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# Optimal Certification Design

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NES 20th Anniversary Conference, Dec 13-16, 2012
Optimal Certification Design (based on the article presented by Sergei Kovbasyuk at the NES 20th Anniversary Conference).
Author: Sergei Kovbasyuk

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### Optimal Certification Design

1. 1. Optimal certication design Sergei KovbasyukEinaudi Institute for Economics and FinanceNES 20th Anniversary Conference, Moscow December 15, 2012
2. 2. Seller pays certication with public ratings. Credit rating agencies: Seller Certier r ∈ {AAA, AAB , ..} t (r )\$ Buyers Buyers interpret ratings µ(.|r ) Critique: rating ination Pagano-Volpin (2009), Grin-Tang (2009). Solutions: 1) mandate t = const , 2) require buyers to pay?
3. 3. Objective: 1. Characterize certication under two regimes: Public contingent fee: t (.) is public knowledge Private contingent fee: t (.) is private to seller and certier 2. Draw policy implications.
4. 4. Model product of unknown quality θ ∼ U [0, 1] monopolistic seller, zero cost mass 1 of buyers buys quantity q at price p buyers utility S (θ , q , p ) = θ q − q γ /γ − pq , γ ≥ 2 certier learns θ , issues m ∈ M and receives t (m) Assumptions: 1. products quality θ is private to the certier 2. certiers objective U (b, m) = λ S (θ , q , p ) + t (m) λ stands for potential litigation costs with buyers, contracts, reputation.
5. 5. Timing M , t is dened. certier learns θ and issues m(θ ) seller observes m and sets p (m) (no commitment) buyers form beliefs µ(θ |m) and buy q (p , m) Perfect Bayesian Equilibrium, M and λ are common knowledge 1. public fee: t (.) is common knowledge 2. private fee: the seller privately proposes t (.) to the certier.
6. 6. Modied revelation principle Lemma 1 For any equilibrium under any {M , t (.)} there exists an outcome-equivalent equilibrium in pure strategies under a grading {G , t (.)} in which the certier announces rating r = EF [θ |θ ∈ gr ] whenever θ is in grade gr . Figure: An example of a grading 0 r=0.2 0.4 r = θ ∈ [0.4, 0.9] 0.9 1 rating r = 0.2 corresponds to a grade g . = [0, 0.4); 0 2 rating r = 0.55 corresponds to a grade g . = 0.55. 0 55
7. 7. 1. Public payment A feasible certication is a grading G and a payment t (.) such that: Buyers: q (r , p ) = arg max EF [S (θ , q , p )|θ ∈ gr ], ∀r ∈ G . (1) q ≥0 Seller: p (r ) ∈ arg max pq (r , p ), ∀r ∈ G . (2) p ≥0 Certier: r (θ ) ∈ arg max [λ S (θ , q (r , p (r )), p (r )) + t (r )], ∀θ . (3) r ∈G Truthtelling: r (θ ) = r , ∀θ ∈ gr , ∀r ∈ G . (4) Limited liability: t (r ) ≥ 0, ∀r ∈ G . (5)
8. 8. Certiers intrinsic bias Suppose: naive buyers, t = const , perfect ratings r ∈ [0, 1]. Proposition 1 Under a xed fee the certier is downward biased: reports θ θ . ∂ [λ S (θ ,q (r ),p (r ))+t ] certiers (IC) requires ∂r = 0 , suppose r = θ ∂ S (.) from the buyers problem ∂ q |r =θ = 0, thus ∂ S r = ∂ ∂ p ∂ ∂ (.) 0. ∂ (.) S (.) p r From now on Bayesian buyers! Corollary 1 Under a xed fee only the uninformative certication is feasible. ∼ Crawford-Sobels uninformative equilibrium for an extreme bias.
9. 9. Certiers intrinsic bias Suppose: naive buyers, t = const , perfect ratings r ∈ [0, 1]. Proposition 1 Under a xed fee the certier is downward biased: reports θ θ . ∂ [λ S (θ ,q (r ),p (r ))+t ] certiers (IC) requires ∂r = 0 , suppose r = θ ∂ S (.) from the buyers problem ∂ q |r =θ = 0, thus ∂ S r = ∂ ∂ p ∂ ∂ (.) 0. ∂ (.) S (.) p r From now on Bayesian buyers! Corollary 1 Under a xed fee only the uninformative certication is feasible. ∼ Crawford-Sobels uninformative equilibrium for an extreme bias.
10. 10. Optimal certication under a public payment Assumption: The seller chooses G , t (.) ex ante. 1 max (pq − t )dF (θ ), s .t .(1), (2), (3), (4), (5). {G ,t (.)} 0 Proposition 2 An optimal certication under a public contingent payment yields a pooling rating with no payments for low qualities θ θ ∗ and full revelation with increasing payments for high qualities θ ≥ θ ∗ .
11. 11. Illustration: λ = 2 , γ = 2 ⇒ U = 1 (θ q − 1 q 2 ) + t 1 2 2 r,t 1 θ∗ r (θ ) 2 t (r (θ )) 0 θ∗ 1θ
12. 12. 2. Private payment (feasible certication) given G publicly known seller privately proposes t (.) certier learns θ and reports r ∈ G given r buyers believe the quality to be in gr . Seller respects (1), (2), (3), (5), BUT she ignores truthtelling (4): 1 max (pq − t )dF (θ ), s .t .(1), (2), (3), (5). (6) {t (.)} 0 A grading G is feasible under a private payment i (4), (6) hold.
13. 13. Rating ination under a private payment Proposition 3 A feasible certication G , t under a private payment has no intervals of perfect revelation: G has a countable number of coarse ratings. Figure: A feasible certication for U = θ q − q 2 /2 + t . 0 θ 1
14. 14. Optimal certication under a private payment Sellers problem: 1 max (pq − t )dF (θ ), s .t .(1), (2), (3), (4), (5), (6). {G ,t (.)} 0 Proposition 4 Under a private payment the uninformative certication with zero payments is optimal : G = 1 and t = 0. 2
15. 15. Social welfare S = E [θ q − pq − q γ /γ|G , t ], Π = E [pq |G , t ]. W = S +Π+λS Proposition 5 Social planner is information loving (prefers perfect revelation). Corollary 1 Public contingent payments dominate xed payments and private contingent payments.
16. 16. Conclusion Certier is endogenously downward biased. Certication is uninformative under a xed fee. Optimal certication under a public payment commands pooling below a threshold, and perfect certication above the threshold. Only imprecise ratings are feasible under private payments (rating ination). Under private payments seller prefers uninformative certication. Policy recommendations: Transparency of certiers fees is benecial Mandating xed fees can be harmful