Designing Supply Contracts: Designing Supply Contracts: Contract Type and Information Contract Type and Information AsymmetryAsymmetry

Authors: C. Corbett, C. S. Tang

Presenter: T.J. Hu

ContentsContents

IntroductionLiteratureModelSupplier's Optimal Supply ContractsComparisonsNumerical ExamplesConclusionsFuture Research

IntroductionIntroduction

The Supply ChainThe Supply Chain

Supplier

Buyer

L(q) w(q)

p(q)

s

cost c

C ~ F(•)

Supplier's Concerns:Supplier's Concerns:

The types of contractsInformation about the buyer's

cost structure

Three Types of ContractsThree Types of Contracts

1. One-part linear contract: w

2. Two-part linear contract: w, L

3. Two-part nonlinear contract: {w(q), L(q)}

Six ScenariosSix Scenarios

Type of contracts Fullinformation

Asymmetricinformation

One-part linear: w F1 A1

Two-part linear: w,L F2 A2

Two-part nonlinear:{w(q), L(q)}

F3 A3

Questions to Answer:Questions to Answer:

What should supplier do when faced with decreased buyer demand?

Value of information about the buyer’s cost structure

Value of more sophisticated contractsWhich of the above two is more valuable?When there is no double marginalization?

LiteratureLiterature1. Supply Chain Management1. Supply Chain Management

2. Economics2. Economics

Supply Chain LiteratureSupply Chain Literature

Deriving optimal ordering policies in the context of a given contract

Deriving optimal contract parameters given the functional form of that contract

Coordination within supply chains, the value of information and various alternative contracting schemes

Selected PapersSelected Papers

Lee, So, and Tang (1998)– Quantify the value of sharing demand information

– Demand follows an AR(1) process Bourland, Powell and Pyke (1996), Cachon and Fisher

(1997), Gavirneri, Kapuscinski and Tayur (1996)– Benefits of information sharing when demand is i.i.d.

Lee and Whang (1996)– Incentive scheme for a multi-echelon supply chain (central

planner, but each echelon uses local information only) Corbett (1996, 1998)

– Asymmetric information leads to to suboptimal outcomes (without central planner)

Selected Papers (Cont’Selected Papers (Cont’dd))

Weng (1995)– Quantifies the value of channel coordination

– Quantity discounts alone are not sufficient to achieve coordination

Corbett and de Groote (1997)– Compares various coordination schemes for a 2-level SC

– Preferences ordering of these schemes for the supplier, buyer and vertically-integrated firm

This paper– Quantifying the value of information and the value of more

complex contracts

Economics LiteratureEconomics Literature

Vertical contracting– Two successive monopolists– Double marginalization

Topics– Comparing total surplus under various

schemes – Contract to mitigate the double

marginalization issue

Selected PapersSelected Papers

Tirole (1988) : The Theory of Industrial Organization F. Machlup and M. Taber (1960)

– Bilateral monopoly, successive monopoly, and vertical integration, Economica, May (1960), 101-119.

Gal-Or (1991a,b)– In general, neither franchise fees nor retail price maintenance

can achieve the integrated solution under asymmetric info

– Equilibrium sometimes achieved with linear pricing and franchise fee contract (two supplier)

Bresnahan and Reiss (1985)– Study the ratio of the profit margins under simple wholesale

price with full information

– How the ratio depends on the convexity of demand function

Contribution of This PaperContribution of This Paper

Combine two strands of theory– building on the basic bilateral monopoly

framework offered in economics– asking the normative and more micro-

level questions more typical of supply chain literature

– measure the cost of sub-optimality (quantification and insights of the differences between the cases)

The ModelThe Model

The Supply ChainThe Supply Chain

Supplier

Buyer

L(q) w(q)

p(q)

s

cost c

)ˆ( ~ˆ cFc

AssumptionsAssumptions

One supplier and one buyerOne productOne period contractDeterministic demandLinear price-demand curve q = a - bpa - b (s+ ) 0F(c)/f (c) is increasing in c

F/fF/f for Normal Distribution for Normal Distribution

-5 -4 -3 -2 -1 00

0.2

0.4

0.6

0.8

1

1.2

1.4

Supplier’s Problem (Supplier’s Problem (SS))

dq

dD

cqcD

cqLqcqwqpqc

cqLcqscqwELw

qb

bq

bb

sqLqw

level,profit n reservatio sBuyer' :

,0),(

,)()()(),( s.t.

