f(x) — The Abstraction of a Startup

Startups, at their core, are abstract functions. They take in time, money, and people as inputs and, if successful, generate shareholder value, jobs, and customer value as outputs.

While Venture Capital is hellbent on using AI and Machine Learning to study the past, my thesis has been simple when it came to studying a startup’s potential. No one really knows. There are signals though, albeit small ones.

I’ve spent the last period through thinking, conversations and writing how to mathematically model a startup company’s output. And that’s led me back to the very core of what a startup venture actually is.

It’s just an abstraction of some numbers.

And right now I’m fully aware people will be disagreeing with me. Hear me out.

Stripping everything back, a startup can be defined as:

f(x)=V

Where…

f(x)

is the startup function, x represents the invested resources, and V is the resulting value. But what happens within this function? Why do some startups thrive while others never reach meaningful outputs?

The Inputs: Time, Money, and People

A startup does not exist in a vacuum. It relies on three fundamental inputs:

1. Time (T) — The non-renewable resource. How effectively a startup utilises time dictates its ability to iterate, learn, and execute before competitors or market shifts make its proposition obsolete.
2. Money (M) — The energy source that fuels operations, product development, and market expansion. The efficient allocation of capital separates sustainable startups from those burning cash without direction.
3. People (P) — The intelligence, creativity, and execution force behind the startup. It’s not just about the number of people but the right people who align with the mission, can pivot when needed, and possess the resilience to handle uncertainty.

Thus, we can redefine our startup function:

f(T,M,P)=V

Where V consists of multiple forms of value:

• Shareholder Value (S) — Returns on investment, market cap growth, or a lucrative exit.
• Jobs Created (J) — Employment opportunities within and outside the startup ecosystem.
• Customer Value (C) — The problem-solving utility provided by the startup’s product or service.

Back to the outputs, with S, J and C it’s possible to expand the function.

Rewriting further:

f(T,M,P)→(S,J,C)

These outputs in rank of importance and attainment have radically changed since the 2008 financial crash. Where exits were plentiful they are now rare, more startups acquiring startups is an exit plan more than a IPO happening.

VC funding has taken a battering in the last five years, the peak 2021 investments are still in the game but the way people worked didn’t really change, reversion to the mean for working from home, everyone had to go back to the office. And most WFH investments were rendered a failure. While Clubhouse may still exist I’ve not heard anyone talk about it in four years.

Investment dynamics have changed, add to that the raise in interest rates and inflation, the era of cheap debt vanished and running a company became far more expensive.

I firmly believe this has had an impact on Venture Capital’s available funds, from a financial point of view it’s easier for a partner to compound daily at 7% annual interest rates for ten years (yes, there are some accounts doing this) than plunge money into a fund that may never see a return.

The Black Box: Execution, Strategy, and Market Conditions

While the inputs and outputs can be mapped, what happens inside the function remains the mystery that founders, investors, and analysts attempt to decode. Several factors influence whether a startup function operates efficiently:

• Execution Efficiency (E): How well the team translates resources into a working product and traction.
• Market Conditions (Mk): External forces such as competition, regulation, and economic trends.
• Product-Market Fit (PMF): The feedback loop between a product and customer demand, ensuring sustainable value generation.

Thus, the refined function incorporates the uncertainty of execution and market influence:

f(T,M,P∣E,Mk,PMF)→(S,J,C)

Optimisation: The Art and Science of Scaling

A startup’s success isn’t just about maximising inputs; it’s about optimising them. Too much capital too soon leads to inefficiency. Scaling prematurely without product-market fit results in failure. The ideal startup function optimises execution efficiency, learns from market feedback, and adapts its resource allocation dynamically.

Investors look for startups that exhibit a high marginal return on incremental investments — where additional time, money, or people produce exponential value rather than linear gains.

It’s worth noting that the world is definitely non-linear.

The Reality: A Non-Deterministic Function

Unlike a pure mathematical function, startups are non-deterministic. Given the same inputs, two startups can yield wildly different results due to variance in execution, unforeseen market shifts, and luck.

However, the principle remains: a startup is a function transforming resources into value. The key to success lies in:

1. Minimising Waste: Reducing inefficient use of time, money, and talent.
2. Maximising Learning: Rapid iteration to refine the function over time.
3. Leveraging Network Effects: Compounding value through users, investors, and partners.
4. Timing the Market Right: Synchronising efforts with the optimal external conditions.

The old first-out-of-the-gate heading for 70% market share have largely vanished. However the need for marquee customers who can provide support in case studies, column inches (or Linkedin video recommendations) cannot be undervalued.

Volatility Shocks

Even the most well-optimised startup function is vulnerable to sudden shocks that can disrupt its trajectory.

These volatility shocks come from various sources: the rapid advancement of AI and automation, which can render products or services obsolete overnight; unexpected regulatory changes that impose new compliance burdens or restrictions; and reputational damage, whether through a public relations crisis or a breach of customer trust.

Reputational damage is the loss to financial capital, social capital and/or market share resulting from damage to an organisation's reputation. (Wikipedia)

Each of these factors introduces unforeseen variables into the startup function, forcing businesses to react swiftly or risk failure.

Successful startups build resilience into their operations by anticipating these shocks, diversifying their risk exposure, and maintaining the agility to pivot when necessary.

The Startup Function as an Experiment

“If you can build and sell then you’re way ahead.” — Mary McKenna, AwakenHub.

Every startup is an experimental function, iterating through cycles of input and output adjustments. Those that master execution and adaptation unlock exponential growth, proving their function as viable, scalable, and valuable.

I run simulations on startups as a service to VCs, Angels and Private Equity. While the data points are proprietary the fundamental black box functions remain core to what I do with ATXGV FutureSight.

We’ll never know but we can model the startups potential outcomes. Running multiple simulations with various small parameter changes can teach an awful lot on the changes a startup could do to increase shareholder value.

The abstraction of a startup is simple: it’s a function. The complexity lies in the execution.

f(T,M,P)→(S,J,C)

Success is not just about running the function, but refining it until the outputs far exceed the sum of its inputs.