Least squares learning in Cournot competition

Least squares learning in

Cournot competition

Stef Hendriks

Bachelor Thesis

University of Amsterdam

Faculty of Economics and Business BSc in Econometrics program Student-number: 10135820 Supervisor: Dávid Kopányi Date: 27-06-2014

1. Introduction

Firms do not always know the full specification of the environment in which they operate. For example firms in a Cournot oligopoly might not know how the price depends on the quantities of production they set. Least squares learning is a natural method firms may use in this situation. We consider a dynamic Cournot oligopoly, where firms use a linear function to estimate their inverse demand function. This estimated function is then used to decided which quantity they set in the next period. The parameters in this function will be updated after each period, until the model converges or until the time periods have reached a maximum.

There are n firms in the model, they all try to maximize their profit by setting their quantities of production. The inverse demand function depends on the total production of all n firms. There are situations where firms might not know the production of all the other firms. In that case they can estimate the inverse demand function with a misspecified function only. The misspecification in this model is that firms only observe production quantities of m other firms, where m is smaller than n.

The main question in this paper is how the number of observed quantities affect the outcome of the model. This question will be studied with analytical and computer simulations. Similar to Brousseau and Kirman (1992) we find that the steady states of the model are a self-sustaining equilibrium. A self-sustaining equilibrium is where the real and the estimated inverse demand function coincide.

The numerical results show that when firms form average expectations about the production quantities of other firms, the more quantities the firms observe, the closer the values of the final quantities lie to each other. Also the average quantities become bigger when m increases and the quantities are more centered around the Nash equilibrium quantity. When firms form naïve expectations about the production quantities of other firms, the quantities will only reach a steady state for m=1. For m=2 the quantities will converge to a two-cycle and for higher values of m the model will not reach a steady state.

A similar kind of misspecified least squares in market competition was analyzed in other papers as well. This paper analyses a similar setup as Kirman (1983) and Anufriev et al. (2013). There are some differences with the papers just mentioned. This paper considers only homogeneous goods in a Cournot oligopoly instead of differentiated goods in a Bertrand oligopoly. The most important difference is that firms observe the action of some other firms in this paper whereas they did not have such information in the other papers. In Kirman (1983) and Anufriev et al. (2013) firms reach a steady state, where in this paper a steady state might not be reached.

Gates et al. (1977) also apply least squares learning, but they use different weighting functions to give more weight to more recent observations. In this paper firms maximize their estimated one-period profit with the production they set in the upcoming period, where in Gates et al the production quantity they set is the weighted average of the previous production quantity and the production quantity that will maximize the estimated one-period profit.

The paper is organized as follows. In the second section the market structure is presented also the Nash equilibrium of the model is derived. The third section shows the method of least squares learning and how this is used in the model. In the fourth section the results from the model are analyzed. The fifth section concludes.

2. Market structure

This paper considers a market with n firms. The firms are competing by choosing their quantities of production. Each firm produces the same homogeneous good. The inverse demand function depends on the total production level of all n firms. The function is given by:

}

0

,

max{

)

(

Q

a

bQ

γ

P

=

−

, (1) where

∑

= = n i i q Q 1

and qi is the total production of firm i. The parameters a and b are both positive. Parameter ϒ determines the convexity of the function. For ϒ=1 the function is linear. The inverse demand function is decreasing and convex when ϒ∈(0,1) and it is decreasing and concave when ϒ>1. Firms have constant and equal marginal costs c. Thus the profit function of firm i is given by:

i i

c

q

q

Q

P

q

)

=

(

)

⋅

−

⋅

(

π

, (2)

where q=(q1…qn) is a vector of production quantities. Firms try to maximize their profit function by varying their quantities, so for a unique maximum the profit function has to be concave.

