Double Marginalization

“Classic double marginalization” result has a single supplier selling a
product to a single retailer, who faces downward-sloping customer demand.
When the retailer doesn’t consider the supplier’s profit margin while
ordering, it will tend to order less than level that would maximize supplier
profits.
 q = quantity retailer orders from supplier, q  0
 p(q) = price at which retailer can sell q units, p(q)  0
There exists a maximum sales quantity q̂ such that pqˆ   0
Over [0, q̂ ], assume p(q) is decreasing, concave, and C
2
 c = production cost per unit for supplier, c  p(0)
 w = (wholesale) price per unit paid by retailer
 Over one period, given c and p(q) are known, game follows:
1. Supplier chooses w
2. Retailer buys q
3. Retailer sells at p(q)
 To analyze this and all subsequent games, follow these steps:
1. Find centralized solution, where a single agent controls all aspects
of supply chain to maximize profits
2. Find decentralized solution, where players make decisions to
maximize individual profits.
3. If (1) and (2) differ, modify profit equations to find a new
decentralizes solution where the behavior more closely follows (1).
1
Double Marginalization
1.
Centrally controlled supply chain
 Profits:
q   pq  cq
As the retailer paying w to the supplier is a transfer of funds within the
supply chain, doesn’t affect the whole chain’s profits
 Since (q) is strictly concave in over [0, q̂ ], there exists an optimal
solution for the chain q which satisfies (q ) = 0:
o
o
 
 
p q o  c  q o p q o  0 (5.1)
2.
Decentralized solution
 Retailer’s profits:
 r q   pq  wq
 Again, profit equation strictly convex, so there exists a q such than
*
 
 
p q*  w  q* p q*  0 (5.1*)
 Since supplier will choose w > c in order to have a profit, comparing (5.1)
and (5.1*) shows that q < q , meaning the retailer will order less than the
*
o
system-wide optimal quantity whenever the supplier makes a profit.
2
Double Marginalization
3.
Investigation
 Marginal cost pricing: setting w = c will allow for q = q , but will leave
*
o
the supplier without any profits
 Two-part tariff: set w = c but charge fixed fee of (q ), then retailer will
o
order q but will see no profits
o
 Profit-sharing contract: select 0    1 where supplier earns (q) and
retailer earns (1 - )(q). Since retailer no longer cares about wholesale
price, will pick q to maximize profits.
o
3
Buy-back Contracts
 Buy-back contract specifies a price b at which the supplier will purchase
unsold goods from retailer. Additionally, assume no supplier receives any
income from returned goods.
 Single supplier and retailer
 q = quantity retailer orders from supplier, q  0
 p = fixed price retailed charges per item, p > 0
 c = production cost per unit for supplier, c  p
 w = (wholesale) price per unit paid by retailer
 (x) and (x) = p.d.f. and c.d.f. of demand on retailer, where (x) C
 Over one period, given c and (x) are known, game follows:
1. Supplier sets w and b
2. Retailer selects amount q to order
3. Supplier produces q units at cost c per unit
4. Demand realized and unsold units returned to supplier
1.
Centralized control
 Profits:
q


 q   cq  p 1   q q   x  x dx 


0
 production costs  expected sales revenues
Again, since w and b represent transfers within supply chain, overall
profit does not depend on them.
4
Buy-back Contracts
 However, this is the traditional newsvendor problem which has an optimal
order quantity q determined by:
o
 
 qo 
2.
p c
p
(5.2)
Decentralized solution
 Retailer profits:
q
q


 r q    wq  p 1   q q   x  x dx   b  q  x   x dx


0
0
 purchase cost  expected sales revenues  expected returns revenue
 If p > w > b,  (q) is strictly concave and has an optimal solution q 
*
r
 
 q* 
p  w
(5.3)
p b
 If w > c and b = 0, a comparison of (5.2) and (5.3) imply that q < q and
*
o
the double marginalization situation occurs.
3.
Investigation
 q = q if w = c, but again, not attractive for supplier
*
o
 (5.3) indicates that increasing b will increase q . In fact, q = q if
*
p  w
p c

p b
p
*
o
(5.4)
Let b̂w be the value of b which satisfies (5.4):
w  c
wc
bˆw  p
 
(q o )
 p  c
Supply chain profits will be maximized when w > c and b = b̂w ,
where b̂w is independent of the demand distribution.
5
Buy-back Contracts
Supplier revenue:
q
 s w, b, q   qw  c   b  x  x dx
0
 sales revenue  expected return cost
If b = b̂w , assume retailer will select q = q :
o




qo
 s w, bˆw, q o  q o w  c   bˆw  x  x dx
0
 s w, bˆw, q o
p
 qo 
w
p c
1

(q o )
q
q
o
 x  x dx
0
o
  x dx
0
As wholesale price increases, supplier’s profits increase. If w = p - ,
where   0, the supplier takes almost all the supply chain profits, but
the retailer will still order q , even as its profit margin shrinks to 0.
o
6
Quantity Discounts

