Viscusi (1978) shows how, in markets with quality uncertainty, perfect certification results in separation from top down due to an unraveling process similar to Akerlof (1970). De and Nabar (1991) argue that imperfect certification prevents unraveling so that equilibria with full separation do not exist. This note shows that, if one considers the buyers' buying decision explicitly, a separating equilibrium with imperfect certification does exist

Strausz, Roland

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Separating Equilibria with Imperfect Certification

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Sonderforschungsbereich/Transregio 15  www.sfbtr15.de Universität Mannheim  Freie Universität Berlin  Humboldt-Universität zu Berlin  Ludwig-Maximilians-Universität München Rheinische Friedrich-Wilhelms-Universität Bonn  Zentrum für Europäische Wirtschaftsforschung Mannheim  Speaker: Prof. Dr. Urs Schweizer.  Department of Economics  University of Bonn  D-53113 Bonn, Phone: +49(0228)739220  Fax: +49(0228)739221                       *Humboldt–University at Berlin,  E-mail: strauszr@wiwi.hu-berlin.de      June 2010    Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. Discussion Paper No. 324 Separating Equilibria with Imperfect Certification   Roland Strausz*  Separating Equilibria with Imperfect CertificationRoland Strausz∗April 15, 2010AbstractViscusi (1978) shows how, in markets with quality uncertainty, perfect certificationresults in separation from top down due to an unraveling process similar to Akerlof(1970). De and Nabar (1991) argue that imperfect certification prevents unravelingso that equilibria with full separation do not exist. This note shows that, if oneconsiders the buyers’ buying decision explicitly, a separating equilibrium with imperfectcertification does exist.Keywords: certification, unraveling, separating equilibriumJEL classification: D82, L151 IntroductionViscusi (1978) shows how, in markets with quality uncertainty, perfect certification resultsin separation from top down due to an unraveling process similar to Akerlof (1970). Deand Nabar (1991) argue that imperfect certification prevents unraveling so that equilibriawith full separation do not exist. The authors, subsequently, conclude in their abstract that∗Contact details: Roland Strausz, Humboldt–Universita¨t Berlin, strauszr@wiwi.hu-berlin.de; I acknowl-edge support from the French-German cooperation project ’Market Power in Vertically Related Markets’funded by the Agence Nationale de la Recherche (ANR) and Deutsche Forschungsgemeinschaft (DFG).1imperfect certification results in “equilibrium outcomes which are very different from thoseimplied by models of perfect certification”. This conclusion casts doubts on the robustness oftheoretical work on certification that, for reasons of tractability, assumes perfect certificationtechnologies (e.g., Biglaiser 1993, Lizerri 1999, Albano and Lizzeri 2001, Strausz 2005, Faure–Grimaud et al. 2009, Stahl and Strausz 2010). This note shows that, if one considers thebuyers’ buying decision explicitly, a separating equilibrium with imperfect certification doesexist. As a consequence, the results from certification models that are based on perfectcertification are more robust to imperfections in the certification technology than the resultsof De and Nabar (1991) suggest.2 SetupTo demonstrate my argument, I follow exactly the setup of De and Nabar (1991), whoconsider a market for a perishable commodity with n potential sellers of one unit of thecommodity. The quality (t) of the product could be either type A or type B.1 Under fullinformation, type A and type B products would be valued as VA and VB respectively by eachof the set of potential buyers, I, where VA > VB > 0. A fraction π ∈ (0, 1) of all suppliers isendowed with type A product. Each supplier knows the true quality level of his product. Inan asymmetrically informed market, a supplier’s decision (d), where d ∈ D = {C,U}, couldbe either to offer his product for sale uncertified (d = U) or get the quality level of the goodcertified (d = C) for an exogenously fixed cost, K < VA − VB, prior to offering it for sale.2The certification process results in a rank (r), where r ∈ R = {1, 2}, for the product whichcould be either 1 or 2. The certification process is such that a type A product gets a rankingof 1 with probability qA while a type B product