. Equilibrium characterization

4. Innovation spillovers

5. Reputational externality

6. Robust extensions

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Motivations

• In the two-sided markets there is incomplete information aboutparticipants’ types⇒ there is room for reputation building.

• Examples (of incomplete information):• Labor market �rm’s productivity

• Educational market university’s quality

• Venture capital market VC’s ability

• Contributions of this paper:• develops a search and matching model where agents of one side of the

market have reputational concerns;

• studies two instances of market failure in venture capital: positivespillovers, and reputation-deal �ow externality.

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VCs’ ability and reputation (Sorensen 2007)A = net impact, C = after adjusting for the sorting

Figure 1: VC’s in�uence disentangled from sorting

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Theoretical contributions

• Developing a dynamic equilibrium model of search and matching inan economy where individuals on one side have symmetricincomplete information about their type.(Shimer and Smith 2000&2005), (Chade 2005), (Damiano et. al 2005), (Hoppeet. al 2009), (Anderson and Smith 2010), (Anderson 2015), (Chade andEeckhout 2017)

• Methodological contribution: matching sets⇒ stopping timeproblems faced by investors⇒ applying the theory of monotonicityto the operators acting on function spaces (Krasnoselskij 1964)

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Contributions (to the economics of venture capital)

• In equilibrium VCs earn reputation for their value-added skills,thereby rationalizing a number of stylized facts:

• Later stage startups attract a wider range of investors (Gompers,Gornall, Kaplan and Strebulaev 2020)

• More reputable VCs exhibit higher tolerance for failure (Manso2011), (Tian and Wang 2014)

• Cost-reducing technological shocks enhances the variety of �nancedprojects: “spray and pray" (Ewens, Nanda and Rhodes-Kropf 2018)

• Extending the baseline model to study two instances of marketfailure:

• Reputation-deal �ow externality among VCs early termination ofstartups; sub-optimal mass of active high ability investors

• Neglect of positive spillovers among startups

• Other related literature: (Silveira and Wright 2016), (Akcigit et. al2019)

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Outline

1. Motivations and contributions

2. Model

3. Equilibrium characterization

4. Innovation spillovers

5. Reputational externality

6. Robust extensions

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Model diagram

• A unit mass of VCs. A known fraction p have high type (θ = 1) while1− p have low type (θ = 0) their types are hidden to EVERYONE.

• Two types of projects q ∈ {a, b} with measures ϕa, ϕb their typesare observable to EVERYONE.

• VCs randomly meet the projects, subject to search friction withquadratic matching technology and frictional rate κ.

• Upon the meeting VC either accepts or rejects the project, thuswaiting for the next chance.

• If the partnership is formed• VC covers the �ow investment cost of c.• A successful outcome with unit payo� arrives with intensity θλq , whereq ∈ {a, b}, and λb > λa.

• VC collects all the revenues, i.e entrepreneur has zero bargaining power(Ueda 2004) and (Hellman and Puri 2002).

• VC has the option to terminate the project.8 / 27

Dynamic timeline

reputation = π meetq-startup

∼ exp. time

reject

�ow cost=c,solve stopping

time prob.

accept

success

failure

π ↑ 1

π ↓

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Outline

1. Motivations and contributions

2. Model

3. Equilibrium characterization

4. Innovation spillovers

5. Reputational externality

6. Robust extensions

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VC strategy

• Reputation is the market posterior belief about the VC’s type πt := P (θ = 1| It).

• When unmatched the VC’s reputation is �xed, and when matched to aq-project:

π̇t = −λqπt(1− πt) before successπt = 1 after success

• What matters in the equilibrium is the matching setM:

(q, π) ∈M⇔ rep. π VC �nds q pro�table.

• this set encodes the decision to accept or reject a project.• it also encodes the decision to terminate or continue the funding.

