Faculty of Economics and Business
Amsterdam School of Economics
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Equilibria under Least Squares Learning in a
Bertrand competition
∗Gijs Egberink
(10098240)
Abstract
In this thesis we drop the assumption that firms have complete knowledge of the market environment and, in particular, the demand for their products. Firms will apply least squares learning to learn about the demand function1. In a homogeneous environment where all firms use the same information set in their estimation, we find that using more information leads to a distribution of the end prices close to the Nash equilibrium. However, firms make less profit than firms that use less information. When firms compete under a heterogeneous market2, we see that firms that use more information make more profit than firms that use less information. When firms have the ability to switch, they end up using the information set that contains the most prices, since this information set performs better most of the time.
Msc in Econometrics
Track: Mathematical Economics
Date of final version: August 31, 2016 Supervisor: Jan Tuinstra
Second reader: D´avid Kop´anyi
Keywords: Location model, least squares learning, switching
∗I am grateful to Jan Tuinstra and D´avid Kop´anyi for their valuable comments that substantially improved
the paper.
1They can only use their own price and sold quantity, and possibly the prices of their competitors 2Where different firms use different, but fixed, information sets
Statement of Originality
This document is written by Student Gijs Egberink who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
1
Introduction
In classical models of competition, there are some assumptions being made on the availability of information, these however might not always hold. For example, firms might not know which factors have an impact on the demand for their product and how large this impact is, such as the impact of their and their competitors’ price on the demand. Learning can play a part in these kinds of situations: agents collect data about the market by documenting their actions and the resulting outcomes and take these into account with their decision-making.
Firms might be able to obtain information about the market. There are limitations on the ability to process information, this explains why some freely available information is not used, or imperfectly used. This forces agents to make choices based on incomplete information. This is known as ”rational inattention”, Sims (2003). In order to gain information on the demand and to obtain a more reasonable model of the market, firms have to make a trade off between information costs and making sub-optimal decisions. The costs may exceed the extra gains made by this information, so firms may operate in an under-informed setting and optimize given the limited information they have.
While agents might not always have full information, they could incorporate learning in different ways to learn about the market environment. Over periods of time they get more data about the market by just observing, they can then do estimations and update their beliefs. There are various methods for modeling learning in the economics literature, mostly applied to game theory. This paper will focus on learning in economic markets, specifically in a Bertrand competition.
Kirman (1975) states that a firm‘s view about the market environment might be wrong in two ways. In the first place, its description of the structure of the system may be correct but it may have estimated parameters that are incorrect. Secondly, the firm could have an incorrect model of the true structure, assigning its prediction errors to its incorrect estimation of the parameters. Kirman studies a simple duopoly problem in which firms are in error in the sense that they specify an incomplete model and add a random error term. In his model, firms use least squares learning to estimate a linear demand curve. With this demand curve, they compute the new profit-maximizing prices given the estimated demand curve. This also results in a new demand that follows from this price. They add both the price and the demand to
the existing data set and new parameters are estimated. In some special cases he finds that, by learning, the firms will get to the same equilibrium as they would have reached if they had been fully informed. But in most cases their learning process leads to a stable point which is different from the true equilibrium. This behavior is rather troublesome, in a way that instead of informing themselves about the true mechanism at work, they estimate an incorrect model in which their behavior is perceived to be optimal.
Anufriev et al. (2013) studies the interaction between gradient (GL) learning and least squares (LS) learning in a Bertrand oligopoly with differentiated goods where firms do not have any knowledge of the demand specification and they use these two methods for determining the prices of their good. They find that, in a pure LS-learning oligopoly, firms move to a so called self-sustaining equilibrium, where expected and actual demand coincide at the price they charge, which we also see in Kirman (1975). Furthermore, different learning methods can coexist, and there is no clear winner in a profit-driven competition between these models. The dynamics are much more complicated under a heterogeneous learning than under homogeneous learning.
Meeuwissen (2015) also studies learning and optimization under incomplete information of the market structure, but he uses a spatial model, where firms have equidistant locations and consumers are uniformly distributed and need to pay transportation costs. In this thesis, we will use a similar setup. We will use a market structure where the influence of the price of a competing firm decreases as a firm is farther away. In a way this is what Meeuwissen does in his thesis, but he puts more emphasis on the distance between firms. Also, we will distinguish more different types of information, to get a better knowledge of the effect of including more information. Meeuwissen uses a switching mechanism which is based on a paper by Eshel et al. (1998) where firms switch if their method leads to a price for which it does not sell anything, whereas we will use a switching mechanism based on Brock and Hommes (1997). This mechanism is based on the performance of both types of learning. The chance a firm will switch is increasing in the performance of the other type of learning. Meeuwissen (2015) finds that when firms act as perceived monopolist, where they only use their own price in the estimation of the demand function, prices converge to an unbounded set of prices. As firms use more information, the prices start to converge to a price closer to the Nash equilibrium. He also finds that when firms that act as perceived monopolists, sometimes outperform firms that use all information.
In this paper, we will take a closer look at Least Squares (LS) learning in a Bertrand market, where an odd number of firms are uniformly spaced on a circle, each producing a symmetrically differentiated, but substitutable good. We will only look at LS learning and will compare the outcomes under different amounts of information used in estimating the demand function. It could be the case that firms do not need a lot of information about the market to reach an optimal equilibrium or a Nash equilibrium. Does using Least Squares Learning lead to an equilibrium in different setups? This question we will answer in this thesis, based on the following sub questions:
• Will using different amounts of information lead to different equilibria in a homogeneous environment?
• How do different learning methods affect each other in a heterogeneous environment? • How do different learning methods affect each other in a heterogeneous environment where
firms have the possibility to switch between information sets?
We find that using more information in the estimation of the demand function leads to a distribution of the prices closer to the Nash equilibrium. However, when firms use less informa-tion, they obtain higher profits. These results are similar as to what Meeuwissen (2015) found. We find that firms that use more information perform better compared to firms that use less information, this is also what Meeuwissen (2015) concludes. When firms can switch between different information sets, most of the time they end up in a situation where firms chose the information that contains the most prices.
This thesis will start in Section 2 with an explanation of the market structure and the different models that will be used. Section 3 will explain the Least Square learning method and give different rules and definitions for convergence. In Section 4, simulations will be run in a homogeneous environment. In Section 5, the results will be shown for simulations in a heterogeneous environment. In Section 6 the results will be discussed from simulations in a heterogeneous environment, where firms have the ability to switch. Lastly, in Section 7 we will discuss our findings.
2
The Market Structure
Consider a market where an odd number of f firms are uniformly spaced on a circle, each producing a symmetrically differentiated, but substitutable good and competing in prices, where f = 2n + 1 and n is a non-negative integer, see Figure 2.1. Consumers can go to each firm. The market structure is fully symmetric, so the firms have symmetric demands and the marginal cost of production is constant, identical for all firms and equal to c > 0. First a generic model will be described, then we will consider two different specifications of this model.
