Coarse Matching and Price Discrimination H. Hoppe, B. Moldovanu, and - - PowerPoint PPT Presentation

Coarse Matching and Price Discrimination

• H. Hoppe, B. Moldovanu, and E. Ozdenoren

Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

introduction

Research Question

Two kinds of agents (“men”, “women”) look for a match. An intermediary can extract transfers and match agents based on their reported types.

▶ randomly? ▶ coarsely? ▶ assortatively?

RQ: How good is coarse matching with two categories for each kind of agent, relative to efficient matching or random matching?

▶ total surplus ▶ agents’ utility ▶ matchmaker’s revenue

Look for lower bounds.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

introduction

Motivation

Extend McAfee (2002)

▶ Obtain lower bounds on surplus in more environments. ▶ Private types mean that the matching must be incentive compatible.

Authors’ motivation: If coarse matching is “pretty good” in the worst case, then (unmodeled) costs of using a finer scheme may offset the benefits. Why do firms offer a “small” menu of qualities?

▶ One reason: a price-discriminating monopolist can get “close” to maximum

revenue with two quality levels.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Model

Model

Men: x ∼ F(x) on [0, 휏F]. Women: y ∼ G(y) on [0, 휏G].

▶ Assume densities f (x), g(y) > 0, and measure 1 of each type. ▶ x, y private information.

Intermediary chooses

▶ Matching rule 휙 : [0, 휏F] ⇉ [0, 휏G] That is, 휙(x) ⊆ [0, 휏G]. ▶ Price schedules pm : [0, 휏F] → ℝ, pw : [0, 휏G] → ℝ. ▶ Implicitly restricts attention to direct mechanisms.

Surplus:

▶ Total surplus xy. ▶ Fixed sharing rule 훼 ∈ [0, 1]. ▶ If man x and woman y match, man gets 훼xy and woman gets (1 − 훼)xy

before transfers to the intermediary

IR: Agents who do not use the intermediary are matched to each other

• randomly. (Q: what happens to deviators when everyone uses the

intermediary?)

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Model

What do matchings look like?

Let 휈M(A) be the measure of men announcing types in A. Define 휈W (.) similarly. A matching 휙 is feasible if 휈M(A) = 휈W (휙(A)) for all (measurable) A ⊆ [0, 휏G] Damiano and Li (2005): Incentive-compatible and feasible matchings partition each group into n bins, match the bins assortatively, and match randomly within bins.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Assortative Matching

Match each man x with the woman 휓(x), where 휓(x) solves F(x) = G(휓(x)) Incentive compatibility: x ∈ arg max

ˆ x

훼x휓(ˆ x) − pm(ˆ x) (1) y ∈ arg max

ˆ y

(1 − 훼)휓−1(ˆ y)(y) − pw(ˆ y) (2) Take FOC: (Can plug in solution and show SOC holds) 훼x휓′(x) − ∂pm(x) ∂x = 0 Lowest pair generates 0 surplus. Hence pm(0) = 0. Therefore, pm(x) = ∫ x 훼z휓′(z)dz (3) Similarly, letting 휑 = 휓−1, pw(y) = ∫ y (1 − 훼)z휑′(z)dz (4)

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Stability

Another solution concept in matching models is stability. In this setting: pick a matching rule 휙(.), and let man x get surplus 훿(x), and woman y get surplus 휌(y). The sharing rule is called “stable” if ∀x, 훿(x) + 휌(휙(x)) = x휙(x) (5) ∀x, y, 훿(x) + 휌(y) ≥ xy (6) Typically, stability means IR and no blocking pairs. Here, stable sharing implies stability.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Stability and efficiency

