Sunday, April 9, 2017

The price adjustment equation and information equilibrium

Ian Wright and I share a love of Mathematica, but after reading his [1] recent post and one that it references I think we might share a framework. In fact, I may want to rewrite the derivation of the information equilibrium relationship using his derivation of his [2] price adjustment equation. Here is Ian's derivation (that I LaTeX-ed up a bit):
We need to translate this price adjustment strategy into something a bit more precise. Let $\Delta s$ be the change in the stock of corn (during a short period of time). And let $\Delta p$ be the change in the price of corn. We want to define $\Delta p$ (the price adjustment strategy) in terms of $\Delta s$ (the indicator of market demand relative to our level of production). 
Clearly, if our prices are huge, in absolute terms, we should make bigger price adjustments compared to when our prices are tiny. So we want $\Delta p$ to be proportional to $p$. We write this as (i) $\Delta p \propto p$. 
And, if our stock decreases then we should raise our prices (and vice versa); or, in other words, (ii) $\Delta p \propto - \Delta s$. 
Finally, we should set the price of corn astronomically high if we’re completely running out of stock; that is, (iii) $\Delta p \propto 1/s$. (This implies that, in the hypothetical situation that our stock reaches zero, then the price of corn is infinity — meaning, quite correctly, that no amount of money can buy corn.) 
Putting (i), (ii) and (iii) together we get the price adjustment equation: 
 $\Delta p = - \eta \Delta s (p/s)$
where $\eta$ is a constant of proportionality. We should give $\eta$ a name. Call it the elasticity of price with respect to excess supply ...
This price adjustment equation can be derived from the information equilibrium condition. I went through this derivation before here in the context of stochastic processes, but just let me repeat it here with the notation changed a bit to match up with Ian's. Let's start with the information equilibrium condition

p \equiv \frac{\Delta d}{\Delta s} = k \; \frac{d}{s}

where $s$ is the stock, and $d$ is the demand for that stock. In the compact information equilibrium notation I sometimes use, I write this condition as $p : d \rightleftarrows s$ (where $p$ is the "detector"). I also usually take the continuous limit $\Delta x \rightarrow dx$. Varying $p$ we obtain:

\Delta p = k \left( \frac{\Delta d}{s} - \frac{d}{s^{2}} \Delta s \right)

\frac{\Delta p}{p} = \frac{\Delta d}{d} - \frac{\Delta s}{s}

In the last line, I re-used the information equilibrium condition (i.e. $p = k \; d/s$). If demand is constant or at least large compared to changes (i.e. $d \gg \Delta d$), then the last line becomes:

\Delta p = - \Delta s \frac{p}{s}

which is Ian's price adjustment equation with $\eta = 1$. The information equilibrium version has $\eta = 1$ because of the original scale invariance it manifests. However, Ian's equation follows from a slightly more general information equilibrium condition:

p \equiv \frac{\Delta d}{\Delta s} = k \; \frac{d^{\zeta}}{s^{\eta}}

which becomes:

\frac{\Delta p}{p} = \zeta \; \frac{\Delta d}{d} - \eta \; \frac{\Delta s}{s}

so that with $\Delta d \ll d$, we have

\Delta p = - \eta \Delta s \frac{p}{s}

Ian's price adjustment equation is both more general (it doesn't require $\eta = 1$) and less general (it doesn't account for changing demand $d$ for stock $s$, i.e. that $\Delta p = f(\Delta d, \Delta s)$). Also, whether or not scale invariance is an important aspect of economics is not known, so the more general equilibrium condition may be better empirically. Ian's derivation points to at least one way the information equilibrium condition may be generalized. 

Actually, Ian asked via Twitter while I was writing this post:
One of the problems when specifying dynamic adjustment is there are too many reasonable equational forms. Does your approach help here?
One could say the information equilibrium approach helps by connecting to underlying information theory thereby restricting the number of possible equation forms. The scale invariance is doing a similar thing (and could be used to build generalizations a la effective field theory (or see here)). It does involve viewing the communication channel the market represents slightly differently. In the traditional view, the price contains information received from supply and demand; in the information transfer view, the "communication" is between supply and demand (usually I put demand as the transmitter and supply as the receiver).

Do we believe in scale invariance or that the market represents an information processing system? If yes, the information equilibrium approach does help control the reasonable forms of equations (actually, this was part of this blog's mission statement). But if we think these other considerations aren't important, then they're just ad hoc restrictions.

