I mean the concept, not the activity. Because it's the most confusing concept in macroeconomics.
Let's take a very simple macro model with no exports, imports, government spending, or taxes. There are four goods: a flow of newly-produced consumption goods C; a flow of newly-produced investment goods I; a stock of antique furniture A; and a stock of money M.
It's a monetary exchange economy. There are three markets: a market for C; a market for I; a market for A; and that is it. There is no market for M. Or rather, all the other three markets are markets for M, because all the other three goods can only be bought and sold for M. Barter is banned.
There is a tabu against using the C and I you have produced yourself. You can only use the C and I you have bought from someone else. So you want to sell the C and I you produce, and buy the C and I someone else has produced.
Just to keep it simple, there are no unsold inventories of C and I. You only produce C and I when a buyer appears with money and wants to buy some of your produce. Think of C and I as services, or goods made to order.
Income Y is defined as C+I. Y=C+I is true by definition.
Saving S is defined as Y-C. So C+S=Y is true by definition.
Therefore S=I is also true by definition. It follows immediately from C+S=Y=C+I, when you subtract C from both sides.
These all refer to actual quantities produced, bought and sold. All of that was accounting. Notice that A and M appear nowhere in those accounting identities.
This is the bit that people may miss: all of that was only true at the aggregate level. At the aggregate level it is true that income from the sale of C+I must equal expenditure on C+I. The aggregate quantity of C+I sold must equal the aggregate quantity of C+I bought. If an apple is sold, it must be bought. But that is not true at the individual level.
Let's repeat the accounting at the individual level.
An individual's income y, from the c+i that individual sold to someone else, can be used to buy c from someone else, or else saved s. What can an individual do with his flow of saving? In this model there are three things he can do with his income [I meant saving]: buy investment goods i; buy a flow of antique furniture delta(a) to add to his existing stock; hoard a flow of money delta(m) to add to his existing stock.
An individual's saving means anything he does with his income except spend it on newly-produced consumption.
Saving isn't a thing, it's a non-thing. It's a residual. It's defined negatively, as not consuming part of your income. So when an individual increases his saving, for a given income, all we know for sure is that he is reducing his consumption. He must be increasing something else, but we don't know what it is he is increasing. We know (in this model) that he must be: buying more investment; buying more antiques; or hoarding more money. But we don't know which. And it really matters which. As I shall show.
That's why "saving" should be abolished.
Having got that accounting out of the way, let's start thinking about Keynesian macroeconomics.
"What is the effect of an increase in desired saving by all individuals?"
What a badly-posed question. We know they all want to buy less consumption goods, but what of those three possible things do they want to do more of? The question does not say. So lets take all three in turn.
1. Investment. Demand for newly-produced consumption goods falls and demand for newly-produced investment goods rises by an equal amount. Total demand for newly-produced goods stays the same. So it's a wash. (Hush, you microeconomists, and Austrians!) No recession.
2. Antique furniture. Demand for newly-produced consumption goods falls, and demand for a flow of antique furniture increases by an equal amount. What happens?
This one's a bit tricky. To keep it simple, let's assume a representative agent model where everyone is identical so there is no trade in antique furniture in equilibrium. If everyone wants to buy more antique furniture, and nobody wants to sell, there's excess demand for antiques. What happens next?
Either the price of antique furniture rises to clear the market or it doesn't. If it rises to clear the market each individual will stop wanting to buy more antique furniture and will reformulate his plans. If it doesn't rise to clear the market each individual will be unable to buy more antique furniture (because none will find a willing seller) and will reformulate his plans. Either way he will reformulate his plans. He either chooses not to buy more antiques or can't buy more antiques, and so must do something else with his income instead.
If he reformulates his plans and decides to increase desired consumption back again there is no recession.
If he reformulates his plans and decides to increase desired investment there is no recession.
If he reformulates his plans and decides to hoard more money, see below.
3. Money. Demand for newly-produced consumption goods falls and demand for a flow of more money increases by an equal amount. Each individual wants his stock of money to be increasing over time.
Money is the medium of exchange in this economy. That makes it very different from antiques. If you want to increase your stock of antiques, you must persuade someone else to sell some of his. It's not something you can do by yourself. If you want to increase your stock of money, you just buy less. Nobody can stop you buying less. There's a recession.
You see? Even in this incredibly simple stylised economy, to talk about the effects of an increase in desired saving is incredibly confusing. All because saving is a non-thing. It can mean any one of three things, even in this incredibly simple stylised economy. And those three things have to be analysed in three very different ways and have very different consequences.
That's why "saving" should be abolished. Talk about something else instead. Talk about a switch away from desired consumption into more desired something else, and specify what that something else is. Because it matters.