A sequential game in quality and price competition

Opleiding: Bsc Econometrie

Universiteit: Universiteit van Amsterdam Begeleider: Tomasz Makarewicz

Bachelorscriptie Econometrie

A Sequential Game

in Quality and Price

Competition

Robbert Huisman, 10249001 24-12-2014

Hierbij verklaar ik, Robbert Huisman, dat ik deze scriptie zelf geschreven heb en dat ik de volledige verantwoordelijkheid op me neem voor de inhoud ervan.

Ik bevestig dat de tekst en het werk dat in deze scriptie gepresenteerd wordt origineel is en dat ik geen gebruik heb gemaakt van andere bronnen dan die welke in de tekst en in de referenties worden genoemd.

De Faculteit Economie en Bedrijfskunde is alleen verantwoordelijk voor de begeleiding tot het inleveren van de scriptie, niet voor de inhoud.

Inhoud

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1. Introduction

In 2007, when the first Iphone was released, Apple had a monopoly in the smartphone market, in which they were selling a high quality product. The quality of a product can be qualified as the amount of features a smartphone has. A good camera, a fingerprint scanner or being waterproof are examples of the features a smartphone can have. After Apple, other companies entered the market of which some started selling high quality smartphones like Apple, while others also started selling low quality smartphones. It appeared that within the market for smartphones, there is also room for companies that produce low quality products. Huawei is one of those companies, which started their sale of smartphones when Apple was selling the Iphone 4. The question that arises here is: why did Huawei decide to enter the market with a low quality product?

In this study, I will show when a company decides to enter a market, given that there is already another company in the market. In the example this is when a company enters the smartphone market when after Apple released the Iphone. The way in which I will research this is by developing a model for a game with two rounds. In these rounds, the companies compete on both quality and price.

In other studies in the field of price and quality competition, researchers have been using different types of models and different assumptions. The model that will be used here is a game theoretic model to investigate price and quality competition. The main question will be: under what circumstances will company B enter the market. A duopoly will be used to

answer this question. A similar approach is used by Tyagi (2000). He also looks at a sequential game in which companies compete for quality. In contrast to this study, he is interested in product placement. So his paper is interested in where a company decides to position itself, whereas I want to find out if a company will actually enter the market. One of the most used assumptions in papers involved with price and quality competition is that a high price is equal to a high quality. This is done because in real life people often associate one with the other. Chioveanu (2012) shows this relation and Banker et al. (1998) also uses this assumption. Higher quality equals higher price is not an assumption that I will make. This is because the cost function is variable, so it can be that one of the two

companies has a higher quality, but with lower costs of making this quality. When this company has lower costs it can ask a lower price.

Besides the assumption of higher price equals higher quality there are also important assumption which involve the consumers. In this study I will assume that there are no inattentive consumers. This assumption will be made for two reasons. Firstly to not let the

2 “cheating” of companies happen, as researched by Armstrong & Chen (2009). Secondly in order to not let consumers buy a product at random, as in the paper of Gans (2002). When the assumption isn notmade, these two reasons can influence the outcome of this study as shown in other studies. So when the consumers will buy a product, all of the consumers will know the exact level of the quality of the products in the market.

Armstrong & Chen (2009) studied the effects of inattentive consumers on price and quality competition. Armstrong & Chen (2009) assumed that some consumers only look at the price of a product and not at the quality. They base the quality of a product solely on the price. If the companies know this, they will use this in their advantage. As Armstrong & Chen (2009) describe it: “a firm may “cheat” and offer a worthless product to exploit these inattentive consumers.” So they will sell a worthless product for a high price. Wolinsky (1983) also wrote a paper on inattentive consumers and he reached the same conclusion as Armstrong & Chen (2009). Both studies also noticed that when the number of inattentive consumers dropped, for instance because of a market transparency policy, the firms will “cheat” less. In another study, Gans (2002) also uses the inattentiveness of consumers. His consumers are not well informed about the quality of the supplier where they will buy the product. His consumers only know after buying the product how high the quality is. The consumers buy the product repeatedly and remember how the quality was at a certain supplier. Every time they have to buy the product again, they will make a decision based on past experiences. In this way, new changes made by a company are not taken into consideration by the

consumer.

Another assumption regarding the consumers is whether they have homogeneous or heterogeneous preferences for quality. The conclusion that I have mentioned from

Chioveanu (2012) is based on consumers having heterogeneous quality preferences. This is because many of the papers within the field of price and quality competition continue on the ideas of Hotelling (1929). He used two symmetric firms and heterogeneous tastes to study price and quality competition. Wolinsky (1983) is another example of someone who uses heterogeneous preferences in his paper. Others, such as Matsubayashi (2007) and Banker et al. (1998) differ from these papers by using homogeneous preferences instead of

heterogeneous preferences. Banker et al. (1998) model consumer demand as: “a linear function of the price and quality levels selected by the two firms.” Therefore, all the

consumers want quality in the same way and their liking of quality is based on the price and quality of the products. This study will follow the model used by Banker et al. (1998) and state that consumers have identical preferences for quality, but using utility instead of consumer demand. Since all consumers have the same taste, they will all want the same

3 product. This means that if one consumer buys A, all consumers will. This leads to a

simplification in which there is only one consumer on the market.

