Apples, Zebras, And Knowledge

allen farrington
10 min readFeb 20, 2021


Part II of the Bitgenstein Serialization

photo by Nathan Dumlao, via Unsplash

Imagine you could see the future in a very peculiar and economically relevant way: you know how everybody will value everything at every point in time. You are certain of it. Would you ever need money? Let’s explore.

If you know how you will value things then you know what you will want to buy at any given point in the future, say Alice’s apples. Now, normally you would need money because Alice might not value whatever you had that you wanted to trade, say zebras. But in this case, you know how Alice will value zebras because you know how she will value everything. If that’s enough to buy the apples, great. If not, find something else she values more highly, say beer, and then go find Bob. Now, if Bob also doesn’t value your zebras, that’s fine too because you can just keep doing this until you close the loop, given you know all these valuations. You can instantaneously grasp the best path from zebra to apple and trade all the way to what you want. If these valuations flap around over time (which you will also know) you can plan to orchestrate your trading across time to get the best rates.

An obvious objection might be that all this excess trading would be a massive nuisance. Maybe you wouldn’t actually need money, but it would still save you time and energy because rather than having to deduce and execute the highest value trading loop in advance of every purchase, you could jump immediately to who valued zebras most, get the money, and go back to Alice. And in fact, you could do even better, because you would sell the zebra when its value peaked, and keep the money for however long happens to pass before wanting to buy the apples. Fair enough, money is still useful.

But now imagine that everybody had this superpower. Everybody knew what everybody else valued at every time. Now would you need money? Let’s ring up Alice and find out. You go to Alice with your zebras to get some apples. Does it matter that Alice doesn’t want zebras? I don’t think it does anymore because Alice now has the superpower too; if she doesn’t want zebras herself, she knows that Quintin does. If she wants Quintin’s quiches, perfect. If not — if she really wants Peter’s pies — then luckily Peter also knows that Quintin wants zebras. If he wants Quintin’s quiches, perfect, if not … what happens is that the previously inconvenient highest value trading loop doesn’t now need to be executed — it merely needs to exist at all in order for everybody to establish the relevant exchange rate between whatever they have and are trying to buy, or whatever they want and are trying to sell. This knowledge would make money moot.

If we were at all tempted to believe that some particularly economically astute individual could, in theory, know all others’ future valuations, we would be forced to admit they also knew all future events as well. How could they know how Alice will value things at an arbitrary point in the future if we don’t know what things there are to value?

This is the key to unwinding the thought experiment. Nobody can know all future events, nor the whims of all perfect strangers. Arguably, we can barely know any future events or the whims of any strangers. We may be reasonably sure of events close to us in time and circumstance, but as we get further away from our own familiarity, further into the future, or both, we obviously can claim to know less and less. Hence, we might be able to guess what we ourselves or others close to us value now, but others we know less well, or anybody at all further and further into the future, we cannot know. If we did know this, we would not need money. But we can’t, therefore money is useful.

What this shows is that a “double coincidence of wants” that makes barter untenable at any worthwhile scale has little to do with “convenience” and is first and foremost a product of knowledge. We can only have a limited appreciation of others’ valuations, and this appreciation diminishes the further removed from us they are in circumstance and in time. And note, this includes our future selves; we do not know for sure what we will value in the future because we do not know what will happen to us in future. Money is useful to us because of economic uncertainty: our fundamental inability to know much at all about what all others think and about what is going to change.

Money would ideally give its users a sense of certainty about the future — a kind of tacit agreement with the rest of the economic network precisely because next to nothing else can be known surely enough to be agreed upon in advance. But if a new money were to emerge, it would very much be challenging the agreement, and hence would be fraught with uncertainty over the period of its emergence. Its very existence would represent a kind of disagreement that would seem confusing and disorienting.

We could point to this engendered uncertainty as clear-cut evidence against the new money, and reason by extension that the necessary network effects will fail to materialise. But this would be to avoid the legitimate question as to whether accepting greater uncertainty in the short run is worth gaining greater certainty in the long run, and if network effects could develop on this basis instead. This would depend on the real-life merits. There would be no textbook answer; no equation to solve; no definitions to check against — particularly not definitions that are only absolutely valid in precisely the case of zero competition, maximal certainty, and static time: a kind of inexplicably non-entropic universe. A new money could not be helpfully described as providing utility as a “medium of exchange” from its beginnings — although, in the long run, being digital certainly wouldn’t hurt its chances in this regard …

What it would seem like if it did seem like a new money were emerging would be a promise of superior certainty in the future, with an admission of increased uncertainty in the present. It would seem as much like a dangerous and exciting narrative as a provable guarantee.

Purchasing Power In A Changing World

photo by Clay Banks, via Unsplash

But how usefully can money fill a demand for certainty? We should think about what it would mean for money to be more or less useful in this way, but attempting to measure this utility brings us to the precipice of a philosophical pickle. Given this utility is understood as an emergent reaction to constant and unpredictable change, we must be careful to choose a measure that is invariant to this change. We don’t want to use a stretchy ruler. We must dig a little deeper lest we fall back into the semantic theory and demand money be a “store of value” without further exploration of what being a store of value could practically consist of.

