Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

UBCEconomics 565

November 24, 2014

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

References

• Brief review: Fox (2009)

• Longer review: Graham (2011)

• Extensive notes: Galichon (2011)

• Identification: Fox (2010b), Galichon and Salanié (2010),and many of the papers below

• Applications• Marriage: Choo and Siow (2006) , Galichon and Salanié(2010)

• Mergers: Uetake and Watanabe (2012), Park (2012),Oktay Akkus and Hortaçsu (2012)

• Venture capital: Sørensen (2007)• Downstream - upstream firms: Fox (2010a)• Medical residents: Agarwal (2012)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

1 Introduction

2 Sørensen (2007)

3 Fox (2010a)

4 Fox and Bajari (2013)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Section 1

Introduction

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Introduction

• Matching: payoffs depend on who matches with whom• Examples:

• Firm mergers• Firm upstream/downstream relationships• Workers and firms• Houses for consumers• Marriage

• Model primitive: payoffs of all potential matches

• Equilibrium: pairwise stability - no couple would preferto deviate

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Introduction

• Structural empirical matching models:• Data on observed matches and their characteristics• Goal: estimate payoff function

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Types of matching• Transferable vs non transferable utility

Matching

One-to-one

UnilateralPreferences(housing)

BilateralPreferences(marriage)

Many-to-one(workers to firms)

Many-to-many(players into teams)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching – theory

• Much more developed than empirical work

• Optimal transportation theory – results imply existenceand (in some cases) uniqueness of optimal matching;and existence, uniqueness, and efficiency of equilibrium

• See Galichon (2011) and references therein

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Section 2

Sørensen (2007)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Sørensen (2007) “How Smart IsSmart Money? A Two-SidedMatching Model of Venture

Capital”

• Fact: companies invested in by more experiencedventure capitalists are more likely to go public

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Sørensen (2007) “How Smart IsSmart Money? A Two-SidedMatching Model of Venture

Capital”

• Question: is this because experiences VCs invest inbetter companies or because experienced VCs’influence adds value to companies?

• Matching model used to distinguish these explanations

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Why matching?

• VCs affect company value by:• Monitoring, management• Providing contacts• Signaling value to other investors

• Prior evidence that companies care about identity ofinvestors; do not simply take best financial offer

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Model I

• Set of investors I, set of companies J

• One-to-many: company has one investor; investormany companies

• Valuation of match Vij• Match correspondence µ• Payoffs: non transferable

• Investor: πi(µ(i)) = λ∑

j∈µ(i) Vij

• Company: πj = (1 − λ)Vµ(j)j

• Equilibrium: pairwise-stability• Opportunity cost of deviating for a pair that is notmatched in µ

Vij ≡ Vµ(j)j ∨ minj′∈µ(i)

Vij′

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Model II

• Opportunity cost of remaining in match

Vij ≡ maxi′∈I:Vi′ j>minj′∈µ(i′ ) Vi′ j′

Vi′j ∨ minj′∈J:Vij′ >Vµj′ j′

Vij′

• µ is stable ⇔ Vij < Vij∀ij ∈ µ ⇔ Vij > Vij∀ij ∈ µ• Define Γµ as set of all valuations such that µ is stable

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Empirical model

• Observe µ, investor and company characteristics Wij,Xij, outcomes IPOij

• Vij = W′ijα + ηij

• Likelihood of matches: P(µ ∈ Γµ − Wα)• Outcome: IPOij = 1{Xijβ + εij > 0}• Assume (ε, η) ∼ N

• Estimate using MCMC

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Conclusions

• This paper: use a matching model to correct forselection

• Focus is not necessarily matching by itself

• Does not look at efficiency of matching or anycounterfactuals related to matching

• Related work:• Park (2012): mutual fund mergers, very similar approach• Uetake and Watanabe (2012): bank mergers followingderegulation, moment inequalities based on matchstability

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Section 3

Fox (2010a)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Fox (2010a) “Estimatingmatching games with transfers”

• Context: Car parts suppliers and automotive assemblers

• Goal: estimate revenues from producing differentportfolios of parts

• Motivating examples:• GM considered divesting Opel – potential loss tosuppliers who would not gain as much from specializing

• Asian assembly plants enter North America – howbeneficial to North American parts suppliers?

