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# Competition Under Price And Quantity Postponement

This paper studies competition under price and quantity postponement where product substituability is included. Supplier who operates as market leader and two retailers who act as follower produce two differentiated products according to the same product family. Price postponement is taken into consideration from a motivation to anticipate product variety while quantity postponement to anticipate demand changing. Game theory is applied to this competition in order to compare between price and quantity postponement effectiveness according to Bertrand and Cournot Stackelberg game. Furthermore product substitutability degrees are included in order to show its effect to postponement competitiveness and as guidance to product development. Demand stochasticity is overcomed by differential game application to both of Bertrand and Cournot Stackelberg game. The first and second simulation results show that Cournot game is superior to highly differentiated product while Bertrand game for product which shares common platform. Moreover the results show that product substitutability gives antipode effects (price, quantity and profit) to Cournot and Bertrand game. The final part of the paper concludes the results and outlines future research direction is discussed.

Keywords: price postponement, quantity postponement, Stackelberg game, Cournot game, product substitutability

## 1. INTRODUCTION

For many years, it has been a common policy for manufacturers to produce in a large batch to keep production cost and ordering cost low. Unfortunately, the current trend in consumer requirements does not support this idea. Consumers wish to be served according to their own â€śspecial needsâ€? and for this reason the variety of products is increasing. Obviously, it makes production lines become busier with frequent setup and down time due to higher product variety. Inline with this idea, a manufacturer needs to make a closer relationship with his opponents in order to keep their market share. Mass customization now has been to be an order qualifier to supply chain.

Kreps and Scheinkman (1983) show that if the firms choose capacities before engaging in Bertrand-like price competition then the Cournot outcome eventuates if the given capacities are at Cournot levels or it should be rationed when the capacity cannot meet market demand. On the contrary, Davidson and Doneckere (1986) argued this investigation and show that alternative rationing rule can eliminate idle capacity because players agree to compete at higher equilibrium capacity.

Either Kreps and Scheinkman or Davidson and Doneckere rationing rule however is difficult to be observed in the real world. One motivation is there are no manufacturers produce homogenous products because all firms are equally placed around a circle which represents a consumer taste spectrum (Salop, 1979). Thus, this paper proposes product substitutability degrees (Spence, 1976) to represent consumer taste spectrum. Lower product substitutability degree describes wider spectrum and vise versa.

Product substitutability inclusion into Cournot and Bertrand-like price competition gives a chance to explore the possibility of cooperation in long-term relationships, such as cooperation between firms in join product development. Lambertini and Cellini (2002) show that join product development.

The effectiveness of the different game strategies has been studied in the context of equal cost function-product firm without product substitution degree (Fujiwara, 2006). Two interesting research questions then arise: (1) how does product substitution degree affects the different game strategy? (2) How does different game strategy affect the optimal price and quantities? These questions are the focus of this paper.

Specifically, we consider a supply chain whiches consist of three parties they are one supplier and two buyer firms whose operate in a monopolistic setting. With regards to both postponement types, Cournot model uses linear quantity function while Bertrand applies linear price function. Supply chain needs to make two sets of decisions: production prices and quantities. Follower can postpone his production decisions to a later time when information on market leader is obtained. Furthermore at the final discussion we can observe both of Bertrand and Cournot Stackelberg games simultaneously by comparing their profit, price level and output (quantity).

The following sections first introduce postponement competition (section 2), where it focuses on features of competition application and game theory. Section 3 describes on postponement modeling with dynamic stackelberg, which is benchmarked against Cournot Stackelberg. Section 4 exhibits results and discussion from problem example. Finally Section 5 explores the opportunity for future research.

## 2. LITERATURE REVIEW

The objective of this section is to give common perception on supply chain strategy and specifically postponement competition application by game theory. In addition to Pine (1993) definition of mass customization, Davis (1987) drives readers to an argument that supply chain strategy for mass customization should be focused on the entity properties. For instance modularity is intended to components standardization while postponement is subjected to reduce lead times by move point of differentiation closer to delivery point (Lee, 1996). Shortly if entities have wide variety so that modularity should be done in order to reduce process variation. In other side, postponement is delivered while components have slime variety in order to reduce lead times. Furthermore Ernst and Kamrad (2004), Mikkola (2004) and Salvador et al (2004) propose a flexible strategy that is reflected by combination of modularization and postponement. In conclusion, product modularity should support postponement policy as well as reduce uncertainty (Swaminathan and Lee, 2003).

