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