All these assumptions will be used in the model which consists of two rounds. In the first round, company A is offering his product with a certain quality and price. Company B does not produce anything in this round. In round 2, company A still offers the same product with the same quality and price. However, in this round company B enters the market. They do this with a product which can differ in quality and price from the product of company A. The aim of this paper is to find out when it is interesting for company B to enter the market and in which cases company A takes the entire market. This will be done given that the consumer is already used to the product of company A. Therefore if the consumer chooses to stay with company A and buy their product again, he will receive extra utility.

The remainder of this paper is organized as follows. Section 2 will present the model of the sequential game, whereas Section 3 shows the results. There it will be shown that there will be two subgame perfect Nash equilibria. In one of them company B will enter the market, in the other one company A takes the entire market in the second round. The conclusion will be given in Section 4.

2. Model

In this chapter I will begin by giving the basic model in 2.1. Modelling the consumer’s

behavior will be done in 2.2. I will show which utility functions are used and what restrictions are made for the parameters of these functions. When the model of the consumer is done, the model for the companies will be shown in 2.3. Just as in the model for the consumers, there are restrictions for the parameters that will be given. These two models will then be used to calculate the possible subgame perfect Nash equilibria. The important calculations and their description will be present in 2.4. The chapter will be finished by giving the conditions that have to be met for the equilibria to hold true in 2.5.

2.1 Basic model

First consider a duopoly market, where company A and company B are competing. These two companies will play a sequential game in which they will compete on both quality and price. There are 2 rounds in the sequential game. In the first round, only company A is a player on the market. They sell their product which has a certain quality KA for a price PA.

4 Figure 1: Sequential game with two rounds.

In the second round, company A will sell the same product for the same price, so again their product will have quality KA and price PA. In this round, company B can also enter the

market. They will sell their product with quality KB and corresponding price PB. This game is

shown in Figure 1.

2.2 Consumer

As stated in the introduction, I will assume that the consumers have homogeneous

preferences for quality, which led to the simplification that there is only one consumer. This consumer will get a specified utility U. If the consumers utility is negative, he does not buy a product. So I will only look at the case where the consumer’s utility is zero or positive. The utility that the consumer gets depends on the level of quality of the product that he buys and the price he has to pay for it. As mentioned, in the first round only company A is in the market and therefore the utility gained by the consumer is:

𝑈𝐴1 = 𝛼𝐾𝐴− 𝛽𝑃𝐴, U1 = max(UA1,0) = UA1 ,

With 𝛼, 𝛽 > 0. The first formula shows the utility gained by the consumer if he buys the product of company A. α shows how much the consumer likes quality, whereas β indicates the disliking of the price of a product. Since company A is the only player on the market in this round, it follows that the consumer will buy product A in the first round.

In the second round, company B can enter the market. This means that both company A and company B can be able to sell their product. In that case, the consumer can choose which one he wants to buy. For the product of company B his utility will only depend on the quality

5 KB and the price PB. However, for the product of company A the consumer will receive extra

utility because of the attachment he has for product A. This means that the utility in round 2 will be as follows:

𝑈𝐴2 = 𝛼𝐾𝐴− 𝛽𝑃𝐴+ 𝛾𝐾𝐴 = (𝛼 + 𝛾)𝐾𝐴− 𝛽𝑃𝐴,

𝑈𝐵2 = 𝛼𝐾𝐵− 𝛽𝑃𝐵, U2 = max(UA2, UB2),

With 𝛾 > 0, being the attachment that the consumer has for buying product A again. 𝛾 is bigger than zero, because if something has a high quality, it is worth more to the consumer and he will be less willing to give it up and change to another product. At the same time, when a product has low quality it is easier to change. In this round company B can enter the market and the consumer will get utility UB2 for buying this product. Which product the

consumer buys depends on the utility it gets from the companies. The consumer will choose the product of the company that will give him the highest utility.

2.3 Companies

Now that it is known how the consumer will react to the products of the companies, the model for the companies can be made. First I will show what the profit of both companies are. Since company A is the only player on the market in the first round, it will definitely make a profit in that round. For the second round it is not known who takes the market. So the profits that company A and B will get are:

𝜋𝐴= 𝜋𝐴1+ 𝜋𝐴2, 𝜋𝐴1= 𝑃𝐴− 𝐶𝐴, 𝜋𝐴2= 𝐼 ∗ 𝑃𝐴− 𝐶𝐴, 𝜋𝐵 = (1 − 𝐼) ∗ 𝑃𝐵− 𝐶𝐵, 𝐼 = {1 𝑖𝑓 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐴 𝑡𝑎𝑘𝑒𝑠 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 2 0 𝑖𝑓 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐵 𝑡𝑎𝑘𝑒𝑠 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 2,

Since there is only one consumer, the quantity that the companies can sell is just one. In this model there will be no fixed costs. The only costs that the companies make depends on the quality of the product they want to sell. The cost functions will be given by a quadratic function of the quality:

6 𝐶𝐵 = 𝛿𝐵𝐾𝐵2.

Where 𝛿𝐴, 𝛿𝐵 > 0. Here 𝛿𝐴 and 𝛿𝐵 are the cost coefficients. If we put this into the profit

functions we get: 𝜋𝐴= 𝑃𝐴− 𝛿𝐴𝐾𝐴2,

𝜋𝐵 = 𝑃𝐵− 𝛿𝐵𝐾𝐵2.

