Thinking out loud: Do government deficits equal private surpluses?

I hear a lot of people say this: "Government deficits equal private-sector surpluses," or "Government deficits equal private savings." For example, Jan Hatzius says this.

Is this true? I'm not immediately sure. It sounds true - after all, how does one save except by lending to another, and how can you lend unless the other borrows? But can I find a counterexample?

Let me think about it.

OK, first suppose there's no government. In that case, if government deficit equals private saving, then private saving must be exactly zero. That means that if total net private savings (assets) begin at 0, they must remain at zero forever.

Is that true?

Suppose that the economy consists of a number of hunters, who never interact with each other. They just hunt deer, eat some of the meat, and store the rest in the form of salted venison in their sheds out back (imagine that salted venison stays edible forever). Over the years, the stock of salted venison grows and grows.

Is it correct to say that total saving in this economy is zero? Well, it seems like the answer is "no", since households are increasing their stock of real assets. There's no debt, no money, and no firms. It seems like real aggregate saving is positive in years when the total stock of salted venison increases.

Can this be right? Do the hunters have some liabilities that I am not counting, to balance out their real assets? I can't think of any. They have no debts. They don't interact with each other at all. I could make up a corporate sector, and pretend that each household owns a "firm", but it seems like the liabilities and assets of each of those firms would exactly cancel out, leaving the private sector (households + firms) with positive total assets, due to the positive assets of households.

So it seems to me that by storing durable goods, it is possible to save by lending not to an external counterparty, but to nature itself. The private sector runs a real surplus every year, and the only deficit is nature's deficit.

Similarly, suppose we measure savings as income - consumption. Well, income (the amount of fresh venison + the amount of salted venison that enters the hunter's possession each year) is definitely more than consumption (fresh venison eaten each year) for each hunter. That seems to me to indicate that real saving is positive for each household, and thus for the private sector as a whole.

I've been talking in real terms so far. But how about nominal terms? To measure surpluses and deficits in nominal terms, we need a unit of account. So suppose that the hunters now are able to interact with each other, and to trade. Suppose that they decide that the unit of account is the pound of fresh venison. Hunters can trade salt for fresh venison, salted venison for fresh venison, etc. The amount of "money" (fresh venison) in the economy is determined randomly, by nature - i.e., some years, there are more deer to be hunted, some years less.

OK, so what about the nominal value of the real assets (salted venison) sitting in the sheds? This is determined by the nominal price of a pound of salted venison - i.e., how many pounds of fresh venison must be exchanged for a pound of salted venison. The total stock of nominal assets in the economy is given by: (lbs of salted venison) * (lbs of fresh venison / lb of salted venison)

Can this rise? It seems to me that it can. Suppose that the stock of salted venison increases, s per the previous example, but the price of salted venison stays fixed (which we can assume because we're just looking at accounting identities, not at the determinants of prices). It seems that aggregate nominal saving is positive.

So it seems to me, at first blush, that there can be net private saving without the existence of a government, and hence without net government deficits. Thus it seems to me, at first blush, that the private sector's surplus need not equal the government deficit.

But this is just "at first blush". I have only thought about this question for less than an hour, and for only about ten minutes if you don't count the time it took to write this post. Furthermore, I am not that knowledgeable about accounting conventions. So I may have missed something in this simple example. If so, please tell me what I missed, in the comment section.


Update: Some people are telling me that "private-sector surplus", as used by Hatzius and others, does not mean "private saving", but rather "private saving minus private investment". In that case, the accounting clearly makes sense, and does not conflict with my example (since storing salted venison is investment).

However, I still have some doubts about whether the true government "deficit" in this latter sense, i.e. govt. investment - govt. saving, can be measured accurately. But that's a topic for another post.