l 2 a slides screening 2 types

ADVANCED MICROECONOMICS II:INCENTIVE THEORY

Lecture 2a: Adverse Selection - Screening With Two Types

Prof. Christian KEUSCHNIGGUniversity of St. Gallen, FGN-HSG

FGN-HSG

November 3, 2014

Christian Keuschnigg (FGN-HSG) L2a: Screening - Two Types November 3, 2014 1 / 22

Introduction

Theory of incentives and information: bilateral contracting

principal wants to control decisions of agentshidden information (adv. selection) vs. hidden action (moral hazard)

Hidden information: w.r.t. agents type (tastes, technology...)1 screening: contract proposed by uninformed party2 signaling: contract proposed by informed party

Examples: of screening problems

regulation and price discrimination, credit rationingoptimal income taxation, labor contracts

This lecture: basic theory, 2 types and many types


1. The Economic Problem

Principal: faces agent with private information of her typeproblem of non-linear pricing by a monopolistic sellerwho faces a buyer with unknown valuation for her product

Buyer: dierent types q, with linearly separable preferences

u (q,T , q) = qv (q)! Tpayment T to seller for quantity q of good, v 0 (q) > 0 > v 00 (q)preference parameter q: 2 types, high or low valuation, qH > qLdemand characteristic q is private information of buyerseller knows only distribution b = Pr (q = qL), 1! b = Pr (q = qH )

Seller: rm with constant unit production cost c > 0,sell q units for a payment T of money, prot amounts to

p = T ! c # qQuestion: what is the best (prot maximizing) contract (T , q)

so that seller (principal) is able to induce buyer (agent) to accept?


michelesalviT PaymentqQualityVValuationthetaPreference Parameter

michelesalvicProduction costPiProfitTRevenue

1. Preference Characteristics

Indierence curves: equal utility (T , q)-bundles of type qi

T = qi v (q)! u, dTdq!!!!du=0

= qi v 0 (q) = MRSi

Single-crossing: Spence-Mirrlees condition, Figure 1

same bundle (T , q)) qH -type has larger valuation MRSisame budget line ) qH -type demands larger quantity

Demand: given payment-schedule Ti = Pqi + F

ui = qi v (qi )! Ti ) qi v 0 (qi ) = P


michelesalviClosed form indifference curve-> all give the sam utility

michelesalviMarginal Rate of Substitutione.g. one bottle more, how much to pay to remain on the same utility level

michelesalviH-typeTheta high, Higher MRSL-typeTheta low, Lower MRS-> the indifference curves only cross once

michelesalviIs an example of a linear payment schedule (linear price)uutilityPPrice of an extra bottle

Figure 1: Single Crossing (Spence Mirrlees Condition)


michelesalviFor h-Type, this is not optimal

michelesalviThis is optimal for h-Type consumers, as they value the consumption of this good high

2. First Best Allocation With Full Information

Social value: of seller-buyer relationship, Si = ui + pi or

ui = qi v (qi )! Ti , pi = Ti ! cqi , Si = qi v (qi )! cqiAgent: with reservation utility (outside value) u

participation (individual rationality) constraint: qi v (qi )! Ti % uu reects agents bargaining power relative to principal

Full information: seller observes buyers type, Fig. 2

seller can oer type specic contracts (Ti , qi ) for each type qi

pi = maxTi ,qi Ti ! cqi s.t. qi v (qi )! Ti % usubstitute Ti into pi , and maxqi pi = qi v (qi )! cqi ! u

FB : qi v 0 (q&i ) = c , T&i = qi v (q

&i )! u


michelesalviu + piUtility plus Profit(both in money values)

michelesalviU_barIs reservation utility (and also a way to specify bargaining power)

michelesalviThe price will gown down until it hits the consumers reservation utility

Figure 2: First Best With Two Part Tari


2. First Best Allocation With Full Information

First best: maximizes social value (total surplus) Si , Fig. 2

rst best reects perfect price discriminationwith type specic contract T &i , q

&i

marginal utility (price) equals marginal costprincipal gets full surplus, no rent to buyer, ui = u

Implementation: type specic 2 part tari Ti = cqi + picharge price c and xed fee pi to extract rent, i.e.

ui = qi v (qi )! Ti = u ) p&i = qi v (q&i )! cq&i ! u

First best: optimal contract maximizes total surplus Siparticipation constraint determines how total surplus is shared


michelesalvi

3. Hidden Information - Simple Solutions

Asymmetric information: seller cant observe typesoer same contracts fT (qi ) , qig to everybody, consider now:(i) linear pricing, (ii) two part tari, (iii) non-linear pricing

Linear pricing: contract species only price p, i.e. T (q) = pqgiven p, buyer i chooses quantity qi to maximize surplus (utility)

