>You can already buy shares in a nations economy - it is called owning currency.

An interesting point! Yes, that is definitely one way that currency could be looked at...

>The big difference from stocks is that most countries issue new shares (dollars, etc) to keep the price stable (indirectly via careful control of interest rates).

Agreed -- but if we were really to analyze things, scrutinize the way things really work at the highest degree of resolution, then we would have to observe that:

a) There are multiple countries/currencies in the world.

b) That each country's/currency's inflation rate is different than all of the other countries/currencies.

c) If each country's/currency's inflation rate is different, this means that each countries's issuance of new shares (and control of interest rates) -- their financial policies, are, by comparison, either more effective, on par, or less effective than those of other countries used for comparison...

Phrased another way:

All countries do not control the rate of inflation of their currencies equally.

Phrased another way:

Some nations are better at controlling their currency's rate of inflation than other nations.

True or not true?

>I am simplifying terribly

As am I! :-)

>but just as shares represent a fraction of a company, cash represents a share of a nation’s economy.

Again, an interesting point!

And yes, I could agree with that!

> Some nations are better at controlling their currency's rate of inflation than other nations.

Manifestly true, for lots of reasons.

The analogy only goes so far because currency is traded for goods, services, and debt all the time whereas shares are usually just traded for currency.