5. A common misconception

Consider the following cost function relating units produced, x, to cost, y.

For each additional item produced, cost increases by 0.5, regardless of the value of x.

A similar sounding statement is

For a 1% change in items produced, costs increase by 0.5%, regardless of the number of items produced.

For example, if costs at 100 items were $500, the costs at 101 items would be $\$500 \times 1.005 = \$502.5$. Also if the costs at 200 items are $707, then the costs at 202 items will be $707 \times 1.005 = 710.54$

So, a one item increase in production at 100 units leads to a change in costs of $2.50.

A two item increase in production at 200 units leads to a change in costs of $3.54, so a 1 item change would increase costs by about $1.77 at 200 units.

This means that the constant percent change cost function is not giving rise to a linear cost function (because the change in y depends on the value of x)!

A slight change in wording leads to a qualitatively different type of cost function. More on this type of cost function in Class 3.