Learning on the choice of price and quantity competition in a model with vertically differentiated goods

Learning on the choice of price and quantity

competition in a model with vertically

differentiated goods

Milan Karsten

27 June 2014

Abstract

In this paper a relationship between market properties and innovation efforts is examined. More specifically the choice for either product, process or both types of innovation based on a firm’s product quality and type of competition is studied. In this study firms can choose to compete on price or quantity. It is shown that firms will want quantity competition, in which the high quality firm prefers process innovation and the low quality firm product innovation. When occasionally a firm chooses to compete on price, the high quality firm will prefer product innovation, and the low quality firm will prefer process innovation.

1

Introduction

A difficult choice that many firms face is that of innovation. In our rapidly evolving economic world, what is the best way to allocate the R&D budget? Typically, two types of innovation are distinguished: product innovation and process innovation. Product innovation improves the firm’s product quality, while process innovation reduces the firm’s production costs. The choice between these two types of innovation of course depends on the specific market. A firm excavating sand will have very little opportunities to product innovate, while process innovation may be very profitable. Although this choice depends largely on the market, there has been much research on finding a more general rule to which type of innovation is most profitable under specific market properties. In this paper two of these market properties are studied.

Research suggests, in a model with 2 firms producing vertically differentiated substitutes, firms prefer quantity competition above price competition, and the high (low) quality firm will prefer process (product) innovation. In this paper, evidence is found to support this property. This conjecture builds on the results of two papers. The first paper is that of Bonanno and Haworth (1998). In a model with two firms, the authors conclude the high quality firm more often chooses product innovation in a Bertrand than in a Cournot model. In a Cournot

model, the high quality firm prefers process innovation. In the Bertrand model firms compete on prices, in the Cournot model on quantities. For the low quality firm the result is reversed.

The second paper is that of Singh and Vives (1994). Singh and Vives offer an appealing framework for combining Cournot and Bertrand competition with innovation. The authors represent Bertrand and Cournot competition by re-spectively price and quantity contracts. If a firm prefers to compete on prices, it chooses for the price contract. The firm can then only set its price. If the firm chooses the quantity contract, it can only choose its quantity to produce. They conclude that with substitutes (complements) firms prefer the quantity (price) contract. This is later explained in more detail.

The present model combines Singh and Vives with Bonanno and Haworth. Two firms produce vertically differentiated substitutes. Both firms can choose to commit themselves to a price or a quantity contract, as in Singh and Vives. They state that firms will prefer the quantity contract. If both firms choose the quantity contract, this will result in a Cournot equilibrium. Bonanno and Haworth state that in Cournot competition the high quality firm will more often prefer process innovation, and the low quality firm will more often prefer product innovation. This leads to the conjecture that in a market with substitutes, firms will want to compete on quantities, and the high quality firm will mostly invest in process innovation, while the low quality firm will mostly invest in product innovation.

This paper will investigate the dynamics of the relation between the com-petitive structure of the market and R&D investment, under learning dynamics. More precisely, the conjecture for firms to compete on quantities, and for the high (low) quality firm to prefer process (product) innovation. To study this conjecture the model is simulated. Evidence is found to support this conjecture in some situations.

The rest of this paper is organized as follows. Section 2 contains an overview of relevant theory. In section 3 the model is developed and a static analysis is done. Section 4 contains the results of the simulation. Section 5 presents the main conclusions and remarks.

2

Theory

To bolster the conjecture about the relationship of the competitive structure of the market and R&D investments, the relevant theory will first be explained more extensively. The paper of Bonanno and Haworth addresses the choice be-tween product and process innovation in a model with vertically differentiated goods. A model with vertically differentiated goods has multiple firms, each producing a different quality product. This difference in quality is assumed to be measurable. The model of Bonanno and Haworth has only 2 firms. In the model product and process innovation respectively lead to higher demands and lower costs. The authors find that in some situations both firms make the same decision on either product or process innovation. But when firms make a

dif-ferent choice under Bertrand competition than under Cournot competition, the high quality firm chooses product (process) innovation in a Bertrand (Cournot) setting. For the low quality firm this result is reversed. In their model firms are allowed to invest in product or process innovation, but firms are not allowed to invest in both. In the model in this paper firms will be allowed to invest any positive amount in both types of innovation, in order to create a more realistic setting.