## 2.1 POSTPONEMENT COMPETITION

Zinn and Bowerzox (1988) differs postponements according to product and process redesign point of view (Lee, 1996). On the contrary Miegham and Dada (1999) proposed six postponement types according to three factors they are capacity, price and quantity. In their paper demand is assumed a function of price so that in that case price can control demand. Meanwhile Biller et al (2005) investigate price postponement effect to quantity and flexibility investment decision according to demand elasticity. The first paper considers to fixed quantity and the second emphasizes on flexible quantity. In addition to both papers, Gilbert and Cvsa (2003) add innovation effort to revenue maximization by postpone quantity or price decision. Similarly, another postponement model is dedicated to product substitutability effect to price and quantity postponement (Bish et al, 2007). This paper considers supply flexibility as a tool to overcome demand uncertainty and keep quantity flexibility. All of those suppose demand can be drawn according to certain functions which is emphasize on price/quantity competition and aside other factors such as decoupling point in push-pull strategy. Push pull strategy needs not only demand information but more emphasize on internal efficiency or costs minimization. In conclusion, two research forts envisage postponement from different perspectives these are operations management (Zinn and Bowerzox, 1988) and economic perspective (Miegham and Dada, 1999).

From all above literature review, this paper studies postponement competition according to both of economic perspective as final goal (price and quantity) and supply chain management perspective by develop a cooperation networks among one supplier and two buyers via game theoretic approach. Previously Miegham and Dada (1999) avoid game theory application because Nash equilibrium does not exist whenever demand is stochastic. On the contrary, game theory is applied in this paper because even demand is stochastic but in this case dynamic Stackelberg game solves this problem by using optimal controlled pricing policy or in other words the effect of demand uncertainty is also considered by this game.

## 3. MODEL DEVELOPMENT

In this section, we propose a study of economic impact of adopting postponement to price and capacity. The objective is to derive findings that will allow us to illustrate results for selecting decision sequence between price and capacity. In what follows we provide a simple analytical framework for evaluating different postponement strategies on revenue basis. The advantage of this approach, in addition to providing comparative results, is that it allows for incorporating decision sequence influencing profit. However, in this research, we only regarding to the development a general framework upon which future work can be based. To focus discussion, consider a supplier that sells a modular component to two retailers who operate based on customized product. From this reason, trade off between service level and efficiency is emphasized on how to decide product quantity and prices according to product standardization degrees. So that how many production quantity and how much its price is the main topic in this research. To gather general understanding for this concept, both postponement concepts will be discussed separately and then general concept will be developed. The following section discusses competition according to Bertrand and Cournot rule.

## 3.1 BERTRAND GAME COMPETITION DECSRIPTION

Consider two competition conditions. Firstly, downstream (retailers) compete according to static Bertrand and Cournot game and secondly downstream and upstream compete according to two stages Bertrand and Cournot game so we can use a duopoly model. It is also assumed that followers are downstream and supplier is the leader. Furthermore follower costs function is assumed equal because they order from the same supplier. Different with previous postponement models this paper intends to compare decision sequence between price and quantity so that the outcome of this paper is in what situation price or quantity postponement is enable.

## Game rules

This two stages Bertrand game adopts backward induction where follower is analyzed first and leader later. In stage 2 suppose followers determine their price as a function of market leader price so that he maximizing his own profits.

(1)

In the first stage, supplier (leader) determines his quantity and price. (2)

Equation (1) and (2) represent our game as backward induction so at first stage Nash equilibrium will be. The intuition behind this situation is simple, market leader of course like to be a monopolist. However, in Bertrand Stackelberg game total profit is calculated from equilibrium price so that quantity tend to be higher if both players agree to increase their product commonality and the consequence is players cooperation desire will increases.

