Prices versus Quantities in a Mixed Oligopoly
28-10-2014
by Rob Goedhart*
Supervisor:
Prof. dr. J. Tuinstra
* Student at the Faculty of Economic and Business at the University of Amsterdam Contact: Robgoedhart93@gmail.com
Master thesis Econometrics, specialization Mathematical Economics Student ID: 10069984
Prices versus Quantities in a Mixed Oligopoly
by Rob Goedhart*
Supervisor:
Prof. dr. J. Tuinstra
Abstract
A mixed oligopoly is an oligopoly where firms face different objective functions. Examples of mixed oligopolies in general are education, health care and banking. This research investigates a mixed oligopoly with one part of the firms using prices as strategic variables, and the other part using quantities. In the current literature of mixed oligopolies, originating from Singh and Vives (1984), it is suggested quantities as strategic variables dominate prices as strategic variables when goods are substitutes (and vice versa for complements). However, the dynamics of pure quantity-setting competition (Cournot competition) can become unstable as the number of firms passes a certain threshold. Using an evolutionary model where firms can switch between the setting and the price-setting strategy we find that, although the quantity-setting strategy is more profitable inside equilibrium, this instability is sufficient to ensure that the price-setting strategy is never completely driven out.
1
Introduction
The current literature on oligopolies revolves mainly around two names: Cournot (1838) and Bertrand (1883). These two forms of competition are both commonly acknowledged, but lead to substantially different predictions. Cournot competition assumes firms use quantities as strategic variable, whereas Bertrand competition assumes firms use prices as strategic variable. Even when considering the exact same market structure, these types of competition lead to different outcomes. An important question therefore is which type of competition gives a better description of market outcomes.
Singh and Vives (1984) investigate a differentiated duopoly where firms can choose whether to use price or quantity as strategic variable. They find that competing in quantities (prices) is a dominant strategy when goods are substitutes (complements). These findings are backed by other research as well, for example Häckner (2000). Other research finds that dominance is strongly dependent on the degree of substitutability between goods (Sakai et al. (1995)), as well as qualitative differences (Matsumoto and Szidarovsky (working paper [1])) . In all this research a two-stage model is considered, where in the first stage firms choose their strategy, and in the second stage they
determine the value of their strategic variable. At first we assume that in the second stage the Nash equilibrium of the corresponding subgame is played, but later on we relax this assumption.
An important finding by Theocharis (1960) is that Cournot-Nash equilibrium becomes unstable under best-response dynamics as the number of firms gets larger. This instability will typically decrease profits and may make the quantity-setting strategy less profitable than the price-setting strategy in the same circumstances. It is unclear whether this instability is indeed large enough to prevent the price-setting strategy from being completely driven out by the quantity-setting strategy. To study this we consider a mixed oligopoly with an arbitrary number of firms. At first we consider two model setups, which differ in the information given to the firm about the other firms. The first model assumes that firms enter the same market each period, so that each firm is aware of the number of price-setters and quantity-setters in their market in every period. In the second model we assume that each period new markets are drawn from a very large population. A certain fraction of this population uses prices as strategic variable, and the other part uses quantities. This structure is similar to
Hommes et al. (2013). We use the first model as a comparison for the second model, in terms of prices, quantities and profits in equilibrium.
Our demand structure builds on the work of Singh and Vives as well as Häckner and Matsumoto and Szidarovsky. In this research we consider only horizontal product differentiation, which means that goods are not of different quality. We also consider goods to be substitutes. Each firm optimizes it's expected profits based on naive
expectations of the other firms' prices and quantities. Since each price-setter (quantity-setter) faces the same optimization problem, we assume a quasi-symmetric system such that each price-setter (quantity-setter) sets the same price (quantity).
After evaluating and comparing the models we perform a stability analysis on the second model, to investigate when the system becomes unstable. This occurs when the fraction of quantity-setters becomes large. In order to investigate the consequences of this instability, we introduce an evolutionary model where the number of price-setters and quantity-setters changes over time, depending on the performance of both
strategies. For this evolution we again consider two different models as a robustness check. Both models yield similar findings.
We find that in our model the quantity-setting strategy is not able to completely drive out the price-setting strategy, even though it is always more profitable in
equilibrium. The instability of the system when the number of quantity-setters is too high is sufficiently large to let the price-setting strategy remain as profitable choice. The dynamics that occur are mostly cyclical switching, similar to the dynamics found by Anufriev et al. (2013), although they consider a different setup. We also find that for smaller markets stability is strongly dependent on the degree of substitutability, but for larger markets this dependence fades.
