Merel Joosten 6137636 Bachelor thesis
New expectation formation rules in Cournot competition
27 June 2012
BSC Econometrics and Operational Research
University of Amsterdam
Index
1. Introduction 1
2. The market structure 3
3. Dynamic Model 4 3.1 Adaptive expectation 5 3.2 Trend extrapolation 8 3.3 Heterogeneous expectations 11 4. Conclusion 15 Appendix 17 Bibliography 24
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1. Introduction
The competition among firms can be modeled in different ways; In some markets firms compete in price with each other, In other markets firms compete in quantity with each other. This paper is about a market where firms compete in produced quantity with each other. This is also known as Cournot competition. The Cournot competition has several characteristics: the quantity the firms produces is decided upon independently, the firms produce simultaneously, the product the firms produce in this market is homogenous.
Theocharis (1960) describes a model with constant marginal costs and a linear demand function. In his model the Cournot adjustment process is applied, which means that sellers in the market believe that their competitors will produce the same quantity in the next period as they produced in the previous periods and they play the best response against these quantities. He argues that in his model there is an equilibrium that is dynamically stable only if there are two sellers in a Cournot competition market. With three sellers in the market there will be finite oscillations. If there are more than three sellers in the market the Cournot solution will always be unstable.
The model in this thesis builds on the paper of Theocharis (1960). The market form that is considered is a Cournot oligopoly with a linear inverse demand function and constant marginal costs. The main difference is that alternative expectation formation rules are applied in the model; adaptive expectations and trend extrapolation. With these new expectation formation rules the dynamics of the process are investigated. So maybe it is possible that in a Cournot competition the equilibrium is stable even if the number of sellers is greater than three. This may be possible if the dynamics of the model are changed with another stabilizing factor for instance a different rules to form an expectation of the production level of the other firms. I investigated whether these expectation formation rules lead to a stable equilibrium of the Cournot competition. So the central question of this research: how does alternative expectation rules affect the dynamics? Further this study investigates what happens with the dynamics of the model if the number of firms is varied in the market. The results that were found is that with both of the new expectation formation rules it is possible that there is stability even with more than three firms that are active in the market. But instability does increase if the numbers of the firms increase.
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There were other papers in the literature that investigated whether the equilibrium can be stabilized. Fisher (1961) questions the results of Theocharis. Fisher argues that the result of Theocharis has limits, since the Cournot solution does not have to be unstable if the number of firms in the market increases. He proposes alternative adjustment processes, but in most of these processes the tendency to instability does increase if the number of sellers increases. For instance one of the stabilizing factors that Fisher mentions is increasing marginal costs.
Canovas et al. (2008) point out an major important issue in a Cournot competition market. This is the fact that if you consider the global dynamics of a market rather than the local stability alone the non-negativity constraint on quantities must be taken into account. This makes the model nonlinear. This is a very important issue, because if a model becomes nonlinear, the dynamics of the process may change. In this case the dynamics become more complicated than in a linear model. Canovas et al. find that firms with constant marginal costs can potentially be infinitely large. So he states that in the Theocharis model you are adding potentially infinitely large firms in the market. This results in destabilization. So Canovas et al. argue that with increasing marginal costs and the non-negativity constrain this will result in a stabilizing situation.
Gates et al. (1977) consider a Cournot duopoly in their research. In their model a firm decides its next production level entirely upon its previous production and profits, since a firm is not capable to observe what the other firms are producing. This research does also use a different method to formulate expectations about the quantity the other firms will produce. Gates et al. (1977) find that with these new expectation formation rules for more than two firms the quantity the firms produces does converge to an equilibrium after two time periods, but this is not the Nash equilibrium.
This research shows in section 2 the market structure that is used in the model. For this market structure the Nash equilibrium and the collusive equilibrium are derived. In section 3 the two new expectation formation rules are introduced. An analytical proof is given to show if there is stability in the market. In this section the dynamics of the model are investigated when some firms use the adaptive expectations rule and the other firms use trend extrapolation rule. In section 4 of this thesis a conclusion is drawn.
