## Thursday, December 1, 2016

### Store of value and medium of exchange are incompatible

 From Wikipedia.

How's that for a bold title? Well, either bold or trivial. Anyway, let's assume information equilibrium is correct. I thought I'd give some reasons for why the debate over bitcoin scaling (H/T Frances Coppola in tweets here, here) is not resolvable. The store of value and medium of exchange purposes of money are incompatible using information equilibrium.

We'll follow the usual information equilibrium argument for money as a medium of exchange here. We'll start with two goods $Y_{1}$ and $Y_2$ and two demands for those goods $X_{1}$ and $X_2$. So we start with $X_{i} \rightleftarrows Y_{i}$ (with IT indices $k_{i}$). Let's introduce money $M$ so that we have

$$X_{i} \rightleftarrows M \rightleftarrows Y_{i}$$

This gives us two (well, four ... two for each $i$) information equilibrium conditions

\begin{align} \frac{dX_{i}}{dM} & = \frac{k_{i}}{k^{(m)}_{i}} \; \frac{X_{i}}{M}\\ \frac{dM}{dY_{i}} & = k^{(m)}_{i} \; \frac{M}{Y_{i}} \end{align}

This follows via the chain rule (no pun intended) and the identity $M/M = 1$. This represents our medium of exchange. You can buy $Y_{1}$ with money and the seller can then buy $Y_{2}$ with that money. In this arrangement, what does store of value mean? It means that the exchange rate for money for $Y_{1}$ is constant ‒ money never buys less of $Y_{1}$ or more. Therefore

$$\frac{dM}{dY_{1}} = \;\text{const}$$

In order for this to be true, we must have $k^{(m)}_{1} = 1$, and therefore $M \propto Y_{1}$. However, you could have picked $Y_{2}$, and if you did, you'd find $M \propto Y_{2}$. Therefore:

$$Y_{1} = \alpha \; Y_{2}$$

where $\alpha$ is some constant. This means that $Y_{1} \rightleftarrows Y_{2}$ with IT index 1 so that $Y_{1}$ and $Y_{2}$ grow at the same rate $R$. You can then, by induction, prove this for any pair of goods $Y_{i}$ and $Y_{j}$.

In order to have money $M$ operate as a store of value and a medium of exchange, it requires the entire economy to basically scale the exact same way. You can't introduce new products. Effectively we're stuck with an economy that is just scaled version of the economy in the past (by a factor of $e^{R \; t}$). (Every good would have to also grow so that there was always the same relative number of units, which would present a problem for bulk goods like blueberries or fluids like beer.) This is connected to the idea that money is something of a physical manifestation of scale invariance (at least in the information equilibrium picture). However the reason the problem exists easy to understand: it's because you can exchange money for one good, and then that money can be exchanged for a different good. Because that simple sequence of transactions is possible, every pair of goods has to grow at the same rate relative to the money stock. That is to say, it's basically because money is a medium of exchange.

And the problem is basically that store of value assumption. There's nothing wrong with an ensemble of markets with different/changing growth rates (like the discussion here, except it uses labor and productivity). With an ensemble, you'd have inflation that decreases over time (like the productivity that decreases over time) ‒ but that is exactly degrading the value of money.

So we have an incompatibility. Money can be a store of value, but only if it's not a medium of exchange. And money can be a medium of exchange, but only if it's not store of value.

1. This is an interesting post. I have been thinking about this independently, and have a different perspective.

First, though, I posted a comment on here recently and it disappeared. Did it get lost or did you delete it? I don’t mind if you delete selected comments as you would need to read them to make the decision to delete them. When I comment here, I am addressing my comments to you. Unless I am explicitly addressing another commenter, I’m not addressing my comments to the wider world using your blog as a mechanism to talk to others about you, so I don’t mind if no-one else sees any comments with which you disagree strongly.

Back to money. “Money” is an odd word. We use it in different ways at different times. I think that the distinction between medium of exchange and store of value is a useful one, but I am coming at the subject from a different angle.

First, we use the term “money” in the same way that we use an abstract concept like “length”. In this sense, length is a dimension in the real world. Money is a dimension in our system for keeping track of certain activities in the real world. I would summarise it as:

Commerce ---> need to keep track of what happens ---> need for counting & accounting ---> need for money as a concept that we use in defining the rules of accounting.

Second, we use the term “money” (also “currency”) as a unit of measure of the accounting concept in the same way that we use a concept like “metre” or “foot”. THE GBP or THE Euro of THE USD are all units of measure. The main differences with a metre and a foot are that the physical measures are constant where currencies change their scales all the time (inflation), and that there is a constant ratio between a metre and a foot but the ratios between currencies also change (exchange rates).

