A Supplier’s Optimal Quantity Discount Policy Under Asymmetric Information

Charles J. Corbett Xavier de Groote

Presented byJing Zhou

Agenda

Introduction The Basic Model: Full Information Optimal Contract Under Asymmetric

Information Comparing the Contracts Discussion

Agenda

Introduction The Basic Model: Full Information Optimal Contract Under Asymmetric

Information Comparing the Contracts Discussion

Motivation

Inefficiency occurs under self-interest

Demand dSupplier Buyer

bs kk

bh

Lot size Q

Setup costs:

Holding cost:

sjb QQQ

Supply Chain

ks + kbhb

Literature Joint economic lot-sizing literature

The supplier wishes to induce the buyer to choose a higher lot size

Quantity discount scheme can achieve the jointly optimal lot size

The supplier has full information about the buyer’s cost structure.

Lal and Staelin (1984) N groups of buyers of different sizes Holding costs, order costs and demand rates vary

between groups but not within groups Optimal unified pricing policy (no closed-form)

In this paper The buyer holds private information about her

holding cost; The supplier knows the prior distribution of the holding cost

Arbitrary continuum of buyer types Demand per unit time is known and constant,

and is not affected by the lot sizing or contracting decision

The supplier can choose not to trade with buyers

Optimal contracts in both full information case and asymmetric information case (quantity discount)

Notation cases considered (full information and

asymmetric information) range of buyer holding cost hb supplier’s prior distribution and density over

hb payment from supplier to buyer, as a

function of hb, under contract i buyer, supplier and joint cost function

(excluding quantity discount) maximum net cost level acceptable to buyer

and supplier (reservation net cost level) buyer, supplier and joint net cost function in

case i, after quantity discount Partial derivative of P(hb) with respect to hb

sb TCTC ,

jsb CCC ,,

AIFIi ,

)(),( bb hfhF

)( bi hP

],[ bb hh

)( bhP

ijisib TCTCTC ,,, ,,

Agenda

Introduction The Basic Model: Full Information Optimal Contract Under Asymmetric

Information Comparing the Contracts Discussion

The Basic Model Two decisions: lot size Q and quantity discount P(Q)

per unit time from supplier to buyer

2/

/)(),(

Qh

QdkkQhC

b

sbbj

Cost Function (excluding discount)

Optimal Lot Size (without contracting)

Net Cost Function (after discount)

Trade No trade

Buyer

SupplierLargest allowable lot size

Supply Chain

2),( Qh

QdkQhC bbbb

b

b

hdk2

QdkQC ss )(

b

sb

hdkk )(2

)()( QpQCs

)(),( QpQhC bb

)(),( QCQhC sbb sb TCTC

bTC

sTC

A Necessary Condition If the joint costs under jointly optimal lot size

Qj exceed , there will be no trade. Therefore, we need

(1)

sbbsb TCTChdkk /)(2

sb TCTC

)(2)(

2

sb

sbbb kkd

TCTChh

Cut-Off Point h*b

Because of the reservation net cost level, , the supplier choose a cut-off point h*

b such that he will choose not to trade with buyers with holding cost hb > h*

b

sTC

The Principal-Agent Framework

The supplier is the principal and the buyer is the agent (adverse selection).

The supplier proposes a “menu of contracts” or discount scheme, specifying the discount P(q) offered for any lot size q.

The buyer decides whether or not to accept the contract. She chooses some order lot size Q if accepts.

The supplier gives the buyer a discount of P(Q).

The Supplier’s Problem

sss

bbbbbbbb

bbbbbq

bbq

ssQPQ

TCQPQCQTC

hhhTCQPQhCQhTC

hhhqPqhC

qhTCQ

QPQCEQTCE

)()()( :IRs

],[ )(),(),( :IRb

],[ )(),(minarg

),(minarg :IC s.t.

)]()([)]([min )(,

Optimal Contract Under Full Information (First-Best)

otherwise

)(2)( if))(,(),(

b

b

bsbjbbjbb

bFI

TCh

dkkhQQTChQhCQhP

Payment from supplier to buyer:

The buyer will choose the joint economic lot size Qj(hb)

The supplier keeps all efficiency gains

Condition (1) guarantees that for all

bbFIjbbbsbFIs

bbFIb

TChTCTCdhkkhTC

TChTC

)()(2)(

)(

,,

,

sbFIs TChTC )(,

],[ bbb hhh

Agenda

Introduction The Basic Model: Full Information Optimal Contract Under Asymmetric

Information Comparing the Contracts Discussion

Assumptions Assumption 1 Decreasing reverse hazard rate:

Many common distributions satisfy this assumption Assumption 2

Rule out the problems caused by the distribution with thin tails

Neither of these assumptions on F(hb) are necessary conditions

0])()([ b

bh hF

hfDb

bbsbbb hkkhfhhF allfor /)(/)(

Recall: The Supplier’s Problem

sss

bbbbbbbb

bbbbbq

bbq

ssQPQ

TCQPQCQTC

hhhTCQPQhCQhTC

hhhqPqhC

qhTCQ

QPQCEQTCE

)()()( :IRs

],[ )(),(),( :IRb

],[ )(),(minarg

),(minarg :IC s.t.