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2. The market structure
We consider a market structure in this model is Cournot competition, so the firms in the market compete in quantity with each other. The products that the firms produce is homogenous and the firms produce simultaneously. The inverse demand function is linear and it is of the form P(Q)=a-b*Q. Where Q ∑ i. The marginal costs are constant for every firm in this Cournot competition market, so MC=c. For the parameters there must hold that a,b and c>0.
There exist an unique symmetric Nash equilibrium in the market structure that is considered. Now the Nash equilibrium will be derived. For the general case the profit of firm i is given by:
(a-b*(qi+ )-c)qi (1)
Where for ∑ (2)
The First Order Condition, the FOC, needs to be calculated if the profit function has to be maximized. For this profit function the FOC is:
(3)
Using equation (3), the reaction functions can be calculated. And since the marginal costs are the same for every firm, firms produce the same quantity in equilibrium: q1=q2=…=qn. So the
general reaction function is: Bi(Q-i) =
(4)
And in the Nash equilibrium the optimal quantity firm i produces is: =
( ) (5)
The corresponding profit for firm i is:
( )
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The collusive outcome is also characterized as well, since it is not certain yet to which outcomes the dynamic model can lead to and the collusive outcome is an important benchmark outcome in market models. The joint profit is given by:
∑ ∑ ( ) = ( ) (7)
So the maximization problem can be solved with the FOC of this profit function:
∑
= (8)
From which Q ∑ i.
Since the firms are symmetric, the focus in on the symmetric collusive outcome, Thus: q1=q2=…=qn. Q =q1+q2+…+qn = Q= n = = (9)
And the corresponding profit in this market is:
( )
(10)
3. Dynamic model
A dynamic model is time based. In this dynamic model the firms compete with each other still in a Cournot competition market but the firms compete now over time. In this dynamic there are expectations on the quantity the other firms produce . The profit function in the dynamic model is:
Πi,t= (a-b*(qi,t+ )-c)qi,t (11)
And to maximize this profit the FOC is needed and in this case the FOC will be:
(12)
In this Cournot competition the Cournot adjustment process does not hold. A Cournot adjustment process is the assumption that the competition in the market produces the same quantity as the produced in the previous period. Instead there are two new expectations formation rules to form the expectation for the quantity the other firms will produce.
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In the first case the expectation formation rule of adaptive expectations is considered:
( ) (13)
The expected quantity of all the firms except firm i in period t, , depends on two variables. The first variable is and this is the expected output level of all the firms except of firm i in the previous period. The second variable is the actual output level of all the firm except of firm i in the previous period. In (1) w is the weight of importance of the expected output level in the previous period for w. For w there holds 0≤w≤1. Firms adjust their expectations in the direction of the previous observed value. If the other firms produced more than expected then the expectations about the quantity the other firms will produce next period is increased. In this case w will be decreased so to observed will become more
important.
In the second case there holds another rule to form the expectations of the other firms in the market. Here the trend extrapolation expectation formation rule holds:
( ) (14)
Here is λ the positive or negative weight of a trend in the expected output level of firm i. If λ∈(0,1) the expected quantity increases with a lower pace. If λ>1 the expected quantity
will explode, since will increase very fast. If λ∈(-1,0) this means that the expected
quantity will alternate with a lower pace between negative and positive increase. If λ<-1 this means that the expected quantity will alternate extremely between positive and negative outcomes and this will explode.
3.1 Adaptive expectations
The next step is to investigate is what happens to steady state of the market if adaptive expectations rule is applied. In the general form of the adaptive expectations rule there holds that:
( )
And in steady state there holds that: and = so:
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(1-w) =(1-w)
=
It is proved that the weight w will drop out the equation. So it does not have any effect in the long run for which w there will be used. Firms play the best response to the expectations of the quantity the other firms produce, so the reactionfunction is:
Bi(Q-i) =
and since in the steady state there holds that: = the
reactionfunction becomes Bi(Q-i) =
. And with further calculation as before the
Nash equilibrium quantity (5) is found. So with adaptive expectations as formation rule to form expectations of the quantity the other firms produce, the Nash equilibrium quantity is reached in a steady state.