Singh and Vives model the interaction between Bertrand and Cournot com-petition by price and quantity contracts. Both firms choose a contract, and the profits are based on this decision. The economic interpretation of the Cournot and Bertrand outcomes is as follows. When both firms commit themselves to supply a quantity, they will choose to produce the Cournot quantity because both firms know the best reaction function of the other firm. The only output level where the best reaction functions intersect is the Cournot-Nash equilib-rium quantity. This is similar for the price contract. The authors conclude that with substitutes firms prefer the quantity contract to keep the prices high. 1

The model of Singh and Vives will be extended in order to introduce product and process innovation. The framework of market interaction in the model in this paper can be described by a two stage game. Firstly firms choose the price or quantity contract, after which equilibria profits are reached as a function of R&D benefits and costs. Secondly the firms invest in product and process innovation in a profit optimizing way. Product innovation can be introduced without changing the equilibria prices or quantities, as Bonanno and Haworth’s model can be shown to be a special case of Singh and Vives’ model, and their results can be applied directly to the extended model.

Introducing process innovation requires some alterations. Singh and Vives consider prices net of constant marginal costs, in order to simplify their model. In Bonanno and Haworth process innovation is expressed as a decrease in marginal costs. Introducing a decrease in marginal costs in the model of Singh and Vives will therefore change the equilibrium prices and quantities, resulting in different profits of the price and quantity contract. This could mean that the result that with substitutes firms prefer the quantity contract does not hold after introduction of marginal costs. However it is reasonable to assume that when non-constant marginal costs are introduced to allow for process innovation and when competing on substitutes, firms will still prefer the quantity contract above the price contract in most of the simulation periods. Because in general Cournot profits are higher than Bertrand profits in a market with substitutes.

Abernathy and Utherback (1975, 1982) investigated the timing of product and process innovation. They present a model where firms initially invest more in product innovation, because large market share gains are attainable. But after a while less market share is to be won with product innovation, and process innovation becomes more profitable. Adner and Levinthal (2001) bring this theory into question. They state that the initial investment in product and

1_{With complements firms want high output levels to reinforce the other firm’s market, and}

process innovation depends on the starting performance of the product. Adner and Levinthal agree that later on process innovation becomes more profitable, but that this does not necessarily mean a decrease in product innovation.

The research of Abernathy and Utherback and that of Adner and Levinthal has mixed results for the conjecture that quantity competition will dominate price competition, and the high (low) quality firm will prefer process (product) innovation. In the model specified in this paper the low quality firm starts with a lower market share, due to a lower quality product. According to Adner and Levinthal the low quality firm will therefore prefer product innovation, at least until market share differences have reduced. This supports the expectation made in this paper, at least during the beginning periods. In later periods, however, Adner et al. say that the low quality firm, with the low initial market share, will not keep preferring product innovation, as the low quality firms’ market share will grow.

There is one antithesis leading from the results of Adner and Levinthal. Assume a market with a high and low quality firm producing substitutes, with high and low initial demands, but now in Bertrand competition. According to Adner and Levinthal the low quality firm with the low market share will prefer product innovation, but Bonanno and Haworth would conclude the low quality firm prefers process innovation. As mentioned before, firms may occasionally choose for the price contract in the model of this paper due to the changed profits by introducing process innovation. In that case a Bertrand competition would arise, and it would be unclear if and in what type of innovation the firms would invest. This paper will analyze these dynamics.

3

Model

In this section the model is discussed in more detail. In section 3.1 it is shown how Bonnano and Haworth’s model is combined with Singh and Vives’ model. Section 3.2 contains the static analysis. In section 3.3 it is explained how the model is adapted for the simulations.

