Heterogeneous innovation with gradient learning
in a vertically dierentiated Cournot market
Olaf Aarts
Student number: 10193839
University of Amsterdam
Thesis Supervisors: Anghel Negriu and Jan Tuinstra
27 June 2014
1 Introduction
The literature about the innovation of rms in Cournot or Bertrand competi-tion has been experimenting with many dierent types of models to nd results that can explain parts of the way these rms compete using innovation. Ca-sual observations reveal that rms normally have a choice between process and product innovations and how much to spent on it (Lambertini and Mantovani, 2010). In their paper it is pointed out that there already has been a lot of dis-cussion about which R&D type is being chosen at what point of the technology life cycle. The traditional approach is that the rms rst do more product in-novation because they have to nd their place with a dominant design, in the high potential market. After this part of R&D competition has been settled, the rm's eorts shift to process innovation because the pro tability of product innovation has then faded. Adner and Levinthal (2001) instead propose a more demand-sided approach in which the great tendency towards product innova-tion during the rst phase depends on the initial performance of the product on the market. Also, the second phase does not necessarily lead to much less product innovation when rms start to invest mostly in process innovation. A possible third stage would be characterized by a balance between those two kinds of innovations. Altogether it is still possible to obtain interesting results about the dynamics of the dierent innovations from inventive setups. What is important in the setup of this research is that on top of analyzing the static equilibrium outcome of previous models, the present work goes further to ana-lyze the dynamic properties of a model where adaptive learning determines the rm's innovation choices, .
In this study, two rms choose the amount that they want to invest in product innovation and process innovation. Product innovation here means that the quality of the product is being improved, and process innovation is a reduction in the variable costs. The products are vertically dierentiated in the
way they were in Bonanno and Haworth (1998), so that when prices are equal, only the rm with the higher quality product enjoys positive demand, see also section 3 for the demand and price functions. Applying Cournot competition (where participants can set their quantities instead of their prices), the rms in the duopoly are optimizing their pro ts by choosing the quantities, qualities and cost reductions of their products. Later, the rms will also be learning about the latter two. The point of using learning dynamics is to study the dynamic properties of the model when rms adapt to an environment of which they form only an approximate and incomplete model, as they do in reality. The learning will be done in the form of gradient learning, so that rms will adjust the variable that they are learning about in the direction in which they are expecting to make more pro t. Because the rms act simultaneously and thus don't know the amounts that the other rm is setting, the rm logically only uses its own functions and values to learn. Gradient learning is therefore a reasonable method in this setting and it has also been applied previously in the literature of oligopolistic markets (for example Anufriev et al., 2013, Arrow and Hurwicz, 1960, Corchon and Mas-Colell, 1996).
In the end the aim is to nd the optimal proportions to invest in product and process innovation to compare them to the dynamics of the innovations. That will be the main result, because those innovations are the rm's novelties and tools to play the Cournot game in a strategic way from several starting points. The rst way of letting the rms choose their optimal innovation amounts is a static one, in which rms have full knowledge about their pro t function. The model will clearly and completely be shown and explained below in section 3, but it is convenient to already have a general understanding of the method used. The model is optimized with a conventional two step system in which. rstly, the innovation variables are optimized for the pro t of each rm, which leads to the innovation amounts expressed in quantities. In step two these quantities are optimized for their pro t functions. These values are optimal in the sense that they have been obtained by rms who have access to all the information.
The rest of the analysis focuses on simulated time-series in which rms will apply gradient learning to optimize their variables again (the quantity, quality and cost reduction variables). They apply gradient learning because in this dynamic part the rms will no longer have full access to the information about their pro t function.
The remainder of this paper is organized as follows. Section 2 deals with the theoretic framework. Section 3 develops the model and the speci c research design. Section 4 contains the analysis and results. Section 5 contains the conclusion.
2 Theoretic Framework
In Bonanno and Haworth (1998) the model contains process and product vation, and nds in a somewhat similar model setup as in this paper which inno-vation is chosen in two dierent kinds of competitions: Bertrand and Cournot.
A result of this study is that the high quality rm in a Cournot competition is more inclined to do process innovation, reducing their variable cost, than the rm in Bertrand competition. Because only Cournot competition is used in this paper, it should be clear that there is a dierence between the behavior of rms in a Bertrand competition and the ones in a Cournot competition. The intuition behind their result is that Cournot competition is less competitive, so that the two rms are less concerned with each other and are more focused on themselves. In that case it seems reasonable for the high quality rm to directly lower their variable costs instead of improving quality which is only pro table through the market mechanism.
A paper that shows shows several similarities with Bonanno and Haworth (1998) is Bacchiega et al. (2011), for instance because it nds amounts of process and product innovation using a cost function with some of the same purposes. Though an important dierence is that this study uses Cournot competition (where Bacchiega et al. (2011) uses Bertrand competition), it diers in more important ways of course: they don't apply gradient learning as they don't use any dynamic time series. Still, their results about the rm's initial e ciency which is in uencing process innovation, which on its turn is in uencing product innovation, are interesting to see.
