Orbiting Firms: Economic Roche Limits

The boundary of the firm remains a perennial and interesting question: where should firms begin, and where should they end? When should a firm be split apart, and when should it absorb another firm?

We've seen in the past where when companies split, their combined value after the split is higher, which is an indication that it’s valuable to break up the company. Vice versa, in some circumstances, when a company acquires another company, the parent company's value increases higher than the previous values combined, indicating that value was generated.

So, is there somewhere a line where this boundary could be more readily seen, defined and enacted upon? In classic economic theory, this is called the “boundary of the firm”, made famous by Ronald Coase. Grossman, Hart, & Moore, for examples, pushed the needle forward in this field, by finding such a boundary with their work on incomplete contracts.

I've explored this question in various ways on my own in the past. From wondering if there's clues in biological cells (when exactly do they choose to split?), to stating that we should incentivize the splitting/forking of cryptocurrencies to enable better coordination.

One recent field of interest is looking at the cosmic concept of the Celestial Roche Limit: it's the point at which an object rips itself apart due to increasing gravitational tidal forces from a nearby object. If it comes close enough, it disintegrates. I found that an interesting idea that one could use to compare and understand how firms might orbit firms in the market.

An object disintegrating from its own tidal forces: https://en.wikipedia.org/wiki/Roche_limit#/media/File:Roche_limit_(ripped_sphere).svg

In economic terms, for example: let's say there exists a manufacturer, and it regularly uses a close supplier. At some point, does it make sense to absorb this firm that's 'orbiting' close by? If we could somehow relate a Celestial Roche limit to this context, we could perhaps paint an estimate, such that, we could say: now, would be a good time for the manufacturer to acquire the supplier.

Whilst the Celestial Roche Limit works only from the outside-in (celestial bodies being ripped apart for coming too close), in economic terms, it could also perhaps point to a context where it makes sense for a company to split up. For example, a marketing department spinning out in order to also take on external clients.

So: can we figure out some boundary of firms from looking at the Celestial Roche Limit? Let's first understand how the Celestial Roche Limit is defined.

Celestial Roche Limit

It's officially defined as: "...the distance within which a celestial body, held together only by its own force of gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction."

The factors to determine this are the size, density, mass, and rigidity of the objects.

The more fluid the secondary object is, the more its tidal forces will rip itself apart. For example, a rubble-pile asteroid will disintegrate sooner than a more rigid asteroid. Disregarding the fluidity, for rigid objects, the derived formula is:

The radius of the primary object multiplied by the density of primary over density of secondary, times 2, all together to the power of 1/3.

Equivalently, because density is mass over volume, it can be also derived as: the radius of the secondary object multiplied by the mass of the primary over the mass of the secondary, times, all together to the power of 1/3.

In the first version, you take the radius of the primary and measure the density of both objects. In the second version, you take the radius of the secondary and the measure the mass of both objects.

So, the important factors are the size, density, and mass of the primary and secondary objects. Once you add rigidity, it becomes more complex, but intuitively, the more rigid, the stronger it is, the less rigid, the weaker it is. So, now, let's run some thought experiments on what each component in this could refer to in economic terms of a firm.

Economic Roche Limits

This is a brainstorm and thought experiment. If you have other ideas or mappings, please share!

Mass = How many resources are there in the firm?
Examples:

Large Mass Firm: eg, a firm with many people, capital, and optionality.
Small Mass Firm: eg, a firm with one person, and working on savings.

Volume = What's the scope of the firm?
Examples:

Large Volume Firm: impacts many parts of the market. Amazon.
Small Volume Firm: narrow in scope, a plumbing business.

Density = Output of the firm (resources/scope)?
High Density Firm: Well capitalised to solve one single solution.
Low Density Firm: Bad tech start-ups trying to do everything.

Rigidity = How reliant are different components in the firm on other sectors of it?

In Coase's terms: How much value is being generated from reduction of transaction costs? A more rigid company is more reliant on all other parts of the company.

Examples:
A fluid firm: amorphous, loosely connected firms. Open source projects, DAOs.
A rigid firm: traditional firms, bound by strong legal contracts, golden handshakes, and recourse/penalties.

With all of this together, we can create a hypothetical equation to determine the Roche Limit of two firms. From Celestial to Economic, we can define this limit as:

Economic Roche Limit1 = Radius of the Primary Firm / (2*(Output of Primary Firm/Output of Secondary Firm))^1/3.

If the secondary object comes within this 'economic distance', it will either break apart on its own and be horizontally or vertically integrated, OR it would be advised to do so by the parent company.

Alternatively:

Economic Roche Limit2 = Radius of the Secondary Firm / (2*(Resources of Primary Firm/Resources of Secondary Firm))^1/3.

These are all vague terms: can we get closer to getting actual, practical numbers? Let’s try.

- Resources can be derived from calculating the value of the assets (including its people).
- Volume of a firm can be derived as how many *disparate* transactions it makes (eg, one client, or a million clients). The radius of a firm is thus: r=((volume/π)(3/4))^1/3.

So, your final answer will get you perhaps a meaningful number, but what does it mean in practice? The next problem is calculating economic distance, in general. Unlike physical time-space, the economic world might be mapped differently. On its own plane of dimension, it has its own gravity wells.

Economic Distance?

The final question then is: how do you calculate economic distance, in general? How close is another firm to another? How do you know that a firm is close to another firm's Roche Limit? Answering that is its own rabbit hole.

An intuitive extrapolation is to look at gravity, which is roughly defined as mass*mass/distance. There is historical research in this area that defined a gravitational force of trade (gdp*gdp/distance). It’s a simple concept: economies close to each other, will likely trade with each other. This is a good estimate for figuring out ‘distance’, but in our modern era, this collapses with the internet. For example, take a games company. I might do all my economic interactions with Steam, Amazon, Facebook, etc, whilst being incorporated in South Africa. Physical distance still matters, but I think there might be a different way to frame it.

I would say: two firm's economic distance is based on the size and frequency of transactions between them. It’s a measure of how close a firm is orbiting to another’s firm gravity well. For example, many firms orbit around Amazon in some form (due to their abundant cloud infrastructure), but it might be in small capacities (a $10 bill per month for a server instance). In some sense, economic distance is taken as a collective perspective on all transactions in the market: somewhat like Google's PageRank. Gravity works the same.

That being said: Roche Limit only matters for objects that are big enough such that its tidal forces rips itself apart when coming too close. Just like how small objects won’t disintegrate before it crashes into another object, my hypothetical, $10/month start-up might be entirely in Amazon’s orbit, but it’s not going to be torn apart by Amazon: or in other words, there’s no reason for Amazon to absorb my firm.

Practicality & Conclusion

This article serves to introduce the concept of an ‘Economic Roche Limit’: the limit at which a company is likely to be absorbed or SHOULD be absorbed by a bigger firm close by. It does so by comparing the Celestial Roche Limit and transposing it to economic factors within firms.

It's not currently practical, and might never be, but I think it's at least an interesting research direction. My gut feeling is that you could probably derive actual, meaningful numbers: creating some model to define firms in terms of its economic mass, volume, and size. Answering this in detail could then give us clues to understand not only when firms should be absorbed, but when they could be split open and have parts of it divested.

At the end of the day, firms orbit each other, and one day, they might find that the company gets ripped apart by its own tidal forces as it approaches a larger company.