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Value functions and the �xed-point• Matching value function v(π, q) non-recursive form :

rv(π, q) = −c+ λqπ [1 + w(1)− v(π, q)]︸ ︷︷ ︸surplus upon

success

−λqπ(1− π)∂πv(π, q)︸ ︷︷ ︸marginal

reputation loss

• Reputation value function w(π) non-recursive form :rw(π) = κϕa [v(π, a)− w(π)]︸ ︷︷ ︸

surplus gainedfrom a-project

χa(π) + κϕb [v(π, b)− w(π)]︸ ︷︷ ︸surplus gainedfrom b-project

χb(π)

• Matching setM⊂ {a, b} × [0, 1]:M = {(q, π) : v(π, q) > w(π)}

〈w〉 〈v,M〉Reputation function Matching variables

Figure 2: Equilibrium feedbacks12 / 27

Equilibrium

Theorem 1: Equilibrium existence and uniqueness

∀{ϕa, ϕb}, ∃ a stationary equilibrium with increasing value functions inπ. This equilibrium is unique for large discount rates.

Regime determination:λa − c > κϕb(λb−c)κϕb+r+λb = Opportunity cost of forgoing the option to wait

0

1

projecttype

π

a[ϕa] b[ϕb]

αHC

(a) high cost matching sets

0

1

projecttype

π

a[ϕa] b[ϕb]

αLC

(b) low cost matching sets

α =c

λb (1 + w(1)) proxy for imperfect learning

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Outline

1. Motivations and contributions

2. Model

3. Equilibrium characterization

4. Innovation spillovers

5. Reputational externality

6. Robust extensions

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Spillovers and endogenous supply of startups

• Interpretations of two types of projects ( λa < λb):• early vs. late stage startups.→ time dimension as 1

λa> 1

λb

• radical vs. incremental innovation→ risk dimension as 1λ2a

> 1λ2b

• Spillovers from early stage small companies late stage businesses.

failed to be internalized in private decisions

• Examples (Mazzucato 2013):• iPhone depends on the Internet; the progenitor of the Internet wasARPANET : a program funded by DARPA in 1960’s.

• Google maps depends on GPS: a US military project called NAVSTAR in1970’s.

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A framework to endogenize• A fraction ζ < 1 of successful early stage projects would translate into

the late stage projects:

κϕbn(1)︸ ︷︷ ︸out�ow from

late stage

= ζλama(1)χa(1)︸ ︷︷ ︸in�ow tolate stage

ϕb =

{ζϕa χa(1) = 1

0 o.w

〈w〉 〈v,M〉

(ϕa, ϕb)

steady state measures

Reputationvalue

Matchingvariables

Figure 4: Equilibrium feedbacks withendogenous mass of projects

• Recall that investment in early stage project takes place i�

λa − c >κϕb (λb − c)κϕb + r + λb

= Opportunity cost of forgoing the option to wait for b-projects16 / 27

Spillover externality

0

c

λb

κζϕa

λa Full investment No investmentInvestment cycles

Figure 5: Investment regimes under endogenous(ϕa, ϕb)

λa − c >

opportunity cost offorgoing the wait︷ ︸︸ ︷κϕb(λb − c)λb + r + κϕb

ϕ̇b = ζλama(1)︸ ︷︷ ︸in�ow

−κϕbn(1)︸ ︷︷ ︸out�ow

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Decentralizing the social optimum I

• What would a hypothetical benevolent planner do? benchmark

• What are the planner’s instruments? At best, the choice of matchingsets→ {(χa(π), χb(π)) : π ∈ [0, 1]}.

• Social optimum:

maxχ

∫ ∞0

e−rt∑

q∈{a,b}

(λqπ − c)Mq(dπ)dt

subject to law of motions for {mq(π), n(π), ϕb} dist. law of motion

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Decentralizing the social optimum II

• Social marginal value functions (to be compared with private marginalvalue functions):

rv∗(π, q) = −c+ λqπ [1 + w∗(1)− v∗(π, q)]− λqπ(1− π)∂πv∗(π, q)

+ ρζλaπ1{q=a}

rw∗(π) =∑q

κϕq [v∗(π, q)− w∗(π)]χ∗q(π)− ρκϕbχ∗b(π)

• Optimal transfers (conceptual, not necessarily implementable from thepublic �nance viewpoint)

• A tax on unmatched investors that choose to invest in late stage projects= ρκϕb

• A subsidy to investors matched with early stage projects = ρζλaπ

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Outline

1. Motivations and contributions

2. Model

3. Equilibrium characterization

4. Innovation spillovers

5. Reputational externality

6. Robust extensions

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Reputation and deal �ow

• Ample evidence that VCs reputation impact their deal �ow e.g(Gompers 1996), (Sorenson and Stuart 2001), (Hsu 2004), (Nanda,Samila and Sorenson 2020).