Si = maxq qi v (q)! pqf.o.c. yields demand functions, D 0i (p) = 1/

"qi v 00i

#< 0

qi v0 (qi ) = p ) qi = Di (p)

and buyer i gets a maximum surplus (utility)

Si (p) = qi v (Di (p))! pDi (p)Expected demand/surplus: sum up over types

D (p) = b #DL (p) + (1! b) #DH (p) ,S (p) = b # SL (p) + (1! b) # SH (p)



Linear pricing: set prot maximizing monopoly price p

p = maxp (p ! c) #D (p)

f.o.c. implies mark-up pricing (note D 0 < 0)

D + (p ! c)D 0 = 0 ) pm = c + D (p)!D 0 (p) > c

Linear pricing: Figure 3, monopoly price implies

ineciently low consumption since qi v 0 (qi ) = p > c ,max. prot by restricting demand and raising p above unit costbuyers are left with rent (area under demand curve) Si (p) > 0,Si =

R q0 p (x) dx ! pq, where

R qi0 p (x) dx =

R qi0 qi v

0 (x) dx = qi v (qi )

Question: can seller do better and extract part of buyers rent?


Figure 3: Linear Monopoly Pricing

Optimality: q + (p ! c) dqdp = 0 , p + q dpdq = c or MR = MC



Two part tari: no explicit discriminationpayment schedule: price p and xed fee Z , same for both types

T (q) = pq + Z

utility ui = maxq qi v (q)! (pq + Z ) = Si (p)! Zby envelope theorem, Si (p) = qi v (q)! pq yields S 0i (p) = !Di (p)

Participation: rm wants to serve both typescan extract maximum fee Z * SL (p) such that uL % 0

Prot: max. p = T ! cq with T = pD (p) + SL (p), i.e. Z = SL (p)p = maxp (p ! c)D (p) + SL (p)

f.o.c. D (p) + (p ! c)D 0 (p) + S 0L (p) = 0, implying mark-up

p = c +D (p) + S 0L (p)!D 0 (p) > c

note D + S 0L = bDL + (1! b)DH !DL = (1! b) (DH !DL) > 0Christian Keuschnigg (FGN-HSG) L2a: Screening - Two Types November 3, 2014 12 / 22


Compare prices: pm > pd > pc = c

monopoly > 2-part > rst best competitive pricesmall price reduction starting from monopoly price pm

1 2nd order negative eect on prot (pm ! c)D (pm),due to envelope theorem, pm maximizes monopoly prot

2 1st order positive eect of lower price on surplus (S 0L = !DL),seller can extract this gain by raising the xed fee Z

small price increase starting from competitive price pc

1 1st order positive eect on prot (pc ! c)D (pc ),2 2nd order negative eect on surplus SL ,

envelope theorem, competitive price pc maximizes surplus

rst best not optimal for monopolist,mark-up pd > c leads to underconsumption

Compare prots: pd > pm > 0, 2-part contract more protable

xed fee Z extracts more surplus from buyers (uL = 0, uH lower)


Figure 4: Optimal Two Part Tari

Remark: qH -type strictly prefers allocation BH to BL,seller could extract even more rents, i.e. from qH -type


4. Second Best - Optimal Nonlinear Pricing

Idea: rm can do better than with a 2-part tariNon-linear contracts: rm cannot observe buyers type

1 oer set of choices [q,T (q)] independent of type2 given T (q), buyer picks outcome yielding highest pay-o

Program: anticipating buyers choices, seller maximizes

p = maxT (q)

b [T (qL)! cqL] + (1! b) [T (qH )! cqH ] s.t.ICi : qi = argmax

qqi v (q)! T (q) for i = L,H,

IRi : ui = qi v (qi )! T (qi ) % 0 for i = L,H.

IC incentive compatibility constraintsIR individual rationality (participation) constraints

Diculty: rm must choose among alternative functions T (q)solve with a 5 step procedure



STEP 1: apply revelation principle (see Mathematical Toolkit)one may restrict large set of schedules T (q) to pair of choices[qi ,T (qi )] made by each type of buyer, use notation Ti = T (qi )need only as many options as there are dierent types

Program: anticipating buyers choices, seller maximizes

p = maxTi ,qi b [TL ! cqL] + (1! b) [TH ! cqH ] s.t.ICH : qHv (qH )! TH % qHv (qL)! TL,ICL : qLv (qL)! TL % qLv (qH )! TH ,IRH : uH = qHv (qH )! TH % 0,IRL : uL = qLv (qL)! TL % 0.ICi : qi -type must prefer own contract (qi ,Ti ) over other

"qj ,Tj

#IRi : each type must be willing to accept and participate

Revelation principle: oers great simplicationreduce choice over innitely many functions T (q) to two options



