1. Assume perfect competition (reasonable assumption for analysis before we move on to other market models, this is the status quo in bitcoins; no regulation, free market entry, etc), no firm invests (I) today unless tomorrows revenue (R) is higher than the investment.

2. Higher I in a prior period results in higher R in the next. At any I < R, firms make a profit the next period, new entrants can enter and invest some (I + small amount) such that R > I + e > I.

3. In perfect competition, I = R because both firms (or any number of n firms) will invest up to a point just below I > R.

4. Limit currency to 1 (divisible infinitely).

5. Population n and amount of total goods and services x are growing. Note that investment is not needed for either of this to grow. Someone might find a new mineral, new population = new labour, etc.

6. Because 4 and 5, then 1/n and 1/x is decreasing. Population n has to spend 1/x on goods each period.

7. Assume perfect information and n buys all x goods produced by firms.

8. Because I = R, firms investing gets a smaller and smaller balance sheet as the time period increases infinitely.

9. Knowing that, and playing the game from 1 to 8, firms hold onto their money instead of investing.

10. 9 is especially apparent if  one firm is starts out the game endowed with a significant proportion of the currency. E.g. If any one firm holds 0.9 currency (arbitrary value), there would be no way any competitor can invest and come out on top because the sum of money = 1.^*

Hence, no investment, which is needed to improve social welfare. In a one economy world with limited currency, money can't expand to attract investments so they obtain more in the next period because everyone knows in the next period, the quantity of money is the same while more n people and more x goods chase the same portion of 1.

^* Note that the current distribution of Bitcoin is also skewed with a smaller proportion holding a significant amount. Read: http://eprint.iacr.org/2012/584.pdf

No math/econ expert by any means and would love to hear your thoughts on this.

Edit:

Simple example:

There ever exists only 2 types of investments.

1) I
2) I>R. Keeping the money makes more sense then investing. So these investments are never made.

The problem with a capped currency is almost all investments fall into category 2 and therefore a bitcoin only world would stifle investment.