Transcript Designing Supply Contracts: Contract Type and Information

Designing Supply Contracts:
Contract Type and Information
Asymmetry
Authors: C. Corbett, C. S. Tang
Presenter: T.J. Hu
Contents
Introduction
 Literature
 Model
 Supplier's Optimal Supply Contracts
 Comparisons
 Numerical Examples
 Conclusions
 Future Research

Introduction
The Supply Chain
s
Supplier
L(q) w(q)
C ~ F(•)
Buyer
cost c
p(q)
Supplier's Concerns:
 The
types of contracts
 Information about the buyer's
cost structure
Three 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 Scenarios
Type of contracts
Full
information
Asymmetric
information
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:





What should supplier do when faced with
decreased buyer demand?
Value of information about the buyer’s
cost structure
Value of more sophisticated contracts
Which of the above two is more valuable?
When there is no double marginalization?
Literature
1. Supply Chain Management
2. Economics
Supply 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 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

d
(Cont’ )
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 Literature

Vertical contracting
– Two successive monopolists
– Double marginalization

Topics
– Comparing total surplus under various
schemes
– Contract to mitigate the double
marginalization issue
Selected 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 Paper

Combine two strands of theory
– building on the basic bilateral monopoly
framework offered in economics
– asking the normative and more microlevel questions more typical of supply
chain literature
– measure the cost of sub-optimality
(quantification and insights of the
differences between the cases)
The Model
The Supply Chain
s
Supplier
L(q) w(q)
cˆ ~ F (cˆ)
Buyer
cost c
p(q)
Assumptions
One supplier and one buyer
 One product
 One period contract
 Deterministic demand
 Linear price-demand curve q = a - bp
 a - b (s+ )  0
 F(c)/f (c) is increasing in c

F/f for Normal Distribution
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-5
-4
-3
-2
-1
0
Supplier’s Problem (S)
max
{ w ( q ), L ( q )}
 (w, L)  Ew(q(c))  s q(c)  L(q(c))
s
 (c, q)   p(q)  w(q)  cq  L(q)   ,c
D  (c, q )  0, c

s.t.
b
q
b
 : Buyer' s reservatio n profit level,

b
b
d
Dq 
dq

Buyer’s individual rationality constraint

Buyer’s incentive compatibility constraint
Sequence 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 (A3)
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 Optimal
Supply Contracts
1. Buyer’s problems
2. Supplier’s problems
Buyer’s Problem (B1, B2)
max b (q)   p(q)  w  c q  L
q
aq


 ( w  c) q  L
 b


In B1, set L=0
 In B2, to buyer, L is independent of q
Solutions for B1, B2
a  b( w  c )
p 
2b
1
*
q  a  b( w  c)
2
*
 
* 2
1
q
2
b  4b a  b(w  c)  L  b  L
*
Comments on the Solutions
1
1) If q  a  b( w  c)  0 or
2
*

1
2
b  4b a  b(w  c)  L  b ,
the buyer simply won ' t order!
1
a
*
2) q  a  b( w  c)  0 implies w   c.
2
b
Thus supplier should not set w to be too high.
*
Buyer’s Problem (B3)


*
*
ˆ
ˆ
ˆ
ˆ
max b (c, c)  L(c)  p ( w(c))  w(c)  c q ( w(cˆ))
cˆ
a 1

 L(cˆ)  b   w(cˆ)  c 
 2b 2

2
For any cˆ, the buyer solves B2 .
He/she then maximizes over cˆ.
Solution for B3
d
FOC : b (c, cˆ) cˆ c  0 
dcˆ
1

L(c)  a  b( w  c)w (c)
2
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 Problems
Optimal Contracts Under Complete
Information: Case F1, F2, F3
Contracts with Full Information
Type of contracts
Full
information
Asymmetric
information
One-part linear: w
F1
A1
Two-part linear: w,L
F2
A2
Two-part nonlinear:
{w(q), L(q)}
F3
A3
Case F1 (SF 1)
max s , F ( w)  w  s q
w
*
1
1
 ( w  s) a  b( w  c)
2

