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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
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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
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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
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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
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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
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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