A unique symmetric Nash equilibrium exists. The first order condition of firm i is given by:

0

1

⋅

−

=

⋅

⋅

−

⋅

−

−

c

q

Q

b

Q

b

a

i γ γ

_γ

_{. (3)}

Since firms are symmetric, we focus on the symmetric Nash equilibrium, where each firm produces the same quantity. Let qi=qN for every i. Then the Nash quantity is than given by:

γ γ

_γ

1 1

₍

₎

)

(

+

⋅

⋅

−

=

₋

n

n

b

c

a

q

_N . (4)

In this paper firms do not have full specification about the market structure, in particular they do not know the inverse demand function. The model assumes that the firms know their own marginal costs. We will consider a dynamic model where firms apply least squares learning about the demand conditions. We discuss the learning process in the next section.

3. Least squares learning

The method firms use in this model is least squares learning. Each firm observes its own production quantity, the production quantities of m other firms and the price they obtain from the previous periods. With these observations they estimate a perceived inverse demand function. With this perceived inverse demand function, they maximize their expected profit. The quantity the firms set in the next period will be determined by the parameter estimates and the expectation about the quantities other firms set. When firms observe the new price obtained by the objective inverse demand function, they update their parameter estimates.

As described before the objective inverse demand function is given by (1). The perceived inverse demand function in period t is given by:

t i n j j ij i t i t i

b

q

x

q

a

q

P

_, 1 , , *

)

(

)

(

=

−

⋅

+

∑

+

ε

= (5)

where xi=(xi,1,…,xi,n) is a vector with m random ones and further only zeros and xi,i is always zero. This vector specifies which production levels firm i can observe. If xi,j =1 then firm i can observe the quantities of firm j set in the previous periods, otherwise it can not. The firms

assume that the inverse demand function is linear. We mainly focus on the case ϒ=1, so the objective inverse demand function is also linear. In this way the effect of the choice of the number of m is easier to analyze, because the only misspecification is the unobserved quantity and not the functional form of the perceived inverse demand function. We will also run some simulations with ϒ≠1.The parameters ai and bi in the perceived inverse demand function are unknown and εi is normally distributed with mean zero. Given the perceived inverse demand function, the firms want to maximize their perceived profit. The quantity they will set is given by:

i n j t j ij i i t i

b

c

q

E

x

b

a

q

⋅

−

⋅

−

=

∑

=

2

]

[

1 , , , (6)

where E[qj] is the expected quantity firm j set in the next period. Firms need to form expectations about the quantities of other firms to determine their production level. We apply two different expectation formation rules in the model. First the naïve expectations:

1 , , ] [q_jt =q_jt−

E , (7)

so firms only look at the previous production quantities of the other firms. The second way is average expectations :

∑

− = − = 1 1 , , 1 1 ] [ t T T j t j q t q E . (8)

After every new observation the firms update their parameter estimates and also the quantity of their production. To obtain the estimated parameters the firm regresses the price on the total quantity it observes. The estimated coefficients are given by the OLS formulas see Heij et al. (2004) for example:

∑

∑

∑

∑

∑

∑

∑

∑

∑

− = Τ = − = Τ = Τ Τ − = Τ = − = Τ − = Τ = Τ Τ + − − + − + ⋅ − − − + − = ₁ 1 1 1 1 1 2 , , 2 , , 1 1 1 1 1 1 1 1 , , , , , ) ( 1 1 ( ) ( 1 1 ) ( ) ( 1 1 )) ( 1 1 )( ( 1 1 ( t n j t n j j ij T i j ij T i t n j t t n j j ij T i T i T j ij T i t i q x q t q x q t q x q Q P t Q P t q x q t b (9) 4

) ( 1 1 ) ( 1 1 1 1 , , , 1 ,

∑

∑

∑

− Τ = Τ − Τ + − ⋅ + − = t n j j ij T i t i T t t i q x q t b Q P t a , (10)

where price P(QT) is obtained from the objective inverse demand function in period T. When the parameter estimates are such that the expected profit of the firm is less than zero, then the firm chooses its quantity qi=0, because the firms have rather zero profit at all than a certain loss. The coefficients of ai and bi can become negative in the model, despite that this would not make any economic sense. If ai or bi is negative, firm i chooses its quantity from the uniform distribution on set S={qєRn

+ :qi>0, Pi(Q)>c, i=1,…, n}. On this set of quantities every firm makes a positive profit.