3.1

The equilibrium

The parameters of the demand curves in Singh and Vives can be extended to be similar to the demand curves of Bonanno and Haworth. The combined demand curves are given by

DH(pH, pL, KH, KL) = 1 − pH KH− KL + pL KH− KL (1) DL(pH, pL, KH, KL) = pH KH− KL − pL KH− KL− pL KL (2) with DHand DLrespectively the demands for the high and low quality firm.2

2

See the appendix for the exact parameter values of the model of Singh and Vives so that the demand curves are similar to Bonanno and Haworth.

The demand functions depend on the prices pH and pL, of respectively the high

and low quality firm. KH is the quality of the high quality firm’s product and

KLis the quality of the low quality firm’s product. Note that KH > KL. In the

model of Bonanno and Haworth the demand functions are multiplied by N, the number of consumers, but for simplification N is normalized to 1. So instead of a finite number of consumers, there is an infinite population with mass 1.

The inverse demand functions are given by

pH(qH, qL, KH, KL) = KH− KHqH− KLqL (3)

pL(qH, qL, KL) = KL− KLqH− KLqL (4)

The cost functions are given by ci = qi∗ (Ai− Ri) + ϵiIi4+ δiEi4

Ii and Eiwith i = H, L respectively are the investments in product and process

innovation. Riis the per unit cost reduction. The innovation costs to the power

4 ensure that solutions will be finite. Ai can be interpreted as the starting

marginal costs. To ensure positive marginal costs, the restrictions Ai > Ri

must hold.

The profit functions are given by

πi= piqi− ci(qi) (5)

with i = H, L .The firms learn to be either a price or quantity setter. The profits will depend on the chosen contract by both firms. It is assumed that after a price or quantity contract is chosen, output and prices are set optimally. The resulting (optimal) profits are functions of the qualities Ki and cost reductions

Ri of both firms and of the efforts of both firms on either type of innovation.

3.2

Static analysis

Because the firms can switch on type of competition, 4 different situations can occur.

1. Both firms compete on quantities. 2. Both firms compete on prices.

3. The high quality firm competes on quantity, the low quality firm on price. 4. The high quality firm competes on price, the low quality firm on quantity. Because the terms in the profits contain qualities to the power 4, the cost to the power 4 are necessary. Otherwise maximization of the profits by setting the innovation efforts goes to infinity. δiand ϵiare set to 0.001, this is necessary for

on the profit. The first two situations have Cournot and Bertrand profits.3

In order to calculate the produced Cournot quantities, one differentiates the high and low quality firm’s profits (5) with respect to qH and qL, and solves for qH

and qL. The inverse demand functions are then evaluated with these quantities

to obtain the prices. These quantities and prices can be entered in (5). The resulting profits depend on qualities, per unit cost reductions and on innovation efforts.

The Bertrand case is similar to the Cournot case, but now profits have to be differentiated with respect to pH and pL to obtain the optimal prices. Solve

again but now for pH and pL. Then the demands can be evaluated with these

prices, and the profits (5) can be evaluated with the prices and quantities. The profits in the last two situations are calculated differently. Because the high and low quality firm’s demand curves differ, the situations are not identical. In the third situation the high quality firm maximizes its profits with respect to qH, where pHis a function of pL, qH, KHand KL, derived from the high quality

firm’s demand function (1). The resulting expression is solved for qH. The low

quality firm maximizes its profit with respect to pL, where qLis a function of pL,

qh, and KL derived from (4), and then solved for pL. Demand (1) and inverse

demand (4) functions can be evaluated with qH and pL to obtain pH and qL.

These can be entered in the profit functions (5).

The last situation is the other way around. The high quality firm maximizes its profits with respect to pH. Its quantity qH is a function of pH, qL, KH and

KL which can be derived from (3). The low quality firm maximizes its profit

with respect to qL. pL is a function of pH, qL, KH and KL obtained from (2).

