Practice Problem

5 The supply and demand equations of a good are given by

The government decides to impose a tax, t, per unit. Find the value of t which maximizes the government's total tax revenue on the assumption that equilibrium conditions prevail in the market.

We conclude this section by describing the use of a computer package to solve optimization problems. Although a spreadsheet could be used to do this, by tabulating the values of a function, it cannot handle the associated mathematics. A symbolic computation system such as Maple, Matlab, Mathcad or Derive can not only sketch the graphs of functions, but also differentiate and solve algebraic equations. Consequently, it is possible to obtain the exact solution using one of these packages. In this book we have chosen to use Maple.

A simple introduction to this package is described in the Getting Started section at the very beginning of this book. If you have not used Maple before, go back and read through this section now.

The following example makes use of three basic Maple instructions: plot, diff and solve. As the name suggests, plot produces a graph of a function by joining together points which are accurately plotted over a specified range of values. The instruction diff, not surprisingly, differentiates a given expression with respect to any stated variable, and solve finds the exact solution of an equation.

The price, P, of a good varies over time, t, during a 15-year period according to P = 0.064;3 - 1.44;2 + 9.6; + 10 (0 < t < 15)

(a) Sketch a graph of this function and use it to estimate the local maximum and minimum points.

(b) Find the exact coordinates of these points using calculus.

Solution

It is convenient to give the cubic expression the name price, and to do this in Maple, we type >price:=0.064*tA3-1.44*tA2+9.6*t+10;

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