2.1 Demand Function
The demand for the product of firm i depends on the price of the goods of all the firms, where the influence of the other firms on the demand for firm i depends on their distance to firm i.
The demand is given by the following equation: Di (pi , p−i) = max α − βpi + 1 2γT n X j=1 γj(pi−j + pi+j) , 0 , (2.1.1) γT = n X j=1 γj, (2.1.2)
where pi is the price of good i, pi−j and pi+j are the prices of respectively the jth neighbor of
firm i to the left and right. For example, pi+2would be the price of the good of the 2nd neighbor
to the right. The parameter γj specifies the relationship between products, it is assumed that
the goods are substitutes, so γj > 0, ∀ j. The influence of the price of the other firms’ goods on
the demand decreases as the firms are farther away, γj+1≤ γj, j = 1, ..., n − 1. The demand is
decreasing in its own price and it is increasing in the price of the competitors’ goods. Note that an increase in γj or n does not make the market more competitive, if all pi+j (except j = 0)
increase by 1 unit, demand for i goes up by 1 unit.
2.2 Equilibria
2.2.1 Nash Equilibrium
The Nash equilibrium is the solution where all firms assume to know the equilibrium strategies of the competing firms. Firms can make the, logical, assumption that all firms maximize their own profit with respect to their own price. If each firm chooses a strategy and no firm can profit by changing strategies while the other firms keep theirs the same, then the current set of prices and the corresponding profits constitutes a Nash equilibrium. There are some restrictions on the parameter values which guarantee that a symmetric Nash equilibrium exists.
Assumption 2.1. The parameters satisfy α − βc + c > 0 and β > 1.
The first part of the assumption ensures that the demand is sufficiently large: demands are positive when each firm sets the price equal to the marginal cost, Di(c, c) > 0. The second part
of the assumption means that the demand for a good strictly decreases if the price of a good from the firm as well as the price of the competitors marginally increase in a symmetric situation. Proposition 2.1 defines the unique Nash equilibrium of our model with demand functions (2.1.1).
Proposition 2.1. Under Assumption 2.1 the model has a unique Nash equilibrium. In this equilibrium, each firm charges the same price pN that is given by:
pN =
α + βc
2β − 1. (2.2.1)
The Nash equilibrium price exceeds the marginal costs. Proof. The profit of firm i is given by:
πi(pi) = (pi− c)(α − βpi+ 1 2γT n X j=1 γj(pi−j + pi+j)). (2.2.2)
The first order condition with respect to pi is:
∂πi ∂pi = α − 2βpi+ 1 2γT n X j=1 γj(pi−j + pi+j) + βc = 0. (2.2.3)
In a Nash equilibrium this equation needs to hold for all firms. Since its a symmetric market, we consider pi= pj. Let p denote the price in a Nash equilibrium. Then (2.2.3) gives
f (p) = α − 2βp + βc + p = 0. (2.2.4) Since (2.2.4) is a simple linear equation, there is a unique solution which we will call pN and is
given by (2.2.1). Note that the corresponding price is larger than the marginal costs as f (c) > 0 (Assumption 2.1), from which we can conclude that pN > c as f (p) is decreasing in p.
From (2.2.1) we can see that the price in a Nash equilibrium is independent of γ and the number of firms, this is due to the fact that the sum of the effects of all the other firms is always the same, by construction. If the costs increase, then also the price increases. Also, pN is decreasing
in β > 1: ∂pN
∂β = −c−2α (2β−1)2 < 0.
2.2.2 Cartel Price
A cartel is an agreement between the competing firms to control the prices in a market, or to exclude entry of a new competitor. Firms can all agree to set a certain price, for example a price that would maximize the total profit of all firms. Firms don’t really have to compete with each other on prices and thus they can set a higher price than they would set in a Nash equilibrium. The price where the total profit is maximized will be called the cartel price. To ensure that the
cartel price is finite, the demand for a good should strictly decrease if the price of a good from the firm as well as the price of the competitors marginally increase. So the right hand term should be lower than β. To ensure this is the case, we divide the right hand term by the sum of the effect of the other firms and set β larger than 1, see Assumption 2.1.
Proposition 2.2. Under Assumption 2.1 the model has a unique cartel price. This is the price pc firms set when they maximize the total profit of all firms, given by:
pc=
α + βc − c
2β − 2 . (2.2.5)
Proof. Firms maximize total profit:
πT = f
X
i=1
(pi− c)Di(p).
The first order conditions with respect to pi is:
∂πT ∂pi = α − 2pib + bc + 1 2γT n X j=1 γi(pi+j + pi−j) + 1 2γT n X j=1 γi(pi+j+ pi−j− 2c). (2.2.6)
Firms operate in a symmetric market, so this condition should hold for all i, so let us consider pi= pj. Let p denote the Cartel price. Then (2.2.6) gives:
f (p) = α − 2pβ + βc + 2p − c = 0. (2.2.7) Since (2.2.7) is a simple linear equation, there is a unique solution which we will call pc and is
given by (2.2.5).
Now we will take a look at two specifications of the general model: The exponential model in Section 2.3.1 and the box model in Section 2.3.2.
2.3 Specifications of the model
2.3.1 Exponential model
The first application of the model that will be used is called the exponential model, where the influence of the price of a competitor will exponentially decrease as the firm is located farther away. In this case γj = γj with γ ∈ (0, 1), which combined with (2.2.1) leads to:
Figure 2.2: Visual representation of the two models Di (pi , p−i) = max α − βpi + 1 2γT n X j=1 γj(pi−j + pi+j) , 0 . (2.3.1) Since Assumption 2.1 and Proposition 2.1 are independent of γ, these still hold. Also pN is the
same as in (2.2.1). For a graphical representation see Figure 2.2.
2.3.2 Box model
The second application that will be used is called the box model. It is similar to the first model, in the sense that the influence a firm has on another firm‘s demand depends on the distance. The difference, however, is that it is more like a binary version of the exponential model. Until a certain distance, called the reach (r), firms have the same influence, after that the firms have a same lower influence. So γj = γh, ∀ j ≤ r, and γj = γl, ∀ j > r, with γh ≤ γl≤ 0. The reach
can’t exceed the number of neighbors, 0 < r < n. This gives the following demand:
Di (pi , p−i) = max α − βpi + γh 2γT r X j=1 (pi−j + pi+j) + γl 2γT n X j=r+1 (pi−j + pi+j) , 0 . (2.3.2) With γT = rγh+ (n − r)γl, the total effect of the neighbors. For a graphical representation see
3
Least Square Learning
The assumption that firms have full information about their environment and thus exactly know the demand for their good is not a very realistic assumption. For a firm to know the demand of a good, the firm has to know the behavior of her consumers. But consumer behavior can be very unpredictable. Firms could try to predict this behavior by analyzing typical buyers of the good they are producing. However, this can be expensive or even impossible and firms may prefer to set their prices under sub-optimal conditions, where the market structure is unknown to them. Instead, firms can use Least Squares Learning. This way, they will learn from the market and they can use this information to get a better understanding of the market structure. This might be a more cost efficient way to analyze the market and its consumers.
