2 the other type of economic model is the monopolistic seller with con

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2.) The other type of economic model is the monopolistic seller with constant marginal costs. The
constant marginal costs are not economically realistic, but it makes it simpler to create a profit
function. For example, suppose that it costs Hazel Farms $2000 per year in mortgage payments
and $3 per pound of hazelnuts produced.
a.) What Hazel Farms' cost function for hazelnuts?
C(x) =
Unlike the blueberry example, most producers have some level of monopolistic power, either
because they are the only producer or due to product differentiation. For example, Apple has a
monopoly on iPhones. Apple is the only producer of iPhones and can therefore set the price
wherever they want it. Of course, there are also Android phones, which do all the same things for
their users. However, these products are differentiated (no pun intended), so that if Apple doubled
the price of the iPhone, some but not all of their buyers would switch to Androids.
Suppose that Hazel Farms has some monopolistic power. This is often revealed by their pricing
strategy: If Hazel Farms charges $10/pound for hazelnuts, they will sell 5000 pounds. If they
charge $15/pound for hazelnuts, they will sell 4000 pounds (losing some but not all of their
customers). If we assume a linear demand function, we can say that for every $1 increase in the
price of hazelnuts, the quantity sold decreases by 200 pounds.
b.) Using the two coordinate points (10,5000) and (15,4000), write down the demand function
(i.e. quantity as a function of price) for Hazel Farms hazelnuts.
Q(p) =
Notice that Hazel Farms sets the price, which then determines quantity. However, in order to
calculate revenue, profit, cost, etc, it is more helpful to have the demand function written as a price
function, i.e, Price (p) is a function of Quantity (x). This function is the inverse of the demand
function.
c.) Write the price function for Hazel Farms hazelnuts. You can find the inverse function of Q (p) or
use the reversed coordinate points (4000,15), (5000,10)./ne.) Now find the profit function л(x) = R(x) - C(x)
П(x) =
f.) Use calculus to maximize Hazel Farms profit function. Remember that when a differentiable
function is maximized, its derivative = 0. Check your answer against the graph of the profit
function below./nTo maximize profit, Hazel Farms should sell
of hazelnuts at a price of $
a revenue of $
pounds
per pound. They will have a cost of
and make $
of profit.
-40000
π(x)
-20000-
Math 241 Lab C
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