This thesis is structured as follows. In the next section the baseline model is explained, with information on the market demand structure. Section 3 contains
information about the models and their equilibria, and makes a comparison between the two in terms of prices, quantities and profits in equilibrium. In section 4 a stability analysis is performed, followed by the evolutionary model in section 5. Finally, in section 6 the results are summarized and some concluding remarks are made.
2
The Market Game
In this section we discuss the baseline model setup and the market structure.
Analogously to earlier research (e.g. Singh and Vives (1984)) we consider the market game as a two-stage game, where in the first stage each firm chooses its strategy (price-setting or quantity-(price-setting) and in the second stage they choose the actual value of their strategic variable. We consider two versions of this model, which differ in the
information after the first stage. In the first model each firm knows the decision of the other firms at the first stage before deciding in the second stage, while in the second model this is not the case. We assume that in the second stage the Nash-equilibrium is played. In order to determine this equilibrium we start with defining our market demand structure.
We use a demand structure based on the one used by Häckner (2000) and Matsumoto and Szidarovszky [1]. We consider an oligopoly that consists of n firms. Häckner uses a general utility function for a representative consumer to allow for n firms producing one product variety each. This utility function is
where denotes the quality of good k and denotes the quantity consumed of good k. The parameter I gives the expenditures on a composite good with price equal to one, that represents all commodities not being considered here. Here represents the degree of substitutability between the goods. We consider the case where goods are imperfect substitutes, The representative consumer maximizes this utility function subject to the budget constraint where m denotes the income and denotes the price of good i. The optimal consumption bundle is then determined by the first-order conditions
The Hessian of (2.1) is a matrix with each diagonal element equal to -1, and each off-diagonal element equal to . It can be shown that this Hessian is negative definite for
(2.1)
, so that the first-order conditions indeed lead to a maximum.1 Solving (2.2) for we find
In our model we focus only on horizontal product differentiation, meaning that the goods are not of different qualities. This means that is the same for each good k. We choose , so that we can write our inverse demand functions more compactly in vector notation as
where p is the vector of prices, q is the vector of quantities, is an
vector of ones, and is an matrix with all diagonal elements equal to one, and all
the off-diagonal elements equal to Note that for the matrix is invertible. Rewriting (2.4) then gives us the direct demand functions
Now suppose that in the first stage, from the n firms in the oligopoly, k firms choose the price-setting strategy and n-k firms choose the quantity-setting strategy. Note that k = 0 (k = n) corresponds to Cournot (Bertrand) competition. First we denote the vectors of prices and quantities of price-setters by and respectively,
and the vectors of prices and quantities of quantity-setters by and respectively. Without loss of generality, assume from now on that the first k firms are price-setters and the last n-k firms are quantity-setters. This means that the first k firms will determine their prices ( while the last n-k firms will determine their quantities . If we define
which is a matrix with each element equal to then we can write the model in terms of the strategic variables more compactly.
1 We have checked it analytically for , and numerically for n = 4, 5,..., 200 and .
(2.3)
(2.4)
(2.5)
Lemma 2.1. The model in terms of prices of the price-setters, and
quantities of the quantity-setters can be written as
where Proof of Lemma 2.1
We want to express the models in terms of the strategic variables and . Equation (2.4) can be written as follows
Now from (2.7) we find that
which can be rewritten as (again using the fact that is invertible for )
Similarly we have
Substituting (2.9) into (2.10), we obtain
(2.7) (2.8) (2.9) (2.10) (2.11)
Given this model it is possible to calculate the profits for the firms. We assume zero production costs, so that the resulting profits simply equal the revenue.
3
Equilibrium
In this section we discuss the equilibrium of two different model setups. At first the situation where k is known to the firms, and second the situation where firms only know the fraction ρ of the population that is a price-setter. After the description of the
equilibria, the equilibria of the two model setups are compared.
3.1 Firms Know the Number of Price-setters
The first model is a two-shot game of a market of n firms with perfect information. At the time of decision making in stage two, each firm is aware of the choices of the other firms in stage one. Thus, each firm knows exactly how many price-setters and how many quantity-setters are participating in the market. Firms also understand the demand structure, and know the values of n and . We assume that firms make a decision based on naive best-responses, which means that each firm best-responds to the quantities and prices set by the other firms. Since the decision problem for each quantity-setter (price-setter) is the same by symmetry, we assume that every quantity-setter (price-(price-setter) sets the same quantity (price). Such a system is called a quasi-symmetric system.
Using the best responses in combination with the market demand structure, we find the Nash equilibrium in the second stage as function of n, and .
