It's an interesting thing to think about: the creation of one or more mandatory investment funds, composed of publicly traded stocks and the like, that are funded from mandatory contributions from all taxpayers. These contributions would provide not only the operating capital to purchase stock but also the management and maintenance costs related to such a large account.
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In the United States, I would not predict that such a program could be put into place in the foreseeable future.
Millions of federal workers who used to have a defined benefit plan are now covered by a contributory (and matched) plan such as you describe. It strikes me that with adjustments to the contribution and match for lower income people, it would provide a desirable alternative to social security. At last the money would be invested, unlike at present; and it would be used for actual economic expansion, unlike at present; and there would be a heritable account with your name on it, unlike at present.
The invested money is far more likely to be used for fat executive bonuses than for economic expansion. I'll explain why.
When a person-- or an account manager-- buys stock, unless they're getting in on a public offering they aren't buying the stock directly from the company whose stock it is. They're buying it from another stockholder who happens to be selling. So, if you have WhizbangCorp stock that you bought at $50 per share, and you sell it to me for $100 per share, then the difference that's left to you after the various brokerage fees and capital gains tax is yours to keep. It doesn't affect the day to day operations of the company in any way. For the most part it doesn't affect the employees either, since there are strict laws prohibiting insider trading that are enforced rigorously on the little people. Executives who are offered stock options are the only ones who benefit from a rise in stock prices, because they can exercise their stock options and sell or take a profit. So the executive bonus program, featuring stock options, is not available to rank and file employees.
Fire-hosing extra money into the markets and requiring the purchase of stock or investment grade securities increases the price of those securities. It also means there's less money circulating for purchases, debt reduction, or the like. Unlike economists, we have to accept the fact that human beings are going to be the way they're going to be, so it's unreasonable to expect *all* the affected people, or even the majority, to curtail their spending when their effective income drops. They're more likely to continue spending but to make use of credit, thereby increasing their debt. The banks benefit from that, but nobody else does.
The current program for federal employees is a kind of a 401(k) matching program: not mandatory at all, but worth doing if you're Mustachian. It is heritable, but from a public policy perspective that can be a weakness. Money for plan administration, oversight, and legislation comes from the leftovers when assets are *not* passed on from one person to the next. That's the dirty little secret of pension plans.