Claim: the stable matching must be assortative. First, show a stable matching must be monotone increasing. Take x′ > x, y ′ > y, and for a contradiction, suppose a stable sharing rule assigns x ↔ y ′ and x′ ↔ y. 훿(x) + 휌(y ′) = xy ′ 훿(x′) + 휌(y) = x′y = ⇒ 훿(x) + 휌(y) + 훿(x′) + 휌(y ′) = x′y + xy ′ xy + x′y ′ ≤ x′y + xy ′ Hence (x′ − x)(y ′ − y) ≤ 0 If x ∕↔ 휓(x), then wlog say x ↔ y > 휓(x). Then, since g(.) > 0, we have G(y) = G(휓(x)) + 휖 for some 휖 > 0. If the matching is monotone increasing, then 휈m([x, 휏F]) ≥ 휈w(휙([x, 휏F])) + 휖. To summarize, if a matching rule is stable, either it is infeasible or it is assortative.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Stable shares

Differentiate 훿(x) + 휌(휓(x)) = x휓(x). Obtain 훿′(x) + 휌′(휓(x))휓′(x) = 휓(x) + x휓′(x) Matching coefficients, 훿′(x) = 휓(x), and 휌′(훿(x)) = x. We know that 훿(0) = 휌(0) = 0. Hence, by the FTC the stable shares are 훿(x) = ∫ x 휓(z)dz (7) 휌(y) = ∫ y 휑(z)dz (8)

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Connection to assortative matching

Proposition

The IC price schedules satisfy pm(x) = 훼휌(휓(x)) (9) pw(y) = (1 − 훼)훿(휑(y)) (10) The net utilities of x and y are 훼훿(x) and (1 − 훼)휌(y). The intermediary’s revenue satisfies min(훼, 1 − 훼)x휓(x) ≤ pm(x) + pw(휓(x)) ≤ max(훼, 1 − 훼)x휓(x) Hence the intermediary extracts half the total surplus if 훼 = 1/2.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Proof of prop 1-1

Total surplus from a pair is u(x, 휓(x)) = x휓(x). Totally differentiate: d dx u(x, 휓(x)) = 휓(x) + x휓′(x) By the FTC, u(x, 휓(x)) = ∫ x

0 휓(z)dz +

∫ x

0 z휓′(z)dz. Hence

훼u(x, 휓(x)) = 훼 ∫ x 휓(z)dz

• 훼훿(x)

+ 훼 ∫ x z휓′(z)dz

• pm(x)

Also, using a change of variables w = 휓(z), we have

1 훼pm(x) =

∫ x

0 z휓′(z)dz =

∫ 휓(x) 휑(w)dw = 휌(휓(x)).

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative Matching

Total Revenue and Surplus

Man x pays Pm(x) = ∫ x

0 훼z휓′(z)dz. After some computation (see appendix),

write Ra

훼 = Ra men + Ra women, where

r a

men = 훼

∫ 휏F 휓(x) [ x − 1 − F(x) f (x) ] f (x)dx Ra

women = (1 − 훼)

∫ 휏F x [ 휓(x) − 휓′(x)1 − F(x) f (x) ] f (x)dx Additionally, the total surplus is Ua = 피(x휓(x)) = ∫ 휏F x휓(x)dF(x)

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Why Coarse Matching?

Perfect (Assortative) matching incurs various transaction costs:

▶ Intermediary: communication (decoding) cost ▶ Agents: evaluation (coding) cost

Agents only need to reveal partial information. In terms of total surplus, the intermediary’s revenue and agents’s welfare:

▶ It is significantly higher than completely random matching. ▶ It may achieve a large proportion of assortative matching.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Coarse Matching Model

훼 ∫ ˆ

y

ˆ xy G(ˆ y)dG(y) = 훼 ∫ 휏G

ˆ y

ˆ xy 1 − G(ˆ y)dG(y) − pc

m

(11) (1 − 훼) ∫ ˆ

x

xˆ y F(ˆ x)dF(x) = (1 − 훼) ∫ 휏F

ˆ x

xˆ y 1 − F(ˆ x)dG(y) − pc

w

(12) ˆ y = 휓(ˆ x) (13) two classes: willing to pay and not willing to pay ˆ x(ˆ y) the lowest type of men (women) who is willing to pay pc

m(pc w).