Lemma 3.1. The Nash equilibrium prices and quantities of price-setters
and quantity-setters respectively, as function of n, and , equal
and the resulting equilibrium profits, equal Proof of Lemma 3.1
In the optimization process, each firms maximizes profits by best-responding to the choices of strategic variables of the other firms in the market. Without loss of generality we evaluate the first price-setting firm in order to determine this best-response function.2 Using the
results of section 2.1 we can write the objective function (profits) of this first price-setter as
where and represent the i'th element of and respectively.
Note that is in this case a function of k, since the resulting
quantity depends on the number of price-setters (and on the prices and quantities of other firms). Maximizing this objective function with respect to yields the following best-response function for the first price-setter
2
We consider a quasi-symmetric system, thus each price-setter faces the same optimization problem. Similar for each quantity-setter.
where the final equality holds under the assumption of quasi-symmetry. Similarly for the first quantity-setter we have the objective function
where and represent the i'th element of and respectively. In
this case not the quantity, but the resulting price is a function of k. Maximizing this objective function over gives us the following best-response function for the first quantity setter
where again the final equality holds under the assumption of quasi-symmetry. In equilibrium we then must have and . Solving (3.2) and (3.4) for these two variables leads to the required result.
Since we want to examine whether using quantities as strategic variables is always more profitable, we have to evaluate whether it is always more profitable to switch from the
(3.2)
(3.3)
price-setting to the quantity-setting strategy. This is the decision that is made in the first stage of the game. To evaluate this we compare the equilibrium profits in the second stage of a firm in a market with an arbitrary number of k-1 other price-setters (of the other n-1 firms, k-1 are price-setters). The firm considered is either a quantity-setter, or the k-th price-setter in that market. Note however that these two possibilities do not constitute the same market equilibrium. The conjecture that suggests that it is always more profitable to be a quantity-setter, given that the market is in equilibrium, is then denoted as
Conjecture 3.1. The quantity-setting strategy yields higher profits than
the price-setting strategy.
for all n, and k. This represents the conjecture that in the two-stage game as described in this section, it is always more profitable to choose the quantity-setting strategy in the first stage.
Numerical simulations suggest that this conjecture holds, as the inequality is true for all possible combinations of n = 1,2,..,100, k = 1,2,..,n and 0.001, 0.002,.., 0.999.
This result has important implications for the two-stage game that we consider. It implies that in the first stage each firm will choose the quantity-setting strategy. In that case, in the second stage the Cournot equilibrium will be the outcome. Note that this result is not dependent on the actual degree of substitutability between the goods. Although a higher degree of substitutability leads to a larger difference in equilibrium profits between the two strategies, it is for the values considered always more profitable to choose the quantity-setting strategy in the first stage. As an illustration, Figure 1 illustrates the conjecture by showing both functions in one graph for two different degrees of substitutability. Next to the conjecture, it also shows that a lower number of price-setters is actually profitable for both the price-setters and the quantity-setters, as their profits are decreasing in the number of price-setters k. We also observe that the difference in profits between the two strategies is larger when the number of price-setters is lower.
The conjecture is in agreement with earlier research (e.g. Singh and Vives (1984), Tanaka (2001)) that suggest that the quantity-setting strategy is more profitable than
Figure 1
(a) (b)
Graphs showing the profit functions (green line) and
for a fixed value of n = 20. The horizontal axes shows k, and the vertical axes shows the profits. The degree of substitutability is equal to (a) 0.25 or (b) 0.75.
the price-setting strategy. Note that this results hold only under the rational assumption that in the second stage always the Nash equilibrium is played. However, the assumption of perfectly rational agents has been put into question by recent literature (originating from Simon (1957)). In earlier research on oligopolies we have also seen that Cournot competition (competition where all firms are quantity-setters) under naive expectations can become unstable as the number of firms n in the market becomes larger ((Theocaris (1960)). In section 4 and 5 we will therefore discuss how this instability influences the results of this section in a dynamic model which does not assume that in the second stage the Nash equilibrium is always played.
3.2 Firms Know the Population Parameter ρ
Since we are looking to introduce an evolutionary model that allows firms to switch between strategies over time, we consider a second model. This model uses a structure similar to Hommes et al. (2013). We assume that there is a very large population of firms where a fraction ρ of the firms uses prices as strategic variables, and a fraction 1- ρ uses quantities. We consider a two-shot game where in the first period many markets of n firms are randomly drawn from this population. Each period new random draws will be made to match firms in markets of size n, and thus the firms are not aware of the exact composition of their market when it comes to number of quantity- or price-setters.
2 4 6 8 10 12 14 16 18 20 0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 2 4 6 8 10 12 14 16 18 20 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x 10 -3
However, they are aware of the fraction ρ of the population of which the draws are made, and thus they do know the probabilities of each possible sample of n firms.