Least squares learning is a learning method that is easy to implement, because all the information that is required is available for free, by observing the market. The information that is required, consists of the past prices and demands, and possibly the prices of the other firms. The only information that is observable from other firms is the price of their good. With least squares learning, firms use an estimated demand function for maximizing their expected profit. The next period they will set their profit maximizing price. With this price, firms will sell a certain amount and they can add these observations to the other price-quantity observations and update their parameter estimates.
In this thesis, two information sets will be distinguished. First is the case where firms are totally unaware of their environment and perceive to be monopolist and they optimize accordingly. The idea behind this is that firms that are far away will not influence their own demand much and that it might be negligible. Second is the case where they are aware that there are more competing firms, but the amount of information they use will vary. This ranges from only taking the direct neighbors into account to taking all the competing firms into account when estimating their demand curve.
3.1 The Learning Mechanism
3.1.1 Perceived monopolist
We consider the case where firms do not know the actual structure of the market. Instead of taking the other firms in consideration, they perceive to be monopolists in the market. Kirman (1983) argues that when the number of firms is high, it requires too much effort to collect all prices, so firms focus on their own price effect and the pricing behavior of the other firms is treated as unobserved error. The perceived demand for each firm i is given by:
Dpi (pi ) = a − bpi + εi, (3.1.1)
where a and b are unknown parameters and εi is an error term, which is assumed to be a
random noise with mean 0. Notice that firm i uses a misspecified model as the true demand is determined by the prices of all the firms. With this estimation of the true demand, firms will determine their price by maximizing their expect profit:
max
pi E[(a − bpi
+ εi)(pi − c)]. (3.1.2)
Resulting from this, the optimal price is: pi = a + bc 2b . (3.1.3) Note that: ∂pi(a) ∂a = 1 2b > 0, ∂pi(b) ∂b = − a 2b2 < 0. (3.1.4)
The optimal price is increasing in a and decreasing in b, (3.1.4). At each time t, firm i knows his own past prices and demands. To estimate the coefficients a and b of the perceived demand function, the firm regresses the past demands on the past prices. All observations have the same weight in the regression. The parameters a and b are estimated by Ordinary Least Squares (OLS), where the estimates are given by:
bi,t= − Pt s=1(pi,s− ¯pi)(qi,s− ¯qi) Pt s=1(pi,s− ¯pi)2 , (3.1.5) ai,t= ¯qi− bi,tp¯i, (3.1.6)
where ai,t and bi,t are parameter estimates of a and b for firm i in period t and ¯pi and ¯qi are the
mean price and sold quantity of firm i over all previous periods until t. To guarantee that the estimated a and b are realistic, firms i supposes that ai,t and bi,t are positive. A ruling will be
added to the LSL as follows:
Random pricing rule. When ai,t > 0 or bi,t > 0 does not hold, firm i chooses a price in the
neighborhood of the previous price. This is, firms draw a price from the uniform distribution on the interval [pi,t− u, pi,t+ u], with u > 0.
There might be the case that the dispersion in the observations starts to settle, due to the convergence of the prices. This can result in the estimation matrix becoming close to singular and leading to imprecise estimations which can result in extremely high prices. Since we assume that it should be clear to firms that this might be an issue, it is reasonable that firms do not follow the pricing rule, but instead choose a price close to the price from the previous period. No jump rule. If (3.1.3) returns a price for firm i that is at least K times higher than the price of firm i in the previous period, then firm i draws a price from the uniform distribution on the interval [pi,t− u, pi,t+ u], with u > 0.
From these assumptions and rulings the price in the next period is given by:
pi,t+1= ai,t+bi,tc
2bi if ai,t, bi,tc > 0 and
a+bc
2b < Kpi,t,
∈ [pi,t− u, pi,t+ u] otherwise.
(3.1.7)
3.1.2 Local competition
In this case firms take the prices of the competing firms into account, this can range from only the closest neighbors to all the competing firms. Often, firms are aware of other firms that are competing in the same market and will therefore influence the price of the good they are selling. Firms have no reason to believe firms to the left of them have a different influence than firms to the right, so firms use the combined prices of firms at the same distance as a regressor.
Dpi = a − bpi+ k
X
j=1
where pi is the price of firm i, pi−j and pi+j are the prices of respectively the jth neighbor of
firm i to the left and right. Agents can vary k ∈ {0, 1, ..., n} and i is an error term, which is
assumed to be a random noise with mean 0. For k = n they estimate the model taking into account the prices of all competitors. In this part we will only look at agents who vary k from 1 to n, because we discussed the case where k = 0 in Section 3.1.1. Similar as to the case of the perceived monopolist, firms determine their price by maximizing their expected profit and play the myopic best response to their competitors’ price. That is, firms maximize their their expected profit under the assumption that their competitors will charge the same price as in the previous period:
pi = a + bc +Pk j=1gj(pi−j + pi+j) 2b . (3.1.9) Note that ∂pi(a) ∂a = 1 2b > 0, ∂pi(b) ∂b = −a −Pk j=1gj(pi−j+ pi+j) 2b2 < 0, ∂pi(gj) ∂gj = pi−j+ pi+j 2bi > 0, (3.1.10)
so the optimal price is increasing in a and gj and decreasing in b. At each time t, firm i knows
his own past prices and demands, as well as the prices of the competing firms. The parameters a, b and gj are estimated using OLS. In this case it is more convenient to write the estimators
in OLS matrix form, because the number of explanatory variables is larger than in the previous case:
Bi = (Xi0Xi)−1Xi0Qi (3.1.11)
where Bi is the vector of parameter estimates and Xi a matrix given by:
Bi = ai,t bi,t gi,1,t .. . gi,k,t , Xi=
1 −pi,0 pi−1,0+ pi+1,0 . . . pi−k,0+ pi+k,0
..
. ... ... . .. ... 1 −pi,t pi−1,t+ pi+1,t . . . pi−k,t+ pi+k,t
and Qi is a vector with all the demands of firm i up until time t. Some restrictions have been
placed on the parameter estimates to make sure they are economically sensible. There are no restrictions on the coefficients regarding the effects of the competing firms, because the effect may be small and therefore insignificant in the regression, so some estimates can be negative where a positive estimate is expected. Resulting from this the random pricing rule and no jump rule apply to firms using local information. So firms set prices at t + 1 according to the following rule: pi,t+1=
ai,t+bi,tc+Pnj=1gj(pi−j+pi+j)
2bi,t if ai,t, bi,t > 0 and
a+bc+Pk
j=1gj(pi−j+pi+j)
2b < Kpi,t,
∈ [pi,t− u, pi,t+ u] otherwise.