)]()([)]([min )(,

Contracting Procedure The buyer knows hb, unobserved by the supplier The supplier offers a menu for any

the buyer announces To ensure that IRb and IRs are satisfied, the supplier can set h*

b such that IRb and IRs are met for all hb <=h*

b

The buyer chooses to accept or reject the contract. Accept (hb <=h*

b): buyer choose buyer and supplier incur net costs per period of

Reject (hb >h*b): both parties revert to their outside alternatives

and incur net costs per period of , respectively

)}ˆ(),ˆ({ bb hPhQ ],[ˆ bbb hhh

)ˆ( and )ˆ( bb hPhQ

)ˆ())ˆ(( and ˆ)ˆ(,( bbsbbbb hPhQC)hP(hQhC

sb TCTC and

Revelation Principle The revelation principle (Laffont and Tirole 1993)

states that if there is an optimal contract for the supplier, then there exists an optimal contract under which the buyer will truthfully reveal her holding cost. Formulate an IC constraint on PAI(hb):

Let FOC be satisfied at and yields IC constraint:

)ˆ()ˆ(2

ˆ

)ˆ(min

ˆ bAIbAIb

bAI

b

hhPhQh

hQdk

b

bb hh ˆ

)())(2

()ˆ( 2 bAIbAI

bbbAI hQ

hQdkhhP

IRb Assume that IRb is always met for any ,

and for IRb becomes

The supplier’s problem

*bb hh

],[ )(())(,( *bbbbbbbbbb hhh)P(hhQChQhTCTC

*bb hh

IRs IRb, IC, s.t.

][)]()(

[

)]([min

**

* ],[),(,

shhbb

shh

bshhhhhPhQ

TCEhPhQdkE

hTCE

bbbb

bbbbbb

Revelation Principle (Con’t)

ly.respective , and costs incurringes,alternativ outside their

revert to playersboth and place takes tradeno ],,[For

.)()(2

)( and )(/)(

)(2)(

by given are schemediscount and sizelot the],,[For

.in increasing is and ),()(

osolution t

theis where with buyers with only trade illsupplier w

then,informatio asymmetricunder schemediscount optimal In the

*

2

*

***

**

bs

bbb

bAIbAI

bbbAI

bbb

bsbAI

bbb

sbbAIbAI

ss

bbb

TCTC

hhh

hQhQdkhhP

hfhFhdkkhQ

hhh

TChhPhQdkTC

hhh

Proposition 1

)()(21)2()(2)(

)()(21)22()(2

21)(

)22()(221)(

)2)((221

2)(

2)(

2)(2)(

: are ],[for levelscost

ingcorrespond andpolicy optimal theuniform, is )(prior When the

,

*,

*,

*

*

bAIbbbbbsbAIj

bAIbbbbbbbsbbAIs

bbbbbsbbAIb

bbbbsbsb

AI

bb

bsbAI

bbb

b

hQhhhhdkkhTC

hQhhhhhhdkkTChTC

hhhhdkkTChTC

TCdhhkkQdkkQhQP

hhdkkhQ

hhh

hF

Proposition 1 (Con’t)

There is a probability 1 - of no trade taking place. For , QAI(hb) is decreasing in hb. In general, cannot be found explicitly as it is

impossible to write PAI(hb) explicitly. Both QAI(hb) and PAI(hb) are strictly decreasing in hb, so

there is a one-to-one mapping PAI(Q) which is increasing. Supplier’s and buyer’s costs are increasing in hb. The buyer’s net costs are equal to at . At , the resulting lot size is equal to the

jointly optimal lot size; as hb increases, the discrepancy between QAI(hb) and QFI(hb) increases.

*bb hh

Some Findings)( *bhF

*bh

*bh

bTC

bh )( bAI hQ

Agenda

Introduction The Basic Model: Full Information Optimal Contract Under Asymmetric

Information Comparing the Contracts Discussion

Proposition 2

],[ )),(,())(,(

ninformatio private has she when decrease costsnet sbuyer' The

][][ :asymmetry

ninformatiounder increase costsnet expected ssupplier' The

],[ ,

:oncoordinati of form nothan efficient more is gcontractin but asymmetry n informatioby reduced is efficiency Global

],[ ),()()()(

:satisfy sizeslot then,informatio asymmetric Under

,,

,,

,,,,

bbbbAIbAIbbFIbFIbb

FIshAIshs

bbbjjFIjAIjbj

bbbbjbFIbAIbb

hhhhQhTChQhTCTC

TCETCETC

hhhTCTCTCTC

hhhhQhQhQhQ

bb

Agenda

Introduction The Basic Model: Full Information Optimal Contract Under Asymmetric

Information Comparing the Contracts Discussion

Discussion

Contribution Consider asymmetric information in EOQ

environment: buyer’s holding cost is not observable to the supplier

The supplier can choose not to trade with buyer with certain large holding cost

Introduce the cut-off point policy

Discussion (Con’t) Limitation

The assumptions on prior distribution are too specific

Lack of necessary proof or explanation of major results or findings

Demand is not affected by the lot sizing or reduction of unit price

Only the uncertainty about buyer’s holding cost is considered. More dimensions can be analyzed.

Thank you!