As a starting point, let us assume it means that a candidate money retains its purchasing power over time; that it will not intrinsically depreciate, like a tool that we need to repair or food that will rot; or that will lose value for more intangible reasons, in the sense of going out of fashion, like clothes or jewellery, or real estate in a declining area.

We immediately run into problems: notice that whether or not any such item “retains its purchasing power” is precisely subject to the change we admit we cannot predict. It is circumstantial and behavioral, and depends on the subjective valuations of other market participants. A depreciated tool may nonetheless become more valuable if a new use for it is discovered and it turns out to be in short supply. Rotted food may command a higher price as compost — perhaps there is a dire compost shortage — or maybe it is discovered that it actually tastes better at a certain maturity. Clothes, jewellery, or real estate might come back into fashion and appreciate just as easily as they once went out.

There is yet another difficulty: surely, we want a failure to maintain purchasing power to capture some way in which the money ends up being valued less relative to what we might want to purchase, but not some way in which everything else ends up costing more? In other words, that the money is less useful, not that everything else is more useful. In the latter case — say a war or a natural disaster that causes all real costs to soar — it hardly seems reasonable to describe the probable resultant price increases in terms of money’s decline in value rather than on things having become much more costly to produce. Taken to an extreme, it seems nonsensical to expect that money’s ability to “store value” will entitle its holder to consume some good that may not even end up being produced — or some quantity of goods greater than everything that is produced.

But it gets even worse: what if an entirely new good or service is invented over the period across which we intend to maintain purchasing power? Has our candidate money’s purchasing power increased infinitely? What if a good stops being produced or sold on the market altogether? Has our purchasing power decreased infinitely? What to make of all this?

It might have first seemed natural to mean specifically that a candidate money “retains purchasing power” if, at t0, it can purchase a given quantity of every other good or service in the market, then, at t1, it can still purchase this same quantity in each case. Upon probing, however, this seems naïve — in the case of a new invention, “infinite” seems an amusing but entirely unhelpful answer. In the case of natural disaster, we can’t expect to purchase the same amount because we can’t expect there to have been produced the same amount, which seems not to be a negative attribute of the money, but rather the money accurately reflecting a negative change in the economic network. Perhaps we could track the changes in a “basket” of goods, but then how do we decide how to weight the inputs and how do we adjust for quality improvements? Any choice will be arbitrary and will better reflect the reality of some economic actors than others.

But even in what we might think of as more “regular” or “normal” economic circumstances, it proves tricky to pin down. Imagine that some good, a widget, hasn’t changed at all between t0 and t1, but that some new invention, a fidget, has superseded it. Nobody wants widgets anymore because fidgets are far better, so widget makers stop producing widgets, and the suppliers to widget makers stop producing widget-making-squidgets too. Now, with nobody making squidgets, it becomes very expensive to make widgets and although far, far fewer are made, their price becomes much higher than at t0, pre-fidget. So, has the money with which all these prices have been quoted “lost purchasing power”?

If you still want to buy widgets, then yes, you have lost desirable purchasing power. If you prefer fidgets, then no, you have gained purchasing power. But crucially, these are connected at root by decisions to shift real, scarce resources from producing one good to another. Hoping for an answer that can be sensibly relative only to our individual demands is myopic because the prices will be dictated by the aggregation of all demands (and for that matter, all supplies — people who sell widgets and squidgets have a stake in this too!). If any demands or supplies change — due to discovery, taste, costs or opportunity costs elsewhere — any circumstance or behavior anywhere — then the allocation of scarce resources to try to match supply to demand will almost certainly change too, and in a way that will wrap back around eventually to affect the original demands and supplies.

Can we grapple with this? And can we coherently describe why it is difficult to grapple with? Even before resolving this, it is worth pointing out that money itself will be subject to these same forces — it will just be a lot less natural to point out whenever there is no legitimate monetary competition because we are using it as our measure in the first place. The binaries of the semantic theory seem to absolutely apply. This money would be a unit of account, and while we can assess how it is accounting for units, it means little to ask if it is doing so well: well relative to what?

But were a new money to emerge, it would naturally have to emerge in a specific place, at a specific time, in a given locality of the economic network. It would not appear everywhere, all at once, and hence could not be helpfully described as providing utility as a “unit of account” from its beginnings — although, in the long run, being global certainly wouldn’t hurt its chances in this regard …

It would present different utilities to different people depending on their circumstances, their behaviors, their beliefs, and its price. Its entry into the web of supply and demand would change the allocation of scarce resources in ways that would wrap back around and eventually affect its own supply and demand. Hoping for an explanation of its value that can be sensibly relative to individual demands is myopic.

What it would seem like if it did seem like a new money were emerging would be widely disparate views, rapid change, and no consensus, with sentiment ebbing and flowing unpredictably as the relative merits of the promise of future certainty from the challenger and the supposed failures of this promise from the incumbent were constantly being weighed and re-weighed by the market.

continue to Part III:

or go back to Part I:

n.b. This is a serialization of my previous trilogy on Bitcoin, economics, and capital markets: Wittgenstein’s Money, The Capital Strip Mine, and, Bitcoin is Venice.

follow me on Twitter @allenf32



allen farrington

I’m an investor. I think about things. I write some of it down.