March 29, 2021 | 12:06

The Mathematical Formula For A Prosperous Life

Math is awesome. A formula in mathematical notation might offer a fresh perspective on the relationships between its constituent terms.

For instance, in geometry, the $ a^2 + b^2 = c^2 $ formula represents the relationship of the length of the hypothenuse to the other two sides' lengths in a right-angled triangle. In physics, the famous $ E = mc^2 $ formula represents the relationship between the mass and the energy in a system’s rest frame.

But what if there would be a formula to elegantly represent the relationship between different concepts in life and our daily activities?

There is such a formula, and in this article, I invite you to find out what it looks like.

Choosing A Starting Point

Whether we like it or not, we need some sort of wealth to live. Usually, it is money since we can buy many necessary things with it, but it is not limited only to money. Therefore, let’s look into economics to see how it defines wealth.

In economic terms, there are two broad categories of activities:

  • consumption activities
  • production activities

For simplicity, let’s agree to represent both production and consumption in money. You work, sell something, or do some other production activity means you gain money. If you engage in a consumption activity, then you spend money on it.

Thus, we have the following wealth relationship between the production and consumption capacity of a person:

$$ Wealth = \frac{production (p)}{consumption (c)} $$

It is brilliant in its simplicity: if the ratio is greater than 1 (i.e., you produce more than you consume), you are good. If, however, your consumption is higher than what you make, you have a problem. Also, $ c $ can never be zero since we have some basic needs like eating for which we need some money.

So we have two formulas so far, and $ p $ and $ c $ are not very informative in themselves. Let’s break down those two.

You Mainly Gain Money By Working

To produce something, you have to put in time and energy. We talked about time previously, and that is a resource we have no control over except how we decide to spend it. Energy or effort, on the other hand, is something we have control over.

Therefore, the production gains from an activity can be written as the following product:

$$ Production = e * t * v $$

Where $ e $ is the energy we put into a job, $ t $ is the time we spend doing it, and $ v $ is the “added value coefficient,” representing how valuable your work is (which translates into how much you are being paid).

Thus, the greater the product of these three variables, the bigger your gains will be. Let’s see some approaches to maximizing the end result:

  1. You can increase it by spending more energy on an activity. Alongside spending more time (see below), that’s what is generally regarded as “work hard.” The downside of this approach is that energy is limited, and you need to regularly “refill your batteries” to keep a high-energy input into a job.
  2. You can increase it by spending in more time on something. For instance, you work overtime and get more money. However, as mentioned earlier, we have a limited amount of time in a day that we should allocate to many different things.
  3. You can increase it by finding a better v coefficient, which means you work at a higher-paid job.

So this is the formula for a single activity, but some people have incomes from many different activities.

To reflect that, let’s introduce a mathematical notation — the sum ($ \sum $).

$$ \sum_{n=1}^{5} 2n = 2*1 + 2*2 + 2*3 + 2*4 + 2*5 = 30 $$

It is straightforward — the sum of numbers 1 through 5 multiplied by 2. Thus, one’s total production can be represented as:

$$ Total Production = \sum_{i} e_{i} * t_{i} * v_{i} $$

The i above represents an activity. Therefore, we take the sum of all jobs we have income for (whatever the number of that jobs), and that gives us the total production we have. To keep things simple, I will use the simplistic notation of $ e * t * v $ for the rest of the article.

Plugging it into our initial formula, we have:

$$ Wealthy Life = \frac{e * t * v} {c} $$

You Consume By Spending Money

Let’s discuss now how to represent consumption. The formula looks similar:

$$ Consumption = e * t * c $$

Where $ e $ is the energy we put into an activity (traveling, watching TV, etc.), $ t $ is the time we spend doing it, and $ c $ is the “cost coefficient,” representing how expensive that activity is.

There is a small problem, though: we have time and energy as variables in both production and consumption equations, and we need to differentiate them somehow.

The “good” news is that we have a limited amount of both time and energy, and if we agree to represent $ 0 <= t <= 1 $ and $ 0 <= e <= 1 $ (a fraction of the whole amount that we have), then we can compute the other part as $ (1 - t) $ and $ (1 - e) $.

For instance, we have 24 hours in a day, which represents a whole unit. Suppose we spend 6 hours on useful work, that’s $ t = 0.25 $. Therefore, we have $ (1 - t) = 0.75 $ or 18 hours of our time that we spend on activities unrelated to useful work.