(3.1.12)
3.2 Convergence and equilibrium
3.2.1 Convergence
When using a misspecified model, prices generally do not converge, see Brousseau and Kirman (1992). Over time, however, the price changes become smaller. Once the price changes are below a certain point, the learning mechanism stops, see Figure 3.1 for a graphical representation of the prices and profits over time.
Definition 3.1. The process has converged if the following holds for price vector p∗ = (p∗1, ..., p∗f):
max i pi,t− pi,t−1 pi,t−1 < δ,
here pi,t is the price of firm i in period t and δ is the convergence threshold.
3.2.2 Self Sustaining Equilibrium
Brousseau and Kirman (1992) have shown that the misspecified least square learning that is considered in this paper does not converge in general. However, over time, the influence of new observations decreases and thus the price changes become smaller over time. So the convergence criterion will be satisfied at some point and the learning mechanism will stop. The resulting prices will be very close to a so called self-sustaining equilibrium. In this equilibrium the
expected and actual demands of a firm intersect, see Figure 3.2. In panel (a) it can be see that the actual and perceived demand functions cross each other in a single point, but the firm did not estimate the correct demand function. Panel (b) shows the true and perceived profit functions. As can be seen, firm 1 maximizes his perceived profit, but the price does not yield the true profit maximum. However, the firm has no reason to believe he is not at the optimum of his profit function. The set of self-sustaining equilibria is infinite.
Definition 3.2. Price vector p∗ = (p∗1, ..., p∗f) and parameter estimates a∗i, b∗i, gi,j∗ (i = 1, ..., f ; j = 1, ..., k) constitute a self-sustaining equilibrium if the following conditions hold for each firm i:
p∗i = argmax pi E[(pi − c)Dip] (3.2.1) Di(p∗) = EDPi (p ∗ i) (3.2.2)
Most of the time the firms use a misspecified model, except when they use full information, since their perceived demand function differs from the actual demand. However, their models might still predict the right demands and thus the prices they charge might be subjectively optimal.
(a) Prices over time (b) Profits over time
Figure 3.1: Time series of prices (a) and profits (b) in a market with 9 firms who all act as a local monopolist. The demand function follows the exponential model and the parameter values are α = 10, β = 1.1, γ = 0.99, c = 2, δ = 0.001 and K = 3. Equilibrium values are: pN = 10.17,
(a) Demand function in SSE (b) Profit function in SSE
Figure 3.2: Demand function (a) and profit function of firm 1 in an SSE a market with 9 firms who all act as a local monopolist. The demand function follows the exponential model and the parameter values are: α = 10, β = 1.1, γ = 0.99, c = 2, δ = 0.001 and K = 3. Equilibrium values are: pN = 10.17, πN = 73.36, pc= 51.00 and πc= 240.10
When this holds for all firms, the model is in equilibrium, because their perceived demand function will not change and thus prices will stay the same. To determine if the equilibrium is an SSE, a test is done to see if the actual demand differs a certain percentage from the perceived demand. When this percentage is below a certain threshold θ, the prices have converged to an self-sustaining equilibrium. In Figure 3.3 we can see a time-series where the actual and perceived demand function converge. To determine if Definition 3.2 holds, the following criterion should hold: max i ( qi,tp − qi,t qi,t ) < θ, (3.2.3)
where qi,t and qi,t are the actual and perceived demands respectively.
3.3 Implementation
Least squares learning is implemented in the following way. For any firm i:
Figure 3.3: The actual and perceived demand of all the firms over time in a market with 9 firms who all acts as a local monopolist. The demand function follows the exponential model and the parameter values are: α = 10, β = 1.1, γ = 0.99, c = 2, δ = 0.001 and K = 3. Equilibrium values: pN = 10.17, πN = 73.36, pC = 51.00 and πC = 240.10
on the set S = {pi ∈ [r1, r2] : pi > c, Di(p) > 0. Here s = 2 + n, with n the number of
neighbors being taken into account.
2. At the end of period 0 the firm uses OLS to estimate the parameters. 3. In period t ≥ 1:
(a) Firms update their prices according to (3.1.7) or (3.1.12).
(b) After realizing their demand, the firm again estimates the parameters using OLS. 4. The process stops when the price difference between two periods is below a certain
per-centage, δ, for all firms.
Note that the firms’ learning processes interfere with each other, because a price change of a competing firm has an impact on the firms own demand. Note that the learning process is more complicated in the initial periods, because prices are more volatile than in later periods, when the learning process slows down.
4
Simulation results for homogeneous markets
In this section, the results for simulations with 100 different initializations will be shown. Each initialization runs until the prices converge according to the convergence criterion, Definition 3.1, or period 1000 is reached. Both the exponential and the box model will be used. Prices have said to converge for δ = 0.0001 and the threshold for a self sustaining equilibrium is θ = 0.1. For all experiments, simulations with all types of information will be used, from perceived monopolist to taking all firms into account. The results are presented in a table with relevant statistics that will be discussed. The columns price and variance show the mean and variance of the end prices of all the firms over all initializations. Change shows the relative decrease in price compared to the method where firms use k − 1 neighbors in their estimation. The min and max values are the overall minimum and maximum of the end prices. The median number of periods it takes to reach an equilibrium is shown under convergence. Dispersion is the mean from the difference between the maximum and minimum end prices or profits in each initialization. Firms operate in a homogeneous environment, meaning all firms use the same estimation model, i.e. they all use the same amount of information. With every experiment done, all the prices converged to a self-sustaining equilibrium. The first part of the experiment is done with the exponential model. Note that an increase in γ or n does not make market more competitive, this way we can make a better comparison between simulations with different settings.
