Process and product innovation in a vertically
differentiated Bertrand oligopoly with OLS learning
Daniel Tobon Arango
27 July 2014
Abstract
This paper looks at a vertically differentiated Bertrand competition where firms have the option to innovate in product and process innovation. Two markets are considered, one where the high quality firm innovates and the other where the low quality firm has the option to innovate. This paper demonstrates that (i) both firms invest more in process innovation than in product innovation; (ii) product and process innovation have a positive (negative) relation for an innovating low (high) quality firm. The paper proceeds with analyzing the innovative behavior of firms when applying least squares learning to learn on profit. This leads to the result that firms end up in a non-unique steady state close to their optimal value.
1 Introduction
Firms that want to invest in R&D have multiple options of innovations from which they can choose. Studies show that many technological firms invest in improved products through product innovation and in cost reduction by choosing process in-novation (Cohen & Klepper, 1996). It is obvious that a firm would like to know the optimal allocation of process and product innovation to determine their optimal strat-egy. Models can be made to find the optimal allocation but it would be to optimistic to assume all market information known. Economists can therefore choose to use learning methods to approximate the adaptive nature of business decision-making. An example is to assume that firms are not aware how the demand for their products depends on the price they charge (Anufriev et al., 2013).
The economic effects of innovation is a popular topic in research because tech-nological changes are major drivers of economic growth and competitive progress
(Dawid, 2006). One of these researches is from Bonnano and Haworth (1998). They studied the relationship between the intensity of competition and the profitability of innovative activity to find out if the incentive to innovate is higher in more competi-tive markets. Other researchers that shined a light on this topic are Bacchiega et al. (2011). They examined a Bertrand duopoly where firms have the option of investing in product improvement or in cost reduction, considering the costs of innovation. One of the reasons to research the economic effects of innovation is its relevance. Studies like that of Bonanno and Haworth (1998) and Bacchiega et al. (2011) can give a good insight in finding optimal innovation strategies and can therefore lead to more economic growth. Finding optimal innovation strategies sounds good but the studies mentioned before assume that firms have full market information. Therefore it is even more interesting to look at firm’s innovative behavior if the market structure is not completely clear to the agents. A good example of a research that studies the behavior of agents when they have to learn on unknown aspects of a market is that of Anufriev et al. (2013). They analyze firms when demand has to be estimated to learn on profit with two types of learning methods. They try to find out if firms eventually choose prices optimally when applying these learning methods.
Following the studies of Bonanno and Haworth (1998) and Bacchiega et al. (2011), this paper also focusses on process and product innovation in a vertically differen-tiated market. Because of the importance of research and development (R&D) this study analyzes the innovative behavior of firms competing on prices and tries to an-swer the following question: what is the optimal allocation of R&D investments in process and product innovation for a Bertrand competitor, considering the costs of innovation? After answering this question in a static analysis the paper continues ex-amining the behavior of firms, now without the assumption of perfect information and full rationality. It assumes that firms are unaware of their profit function, resulting in an estimated profit function. This research analyzes the dynamics of least squares learning to find the estimated profit function and examines if the static equilibrium is reachable in a more realistic context of adaptive firm behavior.
The study of Bonanno and Haworth (1998) also analyzes the innovative behavior of firms in a vertically differentiated market but it mainly focusses on the relationship between the intensity of competition and the choice between process and product. A vertically differentiated market is defined as a market with a high quality firm and a low quality firm where the high quality firm is the only firm that enjoys positive demand if prices are equal. To find the relationship between innovation choices and the intensity of competition they look at both Cournot and Bertrand competitions.
Bonanno and Haworth (1998) argue that a Bertrand competition as a more intense competition than a Cournot competition because the former leads to higher output and lower prices. They derive the Nash-equilibria of both competitions and observe the following results in the case that the high quality firm innovates: (i) both competi-tors choose cost reduction; (ii) both competicompeti-tors choose product improvement; (iii) if the competitors choose differently, the high quality Cournot competitor chooses for cost reduction and the Bertrand competitor for product improvement. If the low qual-ity firm innovates, the last result is reversed. It is hard to reflect the results Bonanno and Haworth (1998) on this study because the firms in this model can choose for both process and product innovation and because this study is limited to a Bertrand competition. Although the results are specific for their research, their model is not. This research uses a model very similar to that of Bonanno and Haworth (1998), this model is explained in the second section.