Can mitigate double marginalization:
 c q  q o

Retailer pays w(q) where w q  
o
 c q  q
Can be shown that retailer will choose q since its marginal cost equals
o
that of the supply chain. Additionally, the supplier will earn a profit
since the average wholesale price is > c.
 Manage operating costs:
If a supplier incurs a fixed cost K for producing any order, each unit
o
costs an average of K /q + c, which is decreasing in q. Quantity
o
discounts encourage the retailer to order more than they would
otherwise (as they don’t see the additional cost).
7
Competition in Supply Chain Inventory Game:
Model
 One supplier (referred to as stage 2 or player 2) and one retailer (stage 1/
player 1)
 Time divided into infinitely many discrete periods
 Consumer demand is stochastic, i.i.d. over all periods
 Sequence of events within a period:
1. Shipments arrive
2. Orders submitted and shipped out
3. Consumer demand is realized
4. Holding and backorder penalties assessed
 Lead time for order’s arrival:
L periods between supplier and its source
2
L periods between supplier and retailer
1
 Any non-negative amount may be ordered
 No fixed costs for placing or processing an order
 Each player pays a constant price for each item ordered
 Holding costs:
Supplier pays h > 0 for each unit in-stock or in-transit Retailer pays
2
h + h per unit in inventory (h  0)
2
1
1
 Backlog:
All orders are backlogged until filled (no demand is turned away):
p = system-wide cost for backlogging an order
 p = retailer’s cost to backlog an order
1
 p = supplier’s cost to backlog an order
2
 + = 1,  ,   0.
1
2
1
2
8
Competition in Supply Chain Inventory Game:
Model
 Demand:
D = random total demand over  periods

 = mean total demand over  periods

  and  = p.d.f. and c.d.f. of demand over  periods, where  (x) is a

continuous, increasing, and differentiable function for all x  0,  
1
 (0) = 0, so positive demand occurs in each period
1
 Local inventory variables for stage i and period t
IT = in-transit inventory between stages i+1 and i
it
IL = inventory level at stage i minus all backorders
it
IP = IT + IL = inventory position
it
it
it
 Policy:
Player i uses a base stock policy of ordering enough items to raise
inventory position plus outstanding orders to level s  [0, S], where
i
S is arbitrarily large
When selecting its base stock level, each player is aware of all model
parameters
After selecting base stock levels, model extended over infinite horizon.
9
Competition in Supply Chain Inventory Game:
Model/Optimal Solution
 Externalities:
1. Retailer ignores supplier’s backorder costs, so tends to carry too
little inventory
2. Supplier ignores retailer’s backorder costs, so tends to carry too
little inventory
3. Supplier ignores retailer’s holding costs so tends to carry too much
inventory (supplier’s average delivery time decreases, raising
retailer’s average inventory)
Optimal Solution
 Optimal solution for the supply chain minimizes the total average cost per
period; it has been shown that a base stock policy produces the optimal
solution. Traditional method allocates cost to firms and then minimizes each
player’s new cost function.


 Gˆ1o IL1t  D1 = retailer’s charge in period t
Gˆ1o  x   h1x  h2  p x
= holding cost for inventory + backorder and order cost
 G1o IP1t  = retailer’s expected charge in period t + L
 
G1o  y   E Gˆ1o y  D L1  1
1

 s1o = retailer’s optimal base stock level found by minimizing G1o  y  :
 
 L1  1 s1o 
h2  p
(5.5)
h1  h2  p
10
Competition in Supply Chain Inventory Game:
Optimal Solution/Game Analysis
G1o  y 
 G1  y  = induced penalty function
o
 

 
G1o  y   G1o min s1o , y  G1o s1o
 Gˆ 2o  y  = supplier’s charge in period t


Gˆ 2o  y   h2 y   1  G1o  y 
= holding cost for inventory + induced penalty
 G2o IP2t  = supplier’s charge in period t
 
