Is it possible to show that the investment = savings identity is not equilibrium but always true? Is it also possible to show that the reason it is always true is that the money saved is the same money that was used for the investment?

 There is actually an interesting and novel way to show that Savings is always equal to Investment for any given time period, that it is not just an equilibrium.  Further, It is also possible to show, using the same model, how money saved is the same money as that money used for investment.  

To do so requires establishing certain definitions  Spending - the transfer of ownership of money from one person or entity to another for a purpose. The purpose would be to acquire goods, services, labor or any other desired outcome. Spending is defined in the most general way possible to include all transfers of ownership of money for a purpose.  Income – the money the recipients of the spending receive, therefore income equals spending. Obviously we are talking about gross income here.  Sales – the sales of the goods, services, labor or some other desired outcome is what the spending is paying for. The amount of sales, therefore equals the amount of spending and therefore equals income as well.  

Therefore spending equals sales equals income.  For any given specified time period then the spending = sales = income occurs. Always true is that sales are paid for by all of the spending in this time period, and all of the current spending creates the current income.  Would we have a viable economy if all the sales were paid for by spending paid for only with money obtained as current income.  

The answer is NO!  

Why?  

Because people will not spend all of their income. They will spend some and save some. And because of the fact that people will save some, the income used for spending will not be sufficient to pay for all the sales.”  

What would have to happen to turn this into a viable scenario?  Simple, some of the sales must be paid for by money acquired in a previous time period, money NOT acquired as part of current income.  John Maynard Keynes referred to the concept of propensity to consume. By this he meant the fraction of current income used for spending. That part of the current income not spent is considered “saved”. Following this, for the purposes of this discussion, let us define the term “Consumption” to mean that amount of spending paid for out of current income.  We know that current income equals total sales, but based on the above definition, we know that the part of spending we are calling consumption here will not be adequate to pay for all the sales,  We need some spending paid for by money acquired in a previous time period.  How much of this spending paid for by money “acquired as income in a previous time period” needs to occur to pay for all the sales?  Well, the amount of spending “paid for out of money acquired in a previous time period” must exactly equal the amount of current income not used for spending, the part of current income that is saved.  So, for the purposes of this discussion, let us “kidnap” and redefine another word, and label the spending paid for by money acquired in a previous time period as “Investment”.  What that means is that the amount of money “Saved” out of the current income is going to equal the amount of “Investment”.  Savings equals Investment. 

If we label consumption as spending paid for out of current income, and investment as spending paid for by money acquired in a previous time period, we have derived the “Savings equals Investment identity”.     

 

With these definitions the “Savings equals Investment Identity” is a true identity, that is easily understood, and is always true “by definition”.

 It is not an “equilibrium” that the economy is always tending towards. It is something that, for any defined time period, will always be exactly true.  This, I believe, is the true genesis of the “Savings equals Investment” identity.  If the reader of this answer is interested, I expand on these concepts in the book called Enlightened Capitalism: A Keynes Primer.  ------------------------------------  Now I would like to take this a step further.  Above we have shown a model where savings equal investment is an identity true for any given specified time period, not just an equilibrium that the economic system tends towards.  

Now I wish to show how using this model we can also show that not only is Savings equal to Investment but the money saved by the end of the given time period is actually the SAME money as the money used for the investment spending.  As defined above spending must be paid for out of current income (consumption)  Or spending must be paid for by money acquired in a previous time period (investment)  Those two categories account for all the spending and all the income.  Using the terms as I have defined  That means that total spending equals total sales equals total income equals “consumption plus investment”  And Savings equals Investment  -------------  Now, as per our definition, consumption spending can only occur using money that has been acquired as income by some person or entity in the current time period.  And investment cannot occur with money acquired as income in the current time period, because otherwise it would meet our definition for consumption..  Although ALL spending can be classified as EITHER consumption spending ( paid for with money acquired as income in this period) OR investment spending (paid for with money previously owned), spending cannot be classified as BOTH consumption or investment at the same time. They are mutually exclusive categories. If spending is categorizable as investment spending it cannot be categorized as consumption spending. If spending is categorizable as consumption spending it cannot be categorized as investment spending. All spending is categorizable as one or the other, but never both at the same time.  

Looking at investment spending, it must be spending paid for using money being spent for the first time in the given time period. If, in the given time period, it had previously been used for spending, then that spending would have caused income for the recipient, and any further spending of that same money, would be spending paid for out of current income and that would make it consumption spending.  

So spending classified as investment must be done with money not previously spent in the time period, money being spent for the first time in the given time period, which means that investment spending of a given amount of money can occur once and only once in that time period. 

Any further spending of that same money counts as consumption.  Once that money is used for investment spending, however, the same money can be used for consumption spending over and over again.  So investment is the spending of money being use for spending and income for the first time, and only for the first time, in the given time period.  

On the other hand, however, the money for consumption spending is not money being used for spending for the first time in the given time, since it must be spending paid for by money already acquired as income (in the current time period). But any money being used for spending must have been spent for a first time or it would never be spent at all. So the only money being used for any spending must be money originally introduced into total spending as investment spending.  

And it is interesting to note that since the money spent as investment is only able to be spent once and still count as investment spending, we can conclude that the value of the amount of actual money used for the investment spending is equal to amount of the investment spending itself.  The same is not true of consumption spending, because the same money can be used repeatedly for consumption spending. 

However, any money used for consumption spending must have been first used for investment spending, in the current time period.  Investment spending is responsible for issuing into spending and income all and any of the money being used for all the spending, investment or consumption.  

After the money is spent once as investment spending, it can be spent over and over again and each time it is spent the amount of the spending gets added to the totals for consumption spending.  Now one assumes that each time a person gets income as a recipient of investment spending or consumption spending, that at least some of them do not spend all of it. That some of them hold on to a chunk of it, they don’t re-spend it all, in the current time period.  

Eventually either one of two things happens to the money being used as spending. It will intentionally have been saved by an income earner, in which case it will no longer be used for spending in the given time period and therefore will count as part of total savings, or some income earner will still possess it because they have not yet had time to spend it. In either case, all of that money, that which was first put into spending as investment spending will still exist.

 It will exist as owned by either the ones who intended to not spend it, to save it, or it will be held by the people who didn’t have time to spend it yet. In either case since the money is held by someone who has not spent it again since receiving it as income (in the given time period). this makes it money received as income which is not again spent, meaning it is income saved.  

This is the same as saying all the money being used for spending is money originally used for investment spending, and all of that money still continues to exists “in someone’s pocket” as savings, at the end of the given time period.  We are not just saying, in this model, that the amount of savings equals the amount of investment, we are actually saying that the money we are calling saved at the end of the time period is the SAME MONEY that was put into the spending as Investment in the current time period. 

It is the same money put into spending as investment, the same money used for any further (consumption) spending. It is the same money that was used for ALL the spending done in this time period.  So it is not just true that the amount of money saved is equal to the money used for investment, We are saying this is true because the money saved is the same money!!! as the money invested.  Another way to say this is that in the process of the economic activity represented by this spending, the actual money was neither created nor destroyed. 

The money used for spending was preserved. What we are calling “savings” is none other than the “preservation” of the investment money,  In my book, Enlightened Capitalism: A Keynes Primer, I address how we can legitimately classify money used in a given amount of spending as “the same money” 

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