Bacchiega et al. (2011) also use a similar model to that of Bonanno and Haworth (1998) but extended their model with costs for process innovation. Where the re-search of Bonanno and Haworth (1998) differs from this paper in many ways, that of Bacchiega et al. (2011) is quite similar. They also consider a Bertrand duopoly where firms can choose both types of innovations simultaneously. The model in their research is set up as a non-cooperative three-stage game where firms find optimal quality innovation in the first stage, process innovation is set in the second stage and the prices are optimized in the third stage. Bacchiega et al. (2011) found that process and product innovation are positive related for the low quality firm but negatively related for the high quality firm. Even though this result sounds plausible for this research as well, it might of course differ. One of the possible reasons for different results is the fact that Bacchiega et al. (2011) only consider the indirect costs of quality difference.
As mentioned before, Anufriev et al. (2013) examined a market where firms do not have full market information. They assume that a firm cannot observe its demand function and should therefore estimate it. Anufriev et al. (2013) analyze different learning rules to learn on firm’s profit. They consider gradient learning, least squares learning and the situation where firms can switch between both learning rules. They find that least squares learners move to a non-unique steady state whereas firms applying gradient learning can converge tot their Nash-equilibrium but when they don’t, they end up in high-period cycles or quasi-periodic cycles. There is endogenous switching in the heterogeneous situation where firms can switch between gradient and least squares learning. Their model illustrates that adaptive learning leads to complex
market dynamics, contrasting the results of the classical static analysis. This paper applies least squares learning as well and examines if the firms in this market also move to a steady state when learning on profit. This paper also tries to examine if a possible steady state lies close to the Nash-equilibrium of the learning firms.
There are several situations that firms encounter and that can be related to the studies that are described above. The studies of Bonanno and Haworth (1998) and Bacchiega et al. (2011) give a lot of information about innovation choices in compe-titions and the findings of Anufriev et al. (2013) give a good insight into the reaction of firms when they do not observe all market information. This study uses the model of Bacchiega et al. (2011) but define product improvement differently which also leads to introducing costs for product improvement. Where Bacchiega et al. (2011) defines product innovation as the difference between the optimal product innovation of both the high and low firms, this paper integrates product innovation in the profit functions. On top of this extension to the model of Bacchiega et al. (2011), the work presented here adds an analysis of the innovation choices of firms when they are bounded rationally.
The remainder of this paper is organized as follows. Section 2 presents the market structure and the model, section 3 solves the three-stage non-cooperative game and analyzes numerical solutions. Section 4 explains the learning process, derives the steady state of both innovations and discusses the behavior of firms applying least squares learning. Section 5 concludes.
2 The model
The model in this paper is based on the model of Bacchiega et al. (2011). In this market the consumers are continually and uniformly distributed over the interval [✓, ¯✓] with a density of one and ✓ = ¯✓ 1 > 0. Parameter ✓ is a measure for the taste of each consumer. The utility of each individual consumer is characterized by U = ✓ki pi
where ki denotes the quality of the product offered by firm i (i = H, L) and piequals
the price charged by firm i. Like Bonanno and Haworth (1998) and Bacchiega et al. (2011) this paper also considers a vertically differentiated market with two firms, a high quality firm (firm H) with quality kH and a low quality firm (firm L) with
products of quality kL. It also means that kH > kL > 0 should be satisfied in the
model. The firms experience different demands because demand depends on quality in this model. Let ✓0be the value of ✓ that corresponds to a consumer that is indifferent
the solution to ✓kL pL= 0, thus ✓0= pkLL. There is also the possibility of a consumer
being indifferent between buying the low quality product and buying the high quality product. The value of taste in this situation, ✓1, is the solution to ✓kH pH= ✓kL pL.