G2o  y   E Gˆ 2o y  s1o  D L2

 s2o = supplier’s optimal base stock level found by minimizing G2o  y 
Game Analysis
 H (s , s ) = player i’s expected per period cost using base stock levels
i
1
2
s and s
1
2
 Best reply mapping for player i is a set-values relationship associating
each strategy s with a subset of the decision space  under the following
j
rules:
r1 s2  
r2 s1  
s
s

min H s , x 
1
  H1 s1 , s2   min H1  x, s2 
2
  H1 s1 , s2  
x 
2
x 
1
A pure strategy Nash equilibrium is a (s , s ) such that each player
*
1
2
*
chooses a best reply to the other’s equilibrium base stock level:
s  r (s ) such and s  r (s )
*
2
2
1
*
*
1
Retailer’s cost function:
11
1
*
2
Competition in Supply Chain Inventory Game:
Game Analysis


 Gˆ1 IL1t  D1 = retailer’s charge in period t
Gˆ1  y   h1  h2  y   1 p y 
 G1 IP1t  = retailer’s expected charge in period t + L
 
G1  y   E Gˆ1 y  D L1 1



 h1  h2  y   L1 1  h1  h2  p 

y
1
x  y  L1 1 x dx
 After firms place orders in period t - L , the suppliers IP
2
2(t-L2)
=s
2
 After inventory arrives in period t, IL2t  s2  D L2 (as retailer has
ordered D L2 over periods [t - L + 1, t]). If s2  D L2  0, the supplier can
2
completely fill the retailer's order for period t and IP1t  s1 . If s2  D L2 < 0,
the order cannot be completely filled and IP1t  s1  s2  D L2
  
H1 s1 , s2   E G1 min s1  s2  D L2 , s1


L2

s2 G1 s1     L2 x G1 s1  s2  x dx
s2
12
Competition in Supply Chain Inventory Game:
Game Analysis
Supplier’s cost function:


 Gˆ 2 IL1t  D1 = supplier's actual period t backorder cost
Gˆ 2  y    2 p y 
 G2 IP1t  = supplier's expected period t + L backorder cost
1
 
G2  y   E Gˆ 2 y  D L1 1

 Hˆ 2 s1 , x   h2  L1  h2 x  G2 s1  min x,0
 
H 2 s1 , s2   E Hˆ 2 s1 , s2  D L2
 h2 

s2
L1
 h2  s2  x  L2  x dx 
0


L2
s2 G2 s1     L2 x G2 s1  s2  x dx
s2
where the first term is the expected holding cost for the units in-transit
to the retailer, the second term is the expected cost for the supplier's
inventory, and the final two terms are the supplier's expected backorder
cost.
13
Competition in Supply Chain Inventory Game:
Game Analysis
Equilibrium Analysis
 Theorem 1:
H (s , s ) is strictly convex in s and H (s , s ) is strictly
2
1
2
2
1
1
2
convex in s .
1
Since the cost functions are strictly convex, each player has a unique
best reply to the other player's strategy.
 Next two theorems show that when a player cares about backorder costs,
each will maintain a positive base stock. Also, as one player reduces its
base stock, the other will increase its base stock, but at a slower rate.
Theorem 2: r (s ) is a function; when  = 0, r (s ) = 0; and when  > 0,
2
1
2
2
1
2
r (s ) > 0 and -1 < r' (s ) < 0.
2
1
2
1
Theorem 3: r (s ) is a function; when  = 0, r (s ) = 0; and when  > 0,
1
2
1
1
2
1
r (s ) > 0 and -1 < r' (s ) < 0.
1
2
1
 Theorem 4:
2
(s , s ) is the unique Nash equilibrium.
*
1
2
*
Figures show he best reply functions, Nash equilibrium, and optimal
solution.
 Theorem 5:
Assuming  < 1, s + s < s + s .
1
*
*
1
2
o
1
o
2
System optimal solution is not a Nash equilibrium whenever  < 1.
1
When  = 1, it may be one under a very special condition.
1
So, competitive selection of inventory policies almost always lead to a
deterioration of supply chain performance.
14
Competition in Supply Chain Inventory Game:
Game Analysis
1.4
1.3
Supplier
reaction
function
Retailer
reaction
function
Nash
equilibrium
1.2
1.1
1
Optimal
Solution
0.9
0.8
0.7
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
 = 0.3, p = 5, h1= h2 = 0.5, L1 = L2 = 1
1.4
1.3
Supplier
reaction
function
Retailer
reaction
function
Nash
equilibrium
1.2
1.1
1
Optimal
Solution
0.9
0.8
0.7
1.9
2
2.1
2.2
2.3
2.4
 = 0.9, p = 5, h1= h2 = 0.5, L1 = L2 = 1
15
2.5
2.6
2.7