Besides these two papers, which are thus using models which are solved according to Cournot or Bertrand competition, some other papers were used to obtain useful considerations for the background of this study. An important feature to mention in this kind of studies is the behavior of the players that are participating. A lot has been written in the literature about how rational they should be, like in Ellison (2006), where it was found that there are three kinds of rationality rules that are being used in studies about industrial organization: simple rules-of-thumb for behavior, explicit bounded rationality for optimizers with cognitive costs, and the psychology approach which even cites evidence for their framework. My paper is an explicit bounds paper because the rms use a learning system which is not optimal but they do not look further for a better one. Nonetheless, they are left free in their behavior concerning other choices. This creates opportunities to construct rules-of-thumb about when rms choose one innovation to invest in and when they spread.
This kind of explicit bounds behavior can easily be integrated into an agent-based model. Dawid (2006) explains why these agent-agent-based models are useful in many cases. For this paper it is an obvious model to use, but it is also important that it behaves well for dynamic processes, which it does. Also, a much richer model could be built and Dawid pleads for a model which is applicable to the real world considering the build-up of knowledge, the freedom of choices and the insecurities that rms deal with. In this paper rms learn indirectly about innovation investments and they enjoy relatively much freedom and therefore also deal with more insecurities in the choice process. This is the case because the rms do not try to predict any of the variables. The dynamics (and even the direction) of quantities, qualities and cost reductions are insecure in that sense.
Anufriev et al. (2013) specify a gradient learning model too, and in their pa-per rms learn about the demand function in a Bertrand oligopoly. This papa-per shows that when rms are adapting their behavior by following qualitatively dif-ferent rules, complex market dynamics emerge. These rules were either gradient learning or LS learning, and rms could constantly choose between them to get information about the prices that would lead to higher pro ts. From this study it is important to notice that the gradient learning parameter that in uenced how big the price change per period was, was very important for the success of that learning method. The parameter therefore had a major in uence in which learning method was chosen by the players in their model. Even though their paper uses Bertrand competition and their set up is quite dierent from the one in this paper, verifying the robustness of results to variations of the parameters seems important.
3 Method Design
3.1 The Model
The vertical dierentiation model used here was rst introduced by Mussa and Rosen (1978) and was also used and clearly explained by Bonanno and Haworth (1998). What they use is, in short, a utility function from which they obtain the demand and therewith the price functions. The utility model consist of N customers with the same income E but dierent quality tastes, denoted by the parameter θ which determines how much they value the quality of a product, k. Letting p be the price of a product, this ends up in the utility function E − p + θk. Here also the H for high quality rms and the L for low quality rms are being introduced, meaning that the rm produces products that are of higher (lower) quality than the other rm, so kH > kL > 0. Then, by using pH,
pL, kH and kL in the utility function, two θ0sexpressed in terms of those four
H and L variables can quickly be found because the quality taste θ is uniformly distributed in the interval (0, 1]. Two θ0s are the solutions for the equations
for which a consumer is indierent between consuming nothing and consuming the lower quality product (resulting in θ0), and for which a rm is indierent
between consuming the lower and the higher quality product (resulting in θ1).
This leads to the demand functions of respectively rm H and rm L DH(pH, pL) = (1 − θ1)N = (1 − pH− pL kH− kL )N DL(pH,pL) = (θ1− θ0)N = ( pH− pL kH− kL −pL kL )N.
Next, the production quantities need to be introduced, which are denoted by qHand qL. The next functions will be expressed in terms of these quantities
of course because of the Cournot competition. The inverse demand functions are given by:
PH(qH, qL) = N kH− kHqH− kLqL N PL(qH, qL) = kL(N − qH− qL) N .
Now the vertically dierentiated market has been integrated in the demand functions, which means that the pro tability of product innovation has been too. The pro tability of process innovation is simply the amount of cost reduction bi, where i = H, L. The costs of the two innovations are adopted in the model
in the same way: the change in costs ai and the change in quality bi are being
squared and multiplied by a parameter and then added to the cost function, so Ci= qi(¯ci− ai) + βi(ai)2+ γi(bi)2.
This means that the quality of a product ki= ¯ki+ bi, where ¯ki is the initial
quality and the costs ci = ¯ci− ai, where ¯ci is the initial cost. All the ki and ci
therefore need to be substituted, but for the sake of clearness it is not written down like that in the pro t functions
πH(qH.qL) = qH( N kH− kHqH− kLqL N − cH) − βH(aH) − γH(bH) (1) πL(qH.qL) = qL( kL(N − qH− qL) N − cL) − βL(aL) − γL(bL). (2)
3.2 The optimizing game
From this model a two-stage game is played to compute the static results. In this game rst the optimal process and product innovation amounts for both rms were found simultaneously. In the second stage of the game the optimal quantities of the rms were found in approximately the same way. The compu-tation of the results was done in reversed order, as usual, because the rms are assuming in the rst stage that the quantities that are found in the second stage have optimized their pro ts. This means that rst the quantities are computed in the second stage to substitute them in the functions used in the rst stage. The game is, more exactly, played in this way:
1. The pro t functions of equations (1) and (2) are maximized over the vari-ables that are to be optimized, aiand bi, by taking the partial derivatives
of the pro t function of rm i with respect to ai and bi
δ δai qi(Pi− ci) − βi(ai) − γi(bi) = 0. δ δbi qi(Pi− ci) − βi(ai) − γi(bi) = 0
By substitution the optimal values for the innovation variables of both rms, a∗
i and b∗i, can be found.
