Thinking some more about the distinction between economic activity and capital in the economy, I came up with this more elaborate version of the flowing water analogy.
Compare economic activity to the flow of water in a pipe. All the working, the production, the buying and selling of goods and services that makes up GDP is represented by the total flow of water in the pipe. Each of us contributes our little bit of water to the stream whenever we work for pay, or whenever we buy some goods or services from someone else. Companies likewise contribute to the stream by producing, buying, selling, paying their workers and so on. Governments add to the stream by spending money on employee salaries, on social programs, on infrastructure etc.
However, governments also subtract part of the stream through taxation. Whenever income or spending is taxed, some of the water is siphoned off. This siphoned off water is what the government uses to add to the stream via salaries, spending etc. At a macroeconomic level then, government involvement in the economy should be a wash, neither adding to nor subtracting from the overall flow of water. We could talk about inefficiencies in government spending and distortions through the tax system as causing some loss, perhaps represented by leaky government pipes losing some of the water they get through taxation; but that is a finer detail and not really necessary for the core of the analogy.
So that is the flow of water, income and expenditure, economic activity. There is another aspect to the macroeconomy though: savings and investment. Whenever you earn money you have the option of spending it - contributing to the flow of water in the pipe - or saving it. Let's represent savings as a reservoir that everyone has (individuals, companies, governments all have their own reservoirs). You can siphon off some of the income you earn into this savings reservoir, reducing the flow of water while building up the amount of stored water in your reservoir. Whenever you use those savings to, for example, buy a car, you are releasing water from your reservoir back into the stream in the economic pipe. So, even if your earnings are constant, your contribution to the economy may not be. If you save money for 6 months instead of spending it, then your economic contribution for those 6 months is less than it would be without savings. When you then spend those savings in a lump sum purchase of a car your economic contribution for that month is much more than it would normally be.
Releasing water from your reservoir doesn't have to be for large purchases. If your income is extremely volatile (say you own a small business that does most of its sales over Christmastime) then you may save some of your income during the good times (siphon water from the flow to add to your reservoir) in order to spend it on day-to-day expenses like food and rent in the quiet times (release water from your reservoir drop by drop).
In economic terms your reservoir is capital. It is distinct from the flow of economic activity, though related to it of course through the mechanism of siphoning off income to accrue capital and spending capital to add to economic activity. On a personal basis, compare having an annual income of $100K to having a bank account with $100K in it. For a company, compare their annual statements of profit and loss with their balance sheet.
Governments too can have such a reservoir of savings. You mostly see this in oil-rich nations with their sovereign wealth accounts but some countries also fund their future pension liabilities with national savings accounts (Ireland had such a thing briefly, back when we had money).
What do you do with your savings, your capital? You can spend it on good and services of course, using your savings from the good times to pay the rent and buy food.
You can exchange it for other capital goods of equal value - e.g. buy a house - which can add to economic activity for that month but leaves you with the same net capital in your reservoir.
You can exchange it for productive capital assets such a a machine to help you do your job better. This will temporarily add to the flow in the pipe just like buying a house, with the added benefit that using the machine will allow you to add a greater amount to the flow in the next few years.
Finally, you can give a bucket of your capital to someone else which they use to fill their reservoir. This last one is often done on the promise that whoever you give it to will give you back a bucketful plus a mugful at some future date. In other words you can loan some person or entity the money at interest. What that person does with the capital is then up to them. Presumably whoever receives the loan can use the capital more productively that you can. They can buy a bigger and better machine to increase their flow so that their total flow is larger even after they subtract what they need to save in order to repay you your bucket plus mugful.
This borrowing and repaying of buckets and mugs of capital/water does not go through the pipes. It is not, by itself, economic activity. This is how you can have trillions of dollars traded on the stock market without it being counted as part of GDP.
What the national government does (or rather should do) with its reservoir is to even out the flow of water through the pipe, to keep the economy on an even keel with neither too much nor too little water flowing. It does this by alternately adding some of the water it siphons off through taxes to its reservoir in good times, and releasing water from the reservoir in bad times. It could in theory keep a constant level of siphoning, of which a portion is constantly added to the reservoir, and manage the flow just by increasing or decreasing/stopping the outflow. Likewise it could keep the outflow constant and just increase or decrease the amount siphoned off so that the reservoir empties and refills in counterpoint to the economic cycle. In reality all modern economies use a combination: in bad times they siphon off less into the reservoir and release more from it, in good times they siphon more into it and reduce the outflow, allowing the reservoir to refill.
The question as to why governments bother to keep the flow evened out is one for another, longer post. For now let's just stipulate that a smooth flow with neither high peaks nor low troughs is a good thing and a worthy goal for governments to pursue.
Which brings us back around to government borrowing. For those countries lucky enough to have an enormous sovereign wealth fund, borrowing is not required. Their reservoirs are full with their own money - earned through royalties on oil most likely - so they can release from their reservoirs whenever they need to, to keep the total flow through the pipe from slowing down during a recession. For most governments, when a recession hits, their reservoirs are close to empty. The only way they can counter the recession is by borrowing some buckets from other people's reservoirs, with a promise to pay back a bucket and a mugful in 3 or 5 or 10 years time, i.e. issue some treasury/national bonds. The bucket brigade springs into action, refilling the government reservoir, which then opens the taps some more to let the money flow out into the pipe. Ideally, the tax income that the government siphons off during the subsequent upturn, is enough to fill all the buckets and mugs that have to be paid back. In practice, in a world of increasing national debt, when the time comes to repay the first bucket and mug, the government borrows that from someone else and promises to pay back a bucket and a bowlful in the future.
So, what government borrowing is doing, is taking capital from those who have it, and spending it, to increase economic activity temporarily. Future taxation, which reduces economic activity temporarily, will be used to accumulate the capital needed to repay the loan. Thus economic activity is kept smooth (or at least smoother than it otherwise would be) while capital is shuffled back and forth in a parallel but separate mechanism.
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