<p>If the marginal output of one factor increases with the amount of the other factor, the two factors are said to be complements in the production. See <a href="https://en.wikipedia.org/wiki/Supermodular_function" rel="nofollow noreferrer">here</a> for the wikipedia link.</p>
<p>This is the case if the production function is supermodular: in case of differentiability this amounts to a positive cross partial derivative:
<span class="math-container">$$
\frac{\partial^2 A}{\partial X \partial Y} &gt; 0.
$$</span>
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. <a href="https://www.jstor.org/stable/2951479" rel="nofollow noreferrer">Milgrom and Shannon, 1994</a>).</p>

CC BY-SA 4.0
2023-07-25T09:53:08.713Z
2023-07-25T09:53:08.713Z
tdm
<p>Consider the following hypothetical situation:</p>
<p>You are producing something where the total amount produced, <span class="math-container">$A$</span>, is equal to the product of two factors, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. Hence <span class="math-container">$A=X*Y$</span>.</p>
<p><span class="math-container">$X$</span> and <span class="math-container">$Y$</span> both cost \$1 to increase by 1.</p>
<p>In this situation, <span class="math-container">$X$</span> is not said to have diminishing returns, because holding other factors constant, <span class="math-container">$X$</span> has linear returns. Increasing <span class="math-container">$X$</span> by 10% will increase output by 10%, and increasing <span class="math-container">$X$</span> by 100% will increase output by 100%.</p>
<p><span class="math-container">$X$</span> is also not said to have increasing opportunity cost, because the opportunity cost of increasing <span class="math-container">$X$</span> by 1 is equal to increasing <span class="math-container">$Y$</span> by 1, regardless of how many <span class="math-container">$X$</span> or <span class="math-container">$Y$</span> you have.</p>
<p>By symmetry, the same arguments are true for <span class="math-container">$Y$</span>.</p>
<p>However, if <span class="math-container">$X=20$</span> and <span class="math-container">$Y=10$</span> (and <span class="math-container">$A=200$</span>), then by reducing <span class="math-container">$X$</span> to increase <span class="math-container">$Y$</span>, for the same cost, you can increase output by having <span class="math-container">$X=15$</span> and <span class="math-container">$Y=15$</span>, so <span class="math-container">$A=225$</span>. Also, by spending \$1 on <span class="math-container">$X$</span>, you would increase output by 5%, while by spending \$1 on <span class="math-container">$Y$</span>, you would increase output by 10%.</p>
<p>In general, increasing one factor will increase the returns of the other factor, and by extension, diminish the <em>relative value</em> of that factor (or increase the opportunity cost in terms of the output, but not in terms of dollars or the alternative purchase).</p>
<p>This situation is very common in video games, and because investing too heavily into either <span class="math-container">$X$</span> or <span class="math-container">$Y$</span> would be inefficient, people will often say that both <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have diminishing returns. However, I've found this to be very misleading, because it implies that <span class="math-container">$A$</span> increases by less, marginally, the more <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> you have, which isn't the case. I'm also not sure if it would be accurate to say <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have increasing opportunity cost, because the cost of both always stay the same.</p>
<p>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?</p>

<utility><definition><marginal><economic-terms>
Is there a name for relative diminishing returns, or relative increasing opportunity cost?
Sepehr Nazari
# Is there a name for relative diminishing returns, or relative increasing opportunity cost?

(Asked by: username_0 on 2023-06-08)

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?

**Tags:** utility, definition, marginal, economic-terms


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## 1 answers

### Answer by username_1 on 2023-07-25. Score: 1 | Accepted answer

If the marginal output of one factor increases with the amount of the other factor, the two factors are said to be complements in the production. See [here](https://en.wikipedia.org/wiki/Supermodular_function) for the wikipedia link.

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](https://www.jstor.org/stable/2951479)).


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