2. In this stage maximization over a single variable per rm is done, which is the quantity qi. This is done in the same way, by taking the
deriva-tive of the rm's pro t function, but with respect to qi(a∗i, b∗i), and then
substitution.
3.3 The learning game
The research then turns to the dynamic part where the two rms repeatedly choose their innovation amounts using the gradient learning method and after that choose their quantities. Gradient learning is a learning method in which the derivative of the pro t function with respect to some variable indicates in which direction the value of that variable will be changed. In this paper, gra-dient learning will be applied to the innovation variables for every time period, which could be seen as a replacement for stage one of the static method. The innovation amounts that emerge from gradient learning are found as follows
ai,t+1= min{ai,t+ λC
δπi∗(ai) δai,t , ¯ci,t(1 + ρC) − 2}, (3) bi,t+1= bi,t+ λK δπi∗(bi) δbi,t (4) where bi,t+1 = ¯ki,t(1 − ρK)if this is smaller than the value of (4). These
dierent possibilities are created so that the costs can not become lower than two, and the qualities can not become lower than zero, see for clari cation of the used functions also see function (5) and (6).
These functions are introducing the parameters λC and λK (discussed in
section 2) and the optimal pro t function π∗
i which was maximized in the former
time period. The dynamic time series of this research is found by repeating the following two-stage game:
1. With gradient learning the new process and product innovation values are found for both rms according to formula (3) and (4) to compute new ¯c0
is and new ¯k0is. These are the equations describing technological
accumulation inside the rm for both rms using new parameters ρK and
ρC:
¯
ki,t+1= ¯ki,t(1 − ρK) + bi,t+1, (5)
¯
ci,t+1= ¯ci,t(1 + ρC) − ai,t+1. (6)
Most of the time series stop after 1000 iterations, see also the results of the dynamic model.
2. Having the new initial values (¯kiand ¯ci) and the innovation amounts, the
pro t functions as in formula (1) and (2) are being lled in. From these pro t functions new optimal quantities are computed in the same way as
in the static model, but if the value of a new quantity is less than zero1,
than the value is adjusted to zero. This adjustment is also being made for the prices if they are below zero. Also, if the pro ts are (after the adjustments) negative or zero at this point, than the pro t is adjusted to zero and the production is being stopped (quantity adjusted to zero), besides that, the innovations are set to standard values so that the cost and quality values should normally be retained, that is: ai,t+1= ¯ci,t+1∗ρC
and bi,t+1= ¯ki,t+1∗ ρK . Finally, an augmenting rule is added to gradient
learning, which is supposed to help a rm make positive pro ts again when it has had zero pro ts in the two latest time periods. This is done in the following way2:
qi= 5 + 2aiβi
4 Results and Analysis
This section starts with the results of the optimizing game from the static model, to be used as a benchmark for the results analysis about learning model. Because this rst section gives results that are optimal, those results are useful to get an understanding of how the rms should behave in the market in terms of innovation and quantity amounts, from dierent initial situations; and how they should change that behavior when the situation changes. Therefore an extensive range of starting points and therewith results are obtained and this gives some useful intuitions and observed generalities. An insight in the duopoly conditions.
4.1 The static model results analyzed
Many dierent starting points are used for the parameters: the initial quality ki
and costs ciand the expensiveness of cost innovation βiand product innovation
γi for both rms. The number of consumers N is constant throughout. The
assignment of dierent initial values to the parameters is build up as follows: - Four dierent combinations of initial quality and cost levels: low
quality with low costs, average quality with low costs, average qual-ity with high costs and high qualqual-ity with high costs. These give some of the possible market circumstances and how advanced and expensive the production of the product is.
- For all four of these combinations some dierences in the expensive-ness of innovation are being set in two gradations, so both product
1 This adjustment and the following ones apply per rm.
2This function has been chosen because the derivative of the pro t function with respect
to aiis equal to the quantity minus the last term of the augmenting function, so that ve is
left over as an impulse for the cost innovation that works eectively because it has helped in many cases but it is not too extreme. This rule helped because increasing costs were a main reason why the rms had negative pro ts.
and process innovation can be either 'expensive' or 'cheap'. If a sort of innovation is expensive for one rm, that sort is always expensive for the other one too, which gives four possible combinations here. - Per case, mainly the initial quality of rm H was varying over a range
of circa ten. In some cases the initial quality of rm L was varied over such a range as well. The initial costs hardly were (except for the dierent levels, namely high and low costs, see above) because of their strong in uence on the outcomes, which means that only a little bit of variation of at most ve would be possible before one of the rms didn't make a pro t anymore. This maximum range remains valid throughout the cases that have been studied in this paper, but increases when the level of quality, costs and the expensiveness of innovation increase. This is also true for the variation range of the quality, that is, varying initial quality could be expanded when I turned to the cases of higher initial quality and costs, as will be made clearer in the following paragraphs.