In a similar way as the situation before, the profits can be found.

In the resulting profits Ki and Ri are chosen optimally by setting Ii and Ei,

this means that Ki= Ki∗ and Ri = R∗i. Where Ki∗ and R∗i are given by

K∗

i = Ki0+ Ii∗

R∗

i = R0i + Ei∗

Here K0

i and R0i mean the starting product quality and per unit cost

reduc-tion. I∗

i and Ei∗are the optimal amounts to be invested in product and process

innovation.

3.3

Simulations

Section 3.3.1 discusses assumptions that are made for computational issues. Section 3.3.2 discusses the learning mechanism, which decides a firm’s choice between contracts in the simulations.

3.3.1 Assumptions for computational issues

In the simulations the quality of the product depends on the quality in the previous period and the product innovation efforts. The per unit cost reduction depends on the reduction in the previous period and on the efforts made in the

3

current period. A discount factor ρ is added to ensure non-infinite quality and cost reductions. Would this discount factor not be added, all innovative efforts would pile up, meaning that at some point the restriction Ai > Ri no longer

holds. These functions are given by Ki,t= Ii,t∗ + Ki,t−1(1 − ρ)

Ri,t = Ei,t∗ + Ri,t−1(1 − ρ)

with i = H, L.

The model in this paper is one of myopic expectations. When maximizing profits by setting innovation efforts at time t firms observe only the quality and per unit cost reduction of the other firm at t-1. In other words, firms expect the innovation efforts of the other firm to be zero when setting innovation efforts, but that may be wrong. Because of this, it is possible for firms to have negative profits. The firms may have expected high benefits from innovation efforts, but when for example the high quality firm becomes the low quality firm, the benefits can be much lower. In that case the firm still has the innovation costs, and makes a loss. This could happen to real firms as well. Firms may expect their product innovation to be more profitable than it turns out to be, resulting in a loss. Another misspecification is that the firms maximize their profits by setting innovation efforts based on whether they were the higher or lower quality firm in the previous period. When by investing in product innovation the low quality firm becomes the higher quality firm, it should maximize the higher quality firm’s profit from the point that its quality is the higher quality. But till the end of the simulation period, the firm acts as if it still is the lower quality firm.

In the simulations innovation efforts have a lower and upper bound. Negative process innovation efforts make no sense, no firm want to increase its variable costs at a price. It is assumed that firms do not want to decrease the product quality, only by not investing the discount factor will lower the qualities. This is not an unreasonable assumption, because a firm first investing in product quality to later lower the quality is rare. Imagine a firm that has had large costs to introduce a higher quality product. After the introduction of the product, it turns out the higher quality product was not a good strategic decision and the firm would have preferably introduced a lower quality product. Now it is most logical for the firm to keep producing the high quality product for some while, because immediately introducing a lower quality product also has a negative influence on the demands and therefore profitability of the high quality product, making the higher quality product an even larger disaster. On top of that the introduction of a lower quality product has high costs. Because the higher quality product is not what the firm had wanted, the firm may choose to introduce a new, lower quality, product sooner than it had planned, but an early stop of the production is not profitable. Therefore non negative product innovation efforts are not too unreasonable. In the simulations a firm can therefore obtain a lower quality product by letting quality decay and then keeping quality stable when the preferred quality is reached.

The innovation efforts also have a upper bound. This is necessary for the model to be stable. The bounds are relatively high, and therefore they place little restrictions on the model, providing for the most realistic outcomes. In-vestments in process innovation are also capped because it is not possible to implement very high investments in cost reductions, while keeping production levels stable. When for example new machines are bought to reduce the costs, it will take time to get them running, and for workers to learn how to use the new machines. With too big changes in production methods, production levels can not be kept stable.