Similar to the first model, each firm best-responds to the quantities and prices set by the other firms. By the same argument as for the previous model, we again consider the system to be quasi-symmetric. However, since in this model the firms are not aware of the exact distribution of their market, they use the binomial probabilities to calculate the expected profits. This is because the profits that come with a specific choice of quantity or price may vary between different markets. We therefore give an indication of the average equilibrium profits by calculating the expected equilibrium profits. The equilibrium price (quantity) of the price-setters (quantity-setters) and their expected profits can be written as a function ofn, and .
Lemma 3.2. The Nash equilibrium prices and quantities of price-setters
and quantity-setters respectively, as function of n, and equal
with
and the resulting expected equilibrium profits equal
Proof of Lemma 3.2
If we again consider the first price-setter, we find that its objective function (the expected profits) equals
where k ranges from 1 to n, since the firm itself is already a price-setter, meaning that the number of price-setters in the market can only be in that range. The function was already used in the previous model and is equal to
Maximizing the expected profits under naive expectations with respect to yields the following best-response function for the price-setters
where the final equality again holds under the quasi-symmetric assumption. Next consider the first quantity-setter. Similar to the first price-setter the objective function of this firm equals
where now k ranges from 0 to n-1 due to the fact that the firm
considered is a quantity-setter, and thus the number of price-setters in the market can only be in that range. In this case not the quantity, but the resulting expected price is a function of k. This function is also
(3.6)
(3.7)
known from the previous model and section 2.1, and is equal to
Maximizing the objective function of the first quantity-setter yields the following best-response function for the quantity-setters
where as before the final equality holds under the assumption of quasi-symmetry. In equilibrium we then must have and . Solving (3.7) and (3.10) for these two variables leads to the required result.
Assuming a sufficiently large population means that one firm switching from one strategy to another does not significantly influence the population parameter . Therefore we compare and similarly as for the previous model setup. Numerical simulations show that as before, the equilibrium profits of the
quantity-setters are higher than these of the price-setters. Thus our simulations suggest that for this model, similarly to the first model, we have
(3.9)
Conjecture 3.2. The quantity-setting strategy yields higher profits than the price-setting strategy.
for all n, and . This represents the conjecture that in the two-stage game as described in this section, it is always more profitable to choose the quantity-setting strategy in the first stage.
3.3 Comparison of Two Model Setups
A next step is to make a comparison of the two different model setups. To do this we evaluate the first model for different values of n, k and , and compare these with their counterparts in the second model with (and the other parameters equal to the first model). We considered all possible combinations of the values n = 5, 10, 20, k = 1,2,..,n and ranging from 0.1 to 0.9 with steps of size 0.1. An important reminder is that the profits of the second model are expected profits, not actual profits. An important finding is that for both the price-setters and the quantity-setters, the expected profits in the second model are higher than the realized profits in their first model counterparts for all cases.
To find an explanation for this, we look into the quantities produced by the quantity-setters and the prices set by the price-setters. We observe that the uncertainty about the distribution of the market causes the quantity-setters to produce a lower quantity for all considered parameter values. The prices set by price-setters yield more complicated differences. It appears that the uncertainty for price-setters in the case of low substitutability (low value of ) causes the price-setters to set a lower price.
However, as grows larger this effect reverses, and the prices in the uncertain situation are higher compared to the situation where k is known. Another finding is that typically the difference between profits of quantity-setters and price-setters is smaller in the model with uncertainty, although there are a few exceptions to this finding. These exceptions only occur for high values of
Since in the model with uncertainty the quantity-setters produce less, and since the profits of quantity-setters and price-setters generally lie closer to each other, it
Figure 2 - Binomial distributions
Stacked binomial probabilities for five different intervals of k/n (shown on the x-axis), for three different values of n, where in all cases.
appears that the uncertainty leads to a more collusive outcome. In the end this is
profitable for all the firms as they end up with higher expected profits. Note that we did not take into account any form of risk averseness, but instead consider risk neutral firms.
Finally, we find that as n grows larger the difference between the two models diminishes. We observe that the differences in (total) profits as well as differences in prices and quantities drop when n is larger. This can intuitively be explained by the law of large numbers, as the sample distribution will converge to the population
distribution. An example of this is shown in Figure 2, where stacked (based on intervals) binomial probabilities are shown for different values of n. It can be seen that as n
increases, the probability of 'extreme' samples (i.e., samples with a fraction of price-setters far away from the population fraction) decreases clearly.