Optimizing the rm's pro ts was done with the parameters as they can be seen in the description of the Figures. It is hardly possible to use lower values than in the cases with low initial parameter values (as discussed), which is convenient because all the other relevant parameter combinations can then be explored by simply systematically increasing these values. Multiplying all of these rel-atively low parameters by for instance three clearly increases the outcomes, in some cases they are (much) more than three times as big, sometimes (much) less, which also changes the ratios between, for example, pro ts and quantities. But the ratios between the outcomes of rm L and rm H per variable type are scarcely dierent, so that most of the upcoming results can be applied for outcomes from much higher starting values too.
As a main result it was found that the innovation variables are normally not smaller than one, while the maxima lie around four, even for starting values that were multiplied with some number. This also means that innovation has relatively less impact when values are increasing because the proportion of the innovations is then much smaller with respect to initial qualities and costs. However, these constant innovation outcomes apply to both rms and both kinds of innovations, so this means the rms never choose for only one kind of innovation, but use both of them to a larger of smaller extent. The only outcomes where a rm sometimes chooses innovation amounts near zero is when its pro ts are very low or just much lower than the pro ts of the other rm, or when innovating is more expensive for that particular innovation type. This last case always shows lower innovation investments than the less expensive case but still holds a great majority of outcomes of ai, bi > 1. Also, in all
of the cases, the optimally chosen innovation amounts of the rms are non-decreasing with the pro ts, see also Figure 1 where kH is varied over a range.
Furthermore, because only the results that gave positive pro ts could be used, it seems more logical that the innovations were never less than or equal to zero. If the innovation outcomes were negative, at least one of the pro ts was always
less than zero. In most cases it is fairly logical that a pro table rm does not choose negative innovations, because there is no advantage in higher variable costs and it certainly seems from the model that lower quality leads to lower pro ts when the other rm's product quality remains the same.
Nonetheless it was clear why the pro ts of a rm became less than zero. Both rms simply have their own advantage: rm L has lower costs and rm H has higher quality by its de nition, and when the rm's initial advantage is too small with respect to the other rm's advantage, it is too disadvantaged and has no pro t. This is true for instance for equal costs due to vertical dierentiation, but normally it occurs sooner, already when kH is getting too
close to kL or when it makes rm H's advantage is too big for rm L to get a
relatively well-proportioned pro t, if any at all. This varies a bit by case but is observed in every series of outcomes, so also in the case that Figure 1 describes: as kH increases the pro ts of rm L go to zero, but when it makes the quality
dierence small enough the pro ts of rm H become zero or even negative. Going through the four dierent combinations of initial quality and cost levels, several other observations can be made. To start with the low quality and low costs case, it has the lowest pro ts of the four, intuitively because there are smaller margins to adjust the variables in, the price for instance. Certainly for rm L the prices are much lower than in the other cases where initial quality and costs are higher; for example when dierent levels for quality and costs are being compared with each other for the same less expensive innovation case βi = γi = 1: the price ranges here from four to ve approximately, while the
prices in the cases with higher quality and costs range from six to eleven. In general it is clear that the higher these initial amounts are, the higher the prices are. The quantities don't show a trend like that and mostly range between the same values in all cases (from approximately three to eight). But that there is less room for variation can also be seen in Figure 2, where for each innovation variable and pro t function there are two lines: one that starts at kH= 13(for
this line cH = 7) and one that starts at kH = 15 (for this line cH = 9). This
setting is used mainly because it shows that giving rm L a bigger advantage by increasing rm H's initial costs is making rm L more competitive at the higher values of kH. Furthermore the graphs are very much alike and only seem
to be horizontally shifted, which may imply a fairly constant and predictable competition when the advantage of a rm is simply increased in this optimizing game. Although this is not true for the price graph of rm H, which is obviously starting at a higher value -but with the same slope- when its costs are higher.
In the case of average initial quality and low initial costs, a much more favorable environment can be observed because these rms would then be more e cient by producing higher quality products at the same initial costs as in the previous case. Certainly for rm L the pro ts have increased a lot and almost tripled. The high quality rm simply has a higher pro t of approximately ten, see also Figure 33, so its pro t isn't multiplied which is logical because its
3Note that this is not a completely fair comparison to Figure 2 which has γi= 1instead of
Figure 1: For average initial qualities and high initial costs
Innovation amounts, pro ts and the equilibrium prices with market shares from varying kH in the
most typical market with average initial qualities and high initial costs. Parameter values: N = 20, kL= 15, cH= 10, cL= 7, βi= 1, γi= 1.
pro t function is very close to a straight line with almost the same slope in all the cases in this static model. Besides that, rm L also constantly has the same typical convex pro t function so that it retains its higher pro t for much
Figure 2: For low initial qualities and low initial costs
Innovation amounts, pro ts and the equilibrium prices with market shares from varying kH in a
market with low initial qualities and low initial costs. Parameter values: N = 20, kL= 10, cL= 5,
βi= 2, γi= 1and with cH= 7for all the graphs starting at kH= 13and cH= 9for all the graphs
starting at kH= 15.
bigger dierences in initial quality. This all comes together in the surprising observation that rm H very constantly -for all βi, γi = 1, 2- starts having a
higher pro t than rm L when the the dierence in initial quality between the two rms exceeds six and a half. This is even true at βi = 1 and γi = 2
when rm H only starts to make a pro t when the initial quality dierence is exceeding four, while at the other three combinations of β and γ rm H already starts to make a pro t when that dierence is around two and a half. Moreover,
the observation that rm H constantly starts having a higher pro t when the quality dierence exceeds one certain value can also be made at the two cases where the initial costs are higher, see table 1.