• In the survey paper (Gompers, Gornall, Kaplan and Strebulaev 2020):23% of VC responded the deal �ow is the most important factor behindthe success.

• sub-sample of low IPO rate investors→ 31%• sub-sample of high IPO rate investors→ 19%

• Aggregate view: higher deal �ow for more reputable ones⇒ lowerdeal �ow for less reputable ones.

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Reputational externality in the meeting rate function• Suppose there is only one group of projects, i.e��ZZλa and λ = λb.• So far, uniform meeting rate⇒ κϕdt. Now, reputational weighting,

i.e:κϕ

ψ(π)∫ψ(π)dF∞(π)

dt = κϕψ(π)

E[ψ(π∞)]︸ ︷︷ ︸µ:=

dt

• Stationary economy→ let investors exogenously enter and leave atthe rate δ.

0 10

1

π

ψ(π)

admissible space of weight functions

Ψ := {ψ : [0, 1]→ [0, 1]|ψ(0) = 0, ψ(1) = 1, ψ′ ≥ 0, ψ′′ ≤ 0}

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(Symmetric) equilibrium vs. the optimum outcome

0 α p 10

m(π)

m(1)

n(p)

n(1)

n(α)

π

dist. of π∞

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(Symmetric) equilibrium vs. the optimum outcome

0 α p 10

m(π)

m(1)

n(p)

n(1)

n(α)

π

dist. of π∞Properties of symmetric equilibrium

• cond (1) steady state µ:µ = E [ψ(π∞)] = M(µ, α).

• cond (2) endogenous separation:α = c

λ(1+w(1))= A(w(1))

• cond (3) endogenous rep. value:w(1) = (r+δ)

−1κϕ/µr+δ+λ+κϕ/µ

(λ− c) =W(µ).

Fixed-point µe = M

µe, αe︷ ︸︸ ︷A ◦W(µe)

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(Symmetric) equilibrium vs. the optimum outcome

0 α p 10

m(π)

m(1)

n(p)

n(1)

n(α)

π

dist. of π∞Properties of symmetric equilibrium

• cond (1) steady state µ:µ = E [ψ(π∞)] = M(µ, α).

• cond (2) endogenous separation:α = c

λ(1+w(1))= A(w(1))

• cond (3) endogenous rep. value:w(1) = (r+δ)

−1κϕ/µr+δ+λ+κϕ/µ

(λ− c) =W(µ).

Fixed-point µe = M

µe, αe︷ ︸︸ ︷A ◦W(µe)

Planner’s optimum:

maxα

rS(µ, α) = (λ− c)m(1) +∫ pα

(λπ − c)m(π)dπ

subject to µ = M(µ, α)23 / 27

Decentralized vs. centralized outcomes

Proposition 2: Equilibrium vs planner outcomes

In the described economy with reputational externality,

(i) there always exists a symmetric equilibrium with 0 < αe < p.(ii) In particular, a local reduction in the termination point αe

increases the social surplus.(iii) Comparative statics: ψ ↑⇒ w(1) ↓, µe ↑ and αe ↑

α∗ αe p0

α

S(α)

0 10

1

ψ1

ψ2

π

ψ(π)

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Outline

1. Motivations and contributions

2. Model

3. Equilibrium characterization

4. Innovation spillovers

5. Reputational externality

6. Robust extensions

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Robust extension

• Economy with short-lived investors, where individuals are subject toexogenous exits and exogenous birth.

