Costs and Profits — Interactive Demo

1. The model

When x units are made, suppose Product A sells at a unit price of

P(x) = a − bx yen.

The cost of producing the x units is

C(x) = F + cx yen.

The revenue is

R(x) = x·P(x).

The profit is

I(x) = R(x) − C(x).

base price a (yen)

price drop per unit b (yen/unit)

fixed cost F (yen)

unit cost c (yen/unit)

production quantity x (units)

2. Explore the graphs

Revenue and cost

Hover or drag on a chart to move x. Profit is the vertical gap R(x) − C(x); the shaded band is where profit is positive.

Revenue R(x) Cost C(x)

Profit

The profit curve is a downward parabola; its vertex is the maximum profit.

Profit I(x)

Table view (every 500 units)

3. The worksheet questions

With P(x) = 1000 − 0.2x and C(x) = 500 + 100x (press the first preset button above):

(a) Find the profit I(x) when x units of Product A are made.

Revenue is quantity × unit price:

R(x) = x·P(x) = x(1000 − 0.2x) = 1000x − 0.2x2
I(x) = R(x) − C(x) = (1000x − 0.2x2) − (500 + 100x)
so  I(x) = −0.2x2 + 900x − 500.

(b) How many units maximize the profit, and what is that maximum profit?

Complete the square (or use the vertex formula x = −B/2A for Ax² + Bx + C):

I(x) = −0.2(x2 − 4500x) − 500
      = −0.2(x − 2250)2 + 0.2·22502 − 500
      = −0.2(x − 2250)2 + 1 012 000.

The coefficient of x² is negative, so the parabola opens downward and the vertex is the maximum: producing 2250 units gives the maximum profit of 1,012,000 yen. Check it on the profit chart above — the marked vertex sits exactly there.

(c) Find the maximum profit if instead C(x) = 500 + 200x.

Press the second preset button and watch the vertex move. Now

I(x) = (1000x − 0.2x²) − (500 + 200x) = −0.2x² + 800x − 500

= −0.2(x − 2000)² + 799 500.

The maximum profit is 799,500 yen, attained at x = 2000 units. A higher unit cost shifts the best quantity down (from 2250 to 2000) and cuts the peak profit.

In general, for P(x) = a − bx and C(x) = F + cx with a > c, the profit I(x) = −bx² + (a − c)x − F is maximized at x* = (a − c)/(2b) with I(x*) = (a − c)²/(4b) − F. Since x counts units, if x* is not a whole number the best integer quantity is one of the neighbors of x* — the demo checks both.

Other source(s)

https://sites.duke.edu/collardwexler/files/2015/01/Monopoly_1.pdf
https://openstax.org/books/principles-microeconomics-3e/pages/9-2-how-a-profit-maximizing-monopoly-chooses-output-and-price