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Math 115 Final Project

# Math 115 Final Project - Math 115 Modeling Modeling...

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Math 115 Modeling Modeling Duopolies and Oligopolies in Economics Victoria Lee Jennifer Sandness Robert Treyes 1

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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. 1 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 2 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 1 2 Victoria Lee Jennifer Sandness Robert Treyes 2
profit, this information allows us to use a simultaneous system of equations to find the Cournot point. 3 Cournot says that even though the firms would like to be at their individual maxima, (which we have marked with a dot on the curve) the optimal choice for the set of firms to make the most profit collectively will be at this point.

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