Question

Diagram 1: Suppose that an activity, x, generates benefits according to the benefit function B(x) - 26x - 0.01x?, and costs according to the cost function C(x) = 2x + 0.01x. Solve for the optimal level of x and draw a simple diagram showing the behavior of the Net Benefit function that corresponds to your solution. Include as much detail as possible, including axis labels, the level of x at which the Net Benefits are maximized and the level at which they are zero. Make sure that your diagram is neat, and that all details are legible.

Answer 1

Net benefit = B - C

Net benefit = (26x - 0.01x2) - (2x + 0.01x2)

Net benefit = 24x - 0.02x2

These NB are maximized at the point dNB/dx = 0

That is,

24-0.04x = 0

x = 600

Also,

Optimal number of x is where B = C

That is,

26x - 0.01x2 = 2x + 0.01x2

24x = 0.02x2

24 = 0.02x

x = 1200

Required graph is below:

The required graph is below: B,C B-C graph 25000 20000 15000 10000 5000 0 © 200 400 600 800 1000 1200 1400 1600 -5000 -10000 + B +C ---NB As it can be seen, optimal number of x is 1200 where B and C curves intersect each other NB is an inverted U-shaped function NB is maximized at x = 600 where difference between B and C is maximum NB is O at x = 1200 where B = C (optimal number of x)