such pricing scheme is incentive compatible.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Coarse Matching Model

cutoff point: ˆ x = EX, and ˆ y = 휓(Ex) EXL = EX − ∫ EX F(x)dx F(EX) , EYL = 휓(EX) − ∫ 휓(EX) G(x)dx G(휓(EX)) total surplus: UEX = ∫ EX ∫ 휓(EX) xy F(EX)dG(y)dF(x) + ∫ 휏F

EX

∫ 휏G

휓(EX)

xy 1 − F(EX)dG(y)dF(x) = EXEY + F(EX) 1 − F(EX)(EX − EXL)(EY − EYL) intermediary’s revenue (fix 훼 = 1/2): REX = [1 − F(EX)]pc

w + [1 − G(휓(EX))]pc m

= 1 2[EX(EY − EYL) + 휓(EX)(EX − EXL)]

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Total Surplus

Note definition 2: Ua = (1 + CCV 2(x, 휓(x)))Ur If the distributions F and G both satisfy:

1

F and G are both log-concave (F, G DRFR)1

2

(1-F) and (1-G) are both log-concave (F, G IFR)2

UEX ≥ Ua + Ur 2 ⇒ UEX ≥ 3 4Ua, UEX ≥ Ur If F and G are both concave and 휓 is convex, then: UEX ≥ 5 4Ur

1decreasing reversed failure rate f (t) F(t) 2increasing failure (hazard) rate f (t) 1−F(t)

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Total Surplus

Proof: F(EX) ≥ E(F(X)) = 1/2 ⇒ F(EX) 1 − F(EX) ≥ 1 Fconcave ⇒ ∫ EX F(x)dx EXF(EX) ≥ 1/2 ⇒ EX − EXL ≥ 1 2EX G concave and 휓 convex ⇒ EY − EYL = EY − 휓(EX) + ∫ 휓(EX) G(x)dx G(휓(EX)) ≥ EY − 1 2휓(EX) EYL ≤ 1 2휓(EX) ≤ 1 2E(휓(X)) ⇒ EY − EYL ≥ 1 2EY

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Intermediary’s Revenue

If the distributions F and G both satisfy:

1

F and G are both concave

2

(1-F) and (1-G) are both log-concave (F, G IFR)

REX ≥ 1 2Ra Intuition: As the distribution of types on the other market side becomes more concave, the mass of potential partners with very low type gets larger, leading to a higher revenue since agents in high class are willing to pay more. If EX ≥ EY and 휓 is convex, then REX

m

≥ REX

w

Intuition: If F and G have the same mean, but G has a higher variance, the chances for men to match with a lower type of women are higher than women, thus men in higher class are willing to pay more.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Agents’ Welfare

W EX = UEX − REX, W a = Ua − Ra Fix 훼 = 1/2, if the distributions F and G both satisfy:

1

F and G are both convex

2

F and G are both log-concave (F, G DRFR)

W EX ≥ W a Proof REX = 1 2[EX(EY − EYL) + 휓(EX)(EX − EXL)] ≤ 1 2[EX(EY − 1 2휓(EX)) + 휓(EX)1 2EX] ≤ 1 2Ur

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Coarse Matching

Agents’ Welfare

W EX = UEX − REX ≥ 1 2(Ua + Ur) − 1 2Ur = 1 2Ua = W a However, W r = Ur = 1 1 + CCV 2(x, 휓(x)) ≥ 1 2Ua = W a

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Application

Lemma

휓 is convex (concave) then Ra

1 ≥ (≤)Ra 1/2. If the asssortative matching function

휓 is convex (concave) and if EX ≥ (≤)EY then REX

1

≥ (≤)REX

1/2.

Proposition

Let F be IFR, G be IFR and concave. Then REX

1

≥ 1

4Ra

• 1. If, in addition, 휓 is

concave then REX

1

≥ 1

2Ra 1.

Proposition

1) Let F and G be IFR and concave, and let 휓 be convex. Then W EX

1

≥ 1

2W a 1 .