At the first stage leader tries to determine his own price based on market situation (this game is a competition with perfect information) for instance market share, competitor capability to handle market demand. The intuition behind this situation is in backward induction outcome does not involve non-credible threats. Leader anticipates that follower react optimally to any action of leader might choose while leader has no any reason to threat because stage 2 is not leader self interests to threat follower. Obviously, this situation will be subgame perfect.

## Notations

Equilibrium quantity, market leader, follower and leader price

Compound and discount factor for dynamic stackelberg game, product commonality

Here it is the form of this game that represents

First game is rounded by Stackelberg game for quantity postponement

Second game is dedicated to price postponement

Profit comparison between quantity and price postponement.

## Bertrand game model

Consider a Bertrand duopoly model with price function for retailers given by (see Gibbons, 2002)

(3)

Whereandis time and form postponements prices from both of market leader and follower respectively and ideally they should be produce in equal amount.

This game put Bertrand dupoly model with the following reasoning. Naturally, customization process may want to choose different (and presumably higher) price because of variety. Supplier, on the other hand should take advantage from the above situation by serves inventory in order to reduce selling price and finally either inventory reduction or higher customization price are however risky for customer loose. One solution that is usually adopted is keep steady and optimum production output by considering market fluctuation and competition. The following model can be one of solutions for previous problems.

Stackelberg model is taken because in this case supply is the most dominant problem so supplier as a market leader more comfort with his position so he emphasizes his strategy to keep his position ahead of others by product availability in market while follower ussualy more innovative in order to fullfil customer requirement. It is also assumed that between leader and follower can observe their manufacturing performance each other so that this is a dynamic game with perfect information. This assumption is adopted because between supplier and retailer employ vertical integration so that both of them can access demand and supply data. In conclusion, this game applies commonality degree in order to attract cooperation between leader and followers.

## Bacward induction

Bacward induction is used to analyze equilibrium price and capacity in order to guarantee Nash equilibrium so that we move first from stage 2 as follow

## Stage 2 Follower decide his price according to leader price

(4)

The first order condition is

(5)

Similarly, the FOC from second product variant is

(6)

Solving these two equations simultaneously, one obtains

(7)

Stage 2 explores price equilibrium between two buyers. This equation shows effort to maximize standard platform utilization by increasing product substitutability value. Furthermore both buyer and supplier can take advantage from this problem because whenever supplier increases his selling price, buyer product price also increases. Price equilibrium from stage 2 then is used to decide supplier price as follow

## Stage 1 Leader decide his own profit function

At the first stage we can find as

(8)

Find c by insert (7) into (8) so we get

(9)

(10)

Stage 1 describes that product substitutability influences supplier price considerably. We can see that supplier price is a concave function of product substitutability (g). Shortly innovation between two buyers increases supplier price also. Finally price decision is used to decide capacity which is postponement until price is issued.

## Capacity postponement decision

Capacity postponement is assumed because Bertrand game players will not try to steal their opponent customer by lower price because their price will simply fall to zero. Moreover they must consider their own capacity and customer demand. Furthermore in economic theory demand is a function for a firmâ€™s product (or service) relates the quantities of a product that consumers would like to purchase and it quantities also might be a function of its price (Truett and Truett, 1984).

From this point on, this game is developed according to Fershtman and Kamien (1987) and Fujiwara (2006) but quantity is variable rather than price.

;; (11)

In equation (11) we recognize s as speed of quantity to go to its optimal value and it is eligible to both followers according to Bertrand duopoly quantity function. Because we assume that both followers have same cost function then both players must have equal quantity.

To solve (11), let us set up a current-value Hamiltonian as

(12)

S.t (11) ,,

Whereis costate variable that is associated with the quantity dynamics and at the following derivation we will recognize s and as compound factor and discount rate.

(13)

(14)

Steady state quantity can be found from (14) as

(15)

We can see that equilibrium quantity is a concave function of price. In conclusion quantity postponement gives significant impact to supplier-buyer supply chain whenever both buyers agree to improve their product commonality.

Below price postponement is given in order to do comparative study between quantity and price postponement.

## 3.2 COURNOT GAME COMPETITION DECSRIPTION

In this model we consider a Cournot duopoly model with price function for retailers given by

(16)

Whereis product variant 1 and 2 quantity respectively and ideally they should be produce in equal amount.