2.4 Equilibria

In order to calculate with the model I will use backwards induction. There will be two cases which I will calculate. First I will calculate the equilibrium when company A takes the market in both rounds. This means that company B does not enter the market in round 2. This happens when the consumer wants to buy the product of company A in the second round instead of the product of company B, so the consumers utility for buying product A is higher than his utility for buying product B. After this I will calculate the equilibrium when company B does enter the market. In that case company A takes the monopoly price in round 1. Here the utility that the consumer gets for buying product B is higher.

The first thing I calculate is the indifference function of company B. This is done by solving for 𝑃𝐵 in 𝑈𝐴= 𝑈𝐵 and gives us Lemma 1: the price of company B is equal to the price of

company A plus the quality of company B multiplied by a constant minus a constant times the quality of company A. This can be interpreted by saying that the price of company B is equal to the price of company A. However, the price of B becomes higher if the quality of company B rises and it becomes lower if the quality of company A rises. The formula with the constants is:

Lemma 1: 𝑷𝑩= 𝑷𝑨+ 𝜶 𝜷𝑲𝑩−

(𝜶+𝜸) 𝜷 𝑲𝑨.

Now that the indifference function is known, I use this function to calculate what the optimal quality is for company B. Since the derivative of the profit function of company B does not rely on the price or quality of company A, the optimal quality of company B is constant in the model as shown in Lemma 2: the optimal quality of company B is equal to the liking of the quality divided by 2 times the production costs times the disliking of the price. This method is repeated for company A. Just as for company B, the optimal quality of company A does not depend on the price or quality of the competitor. It follows in Lemma 3 that the optimal quality of company A is really similar to the optimal quality of company B. The difference is that for company A, the attachment to the product is added to the liking of the quality. Also

7 for A the costs for producing A are used instead of the costs for producing product B. These 2 lemma’s are shown here:

Lemma 2: 𝑲𝑩= 𝜶 𝟐𝜹_𝑩𝜷, Lemma 3: 𝑲𝑨 = (𝜶+𝜸) 𝟐𝜹𝑨𝜷.

When 𝑈𝐴 ≥ 𝑈𝐵, company A can take the entire market. For this to happen they need to make

sure that company B does not have an incentive to enter the market. Consequently, the profit of company B has to be equal to zero. Filling in Lemma 1 and Lemma 2 into the profit function of company B and setting it equal to zero, results in the price of company A and the price of company B. Using the price of company A, I also calculated the profit of company A. Since company A sells its product in two rounds, we have to multiply the profit it gets in the second round by two. These prices and profits are given in proposition 1:

Proposition 1: 𝑷𝑨 = (𝜶+𝜸)𝟐 𝟐𝜹𝑨𝜷𝟐 − 𝜶𝟐 𝟒𝜹𝑩𝜷𝟐, 𝝅𝑨= (𝜶+𝜸)𝟐 𝟐𝜹𝑨𝜷𝟐 − 𝜶𝟐 𝟐𝜹𝑩𝜷𝟐, 𝑷𝑩= 𝜶𝟐 𝟒𝜹_𝑩𝜷𝟐, 𝝅𝑩= 𝟎.

After this I look at what happens when 𝑈𝐴≤ 𝑈𝐵. Lemmas 1 and 2 are still valid in this case.

However, the quality of company A is not correct in Lemma 2 if company A takes the monopoly price. When company A takes the monopoly price, they will make sure that they take the highest price as possible, which is equal to the consumer having zero utility. This leads to the following lemma:

Lemma 4: 𝑲_𝑨𝒎 = 𝜶

𝟐𝜹𝑨𝜷.

In this case I need to make sure that company A takes the monopoly price and does not want to change to price as calculated below. In order to do this, I first calculate the monopoly price for company A, which is:

𝑃𝐴𝑚1= 𝛼2 2𝛿𝐴𝛽2.

Setting the profit for company A only having the monopoly price in the first round equal to the profit by having a market price in both round gives us the price for company A. With this price company B has to compete in the second round. Using Lemma 1 gives us the price for company B:

8 𝑃𝐵 = 𝛼2 8𝛿𝐴𝛽2+ 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 4𝛿𝐴𝛽2,

The profits that both companies have together with their prices can be found in proposition 2: Proposition 2: 𝑷𝑩= 𝜶𝟐 𝟖𝜹_𝑨𝜷𝟐+ 𝜶𝟐 𝟐𝜹_𝑩𝜷𝟐− (𝜶+𝜸)𝟐 𝟒𝜹_𝑨𝜷𝟐, 𝝅𝑩= 𝜶𝟐 𝟒𝜹_𝑩𝜷𝟐− 𝟏 𝟒𝜹_𝑨𝜷𝟐((𝜶 + 𝜸) 𝟐₋𝜶𝟐 𝟐), 𝑷_𝑨𝒎𝟏 = 𝜶𝟐 𝟐𝜹𝑨𝜷𝟐, 𝝅𝑨 𝒎𝟏_{= 𝝅} 𝑨 𝟐 ₌ 𝜶𝟐 𝟒𝜹𝑨𝜷𝟐,

2.5 Conditions

In the previous paragraph I’ve shown which equilibria arise in the model. However, these equilibria cannot both happen. So there are some conditions that have to be met before the equilibria can occur. I will show which conditions have to be met that apply to the cost ratio of the two firms. The cost ratio is equal to the cost of producing one product of company A divided by the cost of producing one product of company B.