This leads to ✓1 = kpHH pkLL. Following Bonanno and Haworth (1998) and Bacchiega et
al. (2011), I assume that that every product in this market is demanded. This leads to the following market demands (Di):
DH(pH, pL) = (1 ✓1) = 1 pH pL kH kL (1) DL(pH, pL) = (✓1 ✓0) = pH pL kH kL pL kL (2)
The marginal production costs for the firms are defined as Ci = ci ci, where ci>
0 and ci 0. The component ci denotes process innovation for firm i (i = H, L)
and ciis equal to the fixed, initial marginal costs of firm i. Adapting to new technology
that reduces a firm’s costs bring costs as well, these innovation costs are defined as ( ci)2where > 0 (Bacchiega et al., 2011). In this vertically differentiated market I
also assume that producing a high quality product is more expensive than producing a low quality product, so cH > cL> 0.
As described earlier, firms also have the choice of investing in product innovation. Product innovation is integrated into this model by defining quality slightly differently than described above. This adjustment leads to the following definition of quality: Ki= ki+ ki with ki 0being quality innovation and ki the fixed, initial quality
of firm i’s product. Besides costs for cost reduction, this model also accounts for innovation costs for quality innovation. These costs are defined as ⌘( ki)2 where
⌘ > 0. Furthermore I do not only assume that a higher quality product has higher production costs, I also assume that both demands are positive when the two products are sold at unit cost (pH = cHand pL = cL). This can only be the case if the following
conditions are satisfied (Bonanno & Haworth, 1998). Where conditions (3) apply to the innovating high quality firm and conditions (4) for an innovating low quality firm. kH+ kH kL> cH cH cL; kL(cH cH) > (kH+ kH)cL (3)
kH (kL+ kL) > cH (cL cL) ; (kL+ kL)cH> kH(cL cL) (4)
The firms play a non-cooperative three-stage game and compete on prices. The next section describes the three-stage game where firms set quality innovation in the first stage, cost innovation in the second and prices in the last stage (Bacchiega et
al., 2011).
3 The three-stage game
The three-stage game is divided into two scenarios. The first scenario discusses the situation where the high quality firm has the choice to innovate and the low quality firm cannot innovate. The other scenario discusses the reversed situation. Section 3.1 analyzes the first situation mathematically and section 3.2 does this for the second situation. Section 3.3 studies the static situation and compares the choices of both firms when initial costs and initial qualities differ.
3.1 Innovating high quality firm
This game consists of three stages, firms determine their optimal quality innovation in the first stage, optimal cost reduction in the second stage and find their optimal prices in the last. As usual, this paper solves the game by backward induction. This section considers the situation where the high quality firm innovates and the low quality firm doesn’t. This means that the following profit functions follow from the information given in the previous section and demand functions (1) and (2).
⇡H(pH, pL, kH, cH) = (pH cH+ cH)(1 pH pL kH+ kH kL ) ( cH)2 ⌘( kH)2 ⇡L(pH, pL, kH) = (pL cL)( pH pL kH+ kH kL pL kL )
The firms set their optimal prices in the third stage so with backward induction, this stage is solved first. The following first and second order conditions of this stage are: ⇡H(pH, pL, kH, cH) pH = 0 ; ⇡L(pH, pL, kH) pL = 0 2⇡ H(pH, pL, kH, cH) p2 H < 0 ; 2⇡ L(pH, pL, kH) p2 L < 0 This results into the Bertrand-Nash equilibrium prices.