Timing can be summarized as follows. First firm i sets randomly qi,1 and qi,2 from the uniform distribution on set S. At the end of period two firm i uses OLS to obtain ai,3 and bi,3. In period 3 the firm uses the perceived model with the obtained parameters to maximize its profit. After observing the price and the quantities, the firm uses OLS again to update ai and bi. This process continues until the quantity change is small enough for each firm, so

|qi,t-qi,t-1|<δ for all i.

When more firms in the market use least squares learning then all the learning processes have an effect on each other. When the quantity of another firm changes that is not in the observation set of firm i, then also the outcome of the objective inverse demand function changes. Firm i will attribute price changes to changes in its own quantity and the quantities of the m firms it observes. Thus firms are learning about an inverse demand function that changes every period. At the beginning of the process the quantities will have a very high variation, in later periods the variation will be smaller and eventually quantities will converge to an equilibrium.

Brousseau and Kirman (1992) show that the steady states in their misspecified model are self-sustaining equilibria. In general misspecified least squares learning will not converge, but the quantity changes become smaller over time. The more observations there are, the less weight each observation has. The final point will be very close to a self-sustaining equilibrium in our model as well. In a self-self-sustaining equilibrium the true and perceived inverse demand function give the same price. In this paper firms use a misspecified model, because the true inverse demand function and the perceived inverse demand function are different. If for all firms the real inverse demand function and the perceived inverse demand function cross each other at the chosen quantity, then the model is in an equilibrium, because firms will set the same quantities in the next period.

Fig.1. Equilibrium (left) and disequilibrium (right) of the model with Least squares learning. Parameter values; a=20, b=0.5,

n=10, m=4, c=1, ϒ=1.

Fig. 1 illustrates an equilibrium (left) and a disequilibrium (right). If the real inverse demand function and the perceived inverse demand function cross each other at qt for all firms, then firms will not change their quantities the next period and at+1 and bt+1 remain the same as in the previous period. This is called the self-sustaining equilibrium (Brousseau and Kirman (1992)).

The figure shows the real and the perceived inverse demand functions, the marginal revenue and the marginal cost. The quantity that maximizes the profit is the quantity where the marginal revenue intersects with the marginal cost. We illustrate this quantity with a vertical line in the plot. For an equilibrium the real inverse demand function and the perceived inverse demand function intersect for the optimal quantity. In a disequilibrium the real inverse demand function and the perceived inverse demand function do not intersect for this optimal quantity. In a disequilibrium the firm will add the latest observation to the sample. In the next period the firm will run a new regression. The new observation will change the parameters in the perceived inverse demand function and the firm will set a new quantity.

4. Numerical Results

Simulation is used to analyze how different choices of the number of observed quantities m influence the final equilibria. Multiple simulations are done for different choices of m. Also different expectation formation rules are applied in the simulations. The model

0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 18 20 MC MR PERCEIVED REAL 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 18 20 MC MR PERCEIVED REAL 6

uses the naïve expectations (8) and the average expectations (9) to analyze if this has different outcomes. Every firm uses least squares learning. There are n=10 in the market. The marginal cost are c=1. The real inverse demand parameters are given by a=20, b=0.5 and ϒ=1. When all quantities reach accuracy of δ=10-5 the simulation stops. So the simulation stops when the absolute difference between qi,t and qi,t-1 is less than 10-5 for all i. The simulation also stops when it reaches 10.000 runs. The initial quantities are drawn from the uniform distribution on set S.

First we simulate the model under naïve expectations.

4.1 Naïve Expectations

Fig.2. Time series of quantities (left) and profits (right) for 10 firms, with m=1 quantity observations, under naïve expectations. Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig.3. Time series of quantities (left) and profits (right) for 10 firms, with m=2 quantity observations under naïve expectations.

Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig.4. Time series of quantities (left) and profits (right) for 10 firms, with m=7 quantity observations, under naïve expectations.

Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig. 2 shows the time series of production levels and profits when firms observe the production level of m=1 other firm. Fig. 2 illustrates that firms converge to a fixed value, but different firms reach different production levels. The plot shows the first 3000 runs, even though the quantities reach accuracy after 5402 runs. In this simulation quantities lie between 0.8 and 5.7 and the profits lie between 1.0 and 7.4. The corresponding Nash equilibrium quantity is 3.45 and the Nash equilibrium profit is 5.97. The real and perceived inverse demand do not coincide. The maximum differnce between the real price and the expected price for a certain firm is 0.32 and the minimum difference is 0.01. Firms do not reach a steady state.

Fig. 3 illustrates the time series of quantities and profits for m=2. The quantities do not reach a steady state, instead the figure shows a two-cycle. The plot shows only the last 10 time periods. Firms react on the quantities of the other firms from the previous period. So when all the firms have low quantities in one period, all firms will react on each other by setting a higher quantity, so that all the firms have high quantities in the next period, all the firms will react now with low quantities and so forth. If we look at the average of the last two time periods we see that the quantities lie between 2.1 and 3.5 and the profits lie between 7.8 and 13.7. The average profit for m=2 always exceeds the Nash equilibrium profit and the profit for m=1.

Fig. 4 illustrates the time series of the quantities and profits for m=7. For m=7 there is no convergence. Also there is no pattern in the time series where for m is 2 there was a certain pattern. If we take the average of the last ten time periods we see that the quantities lie between 0.8 and 6.7 and the profits lie between 0.6 and 8.4. Here we see that the quantities and the profits lie closer together for m=1 and m=2 than for m=7.

The results relate to Theocharis (1960). Theocharis states that when competitors in a Cournot oligopoly believe that the production quantities of the other firms remain unchanged different stability results will occur according to the number of competitors. The cases only hold when the inverse demand function is linear and the marginal costs are constant. When there are two competitors the result is always that they converge to the Nash equilibrium, when the number of competitors is three there will be finite oscillations around the equilibrium and when the number of competitors is bigger than three there will be instability. In the misspecified model the firms can observe the quantity of m+1 firms. When m=2 there are oscillations around the equilibrium, when m is bigger than two there is instability and there is stability for m=1. Thus, our results suggest that the results of Theocharis(1960) hold also when firms do not know the true demand structure and they apply least squares learning.

When firms use naïve expectations to determine which quantity the other firms will set in the next period, there is only convergence when firms observe one other firm. To overcome that firms not reach a steady state for m is bigger than 1, there is also the possibility to look at a nonlinear inverse demand function. The parameter ϒ will be changed in the model. We will run two more simulations, one for ϒ<1 and one for ϒ>1. Theocharis’ results hold under linearity. We might observe convergence to a steady state for higher values of m when the true inverse demand function is nonlinear.

Fig.5. Time series of quantities with ϒ=0.5 (left) and quantities with ϒ=1.5 (right) for 10 firms, with m=3 quantity observations, under naïve expectations. Parameter values: a=20, b=0.5 and c=1.

Fig. 5 illustrates that the change of ϒ has no influence on the convergence of quantities. When ϒ changes only the real inverse demand function becomes nonlinear. So the perceived inverse demand function stays linear. When ϒ is not equal one, then for m=1 the quantities will still reach a steady state and for m=2 quantities will still oscillate. Since the dynamical properties of the model are very different for different values of m, we will not run additional simulations to explore the distribution of the possible outcomes.

When firms form naïve expectations then for m is bigger than 2 the model will not converge for any ϒ.

4.2 Average Expectations

Now we will analyze the model under average expectations. Now the question is if firms do not observe their competitors quantity with naïve expectations but with average expectations (8) will the time series converge for every m. Now firms think that the other firms next quantity will be the average of all the previous quantities the other firms have set. We use the same parameters as before. The main question for the simulations hereby is whether the quantities converge to a steady state, how quickly the quantities converge and how m affects the outcome of the model.

