Does an increase in debt mean a decrease in wealth? There's the accounting question, which should help us keep our heads straight. But there's also an economic question: what is the causal relation between debt and wealth?
Let's start with the accounting. For an individual, with given assets, a $100 increase in debt means a $100 decrease in net worth, or wealth. If we add up all the one million individuals in the country, we get the same result: for a country, with given assets, a $100 million increase in debt means a $100 million decrease in net worth.
For an individual, it makes sense to talk about a $100 increase in debt holding assets constant. But for a country, it does not make sense to talk about a $100 million increase in debt, holding assets constant. Who's holding the debt? The only way we could increase debt by $100 million, and hold assets constant, is if the whole of that $100 million was borrowed from foreigners.
There is no debate about net debt owed to foreigners being a liability to a country, to be subtracted from its net worth. So let's assume a closed economy. Let's also assume no government debt for the moment, and deal with that later.
For a closed economy, every financial instrument (debt, shares, IOU's in general) represents $100 that some entity owes to some other entity. It shows up both as a liability on one balance sheet and as an asset on another balance sheet. If debt rises by $100, then both assets and liabilities rise by $100. Aggregate net debt is always zero. "We owe it to ourselves". To get aggregate net wealth, we add up the values of all the real (i.e. not financial) assets like land, machinery, human capital, houses, etc. (and subtract net debt to foreigners if we are in an open economy).
Financial intermediaries complicate the picture but don't change the bottom line. Without a financial intermediary, Paul owes Peter $100, and Peter's asset cancels Paul's liability. Now Paul owes the bank $100, and the bank owes Peter $100, because they went through a bank, rather than direct lending. The bank doubles the gross assets to $200, and the gross liabilities to $200, but leaves net debt at zero.
So much for the accounting relation between debt and wealth. What about the economic relation? Is there any causal relation between debt and wealth?
If debt increases, there is an increase in both borrowing and lending. When Peter lends $100 to Paul, we have to ask two questions: what would Peter have done with the $100 if he hadn't lent it to Paul; and what will Paul do with the $100 he borrowed from Peter? If the answer to both questions is "consume an extra $100", then there is no effect on aggregate net wealth, because total consumption, and therefore total investment, stays the same. If the answer to both questions is "invest an extra $100 (in real capital)", then there is also no effect on aggregate net wealth, because total investment stays the same.
For a change in borrowing, lending, and debt to cause a change in aggregate wealth, we have to give different answers to Peter's and Paul's questions. If Peter would have consumed the $100, and Paul invests the $100 in real capital, then aggregate wealth increases when debt increases. If Peter would have invested the $100, and Paul consumes the $100, then aggregate wealth decreases when debt increases. In principle, the relation between debt and wealth could go either way. The relation between debt and wealth depends on differences between consuming and investing propensities between lenders and borrowers.
And if someone sets up a bank to intermediate Peter's loan to Paul, then gross debt doubles to $200, with no effect on aggregate investment and wealth. But this is only true if the existence of the bank doesn't change the $100 original loan, and doesn't change the answers to Peter's and Paul's questions. And the ease of borrowing and lending via an intermediary will almost certainly change those answers and increase the amount of original loans. Also, since most investment projects are too large for any individual to finance, it seems reasonable to expect the existence of banks to increase investment loans more than consumption loans.
In principle, there could be a positive, negative, or zero correlation between aggregate (gross) debt and aggregate net wealth. In practice I would expect to see a mostly positive correlation, simply because most investment is done by firms, not households, and financed by borrowing from households. So if investment increases (and wealth increases), debt will increase.
But what about household debt, specifically consumer loans, which by definition are used to finance consumption? If people borrow more to finance more consumption, doesn't that mean that investment decreases, and so net wealth decreases? Not necessarily, because that ignores the lenders. Someone must have lent them the money.
It depends on what caused the increase in consumption loans. Suppose half the population decided to borrow more and consume more. Interest rates would rise, investment would fall, and so net wealth would fall. We would see a negative correlation between consumer loans and aggregate net wealth. But suppose instead that half of the population decided to consume less and lend more. Interest rates would fall, investment would rise, and the other half of the population would be persuaded by the low interest rates to borrow more and consume more. We would see a positive correlation between consumer loans and aggregate net wealth.
The representative agent in a closed economy holds firms' debt, and government debt, as assets. He owns the house he lives in, mortgage-free. He doesn't have consumer debt. He doesn't have any liabilities on his balance sheet. We can't think of consumer debt using the representative agent. Consumer debt is caused by differences between people: some want to borrow; others want to lend. An increase in consumer debt is caused by an increase in those differences. It could be caused by an increased desire to borrow and spend by borrowers; or it could be caused by an increased desire to save and lend by lenders. The first would cause a decrease in real investment and aggregate wealth; the second would cause an increase in real investment and aggregate wealth. We shouldn't expect to see any particular correlation between consumer debt and net wealth.
What about government debt?
At first sight, government borrowing from Peter is the same as Paul's borrowing from Peter. If the government consumes the $100 and Peter would have invested it, then net wealth goes down. If the government invests the $100, and Peter would have consumed it, then net wealth goes up. But it's a bit more complicated, because Peter, as a taxpayer, may realise that he will have to pay higher future taxes to repay the debt to himself.
It's much easier if we break it into two steps: first, the government increases taxes and spending by $100 each; second, the government borrows $100 and cuts taxes by $100. The first step is straightforward. If the government has a higher marginal propensity to consume (rather than invest) than do taxpayers, then the redistribution of income towards the government increases consumption and reduces investment and net wealth. And vice versa if vice versa. The second step has no effects at all if the taxpayer saves all the $100 tax cut to pay for the extra $100 present value in taxes (Ricardian Equivalence). But if the current taxpayer ignores the burden of the debt, he will consume most of the tax cut.
It's easy to keep your head clear on whether or not government debt is a burden. If current taxpayers think it is a burden, and so take offsetting action, then it isn't a burden. If current taxpayers don't think it is a burden, and so don't take any offsetting action, then it is a burden. The true answer is always the opposite of whatever people think is the true answer. Clear?
If you look at it in this way, government debt is just like private debt where the private debtor has uncertain intentions about ever repaying the loan. If I borrow $100, with no intention ever to pay it back, then I am $100 richer, and will increase my consumption. The person who lent me the $100 is $100 poorer, but doesn't know it yet. So perceived aggregate net wealth rises by $100. This is exactly the same as when the current taxpayer believes the national debt is not a burden, and so does not see it as a liability which should reduce his consumption.