When 𝑈𝐴 = 𝑈𝐵, I calculated the profit for company A. In order for the first equilibrium to

happen, company A needs to have a positive profit, otherwise they do not enter the market. This leads to Theorem 1, which shows that the cost ratio of the two companies has to be small in order for company A to take the market in the second round. The exact theorem is shown next: Theorem 1: 𝜹𝑨 𝜹_𝑩≤ (𝜶+𝜸)𝟐 𝜶𝟐 = 𝟏 + 𝟐𝜸 𝜶 + 𝜸𝟐 𝜶𝟐,

If there would be no attachment for company A, this theorem shows that the cost from company A has to be smaller than the costs for company B if company A wants to take the market. Since the consumer does have some attachment for the product of company A, the cost of company A can become higher than the cost of company B as long as it does not exceed the theorem.

For the second equilibrium there is a similar condition. Parallel to the first equilibrium, the company needs to have a positive profit, in this case company B. As company B takes the market in this equilibrium, the cost of company A needs to be high in order for B to capture the market. This is opposite to Theorem 1. Therefore the cost ratio has to be higher than instead of smaller than the formula in this equilibrium, which is equal to:

Theorem 2: 𝜹𝑨 𝜹𝑩≥ ((𝜶+𝜸)𝟐−𝜶𝟐 𝟐) 𝜶𝟐 = 𝟏 𝟐+ 𝟐𝜸 𝜶 + 𝟑𝜸𝟐 𝟐𝜶𝟐,

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3. Results

In this chapter, I will show the results of my calculations. First I will show the results for the companies. These results include graphical projections of the equilibria, in order to show what the equilibria and the conditions look like. After that I will show which equilibria is preferred by company A. Following the preference of company A will be the results for the consumers in which I will explain which equilibrium the consumer prefers.

3.1 Companies

In the previous chapter I calculated the qualities, prices and profits of both companies. These calculations showed that there are two possible subgame perfect Nash equilibria. In the first round company A has to choose which price they will take, the monopoly price or the price to take the market in the second round. After this, he will take the same price in the second round. There company B has the choice. His choice is whether he will enter the market or not. If company A takes the monopoly price and company B enters the market, there is a subgame perfect Nash equilibrium. Also when company A chooses to take the price to take the market in both rounds and B does not enter in the second round there is a subgame perfect Nash equilibrium. This game is shown in figure 2:

Figure 2: The subgame perfect Nash equilibria.

In the previous chapter I showed under which conditions the subgame perfect Nash equilibria can exist. In order to give an impression of when which subgame perfect Nash

10 equilibrium can occur, I made the next graph. In this graph I have taken α=10 and plotted theorem 1 and 2.

Graph 1: Theorem 1, Theorem 2 and their overlap.

The blue part (the top) in the graph is where Theorem 2 is valid, and the grey part (the bottom) is for Theorem 1. The orange part (the middle) is where both theorems overlap. So in that area both equilibria can occur. This result is not only possible for these values of α and γ. There are many more values in which it occurs that there is an overlap in which both equilibria can take place.

3.2 Preference of company A

Now that I’ve shown which equilibria are possible, I will show which of the two equilibria is actually preferred by company A. Company A wants to maximize its profit. So it will want to take the equilibrium that gives the highest profit. Company A will want to choose the first equilibrium, that is taking the market in both rounds, if the criterion in Theorem 3 is met. For this to happen, the cost ratio has to be smaller than the formula shown below in Theorem 3. It means that if the costs of producing product A are small compared to the attachment of the consumer for product A, company A will want to take the market in both rounds.

11 Theorem 3: 𝜹𝑨 𝜹𝑩< 𝜶𝜸+𝜸𝟐 𝜶𝟐 = 𝜸 𝜶+ 𝜸𝟐 𝜶𝟐,

If the formula is the opposite, so when the cost ratio is bigger than 𝛾

𝛼+ 𝛾2

𝛼2, company A will prefer to take use the monopoly price and company A will only be on the market in the first round. It will not matter for company A which equilibrium is chosen if the cost ratio would be equal to 𝛾

𝛼+ 𝛾2 𝛼2.

3.3

Consumer preference

In the last paragraph the preference of company A has been shown. In this paragraph I will focus on the preferred equilibrium of the consumer. The consumer wants to have as much utility as possible. When company A takes the market in both rounds, the consumer will receive utility in both rounds. If company A first takes the monopoly price and afterwards company B takes the market, the consumer will only receive utility in the second round. From that I derive the following utilities:

𝑈𝐸1= 𝛼2 2𝛿𝐵𝛽− 𝛼(𝛼+𝛾) 2𝛿𝐴𝛽 , 𝑈𝐸2= 𝛼2 4𝛿𝐵𝛽+ 1 4𝛿𝐴𝛽((𝛼 + 𝛾) 2₋𝛼2 2),

So the consumer will prefer the first equilibrium when 𝑈𝐸1> 𝑈𝐸2. This happens when the

cost ratio is high. How high it has to be is shown in Theorem 4:

Theorem 4: 𝜹𝑨 𝜹_𝑩> 𝟑 𝟐𝜶 𝟐_{+𝟑𝜸𝜶+𝜸}𝟐 𝜶𝟐 = 𝟑 𝟐+ 𝟑𝜸 𝜶 + 𝜸𝟐 𝜶𝟐,

This means that the consumer will prefer company A taking the market in both rounds if the costs for producing of company A are high. If the costs for company A are high, the quality of company A will become lower. When their quality becomes lower, their price does too, and therefore the consumer will prefer company A to take the market in both rounds. If we take the opposite of Theorem 4, so if the cost ratio is lower instead of higher than the given formula, the consumer will want company B to take the market in the second round. When the cost ratio is exactly equal to 3

2+ 3𝛾

𝛼 + 𝛾2

𝛼2, the consumer is indifferent between the two equilibria and he will not mind who takes the market.

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4. Conclusion

In this paper I looked at a sequential game consisting of two round, in which two companies compete on both price and quality. The aim was to find out when company B would decide to enter the market and in which cases company A took the entire market in both rounds. As it turns out, there are two subgame perfect Nash equilibria. The first one is when company A chooses to take the entire market in both rounds and company B does not enter the market. The other one is when company A takes its monopoly price in the first round and company B then enters the market in the second round. These equilibria only occur under some

circumstances.

Another conclusion that we can make is that company A does have a preference for a certain equilibria. This preference depends on some factors. Not only company A shows that under some circumstances it has a preference. Also the consumer likes one equilibrium more than the other, depending on how much he likes the quality, how much attachment he has for the product and how much he dislikes the price.

These results are similar to the results from Tyagi (2000). Tyagi (2000) researched a sequential game in which two companies would have to choose where they place their business. His results showed that if the company that enters the market has lower costs, he will take the best place. This leads to the first company choosing a less good place in order to optimize his profit in the two rounds. If the company that chooses first has lower costs, he chooses the optimal place, similar to company A taking the price to take market in both rounds.

It may be interesting in further research to investigate how the equilibria become when there are more than 2 companies that compete. This can be done in two ways. Either n companies can enter the market in round two, or there can be n rounds in which one company can enter. Another point which could be researched more is how the equilibria would be if the consumers do not have homogenous preferences and to got even further, if there are inattentive consumers.

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References

M. Armstrong, Y. Chen, (2009) Inattentive consumers and product quality. Journal of European Economic Association 7(2–3):411–422

R.D. Banker, I. Khosla, K.K. Sinha, (1998) Quality and Competition. Management Science 44(9):1179-1192.

I. Chioveanu, (2012) Price and quality competition. Journal of Economics 107: 23-44 N. Gans, (2002) Customer Loyalty and Supplier Quality Competition. Management Science 48(2):207-221

H. Hotelling, (1929) Stability in competition. Economic Journal 39: 41–57.

N. Matsubayashi, (2007) Price and quality competition: The effect of differentiation and vertical integration. European Journal of Operational Research 180: 907–921

R.K. Tyagi, (2000) Sequential Product Positioning Under Differential Costs. Management Science 46(7):928-940

X.H. Wang, B.Z. Yang, (2001) Mixed-strategy equilibria in a quality differentiation model. International Journal of Industrial Organization 12:213–226

A. Wolinsky, (1983) Prices as signals of product quality. Review of Economic Studies 50(4):647–658

Appendix I

Derivations of the Lemmas and Theorems: 𝑈𝐴1 = 𝛼𝐾𝐴− 𝛽𝑃𝐴

𝑈𝐴2 = 𝛼𝐾𝐴− 𝛽𝑃𝐴+ 𝛾𝐾𝐴 = (𝛼 + 𝛾)𝐾𝐴− 𝛽𝑃𝐴

𝑈𝐵 = 𝛼𝐾𝐵− 𝛽𝑃𝐵

With restrictions 𝛼, 𝛽, 𝛾, 𝑈𝐴1, 𝑈𝐴2, 𝑈𝐵 > 0 ,

The cost functions for both companies are: 𝐶𝐴= 𝛿𝐴𝐾𝐴2 & 𝐶𝐵= 𝛿𝐵𝐾𝐵2

With restrictions 𝛿𝐴, 𝛿𝐵 > 0

Since the quantity is equal to 1, the profit functions will become: 𝜋𝐴= 𝜋𝐴1+ 𝜋𝐴2 𝜋𝐴1= 𝑃𝐴− 𝐶𝐴 𝜋𝐴2= 𝐼 ∗ 𝑃𝐴− 𝐶𝐴 𝜋𝐵 = (1 − 𝐼) ∗ 𝑃𝐵− 𝐶𝐵 𝐼 = {1 𝑖𝑓 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐴 𝑡𝑎𝑘𝑒𝑠 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 2 0 𝑖𝑓 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐵 𝑡𝑎𝑘𝑒𝑠 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑛 𝑟𝑜𝑢𝑛𝑑 2