Next we will analyze the stability of the Nash equilibrium. For the adaptive expectations the profit function, the FOC are all based on time now so (11) and (12) hold as the profit function and the first order condition.
The next period quantity of firm i is given by the same formula as before:
(15)
To determine if the steady state is stable the eigenvalues of the Jacobian Matrix at the steady state must be determined. First the model must be transformed to a form that is appropriate for stability analysis: The state variables should depend on the 1-lagged values of the state variables only. The system is characterized by the following equations:
( ) (16)
( ) ( ) (17)
To do so ( ) is transformed in this way:
( ) ( ( )) ( ( ) ). Thus the system can be expressed in the following form:
7 ( ) = ( ) ( ) and with J=( ) (18)
Where matrix is a nxn matrix:
( ) ( ) Matrix is a nxn matrix: ( ) Matrix is a nxn matrix: ( ) ( )
And the matrix is a nxn matrix:
( )
The steady state is stable when the absolute value of the eigenvalues of the Jacobian Matrix are smaller than one.
Proposition 1. To calculate the eigenvalues of the Jacobian matrix above there must hold that: Jx=λx. Where λ are the eigenvalues of the Jacobian Matrix. In this particular matrix it can be shown that for the calculation of the eigenvalues, the only eigenvalues that have to be derived are the eigenvalues of matrix .
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λ=( ) with a multiplicity of (n-1)
λ= ( )( ) with a multiplicity of n this is the case if
And if than w is an eigenvalue of the Jacobian with multiplicity n.
For stability the eigenvalues constructed for matrix must be smaller than one. The first (n-1) eigenvalues are always smaller than 1 since for w there holds that 0≤w≤1. So the eigenvalues λ=( ) will never be bigger than one in absolute value. For the last eigenvalue λ= ( )( ) it does depend on the value of w. For n=2, than for every value of w the eigenvalue is smaller than one for n>2 the eigenvalue is only smaller for certain values of w. For the steady state to be stable there must hold:
│ ( )( ) <1 (n-1)(1-w) < 2 w>1 =
n>2 there holds that if w> the system will still be stable. So with adaptive expectations the stability does increase for the number of firms but the for the values of w . So the instability does increase if the number of sellers in the market increases. In the case of than w is an eigenvalue of the Jacobian with multiplicity n. This eigenvalue is always smaller than one since there holds 0≤w≤1.
3.2 Trend Expectations
The next step is to investigate is what happens to steady state of the market if trend extrapolation rule is applied. In the general form of the adaptive expectations rule there holds that:
( )
And in steady state there holds that: and = so:
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=
You can see that the weight λ will drop out the equation. So it does not have any effect in the long run for which λ there will be used. so the reactionfunction is:
Bi(Q-i) =
and since in the steady state there holds that: = the
reactionfunction becomes Bi(Q-i) =
. And with further calculation as before the
Nash equilibrium quantity (5) is found. So with trend extrapolation as formation rule to form expectations of the quantity the other firms produce, the Nash equilibrium quantity is reached in a steady state.
Next we investigate whether the Nash equilibrium is locally stable. For the trend extrapolation the profit function, the FOC are all based on time now and is the same as in (11) and (12) The next period quantity of firm i is given by the same formula as before:
(19)
To determine if the steady state is stable the eigenvalues of the Jacobian Matrix at the steady state must be determined. First the model must be transformed to a form that is appropriate for stability analysis: The state variables should depend on the 1-lagged values of the state variables only. The system is characterized by the following equations:
( ) (20)
To do so ( ) is transformed in this way:
( ) ( ( )) ( ( ))
( ) (21)
10 ( ) = ( ) ( ) and Jacobian=( ) (22)
Where matrix is a nxn matrix:
( ) ( ) Matrix is a nxn matrix: ( )
Matrix is an Identity nxn matrix:
( )
And the matrix is an nxn matrix that consist only out of zero’s so for we can write =[0].