I personally believe the information theory framing is very useful and that scale invariance may be one of the most important (defining, even) properties of economics as a dynamical system (actually, "money" might be abstractly viewed as something that exists to maintain that scale invariance). However I understand this is probably a minority view. Still, it's not like Ian and I are on completely different pages here. (His simulations also manifest some information equilibrium relationships, and I think we have some of the same ideas about statistical equilibrium.)

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A quick aside: Ian's price adjustment equation could be seen as an information equilibrium relationship in its own right where price is in information equilibrium with (one over) the stock. Ian's equation is:

\frac{\Delta p}{\Delta s} = - \eta \; \frac{p}{s}

which we can say is (taking $s \rightarrow 1/s'$):

\frac{\Delta p}{\Delta s'} = \eta \; \frac{p}{s'}

Which I'd write as $p \rightleftarrows 1/s$ or $p \rightleftarrows s'$ in the compact notation. Dealing with the minus sign is necessary because $\eta$ here represents an information transfer index and is necessarily positive as it the the ratio of the information content of a selection from the distribution of $p$-states to the information content of a selection from the distribution of $s'$ states.

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Ian also says in the post [2] above

There’s a simple moral to this story: price adjustment alone does not solve the coordination problem.

Emphasis in the original. This something I agree with. In the information equilibrium view, and in an ideal market, prices simply measure changing demand and changing supply (we call it a "detector" of information flow and it's a ratio of the changing information in the demand distribution to the changing information in the supply distribution). However, the overall story is much more complex because there is no real guarantee e.g. all of the information in the signal sent by a change in the demand distribution is received by the supply distribution. This results in our information equilibrium condition being only a bound:

p \equiv \frac{\Delta d}{\Delta s} \leq k \; \frac{d}{s}

This is still useful via Gronwall's inequality, but leads to a much more nuanced take on the "information aggregation" problem the market is used to solve versus the "allocation" (coordination) problem the market is used to solve. It also leads to viewing supply and demand diagrams as "bounds" with the real price falling somewhere in the lower triangle:

and (for example) price time series being seen as stochastic paths constrained by the bounds.

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PS On a more speculative topic, information equilibrium might be related to Generative Adversarial Networks in machine learning.


  1. Very interesting as always. I’ve been reading Ian Wright’s blog and I hoped that you would comment on what he has been writing. I have two main comments which I think are relevant to your blog: one where I agree with Ian, and one where I prefer your approach to forecasting.

    First, I agree with Ian’s focus on what he calls ‘historical time’. In GDP-related markets, decisions on supply and demand are taken at different times. Capitalism mostly requires suppliers (entrepreneurs) to make decisions to build factories (strategic supply decisions) and deliver products to the market (operational supply decisions) BEFORE the demand appears in the market. That’s why risk is one of the most important factors in capitalism and why entrepreneurs demand profit as compensation for taking risk. Actual demand mostly just matches against the supply that has already been produced e.g. you cannot refuel your car at a service station unless the supplier has already completed all the processes required to get the fuel to the pump.

    This time imbalance forces suppliers to adopt a process in each planning cycle as follows:

    Forecast demand (often using a mathematical model)

    Plan supply (based on demand forecast, often with another mathematical model)

    Deliver actual supply (based on supply plan)

    Match demand (based on actual demand and actual supply).

    Suppliers can then assess their effectiveness in meeting their own supply plans by comparing planned supply and actual supply. More importantly for you, they can assess their effectiveness at forecasting demand by comparing forecast demand and actual demand.

    It is more intuitive to me to see your information transfer concept in terms of forecast demand and actual demand i.e. the information transfer is from the future to the present. I have said this before but I think it is also consistent with what Ian is saying.

    This also makes it easier to think about information transfer in asset markets and so-called prediction markets. In those markets, the same people often form both supply and demand, so it doesn’t make sense to me to think about information transfer between demand and supply. However, the participants in these markets are also trying to forecast the future e.g. the future profitability of a business or the winner of a sports event.

    In all these markets, information transfer is from the future to the present, and is imperfect as unexpected events happen all the time.

    I think this is also consistent with my limited understanding of the Generative Adversarial Networks material, where the generated distribution is the forecast demand, and the real distribution is the actual demand. It’s also consistent with the slide pack I published last year based on Marx / Keynes / macro circuits.

    The prime reason I would identify as a Keynesian (if I had to give myself one of these labels) is that Keynes talked about economics as a participant in markets rather than as an observer. He tried to understand the essential mechanics of the economy as it unfolds in historical time rather than via backward-looking summary totals where historical time, risk and uncertainty are abolished as in, say, Market Monetarism.


    1. A quick note before a more detailed response; the information equilibrium model can easily be interpreted as information transfer from the future to the present:

    2. Yes, we have talked about information transfer from the future previously but my perception is that you have abandonded that idea.

