designing supply contracts: contract type and information asymmetry authors: c. corbett, c. s. tang...
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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