Table 4.1: Resultant prices for different types of information used in estimating the demand function in a competition with 9 firms. The demand function follows the exponential model and the parameter values are: α = 10.0, β = 1.1, γ = 0.99, c = 2.0, δ = 0.0001, θ = 0.10 and K = 3. Equilibrium values are: pN = 10.17, πN = 73.36, pc= 51.00 and πc= 240.10
k price change variance min max profit convergence price and profit dispersion 0 11.17 - 3.37 4.99 24.36 75.93 38 4.29 8.2
1 10.66 0.046 1.53 7.44 17.22 76.42 44 2.35 5.91 2 10.3 0.034 1 7.17 15.45 73.94 42 1.93 3.57 3 10.23 0.007 0.56 7.61 14.67 73.65 42 1.4 2.17 4 10.17 0.006 0 10.17 10.17 73.36 10 0 0
Table 4.2: Mean of the end parameter estimates in a competition with 9 firm. The demand function follows the exponential model and the parameter values are: α = 10.0, β = 1.1, γ = 0.99, c = 2.0, δ = 0.0001, θ = 0.10 and K = 3. Equilibrium values are: pN = 10.17,
πN = 73.36, pc= 51.00 and πc= 240.10 k α β γ1 γ2 γ3 γ4 0 20.077 1.133 - - - -1 16.848 1.075 0.139 - - -2 14.548 1.101 0.133 0.135 - -3 12.432 1.1 0.128 0.122 0.127 -4 10 1.1 0.127 0.126 0.124 0.123 actual 10 1.1 0.127 0.126 0.124 0.123
Figure 4.1: Histogram of the end prices for different types of information used in estimating the demand function in a competition with 9 firms. Parameter values are: α = 10.0, β = 1.1, γ = 0.99, c = 2.0, δ = 0.0001, θ = 0.10 and K = 3. Equilibrium values are: pN = 10.17,
4.1 Number of firms
First an experiment will be done with 9 firms, as can be seen in Table 4.1. The mean price for the perceived monopolist is substantially higher than the price in the Nash equilibrium. The firms use a misspecified model, so they do not account for the positive effect of the price of the other firms. This results in an overestimation of α. A larger α leads to a higher price, see equation (3.1.4) and (3.1.10). As the firms use more information in their estimation, the price gets closer to the price in a Nash equilibrium. The price dispersion decreases as firms use more information, since the bias in their models decreases and their models are more coherent. When they specify the model correctly, they will set prices equal to the Nash equilibrium price. The mean profit is substantially higher for the perceived monopolist in comparison to the other estimation methods. This is due to the fact that the prices are closer to the Cartel price, in which the joint profit is maximized. However, the profit is more dispersed in this case, whereas the profits are equally distributed in the Nash equilibrium. It takes longer for firms to reach an equilibrium with less information as we can see from the convergence rate. Concluding, the distribution of the end prices monotonically shrinks to the Nash equilibrium price as firms use
Table 4.3: Resultant prices for different types of information used in estimating the demand function in a competition with 15 firms. The demand function follows the exponential model and the parameter values are: α = 10.0, β = 1.1, γ = 0.99, c = 2.0, δ = 0.0001, θ = 0.10 and K = 3. Equilibrium values are: pN = 10.17, πN = 73.36, pc= 51.00 and πc= 240.10
k price change variance min max profit convergence price and profit dispersion 0 10.75 - 2.45 4.95 22.71 74.73 43 3.88 7.86 1 10.51 0.023 1.12 7.33 16.81 75.31 47 2.65 5.39 2 10.35 0.015 0.8 6.99 17.08 74.37 46 2.24 3.79 3 10.24 0.011 0.49 7.83 13.08 73.66 50 1.77 1.98 4 10.24 0 0.59 7.85 14.09 73.69 49 1.58 1.92 5 10.19 0.005 0.39 8.67 13.83 73.38 43 1.3 1.42 6 10.23 -0.004 0.33 8.89 14.69 73.74 36 1.1 1.52 7 10.17 0.006 0 10.17 10.17 73.36 10 0 0
Table 4.4: Resultant prices for different types of information used in estimating the demand function in a competition with 9 firms. The demand function follows the exponential model and the parameter values are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10 and K = 3. Equilibrium values are: pN = 10.17, πN = 73.36, pc= 51.00 and πc= 240.10
k price change variance min max profit convergence price and profit dispersion 0 10.9 - 3.48 6.09 36.75 75.43 40 3.51 8.61
1 10.57 0.03 1.28 7.49 16.36 75.87 44 2.2 4.76 2 10.32 0.024 0.95 7.49 14.25 74.19 44 1.75 3.14 3 10.21 0.01 0.46 8.34 13.27 73.57 38 1.1 1.6 4 10.17 0.004 0 10.17 10.17 73.36 10 0 0
more information, as can be seen in 4.1.
Results for the simulation with 15 active firms are shown in Table 4.3. The results are similar to the previous experiment, so the focus will be on extra insights from adding more firms. The largest difference is that prices do not monotonically decrease as more information is used. For k = 6 we see a slight increase in the mean end prices, this is due to the fact that the effect of firms that are this far away, have very little impact on the demand, so there can be an estimation bias. Profits also slightly increase for k = 1 compared to k = 0.
4.2 Impact of the other firms
Now a simulation with a lower γ is done, see Table 4.4. The number of competing firms is set at 9, since more firms do not give a real different insight. A γ of 0.9 was chosen, so the influence firms have on other firms declines quickly and using more firms in estimation has less importance than for γ = 0.99. The biggest difference with the simulation with γ = 0.99 is that after the direct neighbor is taken into account, using more parameters in the estimation does not have a big impact. This is due to the fact that the influence declines quickly as firms are farther away.
4.3 Box model
Now a closer look will be taken at the box model, see Table 4.5. The parameters are set, similarly as the exponential model, only now the box model is used. The number of firms is still 9, as setting firms any higher did not give any new insight. The first thing that stands out is that after taking 2 firms into consideration, using more firms in the estimation does not have a substantial effect on the mean end price. Note that firms now are close to the Nash equilibrium at k = 2, there is still some price dispersion but the mean price coincide with the Nash equilibrium price. This makes sense, since the reach of the model is equal to two and after that the influence of firms on the demand is substantially lower. However, using more information leads to a lower convergence rate.
4.4 Conclusion
The most important result we found, was that an increase in information leads to end prices closer to the price in a Nash equilibrium. However, when firms use less information, they will end up with a higher profit, but these are more dispersed than when they use more information. It takes longer for firms to reach an equilibrium if they use less information.
Table 4.5: Resultant prices for different types of information used in estimating the demand function in a competition with 9 firms. The demand function follows the box model and the parameter values are: α = 10.0, β = 1.1, γh = 0.9, γl= 0.1, c = 2.0, δ = 0.0001, θ = 0.10 and
K = 3. Equilibrium values are: pN = 10.17, πN = 73.36, pc= 51.00 and πc= 240.10
k price change variance min max profit convergence price and profit dispersion 0 11.05 - 3.88 5.6 28.1 75.12 62 5.29 15.45
1 10.79 0.023 1.53 7.68 18.43 77.49 61 2.63 9.27 2 10.17 0.058 0.18 9.04 11.4 73.36 22 0.44 1.14 3 10.17 0 0.12 9.4 10.94 73.36 20 0.34 0.82 4 10.17 0 0 10.17 10.17 73.36 10 0 0
5
Heterogeneous oligopoly with fixed learning strategies
So far we only looked at cases where all firms use the same amount of information in their estimations. However, firms can have different ideas about the market and their perceived demand ore use different information sets. In this section we consider the interaction between the learning strategies that use different amounts of information. Some firms may act as perceived monopolists, while other firms may use information about all the other firms. Firms cannot change the amount of information that is used. The number of possible compositions is large and increasing in the number of firms, but we will look at two different compositions, see Figure 5.1. The first one, is where firms with the same method are clustered together (a) and the second one, is were the methods are evenly mixed (b). First we will compare all different amounts of information against each other, from this we will see what is interesting to take a closer look at. From now on we will only look at the exponential model, we also did simulations with the box model, but this did not give any new insight.