If 𝑈𝐴 = 𝑈𝐵, the consumer will be indifferent between buying product A or B. In this case

either one of the companies will take the market (companies assume they take the market). Lemma 1: 𝑷𝑩= 𝑷𝑨+ 𝜶 𝜷𝑲𝑩− (𝜶+𝜸) 𝜷 𝑲𝑨 Proof: 𝑈𝐴 = 𝑈𝐵 → (𝛼 + 𝛾)𝐾𝐴− 𝛽𝑃𝐴 = 𝛼𝐾𝐵− 𝛽𝑃𝐵 → 𝛽𝑃𝐵 = −(𝛼 + 𝛾)𝐾𝐴+ 𝛽𝑃𝐴+ 𝛼𝐾𝐵 → 𝑷𝑩= 𝑷𝑨+ 𝜶 𝜷𝑲𝑩− (𝜶+𝜸) 𝜷 𝑲𝑨 Lemma 2: 𝑲𝑩= 𝜶 𝟐𝜹_𝑩𝜷 Proof: 𝜋𝐵 = 𝑃𝐵− 𝛿𝐵𝐾𝐵2= 𝑃𝐴+ 𝛼 𝛽𝐾𝐵− (𝛼+𝛾) 𝛽 𝐾𝐴− 𝛿𝐵𝐾𝐵 2

𝜕(𝑃𝐴+𝛼_𝛽𝐾𝐵−(𝛼+𝛾)_𝛽 𝐾𝐴−𝛿𝐵𝐾𝐵2) 𝜕𝐾𝐵 = 𝛼 𝛽− 2𝛿𝐵𝐾𝐵 = 0 → 𝐊𝐁= 𝛂 𝟐𝛅𝐁𝛃

We can use the same method for finding the optimal quality for company A. Lemma 3: 𝑲𝑨=(𝜶+𝜸)_𝟐𝜹 𝑨𝜷 Proof: 𝑈𝐴2 = 𝑈𝐵 → (𝛼 + 𝛾)𝐾𝐴− 𝛽𝑃𝐴= 𝛼𝐾𝐵− 𝛽𝑃𝐵→ 𝛽𝑃𝐴= (𝛼 + 𝛾)𝐾𝐴− 𝛼𝐾𝐵+ 𝛽𝑃𝐵 → 𝑃𝐴= (𝛼+𝛾) 𝛽 𝐾𝐴− 𝛼 𝛽𝐾𝐵+ 𝑃𝐵 𝜋𝐴= 𝑃𝐴− 𝛿𝐴𝐾𝐴2=(𝛼+𝛾)_𝛽 𝐾𝐴− 𝛼 𝛽𝐾𝐵+ 𝑃𝐵− 𝛿𝐴𝐾𝐴 2 𝜕((𝛼+𝛾) 𝛽 𝐾𝐴− 𝛼 𝛽𝐾𝐵+𝑃𝐵−𝛿𝐴𝐾𝐴 2₎ 𝜕𝐾𝐴 = (𝛼+𝛾) 𝛽 − 2𝛿𝐴𝐾𝐴= 0 → 𝑲𝑨= (𝜶+𝜸) 𝟐𝜹_𝑨𝜷

To get the monopoly price for company A, we set 𝑈𝐴1 equal to 0. This gives:

Lemma 4: 𝑲_𝑨𝒎 = 𝜶 𝟐𝜹𝑨𝜷 Proof: 𝑈𝐴1 = 𝛼𝐾𝐴𝑚− 𝛽𝑃𝐴𝑚 = 0 → 𝑃𝐴𝑚 = 𝛼 𝛽𝐾𝐴 𝑚 𝜋𝐴𝑚= 𝑃𝐴𝑚− 𝛿𝐴𝐾𝐴𝑚2 = 𝛼 𝛽𝐾𝐴 𝑚_{− 𝛿} 𝐴𝐾𝐴𝑚2 𝜕𝛼 𝛽𝐾𝐴 𝑚_−𝛿 𝐴𝐾𝐴𝑚2 𝜕𝐾_𝐴𝑚 = 𝛼 𝛽− 2𝛿𝐴𝐾𝐴 𝑚 _{= 0 →} 𝐾_𝐴𝑚 = 𝛼 2𝛽𝛿_𝐴

If we fill in the results from Lemma 2 and 3 into Lemma 1, we get: 𝑃𝐵 = 𝑃𝐴+ 𝛼 𝛽 𝛼 2𝛿𝐵𝛽− (𝛼+𝛾) 𝛽 (𝛼+𝛾) 2𝛿𝐴𝛽 = 𝑃𝐴+ 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 2𝛿𝐴𝛽2 Theorem 1: 𝜹𝑨 𝜹_𝑩≤ 𝟏 + 𝟐𝜸 𝜶 + 𝜸𝟐 𝜶𝟐 Proof:

Since company A will take the entire market here, 𝜋𝐵 should be equal to 0 so that company

B has no incentive to enter the market: 𝜋𝐵 = 0, 𝑃𝐵 = 𝑃𝐴+ 𝛼2 2𝛿_𝐵𝛽2− (𝛼+𝛾)2 2𝛿_𝐴𝛽2 → 𝜋𝐵 = 𝑃𝐵− 𝛿𝐵𝐾𝐵2= 𝑃𝐴+ 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 2𝛿𝐴𝛽2 − 𝛿𝐵( 𝛼 2𝛿𝐵𝛽) 2 = 0 → 𝑃𝐴=(𝛼+𝛾) 2 2𝛿_𝐴𝛽2+ 𝛼2 4𝛿_𝐵𝛽2− 𝛼2 2𝛿_𝐵𝛽2= (𝛼+𝛾)2 2𝛿_𝐴𝛽2 − 𝛼2 4𝛿_𝐵𝛽2 𝑃𝐵 = 𝑃𝐴+ 𝛼2 2𝛿_𝐵𝛽2− (𝛼+𝛾)2 2𝛿_𝐴𝛽2 = (𝛼+𝛾)2 2𝛿_𝐴𝛽2− 𝛼2 4𝛿_𝐵𝛽2+ 𝛼2 2𝛿_𝐵𝛽2− (𝛼+𝛾)2 2𝛿_𝐴𝛽2 = 𝛼2 4𝛿_𝐵𝛽2

