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This question is rather of economical nature. Let's say we have a function $f(x)$ that describes the result of some action depending on the amount of resource used $x$, for example:

$$f(x)=\frac{300}{100+x}$$

The smaller the result, the better!

EDIT: Consider the interval $0\leq x\leq 200$ for realistic $x$ values.

How would I determine, at which point the improvement becomes not worth the investment of more resource anymore? Is there any information missing, or can this be determined from the curve as it is?

Kagaratsch
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  • Let's say realistic values are $0\leq x\leq 200$. On this interval the value of the derivative undergoes a change in its magnitude of only one order of ten. And it is also monotoneously decreasing. So it is again not very clear where to put the cut... – Kagaratsch Jan 15 '14 at 20:15

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You are missing some information to get an economically justifiable answer. Recall that Pareto efficiency in resource allocation occurs when the marginal cost of using the resource equals the marginal benefit.

In particular, I must assume that $f$ represents the benefit curve, and that the marginal benefit is given by the derivative of $f$.

There is no unique answer to this question without a cost curve. Well, there sort of is an answer, but it implicitly assumes that there is no cost associated with using the resource. In which case you would use as much of the resource as you could, since all consumption would improve your welfare at no cost.

Another option would be to assume that the cost of using $x$ resources would be $x$. Then the marginal cost is $1$, and you would solve for $x$ in $f'(x) = 1$. This does represent a relatively sensible cost curve, but it does not have increasing marginal costs, as most real cost curves do.

Actually, since, as you note in the comments, the marginal benefit $f'$ is always negative, you would never want to enter into the transaction. That is, $x$ is always $0$.

nomen
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  • In this example $f'(x)$ is negative and always $|f'(x)|<<1$ on the realistic interval $0\leq x \leq 200$... What if the first $0\leq x \leq 60$ come free of cost, and for the remaining resource the cost per unit resource is constant? – Kagaratsch Jan 15 '14 at 20:55
  • In that case, you would solve for $x$ in $f'(x) = c$, where $c$ is the cost per unit resource. – nomen Jan 15 '14 at 21:03
  • Thank you! That cleared it up for me. – Kagaratsch Jan 15 '14 at 21:06
  • No problem. I did want to add a comment, though. As you noted, the marginal benefit is much much smaller than the marginal cost, so you shouldn't enter into the transaction. In fact, the marginal benefit is negative. So you presumably should never enter into the transaction. – nomen Jan 15 '14 at 21:11