(a) Clustered (b) Alternating
Figure 5.1: Compositions of the beliefs of the firms used in this section. Different colors represent different learning methods.
5.1 Clustered composition
The first composition we will take a look at is where the firms using the same amount of information are clustered together. First a table is presented where all the different types of information compete against each other, see Table 5.1. This table gives the profits, averaged per amount of information that is used. The first and second value in a cell gives the average profit of firms using the amount of information given in the first row and first column respectively and the third value gives the total average profit. We see that if more information is used in estimating the demand function in a market, everyone makes on average less profit. However, when firms with different ideas of information compete against each other, the one that uses more information is better off compared to the one that uses less information. So firms might want to act as local monopolist, but when the competing firms use some amount of information, they perform worse than their competitor. Note that they still will perform worse if all perceived monopolists would switch to the other method.
Two different situations are highlighted, the first one is where perceived monopolists compete against firms that only take their direct neighbors into account (Figure 5.2) and the second one is where they compete against firms that use full information (Figure 5.3). Taking a look at the first situation, we can see that the firms that use more information, set on average lower
Table 5.1: The mean of the profits at convergence, where all different amounts of information compete against each other. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36. The first and second value in a cell gives the
average profit of firms using the amount of information given in the first row and first column respectively and the third value gives the total average profit.
k 0 1 2 3 4 0 87.8/87.8/87.8 78.7/86.7/82.2 77.8/84.2/80.6 75.4/79.1/77.1 75.8/80.1/77.7 1 86/78.5/82.6 76.9/76.9/76.9 75.1/75.5/75.3 73.6/73.9/73.7 73.5/73.8/73.7 2 80.3/76.1/78.4 74.6/74.4/74.5 74.4/74.4/74.4 73.7/73.8/73.7 73.4/73.6/73.5 3 75.4/73.8/74.7 74.7/74.3/74.5 73.5/73.5/73.5 73.1/73.1/73.1 73.2/73.4/73.3 4 75.6/74.1/74.9 73.5/73.2/73.3 73.8/73.5/73.7 73.4/73.3/73.4 73.4/73.4/73.4
(a) (b)
Figure 5.2: Mean prices and profits at convergence with a clustered composition, where perceived monopolists compete against firms only taking direct neighbors into account. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
(a) (b)
Figure 5.3: Mean profits at convergence with a clustered composition, where perceived monop-olists compete against firms that use full information. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
Table 5.2: The mean of the profits at convergence with varying number of firms acting as a perceived monopolist. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
number of firms with k = 0 0 1 2 3 4 5 6 7 8 9 mean profit (k = 0) - 73.1 73.2 73.4 74.2 75.2 75.6 78.5 84.2 97.4 mean profit (k = 4) 73.4 73.4 73.6 74.1 75.9 78.3 80.3 89 109.4
-mean profit (total) 73.4 73.4 73.5 73.8 75.1 76.6 77.2 80.9 87 97.4
prices, see panel (a). We saw earlier that this was caused by overestimation of α, when less information is used. So if a firm sets a higher price, which has the largest (negative) impact on its own demand, and the competitors set a lower price, then we can expect that the demand for this firm will be lower and thus the profits will be lower. We see this is true in panel (b). It does not matter where a firms is located, firms that have a direct neighbor that uses a different amount of information, do not perform different from firms that do not.
When perceived monopolists compete against firms that use full information, we see a similar effect, however groups look even more homogeneous in this case, see Figure 5.3. Firms using a
Figure 5.4: The mean of the profits at convergence with varying number of firms acting as a perceived monopolist. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
certain amount of information have the same average profit at convergence. Another thing that we see, is that when more information is used, all prices are closer to the prices in the Nash equilibrium compared to a situation where less information is used.
Until now, we only looked at a composition where the number of firms that have a certain belief differs only by 1. Now the number of firms that act as a perceived monopolist, and with it the number of firms that use full information, will be varied, see Table 5.2 and Figure 5.4. When more firms are acting as a perceived monopolist, the mean of the profits over all firms increases, however the firms that use more information perform better than the perceived monopolists. If we look at the situation with 4 perceived monopolists where the perceived monopolist and the firms that use full information have an average profit of 74.2 and 75.9. If a perceived monopolist would switch to using full information, he would make a lower profit on average and if a firm that uses full information would switch to being a perceived monopolist he would also make a lower profit. So no one has an incentive to switch.
5.2 Alternating composition
The second composition we will take a look at, is where the different types are alternately distributed among the firms. Note that since we use an odd number of firms, there will be two firms located next to a firms that uses the same amount of information. Table 5.3 gives the mean profits at convergence per type. Compared to the previous table, we do not see any notable
Table 5.3: The mean of the profits at convergence, where all different amounts of information compete against each other. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36 k 0 1 2 3 4 0 83/83/83 78/85/81.1 76.2/80.5/78.1 75.9/80.3/77.8 74.1/76.1/75 1 79/76/77.7 75.3/75.3/75.3 74.6/75.2/74.9 74.3/74.7/74.5 73.9/74.5/74.2 2 77.2/74.9/76.2 74.4/74/74.2 74.7/74.7/74.7 73.4/73.5/73.4 73.6/73.9/73.7 3 75.2/73.9/74.6 75.1/74.4/74.8 73.7/73.6/73.7 73.4/73.4/73.4 73.3/73.4/73.3 4 77/74.8/76 74.4/73.7/74.1 73.2/73/73.1 73.6/73.5/73.6 73.4/73.4/73.4
(a) (b)
Figure 5.5: Mean prices and profits at convergence with a clustered composition, where perceived monopolists compete against firms only taking direct neighbors into account. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
(a) (b)
Figure 5.6: Mean profits at convergence with a alternating composition, where perceived mo-nopolists compete against firms that use full information. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
differences. We still see that firms that use more information, perform better compared to firms that use less information and that the profit is the largest when all firms act as perceived monopolist. The are no substantial differences in profit levels either if we compare it to the clustered composition.