2)Let F and G be convex and DRFR, and let 휓 be convex. Then W EX

1

≥ W a

1 .

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Price discrimination with quality costs

Consumers are distributed over [0, 1] according to distribution F with f = F ′ > 0. Each consumer demands one unit of the good. Utility of consumer of type v from quality q is vq. Cost of producing y units of quality q is c(q)y.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Revenue maximizing profit function

By the standard mechanism design argument we know that monopolist’s revenue and profit are given by Ra = ∫ 1 q(v) ( v − 1 − F(v) f (v) ) f (v)dv and 휋a

1 =

∫ 1 [ q(v) ( v − 1 − F(v) f (v) ) − c(q(v)) ] f (v)dv Let r be such that ( r − 1−F(r)

f (r)

) = 0. Solution is, q(v) = 0 if v ≤ r c′(q(v)) = v − 1 − F(v) f (v) if v ≥ r Define G(y) = F(q−1(y)) we get a distribution of quality levels where q(v) = 휓(v) valuation (men), quality (women).

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Price discrimination example

In paper’s framework: total revenue of the assortative matching is given by (훼 = 1) Ra

1 =

∫ 휏F 휓(x) ( x − 1 − F(x) f (x) ) dF(x) (from computing 피(pm)) Coarse matching: provide two qualities QL = ∫ EV q(z)dF(z)/F(EV ) and QH = ∫ 1

EV q(z)dF(z)/(1 − F(EV )) and REV 1

= EV (EQ − QL)

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Example

c(q) = q2 and v ∼ U[0, 1]. In this case, q(v) = 0 if v ≤ 1

2 and r = 1/2 and

2q(v) = v − 1−v

1

⇐ ⇒ q(v) = 2v−1

2

G(y) = 1+2y

2

for y ∈ [ 0, 1

2

] which is concave and IFR. Computation yields QH = 1

2, Ra 1 = 1 12, REV 1

=

1 16 that is REV 1

= 3

4Ra

• 1. (note

Prop 9 tells us REX

1

≥ 1

2Ra 1)

Total profit is given by 휋a

1 = 1 24 and 휋EV 1

=

1 32 and 휋EV 1

= 3

4휋a.

W EV =

1 32 > 1 48 = W a (note Prop 10 tells us W EV ≥ W a).

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Proof of Lemma 4

Lemma

휓 is convex (concave) then Ra

1 ≥ (≤)Ra 1/2. If the asssortative matching function

휓 is convex (concave) and if EX ≥ (≤)EY then REX

1

≥ (≤)REX

1/2.

Proof dRa d훼 = ∫ 1 (x휓′(x) − 휓(x))(1 − F(x))dx > 0 if 휓 convex. That is Ra

훼 increasing in 훼.

Now, note EX ≥ EY = E휓(X) ≥ 휓(EX) if 휓 convex (concave) and (EY − EYL) ≥ (EX − EXL) by Lemma 3. REX

1

= EX(EY − EYL) ≥ 1 2 [EX(EY − EYL) + 휓(EX)(EX − EXL)] = REX

1/2

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Proof of Proposition 9

Proposition

Let F be IFR, G be IFR and concave. Then REX

1

≥ 1

4 Ra

• 1. If, in addition, 휓 is concave then

REX

1

≥ 1

2 Ra 1.

Proof.

By Lemma 3 EY − EYL ≥ 1

2 EY if G is concave. This yields

REX

1

= EX(EY − EYL) ≥ 1 2 EXEY = 1 2 Ur = 1 2 ( EX휓(X) 1 + CCV 2(X, 휓(X)) ) = 1 2 ( Ua 1 + CCV 2(X, 휓(X)) ) ≥ 1 2 ( Ra

1

1 + CCV 2(X, 휓(X)) ) CCV 2(X, 휓(X)) ≤ 1 if F and G are both IFR. Ua ≥ Ra

1 in general and if 휓 concave Ra 1 < Ra 1/2,

and since Ua = 2Ra

1/2, Ua > 2Ra 1.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Proof of Proposition 10

Proposition

1) Let F and G be IFR and concave, and let 휓 be convex. Then W EX

1

≥ 1

2 W a 1 . 2)Let F and G

be convex and DRFR, and let 휓 be convex. Then W EX

1

≥ W a

1 .

