Supply Contract Models 2

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 such that

Over [0, ], assume p(q) is decreasing, concave, and C2

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).

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1. Centrally controlled supply chain

Profits:

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, ], there exists an optimal

solution for the chain qo which satisfies (qo) = 0:

2. Decentralized solution

Retailer’s profits:

Again, profit equation strictly convex, so there exists a q* such than

Since supplier will choose w > c in order to have a profit, comparing

(5.1) and (5.1*) shows that q* < qo, meaning the retailer will order less than

the system-wide optimal quantity whenever the supplier makes a profit.

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3. Investigation

Marginal cost pricing: setting w = c will allow for q* = qo, but will leave

the supplier without any profits

Two-part tariff: set w = c but charge fixed fee of (qo), then retailer will

order qo but will see no profits

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 qo to maximize profits.

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:

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 qo determined by:

2. Decentralized solution

Retailer profits:

If p > w > b, r(q) is strictly concave and has an optimal solution q*

If w > c and b = 0, a comparison of (5.2) and (5.3) imply that q* < qo and

the double marginalization situation occurs.

3. Investigation

q* = qo if w = c, but again, not attractive for supplier

(5.3) indicates that increasing b will increase q*. In fact, q* = qo if

Let be the value of b which satisfies (5.4):

Supply chain profits will be maximized when w > c and b = ,

where is independent of the demand distribution.

Supplier revenue:

5

Buy-back Contracts

If b = , assume retailer will select q = qo:

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 qo, even as its profit margin shrinks to 0.

6

Quantity Discounts

Can mitigate double marginalization:

Retailer pays w(q) where

Can be shown that retailer will choose qo since its marginal cost equals

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 Ko for producing any order, each unit

costs an average of Ko/q + c, which is decreasing in q. Quantity

discounts encourage the retailer to order more than they would

otherwise (as they don’t see the additional cost).

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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:

L2 periods between supplier and its source

L1 periods between supplier and retailer

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 h2 > 0 for each unit in-stock or in-transit Retailer

pays h2 + h1 per unit in inventory (h1 0)

Backlog:

All orders are backlogged until filled (no demand is turned away):

p = system-wide cost for backlogging an order

1p = retailer’s cost to backlog an order

2p = supplier’s cost to backlog an order

1 +2 = 1, 1, 2 0.

Demand:

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Competition in Supply Chain Inventory Game:Model

D = random total demand over periods

= mean total demand over periods

= 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

1(0) = 0, so positive demand occurs in each period

Local inventory variables for stage i and period t

ITit = in-transit inventory between stages i+1 and i

ILit = inventory level at stage i minus all backorders

IPit = ITit + ILit = inventory position

Policy:

Player i uses a base stock policy of ordering enough items to raise

inventory position plus outstanding orders to level si [0, S],

where 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.

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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.

= retailer’s charge in period t

= holding cost for inventory + backorder and order cost

= retailer’s expected charge in period t + L1

= retailer’s optimal base stock level found by minimizing :

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Competition in Supply Chain Inventory Game:Optimal Solution/Game Analysis

= induced penalty function

= supplier’s charge in period t

= holding cost for inventory + induced penalty

= supplier’s charge in period t

= supplier’s optimal base stock level found by minimizing

Game Analysis

Hi(s1, s2) = player i’s expected per period cost using base stock levels

s1 and s2

Best reply mapping for player i is a set-values relationship associating

each strategy sj with a subset of the decision space under the following

rules:

A pure strategy Nash equilibrium is a (s1*, s2

*) such that each player

chooses a best reply to the other’s equilibrium base stock level:

s2* r2(s1

*) such and s1* r1(s2

*)

Retailer’s cost function:

= retailer’s charge in period t

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Competition in Supply Chain Inventory Game:Optimal Solution/Game Analysis

= retailer’s expected charge in period t + L1

After firms place orders in period t - L2, the suppliers IP2(t-L2) = s2

After inventory arrives in period t, (as retailer has ordered

over periods [t - L2 + 1, t]). If 0, the supplier can completely fill the

retailer's order for period t and . If < 0, the order cannot be

completely filled and

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Competition in Supply Chain Inventory Game:Game Analysis

Supplier’s cost function:

= supplier's actual period t backorder cost

= supplier's expected period t + L1 backorder cost

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.

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Equilibrium Analysis

Theorem 1: H2(s1, s2) is strictly convex in s2 and H1(s1, s2) is strictly

convex in s1.

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:r2(s1) is a function; when 2 = 0, r2(s1) = 0; and when 2 > 0,

r2(s1) > 0 and -1 < r'2(s1) < 0.

Theorem 3:r1(s2) is a function; when 1 = 0, r1(s2) = 0; and when 1 > 0,

r1(s2) > 0 and -1 < r'1(s2) < 0.

Theorem 4: (s1*, s2

*) is the unique Nash equilibrium.

Figures show he best reply functions, Nash equilibrium, and optimal

solution.

Theorem 5: Assuming 1 < 1, s1* + s2

* < s1o + s2

o.

System optimal solution is not a Nash equilibrium whenever 1 < 1.

When 1 = 1, it may be one under a very special condition.

So, competitive selection of inventory policies almost always lead to a

deterioration of supply chain performance.

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15

OptimalSolution

Nash equilibrium

SupplierreactionfunctionRetailerreactionfunction

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

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

OptimalSolution

Nash equilibrium

SupplierreactionfunctionRetailerreactionfunction

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

= 0.9, p = 5, h1= h2 = 0.5, L1 = L2 = 1

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Supply Contract Models 2