Fig.6. Time series of quantities (left) and profits (right) for 10 firms, with m=1 quantity observations, under average expectations. Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig.7. Time series of quantities (left) and profits (right) for 10 firms, with m=2 quantity observations, under average

expectations. Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig.8. Time series of quantities (left) and profits (right) for 10 firms, with m=7 quantity observations, under average expectations.

Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig. 6 shows the time series of quantities and profits in the case firms observe the quantity of only one other firm. Only the first 20 runs are showed. The model converges to a steady state after 1170 runs. The final quantities vary from 0.8 to 8.0 and the profits vary from 2.2 to 22.3. In this case there are 8 firms which have a higher profit than the Nash equilibrium profit.

Fig. 7 illustrates the time series of quantities and profits when firms observe the quantities of m=2 other firms. The plot shows only the first 20 periods, because after that the plot does not change that much. The figure shows that there is convergence to a steady state. In the first few periods the quantities are volatile, but in later periods the volatility becomes smaller. This is already different under naïve expectations. The quantities in this case reach the predefined accuracy after 5672 periods. The end quantities lie between 1.2 and 5.9. The end profits lie between 1.6 and 7.8. The corresponding Nash equilibrium quantity is 3.45 and the Nash equilibrium profit is 5.97. So in this case some firms have a higher profit than the corresponding Nash equilibrium profit and some firms have a lower profit. To be more precise, three firms have a higher profit and seven firms have a lower profit. The difference between the lowest quantity and the highest quantity is smaller than for m=1 and also the number of firms which profit exceeds the profit of the Nash equilibrium is lower in this case.

Fig. 8 illustrates the time series of quantities and profits for m=7. The first 20 runs are showed. The model runs for 10.000 time periods, because accuracy is not reached in this case. The difference between quantities in time period 10.000 and time period 9999 is of magnitude 10-4. The model for m=7 does converge slower than the model for m=2. For m=7

the quantities lie between 2.8 and 4.8. So for m=7 the final quantities lie closer together than for m=2 and for m=1. Also the profits lie closer together. The profits lie between 4.3 and 7.2. There are two firms which have a higher profit than the Nash equilibrium profit and eight firms which have a lower profit.

After simulating the model for all different values of m from 1 to 9, we can observe a certain pattern. The lower the value of m the faster the model converges and the higher the value of m the smaller is the difference is between the final quantities of all firms. When m is bigger than 2 the model does not reach accuracy within 10.000 runs except for m=8 and m=9. When firms use average expectations the model always reaches a steady state, because for all m the difference between the last two quantities after the model stops is always smaller than 10-4. For naïve expectations it is best for firms to observe the quantity of two other firms, because the average profit is higher for m=2 than for m=1. For average expectations it is best for firms to observe only the production quantity of one other firm. When firms form naïve expectations, the model only converges for m=1.

Previous results are based on a single simulation. More simulations are needed to see if the results hold in general. We will run 100 simulations to get stronger results.

Fig. 9. Scatter plot (left) and histogram (right) of final production quantities for m=1. The red line shows the Nash equilibrium quantity. Parameter values: a=20, b=0.5, ϒ=1,n=10 and c=1 .

Fig. 10. Scatter plot (left) and histogram (right) of final production quantities for m=2. The red line shows the Nash equilibrium quantity. Parameter values: a=20, b=0.5, ϒ=1,n=10 and c=1.

Fig. 11. Scatter plot (left) and histogram (right) of final production quantities for m=7. The red line shows the Nash equilibrium quantity. Parameter values: a=20, b=0.5, ϒ=1, n=10 and c=1.

Table.1. Descriptive statistics for different values of m. Parameter values: a=20, b=0.5, ϒ=1, n=10 and c=1.

M=1 M=2 M=7 qN Average qi 3.39 3.43 3.50 3.45 Average πi 6.84 6.24 5.24 5.97 Median qi 3.11 3.25 3.45 3.45 Convergence time 5957 7781 10.000 - 14

Table. 1 shows a certain pattern for the final quantities and the profits. For higher m the average production quantities become larger and the average profit becomes smaller. Also the model needs more periods to reach a steady state. The median of the quantities shows that for larger m the final quantities lie more centered around the Nash equilibrium quantity. This is because when m becomes larger the more information firms have about the market and they play the best response against more firms. For m=7 all the simulations run for 10.000 periods, but the average difference between qt and qt-1 for t=10.000 is of magnitude 10-4.