))(()())((),(max)}(),({

Buyer’s individual rationality constraintBuyer’s individual rationality constraint

Buyer’s incentive compatibility constraintBuyer’s incentive compatibility constraint

Sequence of EventsSequence of Events

Supplier offers one of the three types of contracts

Buyer (with c) selects the order quantity q or (w(q), L(q))

All sales and financial transactions take place simultaneously

Revelation Principle (Revelation Principle (A3A3))

Reformulating the contracts in terms of c, i.e. optimizing over {w(c), L(c)}

There is an optimal contract under which the buyer will reveal truthfully.

Supplier’s Supplier’s Optimal Supply Optimal Supply ContractsContracts1. Buyer’s problems1. Buyer’s problems

2. Supplier’s problems2. Supplier’s problems

Buyer’s Problem (Buyer’s Problem (BB11, , BB22))

Lqcwb

qa

Lqcwqpqbq

)(

)()(max

In B1, set L=0

In B2, to buyer, L is independent of q

Solutions for Solutions for BB11, , BB22

L

b

qLcwba

b

cwbaq

b

cwbap

b

2*

2*

*

*

)(4

1

)(2

12

)(

Comments on the SolutionsComments on the Solutions

high. toobe toset not shouldsupplier Thus

. implies 0)(2

1 )2

order!t 'simply wonbuyer the

,)(4

1

or 0)(2

1 If )1

*

2*

*

w

cb

awcwbaq

Lcwbab

cwbaq

bb

Buyer’s Problem (Buyer’s Problem (BB33))

2

**

ˆ

)ˆ(2

1

2)ˆ(

))ˆ(()ˆ())ˆ(()ˆ()ˆ,(max

ccwb

abcL

cwqccwcwpcLccbc

.ˆover maximizes then He/she

. solvesbuyer the,ˆany For 2

c

Bc

Solution for Solution for BB33

)()(2

1)(

0)ˆ,(ˆ

:FOC ˆ

cwcwbacL

cccd

dccb

Revelation Principle FOC evaluates at c It tells the supplier how to choose w(•) and L(•).SOC holds in the neighborhood of c.

Supplier’s ProblemsSupplier’s Problems

Optimal Contracts Under Complete Optimal Contracts Under Complete Information: Case F1, F2, F3Information: Case F1, F2, F3

Contracts with Full InformationContracts with Full Information

Type of contracts Fullinformation

Asymmetricinformation

One-part linear: w F1 A1

Two-part linear: w,L F2 A2

Two-part nonlinear:{w(q), L(q)}

F3 A3

Case Case F1 (F1 (SSF F 11))

)(2

1)(

)(max *

, 1

cwbasw

qswwFsw

Note: The Supplier knows the buyer’s optimal order quantity q*

Solution for Solution for SSF F 11

2,

*

*

)(8

1)(

)(2

1

2

1 1

1

csbab

w

csb

aw

Fs F

F

Profit and Profit MarginsProfit and Profit Margins

)(2

1)(

4

1 *** swcsb

acwp

Supplier’s profit and profit margin are double those of the buyer!

1 11 ,

*2*

,)(

2

1)(

16

1Fs FFb

wcsbab

More on Profit and MarginsMore on Profit and Margins

. where

,2

1 is ratio thisgeneral,In

2

pD

pDq

q

q

is a local measure of the curvature of the demand curve

Ref. Bresnahan and Reiss (1985)

Question:Question:

b

*

b,F?

1

Is the buyer’s individual rationality constraint satisfied?

If not satisfied, as we commented before, the buyer won’t order and thus both parties’ profits are zero!

Case Case F2 (F2 (SSF F 22))

bFb

FsLw

cc

Lcwbasw

LqswLw

2

2

,

*

,},{

),(s.t

)(2

1)(

),(max

ObservationsObservations

With complete info about c, supplier can set the rationality constraint to be binding.

He then maximizes the joint profits.

)()2(4

1)(max

)(),(max

2

2

,

**

,},{

cwbacswbab

w

qcspLw

Fjw

FjLw

Solution for Solution for SSF F 22

2,

*

,

****

,

*

2*

*

)(4

1

),0(),(

fee) franchise0 if 0(should

)(4

1

,

2

2

2222

2

2

cwbab

LLw

q

cwbab

L

sw

Fj

bFb

FFFFs

bF

F

Comments on Solutions for Comments on Solutions for SSF F 22

happen! willons transactiNo

.0set simply llupplier wi ,0

0)(4

1

then,)(4

1 If

0)(2

1

**

,

2*

2

**

22

2

2

LsL

cwbab

L

cwbab

csbaqsw

FFs

bF

b

F

InterpretationInterpretation

It’s optimal for the supplier to set the whole sale price equal to his marginal cost and use the lump sum side payment to extract all profits from the buyer in excess of his reservation profit level.