STEP 2: eliminate IRH , not binding in FB, also not in 2nd best

qHv (qH )! TH % qHv (qL)! TL % qLv (qL)! TL % 01st inequality from ICH , 2nd from qH > qL, last from IRL

STEP 3: eliminate ICL which is not binding in rst-bestand check that it is also not binding with 2nd best solution

1st best: ecient demand qi v 0 (q&i ) = c , no rent qi v (q&i ) = T &i ,ICH is violated in FB (1st best) with full rent extraction

qHv (q&L)! T &L = (qH ! qL) v (q&L) > 0 = qHv (q&H )! T &H

ICL is slack in FB with full rent extraction

qLv (q&H )! T &H = ! (qH ! qL) v (q&H ) < 0 = qLv (q&L)! T &L

Result: eliminate ICL, slack in 1st best, check in 2nd bestFig.1: single crossing, high valuation qH , want large quantity qH ,low valuation qL, want small quantity qL, uL falls if qL ! qH .



STEP 4: remaining constraints ICH , IRL binding in reduced problem

p = maxTi ,qi b [TL ! cqL] + (1! b) [TH ! cqH ] s.t.ICH : qHv (qH )! TH % qHv (qL)! TL,IRL : uL = qLv (qL)! TL % 0.

ICH binding, otherwise raise TH without eect on IRLIRL binding, otherwise raise TL, but thistightens omitted ICL, thus check in the end!

STEP 5: substitute Ti from binding ICH , IRL constraints

maxqi

b [qLv (qL)! cqL]+ (1! b) [qHv (qH )! cqH ! (qH ! qL) v (qL)]

1 seller gets full rent from qL-type since IRL binding,2 but must leave informational rent (qH ! qL) v (qL) to qH -type,

informational rent increases with consumption qL of qL-type



Non-linear pricing: 2nd best optimal contracts qoj ,Toj fulll f.o.c.

qHv 0 (qoH ) = c , qLv0 (qoL ) =

cm> c , m , 1! 1! b

b

qH ! qLqL

< 1

if m < 0, then dp/dqL = b [m # qLv 0 (qL)! c ] < 0, implying qoL = 0,i.e. the rm would optimally want to exclude qL-type buyerlarger demand of buyer with high valuation, qoH > q

oL

CHECK suppressed constraints: IRH and ICL satised?1 show IRH is slack, i.e. qHv

"qoH

#! ToH > 0:proof: ICH and IRL binding and qH > qL imply

qHv (qoH )! ToH

ICH= qHv (qoL )! ToL

qH>qL> qLv (q

oL )! ToL

IRL= 0

2 show ICL is slack, i.e. qLv"qoL

#! ToL % qLv "qoH #! ToHproof: use ToH = T

oL + qH

%v"qoH

#! v "qoL #& from binding ICH ,and nd that ICL is equivalent to 0 % ! (qH ! qL)

%v"qoH

#! v "qoL #&Christian Keuschnigg (FGN-HSG) L2a: Screening - Two Types November 3, 2014 19 / 22


Non-linear pricing: f.o.c. for 2nd best optimal contracts

qHv 0 (qoH ) = c , qLv0 (qoL ) =

cm> c , m < 1

larger demand of buyer with high valuation, qoH > qoL

Economic conclusions: illustration Figure 51 no distortion at top: demand of qH -type is rst best, qoH = q

&H ,

but consumption of qL-type is ineciently low, qoL < q&L

2 qH -type gets positive information rent (qH ! qL) v"qoL

#,

qL-type gets a surplus/rent of zero (IRH is slack, IRL is binding)!3 information rent depends on demand qoL of low-type,

reducing allocation qoL relaxes ICH constraint, i.e. makes mimicking by high type less attractive, seller thereby reduces information rent and extracts even more


Figure 5: From 2-Part to 2nd Best SolutionHigh type: IC binding, IR slack; Low type: IC slack, IR binding


Summary

Main insights: when agent has private informationprincipal: gets larger pay-o by clever design of contract menurevelation principle: need no more than one contract per type

Basic results: 1st best, principal can extract all surplus2nd best, can extract only part of agents surplusmust leave information rent to prevent adverse selection,i.e. information rent necessary to satisfy incentive compatibilityeciency at the top, but inecient allocation for lower types

Preview: next lecturesthis lecture: L2a - screening with two typesnext lecture L2b - screening with many types (continuously many)Lecture L3 - applications of screening: there are many!

optimal income taxation, capital markets, regulationsocial insurance, labor contracts, workfare etc.

Literature: Lectures 2a-b based onBolton and Dewatripont (2005), chapter 2, without subsecs 2.3.1-2parts of Laont/Martimort (2002, ch. 2) and Salanie (1999, ch.2)


Two TypesIntroductionProblemFirst BestHidden InfoSecond BestSummary

l 2 a slides screening 2 types

Documents

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