Maximize y, minimize x on logarithmic growth curve

Solution 1:

This is more of an Econ problem than a calc one. The only thing from calc that would help was the idea of marginal profit aka rates aka the first derivative. So referring to your question, if you think about it, of course, you'll make more profit the more you invest which is what the original graph represents. So the real profit-maximizing choice is whatever your max budget is since either way you'll be making more money the more you invest. Rather, like you suspected, in real-life, you'll have more variables than just cost vs profit.

Consider comparing that graph to another graph of a different investment. Now you need to know what amount to invest in each to make the most money. This can be solved.

(^Replace "utility" with "profit." The calculations are the same.)

Or consider having two graphs. One is quantity (inventory, workers, factories, etc) vs marginal revenue (revenue, not profit, per unit of quantity), and the other should be quantity vs marginal cost. The Profit-Maximizing Quantity is the quantity where $MR = MC$.

Or consider this from an econ textbook:

"First consider the upper zone, where prices are above the level where marginal cost (MC) crosses average cost (AC) at the zero-profit point. At any price above that level, the firm will earn profits in the short run. If the price falls exactly on the break-even point where the MC and AC curves cross, then the firm earns zero profits. If a price falls into the zone between the break-even point, where MC crosses AC, and the shutdown point, where MC crosses AVC, the firm will be making losses in the short run—but since the firm is more than covering its variable costs, the losses are smaller than if the firm shut down immediately. Finally, consider a price at or below the shutdown point where MC crosses AVC. At any price like this one, the firm will shut down immediately, because it cannot even cover its variable costs."

Solution 2:

If we interpret your graph as "if I put in $x$ buckos, I will recieve $y=f(x)$ buckos", then you should put in more $x$ if $f(x)-x$ is increasing at $x$. I have drawn the line $y=x$ in red (as the scales used to draw the $x$ and $y$ axes are not the same, the $y=x$ line is not at 45 degrees):

the vertical distance between points on the the green and red curves seems to indeed be maximal at about (20,40). Since $f$ is a nice smooth function, the condition that $f(x)-x$ is increasing is the same as the condition that its derivative is positive, i.e. $f'(x)>1$. Since $f$ is increasing, the best point is when $f'(x)=1$, and then from that point onwards, its a waste of resources. We can find this point graphically -

So it turns out, you should be using $x$ a little bit smaller than $20$.