4
Stability
As we have seen in the section 3, it is in equilibrium always more profitable to be a quantity-setter compared to being a price-setter. However, as mentioned before, in the previous sections we assumed that the markets will always be in equilibrium. This assumption is based on the assumption of rationality so that in the second stage of the
0 10/100 20/100 30/100 40/100 50/100 60/100 [0, 0.2) [0.2, 0.4) [0.4,0.6) [0.6,0.8) [0.8, 1] n = 5 n = 10 n = 50
game always the Nash equilibrium is played. Since recent literature has put perfect rationality of economic agents into question (e.g. Simon (1957)), it is interesting to evaluate what happens outside of equilibrium. Therefore we introduce two dynamic systems based on the two previously used models, and investigate the stability of these systems in this section
4.1 Firms Know the Number of Price-setters
In this section we discuss the stability of the system for the model where the firms are aware of the distribution of their market, and thus know the number of price-setters. The first step is writing the system as a quasi-symmetric dynamic system, which is necessary to reduce it to a 2D system. Remember that quasi-symmetric means that each quantity-setter sets the same quantity, and each price-setter sets the same price. In each period the firms best respond (based on naive expectations) to the quantities and prices in the previous period. This together means that we can write the price of the price-setters in period t+1 ( ) and the quantity of the quantity-setters in period t+1 ( ) as follows
Since the system is linear, we can use the Jacobian of the system to determine the stability. A straightforward computation shows that this Jacobian equals
Since eigenvalues of this Jacobian are complicated functions of n, k and , we will discuss stability of the extreme cases of Cournot competition (k = 0) and Bertrand competition (k = n). In the case of Cournot competition there are no price-setters, so that the first
(4.1)
(4.2)
equation is not relevant. We then only have to consider the bottom right element of (4.3) (the derivative of with respect to , which for k = 0 is equal to
so that we find that Cournot competition is unstable for
In the case of perfect substitutes ( this means that the system is unstable for n > 3. This corresponds to the special case investigated by Theocharis (1960). As can be seen from (4.5), for every value of with there exists a value such that the best-response dynamics in Cournot competition are unstable if . Similarly for Bertrand competition we consider only the top left element of (4.3) (the derivative of with respect to ), which is for k = n equal to
which is always smaller than 1/2 for . This means that Bertrand competition is always stable for these values of , independent of n.
For other cases we use numerical simulations to determine the highest value of k for which the system is unstable (the critical value of k) given n and , where the critical value of k is equal to zero in case the dynamics are always stable given n and .
Repeating this for a large array of n and different values of gives us an idea of what the stability of the system looks like. To illustrate this, we have considered three values of , . For these values we have calculated the critical value of k for a wide range of n. Afterwards we divided the critical value of k by the corresponding value of n in order to be able to make a better comparison with the second model.
An illustration of the stability is shown in Figure 3. The lower region is the unstable region of the system (where there are not enough price-setters to stabilize), while the upper region is the stable region. Note that a value of zero means that
(4.5) (4.4)
Figure 3 - Stability of the System Given k
Indications of the stability of the system for , with n on the horizontal axis and on the vertical axis. The area above the graph is the stable area, whereas the area below the graph is the unstable area. The up and down movement of the graphs is caused by the fact that k is an integer, and thus as n increases, k/n decreases if k does not increase.
dynamics are always stable. It can be seen in the figure that as n increases, the required fraction of price-setters in order to keep the system stable rises. For small values of n there is a clear difference caused by the value of A higher degree of substitutability requires more price-setters to keep the system stable, which is clearly seen around n = 10. As n grows to infinity, the difference caused by different values of diminishes, and the critical value of k/n required for stability converges to a value slightly below 0.25. Note that the value of n for which the graphs start to move upwards are not an exact match to the critical values based on Cournot competition found through (4.5), which are for equal to n > 9, 5, 3 respectively. This is because we list k/n with k the last value for which the system is unstable. Thus, the graph will start to move upwards when the system is unstable for k = 1.
0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25
4.2 Firms Know the Population Parameter ρ
In this section we consider the model where firms know the population parameter ρ. Similarly as for the previous model we use the assumptions of a quasi-symmetric system in combination with naive best-responses to write the dynamics as a 2D-system. This leads us to the following dynamics of the price set by price-setters (4.7) and the quantity set by the quantity-setters (4.8)
We then construct the Jacobian, which is equal to
where (4.7) (4.8) (4.9)
As a check we can again consider pure Cournot competition ( and pure Bertrand competition ( . For the case of Cournot competition we obtain (as had to be the case) the exact same restriction on n for stability as in the previous model, since the bottom right element of (4.9) for simplifies to (4.4). Similarly for Bertrand competition we find that for the top left element of (4.9) simplifies to (4.5),
meaning that we find the same results as in the previous model, which has to be the case in these pure forms of competition.