It is similar with energy. If we represent energy as a value between 0 and 1, which is the fraction from the total amount of energy, then $ 1 - e $ is the remaining amount.

Unlike in production, in the case of consumption, you can decrease it in two ways only:

  1. Spending less time on an activity. For instance, watch less TV or spend less time on social networks.
  2. Finding a smaller $ c $ coefficient, which means spending less money or buying cheaper products.

You cannot decrease the consumption by spending less energy since, although it is a renewable resource, you need to spend it on many different activities (including “consumption” ones) to have a healthy body and mind. And this is where it starts to get interesting.

It Is All About The Right Balance

Let’s see how our formula looks like now. I have switched “Wealthy” for “Prosperous” since once we have factored in our energy and time, it broadened the scope from wealth to our lifestyle:

$$ Prosperous Life = \frac{e * t * v} {(1 - e) * (1 - t) * c} $$

This equation gets very close to how life works, and the relationship it defines between energy, time, and money is astounding.

If you give more of your energy to your work, you will have less of it for doing other active leisure activities and vice-versa. And if you spend more time at your job, you will have less time to do other stuff and vice-versa.

Let’s see some more elaborate relationships it encodes:

  • If you have a well-paid job (high v coefficient), you can spend less time and energy on it, saving more time and energy for leisure activities.
  • Want even more money? Put in more time or effort, but then you’ll have less of those for your family time and other activities.
  • Conversely, have a low-paying job (low v coefficient)? You’ll have to put in more time and effort to make up for your consumption.
  • Do you have a minimalist lifestyle, having few possessions? You don’t need a high income then, which might also save you more time and energy you can then spend on leisure.
  • Want luxury stuff? You’ll have to put lots of energy and time into your job to be able to buy it.
  • Want to spend more time with your family? You’ll need to either cut out some other costs or find a job with a good v coefficient.

The list can go on. The way we manage our time and energy, alongside the price tag on some of our purchases, impacts our lifestyle. Not that this is a huge discovery, but the formula portrays it quite well.

Therefore, all you need to do is find that balance where you have sufficient income without spending all your time and energy at your job. And this leads to the final improvement on the equation.

Work For Your Money So That Later It Can Work For You

The $ e * t * v $ represents your active income. You have to go to work and put in time and effort. Another way to generate wealth is through passive income, and that is currently missing from our equation.

There are several ways to generate passive income. However, I will focus on investments since it is one of the most common ways to gain income passively.

$$ Passive Income = i * r $$

The i variable is the invested amount, and r is the return on investment coefficient, which can be positive or negative.

Thus, we have the following formula:

$$ Prosperous Life = \frac {e * t * v + i * r} {(1 - e) * (1 - t) * c} $$

And this is the mathematical formula of a prosperous life. The sum of all your active and passive incomes with all the energy and time you put into those activities divided by the time, energy, and cost of the activities besides work makes up your lifestyle.

If you have enough passive income, you will have plenty of time and energy for leisure activities. Otherwise, you will have to balance your time and effort to gain sufficient income to cover your expenses.

There are lots of tradeoffs you will make, though, and some non-work-related activities (e.g., learning new stuff) might help you increase your income later. Likewise, spending energy to go to the gym can help you feel better and stay healthy, even though it does not help you make more money directly.

Implications On Happiness

In the end, I’d like to say a few words about happiness.

Although this equation does not define happiness per se, it describes a precondition for it:

$$ \frac {e * t * v + i * r} {(1 - e) * (1 - t) * c} > 1 $$

If you produce more than you consume, you have everything you need to be happy.

And it doesn’t really matter how big that ratio is. It doesn’t matter if it is 1.1, 1.5, 2, or even more. If it is greater than 1, you can be happy. Besides that, human nature is fickle, and it will always strive to keep the ratio at around 1.

It means that if you are low on finances, your expenses and consumption will decrease because you will have less money and time to spend since you’ll be working.

But it holds true for the opposite as well. If your income is increasing, your expenses will adjust as well. You’ll start buying more expensive things and spending more on other activities.

Therefore, to be happy, you need enough money to cover the necessary expenses. And then there is our perception that defines what “necessary” really is and whether it is enough or not. But that is an entirely different story.

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