Proof 1) UEX = EXEY + F(EX) 1 − F(EX) (EX − EXL)(EY − EYL)

F(X) ≥ 1/2 by L2

≥ EXEY + (EX − EXL)(EY − EYL)

algebra

= REX

1

+ EY (EX − EXL) + EYLEXL

EX − EXL ≥ 1/2EX by L3

≥ REX

1

+ 1 2 EXEY + EYLEXL

algebra

= 3 2 REX

1

+ 1 2 EXEYL + EYLEXL = ⇒ UEX > 3 2 REX

1

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Price Discrimination

Proof of Proposition 10 continued

Now, W EX

1

= UEX − REX

1

2 3 UEX > REX 1

≥ 1 3 UEX

McAfee

≥ 1 3 1 2 (Ua + Ur)

Ur ≥ 1/2Ua if F and G IFR

≥ 1 6 (Ua + 1 2 Ua) = 1 4 Ua

Ua/2 > Ra

1 if 휓 cvx

≥ 1 2 W a

1

2) REX

1

= EX(EY − EYL)

L3, F and G cvx

≤ 1 2 EXEY = 1 2 Ur This implies that W EX

1

= UEX − REX

1

≥ 1 2 (Ua + Ur) − 1 2 Ur = 1 2 Ua

Ua/2 > Ra

1 if 휓 cvx

≥ W a

1

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Conclusion

Conclusions

Simple matching schemes can work under incomplete information, and are not too far from optimal, even in the worst case.

▶ But how easy is it to satisfy the statistical assumptions? ▶ Horizontally differentiated goods?

... we focus on clearly suboptimal mechanisms, while identifying settings where such mechanisms are very effective (and thus may become optimal once transaction costs associated with more complex mechanisms are taken into account. Foundations for costs of complexity?

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

Assortative matching revenue: men’s share

Man x pays Pm(x) = ∫ x

0 훼z휓′(z)dz. Integrating by parts,

pm(x) = 휓(x)x − ∫ x 휓(z)dz Take the expectation over all men: Ra

men =

∫ 휏F Pm(x)f (x)dx = 훼 ⎛ ⎝ ∫ 휏F 휓(x)xf (x)dx − ∫ 휏F ∫ x 휓(z)dzf (x)dx

• ⎞

⎠ ∫ 휏F ∫ 휏

z

f (x)dx휓(z)dz = ∫ 휏F (1 − F(z))휓(z)dz = Ra

men = 훼

∫ 휏F 휓(x) [ x − 1 − F(x) f (x) ] f (x)dx

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32

SOC for assortative matching

Plugging in the solution to the FOC, man x solves arg max

ˆ x

훼 ( x휓(ˆ x) − ∫ ˆ

x

z휓′(z)dz ) =훼(x − ˆ x)휓(x) + 훼 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ˆ x휓(ˆ x) − ∫ ˆ

x

z휓′(z)dz

• ˆ

x휓(ˆ x)− ∫ ˆ

x 0 휓(z)dz

⎞ ⎟ ⎟ ⎟ ⎟ ⎠ =훼(x − ˆ x)휓(ˆ x) + 훼 ∫ ˆ

x

휓(z)dz d dˆ x :훼(x − ˆ x)휓′(ˆ x) d2 dˆ x2 : (x − ˆ x)휓′′(ˆ x)

• =0 at x=ˆ

x

−휓′(ˆ x) < 0 The above shows that at reporting ˆ x = x gives a local maximum. Since the solution to the FOC is unique, we need to rule out only reporting 0 or reporting 휏.

• H. Hoppe, B. Moldovanu, and E. Ozdenoren ()

Coarse Matching and Price Discrimination Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap / 32