p⇤H( kH, cH) = (kH+ kH)(2(kH+ kH) kL+ 2(cH cH) + cL) 4(kH+ kH) kL p⇤L( kH, cH) = 2cL(kH+ kH) + kL(kH+ kH+ cH cH kL) 4(kH+ kH) kL
Knowing the optimal prices, the game continues into finding the optimal choices for process and product innovation, given the Bertrand-Nash equilibrium prices. Because this scenario assumes that the high quality firm is the only firm that can innovate, the high quality firm chooses its innovation in the first and second stage and the low quality firm considers this as a given. This leads to the following first and second order conditions. ⇡H(p⇤H( kH, cH), p⇤L( kH, cH), kH, cH) cH = 0 2⇡ H(p⇤H( kH, cH), pL⇤( kH, cH), kH, cH) c2 H < 0
The values of cH for which the profit function of firm H satisfies the previous
conditions are maxima, given the Bertrand-Nash equilibrium prices of both firms. These solutions, denoted c⇤
H are a function of kH and therefore are used to find
the solution to the last stage, the optimal quality innovation. The following conditions yield these solutions ( k⇤
H). ⇡H(p⇤H( kH, c⇤H( kH)), p⇤L( kH, c⇤H( kH)), kH, c⇤H( kH)) kH = 0 2⇡ H(p⇤H( kH, c⇤H( kH)), p⇤L( kH, c⇤H( kH)), kH, c⇤H( kH)) k2 H < 0 As mentioned in the previous section, conditions (3) must be satisfied in a mar-ket where firm H innovates. If these conditions and the described first and second order conditions are satisfied, both firms find the Bertrand-Nash equilibrium prices (p⇤
H, p⇤L) given the optimal innovation choices ( c⇤H, kH⇤) of the high quality firm.
The following section describes the three-stage game in this market if the low quality firm is the only firm that can innovate.
3.2 Innovating low quality firm
The three-stage game of the market with an innovating low quality firm is similar to the game with an innovating high quality firm. The profit functions in this case are:
⇡H(pH, pL, kL) = (pH cH)(1 pH pL kH kL kL ) ⇡L(pH, pL, kL, cL) = (pL cL+ cL)( pH pL kH kL kL pL kL+ kL ) ( cL)2 ⌘( kL)2
As in the previous section, the first step is to determine the optimal Bertrand-Nash equilibrium prices. ⇡H(pH, pL, kL) pH = 0 ; ⇡L(pH, pL, kL, cL) pL = 0 2⇡ H(pH, pL, kL) p2 H < 0 ; 2⇡ L(pH, pL, kL, cL) p2 L < 0
If the low quality firm innovates and the high quality firm does not, the following Bertrand-Nash equilibrium prices are found.
p⇤H( kL, cL) = kH(cL cL 2(kL+ kL) + 2(kH+ cH)) 4kH (kL+ kL) p⇤L( kL, cL) = (kL+ kL)(kH+ cH (kL+ kL)) + 2kH(cL cL) 4kH (kL+ kL)
With the first step solved, the next step is to find the optimal cost innovation for the low firm. These solutions are found by solving the first and second order conditions below. ⇡L(p⇤H( kL, cL), p⇤L( kL, cL), kL, cL) cL = 0 2⇡ L(p⇤H( kL, cL), pL⇤( kL, cL), kL, cL) c2 L < 0
The optimal values of cL, c⇤L can now be used to find the optimal values of
quality innovation ( k⇤
L) for the innovating low quality firm. These are found with
the first and second order conditions below.
⇡L(p⇤H( kL, c⇤L( kL)), p⇤L( kL, c⇤L( kL)), kL, c⇤L( kL)) kL = 0 2⇡ L(p⇤H( kL, c⇤L( kL)), p⇤L( kL, c⇤L( kL)), kL, c⇤L( kL)) k2 L < 0
With all these first and second order conditions, the set of optimal values for all variables (p⇤
H, p⇤L, kL⇤, c⇤L) can be derived if conditions (4) are satisfied.
With the derivation of all conditions that must be satisfied in both games, the next step is to analyze the static situation of process and product innovation of both firms if they can innovate. The next section compares the choices of innovation numerically and also shows whether a high quality firm behaves differently than a low quality firm
if it has the option to innovate.
3.3 Comparative statics of process and product innovation
To illustrate innovative behavior of both firms, this section shows numerical results of the three stage game that was described in sections 3.1 and 3.2. This is done by deriving the Nash equilibria for multiple initial values and comparing the behavior of both firms. This comparison considers both scenarios, a market with an innovating high quality firm and a market with an innovating low quality firm. To give insight into the behavior of both firms the game is repeated multiple times with different initial values of both marginal costs and initial quality. The values that are used in this analysis are chosen in such a way that conditions (3) are satisfied in a market with an innovating high quality firm and conditions (4) are satisfied in a market where the low quality firm can choose to innovate.