This game put Cournot dupoly model with the following reasoning. Naturally, price is not a competitiveness objective but market share. For this game, both buyers effort to produce common product is undermined. One solution that is usually adopted is product differentiation in order to get larger market so that cooperation between two product variants at certain commonality degree is recomended. The following model can be one of solutions for this problem.

Stackelberg model is taken because mostly market leader in this game is modular component supplier and they tend to produce standard product platform in a large batch. It is assumed that between leader and follower can observe their manufacturing performance each other so that this is a dynamic game with perfect information.

## Game rules

This two stages Cournot game adopts backward induction where follower is analyzed first and leader later. In stage 2 suppose followers determine their price as a function of market leader price so that he maximizing his own profits.

(17)

In the first stage, supplier (leader) determines his quantity and price. (18)

Equation (1) and (2) represent our game as backward induction so at first stage Nash equilibrium will be. The intuition behind this situation is simple, market leader of course like to be a monopolist. However, in Cournot Stackelberg game total profit is calculated from equilibrium quantity so that price tends to be higher if both players compete according their own unique feature and the consequence is player cooperation desire decreases.

## Game 2: Price postponement

This game will decide equilibrium capacity first before price and it will be run under backward induction as follow

## Stage 2 Follower decide his capacity according to leader capacity

(19)

By assuming equal costs function and both players try to cooperate by produce products with a certain commonality degree so (19) can be modified according to Cournot duopoly inversion (Spence, 1976) as follow

(20)

Equation (20) describes that total revenue consists of total profit for two followers minus their total costs then the first order condition for (20) is

(21)

Similarly the FOC for second product variant is

(22)

We can solve (21) and (22) simultaneously to be

(23)

## Stage 1 Leader decide his own profit function

At the first stage we can use (23) in order to calculate leader profit as follow

(24)

FOC of (24) according to c then we have

(25)

Then (25) can be rewritten as

(26)

## Price postponement decision

Price postponement is taken by an assumption that Cournot game players trust to their opponent that they will keep their capacity at constant value (Trett and Truett, 1984). There are some ctirics for this method such as it is hard to believe Cournot assumption and in duopoly players should get together their price and quantities. The result of this game usually higher price with lower equilibrium output and this paper investigates their opportunity to be improved by an assumption that at infinite time both players will achieve a steady state price. From this point on, this game is developed according to Fershtman and Kamien (1987) and Fujiwara (2006) as follow

;; (27)

In equation (27) s is a speed value of price to go to its optimal value. Equation (27) also eligible to both suppliers (leader and follower) because Cournot duopoly price definition is a function of its constituents quantities they are leader and follower. In other words, both players adopt equal price policy.

Finally price dynamic can be derived by insert (26) into (27) as follow

(28)

To solve (28), let us set up a current-value Hamiltonian as

(29)

S.t (28),

Whereis costate variable that is associated with the price dynamics and at the following derivation we will recognize s and as compound factor and discount rate.

(30)

(31)

Solve (30) and (31) simultaneously then equilibrium price is finally founded as

(32)

## Result and Discussion

Previous section discusses about how to optimize quantity and price according to two different postponement strategies. It is quiet different with previous approaches by assuming identic costs function (Fujiwara, 2006; Fershtman et al, 1987) according to price postponement without product substitutitability degrees while in this paper we differ both players according to quantity and price postponement. Moreover both postponement strategies are also treated with different solutions, Cournot and Bertrand game. Firstly, we intend to compare between common Stackelberg and dynamic stackelberg game to see their unique feature.

(Please attach table 1 approximately here)

(Please attach figure 1 approximately here)

(Please attach figure 2 approximately here)

Those figures and table will be used to answer our previous research questions as follow

Question 1: how does product substitution degree affects the different game strategy?

Analysis

Figure 1 and 2 exhibits comparison between Cournot stackelberg versus Bertrand stackelberg from quantity and price at various substitutability degrees. Generally, it is shown that product substitutability degree gives reverse effect to Bertrand and Cournot game. Moreover Cournot game is better in highly differentiated products than Bertrand game and on the contrary this game ineffective at highly compatible product. Furthermore Cournot game gives higher profit to retailers while lower to supplier. In conclusion, Cournot game is better to be applied whenever supplier is not a part of supply chain or common platform does not exist.