Note: The Supplier knows the buyer’s
optimal order quantity q*
Solution for SF 1
a 1
w 
 ( s  c)
2b 2
1
2
*
s,F1 (wF1 )  8b a  b(s  c)
*
F1
Profit and Profit Margins
1
1
2
*
b,F1  16b a  b(s  c)  2 s,F1 (wF1 )
*
1 a
 1 *
p  w  c    ( s  c)  ( w  s )
4 b
 2
*

*
Supplier’s profit and profit margin are
double those of the buyer!
More on Profit and Margins
1
In general, this ratio is
,
2 
where   q

2
q
D p
Dq p
.
 is a local measure of the curvature of the
demand curve
 Ref. Bresnahan and Reiss (1985)
Question:

?
b,F b

*
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 F2 (SF 2)
max s , F ( w, L)  w  s q  L
*
{ w , L}
2
1
 ( w  s) a  b( w  c)  L
2
s.t

b , F2
(c, c) b

Observations

With complete info about c, supplier can
set the rationality constraint to be binding.
 He then maximizes the joint profits.
max  j , F ( w, L)  ( p  s  c)q 
*
{ w , L}
*
2
1
max  j , F ( w)  a  b( w  2s  c)a  b( w  c)
2
w
4b
Solution for SF 2
wF*2  s,
1
2
L  b  a  b( w  c) 
4b
(should  0 if q *  0  franchise fee)

*
F2


*
s , F2
*
b , F2
(wF*2 , L*F2 )   L*F2 ( 0),
b

1
2
 j ,F2  4b a  b(w  c)
Comments on Solutions for SF 2
1
w  s  q  a  b( s  c)   0
2

1
2
a  b(w  c)   b , then
If
4b

1
2
*
LF2  b  a  b( w  c)   0
4b
*
F2

*
*
s , F2
  L*F2  0, supplier wi ll simply set L  0.
 No transacti ons will happen!
Interpretation

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 F3 (SF 3)
 Superset
of F2
 F2 is optimal given full info on c:
buyer only gets minimum level
 Value of addition flexibility is 0
  results carry over from F2
Supplier’s Problems
Optimal Contracts Under Asymmetric
Information: Cases A1, A2, A3
Contracts with Asymmetric
Information
Type of contracts
Full
information
Asymmetric
information
One-part linear: w
F1
A1
Two-part linear: w,L
F2
A2
Two-part nonlinear:
{w(q), L(q)}
F3
A3
Case A1 (SA1)

max E[s , A ( w)]  E w  s q
w
1

c
c

*

1
( w  s) a  b( w  c)dF (c)
2
Note: The Supplier knows the form of the
buyer’s optimal order quantity q*
Solution for SA1
a 1
w 
 ( s  E[c])
2b 2
1
2
*
s, A1 (wA1 )  8b a  b(s  E[c]) 
*
A1
Profit and Profit Margins
1
2
b, A1  16b a  b(s  2c  E[c]) 
*
1 a

p  w  c    ( s  2c  E[c]) 
4 b

*
*
1 a

w  s    ( s  E[c]) 
2 b

*
A1

Supplier has incentive to induce the buyer
to reveal his true cost c. (???)
Question:

Is the buyer’s individual rationality
constraint satisfied, i.e.


?

b,A
b

*

1
If not satisfied, the buyer won’t order.
1
Also need q  a  b( s  2c  E[c])   0
4
*
Case A2 (SA2)
max E[s , A ( w, L)]  E[w  s q  L]
*
{ w , L}
2
1


  ( w  s ) a  b( w  c)  L dF (c)
c
2



1
2
s.t.
b, A2 (c, q)  4b a  b(w  c)  L  b
c
Observations

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
1
*
*
b b, A2 (c , q)  4b a  b(wb, A2  c )  Lb, A2

Solution for SA 2
FOC  w*A2  s  c  E[c],
1
2
L  b  a  b( s  2c  E[c]) 
4b
*
1
*
*
*
E s , A (wA2 , LA2 )   LA2  c  E[c]a  b( s  c ) ,
2
2

*
A2



*
b , F2
b

Remarks
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

1
q  a  b( s  2c  E[c])   0
2
*
Question:

When and what if the expected
supplier’s profit is zero?