The eigenvalues of the Jacobian (4.9) under mixed competition are again complicated functions of n, and . In order to investigate stability we therefore use a numerical approach. Similarly as for the previous model, we consider
. We let n range from 1 to 100, and range from 0.01 to 0.99 with steps of 0.01. Figure 4 illustrates for each n the highest value of for which the dynamics are unstable. When dynamics are always stable for a particular value of n, the critical value of is shown as zero. It can be seen that as n grows larger, the value of for which the dynamics become stable tends towards a value slightly below 0.25, with only a small difference based on Differences between the three different degrees of substitutability are more noteworthy in the movement for low values of n. The higher the degree of
Figure 4 - Stability of the System with Population Parameter
Indications of the stability of the system for , with n on the horizontal axis and on the vertical axis. The area above the graph is the stable area, whereas the area below the graph is the unstable area.
substitutability, the higher needs to be to have a stable system for low values of n. This result is intuitively clear, since a higher degree of substitutability means that firms are more dependent on each other, and thus their best-responses will be more narrowly connected. In the extreme case of , changes in other firms' strategies would have no impact on the decision of the firm.
Note that, in contrast with the previous model, we find that the values of n for which the graphs start to move upwards are in this figure in agreement with the critical values of n that we find from (4.5) (n > 9, 5, 3 for respectively). This is because in this case the steps with which increases are sufficiently small, whereas in the previous model we had to deal with an increase of the integer k.
4.3 Comparison of the Two Models
After investigating the stability of both models, we can conclude that the stability has similar forms. In both cases a higher value of increases the unstable region (and thus decreases the stable region), but this effect diminishes as n grows to infinity. Also, for both cases we see a larger instability for the higher values of when n is low. Next to this, if we take a closer look at the convergence in both models, we see that the fraction of price-setters required to stabilize the system is in both cases slightly below 0.25. An
0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
intuitive explanation is as we have seen before, based on the law of large numbers. As n grows to infinity, the fraction k/n of price-setters in each particular market will
converge to . Thus, as n grows to infinity the second model converges to the first model, since the uncertainty about the market composition diminishes.
5
Evolutionary Population Parameter
Previously we have assumed that the population parameter remained unchanged over time. However, as performance of the two different strategies is known to the firms, this assumption does not seem very realistic. The next step is thus to let this parameter depend on the past performance of the strategies, so that better performing strategies are more likely to be used. In the first part of this section we discuss how we implement this, and what the results of this implementation are. In the second part we will discuss the sensitivity of the system based on initial values.
5.1 The Evolutionary Models
To implement an evolutionary population parameter we consider two dynamics. The first dynamic is based on Anufriev and Hommes (2012), and the second dynamic builds on replicator dynamics (Droste et al. (2002)). For the first dynamic, we start by defining a performance measure for time t, denoted by . This performance measure represents the performance of the price-setting strategy compared to the quantity-setting strategy, and it is based on past profits of both strategies. We use a performance measure of the following form
where represents the memory of the performance rule. Next, we calculate the population parameter on time t, , with the following formula
(5.1)
where represents the idea that not all firms directly update their strategy. The parameter represents the intensity of choice, as a higher value of this parameter causes firms to switch to the best performing strategy more rapidly.
For this model for we find two outcomes. The first outcome is that converges to a steady value that is slightly below 0.5. This means that the quantity-setting strategy is slightly more profitable than the price-quantity-setting strategy, but not enough to completely drive it out. However, this is also caused by the choice of . If the value of is low, the difference in profits will not be enough to drive down further. Higher choices of lead to the second outcome, where we observe endless fluctuations of . As the quantity-setting is more profitable in most cases, will be driven
downwards. However, as we have shown in section 4, the system becomes unstable as gets lower. At some point this instability causes the price-setting strategy to be more profitable. This results in going upwards again, and this cycle repeats. Examples of the two outcomes are shown in Figure 5. Note that the chosen values of are rather high (33000 and 43000). This is because the profits of the firms (and thus the performance measure) are low, and thus we require a sufficiently large value of to compensate for this. Scaling the profits and reducing the value of would lead to the same results.
To illustrate the effects of the parameters, some examples are shown in Figure 6. As can be seen in (a) and (b), a larger value of will make the system less volatile. In (c) and (d) it can be seen that an increase of will make changes in happen more slowly and smoothly. Finally, (e) and (f) show that an increase in can create heavy
fluctuations, as switching will happen more intensely.