Conclusion

Product substitution degree gives different significant impact to both of Cournot and Bertrand game. Figure 1 and 2 depict lower substitutability gives advantage to Cournot and on the contrary higher substitutability is preferred by Bertrand game. Cournot game can be explained as if customer has more options for product variant then they will deal with a product based on product configuration. Price postponement had better to be applied at highly differentiated products because they split market sharply according to each product unique features. Different case if product variants does not exist then consumerâ€™s decision will depend on product availability so that producer bargaining position will decrease and finally producer should postpone his production quantity until he receives exact demand information because market is differentiated just according to product availability.

Question 2: How does different game strategy affect the optimal price and quantities?

Analysis

Question 2 is figured out through figure 1 and 2 which show how different game strategies influence price and quantity decisions. Bertrand game gives more opportunity to leader and follower to gain more profit through higher product and material price at higher product substitutability degrees while Cournot game gives the reverse effect as it depicted by figure 1 and 2. Furthermore figure 2 exhibits Cournot game affects more significantly to price and quantities than Bertrand game. In short, both games should be applied according to product design in order to get optimal price and quantities.

Conclusion

Cournot and Bertrand game give reverse effect to aggregate profit. Figure 1 and 2 depicts how we should apply both games according to our product design in order to achieve optimal profit. Bertrand game gives more advantage to the entire supply chain because among two retailers and supplier share more equal profit than Cournot game. On the contrary Cournot game gives significant profit to retailers and less to supplier. This situation can be explained as Cournot game traditionally is a quantity competition so retailers will order as much as possible in order to dominate market share and as a consequence this situation will reduce product price in market. On the contrary Bertrand game is a price competition so retailers will decide their order according to price equilibrium even they cannot flood market with their products. Furthermore product substitutability also gives additional consideration to retailer decision. In conclusion, Cournot game is better to be applied to unique product because it can maintain its price higher while Bertrand game is intended to common product for price stability.

## 5. Further research and conclusion

This paper proposes an alternative method to apply Bertrand and Cournot game to quantity and price postponement according to profit maximization. Both postponement types are explored in order to investigate their compatibility to product differentiation and standardization. This paper proposes dynamic Bertrand and Cournot Stackelberg games with regards to product substitutability degree.

It is shown that the generic model derived is consistent with research questions and important from academic perspective as it utilizes a generic model of multistage price and quantity postponement competition. Particularly, this paper offers a comprehensive solution of both types of postponement according to Dynamic Stackelberg game. Even this paper just proposes a theoretical modeling it is also possible to apply at real situation because this paper accommodates common demand function as it is widely used in economic theory. Furthermore there is a great chance to improve it by moving from duopoly to opligopoly competition where there are more than two monopolists.

For management implication point of view, this paper gives an insight about coordination mechanism for both of market leader and follower. In real business situation there is no pure competition but some degree of cooperation that is so called coopetition. Example for Bertrand game is join production between Cakra Kembar and Kereta Kencana wheat flour for some markets. Both products are manufactured by Bogasari Flour Mills Surabaya Indonesia which is the biggest wheat flour mills in the world. Both products share common wheat grain contents. Kereta Kencana and Cakra Kembar share their market in order to keep their price so in this case they are managed as Bertrand game. On the contrary, Semar which is produced by Bogasari Surabaya fights with Pena Emas which is produced by Sriboga Ratu Raya Semarang according to Cournot game. Both products have much different product features and produce as much as possible wheat flour to market even they must suffere from price reduction.

For future research direction, oligopoly model is considered to be developed according to future market demand that is determined by how close customer requirements is meet so that in future oligopoly model quantity and price can be replaced with some parameter such as inventory and lead times. From this result, a sequence between lead times and inventory can be determined and the outcome will be a decision which one more important for a company, agility or efficiency so that the outcome can be used by top management to compose their business strategy. Finally, the future research should accommodate strategic and tactical level alligment in order to develop comprehensive decision analysis.