Optimization of univariate functions
Many problems in economics require identification of an optimal outcome. This task is achieved by maximizing or minimizing some economic entity. For example, a firm may seek to maximize profit π by choosing the optimal level of output Q. Since π is the difference between total revenue (TR) and total cost (TC) we can write
π(Q)= TR(Q)− TC(Q)
where π(Q) is known as the objective function and Q is referred to as the choice variable. The objective function π(Q) is the function that we seek to optimize. Other problems in economics require optimizing across two or more choice variables: one such example would be to minimize a cost function with respect to capital and labor. Another example would be to maximize a profit function with respect to capital, labor and land. In this chapter we will confine ourselves to examining objective functions with one variable, otherwise known in calculus as univariate functions. Finding extreme values of univariate functions is one of the most widely known and applied areas of calculus.