The Optimality Equation

Optimal Allocation

Going to an extreme (all-in on one side) is usually a bad idea.

But it's not always about being "perfectly balanced" either.

Real progress is about optimal allocation under constraints.

A simple equation (Quality × Quantity)

Let $\text{quality} = x$ and $\text{quantity} = y$ .

You'll usually have a fixed budget of energy or time which means when you increase $x$ you will decrease $y$ , this constraint can be demonstrated by this equation:

x + y = 100

An output that you want is usually the product of the two (i.e. quality x quantity):

\text{output} = xy

Take a look at these output examples:

$x=90,\,y=10 \rightarrow \text{output}=900$ (Perfectionism, low volume)
$x=60,\,y=40 \rightarrow \text{output}=2400$
$x=50,\,y=50 \rightarrow \text{output}=2500$ (Maximum Output)
$x=20,\,y=80 \rightarrow \text{output}=1600$
$x=1,\,y=99 \rightarrow \text{output}=99$ (Spamming garbage)

You can see that extremes fail here.

Pushing $x$ too high forces $y$ too low, and the output drops.

Notice that $y = 100 - x$ , therefore the output becomes a function of $x$ :

f(x) = x(100-x) = 100x - x^2

Setting the derivative to zero $\left(f'(x) = 100 - 2x = 0\right)$ gives us $x=50$ .

Meaning: if output is $xy$ (weights are equal), an equal split is optimal.

Raising the base

If you want both quality and quantity to increase, you can't do it while $x+y=100$ . You have to increase the base constraints:

x+y=B \quad \Rightarrow \quad B \uparrow \Rightarrow x \uparrow \text{ and } y \uparrow

This is what "improvement" really is. You aren't just reallocating effort; you are expanding the budget through better tools/skills and higher leverage.

When quality matters more

Sometimes quality can have more weight than quantity. We can model this by changing the output equation like so:

\text{output} = x^2y,\quad x+y=100

Here, quality ( $x$ ) is squared, making it "twice as important" as quantity ( $y$ ).

Substitute $y=100-x$ and differentiate:

f(x) = x^2(100-x) = 100x^2 - x^3

f'(x) = 200x - 3x^2 = x(200 - 3x)

Setting this to zero gives the critical point:

x=\frac{200}{3} \approx 66.7

y \approx 33.3

Notice the ratio:

x = 2y

Because quality was squared $\left(x^2\right)$ and quantity was linear $\left(y^1\right)$ , the optimal allocation is a $2:1$ split. The math mirrors the priority.

It is not "balanced" (50/50), but it is optimal.

More than 2 variables

Real life usually have more than just $2$ variables:

x+y+z = B

But the principle still holds:

Don't maximize one variable at the expense of zeroing out the others.
Align allocation with impact. If $x$ has the highest exponent (impact), allocate the most resources there, but do not neglect $y$ and $z$ .

\textbf{Don’t chase extremes. Find the optimal allocation, then raise the base.}