Introduction
In economics, two market forms are most well known: perfect competition and
monopoly.
Perfect competition occurs when there are many small firms, while a
monopoly has only one firm that controls the entire market.
Then, there is another form
of market called a duopoly, which is dominated by two firms.
Though a duopoly is an
intermediate step from a monopoly towards perfect competition, it becomes clear that a
duopoly is more complicated than either of the extreme cases.
In perfect competition, the
price of an object is set independently of any contributing firm, while a monopoly can set
its prices extremely high because it is the only company selling that object.
Thus, firms
make little to no profit when in perfect competition, but earn a large amount of profit
when in a monopoly.
However, a competing firm in a duopoly neither sets the price nor
is independent of the price and achieves a profit somewhere in-between.
Thus, the
duopoly is the most interesting form to model.
The Cournot Model
The first assumption we will make is that the price is determined solely on
quantity.
We will set up the price of the good, p, by taking the inverse of the demand.
P
= 1/(x+y) where x and y denote the quantities that each competing firm supplies.
This is
not a perfect way to estimate the price because when the quantity supplied is quite large,
the price approaches 0 and when neither firm produces the price tends to infinity.
However, it does capture the general trend which is that supply and price are negatively
correlated.
The next assumption that we will make is that the firms each have constant
marginal costs.
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The profits of the duoplolists are then, price * quantity – marginal cost *
quantity, or:
U(x,y) = x/(x+y) - ax
V(x,y) = y/(x+y) – bx
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We will assume that each firm is in the market to make profit, so they will make
their decisions on how much to supply based on a profit maximization technique.
Because profit curves are convex, the first firm will take the partial derivative of U with
respect to x and set it equal to zero, the second firm will look at V with respect to y.
Once this is done, we solve for the quantity of one firm based on the quantity of the other,
which is called the reaction curves.
If we use the second derivative test we can see that these are maxima if x and y are
positive, which is a reasonable condition.
The shape of these reaction curves is that they
start at 0, at the origin, hit the maxima and then drop off to 0 again.
The firms know each
other’s reaction curves and know that their competitor will be trying to maximize their
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Victoria Lee
Jennifer Sandness
Robert Treyes
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