Table 1: The approximate dierences in initial quality at which πH> πL
As kHis being increased, rm H is earning higher pro ts and rm L is earning lower pro ts. At the
level of kH where these pro t functions intersect, a certain dierence between the initial qualities
is reached and this dierence is shown in the table. Parameter values: N = 20, cH= 9and cL= 5
in the low costs case and cH= 10and cL= 7in the low costs case.
In the last case that will be discussed, the initial quality and initial costs are high which seems to give bigger margins to adjust the variables in, opposite to the low quality and cost case of course. This is intuitively reasonable because changing a parameter has proportionally less in uence when larger numbers are used, which can directly be observed by the fact that in this case the pro ts don't become negative unless the rm's cost or quality advantage is almost zero. It is also shown through the slightly higher innovation outcomes where, as in most of the considered cases, the process innovation of rm L is on average the highest which makes sense again because generally rm L innovates more and both use more process innovation than product innovation. For these dierences it hardly seems to matter how high the initial values of quality and costs are, which is not as expected by Bonanno and Haworth (1998) because a result of them is that the more intense a competition is, the less the high-quality rm will opt for process innovation. The competition becomes more intense as the pro ts are lower as in the case with low initial costs and qualities, but if the innovation amounts of rm H changed relatively to rm L's when the competition was more intense, then it was rm H that innovated relatively more in both types of innovation in the more competitive case. Otherwise, the rms do opt more for process innovation in general and they innovate more in less competitive cases, which is in line with their results because the Cournot market is not the most competitive one that can be chosen. Anyway, comparing the results might be problematic because the innovations can be chosen simultaneously instead of binary and the competitiveness may not be very dierent per case (of dierent quality and costs levels) in this model.
What clearly matters at every level is the amount of the pro t; or in other words, the bigger a rm's initial advantage relative to the other one's advan-tage is, the bigger the innovation amounts. This can be explained by arguing that higher pro ts give a rm the opportunity (the money) to improve itself
Figure 3: For average initial qualities and low initial costs
Innovation amounts, pro ts and the equilibrium prices with market shares from varying kH in a
market with average initial qualities and low initial costs. In comparison to Figure 2, not only have the initial qualities changed, the expensiveness of innovation has too. Parameter values: N = 20, kL= 15, cH= 9, cL= 5, βi= 2, γi= 2.
and therewith its pro ts, so with a negative relation between competition and innovation as defended by Schumpeter (1943). On the other hand it seems im-portant to improve your disadvantageous initial costs and quality to be more competitive and earn higher pro ts. In this model the rst argument clearly -and constantly- weighs heavier.
Besides these factors, the expensiveness of innovating has an important in- uence on the innovation amounts, as is to be expected. When βi = 2process
innovation amounts are lower and when γi= 2product innovation amounts are
lower than when they equal one. When only one of the two sorts of parameters is equal to two (for both rms), then the innovation type that is less expensive does not bene t by getting higher values, see for instance Figure 4 where aistays
almost exactly the same although the product innovation has become more ex-pensive. Moreover, the price equilibria and the quantities are very similar in both cases as well, suggesting that the rms behave almost the same but with less innovation of the expensive type and altogether with a lower pro t. These similar equilibria have also been observed at the other levels of initial quality and costs, as has the slight increase in the price of rm L and the slight decrease in the quantity of rm H in cases where the innovation expenses are speci ed oppositely and βi = 2 instead of γi = 2. So when the process innovation gets
more expensive, both rms respond in a clearer way with their equilibria than when product innovation becomes expensive. But the innovation amounts are still only adjusted with a relatively small value and the other, less expensive innovation type still doesn't increase.
The analysis of the outcomes of this static model altogether gave a lot of results that should be helpful to understand why rms are acting as they do in the dynamic model. Also keeping these results in mind when looking for relevant starting values for the variables and parameters in that model is important.
4.2 The dynamic model results analyzed
Also for this part of the analysis, results were obtained from many dierent starting points. However, this was not done in the same systematic way as in the previous part because, except for the three parameters γH,γL and ρK, all
of the variables and parameters did not have an in uence on the outcome of the simulations in which both rms made a pro t. This is clearly very dierent from the static model, where the optimal outcomes were sensitive for every ad-justment in the initial values of qualities and costs. In this dynamic model, also initial values for quantities and innovation amounts can be adjusted, but this has no in uence as well, just like the parameters of the process innovation. To be sure of this result, I rst varied all of these initial variables4and parameters
over a range from near zero to two times their standard value keeping all other values constant, then I did this again but with more variables that were being adjusted per simulation. After trying many combinations of that, also some
4For clarity, the initial variables and parameters are: ci, ki, qi, ai, bi, βi, γi, λC, λK, ρC
Figure 4: Innovation at dierent expenses for high initial qualities and high initial costs
Innovation amounts from varying kH in a market with high initial qualities and high initial costs.
