In the last module we looked at a model of price determination that is the darling of free market economists. Prices were set by the laws of supply and demand with no collusion among either producers or consumers. In this module we look at a different marketplace, one that requires a different kind of model. In this marketplace there is only one producer and she can choose whatever price she wants. She has done a lot of market research and she knows the demand function, D(p) -- that is, she knows how many units will be sold at each possible price.
She is naturally interested in maximizing her profit -- setting the price so that her total profit is as high as possible. Notice there are two countervailing factors at work. Her total profit is the product of her unit profit and the number of units sold, or the demand. If she sets a high price she will increase her unit profit at the expense of the number of units sold. If she sets a low price she will increase the number of units sold at the expense of the unit profit.
This is a very common situation -- we want to maximize something -- in this case profit -- and we have control over one or more parameters -- in this case price.
Besides being interesting and important, these problems can be used at various points in the curriculum to exercise different mathematical skills. We begin with linear demand functions because they are easy to work with. Linear demand functions lead to total profit functions that are quadratic and these functions are easy to maximize. See Maximizing a Quadratic Function. We also discuss calculator or computer algebra system based methods of maximization in the CAS windows. These allow us to work with more reailistic demand functions. Calculus is another particularly powerful tool for studying optimization and the problems we are studying here make great applications in a Calculus class.
The first set of problems below all use the same demand function
They look at some of the same questions we examined in the last module but now in this different setting -- a monopoly rather than a free market.
Suppose that the manufacturing cost of the product in which we are interested is c so that the profit per unit at the price p is
and the total profit is
For the remaining problems we look at the demand function
You may want to use your CAS window to solve these problems.