Case Case F3 (F3 (SSF F 33))

Superset of F2F2 is optimal given full info on c:

buyer only gets minimum levelValue of addition flexibility is 0 results carry over from F2

Supplier’s ProblemsSupplier’s Problems

Optimal Contracts Under Asymmetric Optimal Contracts Under Asymmetric Information: Cases A1, A2, A3Information: Cases A1, A2, A3

Contracts with Asymmetric Contracts with Asymmetric InformationInformation

Type of contracts Fullinformation

Asymmetricinformation

One-part linear: w F1 A1

Two-part linear: w,L F2 A2

Two-part nonlinear:{w(q), L(q)}

F3 A3

Case Case A1 (A1 (SSAA11))

)()(

2

1)(

])([max *

, 1

cdFcwbasw

qswEwE

c

c

Asw

Note: The Supplier knows the form of the buyer’s optimal order quantity q*

Solution for Solution for SSAA11

2,

*

*

])[(8

1)(

])[(2

1

2

1 1

1

cEsbab

w

cEsb

aw

As A

A

Profit and Profit MarginsProfit and Profit Margins

])[(2

1

])[2(4

1

*

**

1cEs

b

asw

cEcsb

acwp

A

Supplier has incentive to induce the buyer Supplier has incentive to induce the buyer to reveal his true cost to reveal his true cost cc. (???). (???)

2*

,])[2(

16

11

cEcsbabAb

If not satisfied, the buyer won’t order.

Question:Question:

b

*

b,A?

1

Is the buyer’s individual rationality constraint satisfied, i.e.

0])[2(4

1 need Also * cEcsbaq

Case Case A2 (A2 (SSAA22))

2

2

,

2

*

,},{

)(4

1),( s.t.

)()(2

1)(

][]),([max

Ab b

c

c

AsLw

Lcwbab

qc

cdFLcwbasw

LqswELwE

ObservationsObservations

For any given w, the supplier will always choose the lowest L that still satisfies the buyer’s rationality constraint.

b(c) is decreasing in c necessary and

sufficient to set b( ) = infc b(c) b-

*,

2*,, 222

)(4

1),( AbAbAbb

Lcwbab

qc

Solution for Solution for SSA A 22

bFb

AAAAs

bA

A

csbacEcLLwE

cEcsbab

L

cEcsw

*

,

****

,

2*

*

2

2222

2

2

,)(][2

1),(

])[2(4

1

],[FOC

RemarksRemarks

If =E(c)=c, case A2 reduces to F2.The information asymmetry means

the supplier must now offer a larger side payment (or less franchise fee) than in F2, to meet the “worst-case” buyer’s min profit requirements.

Effectively, need

0])[2(2

1* cEcsbaq

Question:Question:

When and what if the expected supplier’s profit is zero?

)(][2

1

])[2(4

1),( 2***

, 222

csbacEc

cEcsbab

LwEbAAAs

Case Case A3 (A3 (SSA A 33))

Using buyer’s optimal order quantity q* and FOC from B3

ccc

cwcwbaL

LcwbaswE

LqswEE

bb

AsLw

,),(

,)(2

1 s.t.

)(2

1)(

])[(][max *

,)}(),({ 3

Euler’s Equation:Euler’s Equation:Necessary ConditionsNecessary Conditions

.0

:equation sEuler' esatisfy th should

,)(,)( where

,)',,()(

sfunctional theof points criticalSmooth

'

10

1

0

yy

x

x

Fdx

dF

bxyaxy

dxyyxFyJ

Constrained Problems: LagrangianConstrained Problems: Lagrangian

0)',',,,(

and ,0)(

,0)(

:satisfy should

,)(,)(,)(,)( where

0)',',,,( s.t. ,)',,',,(),(

sfunctional theof points criticalSmooth

'

'

11001100

1

0

zyzyx

xFdx

dF

xFdx

dF

zxzzxzyxyyxy

zyzyxdxzzyyxFzyJ

zzz

yyy

x

x

Solution for Solution for SSA A 33

)ˆ(

)ˆ()ˆ(*

3 cf

cFscwA

Based on our assumption, the optimal w is increasing in , whereas in earlier cases, it is decreasing in or E[c].