3.3.2 The learning mechanism

A firm’s choice between the price or quantity contract depends on a learning mechanism. The learning mechanism used is similar to the switching mecha-nism in Anufriev, Kopányi and Tuinstra (2013) to determine the type of learning behavior. The authors expanded the model of Roth and Erev (1995), which is based on reinforcement learning. A probability is defined for choosing either the price or quantity contract. This probability is based on the performance mea-sure. Let the performance measure of the price (quantity) contract be denoted by M Pi,t,(M Qi,t) with i = H, L.

M Pi,t

!

(1 − w)M Pi,t−1+ wπi,t if i chose price contract at t

M Pi,t−1 otherwise

wis the weight of the last observation. The performance measure for the quan-tity contract is constructed analogously. It is basically a weighted average of past profits when the firm used the particular contract. With this performance measure, the probability that either the price or quantity contract is chosen is calculated. The probability that the firm choses the price contract is given by

P_i,t+1price= (1 − 2η)exp(ω(MQi,t1−M Pi,t))+1 + η

Here η is the probability of experimentation and ω ! 0 measures the sensibility to a difference in the performance measure. A ω of infinity means that the contract with the highest performance measure will be chosen with probability (1 − 2η), a ω of zero means firms randomize between both contracts. With this definition of the probability, the choice is based on the difference in performance measure with probability (1−2η) and on randomization with probability 2η. The chance that firm i uses the quantity contract at period t is given by 1 − Pi,tprice.

The next enumeration describes how a contract is chosen in each period. 1. In the first period the type of contract chosen by each firm is randomly

drawn.

2. In period 2, firms try the other contract; price contract firms switch to a quantity contract and vice versa. This is necessary to have a nonzero starting performance measure.

3. From period 3 and on the probability for firm i to choose any contract depends on PP rice

i,t .

4. The simulations stop when a chosen number of periods is reached, or when the restrictions that the variable costs can not be negative are violated.

4

Results

For all simulations, the discount factor ρ is 5%. The experimentation parameter ηis set to 2%. This way firms experiment on the outperformed contract enough to stabilize the model, but the experimentation is not so structural that it would affect long term results. The sensitivity to performance measure ω is 0.01.

In section 4.1 timelines of the model are described. It illustrates and explains why and when in the model firms make a specific decision on innovation. Section 4.2 addresses the choice between product and process innovation.

4.1

Timeline analysis

Figure 1 shows a typical timeline of 1000 periods with the quality of the 2 firms. In this simulation starting variable costs are AH = 250, AL= 250. The

high quality firm starts with a quality of qH = 65, and the low quality firm

with a quality of qL = 50. Starting cost reduction RH and RL are set to zero.

Innovation efforts are capped at 50 for both product and process innovation. With these bounds both companies can almost double their starting quality in a single period, which is a rather high bound that does not restrict the model too much. Also a cap on the cost reduction of 50, which is 20% of the starting variable costs, is a reasonable assumption as too big changes in production method can not be made while keeping production levels stable.

For the numerical optimization software used to maximize the profits by setting the innovation efforts it is required to provide begin values. For these

Figure 1: Firms’ product quality over time.

Figure 2: Firms’ cost reduction over time.

simulation the beginning innovation efforts are set to 35 for both product and process innovation. If these beginning innovation efforts are set anywhere in the range from about 0 to 20, the firms will choose to invest zero in each period. When the starting values are set anywhere between 20 and 50, the firms will choose to invest. Setting starting values at 20 or at 50 does not matter for the results, the firms will invest the same amounts. This is caused by two different local maxima in the profit function. The local maxima where firms invest has higher profits and therefore firms prefer to invest over not investing. On top of that not innovating is not realistic in the modern economic world. Not innovating will eventually drive a firm out of business.

In this simulation the firms tend to prefer the quantity contract, the two firms choose the price contract in only 103 out of 2000 situations. This is 5.15% of the total situations the firms can choose for a contract. 5.15% minus the experimentation factor of 2% means that the firms choose the price contract voluntarily in 3.15% of the simulations. Firms mainly choose the price contract when they have had big innovation costs on the quantity contract, which lowers the quantity contract performance measure drastically. After a while they go back to quantity competition when they had a few quantity contracts due to experimentation in periods in which no investments were made.