Fig. 9, 10 and 11 show the final quantities firms will set after the simulations stop. In this case the model does 100 simulations. These graphs give a good impression of the effect of m on the outcome of the model. For m=7 the histogram is more peeked than for m=1 and m=2. The scatter plots show that the final quantities lie closer together for higher m. Note that the values on the y-axis differ for the different plots. On the x-axis from the scatter plot every firm has its own point, so for the first simulation qi is given for 1 till 10 and for the second simulation qi is given for 11 till 20 etc. The x-axis from the histogram gives the quantities that will be reached in the steady state, the y-axis gives the occurrence of all the 100 simulations. The plots also show that the final quantities lie around the Nash equilibrium quantity. The value of m has no effect on this.

Fig. 9, 10 and 11 show the outcome of the model when firms observe the quantities of the other firms with average expectations. Now we will observe how the results change if firms use naïve expectations to observe the quantities. Only for m=1 the two different models can be compared, because for m=1 both the naïve and the average expectations converge.

Fig. 12. Scatter plot (left) and histogram (right) of final production quantities for m=1, when firms observing with naïve

expectations. The red line shows the Nash equilibrium quantity. Parameter values: a=20, b=0.5, ϒ=1,n=10 and c=1.

Fig. 9 and Fig. 12 will be compared. Fig. 12 does not show immediately that much difference with Fig. 9. The average production quantities are 3.46 and the average profit is 5.95, which lie both really close to the Nash equilibrium values. The average number of periods firms need to reach a steady state is t=9999. The median of the quantities is 3.23. As we can see with the naïve expectations the firms quantities and profit lie closer to the related Nash equilibrium values than when firms use average expectations. When firms observe the quantity of only one other firm, they would rather choose average expectations, because on average they would make more profit, than when they use naïve expectations.

5. Concluding Remarks

This paper demonstrates that with average expectations the more individual quantities firms can observe, the closer the steady states of the production quantities lie to the Nash equilibrium. The less individual quantities firms observe the faster firms reach a steady state. There is a difference in the results when firms form naïve expectations instead of average expectations. For naïve expectations the quantities will not always reach a steady state.

This paper analyzes how the number of observed quantities affect the outcome of the model. The model uses least squares learning in a Cournot oligopoly with n firms and homogenous goods. Firms do not know the real inverse demand function. The firms use a estimated inverse demand function to determine the quantities.

Three different setups have been analyzed. The first setup is when the real inverse demand function is linear and the firms form naïve expectations. When firms observe the production level of only one other firm the quantities will converge, when firms observe the production levels of two other firms the quantities will reach a two cycle and when firms observe the production levels of three or more other firms the quantities do not converge. Theocharis’ (1960) paper is used for the economic theory behind this. When the inverse demand function is linear, the marginal cost is constant for every firm and the firms believe that the production quantities of the other firms remain unchanged, the model will not converge when there are more than 3 competitors in the market.

The second setup is when the real inverse demand function is nonlinear and the firms form naïve expectations. The perceived inverse demand function is still linear. We run a simulation where firms observe the production levels of three other firms. Here the quantities also do not converge. When firms observe the production levels of two or less other firms the

quantities still converge. We conclude that Theocharis’ theory is objected on the perceived inverse demand function in this case.

The third setup is when the real inverse demand function is linear and the firms form average expectations. We find that the more quantities the firms observe, the more the quantities are centered around the Nash equilibrium quantity. This is because when the number of quantities the firms observe increases the more information firms have about the market and they play the best response against more firms. Also the less quantities the firms observe the faster the model converges. Firms make more profit on average, when firms observe less quantities.

