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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
Duopolies – interactive models
or
Qi=1−Q j
1
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 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.
Duopolies – interactive models
2
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
Duopolies – interactive models
3
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
