Supporting document to “Inequity aversion in a Joint
Economic Lot Sizing environment with asymmetric
holding cost information”
9. Derivation of optimal contract parameters
Formelabschnitt 9


n
f 
n 1
f 
max   i 1 i  pi   wi    d   i 1 1   i   pi   wi 1 
  d  1   n   pn  P S  d 
qi 
qi 1 


h
h
h
n 1




 i1 i,i 1   wi  ti   d  2i qi  wi 1  d  2i qi 1       wn  tn   d  2n qn  R  d  
    t
n
i 1
i
l
 ti   i (ti  th )    i  2  j i ij  q j  qi 
n 1
(9.1)
h
h
L  i  pi  f  d (1   i 1 )  pi 1  f  d


 i ,i 1  i  i 1,i  i 1   j i ij  0 , i  1,..., n  1
2
2
qi
qi
qi
2
2
(9.2)
h
h
L  n  pn  f  d (1   n 1 )  pn 1  f  d
n 1


 n 1,n  n 1    n   i 1 in  0
2
2
qn
qn
qn
2
2
(9.3)
L
  i  pi  d  1   i 1   pi 1  d  i ,i 1  d  i 1,i  d  0 , i  1,...., n  1
wi
(9.4)
L
  n  pn  d  1   n 1   pn 1  d  n 1,n  d    d  0
wn
(9.5)



L i
f 
f 

 pi   wi    d  i  pi   wi 1 
  d  i ,i 1  d   i  i  0
ti ti
qi 
ti
qi 1 


, i  1,..., n  1
(9.6)


L  n
f 

 pn   wn    d  n  pn  P S  d    d   n  n  0
tn tn
qn 
tn

L
i ,i 1
h
h


    wi  ti   d  i qi  wi 1  d  i qi 1   0
2
2


, i  1,..., n  1
(9.7)
(9.8)
h
L


    wn  tn   d  n qn  R  d   0

2


(9.9)
L
 tl  t i  0
 i
(9.10)
, i  1,..., n
1
L
 ti  t h  0
i
, i  1,..., n
(9.11)
L
 q j  qi  0
ij
(9.12)
L
L
L
L
L
ij ,
qi  0,
wi  0, ti  0,
i.i 1  0,
ij
qi
wi
ti
i ,i 1
L
L
L
L
  0,
 i  0,
i  0,
ij

 i
i
ij
(9.13)
Let pˆ i denote the probability that contract Ai is chosen, i.e.
pˆ i   (ti ) pi  (1   i 1 ) pi 1 , i  2,..., n
pˆ1   (t1 ) p1
From (9.5) follows
pˆ n  d  n 1,n  d    d  0
(9.14)
And it follows directly that
  pˆ n  n 1,n
(9.15)
From (9.4) follows
i ,i 1  pˆ i  i 1,i
(9.16)
Inserting (9.16) into (9.15) for all i  1,..., n  1 gives
  pˆ n  pˆ n1  ...  pˆ1  p1  p2  ...   n  pn  1  (1   (tn )) pn
(9.17)
Inserting (9.17) into (9.15) gives
n1,n  1  pn  1  n1   pn1
(9.18)
And it follows from (9.16) that
2
n 2,n 1  n 1,n  pˆ n 1  1  (1   n ) pn  pˆ n  pˆ n 1
....
i 1,i   k 1 pk   (ti 1 )  pi 1
i 2
(9.19)
i ,i 1  1  (1   n ) pn  pˆ n  ...  pˆ i 1   k 1 pk   i  pi   k 1 pˆ k
i 1
i
1,2  1  p1
Inserting (9.17) and (9.19) into (9.3) and (9.2) gives
qi (ti , ti 1 ) 
hi 

2 f d

 k 1 pk  i 1  pi 1   hi  hi 1    j i ij
i 2
 1    p
i 1
i 1
(9.20)
  i pi 
If the implementability condition is not binding in the optimal menu of contracts, then  ij  0 and the
optimal order sizes result from
qi 
hi 