As in the previous section, we will now take a closer look at two specific situations, where perceived monopolists compete against firms that only take their direct neighbor into account and against firms that use all information. Figure 5.5 shows that firms that take their direct neighbor into account perform better than firms that do not. The reason for this is the same as we discussed before, perceived monopolists set a higher price and this results in a higher demand for the firms that only take their direct neighbor into account. The location of the firm does not have an effect on the profits, because almost all firms are next to a firm with different beliefs, except for two. Figure 5.6 gives the mean profits at convergence in a market where perceived monopolists compete with firms that use full information. When in a market, more information is used, the prices goes towards the price in a Nash equilibrium and the overall profits go down. Now we will look at the average profits of convergence when we vary the number of perceived monopolists, see Table 5.4. The are no substantial differences in the mean profits levels at con-vergence with the clustered composition. So the location of the perceived monopolists do not have have substantial influence on the mean profits, however the number of perceived monop-olists do. So we see that the mean profit increases as more firms act as perceived monopolist, but the firms that use full information perform better compared to the perceived monopolists.
If in a heterogeneous oligopoly with fixed learning strategies more information is used in a
Table 5.4: The mean of the profits at convergence with varying number of firms acting as a perceived monopolist. Parameters are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10, K = 3, pN = 10.17 and πN = 73.36
number of firms with k = 0 0 1 2 3 4 5 6 7 8 9 mean profit (k = 0) - 73.1 73.3 73.1 74 75 77.1 80.3 85.5 97.4 mean profit (k = 4) 73.4 73.4 73.6 73.7 75.2 77.7 84 93.7 115.3
market, everyone makes on average less profit. However, when firms with different amounts of information compete against each other, the one that uses more information has a higher profit than the on that uses less information. Based on these findings, we expect that if firms have the ability to switch, they will choose to use more information in their estimation, which leads to higher profits. However, since the profits are lower when everyone uses more information, compared to a situation where everyone uses less information, it could be that firms switch back to using less information. We consider the following possibilities
• All firms choose to use the least amount of information. • Cyclical switching between learning strategies.
• No clear pattern as the differences in profitability disappear.
Now we will add a switching mechanism to the heterogeneous environment to test if our expec-tations are correct.
6
Endogenous switching between learning strategies
Now we will extend the previous heterogeneous model allowing for endogenous switching between the two types of information. To decide which information set will be used, firms take into account their performance. The probability of choosing a specific information set is positively related to the past profits realized with each mechanism, firms consider their own and their direct neighbors past profits for evaluating performances. In this section we will focus on firms that act as a perceived monopolist and firms that use all prices. We also ran simulations where we let firms who use their own price and the price of their direct neighbor compete with perceived monopolists and firms who use full information, but this gave similar results. We will call these different amounts of information, information sets.
6.1 The switching mechanism
This switching mechanism is based on the switching mechanism as in Brock and Hommes (1997). It is assumed that that there is a chance ρ that firms have the opportunity to switch and firms can only chose between two types of information, information set 1 and information set 2. Every
firm i has a performance measure for both information sets, which depends on their own and their neighbors past profits. Let li,tM 1(li,tM 2) denote the performance of using information set 1 (2). The performance measure is the weighted average of past profits generated by that information set, where the weights decay geometrically. These measures determine the probability firm i applies a certain information set. The performance measure is updated in each period as follows:
li,t+1M =
(1 − w)lMi,t + w¯πi,tM if firm i or neighbors use information set n, lMi,t otherwise,
(6.1.1)
where w ∈ (0, 1] is the weight of the most recent profit and M ∈ {M 1, M 2} indicates the information set used in estimation. ¯πM
i,t is the mean profit of firm i and his direct neighbors
using information set n in time t: ¯ πMi,t = P1 j=−1IM(Mj)πMj,t P1 j=−1IM(Mj) , IM(Mi) = 1 if Mi = M, (6.1.2)
where Ii,M is 1 if firm i uses information set M and zero otherwise. Performance measures
are weighted averages of past profits by the given information sets, where the weights decay geometrically. From these performance measures, a probability of applying information set M is given as: Pi,t+1n = λ + (1 − 2λ) e φln i,t+1 eφl1i,t+1+ eφl2i,t+1 , (6.1.3)
where φ ≥ 0 describes how sensitive the firms are to differences in the performance measures and λ is the probability of experimentation. For a higher φ, the probability of choosing the information set with the higher performance increases. For φ = 0, firms chose with equal probability of 50% for both information sets. For φ = ∞, the firms chooses the information set with the highest performance with probability 1 − λ. The model is implemented as follows:
1. A number of starting prices, {pi,−s, ..., pi,−2, pi,−1}, are drawn from a uniform distribution
on the set S = {pi ∈ [r1, r2] : pi > c, Di(p) > 0. Here s = 2 + n.
2. In period 1, k randomly chosen firms apply information set 1, the other firms use in-formation set 2. Prices are determined by the learning mechanisms discussed in Section 3.
3. In period 2:
(a) Firms set the prices using the other information set: firms using information set 1 in period 1 use information set 2 and vice versa. Prices are determined by following the learning mechanism and the initial performance measures are calculated.
(b) Firms choose an information set for the following period: firm i applies information set 1 with probability PM 1
i,3 .
4. In period t ≥ 3:
(a) Prices are determined by the two information set and the performance measures, lM 1i and lM 1i , are updated.
(b) Firm i chooses information set 1 for period t + 1 with probability Pi,t+1M 1 . 5. The process ends when the predefined time limit T3 is reached.
To get a better understanding of the simulations it is important to see if they converge to a certain distribution, two characteristics are defined for this, namely fracturing and clustering. Fracturing is the percentage of firms using a certain information set. We compute fracturing, Vt, as a weighted average: V0 = v0, Vt= (1 − ψ)Vt−1+ ψvt∀ t > 0, vt= NM,t Ntotal,t , (6.1.4)
where NM is the number of firms using information set M and Ntotal the total number of firms
in period t. The coefficient ψ is the degree of weighting decrease. A higher ψ discounts older observations faster4. So Vtmeasures the fraction of firms using rule M . With clustering we want
to check if firms cluster or group together over time, we do not distinguish different information sets here. We calculate a clustering value, C, at the end of each simulation. This value gives
3In this Section we will focus on using T=1000. 4In this Section we will focus on using ψ = 0.2.
each firm a value, which is 0 if both his neighbors use a different information set, 0.5 if only one uses the same information set and 1 if both neighbors use the same information set as he does:
C = f X i=1 ci, ci= 1 2(IM(Mi−1) + IM(Mi+1)), IM(Mj) = 1 if Mj = Mi, (6.1.5)
where ci is the cluster value per firm and C is the cluster value per simulation. If there is almost
no clustering, if no firm has a neighbor using the same information set, the cluster value is 0. Note that this value can not be zero for an odd number of firms, which is the case in this thesis. If there is one big cluster, so all firms use the same information set, the cluster value is equal to f , so 0 ≤ C ≤ f .