The analysis can be extended in multiple ways. First we can introduce a weighting function, so that older observations have less weight than the newer observations. Older observations have less information about the inverse demand function than the newer observations, because they are more volatile. Another adjustment can be that firms do not use least squares learning in their model, but another learning method, for example gradient learning. Also interesting is how the outcome of the model changes when there is Bertrand competition instead of Cournot competition and what happens if the goods are not homogenous but heterogenous. Furthermore, it might be interesting to change the market environment in a more complex one, such that firms do not only make decisions about the quantity they set, but also decisions about location and if they invest or not.

Acknowledgments

I am grateful to Dávid Kopányi for his help and suggestions that improved the quality of this paper.

Appendix A: Derivation of the best response quantity (6)

First we consider the expected profit of firm i. Where the expected price is given by (5).

Then we will remove the brackets.

We will derive the first order conditions

Then we will move all the terms with qi to one side

We can see because the firms can not produce negative quantities that bi>0, and because bi>0 and qi>0 ai>bi>0 and ai>c. These constraints are used in the model.

i n j j j i i i i n j i i j j i i i n j j j i i i i i n j i i j j i i i i i i i n j i i j j i i i i i

b

c

q

E

x

b

a

q

q

b

c

q

E

x

b

a

c

q

E

x

b

q

b

a

FOC

q

c

q

q

E

x

b

q

b

q

a

q

c

q

q

E

x

q

b

a

⋅

−

⋅

−

=

⋅

⋅

=

−

⋅

−

=

−

⋅

−

⋅

⋅

−

⋅

−

⋅

⋅

−

⋅

−

⋅

=

⋅

−

⋅

⋅

+

⋅

−

=

∑

∑

∑

∑

∑

= = = = =

2

]

[

2

]

[

0

]

[

2

:

]

[

]))

[

(

(

1 , 1 , 1 , 1 , 2 1 ,

π

π

18

Appendix B: simulations for 2 firms in the market

Fig. 13. Final production quantities where n=2 in histogram for m=1 (left) and m=0 (right), when firms observing with average

expectations. The red line shows the Nash equilibrium quantity. Parameter values: a=20, b=0.5, ϒ=1 and c=1.

M=0 (AE) M=0 (NE) M=1 (AE) M=1 (NE)

Average qi 11.47 11.47 12.67 12.67 Average πi 86.37 86.37 80.22 80.22 Median qi 12.58 12.58 12.67 12.67 Convergence time 1209 1209 2073 25

Table.2. Descriptive statistics for different values of m with average expectations(AE) and naïve expectations(NE). Parameter

values: a=20, b=0.5, ϒ=1, n=2 and c=1.

In addition to the paper so far, we also simulate the model for a market where there are only two competitors. For n=2 we expect that for every m (m=0 and m=1) the quantities will reach a steady state with naïve expectations.

Fig. 13 illustrates the quantities in the steady states for 100 simulations with average expectations. Table. 2 gives some important values for the final quantities for both expectation rules. When m=0 the outcome of the model is the same for naïve and average expectations. For m=0 the firms do not observe the quantities of the other firm, so in this case it does not matter what the expectation of the other firms’ quantity is. We see that when firms do not observe the other firms’ quantity the average quantities are smaller than the

Nash equilibrium quantity (12.67) and the average profit is higher than the Nash equilibrium profit (80.22).

When firms observe the other firms’ quantity there is a difference for the different expectations. For m=1 the Nash equilibrium is the unique steady state of the model under both expectation rules. Also the average profits are equal to the Nash equilibrium profit (80.22). But when we observe the two histograms there is a clear difference. When firms use average expectations we see two peeks in the histogram, where for the naïve expectations all the final quantities lie very close to the Nash equilibrium quantity. This is because the average expectations use all the past quantities. This means that the observed quantities in the begin of the simulation also have an influence of the expectated quantity in the next period. The observations in the begin of the simulation have a high votality, but they have just as much weight as the last observation. The naïve expectations also reach faster a steady state than the average expectations.

