Ideas are underrated Part I: A mathematical model of Entrepreneurship

Every great business venture starts with an idea. But in order for a business to come into life, the idea needs to be implemented first. This article sums up the different incentives for entrepreneurs to implement their ideas, versus the obstacles that deter them from doing so, in the form of a utility model. In Part II of this two-article series, I will demonstrate how we can facilitate the implementation of ideas, based on this model.

Positive Incentives

Naturally, the biggest incentive to become an entrepreneur is a financial one. The entrepreneur wants to build up a successful company. Especially in the cryptosphere, we measure the success of a company by its market capitalization (hence the word “capitalism”). The public trading of shares or tokens gives a company a valuation (v) for the sum of all its tangible and intangible assets.

For startups that are not yet publicly traded, a valuation is established and verified through funding. With each successful funding round and with each onboarded co-founder, the entrepreneur sells equity (e), while the valuation of his company rises. In general, the product v * e should increase over time. If this is not the case, our fictional entrepreneur is doing something wrong.

One problem for entrepreneurs is that launching a startup is a risky business. Let’s face it, most startups aren’t successful and will fail during the first two years. How big the percentage of startups that fail is depends on who you ask and common folk wisdoms such as 90% or 80% vary in terms of how “failure” is defined. Of course, the economic branch a startup is located in has an influence here as well.

An entrepreneur who considers to implement an idea or not must come to terms with the fact that there is a high chance of failure. Therefore, v * e must be discounted by another percentage (s) that denotes both the degree and the probability by which the entrepreneur projects his startup to be successful.

Not everybody has entrepreneurial spirit though. Not everybody likes to take the risks associated with launching a startup, but most importantly, entrepreneurs need the boldness to give up on their regular salary and run a business instead. This concept is also known as time-preference. Generally, people prefer to receive the reward of their work immediately. An entrepreneur foregoing his salary therefore needs a low time-preference, which allows him to defer his gratification to a later point in time, when his business has grown enough.

In order to account for this, we add another factor (x) to our model. Factor x denotes the percentage at which a given person ranks among the general population, including time-preference, risk-taking, and entrepreneurial spirit into the rating. For example: If you rank higher than 75% of the population and lower than 25% of the population, you have an x of 0.75.

Finally, we want to smoothen out the curve at which x affects our model. For people who rank high on the x scale, the impact should not be as much as for people further down the scale. People at the bottom end should therefore only have a sufficient incentive to implement their ideas, if they perceive them as absolutely disruptive. Therefore, only the square root of x will go into our model.

Since these values (except x) will change over time, we want to give them a time-sensitive function. Putting all the factors together, the positive incentives (u) at any monthly time-point t, are calculated with:

u(t) = v(t) * e(t) * s(t) * √x

Negative Incentives and complete model

It is in the nature of things that, when launching a business, the entrepreneur will only get a miniscule monthly income (if any), compared with what he could earn in a salaried position. In our model, we denote this difference with Δ. The positive incentives are offset by the product Δ * t, therefore the total utility a potential entrepreneur can draw out of implementing an idea is:

u(t) = v(t) * e(t) * s(t) * √(x) — Δ * t

Break-Even Points

Now of course the question is, what can we do with this model? For example, we could set the whole model (excluding factor x) to zero and resolve for t:

v(t) * e(t) * s(t) — Δ * t = 0

t = ( v(t) * e(t) * s(t) ) / Δ

This way, when modeling the growth of his business on a month-by-month basis, the entrepreneur can project how much time (or rather, how many funding rounds) it takes him to break even financially. This helps him estimate for how long he must plan to defer his salary.

Another thing we can do is solve the equation for x:

v(t) * e(t) * s(t) * √x — Δ * t = 0

x = ( (Δ * t) / ((v(t) * e(t) * s(t)) ) ^ 2

This gives us an idea about how high a person must rank among his peers in terms of entrepreneurship, in order to implement the idea. Conversely, 1 — x is the chance that an idea, when it is thought up by some member of the population, gets implemented.

I can’t stress enough that free market capitalism thrives on business ideas being implemented, but entrepreneurship remains a challenging and risky task. In Part II of this article series, I will illustrate some ways the obstacles that leave ideas unimplemented can be reduced.

Tobias W. Kaiser is a Research Associate at a-Qube. He is specialized on Tokenomics, decentralized business models, and Game Theory.