What is the duopoly Nash-Cournot equilibrium if the market demand function is Q=1000-1000 p and

What is the duopoly Nash-Cournot equilibrium if the market...

Key Concepts

Marginal Cost

The cost of producing one additional unit of output. In competitive models such as the Cournot duopoly, marginal cost is a key parameter because firms set output levels where marginal cost equals marginal revenue to maximize profits.

Reaction Function

A mathematical representation of a firm's optimal output decision as a function of the output chosen by its competitor. Reaction functions are central to determining equilibrium in Cournot models because they capture the interdependence of firms' decisions.

Market Demand Function

An equation that describes the relationship between the total quantity demanded by consumers and the market price. This function is crucial for determining the price level and thus influences each firm's revenue and profit maximization decisions.

Cournot Competition

A model where firms compete by choosing quantities rather than prices. Each firm's optimal output decision depends on the quantity produced by its competitor, and the equilibrium is achieved when each firm's output decision is a best response to the other’s.

Duopoly

A market structure in which two firms dominate and must consider the actions and reactions of the other when making strategic decisions. This concept is fundamental when analyzing competitive behavior in markets with only two key players.

Nash Equilibrium

A solution concept in game theory where no player can benefit by unilaterally changing their strategy given the strategies of others. In the context of a duopoly, it represents the point where each firm's output level is optimal considering the output of its rival.

Transcript

00:01 So while seeing the question let's put a move to the answer so let's see here that the first one is coronate equation which is pi 1 is equal to p minus mc q1 so here is pi 1 is equal to 150 minus q1 minus q 2 minus 60 q 1 so pi 1 is equal to 150 minus 60 minus q2 so q1 minus q1 is squared so pi 1 is equal to 90 minus q2 so q1 minus q1 to the power 2 so br br1 so do you pi 1 by do you q1 is equal to 90 minus q2 minus 2 q1 is equal to 0.

01:49 So q1, br is equal to 90 minus q2 by 2.

01:59 So similarly, q2, br is equal to 90 minus q1 by 2.

02:17 So solving to b r 2 q1 is equal to 90 minus 45 minus 0 .5 q1 so 2 q1 is equal to 45 plus 0 .5 q1 so 1 is equal to 45 plus 0 .5 q1 so 1 .1 is equal to 45 so thus q1 is equal to to q2 is equal to 45 by 1 .5 is equal to 30 so q1 q2 is equal to 30 30 and the second one is starkler's equation so f1 move first so max pi 1 s t q2 b r so pi 1 is equal to 90 minus q 2 br so q1 minus q 1 to the power 2 which is equal to 90 minus 45 minus 5 q 1 so q 1 minus q 1 to the power 2 so pi 1 minus q 1 minus q 1 to the power 2 so pi 1 is equal to 45 plus 0 .5 q1, q1 minus q1 to the power 2 which is equal to 45 q1 minus 0 .5 q1 to the power 2.

05:26 So do you, pi 1 by 2 u, q1 is equal to 45 minus q1 is equal to 0.

05:49 So q1 sl is equal to 45 and q2 sf is equal to 45 by 2 is equal to 22 .5.

06:12 So so iso pi curve so pi 1 is equal to 45 to the power 2 minus 0 .45 to the power 2 is equal to 100012 .5 so then iso pi curve is equal to 112 .5 is equal to 90 minus q2 q1 minus q1 to the power 2 so in start berg equation is o curve of fum 1 is tangent to b r 2 so here is the full solution of the equation that so the first point is score not equation so pi 1 is equal to p minus m c q 1 5 1 is equal to 150 minus q 2 minus 6t so q 1 5 1 is equal to 150 minus 60 minus q 2 so q 1 minus q 1 to the power 2 so pi 1 is equal to 90 minus q 2 so q 1 minus power 2 so b r 1 so do you pi 1 by do you q1 is equal to 90 minus q 2 minus q 1 is equal to 0 so q 1 br is equal to 90 minus q 2 by 2 similarly q 2 br is equal to 90 minus q 1 by 2 solving to br 1 2 q 1 is equal to 90 minus 45 minus 0 .5 q1 so 2 q1 is equal to 45 plus 0 .5 q1 so 1 .5 q1 is equal to 45 thus q1 is equal to 45 by 1 .5 is equal to 30 so q1 comma q2 is equal to 30.

09:21 So the second point is a star is starkle birth equation f1 move first so max pi1 s plus q2 br so pi 1 is equal to 90 minus q2 br so q1 minus q1 to the power 2 which is equal to 90 minus 45 minus 0 .5 q1 so q1 minus q1 to the power 2 so 5 1 is equal to 45 plus 0 .5 q1 so q1 minus q1 to the power 2 which is equal to 45 q1 to the q1 to the power 2 so do you pi 1 by do you q1 is equal to q1 is equal to 0...

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