Third, we use the term “money” as a measurement of the accounting concept in terms of the unit of measure. I have €10. I spend €4. I now have €6. In this sense, the money I HAVE is the purchasing power I possess. The money I spend is the purchasing power I USE.

Economists use the term “medium of exchange” to reflect money’s USE in spending. They use the term “store of value” to reflect the HAVING (possession) of money.

In this sense, it might be simpler to think of money as purchasing power, and to avoid the term money. Purchasing power has two states from the perspective of the individual: used and unused (I don’t think these are quite the right words but they’ll do for now). UNUSED purchasing power is the money I HAVE. USED purchasing power is the money I SPENT. In macro, we also need to realise that if I USE my purchasing power by buying a bicycle from you, my USE of purchasing power results in you HAVING increased purchasing power. That’s because money is conserved under exchange.

There is no doubt that we both HAVE money and that we USE it, so although I might prefer different terminology, I’m on the side of the economists here.

If money wasn’t a store of value, we might imagine a world where you could use money to buy things today, but that tomorrow the money would have no value.

I think that scenario would result in panic / herding behaviour. People would do anything to get their money out of the bank and then exchange it for something else which WOULD have value tomorrow. Money goes wrong when people imagine that it will have no value tomorrow. That’s what happened in the 1920s.

1. I think long comments tend to get sent to the spam folder automatically (I don't think I can control the settings with blogger). Most of the time I get an email notification for comments, so if it doesn't appear, I can fish them out.

I saw your earlier comment when I went to fish this one out of the spam folder. I swear it's not my fault!

Let me think about the substance of your comment some more before responding to it.

2. I generally agree with what you are saying here -- and that it is important to understand money as a unit among it's other roles like exchange.

I think this is the crux of it:

"The main differences with a metre and a foot are that the physical measures are constant where currencies change their scales all the time (inflation), and that there is a constant ratio between a metre and a foot but the ratios between currencies also change (exchange rates)."

I agree that this means money is a very different measurement of value from other units of measurement of other quantities as commonly experienced.

I think there is a resolution to this, however. We may have jumped the gun regarding what money measures. We think it measures value, but really it is a unit of information.

If we take two dollars from 1920 it would buy you a sweater in 1920 (per here). In 2016, two dollars will buy you an Amazon "kindle single" e-book. The value proposition here is really not compatible, but maybe the information entropy of the distribution of that kindle single across the 2016 population is equivalent to the information entropy of a sweater across the 1920 population. In that sense a "dollar" still represents the same "unit", but it's information, not value. The details of the object costing two dollars does not matter -- it's a matter of how it is allocated among possible states. (For some visuals, see here.)

3. Ok. Don’t worry about the publishing delay. I was concerned that I had upset you as my previous comment was robust in asking what you are trying to achieve.

From the commenter’s perspective, the comment appears to be published correctly (I always check). However, if you come back to check for a reply, the comment has disappeared. This makes the commenter believe that it has been deleted after it was published.

It is bad system design. If the system intends to send the comment to spam it should tell the commenter and say that the comment will not be published until the blog owner has taken some action. That’s not the blog owner’s fault.

In future, I will send a short comment about spam if a longer comment has “disappeared”.

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4. In this post, I suspect we mostly agree although we are using different words. Many of the problems here arise from very basic vocabulary.

For example, I agree that money is information but I use the word “information” in a different way to you so I doubt we mean the same thing. I think of information as something that is stored in an all-knowing logical computer system. My version of “information” contains meaning and is recorded in the all-knowing computer system as a way of keeping track of things (accounting in its most general sense). We all communicate with the system by sending and receiving logical messages which also contain meaning.

You use information in a different way that I can’t quite pin down. Your analogy seems to be a logical telecoms network which transports my logical messages between different logical computer systems but, even then, you don’t seem to be talking about the meaning of the message, just its existence.

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5. Note also that the basic monetarist model

MV = PQ

conceives of money only in its USE. The V term tells us how often the money is used. The concept of money as a STORE does not exist.

In the basic two-sector Keynesian model

I + C = C + S

The C terms represent the equivalent of the monetarist model where money is USED for exchange between businesses and households.

The S term (saving) represents money as a STORE. That is why it causes so much confusion. People who think like a monetarist often use “putting money under the bed for safe keeping” as an example of “saving” but this is wrong and is a great “tell” that someone does not understand the Keynesian model.

There is no equivalent of Keynes’ I term in the monetarist model either. This term is where assets, good and services are created. Monetarists never talk about this, so it appears to a Keynesian that Monetarists don’t think that the creation of assets, goods and services is important. Nevertheless, they can be bought and sold!

Any logic (or mathematics) built on top of these two models will diverge because of these fundamental differences.

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6. I have published three short comments in the last few minutes, not including this one. The first has already disappeared.

7. Hopefully that fixed it.

I have no idea what heuristics blogger uses. One is "long", another is "more than 1 link", but it's not obvious.