Parameter values: N = 20, kL= 20, cH= 10, cL= 7and two dierent combinations of innovation
initial settings have been used with many variables having completely dierent ratios relative to each other than originally chosen or used in the static model. In the cases with positive pro ts the values were very steadily converged after 1000 iterations, but not in the cases where at least one rm had zero pro ts. In any case, when the augmenting rule is disabled, a rm that has zero pro ts at some point does not get positive pro ts in that simulation again, which surprisingly means that in most cases the other rm has zero pro ts in the end as well, instead of taking advantage of it. This seems to be because of the costs that keep increasing exponentially for both rms when one rm has zero pro ts. This is strange behavior, but it is the way the model works when the quantities are zero and therewith the derivative of the pro t function with respect to aibecomes and stays negative. Those cases do not present reality in
a good way in this model, but when the augmenting rule is enabled, at least one rm normally5gets a positive pro t again and the variables do not only increase
or decrease during the rest of the simulation (as the costs did). When one rm gets positive pro ts again a number of variables keep uctuating because of the impulse that the quantity keeps getting. The rm that doesn't make any pro ts has uctuations just above and at zero, and the other rm is in uenced by it as well and uctuates a relatively little bit in the positive values, see also Figure 5, where rst rm L has a positive pro t and then rm H, because of the initial advantage in quality, which is being varied. This is also the only way in which qi and most of the other variables (except for the aforementioned
exceptions) can have any in uence on the outcome of the simulation, so they can make the simulation fail because these are less realistic simulations where the augmenting rule had to intervene but where still only one rm has a positive pro t. Therefore the simulations with zero pro ts will not be researched any more than this, although it is important to remember that a zero pro t result can be because of initial values of the variables.
Continuing with the dynamics of the simulation, it was as expected that the dynamics are in uenced by all initial variables and parameters. When the simulation was stopped after 100 iterations, the variables were still converging in most cases. Doing such simulations for a range of dierent initial values of, for example, kH, most of the outcomes were already in the same range as they
would be after 1000 iterations, but they did not show a smooth connection, as can for example be seen in Figure 6. This discontinuity means that obtaining unambiguous observations from the dynamics is harder, but in general it be-came clear that in a smooth simulation without any points with zero pro ts variables were moving only in one direction during a simulation with hardly any uctuations. Only during the rst 50 time periods the lines of the two rms uctuate and alternate for the highest value. Once one rm, which is always rm L when both pro ts are positive, is too disadvantaged, all the lines become more and more distanced from each other and converge to their end value. It is also possible that one rm is so disadvantaged that its pro ts become zero,
5Normally here means that the initial values lie within a certain range of around three
times as big to three times as small as the values that may be seen as normal from the static model
Figure 5: The eect of varying the initial quality of rm H on zeroing pro ts
The upper graph shows how dierent initial values of kH in uence if a rm earns pro ts after
1000 iterations and that this eects for instance the price. The lower time series show what a zero pro t simulation can look like and how the augmenting rule gives rm L positive pro ts but also keeps uctuating qH.Initial parameter values: N = 20, kH = 2(varied in the upper graph),
kL = 15,cH = 8, cL = 7, qH = 8, qL = 7, ai = 1.5, bi = 2, βi = 1, γH = 0.8, γL = 0.6,
λK= λC= 0.1, ρK= 0.15, ρC= 0.2.
but it is then being helped by the augmenting rule to make the lines uctuate a little again after which both rms nd their same end values anyway. This is similar to the static model in the way that a rm can be disadvantaged because its quality is too low while its costs are too high relatively to the other rm. But the dynamics are clearer in its results because the pro ts of rm H are always at least two times as big, if they both make a pro t.
Now that a big part of the dynamics and reasons for the kind of results that were found have been discussed, the values of the outcomes themselves can be further discussed. Firstly it is important to nd good initial values for the parameters that have an in uence on the results: γH,γL and ρK. These set how
expensive product innovation is and how much the product quality is decreased every time period. The fact that these parameters have an in uence on the outcomes, even though many initial variables don't, is not surprising because the model can not adjust its values by de nition, but it can and does for the variables so that their values, along with the pro ts and prices, are the same in the end too. Also the reason for the fact that the betas and the rho and lambda of the process innovation have no in uence is clear, namely that the costs always end in the minimum value that they can become, which is why the model has been set up in the way that ci≥ 2is always true (see section three). This results
in a0
is that are always the same as well, which gives the parameters that are
Figure 6: Quality, pro ts and prices with no smooth connection after 100 iterations
Graph showing that quality, pro ts and prices of both rms follow very speci c dynamics, because after 100 iterations one can not observe a smooth change from varying, in this case, kH.Initial
parameter values: N = 20, kL= 15,cH = 8, cL= 7, qH = 8, qL= 7, ai = 1.5, bi= 2, βi= 1,
γH= 1.5, γL= 1, λK= λC= 0.1, ρK= 0.1, ρC= 0.2.
ci= 2are always reached within 30 time periods for all the initial values that I
have tried: the costs either dropped dramatically at some point until they got to two or, with the augmenting rule disabled and if pro ts became zero, they kept increasing exponentially. Finally, λK having no in uence is harder to explain
because the model can not adjust its values and it is not made super uous like the process innovation parameters. Just like the variables, it only has in uence on whether pro ts of one or both rms become zero, so it can make a successful simulation result impossible. But unlike the variables, it does not even seem to have an in uence on the dynamics when both pro ts stay positive. Letting λK vary over a range, hardly any dierence can be observed between even the
starting values of most variables, see also Figure 7 where the dynamics were very similar until λK = 1.05and suddenly the dynamics uctuate rapidly. The
larger the values of the lambda, the smaller the chance that both rms make a pro t. I call it a chance because a whether or not having a zero pro t result is also very dependent of values of all the other variables and parameters.
