Course:MATH110/Archive/2010-2011/003/Notes/Optimization/Problem 12

Question

In economics, you can model a company's profit of selling a specific good by studying its cost curve and its revenue curve. Both curves are functions of the quantity of items being produced. The profit is then simply defined as the difference between the revenue and the cost.

For example, consider a company selling a good for which the cost function $C(q)$ and the revenue function $R(q)$ are given by

$C(q)=1+6q-3q^{2}+q^{3}$

$R(q)=45q-15q^{2}+2q^{3}$

where $q$ is the number of items produced (in millions of units) and both costs and revenue functions are in millions of dollars. (For those in the know, the revenue function is itself constructed from a demand curve linking the price per item $p$ to the quantity produced $q$). Hence we get the profit function

$P(q)=R(q)-C(q)$

Recall also that the marginal cost is the derivative of the cost and the marginal revenue the derivative of the revenue (as we've seen in class, it's slightly more tricky than this because of some units issue, but let's move on).

• In the above example, find the maximum quantity that the company should produce in order to maximize its profit. We'll call that maximum quantity $q_{max}$.
• Sketch the graph of the marginal cost and the marginal revenue on the same pair of axis. These two curves intersect, compute the coordinates of the intersection point(s).
• What do you observe, can you explain what is going on with your understanding of calculus.
• Some economics textbook state that the maximal profit is achieved for a production level (the quantity produced) where the marginal cost and the marginal revenue intersect. Explain how this idea is a little un-precise, reword it more accurately and explain how this idea is mainly calculus and nothing more.