• General type space for project qualities, q ∼ φ(dq), and inconclusivebreakthroughs, i.e λq(θ) not necessarily equals θλq :

• λq(θ) is increasing in q.

• Increasing di�erences, i.e for every q′′ > q′:

λq′′(H)− λq′′(L) ≥ λq′(H)− λq′(L) proof sketch

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Conclusion

• Characterization of matching sets in the unique stationaryequilibrium with increasing value functions in reputation:

• Higher tolerance for failure in more reputable VCs.• Higher search frictions could save the market from breakdown when

there is spillover from early- to late-stage projects.

• In presence of reputational externality:• Early termination of startups.• Size of active high-ability investors becomes ine�ciently small.• Strengthening the reputational externality (ψ ↑) leads to lower

tolerance for failure (αe ↑).

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Model diagram

High typeθ = 1 : p

Low typeθ = 0 : 1− p

Investors

Incomplete information⇒ Reputation formationπt = P (θ = 1|It)

Type-bq = b : ϕb

Type-aq = a : ϕa

Projects

Who matchesto whom?

s.t search frictions

(q, π) ∈M

Success rateθλq

Back to the model1 / 5

Non-recursive form of the value functions

• Matching value function:

v(π, q) = supτ

{E

[e−rσ − c

∫ σ0

e−rtdt+ e−rσw(πσ);σ ≤ τ]

+ E

[−c∫ τ0

e−rtdt+ e−rτw(πτ );σ > τ

]}• Reputation value function:

w(π) = supM

{E[e−rmin τqv(π, arg min τq)

]: (q, π) ∈M

}Back to the recursive forms

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Distributional law of motions

ṁq(π) = − λqπmq(π)︸ ︷︷ ︸successful projects

out�ow

+κϕqn(π)χq(π)︸ ︷︷ ︸in�ow of

recent matches

+λq∂π (π(1− π)mq(π))︸ ︷︷ ︸net learning in�ow

ṅ(π) = −∑q

κϕqn(π)χq(π)︸ ︷︷ ︸out�ow of

recent matches

ϕ̇b = ζλa

(ma(1) +

∫πma(π)dπ

)︸ ︷︷ ︸

spillover from successful earlyto late stage projects

− κϕb(n(1)χb(1) +

∫n(π)χb(π)dπ

)︸ ︷︷ ︸

out�ow due torecent partnerships

.

Back to planner’s problem

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Proof sketch for the general type space I

• Projects’ type space q ∼ φ(dq). Assume λL(q) ≤ λH(q) ≤ λ.

• Stopping time problem:reputationfunc. w(π)

Stopping timeproblem

Stopping timeproblem

matching valuefunc.=Tqw

• Fixed-point mapping w = Aw:

[Aw](π) = sup{∫

B[Tqw](π) φ(dq)1 + κr φ(B)

: B ⊂ Supp(φ)}

• Let L1[0, 1] be the underlying space for w, endowed by the partialorder % induced by L1+[0, 1].

• Properties of A:• Positivity: Aw % 0, for every w % 0.• Monotonicity: if w2 % w1 ⇒ Aw2 % Aw1.• Every �xed-point of A is order-bounded by λ/r.

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Proof sketch for the general type space II

• Let 〈0,λ/r〉 :={f ∈ L1+[0, 1] : 0 - f - λ/r

}be the order interval,

then A : 〈0,λ/r〉 → 〈0,λ/r〉 � Knaster–Tarski theorem �

• Use a coupling argument along with the Strassen’s theorem to showA maps increasing functions to increasing functions.

• Construct the sequence w0 = 0 and wn = Awn−1⇒ wn ↑ w∞pointwise, where w∞ is increasing.

• Showing the L1 continuity: ‖Awn −Aw∞‖1 → 0, therefore

w∞ = Aw∞.

• Using the complementarity in λq(θ) to show ∃ α, the lowest boundarypoint ofMb, where b = sup{Supp(φ)}.

back

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Motivations and contributionsModelEquilibrium characterizationInnovation spilloversReputational externalityRobust extensionsAppendix