1
2


E s , A (w , L )  b 
a  b( s  2c  E[c])
2
4b
1
 c  E[c]a  b( s  c ) 
2
*
*
A2
*
A2

Case A3 (SA 3)
max E[s , A ]  E[( w  s )q *  L]
{ w ( ), L ( )}
3
1


 E ( w  s ) a  b( w  c)   L 
2


1

s.t.
L  a  b( w  c) w , c
2
 (c, c)   ,c

b

b
Using buyer’s optimal order quantity q*
and FOC from B3
Euler’s Equation:
Necessary Conditions
Smooth critical points of the functional s
x1
J ( y )   F ( x, y, y ' )dx,
x0
where y ( x0 )  a, y ( x1 )  b,
should satisfy th e Euler' s equation :
d
Fy  Fy '  0.
dx
Constrained Problems: Lagrangian
Smooth critical points of the functional s
x1
J ( y, z )   F ( x, y, y ' , z , z ' )dx, s.t.  ( x, y, z , y ' , z ' )  0
x0
where y ( x0 )  y0 , y ( x1 )  y1 , z ( x0 )  z0 , z ( x1 )  z1 ,
should satisfy :
d
Fy  Fy '   ( x) y  0,
dx
d
Fz  Fz '   ( x) z  0, and
dx
 ( x, y , z , y ' , z ' )  0
Solution for SA 3
F (cˆ)
w (cˆ)  s 
f (cˆ)
*
A3

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 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 Prior
w (c )  s  c  c
*
A3
1
L (c)  ca  b( s  c  c)  k
2
*
A3
Special Case A3
d
(Cont’ )
1
a 1
w (q)   s  c    q
2
b b

1
2
*
2
a  b( s  c)  4q  k
LA3 (q)  b 
8b
*
A3



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
Comparisons
The Impact of Buyer’s Cost on
the Supplier’s Profit Margin
Buyer’s cost c  
Buyer’s profit margin mb  
Buyer orders less 
Supplier’s profit 
 How should the supplier respond?

Supplier’s Response



1a 1
 (c  s )
c
In case F1, A1, A2: ms , F1 
2b 2
– sacrifices margin
1a 1
for volume
ms , A1 
 ( E[c]  s )  E[c]
2b 2
In F2 and F3:
ms , A2  c  E[c]
 E[c]
– insensitive
In A3:
– sacrifices volume
for margin
ms , F2  ms , F2  0
ms , A3
F (c )

f (c )
c
“Effective” Wholesale Price

More precisely, one should take the side
payment into account and evaluate the
“effective” unit wholesale price as follows:
L
w  w
q
e
The Value of Info to the Supplier
 F1 , A1 s
b
 E[s , F s , A ]  Var (c)
1
1
8
*
*
 F2 , A2 s  E[s , F s , A ]
*
*
2
 F2 , A2 s

2
b
1
 Var (c)  c  E[c]a  b( s  c ) 
4
2
 2  F1 , A1 s
The value of information is (significantly) greater
when the supplier has the flexibility to offer twopart contracts.
The Value to the Supplier of
Offering Side Payments
 F2 , F1 s
 A2 , A1 s
 F2 , F1 s


1 a  b( s  c) 
 b 
8
b
2
 1 a  b( s  2c  E[c ]) 
 b 
8
b
  A2 , A1 s

2
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 of
Contracting Flexibility
Value 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
Contracting Flexibility (Cont’d)
F1
A1
F2
A2
Numerical Examples
Conclusions
Conclusions

Under full information, a supplier will
decrease his wholesale price in
reaction to a buyer cost increase,
maintaining the volume while
sacrificing margin.
Conclusions

d
(Cont’ )
Under asymmetric information,
however, the supplier may do the
opposite: increase average
wholesale price, thus maintaining
margin while sacrificing volume.
Conclusions

dd
(Cont’ )
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
ddd
(Cont’ )
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 twopart contracts rather than one-part
contracts is decreasing in pricesensitivity b.

Future 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:




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:
Contract Type and Information
Asymmetry
Authors: C. Corbett, C. S. Tang
Presenter: T.J. Hu