Simple differentiation / economics marginal cost question

Question

This seems like a very simple question, so I'm sure I'm doing something stupid here, but I'm not quite getting my head around the following question:

I have a total cost function:

$C = 5x^2 +15x + 200$

I've been asked to differentiate it to get the marginal cost, giving:

$dC/dx = 10x +15$

Then I've been asked to evaluate it at $x = 15$, giving $dC/dx = 165$

The part I'm struggling with is that I'm then asked whether this marginal cost, 165, would be the cost of the 16th unit. Given the point of marginal cost is that it is the cost of one additional unit the answer would intuitively seem to be yes, and indeed the answer given by the textbook is yes. However when I evaluate the initial total cost function at $x = 15$ and $x = 16$ I get:

Total cost of 15 units $= 5(15)^2 +15(15) + 200 = 1550$

Now if the cost of the 16th unit is 165 surely the total cost of 16 units should be $1550 + 165 = 1715$, but:

Total cost of 16 units $= 5(16)^2 + 15(16) + 200 = 1720$

This would seem to imply the the cost of the 16th unit is 170, where is the difference of 5 coming from? I presume I'm making some stupid basic arithmetic error here but I've been over it a few times and don't see it. Either that or these values are right and I just don't quite get the whole thing. I've tried for other values of x and always get a difference of 5. If anyone could point out where I'm going wrong that would be much appreciated.

Answer 1

Here's what I think (although I may be wrong). Usually, undergraduate economics textbooks try to give you intuition and sometimes are not so rigorous when it comes to the math. The 16th unit will be \$165 only if the cost function is linear otherwise the cost of the 16th unit is $\textbf{approximately}$ \$165. Mathematically, the change in cost is from (16 - $\varepsilon$)th unit to the 16th unit. To get exactly \$170, you would need to integrate the marginal cost from 15 to 16 i.e. $$ \int_{15}^{16} \frac{dC}{dx}dx=\int_{15}^{16}(10x+15) dx=170 $$ Also, the difference that you get of 5 comes from, $$ \textrm{cost of (x+1)th unit} - \textrm{cost of xth unit} =(5(x+1)^2+15(x+1)+200 )-(5x^2+15x+200 )\\ = 10x+20=(10x+15)+5 =\frac{dC}{dx} +5 $$

Answer 2

The way in which you compute the marginal cost of the next unit is based on the idea of first derivative of the cost function. When you calculate the marginal cost (the cost of an additional unit) you calculate how the cost function varies if you add another unit of input. But the point is that you are calculating the variation of your cost function using the differential of your function at one point. And you can think the differential as the linear approximation of the function at a point. Hence the difference between your results is due to the approximation error, which comes from the fact that you are using a linear approximation to the cost function instead of the cost function itself.

Hope it helps.