So in the equilibrium, the companies have the following price and quality: 𝑃𝐴= (𝛼+𝛾)2 2𝛿_𝐴𝛽2− 𝛼2 4𝛿_𝐵𝛽2, 𝐾𝐴 = (𝛼+𝛾) 2𝛿_𝐴𝛽 𝑃𝐵 = 𝛼2 4𝛿_𝐵𝛽2, 𝐾𝐵 = 𝛼 2𝛿_𝐵𝛽

This will lead to the following profits and utility: 𝜋𝐴= 𝑃𝐴− 𝛿𝐴𝐾𝐴2= (𝛼+𝛾)2 2𝛿𝐴𝛽2 − 𝛼2 4𝛿𝐵𝛽2− 𝛿𝐴( 𝛼+𝛾 2𝛿𝐴𝛽) 2 =(𝛼+𝛾)2 4𝛿𝐴𝛽2 − 𝛼2 4𝛿𝐵𝛽2 and 𝜋𝐵= 0 Since company A sells their product also in round 1, their total profit will be: 2 ((𝛼+𝛾)2 4𝛿_𝐴𝛽2− 𝛼2 4𝛿_𝐵𝛽2) = (𝛼+𝛾)2 2𝛿_𝐴𝛽2 − 𝛼2 2𝛿_𝐵𝛽2

In order for company A to take the entire market they need to have a positive profit, so 𝜋𝐴≥ 0 → (𝛼+𝛾)2 4𝛿𝐴𝛽2 − 𝛼2 4𝛿𝐵𝛽2≥ 0 → (𝛼+𝛾)2 4𝛿𝐴𝛽2 ≥ 𝛼2 4𝛿𝐵𝛽2→ δB δA≥ α2 (α+γ)2 𝜹_𝑨 𝜹_𝑩≤ 𝟏 + 𝟐𝜸 𝜶 + 𝜸𝟐 𝜶𝟐 Theorem 2: 𝜹𝑨 𝜹_𝑩≥ ((𝜶+𝜸)𝟐−𝜶𝟐 𝟐) 𝜶𝟐 = 𝟏 𝟐+ 𝟐𝜸 𝜶 + 𝜸𝟐 𝜶𝟐 Proof:

The other possibility is when company B enters the market. This happens when 𝑈𝐴 < 𝑈𝐵. In

this case, company A will decide to ask the monopoly price. Both companies still have the same optimizing qualities:

𝐾𝐴𝑚 = 𝛼 2𝛽𝛿𝐴, 𝐾𝐵= 𝛼 2𝛿𝐵𝛽 𝑈𝐴1 = 𝛼𝐾𝐴𝑚− 𝛽𝑃𝐴𝑚1= 0 → 𝛼 𝛼 2𝛽𝛿_𝐴− 𝛽𝑃𝐴 𝑚1_{= 0 → 𝑃} 𝐴𝑚1= 𝛼2 2𝛿_𝐴𝛽2 With this price I can calculate the monopoly profit:

𝜋_𝐴𝑚1= 𝑃_𝐴𝑚1− 𝛿𝐴𝐾𝐴𝑚2

The profit that company A would have if they would have the entire market in both rounds is: 𝜋𝐴2= 2𝑃𝐴2− 2𝛿𝐴𝐾𝐴2

Company B should set his price in such a way that company A is indifferent between having a monopoly in the first round and having the entire market in both rounds.

𝜋𝐴𝑚1= 𝜋𝐴2→ 𝑃𝐴𝑚1− 𝛿𝐴𝐾𝐴𝑚2 = 2𝑃𝐴2− 2𝛿𝐴𝐾𝐴2→ 𝑃𝐴2 = 𝑃_𝐴𝑚1+𝛿𝐴(2𝐾𝐴2−𝐾𝐴𝑚2) 2 = 1 2( 𝛼2 2𝛿𝐴𝛽2+ 𝛿𝐴(2 ( 𝛼+𝛾 2𝛿𝐴𝛽) 2 − ( 𝛼 2𝛽𝛿𝐴) 2 )) = 1 2( 𝛼2 2𝛿𝐴𝛽2+ (𝛼+𝛾)2 2𝛿𝐴𝛽2 − 𝛼2 4𝛿𝐴𝛽2) = 𝛼2 8𝛿𝐴𝛽2+ (𝛼+𝛾)2 4𝛿𝐴𝛽2 𝑃𝐵 = 𝑃𝐴2+ 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 2𝛿𝐴𝛽2 = 𝛼2 8𝛿𝐴𝛽2+ (𝛼+𝛾)2 4𝛿𝐴𝛽2 + 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 2𝛿𝐴𝛽2 = 𝛼2 8𝛿𝐴𝛽2+ 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 4𝛿𝐴𝛽2 𝜋𝐵 = 𝑃𝐵− 𝛿𝐵𝐾𝐵2= 𝛼2 8𝛿𝐴𝛽2+ 𝛼2 2𝛿𝐵𝛽2− (𝛼+𝛾)2 4𝛿𝐴𝛽2 − 𝛿𝐵( 𝛼 2𝛿𝐵𝛽) 2 = 𝛼2 4𝛿𝐵𝛽2− 1 4𝛿𝐴𝛽2((𝛼 + 𝛾) 2₋𝛼2 2)