gets a ranking of 1 with probability qB with1 ≥ qA > qB ≥ 0. Certification is perfect when qA = 1 and qB = 0. The buyers, i ∈ I,who are all alike, do not observe the true quality level of any product before purchase. Theydo, however, know the distribution of types (π) and the characteristics of the certification1As in De and Nabar (1991), I consider only two types, but results extend to more than two types.2Clearly, for K > VA − VB certification is too costly to be helpful.2process (qA, qB). In addition, they observe each supplier’s decision d and, for those supplierswith d = C, also the certifier’s ranking r.De and Nabar introduce P (U) as the price of an uncertified good and P (C, r) as the priceof a good that is certified with rank r. They assume that ”all economic agents are risk-neutraland all markets are competitive” and I follow their idea that, ”in a competitive market, theprice the buyers pay for a product would equal the value they expect from it. In other words,in such a market, P (C, r) = VAρ(A|C, r)+ VBρ(B|C, r) and P (U) = VAρ(A|U) + VBρ(B|U),where ρ indicates market posterior beliefs”.De and Nabar, subsequently, treat supply and demand in a reduced form and, implicitly,assume that when the price for a product equals the value that buyers expect from it, allgoods on the market are bought. It is here that I extend the analysis and explicitly introduceα(P, U) as the probability that buyers buy an uncertified good with a price P and α(P,C, r)as the probability that an uncertified good is sold when the price is P .Because a supplier with quality t has probability qt of obtaining rank r = 1, the expectedrevenue for a supplier of quality t who certifies isW (t, C, P ) = qtP (C, 1)α(P (C, 1), C, 1)+ (1− qt)P (C, 2)α(P (C, 2), C, 2)−K.His revenue from selling the good uncertified is W (t, U, P ) = P (U)α(P (U), U). A com-parison to De and Nabar confirms that the extension incorporates their framework forα(P (C, 1), C, 1) = α(P (C, 2), C, 2) = α(P (U), U) = 1.Based on Kreps and Wilson (1982), De and Nabar define a Nash Sequential Equilibrium(NSE), as an ordered triple (d∗t, P ∗, ρ) satisfying the three conditions supplier rationality,buyer competitiveness, belief consistency. Because I consider the buyers’ buying behaviorexplicitly, I need to extend these conditions by buyer rationality. Hence, I define a NashSequential Equilibrium (NSE) as a combination γ∗ = (d∗t , α∗, P ∗, ρ∗) that satisfies:(I’) supplier rationality: d∗t∈ argmaxd{W (t, d, P )}, d ∈ [0, 1], t ∈ T .(I”) buyer rationality: α∗(P, U) ∈ argmaxαα[VAρ(A|U)+VBρ(B|U)−P ] and α(P,C, r) ∈argmaxαα[VAρ(A|C, r) + VBρ(B|C, r)− P ], t ∈ T , r ∈ {0, 1}.3(II) buyer competitiveness: P ∗(C, r) =∑Vtρ(t|C, r), P∗(U) =∑Vtρ(t|U).(III) belief consistency: If C or U is an equilibrium strategy, the posterior beliefs of eachbuyer i, ρ∗(t|C, r) or ρ∗(t|U), are determined by Bayes rule. If C or U is an off-equilibriumstrategy, then ρ∗(t|C, r), ρ∗(t|U) ∈ [0, 1].3 ResultsIn this note, I am interested in the existence of a separating equilibrium in which only sellerswith the high quality A certify, i.e., (dA, dB) = (C,U). Viscusi (1978) already noted theexistence of such an equilibrium when certification is perfect but not too costly. With perfectcertification, the sellers’ separation strategies (C,U) together with the belief consistencyrequirement (III) pin down the beliefs ρv(A|C, 1) ≡ 1 and ρv(A|U) ≡ 0, but leave the beliefρv(A|C, 2) undetermined. Hence, ρv(A|C, 2) ≡ 0 is consistent with the belief consistencyrequirement. Due to the buyer competitiveness requirement (II), the belief structure ρv(.|.)implies P v(U) ≡ VB, Pv(C, 1) ≡ VA, and Pv(C, 2) ≡ VB. Given these prices, the separatingstrategy (dsA, dsB) is consistent with supplier rationality requirement (III). Hence, in theframework of De and Nabar (1991) there exists a separating equilibrium when certificationis perfect. With appropriate choices of α(., .), the separation equilibrium also exists in theextended framework.Proposition 1 (Viscusi 1978) With perfect certification (qA, qB) = (1, 0) there exists anequilibrium that sustains the separating strategy (dA, dB) = (C,U).Proof: It is straightforward to check that the combination γv ≡ (dvt, αv, P v, ρv) withαv(P, U) ≡ 1 for P ≤ VB and α(P, U) ≡ 0 for P > VB, α(P,C, 1) ≡ 1 for P ≤ VA andα(P, U) ≡ 0 for P > VA, α(P,C, 2) ≡ 1 for P ≤ VB and α(P, U) ≡ 0 for P > VB satisfiesthe conditions (I’), (I”), (II), and (III) and is therefore an NSE. Q.E.D.De and Nabar (1991) argue that there does not exist a separation equilibrium whencertification is imperfect. I concentrate on the case qA < 1 and qB ∈ [0, qA), where theargument of De and Nabar is clearest. In this case, the belief consistency condition (III)4implies not only that ρ∗(A|C, 1) = 1 and ρ∗(B|U) = 0, but also that ρ∗(A|C, 2) = 1.Hence, the buyer competitiveness condition (II) implies P ∗(C, 1) = P ∗(C, 2) = VA andP ∗(U) = VB. Because De and Nabar, effectively, assume α(P (U), U) = α(P (C, 1), C, r) =α(P (C, 2), C, r) = 1, we have W (A,C, P ) = W (B,C, P ) = VA − K and W (A,U, P ) =W (B,U, P ) = VB. Due to K < VA − VB, it follows that also the seller of quality qB has astrict incentive to certify. This leads to the following result.Proposition 2 (De and Nabar 1991) For any qA < 1, there does not exist an equilibriumwith α(P (U), U) = α(P (C, 1), C, r) = α(P (C, 2), C, r) = 1 that sustains the separatingstrategy (dA, dB) = (C,U).Proposition 2 suggests that the existence of a separating equilibrium crucially dependson perfect certification and is not robust to imperfections, however small, in the certificationtechnology. Yet, because the buyers at the prices P ∗(C, 1), P ∗(C, 2) and P ∗(U) are actuallyindifferent, the buying behavior α(P (U), U) = α(P (C, 1), C, r) = 1 and α(P (C, 2), C, r) = 0is also consistent with buyer rationality (I’). In this case, we have W (A,C, P ) = qaVA −Kand W (B,C, P ) = qBVA −K so that we have a separating equilibrium when qBVA − VB <K < qAVA−VB. If we define q¯ ≡ (K +VB)/VA so that, due to K < VA−VB, we have q¯ < 1,we obtain the following result.Proposition 3 For imperfect certification (qA, qB) with qB < q¯ and qA > q¯ there exists anequilibrium with α(P (U), U) = α(P (C, 1), C, r) = 1 and α(P (C, 2), C, r) = 0 that sustainsthe separating strategy (dA, dB) = (C,U).Proof: From qB < q¯ and qA > q¯ it follows that qBVA − VB < K < qAVA − VB so thatunder the buyers’ buying behavior α and prices P ∗(C, 1), P ∗(C, 2) and P ∗(U), the sellerof quality VA has a strict incentive to certify, whereas the seller of quality VB has a strictincentive not to certify. To complete the proof, the buyers’ rationality condition pins downthe buying behavior α for all remaining prices. In particular, α(P, U) = 1 for P < VBand α(P, U) = 0 for P > VB, α(P,C, 1) = 1 for P < VA and α(P, U) = 0 for P > VA,α(P,C, 2) = 1 for P < VA and α(P, U) = 0 for P > VA. Q.E.D.54 ConclusionWhen buyers and sellers are price takers, a separation equilibrium (C,U) exists when theimperfection is not too extreme. In particular, an equilibrium exists when the certifica-tion technology is close to the perfect one (qA, qB) = (1, 0). In this sense, the separationequilibrium of Proposition 1 is robust to imperfections in the certification technology.The robustness result does not only hold when buyers and sellers are price takers, butalso when, as in Stahl and Strausz (2010), the privately informed seller sets prices. In thiscase, the seller’s price is a signal about his quality and the buyer’s belief will depend on it.The competitive prices P ∗(C, 1) = P ∗(C, 2) = VA and P∗(U) = VB of Proposition 2 and 3can be sustained with the out–off–equilibrium belief that buyers interpret any other price asan indication that quality is qB.LiteratureAlbano, G. and A. Lizzeri, 2001, Strategic Certification and Provision of Quality, Interna-tional Economic Review 42, 267–284.Biglaiser, G., 1993, Middlemen as Experts, RAND Journal of Economics 24, 212–223.De, S. and P. Nabar, 1991, Economic implications of imperfect quality certification, Eco-nomics Letters 37, 333-337Faure-Grimaud, A., E. Peyrache, and L. Quesada, 2009, The ownership of ratings, RANDJournal of Economics 40, 234–257.Lizzeri, A., 1999, Information Revelation and Certification Intermediaries, RAND Journalof Economics 30, 214–231.Stahl, K. and R. Strausz, 2010, Certification as an inspection versus a signalling device,mimeo Humboldt–Universita¨t.Strausz, R. (2005), Honest certification and the threat of capture, International Journal ofIndustrial Organization 23, 45-62Viscusi, W., 1978, A note on ”lemons” markets with quality certification, The Bell Journal6of Economics 9, 277-2797

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Separating Equilibria with Imperfect Certification

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