Supply Chain Contracts Gabriela Contreras Wendy O’Donnell April 8, 2005

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems and open questions

A contract provides the parameters within which a retailer places orders and the supplier fulfills them.

Example: Music store Supplier’s cost c=$1.00/unit Supplier’s revenue w=$4.00/unit Retail price p=$10.00/unit Retailer’s service level CSL*=0.5

Question What is the highest service level both the supplier and retailer can hope to achieve?

Example: Music store (continued) Supplier’s cost c=$1.00/unit Supplier’s revenue w=$4.00/unit Retail price p=$10.00/unit Supplier & retailer’s service level CSL*=0.9

Characteristics of an Effective Contract: Replacement of traditional strategies No room for improvement Risk sharing Flexibility Ease of implementation

Why? Sharing risk increase in order quantity increases supply chain profit

Types of Contracts: Wholesale price contracts Buyback contracts Revenue-sharing contracts Quantity flexibility contracts

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems & open questions

Example: Ski Jacket Supplier Supplier cost c = $10/unit Supplier revenue w = $100/unit Retail price p = $200/unit Assume: –Demand is normal(  –No salvage value

Formulas for General Case 1. E[retailer profit] = 2. E[supplier profit] = q(w-c) 3. E[supply chain profit] = E[retailer profit] + E[supplier profit]

Results: Optimal order quantity for retailer = 1,000 Retail profit = $76,063 Supplier profit = $90,000 Total supply chain profit = $166,063 Loss on unsold jackets: –For retailer = $100/unit –For supply chain = $10/unit

Optimal Quantities for Supply Chain: When we use cost = $10/unit, supply chain makes $190/unit Optimal order quantity for retailer = 1,493 Supply chain profit = $183,812 Difference in supply chain profits = $17,749

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Buy-Back Contracts Supplier agrees to buy back all unsold goods for agreed upon price $b/unit

Change in Formulas: 1. E[retailer profit] = 2. E[supplier profit] = q(w-c) 3. E[overstock] = + bE[overstock] – bE[overstock]

Expected Results from Buy-back Contracts for Ski Example

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Revenue-sharing Contracts Seller agrees to reduce the wholesale price and shares a fraction  of the revenue

Change in formulas E[supplier profit]= (w-c)q+  p(q-E[overstock]) E[retailer profit]= (1-  )p(q-E[overstock])+v E[overstock]-wq

Expected results from revenue- sharing contracts for ski example Wholesale Price w Revenue- sharing Fraction,  Optimal Order Size Expected Overstock Retail Expected Profit Supplier. Expected Profit Expected Supply Chain Profit $ $124,273$ 59,429$183,702 $ $ 84,735$ 98,580$183,315 $ $ 45,503$136,278$181,781 $ $ 7,606$158,457$166,063 $ $110,523$ 71,886$182,409 $ $ 71,601$109,176$180,777 $ $ 33,455$142,051$175,506

“Go Away Happy” “Guaranteed to be There”

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Quantity-flexibility Contracts Retailer can change order quantity after observing demand Supplier agrees to a full refund of  q units

Quantity-flexibility Contract for Ski Example

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Contracts and the Newsvendor Problem One supplier, one retailer Game description: Accept Contract? Q Production Transfer payments End Y N Demand Recognition Product Delivery

Assumptions Risk neutral Full information Forced compliance

Profit Equations  r = pS(q) – T  s = T – cq  q  = pS(q) – cq =  r +  s p= price per unit sold S(q)= expected sales c= production cost Proof:

Transfer Payment What the retailer pays the supplier after demand is recognized T = wq w = what the supplier charges the retailer per unit purchased

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Newsvendor Problem Wholesale Price Contract Decide on q, w

Let w be what the supplier charges the retailer per unit purchased T w (q,w)=wq

Retailer’s profit function  r = pS(q)-T

Supplier’s Profit Function  s = (w-c)q

Results: Commonly used Does not coordinate the supply chain Simpler to administer

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Buy-back Contracts Decide on q,w,b Transfer payment T = wq – bI(q) = wq – b(q – S(q))

Claim A contract coordinates retailer’s and supplier’s action when each firm’s profit with the contract equals a constant fraction of the supply chain profit. i.e. a Nash equilibrium is a profit sharing contract

Buy-back contracts coordinate if w & b are chosen such that:

Recall:  r = pS(q) – T  r = pS(q) – wq – b(q – S(q)) = (p – b)S(q) – (w – b)q  =  (q)

Recall:  s = T - cq  s  – cq wq – b(q – S(q)) = bS(q) + (w – b)q – cq  q)

Results Since q 0 maximizes  (q), q 0 is the optimal quantity for both  r and  s And both players receive a fraction of the supply chain profit

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Newsvendor Problem Revenue-Sharing Contracts Decide on q, w, 

Transfer Payment T r = wq + pS(q) (1- 

Retailer’s Profit  r =  pS(q)- T For  Є (0,1], let  p= p w= c  r=  (q) 

Similar to Buy-Back From Previous Slide:  r (q,w r,  )=  (q) Recall from Buy-Back:  r (q,w r,b)=  (q)

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems

Quantity-flexibility Contracts Decide on q,w,  Supplier gives full refund on  q unsold units i.e. min{I,  q}

Expected # units retailer gets compensated for is I r Proof:

Retailer’s profit function  r = pS(q) – wq + w

Optimal q satisfies: w = p(1 – F(q)) 1 – F(q) + F((1 –  )q)(1 –  ) If supplier plays this w, will the retailer play this q?

Only if retailer’s profit function is concave As long as w < p and w > 0

Supplier’s profit function  s = wq – w What is supplier’s optimal q?

Key result The supply chain is not coordinated if (1 –  ) 2 f((1 –  )q 0 ) > f(q 0 ) q 0 is the minimum

Result Supply chain coordination is not guaranteed with a quantity- flexibility contract Even if optimal w(q) is chosen It depends on  & f(q)

Summary You can coordinate the supply chain by designing a contract that encourages both players to always want to play q 0, the optimal supply chain order quantity

Outline Introducing Contracts Example: ski jackets –Buy-back –Revenue-sharing –Quantity-flexibility Newsvendor Problem –Wholesale –Buy-back –Revenue-sharing –Quantity-flexibility Results for other problems and open questions

Newsvendor with Price Dependent Demand Retailer chooses his price and stocking level Price reflects demand conditions Can contracts that coordinate the retailer’s order quantity also coordinate the retailer’s pricing? Revenue-sharing coordinates

Multiple Newsvendors One supplier, multiple competing retailers Fixed retail price Demand is allocated among retailers proportionally to their inventory level Buy-back permits the supplier to coordinate the S.C.

Competing Newsvendors with Market Clearing Prices Market price depends on the realization of demand (high or low) & amount of inventory purchased Retailers order inventory before demand occurs After demand occurs, the market clearing price is determined Buy-back coordinates the S.C.

Two-stage Newsvendor Retailer has a 2 nd opportunity to place an order Buy-back Supplier’s margin with later production < margin with early production

Open Questions Current contracting models assume on single shot contracting. Multiple suppliers competing for the affection of multiple retailers Eliminate risk neutrality assumption Non-competing heterogeneous retailers