The values that are used in the comparison are displayed in Table 1. The rows of table 1 correspond with each graph of figure 1 and the columns indicate the initial values. Figures 1a and 1b show the innovation choices for different quality values and figures 1c and 1d illustrate the choices for different values of marginal costs. All the figures show four plots, each one represents the innovation choices for both firms and both types of innovation. The solid lines (kh and kl in figure 1a) represent quality innovation for respectively the high quality firm and low quality firm, the dashed lines illustrate process innovation for both firms. Figure 1a shows the choice of an innovating firm for different initial quality differences if the values of kL, cH and cL
are held constant. In figure 1b, kL ranges from 15 to 50, keeping all other parameters
constant. Figure 1c varies marginal costs of firm H and figure 1d varies the costs of firm L. The values of and ⌘ are both set to 0.5 in this analysis.
kH kL kH kL cH cL cH cL i ii iii iv v vi a [60,100] 50 [10,50] 16 8 8 b 60 [15,50] [10,45] 12 3 9 c 60 40 20 [12,26] 6 [6,20] d 60 40 20 20 [1,13] [7,19]
Table 1: Initial values for figures 1a-d
As mentioned above, figures 1a and 1b give an illustration of the innovation choices of both firms for different values of kH (kH 2 [60, 100]) and different values of kL
illustrate that the high quality firm prefers cost innovation above quality innovation in nearly all cases. Other results of this analysis are that the value of quality innovation seems independent of initial quality difference and that cost innovation of firm H has a positive and concave relationship with initial difference. An intuitive explanation for these observations is the expectation that difference in quality can work as a substitute for quality innovation. This means that the need of innovation is less high for firm H when the initial quality difference is larger which can be seen in figure 1a and 1b. Because the high quality firm does not have a high incentive to innovate in quality it will therefore choose for more cost innovation if the difference in quality is higher. This also explains the positive relationship between cost innovation and initial quality difference.
(a) Initial quality of firm H varies (b) Initial quality of firm L varies
(c) Initial marginal costs of firm H vary (d) Initial marginal costs of firm L vary
Figure 1: Relationship between initial differences and innovation choices The second scenario describes the situation where the low quality firm has the option to innovate. Figures 1a and 1b show the innovation choices for a low quality firm with difference in the quality of firm H (figure 1a) and of firm L (figure 1b). These graphs show that a low quality firm has different innovative behavior than the high quality firm. The figures show that the low quality firm innovates less in quality
improvement than the high quality firm. Again, this can be explained by the high initial difference in quality. The low quality firm would have to invest drastically in quality in this situation to cope with the difference. It is clear that choosing drastic innovation would bring high costs and therefore it would be less interesting for the low quality firm to innovate in quality. Therefore it would be more interesting for a low quality firm to invest in cost reduction. Although this type of innovation would have the preference of firm L, the graphs of cost innovation flatten from a certain point. This is due to the costs exceeding the benefits of this type of innovation when the quality differentiation is higher.
The next step is to look at innovation choices when the initial differences in marginal costs of both firms vary. Figures 1c and 1d show the results of this analysis. The following two results can be noticed from these figures: (i) a low quality firm chooses to innovate more in both types of innovations if the initial difference in costs is bigger; (ii) the high quality firm has reversed behavior, it innovates less when cost difference is bigger.
This analysis proceeds with the comparison between product and process innova-tion as percentages of their initial values. The values that are used for this comparison are the same values as displayed in table 1. Figure 2a shows the combinations of pro-cess and product innovation as a percentage of their initial value with all initial values constant except for kH (row a, table 1). In this figure the graph of the high quality
firm moves to the right as quality difference gets bigger. The graph of firm L moves to the left as quality difference gets bigger. Figure 2b demonstrates all combinations of process and product innovation as a percentage of their initial value with all values equal to the values in row c, table 1. The graphs of both firms move to the left when initial costs increase in figure 2b.