As soon as the difference in quality gets small enough due to the discount factor, the lower quality firm tends to maximally increase its quality to get a bigger marketshare. Each of these periods the firm invests the upper bound of 50 in product innovation. In this period the firm has negative profits, as the firm is the low quality firm at the time of innovation, but becomes the high quality firm after the innovations. The high quality firm has less benefits from the quality improvement, and still has big innovation costs. This comes close to reality, as high investment costs and thus negative profits are generally followed by higher profits. It is seen that after about 100 time periods, both qualities converge to a 50-cycle. Adding more periods to the timeline does not change the cycles, and are therefore left out. The system will never reach a complete steady cycle due to the experimentation of the firms.

Figure 2 shows the process efforts of the same simulation. The cost reduction also reaches a 50-cycle. The process innovation efforts perfectly coincide with the product innovation efforts. Still, the lower quality firm competing on quantity invests more in product innovation than in process innovation. In a couple of periods an outlier can be seen. In these periods the moment of innovation for the low quality firm is in the same period as a price contract due to experimentation. Only when a price contract for the low quality firm coincides with the period the low quality firm wants to invest the upper bound of 50 is invested in cost reduction. After this happens, cost reductions decrease by the discount factor till the system is back in the 50-cycle. Apparently investing in process innovation is more profitable for a low quality firm in price competition than in quantity competition. More on that subject in section 4.2.

With a different proportion of starting variable costs compared to starting qualities, different levels of process innovation efforts can be seen. Figure 3 and 4 again show timelines with the product quality and cost reductions of the two

Figure 3: Firms’ product quality over time with high starting variable costs.

Figure 4: Firms’ cost reduction over time with high starting variable costs.

firms with the same starting variables, except for the starting variable costs Ai.

These are now set to AH = 1000 and AL = 1000. It can be seen that the low

quality firm still invests the maximum of 50 in product innovation, but now the lowest quality the firms reach is slightly higher. Therefore the discount factor now decays the quality more in absolute value. Because of this the system now converges to a 43-cycle. The process innovations again coincide with the product innovation, but now less is invested. In the previous simulation the firm invests about 30 each of the periods it innovated while it competed on quantity, now it invests only about 22. This means that process innovation is less profitable now. The low quality firm can now decrease its variable costs by a smaller percentage than before for the same costs, and therefore chooses to invest less in process innovation.

4.2

On the choice of product or process innovation

The fact the low quality firm chooses to invest less in the second simulation means that it is important to find a appropriate ratio of the starting product qualities compared to the starting variable costs, in order to find which type of innovation is preferred. Bonanno and Haworth state that in some cases firms choose to invest only in product innovation or process innovation no matter the type of competition. In other cases firms make a different choice in quantity competition than in price competition, and that is where this paper is mostly interested in. Therefore different starting values have been tested.

With the first simulation of section 4.1 the firms make different decisions when competing on quantity than on price. Therefore 250 repetitions of the first simulation of section 4.1 with the starting variable costs of 250 are executed. Table 1 on the next page shows the average innovation efforts per period based on competition type and product quality.

A firm that committed itself to the quantity contract and is the low quality firm before investing in innovation will on average invest 1.968 per period in

Firm competing on Firm’s quality Innovation Average per period innovation

Quantity High Product 0.032 (0.129)

Quantity High Process 0.042 (0.158)

Quantity Low Product 1.968 (0.518)

Quantity Low Process 1.211 (0.325)

Price High Product 0.451 (0.317)

Price High Process 0.446 (1.082)

Price Low Product 8.387 (3.802)

Price Low Process 10.204 (4.769)

Table 1: Average innovation efforts per period based on type of competition and product quality with standard deviation in parentheses. The low quality firm competing on quantities will generally prefer product innovation, as its average per period product innovation efforts are higher. n=250*1000 periods.

improving its quality and only 1.211 per period in process innovation. One should not interpret these results in absolute value. The average per period innovation efforts should be seen relative to the total average quality and total average cost reduction, and more importantly relative to the innovation efforts in the other type of innovation.

