COURNOT DUOPOLY – CALCULATIONS Jüri Eintalu
1. Assumptions of the model Qi P
- Quantity of production of the Firm i - Price
It has been assumed that both firms are using the same, optimal price. The following linear demand function has been assumed: P=2−(Q1 +Q2) It has been assumed that there are no fixed costs. It has been assumed that the marginal costs are constant and identical for both firms: c=1 Thus, a symmetrical model has been used.
2. Profits Πi
- Profit of the Firm i
Π i =P⋅Qi−Q i
Π i =[1−(Q1+Q 2 )]⋅Qi Π i =0
if
profits Qi=0
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or
Qi=1−Q j
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COURNOT DUOPOLY – CALCULATIONS Jüri Eintalu 3.
Reaction functions
We shall write the profits in the following polynomial form:
Π i =(1−Q j )⋅Qi−Q2i i≠ j
always assumed!
The reaction function shows the best response - the quantity of production, maximizing one's profit, given the quantity of production of the competitor. To calculate the maximum point, we calculate the derivative of the profit and equate it with the zero (in the present special case of parabolic functions the derivative can be avoided indeed): ∂Π i =1−Q j−2Qi ∂Qi 1−Q j−2 Qi=0 1 Q Q i (Q j )= − j 2 2
reaction function of the Firm i
1 Q R1 : Q1 (Q2)= − 2 2 2 1 Q R2 : Q 2 (Q 1)= − 1 2 2
reaction function of the Firm 1 reaction function of the Firm 2
On the graph, we have to use the inverse function of the reaction function R1 : R1 : Q2 (Q1)=1−2 Q1 Note also that if Qi⩾1 , then the reaction function is Q j (Qi )=0 , which doesn't follow from the calculation with the derivative above.
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COURNOT DUOPOLY – CALCULATIONS Jüri Eintalu
4. Nash equilibrium In the Nash equilibrium, both quantities of production {Q1 N ,Q 2 N } are the best responses to each other. Thus, on the graph, the Nash equilibrium is the intersection point of the reaction functions. (Indeed, this must be one of the equilibria, if there happen to be many.) To calculate the Nash equilibrium algebraically, one has to solve the system of two linear equations. However, our model is symmetrical, therefore, we can assume that Q1 N =Q2 N =QN and it is sufficient to solve only one equation 1 Q QN = − N 2 2 which gives us the result Q1 N =Q2 N =QN =
5.
1 3
Nash equilibrium
Cartel agreement
Because of the symmetry of our model we shall assume a fair, symmetrical cartel agreement (no complicated game of Bargaining will be assumed here): Q1 C =Q2 C =Qc Then, both firms have always equal profits
Π (QC )=Q C −2Q 2C and it is possible to maximize these profits simultaneously. dΠ =1−4 QC dQ C Q1 C =Q2 C =QC =
1−4 QC =0 1 4
which gives the result
cartel agreement
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COURNOT DUOPOLY – CALCULATIONS Jüri Eintalu 6.
Nash equilibrium and cartel agreement
Note that in the case of cartel agreement the quantities of production are lower than in the case of Nash equilibrium: QC
7.
Graph
COURNOT DUOPOLY
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COURNOT DUOPOLY – CALCULATIONS Jüri Eintalu 8.
Nash equilibrium as the origin of coordinates
As I see it, if to choose the Nash equilibrium as the new origin of coordinates, then the calculations and formulae are becoming illuminating and simple. Let us define X =Q1−QN =Q1−
1 3
Y =Q2−QN =Q2−
1 3
Now, the reaction functions can be presented as follows: R1 :
Y X (Y )=−( ) 2
reaction function of the Firm 1
R2 :
Y ( X )=−(
X ) 2
reaction function of the Firm 2
9.
Dynamic game
Let us assume that the players are making their moves sequentially, by turns, each one at each step maximizing one’s temporary profit. Then, after at maximum 2 steps the game is surely on one of those reaction function segments, which are intersecting in the Nash equilibrium. Suppose that the corresponding coordinate of the Firm 1 is X and that Firm 2 is to move. Then, each of the further steps is described by the reaction functions given above. After 2n steps we will have the following coordinates: X 2 n=
X 22 n
Y 2 n=−[
X 2
(2 n−1 )
]
Obviously, this sequence converges to the Nash equilibrium.
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COURNOT DUOPOLY – CALCULATIONS Jüri Eintalu
10. Static game It can be proved that the strategy Qi=
1 2
( X=
1 1 or Y = ) dominates every higher strategy 6 6
1 . Then, in turn, the lower bond is given by the reaction function formula above. After that, 2 the next upper bound is again given by that same formula. Actually, this result is not so easy to prove as it might seem. Often, a mistaken definition of the dominated strategy has been used. In the present case, it accidentally gives the right result. After an infinite sequence of iterated eliminations of dominated strategies, the remaining, allowed area converges to the Nash equilibrium. Q i>
11. Literature Gibbons, Game Theory for Applied Economists Chapter "1.2A Cournot Model of Duopoly" - I started my calculations from this chapter and tried to develop them towards generality and mathematical simplicity.
Dupolies – Game Theory – interactive models by the Author: http://tube.geogebra.org/b/1423219#
2015 July 20 DO NOT KLICK
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