# Is there a name for relative diminishing returns, or relative increasing opportunity cost?

Consider the following hypothetical situation:

You are producing something where the total amount produced, $$A$$, is equal to the product of two factors, $$X$$ and $$Y$$. Hence $$A=X*Y$$.

$$X$$ and $$Y$$ both cost \$1 to increase by 1. In this situation, $$X$$ is not said to have diminishing returns, because holding other factors constant, $$X$$ has linear returns. Increasing $$X$$ by 10% will increase output by 10%, and increasing $$X$$ by 100% will increase output by 100%. $$X$$ is also not said to have increasing opportunity cost, because the opportunity cost of increasing $$X$$ by 1 is equal to increasing $$Y$$ by 1, regardless of how many $$X$$ or $$Y$$ you have. By symmetry, the same arguments are true for $$Y$$. However, if $$X=20$$ and $$Y=10$$ (and $$A=200$$), then by reducing $$X$$ to increase $$Y$$, for the same cost, you can increase output by having $$X=15$$ and $$Y=15$$, so $$A=225$$. Also, by spending \$1 on $$X$$, you would increase output by 5%, while by spending \\$1 on $$Y$$, you would increase output by 10%.

In general, increasing one factor will increase the returns of the other factor, and by extension, diminish the relative value of that factor (or increase the opportunity cost in terms of the output, but not in terms of dollars or the alternative purchase).

This situation is very common in video games, and because investing too heavily into either $$X$$ or $$Y$$ would be inefficient, people will often say that both $$X$$ and $$Y$$ have diminishing returns. However, I've found this to be very misleading, because it implies that $$A$$ increases by less, marginally, the more $$X$$ and $$Y$$ you have, which isn't the case. I'm also not sure if it would be accurate to say $$X$$ and $$Y$$ have increasing opportunity cost, because the cost of both always stay the same.

Is there a name for this kind of factor, and if not, how would you describe it succinctly to get across the idea that having too much of one factor is less optimal than having a balanced amount of both?

This is the case if the production function is supermodular: in case of differentiability this amounts to a positive cross partial derivative: $$\frac{\partial^2 A}{\partial X \partial Y} > 0.$$ There is a huge literature in economics on supermodularity and complementarity as it is tightly related to comparative statics and provides useful equilibrium existence results (e.g. Milgrom and Shannon, 1994).