(a) Initial quality of firm H varies (b) Initial marginal costs of firm H vary
Figure 2a shows that the relative product innovation of firm H decreases as the initial quality difference gets bigger. It also illustrates that process innovation as a percentage of the initial costs are bigger. This confirms the conclusion that firm H allocates its innovation choices more towards process innovation when the quality difference is higher because there is less need for quality improvement. It also shows that the low quality firm lowers its innovations when the quality difference is higher. Figure 2a confirms the results of Bacchiega et al. (2011). They concluded that process innovation fosters (hinders) product innovation for the low (high) quality firm. This figure shows the positive relation between process and product innovation for the low quality firm and a negative relation between both types of innovation for the high quality firm.
4 Least Squares learning
In this section I assume that the profit function of the firms are unknown that is, the firms do not know how their profit reacts on innovation. To learn on profit an estimated profit function is made. With this perceived profit function, the method of recursive least squares (Evans & Honkaphja, 2001) and simulation this paper looks if the firms move to a steady state when applying least squares learning (Anufriev et al., 2013) and if they do, if this steady state approaches the Bertrand-Nash equilibrium so the optimal choices of innovation.
4.1 The learning mechanism
With least squares learning the firms estimate their next innovation choices based on past innovation choices and maximize an estimated profit function. The perceived profit function is a non-linear function of product innovation, process innovation and the squares of both. The parameters of this least squares function are estimated in each period and these determine the innovation choices of the next period. The estimated profit function for firm i (i = H, L) is defined as follows.
⇡Pi = ↵i+ i ki+ i ci µi( ki)2 ⌫i( ci)2+ ✏i (5)
Where ↵, , , µ and ⌫ denote the unknown parameters and where ✏ is an error term with mean zero. Of course this profit function is misspecified because the real functions also depend on prices, costs and demand. The quadratic terms of the costs of innovation adjust for this misspecification. For obtaining the estimates of the
unknown parameters, recursive least squares as described in Evans and Honkaphja (2001) is used. Let ˆ↵i, ˆi, ˆi, ˆµi and ˆ⌫i denote the OLS estimates for firm i and let
ˆi,t be the vector of these estimates at time t. Furthermore the horizontal vector of independent variables at time t is called i,t and is defined as ⇡i,tP = i,t i,t+ ✏i,t.
Then the solutions for these estimates are as follows (Evans & Honkaphja, 2001). ˆi,t= ˆi,t 1+ 1
tRi,t
0
t(⇡i,t i,tˆi,t 1) (6)
Where Ri,t is defined as:
Ri,t = Ri,t 1+
1 t(
0
i,t i,t Ri,t 1) (7)
Here ⇡i,t denotes the profit of firm i at time t. As mentioned before, the firms
determine their innovations based on the OLS estimates of the previous period1.
They determine the innovation for the next period by maximizing its expected profit function defined as E[⇡P
i,t+1] = i,t+1ˆi,t. The innovation choices are found with the
next conditions. E[⇡P i,t+1] ki,t+1 = 0 , E[⇡ P i,t+1] ci,t+1 = 0 This leads to the following solutions.
ki,t+1 = i,t
2µi,t
, ci,t+1 = i,t
2⌫i,t (8)
Because there are five parameters to be estimated in this model, the firms imple-menting least squares learning choose their innovation choices randomly for the first five periods. This is the start of the following process that firms experience:
1. ki,t and ci,t are randomly drawn from the uniform distribution for t = [1, 5].
2. The innovating firm obtains its OLS estimates at the end of period 5 by applying formulas (32) and (33).
3. From period 6 on the firms start to determine their innovations with formula (34).
4. After updating their profit the firms also update the OLS estimates with for-mulas (32) and (33).
1If
i= ( i,1, ..., i,5)0is of full rank and if ⇡i= (⇡i,1, ..., ⇡i,5)0the initial value of ˆiis equal to ˆi= ( 0
i i) 1 0
5. The process stops when | ki,t ki,t 1| < or | ci,t ci,t 1| < where
> 0 is a threshold value.
In the next section this process is applied and the results are demonstrated to evaluate if a firm’s innovation choices move towards a steady state when applying least squares learning.