However, under the evolution described in (5.2) will always converge to when each of the strategies is equally profitable. Since we would like to capture the idea that will remain constant in that case (people will not change their strategy if both strategies are equally profitable), instead of converging to 0.5, we adapted the evolution of to follow the exponential replicator dynamics, similar to Droste et al. (2002). Using the same performance measure as before, we now determine by
This function has the property that as long as the price-setting strategy is more (less)
Figure 5 - Evolution of
(a) (b)
Examples of the described behavior, with on the vertical axis and the period t on the horizontal axis. The only difference is the value of . Parameter values are equal to n = 10,
, and (a) and (b) . Initial values are
profitable than the quantity-setting strategy, the value of will slowly move up (down). Note however that the extreme cases of and are steady states. This can lead to problems in the initial state of the process if is high. In order to deal with this we consider two possible solutions. The first solution is to lower the value of , and increase the number of time periods. This can be done without loss of generality, since the time is not an absolute measure here. The general behavior of the dynamics thus remains representative. The second solution introduces a small noise, such that steady states are not available. Evolution according to the dynamics with noise is described by
where we choose to be uniformly distributed between 0 and 0.05 to represent a small but sufficient shock.
Findings using the exponential replicator dynamics for the population parameter are in agreement with the previous findings for both of the mentioned solutions. Starting in a stable state, with sufficiently large, the quantity-setting strategy will be most profitable and thus lead to a decrease of . However, as the value of becomes
0 50 100 150 200 250 300 350 400 450 500 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (5.4)
Figure 6 - Effects of parameters
(a) (b)
(c) ( ) (d) ( )
(e) (f)
Shown are the effects of changes in the different parameters. All graphs use n = 10, ,
and unless states otherwise. Initial values are 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
lower the instability of the system causes the price-setting strategy to be most profitable. As a result the value of drastically increases again to re-stabilize the system, and the cycle repeats. Although we do notice that the price-setting strategy will not be completely driven out, instability alone is not sufficient to make this strategy most profitable. Good examples of these dynamics are shown in Figure 7. We added the critical value for instability (given the parameter values, as calculated in section 4) in the figure and observe that it is higher than the turning point for the values considered. This is the case for other combinations of parameter values as well. As it seems, even when the system is unstable it is still possible that the quantity-setting strategy is most
profitable. However we always observe a turning point where this is no longer the case, and the system moves back to a stable state to restart the cycle.
In the model with noise, as described in (5.4), a low value of will lead to a fraction of price-setters slightly below 0.5. This is similar to the findings when using the model described in (5.2), and it happens because the noise is in that case too large compared to the change caused by the difference in profits, so that the fraction of price-setter will remain somewhere slightly below 0.5. This can again be limited by reducing the value of the shock further (e.g. to a uniform(0, 0.01) distributed variable), but the general idea remains the same. An example is shown in Figure 8.
5.2 Sensitivity Analysis
Although we find only two possible outcomes for both models, the models are sensitive dependent on the initial conditions and chosen parameter values. These dependencies are already shortly mentioned in the previous section. For the model in (5.2), we have seen that a value of that is low will lead to a convergence of the population parameter to a value close to 0.5. This can also happen if the value of becomes large. Next to this the model is sensitive to changes in initial conditions as is shown in Figure 9. A small change in the value of (0.010 to 0.011) leads to clear changes in the long term value of . The same holds for small changes in initial values and . However, the structure of the evolution does not change because of this, as the shape of the graphs over time is similar. When and are chosen so that the dynamics are stable, the initial values , and do not influence the long run outcome. This is also true for the
Figure 7 - Exponential Replicator Dynamics
(a) (b)
Typical dynamics using the exponential replicator function (a) (5.3) or (b) (5.4) for The
vertical axis shows , and the horizontal axes shows the period t. The red line indicates the critical value of ρ for the given parameter values, as calculated in section 4. Parameter values are n = 10, , and . Initial values are
Figure 8 - Exponential Replicator Dynamics with noise
The evolution of the fraction of price-setters under dynamics described in (5.4), when the value of is 'too low'. The vertical axis shows , and the horizontal axes shows the period t. The red line indicates critical value of ρ for the given parameter values, as calculated in section 4. Parameter values are n = 10, , and . Initial values are 0 500 1000 1500 2000 2500 3000 3500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 800 900 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 800 900 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
memory parameter , since this has no influence on the equilibrium of the system.3 This result is shown in part (c) of Figure 9.
The model in (5.3), which uses replicator dynamics, also has some issues with choices of parameters and initial values. As is already mentioned, having a large value of will result in the population parameter ending up in a steady state of or , depending on the initial conditions. We consider the two solutions to this that are explained in section 5.1. At first we lower the value of , and for the second solution we introduced a noise term as in (5.4)
Using a sufficiently low value , so that we do not end up in a steady state at either or , we find similar results as for the previous model. Changes in initial values do to some extent change the long term value of , but the shape of the evolution remains the same. The changes appear to be of less impact than for the first model, but note however that this is partly caused due to the lower value of . This makes the fluctuations based on initial conditions smaller, which leads to smaller deviations in long term behavior. An example is shown in Figure 10, where again the initial value is slightly changed (in this case from to ). Another example shown in the figure is a slight change in the initial value (from to
).