In the table can be seen that the high quality firm competing on quantity will invest only very little, with a slight preference for product innovation. A low quality firm competing on quantity will prefer product innovation. When competing on prices which happens mostly due to experimentation, the high quality firm will slightly prefer product innovation, but with little difference and a high standard deviation. The low quality firm competing on prices slightly prefers process innovation. It can be seen that the low quality firm makes a different decision in price than in quantity competition. The results of the simulation are in line with the results of Bonanno and Haworth.

In reality innovative abilities depend mainly on the investment made, but there comes a point that more investments will be less effective. This property is taken into account in the model by the costs of innovation to the power 4. That is why applying more restrictions on innovative abilities does not seem in accordance with reality. The model of Bonanno and Haworth does not assume that innovative abilities are limited, and therefore the assumption made in this model of relative unrestricted innovative abilities does not conflict with the assumptions of the model of Bonanno and Haworth. Hence the results of this paper can be compared with the results of Bonanno and Haworth. The model of this paper is not very suited to model a market with low innovative abilities, as in a market with little opportunities to innovate placing restrictions on innovative abilities may be very realistic. One should thus refrain to apply the results of this model to such a market.

In the simulation the quantity contract is used in 93.9% of the periods. This is in line with the results of Singh and Vives. They state that firms will prefer the quantity contract. Even tough the model of Singh and Vives was extended to

allow for process innovation in this model, what changed the equilibrium prices and quantities and therefore the profits, the quantity contract is still preferred.

5

Conclusion

In this paper firm’s decisions on investing in either product or process innova-tion have been simulated in a model with vertically differentiated goods. The results of the simulation are in agreement with the results of Singh and Vives. When firms can choose whether they want to compete on prices or quantities, the firms will prefer to compete on quantities. On the choice of product or process innovation, the results of this study are in line with Bonanno and Ha-worth’s results. When competing on quantity, the high quality firm slightly prefers process innovation, and the low quality firm will prefer product innova-tion. When the price contract occurs due to experimentation, the high quality firm will prefer product innovation, and the low quality firm will slightly prefer process innovation.

There are some remarks on the model of this paper and therefore on the robustness of the results. In the simulations innovative abilities are very un-bounded. This is a suitable assumption for markets with high innovative pos-sibilities. In markets with low innovative possibilities on the other hand, this model may not be so realistic. Firms may act differently in their choice on in-novation when the possibilities are small, and therefore the results of this study are not valid for such markets. Furthermore the firms have myopic expectations in the model of this paper. Both firms maximize their profits by setting inno-vation levels, in which they assume the other firm’s quality and cost reduction at period t to be equal to period t-1. In other words, both firms assume the other firm will not invest in the current period, which is not a very realistic as-sumption. On top of that the firms do not take into account that by innovating they may become the higher quality firm, resulting in different profits. Another assumption that is not in line with reality, is that after both firms choose to compete on prices or quantities the firms set prices and quantities optimally, resulting in the Nash-equilibrium. This means that it is assumed that both firms know both demand and cost functions exactly, which is not realistic.

Further research on innovation could try to tackle these assumptions. One could for example look at a model with rational expectations, where firms maxi-mize the profits taking the other firm’s innovation efforts into account. Another research subject could be whether firms act differently when they are relatively bounded in their innovative abilities. One could also look at a model where as soon as the low quality firm becomes the higher quality firm, it immediately switches to innovating as the high quality firm in the same period.

Acknowledgements

Special thanks to Anghel Negriu for guidance in the research project and many useful remarks on this paper.

References

[1] Abernathy, W. J., and J. M. Utherback. (1975). A dynamic model of process and product innovation. Omega 3, 639–56.

[2] Abernathy, W. J., and J. M. Utherback. (1982). Patterns of industrial inno-vation. Readings in the Management of Innovation, 97–108.