4.2 Results
To illustrate the dynamics of process and product innovation for an innovating firm applying least squares learning the model described in the previous section is sim-ulated. The simulation is done for both situations so for an innovating low quality firm and for an innovating high quality firm. The values for the fixed variables in these simulations are kH = 60, kL= 50, cH = 16and cL = 8. To check whether the
innovating firms move to an equilibrium the simulations are repeated ten times for t = 20. The threshold in these simulations is set to = 10 7. For these initial values
of kH, kL, cH and cL the Bertrand-Nash equilibrium values of process and product
innovations for the high quality firm are ( k⇤
H, c⇤H) = (0.244, 0.228). The optimal
innovation values for the low quality firm are ( k⇤
L, c⇤L) = (0.094, 0.358).
(a) Product innovation (b) Process innovation
(a) Product innovation (b) Process innovation
Figure 4: Time series of innovation choices for 10 LS learning low quality firms Figures 3 and 4 illustrate the time series of 10 least squares learners with a low and high quality product. The figures show that innovation choices are more volatile in the first five periods but settle down shortly after. The movements in the first five periods can be explained by the fact that the firms set their innovation randomly for the first five periods. After these first periods the choices of innovation move to a steady state really quickly. Product innovation choices of the firms in figure 3 move to a steady state in the range of approximately 0.2 to 0.3 and process innovation of approximately 0.15 to 0.25. This implicates that high quality firms that learn on profit with least squares learning move to a steady state close to their optimal choices. Figure 4 also demonstrates this for the low quality firms. The range of product innovation for these firms is 0.04 to 0.1 and are between 0.28 to 0.4 for process innovation. Firms moving to a steady state is in line with the conclusion of Anufriev et al. (2013). Firms in their research moved to a non-unique self-sustaining equilibrium where actual and perceived demand coincide at a price charged by the firms. Figures 3 and 4 illustrate these same findings; the perceived and actual profit of a firm coincide at levels of innovation that are close to optimal for the innovation firms.
5 Conclusion
This study examined the dynamics of process and product innovation of Bertrand competitors in a vertically differentiated market with a model comparable to the model of Bonanno and Haworth (1998) and Bacchiega et al. (2011). Bonanno and
Haworth (1998) also looked at process and product innovation in a vertically differen-tiated market but shined a light on the relationship between intensity of competition and innovation choices whereas this research studied the optimal innovation choices for an innovating firm in Bertrand competition. Bacchiega et al. (2011) expended this model with process innovation costs. Both studies defined product innovation as the difference between the quality of firm H and the quality of firm L. The model of this paper distinguishes itself because it defines product innovation explicitly and because it adds the costs of innovating in quality. This model led to two main conclusions: (i) both firms invest in both types of innovation but process innovation exceeds product innovation; (ii) product and process innovation have a positive (negative) relation for an innovating low (high) quality firm.
After finding the optimal allocation of process and product innovation for multiple initial values, this study also examined the behavior of firms without full market infor-mation. Anufriev et al. (2013) estimated a demand function and studied the behavior of firms when learning on profit with least squares learning and found that firms end up in a non-unique self sustaining equilibrium close to their Nash-Equilibrium. This paper also used least squares learning by applying recursive least squares (Evans & Honkapohja, 2001) to learn on profit. Where Anufriev et al. (2013) perceived an estimated demand function, this paper estimated a profit function with innovation as variables. The method used in this study found the same results as Anufriev et al. (2013) even tough the functions and models were different. This means that firms in this market choose their innovations close to their Bertrand-Nash equilibrium when applying least squares learning.
Although this study gives insights in the dynamics of process and product innova-tion and in the behavior of firms there are many other interesting factors that could be considered in future research. This research limited its model to process and product innovation but there are other types of innovations that are interesting to consider. It can also be interesting to consider spill overs in marginal costs (Bacchiega et al., 2011) or to integrate preference for product variety in the model (Rosenkranz, 2003). Not only can other researchers expand the model but there are also other learning methods to consider in this market, for example gradient learning (Anufriev et al. 2013).
Acknowledgement
References
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