2 f d

 k 1 pk   i1  pi1   hi  hi1 
i2
 1    p
i 1
i 1
i  1,..., n
(9.21)
  i pi 
Yet, if bunching occurs in the optimal menu of contracts, then at least one  i , j  0 . We define qij as
the order sizes that are bunched, i.e., qij  qi  qi 1  ...  q j j  i . In this case, the necessary KKT
condition (9.2) changes to:
(1   z 1 )  pz 1  f  d
h
h 
L
j   p  f d
  z i  z z 2

 z , z 1  z  z 1, z  z 1   0 , i  1,..., n  1
2

qij
qij
qij
2
2 

Inserting (9.19) and rearranging gives
qij 
hj 

2 fd

(9.23)
 k 1 pk  i 1  pi 1   h j  hi 1 

i 2
j 1
k i
pk   1   i 1  pi 1   j  p j 
Given the optimal contract parameters that can be determined via the algorithm stated in Section 3m
the optimal wholesale prices result from (9.9)
wn AI  R  tn 
hn
 qn AI (tn )
2d
(9.24)
and (9.8)
3
(9.22)
wi  wi 1  ti 
hi
 qi 1  qi 
2d
(9.25)
As long as the additional incentive is an interior solution (i.e., as long as  i , i  0 ), the optimal
value for ti can be obtained by solving the equation system resulting from inserting (9.19) into (9.6),
i.e.,

 f
 i
f
 pi    wi  wi 1   

ti
 qi 1 qi


  


i 1
k 1

pk   i  pi  0 i  1,..., n  1
(9.26)
And for inserting (9.24) and (9.25) gives li which denotes the marginal revenues/costs of increasing ti .
li 


i pi 
h
i 1

   ti d  i  qi  qi 1   fdqi 1  fdqi 11   i  pi   k 1 pk  0 i  1,..., n  1 (9.27)
ti d 
2

Respectively, t n can be obtained by inserting (9.17) into (9.7)


 n
h
f
 pn  R  tn  n  qn   P S    n pn  1  pn
tn
2d
qn


(9.28)
Note, that the equations (9.26)/(9.27) are uni-modular and have therefore at most two roots.
Proof:
 2 i
The first derivative w.r.t. ti while considering 2  0 gives:
 ti
 h q
 p

q
h q
q 
l
  i  i  d  i  pi  fdqi 12  i 1  i i 1   i i  fdqi 2  i 
ti
ti d
ti
ti
2 ti  2 ti
ti 
(9.29)
Let the first term,
x1  
 i pi


  d  i  pi  2  i  pi  0
ti d
ti
ti
(9.30)
denote the effect of changing ti if the order sizes are not adjusted (e.g., the order sizes are calculated
with    ), where li  x1  x2 . (9.30) is constant and negative, i.e.,
(9.27) would be strictly
monotonically decreasing if the order sizes were not adjusted. However, in the optimal solution, the
order sizes qi and qi 1 will only be adjusted, if the marginal revenues of increasing the order size are
4
larger than the marginal cost. Thus, it follows that adjusting the order sizes can only have a positive
contribution in (9.27) and it follows that there can be at most one maximum.
Let x2 denote the effect of the order size adjustment, i.e.,
x2   fdqi 12 
qi 1 hi qi 1  hi qi
q


 fdqi 2  i
ti
2 ti  2 ti
ti

0

(9.31)
The effect of the order size adjustment is positive (which follows directly from
 fd hi
  fd hi 
 qi 1   
 qi   0 ) and this positive effect becomes larger with increasing ti which

q
2
q
 i 1
  i 2 
follows directly from (9.31). Thus, it follows that (9.27) is unimodular (a monotonically increasing
positive term, (9.31), is added to a constant negative term, (9.30), and thus there are at most two roots
for each equation in (9.29) satisfying the necessary KKT conditions. If there are exactly two roots,
then the first necessary condition is a local maximum (the marginal revenues of increasing ti are larger
than the marginal costs), and the second root is necessarily a local minimum.
Assume the case with two roots (y1 and y2). For ti < y1 the marginal revenues (i.e., incentivizing
choosing a larger order size) are larger than the marginal costs (i.e., the informational rent to the
buyer). Thus, for a local maximum the supplier should increase t i to y1. For y1<ti<y2, the marginal
costs are higher than the marginal benefits, and increasing t i to y2 will yield a local maximium.
Increasing ti further contributes positively to the objective function. Thus, if the additional benefits
gained by increasing ti from y2 to th offset the cost occurred from increasing ti from y1 to y2, then the
global optimum will be th, otherwise it will be y1.
5