6.2 Results
To get a better understanding of the switching model, first some simulations will be done with different settings to see which settings are interesting to investigate further. The two methods that firms can choose between will be where they act as perceived monopolists versus where they have full information, because the differences are the largest here. Firms compete in a market that follows the exponential model. The parameters that will be varied are the chance a firm has the opportunity to switch (ρ), the chance of experimentation (λ) and the impact of a better performance (φ). Table 6.1 shows the chosen settings as well as the mean of the equilibrium price, the mean profit, the mean of the cluster value and the mean of the fractures at convergence of all the simulations. To check which information set a firm uses at convergence, we take the information set they used most often in the last 100 periods. We still see that the perceived monopolists perform worse compared to the firms that use full information. There does not seem to be any noticeable differences fracturing among the different settings, when φ = 25 there is a small increase in clustering. We do see an increase in clustering in (h), this is because almost all simulation converge to a situation where all firms use the same information set and thus you could say that it is one big cluster, which we will discuss later.
Table 6.1: Results of the simulations with different settings for the parameters and the parameter values are: α = 10.0, β = 1.1, γ = 0.9, c = 2.0, δ = 0.0001, θ = 0.10 and K = 3. Equilibrium values are: pN = 10.17 and πN = 73.36
a b c d e f g h ρ 0.1 0.1 0.1 0.1 1 1 1 1 λ 0.05 0.05 0 0 0.05 0.05 0 0 φ 1 25 1 25 1 25 1 25 price perceived monopolist 9.8 8.9 9.6 8.2 9 8 8.5 4.3 full information 10.2 10 10.3 10.2 10.2 10.2 10.1 8.5 total 10 9.4 10 9.2 9.6 9.1 9.3 6.4 profit perceived monopolist 70.6 63.9 69 58.8 65.2 58 61.5 30.2 full information 73.9 72.2 74.5 73.2 73.4 73 72.7 61.4 total 72.2 68 71.8 66 69.3 65.5 67.1 45.8 clustering 5.1 6.1 5.2 6.8 5.4 6.9 5.9 8.4 fracturing 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3
Figure 6.1 shows the fracturing of the mentioned settings with 9 firms. What directly stands out is that φ has the largest impact on the distribution of the information sets. When φ is low, the distribution of the information sets over all simulations is close to a Normal distribution with mean 3. This makes sense, because the chance a firm chooses a better performing information set, in this case full information, is relatively low, however not negligible, because then we would expect the mean number of firms acting as perceived monopolist to be around 4 or 5. When φ is high, there is a tendency towards lower values thus firms use full info more often. In the case that λ is equal to zero, we see that firms converge most of the time to a situation where they all use full information. At first we expected that a higher φ would end in a situation where all firms use full information, as this is the better performing information set when it competes against a perceived monopolist. However, we saw in the previous section, that while the performance of
perceived monopolist is worse compared to firms that use full information, the mean profit over all the firms decreases as more firms use full information. So once more firms use full information, it might be beneficial for all firms to switch back to acting as a perceived monopolist and then firms might switch back to using full information as this information set performs better and so on; there might be cyclical switching between information sets. After we discuss the impact of ρ and λ we will go deeper into this. The impact of ρ is low, a higher ρ flattens the distribution a little bit, as firms can switch more often. As firms that can switch more often are more likely to end in convergence or a situation with cyclical switching. The impact of λ is mostly visible in the case where ρ = 1 and φ = 25, in this case almost all simulations converge to a situation where all firms use full information. In all the other cases we see a more evenly distribution. When there is no experimentation and all firms converge to a certain method, it is harder for firms to compare their current information set to the other, so they are more likely to keep using their current information set as they chose this since it was the best performing information set. This also shows that performance differences become small when firms do not get locked up with a certain method.
Figure 6.2 shows a time-series per situation and Table 6.2 shows the distribution of the different situations. We can see that λ has a big impact on how the fracturing of a simulation goes. When firms can experiment, they are more likely to end up in a situation where they oscillate around a point and when firms can not switch, they are more likely to converge to a situation where all firms use the same information set. There are more cases of cyclical switching if firms can experiment.
Now we will take a closer look at the situation where ρ = 1, λ = 0.05 and φ = 25 (f ) and ρ = 1, λ = 0.00 and φ = 25 (h) to see if it is indeed the case that there is some cyclical switching.
Table 6.2: Distribution of the different situations
Setting f h
Convergence 13 81 Cyclical switching 26 9 Oscillation around a point 61 10
It would be realistic to assume that firms can always use more or less information and they would choose the best performing information set. We want to look at how the number of firms using a certain information set, develops over time. We assume that firms can converge to a situation where all firms use the same information set, a situation where there is cyclical switching and a situation where firms keep oscillate around a certain point. Because a simulation needs some time periods to get to a certain point where we can see if there is convergence or some switching, we will only take a look at the last 500 periods. For each simulation we calculate the variance of the fracturing value V and based on this we divide the simulations in the situations mentioned
(a) (b)
Figure 6.2: Example of a time-series of the fracturing per situation for λ = 0.05 (a) and λ = 0 (b)
before as follows:
• Convergence; if the variance of the last 100 periods is smaller than 0.1.
• Oscillation around a point; if the simulation does not converge and if the variance of the last 500 periods is below 0.17.
• Cyclical switching; if the simulation does not converge and if the variance of the last 500 periods is larger than 0.17.
7
Discussion
In this thesis we dropped the assumption that firms have complete knowledge about the market environment. We introduced Least Square learning, firms use their own price and the price of their competitors to get a better understanding of the market environment. Initially all firms used the same amount of information in their estimation. In this heterogeneous environment we found that if firms use more information the distribution of the prices at convergence became closer to the Nash equilibrium, when firms used all the prices of all the firms in their estimation, all prices at convergence were the same as in a Nash equilibrium. However, firms that act as a perceived monopolist had on average higher profits, but the differences in profits between the firms were higher. Then we looked at the situation were the firms competed in a heterogeneous environment with fixed information sets. We concluded that firms that used more information performed better than firms firms that used less information. However, the mean of the total profit over all firms decreased as more information is used. After this we introduced a switching mechanism. When firms are allowed to switch, we saw that firms were more likely to use more information in their estimation, this is what we expected. However, they did not always end up using all the information, this is due to the fact if all firms use less information, they end up with higher profits overall, also because differences in profitability might become small.
This thesis was setup to deal with the unrealistic assumption that firms have complete knowledge of the market environment. However, in this thesis we apply a market structure, where the demand is solely based on the prices of all the firms, what in itself also might be an unrealistic assumption. When firms used all prices to estimate the model, they estimated
the model correct with very little bias, for future research the demand function could also be influenced by some other factors. At last, when we implemented the switching, we let firms choose between only two information sets. This could be expanded by letting firms choose between all available information sets.
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