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Fox (2010a) “Estimatingmatching games with transfers”

• Data: part suppliers and assembler matches• No observed prices

• Approach: use equilibrium conditions of matchingmodel to identify revenue function

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Model I

• Two-sided, many-to-manyUpstream Downstream

Example Parts AssemblersCharacteristics u ∈ U d ∈ DTransfer t −t

• Full match: ⟨u, d, t⟩• Physical matches M = {⟨u1, d1⟩, ..., ⟨rn, dn⟩}

• Mu = set of d matched with u• Md = set of u matched with d

• Revenue functions rup(Mu), rdown(Md)• Equilibrium: pairwise stability ⇒

rup(Mu1) + tu1,d1 ≥ rup (Mu1 \ ⟨u1, d1⟩ ∪ ⟨u1, d2⟩) + tu1,d2

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Estimator I

• Add pairwise stability conditions to eliminate transfers:

rup(Mu1) + rdown(Md1) + rup(Mu1) + rdown(Md1) ≥≥rup (Mu1 \ ⟨u1, d1⟩ ∪ ⟨u1, d2⟩) +

+ rdown (Md1 \ ⟨u1, d1⟩ ∪ ⟨u2, d1⟩

)+

+ rup (Mu2 \ ⟨u2, d2⟩ ∪ ⟨u2, d1⟩) ++ rdown (

Md2 \ ⟨u2, d2⟩ ∪ ⟨u1, d2⟩)

• Use to form maximum score estimator• Assume r(M) = Z(M)′β• Then inequalities ⇔ X′

u1,u2,d1,d2β ≥ 0• Objective:

QH(β) = 1H

∑

h∈H

∑

u,d

1{X′u1,u2,d1,d2β ≥ 0}

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Empirical specification I

• Data from SupplierBusiness on 1252 suppliers, 14assemblers with 52 brads, 392 models, and 52492 partsin 187 component categories

• Data includes mainly North American and Europeanfirms

• Observation:⟨ u︸︷︷︸

part supplierFederal-Mogul

, d︸︷︷︸car model

Fiat

, l︸︷︷︸part

front pads

⟩ in market

h︸︷︷︸componentdisk brakes

• Revenue function:• rup(M) = Zup(M)′βup

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Empirical specification II

• Zup(M) = measure of specialization of supplier in itsmatches, specifically HHI of parts across

• Assemblers, brands, models, continents

•

• rdown(M) = Zdown(M)′βdown

• Zdown(M) = measure of specialization of parts in itsmatches, specifically HHI of suppliers across

• Assemblers, brands, models

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Results

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Results

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Results

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Other applications andextensions I

• Identification: Fox (2010b)

• Unobserved heterogeneity: Fox and Yang (2012)• Matching maximum score estimator of Fox (2010a) usedin

• Fox and Bajari (2013): FCC spectrum auction – no tradesafter auction implies pairwise stability

• Oktay Akkus and Hortaçsu (2012): bank mergers,matching with observed transfers

• Levine (2009): pharmaceutical marketing firms anddrugs

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Section 4

Fox and Bajari (2013)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Fox and Bajari (2013)“Measuring the Efficiency of an

FCC Spectrum Auction”

• Estimates an auction model using pairwise stability• C block FCC spectrum auctions 1995-1996

• Simultaneous ascending auctions for 480 geographicareas

• Theory & evidence from other FCC auctions suggestscollusion

• Goal: estimate distribution of valuations & andallocative efficiency

• Identifying assumption: allocation of licenses ispairwise stable in matches, that is, an exchange of twolicenses by winning bidders must not raise the sum ofthe valuations of the two bidders

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Background

• C block 1900 MHz spectrum used for mobile phones• Auction format:

• Multiple rounds• Each round, simultaneously submit bids (or not) on all480 regions

• Auction ends when no more bids placed on any item ina round

• Lasted 185 days

• Only new carriers participated (small business discount)

• 255 bidders, 85 winners, most either went bankrupt ormerged with incumbent carriers

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Suggestive evidence ofintimidatory collusion

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Valuations

• a = 1, ..,N bidders, j = 1, .., L licenses• Profit of bidder a from J ⊂ L

πa(J) −∑

j∈J

pj

• Parameterization:

πa(J) = πβ(wa, xJ)︸ ︷︷ ︸±1·eliga·(

∑jinJ popj)+β′complemj

+∑

j∈J

ξj +∑

j∈J

εa,j

• π(w, x) and ξj common knowledge of bidders, ξjunobserved by econometrician

• ε i.i.d., private for bidders, unobserved byeconometrician

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Measuring complementarities

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

• Each bidder makes a payment before the auction beginsfor initial eligibility. A bidder’s eligibility is expressed inunits of total population. A bidder cannot bid on apackage of licenses that exceeds the bidder’s eligibility.

