How Not To Be Misled By Mathematical Identities

In the last 3 weeks I’ve outlined the income-expenditure identity and the saving-investment identity, and described the conclusion of orthodox economics that national saving equals investment because savings are the source of investment through intermediation by banks and financial markets.

Last week I pointed out that this simply isn’t true – banks do not intermediate savings at all, and financial markets may intermediate saving to borrowers, but this does not necessarily become investment.  The exact identity between these two values cannot therefore be because of these systems.

But there is a much more fundamental flaw in the conventional understanding of this identity.  The model with which we logically derived the saving-investment identity over previous weeks makes no mention of financial markets, banks or the monetary system.  These systems are therefore irrelevant to the saving-investment identity because they are not part of the model and definitions by which the identity is derived.

If we imagine a very simple economy that only uses cash, and has no banks, financial markets or electronic means of payments, where all savings are stored physically as cash by savers, saving would still equal investment (this, in fact, is the first model I presented at the start of the blog).  Everything about our GDP calculations would still apply to this economy.  Therefore the reason that national saving must equal investment simply cannot be because of intermediation in financial markets because this identity would still hold even if financial markets didn’t exist.

If you cannot see the logic of this, read the paragraph above again.   It’s very straightforward.

In mathematics, an identity is distinguished from an equation in that it is a relation that is always true.  An equation has a specific solution (or set of solutions) and is only true when this solution applies, but an identity is always true.  In the case of the identities we are looking at, we’ve defined our terms and then derived the resultant identities using algebra.  The identities are true based on the original definitions of the terms involved.

This is different to a scientific law, which is derived from repeated observation.  For example, when Georg Ohm derived “Ohm’s Law” in 1827, that Volts = Amps x Resistance, he established the result based on experimental testing rather than pure logical deduction.  Ohm’s law is now a staple of any basic science curriculum, but at the time he was derided for using experimentation rather than logical deduction to produce his law.

By contrast, if you take the formula Speed = Distance / Time, this isn’t a scientific law that had to be discovered and verified experimentally.  It is a statement that is true because we define speed as the distance travelled per unit of time.  The same is true for the saving-investment identity, but in a more convoluted manner: we are not choosing to define national saving as being the same as investment, but as a result of the way that we have chosen to define saving and investment, it turns out that this identity does in fact hold.  So in a roundabout way the identity is true by definition.

And therefore an understanding of what it is telling us needs to be found in those definitions.  Financial markets are not part of the model so they cannot provide the explanation, and this identity would hold in a world without financial markets.

Now this might seem like a nit-picking point, but I quoted Mankiw last week asking the rhetorical question, “What mechanisms lie behind this identity?” and giving the answer, “the financial system”.  You should by now instantly see his error – the financial system is not part of the model by which we derive the identity, so cannot be used to explain it.

Mankiw is just plain wrong.  And remember, this isn’t a quirk of Mankiw – his textbook is standard mainstream macro, taught in introductory courses the world over.  How can an entire academic field, an entire profession, make such an obvious error?  How is it that thousands of presumably intelligent economics undergraduates year after year fail to spot it?

I believe you have 2 errors compounding on each other here.  First of all, orthodox economists incorrectly believe that banks and financial markets take savings and simply intermediate them to investment so this leaps to their minds as an obvious explanation of the saving-investment identity.  And this then leads them to making the error of offering an underlying causal explanation to the identity that is not part of the definitions of the terms in the model.

Am I making too much out of this?  Well you can decide for yourself over the next few weeks.  When I was first presented with the standard explanation of the saving-investment identity on an on-line economics course I could instantly see that it was flawed.  Trying to work out what is really going on here has itched away in my brain for years.  The train of thought this set in motion has led me to the most recent, cutting-edge papers on the seismic shifts in financial markets that have occurred in the last 15 years.

These papers are the really exciting bit, and if I could start the blog again I would actually start with them – but I hadn’t even read these papers when I started the blog.   I found them because I was answering the questions raised by this identity.  So in the next few weeks I will cover the key concepts that these questions led me to, which will provide the framework for understanding what is going on in financial markets right now as described in these papers.  And then the significance of everything in the blog will become clear (hopefully).

Leave a Reply

Your email address will not be published. Required fields are marked *