For both the average and the naïve expectations we can conclude the same as mentioned earlier in this paper. When m is bigger the final quantities lie closer together and closer to the Nash equilibrium quantity. For smaller m the average quantities will be smaller and the average profits will be bigger. The only thing different from mentioned earlier in the paper is that for naive expectations the quantities will reach faster a steady state for bigger m. For average expectations the firms will still reach faster a steady state for smaller m.

Appendix C: simulations for Q for 6 firms in the market

Fig.15. Histogram for Q (left) and scatter plot for individual quantities where n=6 for m=1. Parameter values: a=20, b=0.5, ϒ=1

and c=1. 27 28 29 30 31 32 33 34 35 36 37 0 2 4 6 8 10 12 14 0 100 200 300 400 500 600 0 2 4 6 8 10 12

quantities in final stage

quant it ies firm# Occu re nce Q 20

Fig.16. Histogram for Q (left) and scatter plot for individual quantities where n=6 for m=2. Parameter values: a=20, b=0.5, ϒ=1 and c=1.

Fig.17. Histogram for Q (left) and scatter plot for individual quantities where n=6 for m=4. Parameter values: a=20, b=0.5, ϒ=1

and c=1. M=1 M=2 M=4 QN Convergence time 5445 8354 9299 - Average Q 31.83 32.53 32.46 32.57 Median Q 31.93 32.52 32.37 32.57 Variance Q 2.51 0.58 0.11 0

Table.3. Descriptive statistics for different values of m. Parameter values: a=20, b=0.5, ϒ=1, n=6 and c=1.

In the paper we have seen that when m increases the individual quantities lie more centered around the Nash equilibrium quantity. If we look at the scatter plots for the individual

29 30 31 32 33 34 35 0 2 4 6 8 10 12 14 16 18 0 100 200 300 400 500 600 2 3 4 5 6 7 8 9 10 11

quantities in final stage

quant it ies firm# 32 32.5 33 33.5 34 34.5 0 2 4 6 8 10 12 14 16 0 100 200 300 400 500 600 4 4.5 5 5.5 6 6.5 7 7.5

quantities in final stage

quant it ies firm# Occu re nce Occu re nce Q Q 21

quantities in Fig. 15, Fig. 16 and Fig. 17 we can see that this is still the case. Note the difference in the values on the y-axis for the different values of m.

If we look at the summation of the individual quantities for all the simulations we observe something different. Table. 3 shows some important descriptive statistics. The average Q and the median of Q show that for m=2 the summation of the quantities lie more centered around the summation of the Nash equilibrium quantities. Fig. 15, Fig. 16 and Fig. 17 illustrate also the histograms for Q. We observe that for bigger m the histogram of Q is more peeked. In the histograms we can also see that for m=2 the summation of the quantities lie more centered lie more centered around the summation of the Nash equilibrium quantities. Table. 3 also shows the variance of Q. We can see that for bigger m the variance decreases.

References

Anufriev, M., Kopányi, D., Tuinstra, J. (2013).”Learning cycles in Bertrand competition with differentiated commodities and competing learning rules.” Journal of Economic Dynamics

and Control, 37: 2562-81

Brousseau, V., Kirman, A. (1992). “Apparent convergence of learning processes in mis-specified games.” In: Dutta, B., Mookherjee, D., Parthasarathy, T., Raghavan, T., Ray, D., Tijs, S. (Eds,), Game theory and Economic Applications, Springer-Verlag

Gates, D. J., Rickard, J. A. and Wilson, D. J. (1977). “A Convergent Adjustment Process for Firms in Competition.” Econometrica, 45:1349-64

Heij, C., de Boer, P., Franses, P. H., Kloek, T., van Dijk, H. K. (2004). Econometric Methods with Applications in Business and Economics

Kirman, A.(1983). “Mistaken beliefs and resultant equilibria.” In: Teglio, A., Alfarano, S., Camacho-Cuenca, E., Ginés-Vilar,M., Managing Market Complexity: The Approach of Artificial Economics, Springer-Verlag, pp. 155-166

Theocharis, R,D. (1960). “On the stability of the Cournot Solution on the Oligopoly problems.”

The Review of Economic studies, 27: 133-34