• geomcomplemJ =∑

i∈J popi

∑j∈J\{i}

popipopj

distδi,j∑

j∈L\{i}popipopj

distδi,j

• travelcomplemJ =∑

i∈J popi∑

j∈J\{i} trips(origin i, destination j)∑j∈L\{i} trips(origin i, destination j)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Pairwise stability

• Pairwise stable in matches:

πa(Ja)+πb(Jb) ≥ πa ((Ja \ {ia}) ∪ {ib})+πb ((Jb \ {ib}) ∪ {ia})

• Evidence for:• Often satisfied in experimental data• Swaps did not occur after auction• Holds in theoretical models of ascending auctions withcollusion: Brusco & Lopomo (2002) andEngelbrecht-Wiggans & Kahn (2005) and Milgrom (200)

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Estimation

• Similar to Fox (2010a), but without transfers

• Objective function = sum of indicators for pairwisestability inequalities

• Fixed effects drop out of pairwise stability conditionsfrom differencing

• Inference through subsampling

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Interpretation

• SD of elig · (∑

pop) 0.029, SD of geocomplem is 0.024

• βgeo = 0.32, so geocomplem 32% as important aspopulation

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

• 6.7 and 9.8 are implausibly large — imply increasingcomplementarity is worth as much as having 6 timesthe population in the area

• Model with prices implies value of nationwide license is$360 billion, but total bids where $10 billion, annualrevenues in 2006 were $113 billion

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Agarwal, Nikhil. 2012. “An Empirical Model of the MedicalMatch.” URL http://www.people.fas.harvard.edu/~agarwal3/papers/AgarwalJMP.pdf.

Choo, Eugene and Aloysius Siow. 2006. “Who Marries Whomand Why.” Journal of Political Economy 114 (1):pp. 175–201.URL http://www.jstor.org/stable/10.1086/498585.

Fox, Jeremy T. 2009. “Matching models: empirics.” In TheNew Palgrave Dictionary of Economics, edited by Steven N.Durlauf and Lawrence E. Blume. Basingstoke: PalgraveMacmillan. URL http://www.dictionaryofeconomics.com/article?id=pde2009_M000426.

———. 2010a. “Estimating Matching Games with Transfers.”Working paper, University of Michigan. URLhttp://www-personal.umich.edu/~jtfox/working-papers/matching_fox.pdf.

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

———. 2010b. “Identification in matching games.”Quantitative Economics 1 (2):203–254. URLhttp://dx.doi.org/10.3982/QE3.

Fox, Jeremy T. and Patrick Bajari. 2013. “Measuring theEfficiency of an FCC Spectrum Auction.” AmericanEconomic Journal: Microeconomics 5 (1):100–146. URLhttp://www.aeaweb.org/articles.php?doi=10.1257/mic.5.1.100.

Fox, Jeremy T. and Chenyu Yang. 2012. “UnobservedHeterogeneity in Matching Games.” Working Paper 18168,National Bureau of Economic Research. URLhttp://www.nber.org/papers/w18168.

Galichon, Alfred. 2011. “Theoretical and Empirical Aspects ofMatching Markets.” URLhttp://alfredgalichon.com/wp-content/uploads/2012/10/MatchingnotesColumbia.pdf.

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Galichon, Alfred and Bernard Salanié. 2010. “Matching withtrade-offs: revealed preferences over competingcharacteristics.” URL http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1640380.

Graham, B. 2011. “Econometric methods for the analysis ofassignment problems in the presence of complementarityand social spillovers.” Handbook of social economics1:965–1052. URL http://128.32.105.3/~bgraham/Published/HandbookOfSocialEconomics_1B_2011/BSG_HandbookOfSocialEconomics_1B_2011.pdf.

Levine, Anna A. 2009. “Licensing and scale economies in thebiotechnology pharmaceutical industry.” URLhttp://apps.olin.wustl.edu/workingpapers/pdf/2010-11-004.pdf.

Matching

Paul Schrimpf

Introduction

Sørensen(2007)

Fox (2010a)

Fox and Bajari(2013)

References

Oktay Akkus, J. Anthony Cookson and Ali Hortaçsu. 2012.“The Determinants of Bank Mergers: A RevealedPreference Analysis.” URLhttps://docs.google.com/uc?export=download&id=0B9xl-vM6JOoDMjVtVXhiNWt5bjA.

Park, Minjung. 2012. “Understanding Merger Incentives andOutcomes in the US Mutual Fund Industry.” URL http://faculty.haas.berkeley.edu/mpark/MergerPaper.pdf.

Sørensen, Morten. 2007. “How Smart Is Smart Money? ATwo-Sided Matching Model of Venture Capital.” TheJournal of Finance 62 (6):pp. 2725–2762. URLhttp://www.jstor.org/stable/4622352.

Uetake, Kosuke and Yasutora Watanabe. 2012. “Entry byMerger: Estimates from.” URLhttp://www.kellogg.northwestern.edu/faculty/watanabe/htm/EntryByMerger.pdf.