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Cournot and Stackelberg: How to Solve - YouTube
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hello in this video we're gonna solve
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for the Carnot and Stackelberg
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equilibrium we'll start with the two
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firm Carnot the inverse market demand is
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given as follows P is the market price
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and Q is the market quantity where the
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market quantity is split between firm 1
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and firm 2 if we were to substitute in
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for Q would put in Q subscript 1 plus Q
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subscript 2 to get this result firm ones
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output is given by Q subscript 1 firm
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2's output again it's given by Q
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subscript 2 we're gonna sumed that each
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firm has a constant marginal cost equal
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to two dollars so MC equals 2 if the
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firm's had different marginal cost
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structures we would still follow the
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same pattern in finding the equilibrium
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output in price in this market I'm going
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to start with firm 1 so again firm 1 we
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have the inverse market demand and firm
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one's marginal cost we're gonna
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calculate firm ones total revenue or
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revenue first so revenue is price times
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quantity so for price I'm gonna plug in
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the inverse market demand and that's
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what you see here in brackets and then
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I'm just gonna simplify a little bit of
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what's in bracket so minus 2 times Q
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subscript 1 minus 2 times Q subscript 2
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we get this result and then multiplying
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this Q subscript 1 throughout what's
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presented in the brackets we now get
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this and our next step is to get
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marginal revenue for firm 1 which is
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going to be the partial derivative of
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the revenue function with respect to
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firm ones output so 26 Q subscript 1 is
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now 26 minus 2 Q subscript 1 squared
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simplifies to minus 4 Q subscript 1 and
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then there our last term here the
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partial derivative a nap with
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respect to Q subscript one leaves us
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with minus 2 Q subscript 2 to maximize
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profits we'll set marginal revenue equal
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the marginal cost as I said the marginal
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cost for firm 1 is 2 if we was some
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other number we would just plug whatever
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that number is into the right-hand side
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of this equation so once we have this I
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will solve for firm ones output so
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moving some things around 26 minus 2 is
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where 24 is coming from and now dividing
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through by 4 we get firm ones reaction
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function when both firms have identical
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marginal cost and only when both firms
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have identical Marshall cost structures
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the reaction functions are going to be a
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mirror image of one another for so firm
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twos reaction function will be the
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mirror image of firm ones so just
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reverse and basically just change these
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subscripts where you see a 1 you plug in
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a 2 where you see it to you plug in a 1
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and that'll be firm twos reaction
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function so that is a shortcut method to
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get firm twos reaction function when
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both firms have identical marginal cost
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structures nevertheless I will solve for
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firm twos reaction function so looking
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at this print from the perspective of
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firm to firm 2 faces the market inverse
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demand and has marginal cost also equal
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to 2 we're gonna get firm twos total
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revenue price times quantity so for the
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price I plug in the inverse market
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demand
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I will simplify simplify some more I'm
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going to take the partial derivative of
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the revenue function with respect to
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firm 2's output this will be the
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marginal revenue a firm 2 and will set
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marginal revenue equal to marginal cost
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and now we'll solve this for firm twos
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output moving some things around
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dividing through by four we get firm 2's
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reaction function and again it is a
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mirror image of firm ones reaction
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function so we have two reaction
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functions firm 1 and firm twos we have
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two equations and two unknowns so I will
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substitute firm twos reaction function
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into firm ones reaction function so
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here's firm ones reaction function and
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for firm 2 where I have a Q subscript 2
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I'm not plugging in 6 minus one-half Q
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subscript 1 now we're gonna just solve
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this for Q subscript 1 so minus 0.5
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times 6 is -3 minus 0.5 times minus 0.5
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gives us the plus 0.25 and that's
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multiplied by the Q subscript 1 6 minus
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3 is just 3 and then subtracting minus
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0.25 Q subscript 1 from both sides we
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have this now on the left hand side and
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so 1 minus 0.25 leaves us with 0.75 Q
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subscript 1 and dividing through by 0.75
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firm 1 will produce 4 units of output
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firm twos output we take firm 2 3 action
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function and we just plug in 4 for Q
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subscript 1 and firm 2 will also produce
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4 units of output so when both firms
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have the same cost structure each firm
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will produce the same amount of output
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to get the price that each firm will
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sell its product for we go back to the
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inverse market demand and we're just
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going to plug in our values for firm
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ones output and firm 2's output so doing
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that simplifying
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the output will sell for $10 a piece
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in this to farm cournot model alright
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let's move on to stackelberg we're gonna
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assume firm 1 is to lead or firm 1 sets
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its output first firm 2 sees firm ones
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output and then we'll set its output so
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same kind of setup here same inverse
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market demand we can assume each firm
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has the same marginal cost structure I'm
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going to we're going to need this
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information
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this is firm 2's reaction function when
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firm 1 sets its output it's going to
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consider how firm 2 will respond to that
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output so that's why I have firm 2
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reaction function here so once again
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we're going to get firm ones revenue
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price times quantity for price I
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substitute in the inverse market demand
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I'm going to simplify that a little bit
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and then what I'm going to do here where
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I have this Q subscript 2 I'm gonna plug
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in firm twos reaction function okay so
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for Q subscript 2 i plug in firm twos
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reaction function and now we're just
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going to simplify this some more minus 2
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times 6 is minus 12 minus 2 times
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negative 0.5 is going to be plus 1 plus
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Q subscript 1 simplifying that some more
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26 minus 12 is 14 minus 2 Q subscript 1
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plus Q subscript 1 is just minus 1 Q
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subscript 1 and simplifying some more
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after multiplying this Q subscript 1
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through what's in brackets we're going
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to get marginal revenue for firm 1 and
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this would just be 14 minus 2 Q
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subscript 1 we set marginal revenue
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equal to marginal cost
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12/2 firm one will produce six units of
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output
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how much will firm to produce take this
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six and plug it back into firm twos
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reaction function and we'll see that
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firm 2 produces three units of output
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with a Stackelberg model if both firms
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have the same cost structure the firm
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that goes first will produce twice as
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much output as the following firm and
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finally to get the price in this market
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evaluate the inverse market demand at
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the total amount of output in this
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market six plus three and we'll see that
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the price in this market is eight
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dollars okay that's it I hope you found
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this video helpful
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