The range of values for which the three 'in uencing parameters' give positive pro ts for both rms at the end of the simulations is relatively small, because the gammas can not become two (like in the static model) and the product innovation rho can not become 0.5 without making the pro t(s) zero. But the nearer their values come to zero, the higher the pro ts become, which is very logical because these parameters represent negative factors for a rm. Nonethe-less, how soon pro ts become zero is also strongly in uenced by starting values
Figure 7: Dynamics for slightly dierent λK
Time series in the upper panel converge smoothly to positive pro t results with λK= 0.95and
does this very similar for lower values too, where the time series in the lower panel show how a small change to λK= 1.05can be of great in uence for the dynamics and results. Initial parameter
values: N = 20, kH = 20,kL= 15, cH = 8, cL= 7, qH = 8, qL= 7, ai= 1.5, bi = 2, βi= 1,
of other variables and parameters, as discussed earlier, but to be consistent with the static model, I chose the values for those variables and parameters similar to the average levels in the static model. Also, I want to prevent zero pro ts and pro ts that are more than twice as high than normally in the static model. In that way a somewhat fair comparison could be made, although some clear dierences will be discussed that seem to make a comparison fairly arti cial, like rm H always having a higher pro t than rm L in the cases that they both make a positive pro t. Preventing disproportionately high pro ts therefore only has to be done for rm H; also to let both rms stay somewhat competitive, which is more interesting for this research. Giving rm L an advantage can only be done though by making its production innovations less expensive than rm H's. Naturally, giving both rm the same values for gamma gives rm L zero pro ts in much more cases, and ρK would have to be a relatively low
value. Moreover, the outcomes of the costs are the same for both rms (always two), but rm H still retains its higher quality advantage. In Table 8 dierent ratios between the gammas can be observed, and for the settings γL= 0.6and
ρK = 0.1the minimal gamma ratio γL/γHbetween the two is 0.6, before pro ts
of rm H become zero because it is too disadvantaged. If ρK = 0.2, rm L
would have zero pro ts up to γK = 0.9 and rm H would have zero pro ts for
higher values of that gamma, so this rho is obviously too radically high in this setting. Also higher values for the gammas, for instance γL = 1 , give hardly
any positive pro ts for both rms: only for γH = 1.6both rms make pro ts,
so with the gamma ratio just below 0.6. This ratio is therefore an eective one, because it gives rm L a maximum advantage while it gives positive pro ts in many cases that have been tried out as well.
Table 2: Changing the gamma ratio by varying γK
For a xed value of γL= 0.6, the gamma ratio changes when γHis being varied. Initial parameter
values: N = 20, kH = 20, kL = 15, cH = 8, cL = 7, qH = 8, qL = 7, ai = 1.5, bi = 2,
βi= 1,γL= 0.6, λK= λC= 0.1, ρK= 0.1, ρC = 0.2.
To nd a range of product innovation rhos that are eective for the simula-tion as well, I have, for dierent values of ρK, varied the gammas over a range
while keeping the gamma ratio constant: γL/γH = 0.6, see Table 9. Here it
can be seen that -for the normal initial settings as discussed before- for bigger rhos, the pro ts become zero for very low values of gamma already, and for smaller rhos the pro ts for the most common gammas are disproportionately high. Moreover, the pro ts increase considerably when the rhos decrease, but the ratio between the pro ts of the rms stays approximately the same for both dierent values of rho and dierent values of the gammas: rm H's pro ts are just more than three times as high. So the only way this pro t ratio can be really changed is by adjusting the ratio between the gammas which could be observed in Table 8. The quantities are very steady as well, for almost all con-siderable combinations of the in uencing parameters, even for dierent gamma ratios. Oppositely, the prices are changing a lot with the dierent values of the gammas and the rhos, but the prices don't respond as much to the changes in the gamma ratios in Table 8, although they still respond more there than the quantities do. This is to some extent similar to the static results, where the quantities were never bigger than nine, while the prices lay in a bigger range of values. This could be because the quantities are being optimized and are more costs related than the prices, which are more product quality related in the pro t function. It can be observed in the derivatives of the pro t function as well, where the a0
isare very dependent on the quantities and the b0isare very
dependent on a function that looks similar to the price function. Furthermore, it is notable that these b0
is are just like the prices much more exible when
the values of the gammas are varying than when the gamma ratio is varying, although they are not exible at all when the product innovation rho is varying and the gammas stay exactly the same, see Table 9.