[3] Adner, R. and D. Levinthal. (2001). Demand Heterogeneity and Technology Evolution: Implications for Product and Process Innovation. Management Science 47,611-628.

[4] Anufriev, M., Kopányi, D. and J. Tuinstra. (2013). Learning cycles in Bertrand competition with differentiated commodities and competing learn-ing rules. Journal of Economic Dynamics & Control 37, 2562–2581. [5] Bonanno, G., and B. Haworth. (1998). Intensity of competition and the

choice between product and process innovation. International Journal of Industrial Organization 16, 495–510.

[6] Roth, A. and I. Erev. (1995). Learning in extensive-form games: experimen-tal data and simple dynamic models in the intermediate term. Games and Economic Behavior 8,164–212.

[7] Singh, N., and X. Vives. (1984). Price and quantity competition in a differ-entiated duopoly. RAND Journal of Economics 15, 546–554.

Appendix

By combination of the model of Singh et al. and Bonanno et al. the demand functions for the high and low quality firm are given by

DH(pH, pL) = a1− b1pH+ cpL with a1= 1, b1= KH−K1 L, c = 1 KH−KL DL(pH, pL) = a2+ cpH− b2pL with a2= 0, b2= KH−K1 L + 1 KL, c = 1 KH−KL Elaborated the demand functions look like this:

DH(pH, pL, KH, KL) = 1 − _KHp_−KH _L +_KHp_−KL _L

DL(pH, pL, KH, KL) = _KHp_−KH _L −KHp−KL L −

pL

KL

In the model of Bonanno et al. the demand functions are multiplied by N, the number of consumers, but for simplification N is set to 1. So instead of a finite number of consumers, there is an infinite population with mass 1.

pH(qH, qL) = α1− β1qH− γqL

with α1= KH, β1= KH,γ = KL

pL(qH, qL) = α2− γqH− β2qL

with α2= KL, β2= KL.

Or elaborated this simplifies to

pH(qH, qL, KH, KL) = KH− KHqH− KLqL

pL(qH, qL, KH, KL) = KL− KLqH− KLqL

The expressions for the profits in the 4 different cases are as follows. When both firms compete on quantities the profits are given by πH = (KH(2AH−AL−2KH+KL−2RH+RlL)) 2 (−4KH+KL)2 − 0.001I 4 H− 0.001EH4 πL =(−2ALKH+AHKL+KHKL−KLRH+2KHRL) 2 (KL(−4KH+KL)2) − 0.001I 4 L− 0.001E4L

Similarly, when both firms compete on prices the profits are given by πH = (2AH KH−ALKH−2K2H−AHKL+2KHKL−2KHRH+KLRH+KHRL)2 ((KH−KL)(−4KH+KL)2) −0.001I 4 H− 0.001E4 H πL =(KH(2AL KH−AHKL−ALKL−KHKL+KL2+KLRH−2KHRL+KLRL)2) ((KH−KL)KL(−4KH+KL)2) −0.001I 4 L− 0.001E4 L

When the high quality firm competes on price, and the low quality firm on quantity, the profits are given by

πH = (2AH KH−ALKH−2K2H−AHKL+2KHKL−2KHRH+KLRH+KHRL)2 (KH(4KH−3KL)2) −0.001I 4 H− 0.001E4 H πL =((KH−KL)(−2ALKH+AHKL+KHKL−KLRH+2KHRL) 2₎ (KH(4KH−3KL)2KL) −0.001I 4 L−0.001E4L

In the reversed situation, when the high quality firm competes on quantity and the low quality firm on price, the profits are given by

πH = ((KH−KL)(2AH−AL−2KH+KL−2RH+RL) 2₎ (4KH−3KL)2 − 0.001I 4 H− 0.001EH4 πL =

(2ALKH−AHKL−ALKL−KHKL+K2L+KLRH−2KHRL+KLRL)2

((4KH−3KL)2KL) − 0.001I

4 L−

0.001E4 L