Coase highlights, as did Adam Smith, that the firm makes sense in only in a wider economic environment. Let me immediately jump to the situation of firms in a wider economic context. There's a fairly easy, though not initially very useful way to model them.
How many companies are there in the world? That's clearly a temporally bounded question. Right now (April 2014) a casual search on the web reveals that there are 120 million companies in the world, of which 45,000 are listed on various exchanges, and, separately, about 65,000 are classed as multi-national. (I got this from Quora). Forbes tells us that there are 147 companies which 'control everything'. While I appreciate the journalistic hyperbole, it isn't going to be too much of a surprise to me that there's a power law distribution in ownership of the companies in the world, and in their size. This is based on research at the Swiss Federal Institute of Technology, which performed an ownership analysis of 37 million of the hypothesised 120 million and discovered that 147 companies owned 40% of the net value represented in the 37 million. Apparently 737 companies in their research owned 80% of the value.
There are N humans and M companies. Companies can be owned by humans or other companies. The final owners of all value in all companies are always humans, though in specific cases, the mth company is owned by a number of humans and a number of other companies. The only constraint at the individual company level is that a company cannot own itself.
A single 32 bit word in a computer can represent over 4 billion distinct things. This would be more than enough to represent distinct companies. But let us be generous. Let us say we are looking to capture the ownership relationship. With 7 billion people on the planet (a large fraction of whom own nothing), together with our 120 million companies, we'd like more than 4 billion distinct things. We could just represent each human or company by a 64 bit word, and we'd be good to go for millennia. If we plan to model it on computers, we might want to economise. We could do this by assuming that no more than 4 billion people own shares. This is eminently reasonable for this year. That's 4 bytes per human/company. Pew research recently (2013) claimed that 53% of Americans have no stocks, even including retirement account holdings. This from the world's leading stock owning nation.
Let's embed 120 million companies as a series of 120 million 32 bit words, and worry about the ownership identities later, knowing that our 32 bit word is probably more than capable of capturing this. That's 457 megabytes. This can model all the companies in the world. With 27 Gigabytes of memory I can start to model all the people and all the companies in the world, just in the memory of a computer costing less than £1,500.
What, in addition to its identity, would we like to minimally model with a company? For me, two things which jump out above all others are who owns it, and how valuable it is. Or put another way, what the company does isn't being modelled at this stage. Let me get a handle on this. Upper limits first. The current high water mark was Apple, a while back, which topped $463,000,000. Companies are said to be bust if their net equity is negative, so I could imagine again a 32 bit word could capture this number, just about. Precise values are not needed. Nor negative numbers. So make it integral, giving values from 0 to 4,294,967,295. If we say the number represents the value in thousands of dollars, this is a fair compromise to getting small values but capturing larger ones. The maximum value in any one company would top out at 4.2 trillion dollars, which is plenty of headroom above 0.4 trillion. That's another 457 Mb for the value of each company.
Finally, the ownership relation. This has the possibility to be a big one. For example, Apple has by implication of its market cap divided by its share price about 890 million shares of a free float. Clearly, at worst, this could be distributed to 890 million separate individuals. Times 120 million companies, worst case, this is quite intractable if your goal is to have it all in RAM. In a couple of years it ought to require no cleverness and we will be able to model each share as a separate object. I think it might be useful to own fractions of a share, a share of course already being a fraction of ownership of the value of the company.
In the real world, a company has a register of holders of shares, and from the individual perspective, an owner of a collection of shares has their own list of the things owned and their amounts. Having both is redundant information, though it might allow faster two way lookup (what companies does this person own and who owns this company). Also a company can be modelled as a unitary whole, with fractional ownership. That is, the number of shares per se is not essential. All companies are fully owned by someone. Transformations of the ownership unit will be applied if a dilution event occurs.
The company will have a list of N owners, together with their fraction. Ownership can change.
Later, I think it is possible using techniques for automatic classification to encode the similarity between companies as patterns in the 32 bit word. This will allow industry modelling. Finally, it is possible to imply a power law distribution of company value and distribution of ownership using real data.
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