For the replicator dynamics with noise as in (5.4) we show the dependence on initial conditions in Figure 11. Note that we used the same values for the noises in the comparison. For values of where the system is unstable, small changes in initial values again cause differences in long term outcomes. When is sufficiently small this
dependence fades, although the long run outcome is no longer independent of as was the case in the model of (5.2). This can be clearly seen in Figure 11 (c) and (d).
Although we observe that the long term outcomes in all models can depend on initial conditions, these conditions do not influence the structure of the evolution of the population parameter. For all models we observe two possible outcomes, which are either a constant cyclical switching or a stable steady state slightly below 0.5. Which of this two outcomes is observed depends on the parameter values of and , rather than the initial values.
3 This follows from solving equation (5.2) for
Figure 9 - Sensitivity of the First Model
(a) (blue) and (red).
(b) .
(c) (blue) and (red), .
Shown is how a small change in initial value(s) of (a) , (b) or (c)
changes the evolution of . Used parameter values are n = 10, , and
, and initial values are unless
stated otherwise. 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1
Figure 10 - Sensitivity of replicator dynamics
(a) (blue) and (red).
(b) (blue) and (red).
Sensitivity of the replicator dynamics as in (5.3) for changes in initial values of (a) or (b) . Parameter values are equal to n = 10, , and , and initial values are . 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Figure 11 - Sensitivity of Replicator Dynamics
(a) (blue) and (red).
(b) .
(c) (blue) and (red), .
(d) and (red), .
Sensitivity of the replicator dynamics as in (5.4) for changes in initial values of (a) , (b) , (c) or (d) (stable dynamics). Parameter values are
equal to n = 10, , and , and initial values are
. 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1
6
Concluding Remarks
In this research we establish that instability caused by competition in quantities is sufficiently large to allow the price-setting strategy to remain in a mixed oligopoly. We observe a cyclical switching between both strategies in most cases, but the quantity-setting strategy is never able to drive out the price-quantity-setting strategy. In current literature on mixed oligopolies, originating from Singh and Vives (1984), it is suggested quantities as strategic variables (Cournot competition) dominates prices as strategic variables (Bertrand competition) when goods are substitutes (and vice versa for complements). However, the dynamics of Cournot competition under naive best-response dynamics can become unstable as the number of firms passes a certain threshold. In a mixed oligopoly where some firms use prices as strategic variables, and some firms use quantities as strategic variables, this instability can make the price-setting strategy more profitable than the quantity-setting strategy.
In the setting with an evolutionary value of this is shown in the cyclical
switching between strategies. When the system is stable, the quantity-setting strategy is more profitable, causing more firms to switch to that strategy. However, as more firms choose to do so the system becomes unstable, which is in accordance with Theocharis (1960) findings on pure Cournot competition. At a certain point (not necessarily directly after becoming unstable) the instability is large enough to make the price-setting
strategy more profitable, causing firms to switch back towards that strategy again. We considered two different models for , and both lead to the same results. Even though we find that in equilibrium the quantity-setting strategy is always more profitable, outside equilibrium it is not able to drive out the price-setting strategy. Although our models are sensitive to changes in initial conditions, these initial conditions have no impact on the general idea of the evolution. The quantity-setting strategy is in no case able to drive out the price-setting strategy outside equilibrium. Another finding is that stability of the system for smaller markets is dependent on the degree of substitutability, but for larger markets this dependence is no longer relevant.
In this research we have considered substitute goods only. Next to that we have assumed that there are no qualitative differences between goods. It would be a great addition to evaluate the dynamics allowing for complements and qualitative differences as well. Next to this our demand structure is linear, whereas a non-linear structure might lead to different results. Another addition could be to investigate more than only
naive best-response dynamics. Commonly used best-responses are based on (for example) adaptive expectations rather than naive ones. This will lead to different (and perhaps more stable) dynamics.
Next to that it would be interesting to evaluate the described dynamics in an experimental setup. In such an experiment, the participants would have to be informed about the market demand structure. After an introduction, they will have to determine a price or quantity, whichever they prefer. Their profits are then calculated based on their input. Such an experiment could be continued for 50 or 100 periods, and it would then be possible to see the percentage of participants using each strategy, as well as their average profits. Interesting is then to see whether or not the theoretical predictions of higher Cournot profits, as well as the decreasing number of price-setting agents, will be realized and to what degree.
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