      If I were marketing ITE I would put information transfer from the future right at the heart of the matter. Markets are not perfect because they rely on forecasting the future and we can never know the future apart from extremely simple systems such as planets orbiting stars.

      That is why risk and uncertainty should also be central to economics. They arise because we can't forecast the future accurately. Nevertheless, we must make decisions without knowing the future. That is why capitalism values the people who are prepared to make decisions about the future and accept the ensuing risk.

    3. I haven't abandoned it at all. It just has limited consequences for the mathematics since I generally deal with continuous time so it is infinitesimally in the future.

      You're probably right, though, in terms of explaining things. It probably makes much more sense for most people to think of the "demand distribution" as the "future distribution of supply" that can sometimes be in error (non-ideal information transfer).

      I think that picture helps understand the "expectations operators $E_{t}$" (as I discuss here).

      In a sense, you can think of this as an example akin to the "Heisenberg picture" vs "Schrodinger picture" in quantum mechanics. Both representations are equally valid. The first attributes the time dependence to the quantum operators, the second to the quantum states. Both are actually equivalent (just like the temporal flow of info vs "spatial" flow of info), but sometimes one is more intuitive than the other. You are saying the temporal one is more intuitive, and I think you've convinced me that is probably true.

    4. Also, I always refer to the temporal picture if I'm ever asked about how I know information flows from "demand" to "supply" (which tells us things like the observed price being less than the "ideal" price instead of greater).

  2. Second, I don’t agree with Ian’s view of price adjustment which is what you are exploring in detail here. Ian identifies the problem himself when he says (in the Twitter quote): “One of the problems when specifying dynamic adjustment is there are too many reasonable equational forms”.

    When economists talk about a business’s expectations, they mostly mean its demand forecast. The demand forecast drives supply. However, businesses also expect that their own expectations will be wrong. Hence, they expect that they will have to adjust their plans as they are being implemented. One of the key characteristics of effective businesses is their agility in dealing with unexpected events. These events could be anything e.g. a fire at a factory; a supply problem with raw materials; a new product from a competitor; as well as a deviation from forecast demand.

    Businesses can react to operational problems in many ways and over different timescales.

    Some businesses will indeed adjust prices to deal with unexpected inventory issues. For example, hotels will normally reduce the price of tonight’s non-booked rooms as they will not be able to sell tonight’s rooms tomorrow for any price at all. Airlines also cannot sell a seat for today’s flights tomorrow. However, they tend to overbook flights to compensate, and have various techniques for solving the resulting inventory issues.

    In these cases, unsold inventory immediately becomes waste so adjusting the price, as suggested by Ian, or paying passengers to take another flight, can be viable options. However, reducing price below cost means taking a loss, so it is always a last resort and viable only as an exception.

    Other businesses will adjust future production. When Apple introduces a new product, demand may be very high. However, Apple does not increase the price until demand reduces to supply. Instead it increases production and fulfils all demand, albeit with a time delay.

    Yet further businesses re-engineer their entire business models to deliver product only where there is actual demand. This prevents inventory problems from arising at all. Typically, this happens in markets where the product is expensive and / or customised e.g. cars, where unsold inventory represents a very high unwanted cost and risk. Make-to-order processes often depend on sophisticated computer systems which integrate the whole supply chain, so this is a good example of how businesses evolve more sophisticated mechanisms over time for dealing with unexpected events.

    The problem with including this type of realistic logic in a mathematical model is that the model would need to be very complex to approximate reality. Even then, it would need to change as businesses evolve more sophisticated adjustment mechanisms. This is the same problem as with the Post-Keynesian stock-flow consistent models, where realism also results in complex and fragile models. On the other hand, including some adjustment processes, but not others, would be arbitrary but neither simple nor realistic.

    I prefer your simple approach to forecasting as it avoids these problems. Ian’s material (and mine) is better at identifying underlying logic and complexities. It is too complex to model mathematically for macro forecasting. On the other hand, I think it is better in terms of using underlying logic to explain economics to non-specialists, to link micro and macro, and to suggest possible policy proposals.

    1. I think this is another case where the "maximum entropy" view helps. The price adjustment equation that I derived from information equilibrium above does not hold for a single firm or good, but only over an ensemble of goods or, say, a whole industry. Individual firms might not change their price at all. However, the average cost of a "smart phone" (Android/iOS) will adjust based on global supply and demand.

      I am not sure, but I think Ian's example has one corn firm and one iron firm. The information equilibrium picture would have one corn industry (with many firms) and one iron industry (again, with many firms).