How to Solve a Cournot Oligopoly Problem - YouTube

hello, in this video I am going to demonstrate Cournot. Cournot is a model

where we have two firms producing identical Goods. they're

competing by setting output and this output is set simultaneously, so the

market demand conditions are given by this inverse demand of price equals 100

minus Q, where Q equals the output of firm 1 plus the output of firm 2. I'm

going to go ahead and make that substitution here into the inverse

demand. I'm going to rewrite it now as 100 minus Q 1 plus Q 2 and let me just

simplify that

and we get the following expression. so from firm one's perspective

total revenue is price times the amount of output produced by firm one. so let's

get firm one's total revenue equation

what you see I did here was substituting the inverse demand equation in for P and

that's going to be all then multiplied through by the output of firm 1, Q

subscript 1, this will simplify down to 100 Q subscript 1 minus Q subscript 1

squared minus Q subscript 1 times Q subscript 2. we want to get marginal

revenue because it's going to eventually lead into profit maximization now let me

put some subscripts here on the total revenue and marginal revenue. to

get marginal revenue we will take the derivative of the total revenue function

with respect to the output of firm 1. technically, a partial derivative and we

get the following result.

okay, and let me just rewrite that over here so we got marginal revenue. now

we're going to set firm one's marginal revenue equal to marginal cost. marginal

cost will come from the total cost equation. each firm we're assuming has

the same cost structure and we get a marginal cost of 40 so let me go on to

the next page and I'm just going to set marginal revenue equal to marginal cost

and now after we do this we want to simplify this down and solve for the

output of firm 1

and there you go you have the output of firm 1 equaling the equaling 30 minus

the output or half the output of firm 2 this is referred to as firm ones

reaction function ok it gives us the best response for firm 1 in terms of

output for any output level decided by firm 2

ok now our next step is to do basically the same thing but now from firm 2's

perspective let me go go up here so from firm 2's perspective firm 2 facing the

same market demand conditions

so nothing is changed here we want to get firm two's total revenue so just

like we did for firm one we're going to substitute the inverse demand into the

total revenue equation for P but this time we're going to multiply by firm

twos output the amount of outputs sold by firm two

we are going to simplify this down

and we get the following result to get marginal revenue I'm going to take the

partial derivative of the total revenue function with respect to the output of

firm 2 and we get 100 minus 2 Q subscript 2 minus Q subscript 1 let me

go on to the next page but what we're going to do on the next page it just

simply set marginal revenue for firm 2 equal to marginal cost

and doing that we get

the following result and this will simplify nicely to something that looks

like this and what we got here is firm two's reaction function and let me just

remind us firm ones reaction function is basically the mirror image of firm twos

reaction function and this will always be a property here of Curnow if you're

dealing with identical cost structures for both firms we've got two reaction

functions what we want to do is to substitute one of these reaction

functions into the other so what I'll do is I'll take the reaction function for

firm 1 and I'll substitute into it the reaction function of firm 2 and now

we're going to simplify this

and we get 3/4 the output of firm 1 equals 15

and in this case we are going to get 60 divided by 3 or 20 units of output so

firm one should produce 20 units of output in order to maximize profits in

this Cournot competition. for firm 2 to get firm two's output we can just simply

take this 20 here and plug it into firm twos reaction function so doing that let

me go over here then if firm 1 wants to produce 20 units of output let's see

what firm 2 wants to do and we see that firm 2 also wants to produce 20 units of

output and this is no accident in cournot once again if you have the

firm's having equal cost structures each firm will produce the same amount of

output next thing we should get is the market price so the market price P

equals what was our inverse demand it was a hundred minus Q and so in this

case a hundred minus 40 the total industry output is Q 1 plus Q 2 so we

get a market price of $60 and that is how you solve or a car no problem let me

know if you have any questions