Comparing this dynamic model with the static model, I have observed im-portant dierences like the bigger pro ts for rm H in the simulations, both relative to rm L and relative to rm H in the static model. Most importantly, the static model is clearly much more sensitive for changes in the initial val-ues: all the variables that have been used in the static model did not have an in uence on the outcomes in the dynamic model. The mechanism that a rm innovates more when its pro ts are high is always stronger than the one that rms innovate more when innovating is less expensive, because in none of the considered cases does rm L innovate more in product innovation than rm H, while it was always less (or equally) expensive for rm L. In between a big dierence and a similarity is the fact that the costs always got to the minimum possible value: process innovation has been used more in the optimizing model as well, but did not go to zero by default. The dierence is obviously that in the dynamic model the rm gets much more chances to innovate and can therefore decrease its costs every time period. Letting the costs decrease more every time period had, as stated, no eect until the pro ts became zero. Changing the way that rms pay for innovating, like making process innovation more expensive as the costs become lower, was not according to the static model; and it was not possible to complicate the static model. But many other similarities could be observed for the most eective values of the in uencing parameters. Firstly, the range of values were very similar for the quantities, product innovation amounts,
Table 3: Varying γK for the same gamma ratio and three dierent ρ0Ks
For the upper left table ρK = 0.125, for the upper right table ρK= 0.100and for the lower left
table ρK= 0.075. In every table γL= 0.6γH,where γHis being varied. Gives an overview of the
outcomes of all the variables for several combinations of parameter values, and for which of these combinations positive pro ts can be achieved. Initial parameter values: N = 20, kH= 20,kL= 15,
prices and to a lesser extent also the qualities. For the costs this is not the case of course, but the contrary is not true either, because the costs and process innovation amounts are simply not comparable. Secondly, the rms always in-novated in both types of innovation with positive innovation amounts and they are mostly higher than one when both rm's pro ts are positive. Thirdly, vary-ing one variable (or two variables with the same ratio like with the gammas), the pro ts were always either only non-decreasing or only non-increasing, along with the other outcomes: the qualities, quantities, prices and product innovation amounts of both rms increased and decreased when the pro ts did. Fourthly, the very logical mechanism that when a rm is too disadvantaged, it doesn't make any pro ts. When one aspect is the same for both rms but the other aspect is much more advantageous for one of the rms, the other rm hardly ever made a pro t. Fifthly, more expensive product innovation made the values of bi lower, although this was probably more an eect of the decreasing pro ts.
Lastly, from the dierent levels of initial costs and quality in the static model it became clear that rms easier made a pro t (for bigger ranges of other initial variables) and this is also true for the dynamic model. Initializing higher quality (or costs) while keeping the quality ratio between the rms equal gave positive pro ts for both rms for a larger range of values of the gammas, see for instance Table 4.
Table 4: Varying kH with the quality ratio between the rms equal
For higher values of the initial qualities both rms make positive pro ts. The ratio between the initial qualities is kH/kL = 1.10. Initial parameter values: N = 20, cH = 8, cL = 7, qH = 8,
5 Conclusion
The main answer that this paper gives about innovation amounts in a dier-entiated Cournot market setting is that rms should always innovate according to the optimal results, and that gradient learning is a good model to show that particular result with simulations. The rst note that should be made for this conclusion is that the process innovation can and should be modeled in a dif-ferent way if the research has the capacity to do that, because the dynamics for the costs and the improvement of the costs is hard to compare and judge in this way.
It has become clear though with gradient learning and from optimal results that a rm in this setting rather innovates in its process than in its product when both types of innovations are similarly valued in the market. The proportions in which they innovate are surprisingly inconsistent, certainly in the optimal method, because the innovation amounts were did not come near zero and did not increase proportionately with the pro ts, but stayed within a range of values that never exceeded a certain maximum, even for much higher initial qualities and costs. The innovation amounts normally increase if the pro ts increase due to an adjustments to an initial variable, and with gradient learning this relation is more sustainable than it should be because then its ratio (between the pro ts and the product innovation) stays approximately the same.
Nonetheless, a lot of observations about how rms innovate under dierent circumstances were hard to make in the non-sensitive model with gradient learn-ing, but could successfully be made in the static model. This means that there is no (robust) comparison to a dynamic model, but a lot can be learned from the observations themselves. How rms in a duopoly like this behave and how the mechanisms work in it has been researched in this paper as well and can be used to draw more conclusions than just about the innovations, which may be helpful for future researches about this sort of models. Many observations form conclusions for themselves, but generally speaking what is most noticeable is that the simulation results hardly responded to changes in the initial values, whereas most of these results were still very comparable to the optimal ones under certain conditions for the three gradient learning parameters that were the only ones to have an in uence on the simulation results. These were 'how expensive product innovation is per rm' and 'how much the product quality decreased every time period'. Many observations made from the static model could be made for the dynamic model as well. This is all fairly surprising be-cause of the simulation's insensitivity and it makes gradient learning a better method for this model than one would say principally by looking at the partially extreme results.
Still, the method may be to practical and imprecise to nd results and therewith propositions that elaborate on the more speci c issues like why rms behave as they do. Altogether should these conclusions and observations be a guidance and a comparison with those issues and the general mechanisms of heterogeneous innovations in this setting has become more clear.
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