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# Considerations Of The Relationship Between Price Elasticities And Expectations In Price Formation

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Posted on 01 April 2020 by Philip Pilkington Article of the Week from Fixing the EconomistsAs I have already written in my introduction just after I had sent off the final draft of my paper I noticed a rather glaring error. This error can be best understood by comparing equation 1.10 and equation 2.10 from my paper both of which I reproduce below.Please share this article - Go to very top of page, right hand side, for social media buttons.(Note that a guide to the algebraic terms used can be found at the very beginning of my paper).Note that equation 1.10 represents how the price of what I call in the paper a ‘pure’ financial asset - say, a company bond - is set, while equation 2.10 represents how the price of what I call in the paper an ‘impure asset’ - that is, one that has

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posted on 01 April 2020

by Philip Pilkington

Article of the Week from Fixing the Economists

As I have already written in my introduction just after I had sent off the final draft of my paper I noticed a rather glaring error. This error can be best understood by comparing equation 1.10 and equation 2.10 from my paper both of which I reproduce below.

(Note that a guide to the algebraic terms used can be found at the very beginning of my paper).

Note that equation 1.10 represents how the price of what I call in the paper a ‘pure’ financial asset - say, a company bond - is set, while equation 2.10 represents how the price of what I call in the paper an ‘impure asset’ - that is, one that has a financial and a real component (say, oil or gold) - is set. The difference is rather obvious. In equation 1.10 all that matters is (a) the price expectations variable in its complete form, that is the series of variables in the middle of the right-hand side of the equation and (b) the supply variable in its complete form, that is the series of variables at the very end of the equation. The intuitive meaning of this is that the price of a pure financial is set only based on expectations and increases or decreases of a supply of this asset.

Equation 2.10 tells a different story. First of all, we have two expectation terms. The first is the same as the one found in equation 1.10. This denotes the amount by which investors think that the price will rise purely due to financial considerations. So, let’s say that there a large group of speculators who think that gold is going to rise in price as people become more anxious about central bank actions. This will be reflected in this term. The second term, however, refers to the extent to which investors think that the price will rise due to real market considerations. So, let’s say that there a group of speculators that think that the real demand for gold is going to increase because a new technology has emerged in some line of production that requires gold to work or, alternatively, say that gold is becoming more fashionable in the jewelery market.

In reality, these two terms have a great deal of overlap. Imagine, for example, that the demand for gold increases suddenly in India because of fears of inflation. Generally we would tend to attribute this increase in demand to the financial component of our equation. In India, however, while gold is used as a store of value the manner in which this is done is cultural - brides, for example, are endowed with gold jewelery as part of their dowry. But the distinction between whether the jewelery is thought of as an asset and whether it is thought of as a decorative product is murky to say the least. As The Jewelery Company website notes,

Gold jewellery is simultaneously a status symbol and instrument of adornment as well as an investment.

I think, however, that despite these complications it is didactically useful to separate notionally financial demand from real demand. It allows us to conceive of the fact that markets for ‘impure assets’ are part determined by what investors think is taking place on the financial side of the market and part determined by what is taking place on the real side.

The real problem that I discovered, however, is in the price elasticity terms that are used in these equations as represented by the lower-case delta (the Greek letter that looks like a small ‘d’). What do these terms mean? Well, think about it this way: if 100 new assets are introduced in a market in which the price is \$100 per asset what effect does this have on the price? This will depend on a number of different factors such as the stock of assets outstanding in the market at any given point in time and the effect this introduction of new assets has on expectations.

So, we include a price elasticity term. Let’s say that the introduction of 100 new assets onto the market - which will be represented in the equation as setting the qZ term to 100 - has the effect of decreasing the price by \$10. We can then set the price elasticity term at 0.1 and we will get precisely this outcome.

The problem here is twofold and interrelated. First of all, the price elasticity term does not work in the same way for the price expectations variables - that is, the variables denoted by Pes - as it does for quantity variables like qZ. This is because price expectations variables are self-fulfilling. If the market thinks that the price is going to rise by 20%, given a few qualifications regarding market confidence and profit expectations, the market will basically rise by 20% as investors pile in. Whereas in the case of quantity a certain amount of an asset is created or destroyed and the price effect responds to this creation or destruction. We might say, then, that the price expectations variables are active while the quantity variables are passive.

Secondly, the price elasticity term is actually intimately tied up with the confidence term, which is represented in the equation by the small-case Greek letter gamma which looks like a small-case y. As I just said the amount by which a market price will respond to an increase/decrease in the quantity of an asset will depend on how this increase/decrease affects expectations. This means that there is a relationship between the confidence term and the price elasticity term.

This does not, however, mean that they are identical. The price elasticity term is an outcome of the confidence term, but the confidence term is not fully determined by the price elasticity term. Intuitively this is because there are things that can affect confidence that have nothing to do with increases and decreases in the quantity of an asset. A company, for example, might increase their amount of shares outstanding substantially but because investors expect that this company is going to use the funds for what appears to be an investment drive with great potential expectations may be positively reinforced by this increase in share issuance, thus not only offsetting the quantity increase but even causing the price to rise.

My mistake was to include the price elasticity term in the components of the equation that deal with price elasticities. In actual fact, the price elasticity term should only be included where expectations are not playing a role; that is, it should only be included in the last two components of equation 2.10 - those which deal with quantity supplied and real quantity demanded respectively.

This, however, leads to a further problem. Namely that these two variables cannot really affect the market price in the case of pure and impure assets until the following period unless they outweigh the speculative effects caused by the price expectations term. Put simply: in any given period it does not matter how much real supply and demand increase or decrease, rather all that matters is expectations unless the real supply and demand outstrip the speculative activity brought about by the expectations. Expectations, in a sense, ‘override’ real considerations unless these real considerations outweigh the expectations; so, even if the market is suddenly flooded with an asset its price will still rise if investors think that it will rise. It is only in the next period that these increases or decreases have any real effect on the price and this is only insofar as they affect expectations.

In a sense my equations were not radical enough. I was still working on the assumption that real supply and demand actually had a first order effect on price. But it is now clear to me that they usually only have a second order effect - i.e. they can only effect price by way of the effect they have on expectations. In order to deal with this we have to set a lag on these variables and then tie them to the price expectation variables in the period that follows them. But I think that this post is long enough, so I will save that for another day. The more I think about it, the more complicated it becomes to represent these dynamics in a neat manner.