To investigate whether the steady state of the system is stable, there must be calculated if the absolute value of the eigenvalues of the matrix are smaller than one and since the last part of the diagonal matrix is a matrix that consist only out of zero’s, the matrix is not a nonsingular matrix. So for this part the eigenvalues are zero and the only part of the Jacobian matrix there has to be looked at is the part of .
Proposition 2 To calculate the eigenvalues of the Jacobian matrix above there must hold that: Jx=μx. Where μ are the eigenvalues of the Jacobian Matrix. In this particular matrix it can be shown that for the calculation of the eigenvalues, the only eigenvalues that have to be calculated are the eigenvalues of matrix .
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=( ) with a multiplicity of (n-1)
= ( )( ) with a multiplicity of n
For stability the eigenvalues constructed for matrix must be smaller than one in absolute value. The first (n-1) eigenvalues =( ) are always smaller than one in absolute value if -3<λ<1. For the last eigenvalue it does depend on the value of n and λ together, for n=2 the same eigenvalue appears so -3<λ<1 must hold to be stable. After that μ is only stable for certain values of λ in combination for certain values of n. The last eigenvalue = ( )( ) is only smaller than one in absolute value if there holds:
│ ( )( ) <1 ( )( )
<1+
So ( )( ) ( )( ) ith trend extrapolation the stability does increase for the number of firms for certain values of λ. The range of λ becomes smaller if n increases, so the instability does increase if the number of sellers in the market increases.
3.3 Heterogeneous expectations
The mixed case the first m firms use the adaptive expectations. While the last (n-m) firms use trend extrapolation. This is very interesting since it is very realistic that in a market where more than one firm operates, that the firms will operate in different ways. So it is possible that these firms use different expectation formation rules. We will first determine the steady state of model with the heterogeneous firms. For both the expectation formation rules we have seen previously that in the steady state there holds:
=
Firms play the best response to the expectations of the quantity the other firms produce, so the reactionfunction is:
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Bi(Q-i) =
and since in the steady state there holds that: = for both rules the
reactionfunction becomes Bi(Q-i) =
. And with further calculation as before the
Nash equilibrium quantity (5) is found. So the Nash equilibrium remains the steady state under heterogeneous expectations as well.
Next we investigate whether the Nash equilibrium is locally stable
The next period quantity of firm i is given by the same formula as before:
(19)
To determine if the steady state is stable the eigenvalues of the Jacobian Matrix at the steady state must be determined. First the model must be transformed to a form that is appropriate for stability analysis: the state variables should depend on the 1-lagged values of the state variables only. The system is characterized by the following equations:
= ( ( ) ) i=1,…,m
( ) i=1,…,m
= ( ( )) i=m+1,..,n
The state variables should depend on the 1-lagged values of the state variables only. Now the state variables are and
Thus the system can be expressed in the following form:
( ) = ( ) ( )
with the Jacobian=(
)
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In this Jacobian matrix there are 16 matrices, 5 of these matrices only consist out of zero’s. This are the matrices: M,P,U,X,Y. The other matrices are of the following forms:
Where matrix is a mxm matrix: Matrix K is a mx(n-m) matrix
( )( ) K= ( )( )
Matrix L is an mxm matrix: Matrix N is is a (n-m)xm matrix
( ) N= ( )( ) Matrix O is a (n-m)x(n-m) matrix: ( )( )
Matrix Q is a (n-m)xn matrix the first m columns are filled with . The last (n-m) are filled with and zero on the diagonal:
( )
Matrix R is a mxm matrix: Matrix S is a mx(n-m) matrix
( ) ( ) S=( ) ( )
14 Matrix T is a mxm matrix: ( )
the matrix V and W together form a nxn identity matrix; The upper part of V is the m×m identity matrix, the lower part consists of zero’s. Similarly, the lower part of W is the (n-m)×(n-m) identity matrix, the upper part consists of zero’s only.