From FOC, L is also increasing in .

Buyer’s TradeoffBuyer’s Tradeoff

Accepting a higher lump sum payment and a higher unit whole sale price versus

Accepting a lower lump sum payment and a lower unit whole sale price.

Special Case A3: Uniform PriorSpecial Case A3: Uniform Prior

kccsbaccL

ccscw

A

A

)(2

1)(

)(

*

*

3

3

Special Case A3 (Cont’Special Case A3 (Cont’dd))

The unit whole sale price can be interpreted as the average of a constant part and a part decreasing in q, illustrating how w decreases with quantity. Compare: p=a/b - q/b

kqcsbab

qL

qbb

acsqw

bA

A

22*

*

4)(8

1)(

1

2

1)(

3

3

ComparisonsComparisons

The Impact of Buyer’s Cost on The Impact of Buyer’s Cost on the Supplier’s Profit Marginthe Supplier’s Profit MarginBuyer’s cost c

Buyer’s profit margin mb Buyer orders less Supplier’s profit

How should the supplier respond?

Supplier’s ResponseSupplier’s Response

In case F1, A1, A2:– sacrifices margin

for volume

In F2 and F3:– insensitive

In A3:– sacrifices volume

for margin ccf

cFm

mm

cEcEcm

cEscEb

am

cscb

am

As

FsFs

As

As

Fs

)(

)(

0

][ ][

][ )][(2

1

2

1

)(2

1

2

1

3

22

2

1

1

,

,,

,

,

,

““Effective” Wholesale PriceEffective” Wholesale Price

More precisely, one should take the side payment into account and evaluate the “effective” unit wholesale price as follows:

q

Lwwe

The Value of Info to the SupplierThe Value of Info to the Supplier

sAFsAF

AsFssAF

AsFssAF

csbacEccVarb

E

cVarb

E

1122

2222

1111

,,

*

,

*

,,

*

,

*

,,

2

)(][2

1)(

4

][

)(8

][

The value of information is (significantly) greater when the supplier has the flexibility to offer two-part contracts.

The Value to the Supplier of The Value to the Supplier of Offering Side PaymentsOffering Side Payments

sAAsFF

bsAA

bsFF

b

cEcsba

b

csba

1212

12

12

,,

2

,

2

,

])[2(

8

1

)(

8

1

As demand becomes more price-sensitive, the absolute penalty from using only wholesale price without side payments decreases.

The value of contracting flexibility is greater under full information.

Value of Information v.s. Value Value of Information v.s. Value of Contracting Flexibilityof Contracting FlexibilityValue of information increases with

b, while value of contracting flexibility decreases with b. Therefore,

In more price-sensitive environments, supplier should focus more on obtaining info about the buyer’s costs.

Value of Info v.s. Value of Value of Info v.s. Value of Contracting Flexibility (Cont’Contracting Flexibility (Cont’dd))

F1F1

F2F2

A1A1

A2A2

Numerical Numerical ExamplesExamples

ConclusionsConclusions

ConclusionsConclusions

Under full information, a supplier will decrease his wholesale price in reaction to a buyer cost increase, maintaining the volume while sacrificing margin.

Conclusions (Cont’Conclusions (Cont’dd))

Under asymmetric information, however, the supplier may do the opposite: increase average wholesale price, thus maintaining margin while sacrificing volume.

Conclusions (Cont’Conclusions (Cont’dddd))

The value to the supplier of obtaining better information about the buyer’s cost structure increases with the variance of the supplier’s prior distribution about that cost parameter and with price-sensitivity of demand.

Conclusions (Cont’Conclusions (Cont’dddddd))

The value of better information is greater when the supplier can offer two-part contracts rather than only one-part contracts, and

The value of being able to offer two-part contracts rather than one-part contracts is decreasing in price-sensitivity b.

Future ResearchFuture Research

……..

In many contracting situations, the supplier starts in case A1: – offering a simple linear wholesale price – with no side payment– without knowing the buyer’s cost

structure

Questions to Answer:Questions to Answer:

When should the supplier focus on obtaining better information about the buyer’s cost structure?

When should he offer more sophisticated contracts?

How would the results change if we introduce stochastic price-sensitive demand?

What changes if the the supplier cannot observe the price-sensitivity parameter b?

Designing Supply Contracts: Designing Supply Contracts: Contract Type and Information Contract Type and Information AsymmetryAsymmetry

Authors: C. Corbett, C. S. Tang

Presenter: T.J. Hu