For company B to take the market they need to have a positive profit, so: 𝜋𝐵 ≥ 0 → 𝛼2 4𝛿_𝐵𝛽2≥ 1 4𝛿_𝐴𝛽2((𝛼 + 𝛾)2− 𝛼2 2) → 𝜹𝑨 𝜹𝑩≥ ((𝜶+𝜸)𝟐−𝜶𝟐 𝟐) 𝜶𝟐 = 𝟏 𝟐+ 𝟐𝜸 𝜶 + 𝜸𝟐 𝜶𝟐 𝜋_𝐴𝑚1= 𝜋_𝐴2_{= 𝑃} 𝐴𝑚1− 𝛿𝐴𝐾𝐴𝑚2= 𝛼2 2𝛿_𝐴𝛽2− 𝛿𝐴( 𝛼 2𝛽𝛿_𝐴) 2 = 𝛼2 4𝛿_𝐴𝛽2 So the prices and profits are:

𝑃𝐴𝑚1= 𝛼2 2𝛿_𝐴𝛽2 , 𝑃𝐴 2₌ 𝛼2 8𝛿_𝐴𝛽2+ (𝛼+𝛾)2 4𝛿_𝐴𝛽2, 𝜋𝐴 𝑚1_{= 𝜋} 𝐴2= 𝛼2 4𝛿_𝐴𝛽2 𝑃𝐵 = 𝛼2 8𝛿_𝐴𝛽2+ 𝛼2 2𝛿_𝐵𝛽2− (𝛼+𝛾)2 4𝛿_𝐴𝛽2, 𝜋𝐵= 𝛼2 4𝛿_𝐵𝛽2− 1 4𝛿_𝐴𝛽2((𝛼 + 𝛾) 2₋𝛼2 2)

Appendix II

18 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 𝟑: 𝜹𝑨 𝜹𝑩< 𝜶𝜸+𝜸𝟐 𝜶𝟐 = 𝜸 𝜶+ 𝜸𝟐 𝜶𝟐 Proof:

Company A will prefer taking the market in both rounds, so equilibrium 1, if: 𝜋𝐴𝐸1> 𝜋𝐴𝐸2→(𝛼+𝛾) 2 4𝛿_𝐴𝛽2− 𝛼2 4𝛿_𝐵𝛽2> 𝛼2 4𝛿_𝐴𝛽2→ 𝛼𝛾+𝛾2 4𝛿_𝐴𝛽2> 𝛼2 4𝛿_𝐵𝛽2→ 𝜹𝑨 𝜹𝑩< 𝜶𝜸+𝜸𝟐 𝜶𝟐 = 𝜸 𝜶+ 𝜸𝟐 𝜶𝟐

Appendix III

The utility that the consumer gets in the first equilibrium is: 𝑈𝐸1= 𝑈𝐴1+ 𝑈𝐴2 = (2𝛼 + 𝛾) ( (𝛼+𝛾) 2𝛿𝐴𝛽) − 2𝛽 ( (𝛼+𝛾)2 2𝛿𝐴𝛽2 − 𝛼2 4𝛿𝐵𝛽2) = 𝛼2 2𝛿𝐵𝛽− 𝛼(𝛼+𝛾) 2𝛿𝐴𝛽 In the second equilibrium this is:

𝑈𝐸2= 0 + 𝑈𝐵 = 𝛼 α 2δ_Bβ− 𝛽 ( 𝛼2 4𝛿_𝐵𝛽2− 1 4𝛿_𝐴𝛽2((𝛼 + 𝛾) 2₋𝛼2 2)) = 𝛼2 4𝛿_𝐵𝛽+ 1 4𝛿_𝐴𝛽((𝛼 + 𝛾) 2₋𝛼2 2)

The consumer will then prefer company A taking the market in both rounds when:

𝑈𝐸1> 𝑈𝐸2→ 𝛼2 2𝛿𝐵𝛽− 𝛼(𝛼+𝛾) 2𝛿𝐴𝛽 > 𝛼2 4𝛿𝐵𝛽+ 1 4𝛿𝐴𝛽((𝛼 + 𝛾) 2₋𝛼2 2) → 𝛼2 4𝛿𝐵𝛽> 1 4𝛿𝐴𝛽((𝛼 + 𝛾) 2₋𝛼2 2 + 𝛼(𝛼 + 𝛾)) → 𝜹_𝑨 𝜹_𝑩> 𝟑 𝟐𝜶 𝟐_{+𝟑𝜸𝜶+𝜸}𝟐 𝜶𝟐 = 𝟑 𝟐+ 𝟑𝜸 𝜶 + 𝜸𝟐 𝜶𝟐