Channel Coordination under Competition A Strategic View

Transcription

1 Channel Coordination under Competition A Strategic View Guillermo Gallego Columbia University Masoud Talebian University of Newcastle CARMA Retreat Newcastle Aug 18, 2012

2 Competition Suppliers Market

3 Competition Suppliers Market Bertrand model considers oligopolistic firms which compete with price. Gallego and Hu [2007] consider the same setting, but firms are allowed to dynamically change their prices.

4 Coordination Market Supplier Retailer

5 Coordination Market Supplier Retailer A monopolistic supply chain is not coordinated with simple wholesale pricing contracts, due to several factors including: Double marginalization: Spengler [1950],... Not enough sales effort: Cachon and Lariviere [2005], Taylor [2002], Krishnan et al. [2004],...

6 Competition and Coordination Suppliers Market Retailer

7 Competition and Coordination Elmaghraby [2000]: If suppliers submit curves that reflect volume discount, then solving for the least-cost (set of) suppliers given the submitted curves is not guaranteed to be a simple task. Cachon [2003] More research is needed on how multiple suppliers compete for the affection of multiple retailers, i.e., additional emphasis is needed on many-to-one or many-to-many supply chain structures. Suppliers Market Retailer

8 Model Suppliers Market Retailer Contracts: general format quantity discounts Pricing: price taker or price setting retailer Method: non-cooperative non-zero-sum game theory

9 Definitions π(x) = i x ip i (x) e(x) pi (x) : the price of product i when selling x units. e(x) : the sales cost when selling x units.

10 Definitions π(x) = i x ip i (x) e(x) pi (x) : the price of product i when selling x units. e(x) : the sales cost when selling x units. c i : capacity of supplier i. c = c i c i = c c i

11 Definitions π(x) = i x ip i (x) e(x) pi (x) : the price of product i when selling x units. e(x) : the sales cost when selling x units. c i : capacity of supplier i. c = c i c i = c c i r(c) = max{π(x) : x [0, c]}. s(c) = arg max{π(x) : x [0, c]}.

12 Definitions π(x) = i x ip i (x) e(x) pi (x) : the price of product i when selling x units. e(x) : the sales cost when selling x units. c i : capacity of supplier i. c = c i c i = c c i r(c) = max{π(x) : x [0, c]}. s(c) = arg max{π(x) : x [0, c]}. r i (c) = r(c) r(c i )

13 Definitions T i : the contract between supplier i and the retailer.

14 Definitions T i : the contract between supplier i and the retailer. π(x T ) = π(x) T i (x i ). r(c T ) = max{π(x T ) : x [0, c]}. s(c T ) = arg max{π(x T ) : x [0, c]}.

15 Sub-modularity Assumption r is submodular, i.e., has decreasing difference. (r(x y) r(x) r(y) r(x y)).

16 Sub-modularity Assumption r is submodular, i.e., has decreasing difference. (r(x y) r(x) r(y) r(x y)). While supermodularity is preserved under maximization (Topkis [1998]), submodularity is not generally preserved under maximization. Maximizing a submodular function in general is NP hard.

17 Multi-modularity Lemma r(c) is submodular if: For all i = 1,..., m: p i (x) is anti-multimodular, For all i = 1,..., m: p i (x) is decreasing on x j for all j = 1,..., m, e(x) is multimodular. Multimodularity can be considered as a stronger assumption than submodularity. Multimodularity results in component-wise concavity. For Multimodularity definition and properties refer to Hajek [1985] and Li and Yu [2012].

18 Equilibrium Theorem In a competitive setting with sophisticated contracts, a set of contracts are Nash equilibrium if and only if they are in the following format: 0 for x = 0 Ti r i (c i + x) for 0 < x < s i (c) (x) = r i (c) for s i (c) x min j i s i (c j ) r i (c) for min j i s i (c j ) < x c i In addition, all Nash equilibria result in a coordinated chain with unique profit split as: Ti (s i (c T )) = r i (c) = r(c) r(c i ) r(c T ) = r(c i ) (m 1)r(c)

19 Franchise contracts Proposition A franchise contract is equilibrium if and only if it is in the following format: T i (x) = r(c) r(c i ) for 0 < x

20 Marginal-units discount contracts Proposition A marginal units discount contract is equilibrium if and only if it charges price p for units less than threshold l, and 0 for units over the threshold, where: r c i (c i ) p and l = r(c) r(c i ) p

21 All-units discount contracts Proposition An all units discount contract is equilibrium if and only if it charges price p if order is less than threshold l, and, r(c) r(c i ) s i (c i ) if order is more than or equal to threshold, where: r c i (c i ) p and l = min j i s i (c j )

22 Example m = 2 c = [5, 10] Fully substitutable products Market of size 10 Fixed price of 20 π(x) = 20(x 1 + x 2 ) (x 1 + x 2 ) 2.01x1 2 x 2

23 Example: Contracts Contracts ($) Supplier 2 Supplier 1 Capacity (units)

24 Comparative statics Proposition As a supplier s capacity increases, her profit increases, the competitor suppliers profits decrease, and the retailer s profit increases.

25 Example: Sensitivity to first supplier s capacity Profit($) Total Chain Supplier 2 Retailer Supplier 1 Capacity (units)

26 Example: Sensitivity to second supplier s capacity Profit($) Total Chain Retailer Supplier 1 Supplier 2 Capacity (units)

27 Capacities Proposition Under sophisticated contracts capacity buildings are coordinated, i.e., the equilibrium capacities are given by c i (c i ) = arg max c i {r(c i + c i )}

28 Wholesale Pricing Theorem In competitive setting with wholesale prices, there exists a Nash equilibrium, in which the chain is uncoordinated unless suppliers capacities are less than a threshold.

29 Example: Win-lose analysis Supplier 1 Supplier 2 Retailer

30 G. P. Cachon. Supply chain coordination with contracts. In S. Graves and T. de Kok, editors, Handbook in OR/MS: Supply Chain Management. North-Holland, G. P. Cachon and M. A. Lariviere. Supply chain coordination with revenue-sharing contracts: Strenghts and limitations. Management Science, 51(1):30 44, W. J. Elmaghraby. Supply contract competition and sourcing policies. Manufacturing & Service Operations Management, 2: , G. Gallego and M. Hu. Dynamic pricing of perishable assets under competition. Working Paper, IEOR Department, Columbia University, NY, B. Hajek. Extremal splittings of point processes. Mathematics of Operations Research, 10(4), H. Krishnan, R. Kapuscinski, and D. Butz. Coordinating contracts for decentralized supply chains with retailer promotional effort. Management Science, 50(1):48 63, Q. Li and P. Yu. Multimodularity and structural properties of stochastic dynamic programs. working paper, 2012.

31 J. J. Spengler. Vertical integration and anti-trust policy. The Journal of Political Economy, 58(4): , T. Taylor. Supply chain coordination under channel rebates with sales effort effects. Management Science, 48(8): , D. M. Topkis. Supermodularity and complementary. Princeton University Press, 1998.