(V ) ( )
The general case with n firms turns out to be very complicated, the eigenvalues cannot be determined analytically. Therefore we focus on simpler cases and calculate the corresponding eigenvalues using Mathematica.
First we consider n=2 and m=1, that is there are two firms on the market, one of them uses adaptive expectations while the other one uses trend extrapolation. Here μ=0 is the only eigenvalue with multiplicity of 1. So for two firms in the market there is always stability in the steady state and it does not depend on the weights w or λ.
Next we consider n=3 and m=2. Thus, there are 3 firms on the market and the first 2 use adaptive expectations. Here μ=0 with a multiplicity of 3 and μ= with multiplicity 1. The absolute value of the of the last eigenvalue is smaller than one if <1 and this is the case for -3<w<1. However for w there holds 0≤w≤1. So this eigenvalue is always smaller than one. So in this market the stability only depends on the value of w that is determined by the two firms that use adaptive expectations to form expectations.
Finally we consider n=3 and m=1. Thus, there are 3 firms on the market and only the first one uses adaptive expectations. The eigenvalues that belong to this Jacobian are μ=0 with a multiplicity of 2 , μ= ( √ ) with multiplicity of 1 and
μ= ( √ ) also by multiplicity of 1. The absolute value must be smaller than one. For the condition ( √ ) reduces to <λ<3-2√ and
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and ( √ ) leads to <3-2√ . The Nash equilibrium is stable only if <λ<3-2√ .
Conjecture By comparing the different values for n from above and the other in the Appendix we conclude that if n increases the eigenvalues do not change, only for the multiplicities it holds that:
For μ= the multiplicity is (m-1) where m is the number of firms using adaptive expectations.
For μ= ( √ ) the multiplicity is (n-m-1) where (n-m) is the number of firms using trend extrapolation
For μ= ( √ ) the multiplicity is (n-m-1) where (n-m) is the number of firms using trend extrapolation.
And for μ=0 with multiplicity is n if m≥(n-m) and (n-1) if m>(n-m)
So if n stays fixed but the number of firms who use adaptive expectations and firms who use trend extrapolation changes, the number of eigenvalues stays the same. However the multiplicity of the eigenvalues; μ= , μ= ( √ ), μ= ( √ ) do change.
There is one problem with the results that we found. Mathematica could not find all the eigenvalues, therefore the conditions that we derive are necessary but not sufficient for stability. For stability the number of eigenvalues that are found must be equal to the multiplicity. And this condition does not hold here.
4. Conclusion
In this thesis there is researched what happens with a Cournot competition market, if there are two new rules to form expectations on the quantity the other firms in the market produce. The two expectation formation rules that were applied were adaptive expectations and trend
extrapolation. The main question was what happened to the dynamics of this model with these two new expectation formation rules.
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The market form in the model was Cournot competition. The inverse demand function was linear and the marginal costs were constant. For the adaptive expectations firms adjust their expectations in the direction of the previous observed value. If the other firms produced more than expected then the expectations about the quantity the other firms will produce next period is increased. For the trend extrapolation rule firms adjust their expectations in the assumption that there is a positive or negative trend in the quantity the other firms will produce the next period.
The main result that was found in this thesis is that with these two new expectation formation rules it is possible that there is stability in the steady state for more than three firms in the market. For the adaptive expectations there is stability for n firms if w . For the trend extrapolation rule there is stability for n firms if ( )
( )
( )
( ). As you can see if
the number of firms increase, the instability does increase. For the mixed case the stability does not depend on n any more only on the values of the weights. The number of firms only influence the multiplicity of the eigenvalues. There is one problem with the results that we found in the model with heterogeneous expectations. Mathematica could not find all the eigenvalues, therefore the conditions that we derive are necessary but not sufficient for stability. For stability the number of eigenvalues that are found must be equal to the multiplicity. And this condition does not hold here.