8. Regarding the definition of information, yes, I am using it slightly differently than the colloquial definition -- I am using the technical mathematical definition of information entropy that is used in the theory of communication networks.

In a sense, it might be better to think of the technical definition as potential information or even probability. What it tries to quantify is whether the information that you're thinking of (the normal definition) could potentially move from one place to another.

For the probability distribution of demand to be equal to the probability distribution of supply, so that for a large number of events, supply is approximately equal to demand information (in the colloquial sense you're using) had to flow from one to the other. Some kind of information had to move around to get blueberries to the store where I buy them.

For example, if I roll two dice and you roll two dice, you will have to send me (at worst) about 5.2 bits of information = 2 log2(6) for me to change my dice to match up to yours. I could have rolled the same roll and not have to do anything. I could have rolled something different.

The information equilibrium framework is saying that that process of matching dice is basically the process of supply meeting demand, and in the post above the information allowing the dice to match up can be carried by money.

2. Well if money is not a Store of Value then you can't use it in the future. Bitcoin's problem is that *it* can't do both, but that's bitcoin's problem not money's.
Personally I think money as tokens of Trust from society to the holder

1. I am not sure what this means. "Store of value" above means zero or low inflation. It's not a true binary, but the ideal of zero inflation money is impossible per the argument above (inside the information equilibrium framework).

2. Even at 10% inflation money still stores value, it just decays faster.

3. Now I'm even more confused as to what you mean.

A store of value with a decaying value is kind of an oxymoron ...

4. Is there anything in nature that does not decay?

3. Economic Models generally either take into account either multiple agents or time, in the first money is payment system in the second view its store of value. So you can't *use* money's both aspects in a model. But again that's the model's problem

1. I'm not sure I understand what this means. It's definitely not a restriction on actual models; you can easily define a model with a money "field" m(k, t) where k is the k-th agent and t is time.

2. Sure you can define it, but when it comes to solving it you make assumptions as in your analysis above all transactions happen in a single moment. The time dimension is dropped, so money is just a transaction balance measure, you could as well use pebbles and the model would still be correct. That pebbles is a payment system. What if you don't have M, but Mi. Money now not the same as Money after

3. If you're allowing for transactions to occur at a single moment, then there will be an (instantaneous) inflation rate associated with each single moment as well (money value has a magnitude and rate of change). Therefore both aspects (store of value, medium of exchange) will be "used" at each moment in time.

4. Ordering of transactions is only needed in a model for payment system to work, if you disregard the in between time, ie drop time dimension, you simply maximize money velocity and thus inflation rate effectively killing "Store of Value", and "Prove" that money is not a store of value.
If on the other hand you Hard Code ZIR (Remove time from money) which is a violation of natural laws, like a perpetual motion machine. You a either make it priceless, unfit as an exchange medium for decaying goods, so no Payment System yes.
But this is false dichotomy in my mind, you cant possibly say that at 10% inflation for example money is not a Store of money, like saying that a Battery that leaks charge at 10% annually is not a store of power.
Money is also as a Product, that has demand, and has it's own growth rate, and is managed.

4. Nice post

I agree that the two concepts are at odds.

I think an important part of it is because money has to have an issuer. Someone (or some entity controlled by a collection of people) makes a decision as to what money will be. In the US its \$, in Europe its Euro etc etc . The issuer has an interest in what the money is used for. Money is a type of social control mechanism for them. And the nature of the issuer (public or private) affects the mechanism of control
Private money issuers, like banks, decide who they will extend credit to and what they will ask in return. In a large part these issuers determine the prices of what they lend for. Mortgage lenders have an oversized influence on the price of real estate. But they are also interested in a return. Their rate of interest is both a cost to one sector (most of us who borrow or use credit ) and a return to another (bank executives)

Govts, through taxation policy and other fiscal decisions, end up influencing the spending power of its citizens, which can affect overall price levels. Very targeted tax policy can change the price of a specific good (think gasoline or cigarette taxes)
Govts arent interested in a monetary return like banks are, they are interested in influencing a service for them (military conscription or building a road say) or changing a behavior like smoking which is an expense for them.

Both entities are using (issuing) the same dollar and have conflicting goals. Private issuers are really interested in keeping the unit of measure relatively constant so they are getting paid back in relatively the same terms while the public issuers are only looking for real returns ( an hour of labor or some other service). They will pay whatever they have to pay for whatever service they need. Its this conflict between private and public issuance that is at the heart of the conflict you describe I believe.

1. In this, you describe the tension between store of value and medium of exchange as a behavioral relationship. Which may well be true.

However, that is different from what I am saying ... I am saying that the two uses are inconsistent with money ideally mediating information transfer between supply and demand. If it is a tool that matches supply with demand, then it cannot simultaneously have constant value.