Nonlinear Price Function

Consider the price-demand relationship, in which price-demand is not a linear function.

Assume that the fixed and variable costs are $10,000 and $100, respectively. 

Questions

1. Calculate the demand that maximizes profit, maximum profit, and breakeven points

2. Can the maximum revenue be calculated? Why or why not? 

Solution Using Calculus

None of the formulas for the linear price function are applicable in this case. Therefore, we will need to apply basic calculus methods to solve this problem.

D is imaginary; there is no real value of demand for which the maximum revenue can be calculated.

Solution Using MS Excel

Figure 2-9 shows the solution using MS Excel.

While the manual solution using calculus provides a single optimum point for price, demand, cost, revenue, and profit, solving the problem using Microsoft Excel offers even deeper insights, including how cost, revenue, and profit change over a wide range of demand values. As the calculus solution indicates that no real maximum revenue exists for this problem, a better visual can be seen through the graphs developed by the MS Excel solution.

Note that the unit price/demand is plotted along the secondary Y-axis, with values in hundreds, while the cost, revenue, and profit are plotted on the primary Y-axis, with values in millions.