The results that were found do hold in every Cournot competition with a linear demand function and marginal costs. However it is possible that for other expectations formation rules the steady state does not stabilize. Further research is needed in the model with heterogeneous expectations. It can be interesting to investigate why the number of eigenvalues are not the same as the multiplicity of the matrix. Another thing that can be interesting to investigate is what happens with the dynamics of the market if the marginal costs are not constant. In the literature results show that with increasing marginal costs a market stabilizes. It may be interested to investigate a Cournot competition with these increasing marginal costs and the expectation formation rules that were investigated in this thesis. Maybe this combination increases the stability.
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Appendix 1 Proposition 1
To calculate the eigenvalues of the Jacobian matrix there must hold:
Jx=λx (A1)
So in matrixform this transforms into: (
) ( ) ( ) = ( ) ( ) ( ) (A2) Now there are two equations:
(A3)
(A4)
From (A4) we get: ( )
( ) (A5)
Now using the expression for in (A3) gives:
( ) (A6)
And since is a diagonalmatrix and the diagonal entries of are zero =0 so = . This implies that is an eigenvalue of So in the proof above it is shown that the only eigenvalues that needs to be calculated are the eigenvalues of and the vector x can not be zero. If x=0 than Ax=λx=0. This has infinite solutions for λ, since x=0 will always give this solution.
is a non-singular matrix since has n independent columns. So if the absolute value of the eigenvalues of are smaller than one than the steady state will converge to the
equilibrium. To calculate the eigenvalues of :
( )(
)
And to calculate . For this formula there are n different equations:
( )(∑ )
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( )
(∑ ) (A7)
For these n equations it is possible to rewrite these equations (n-1) times into:
( )(∑ ∑ ) ( ) ( )( ) ( ) λ= ( ) ( ( ) )= ( ) ( ) ( ) λ=( ) (A8)
With the multiplicity of (n-1) times. And there is one last eigenvalue that can be constructed adding up the n equations (A7). Here all the equations are added up so:
( ) ((n-1) (∑ )) ( ) ( ) ((n-1) (∑ )) (∑ ) λ= ( ) (( ) (∑ )) ∑ ) λ= ( )( ) with a multiplicity of 1. For we specify:
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Appendix 2
To calculate the eigenvalues of , must be a non singular matrix. In this Jacobian Matrix is a non-singular matrix since al the n columns are independent. And in matrix are easy to construct and are all real. To calculate the eigenvalues of , is rewritten as:
( ) ( )
And to calculate these eigenvalues there must hold . And are the eigenvalues (since λ is already used in this formula. For this formula there are n different equations:
( )(∑ )
( )(∑ )
(A9)
For these n equations it is possible to rewrite these equations (n-1) times into:
( ) (∑ ∑ ) ( ) ( ) ( ) ( ) = ( ) ( ) ( )= ( ) ( ) ( ) =( ) (A10)
With the multiplicity of (n-1) times. And there is one last eigenvalue that can be constructed out of the n equations (A9). Here all the equations are added up so:
( ) ((n-1) (∑ )) ( ) ( ) ((n-1) (∑ )) (∑ ) = ( ) (( ) (∑ )) ∑ )
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= ( )( ) with a multiplicity of 1. (A11)
Appendix 3 Conjecture
The matrix is Formed with the script:
This is the case of n=3 with 2 firms who use trend extrapolation rule and 1 uses adaptive rule. Below there holds n=4 adapt=2 trend=2
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Below n=5 trend=3 adapt=2
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Here n=5 adaptive=3 and trend=2
This is the case for n=2 and one firm uses adaptive expectations an the other trend extrapolation
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Bibliography
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D.J. Gates, J.A. Rickard and D.J. Wilson(1977) A convergent adjustment process for firms in competition. – Econometrica Vol 45, 6, 1349-263
R.D. Theocaris (1960) On the stability of the Cournot Solution on the Oligopoly problem-
review Vol. XXVII, 133-134
