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In theory the price of a stock has no economic meaning, as only the
size of your fraction of the underlying business matters.
1,000 shares at $1000 each is the same as 1,000,000 shares at $1 each.

But, there are several effects which remain.
First, most exchanges delist any stock which trades below $1 for any length of time.
This is the explanation for the large number of share consolodations after the tech bubble burst.

Second, margin requirements for shares trading under $5 are often
different for those over $5, as are shorting rules sometimes. I believe
these are purely the discretion of the brokers, not regulatory.
As an example, lots of brokers won't let you short anything under $5 at all.

It costs more in commissions to trade 1,000,000 shares at $1 than it
does to trade 1,000 shares at $1000. Also, the for certain types of
trades, the minimum trade is a board lot (100 shares for anything but
Berkshire Hathaway, for which it's 10 shares). For example, you can't
do a limit order at the NYSE at less than a board lot and expect the
order to get filled promptly and properly. Neither of these things
make economic sense, but they exist. One one hand, you have a
pressure to keep prices high to keep commissions reasonable for the buyers.
Conversely, it takes a whole lot more money to buy 100 shares of something
trading at $4000 per share than it does to buy 100 shares of something
trading at $40, and a lot of companies want a substantial fraction of
the investing population to be able to buy shares (a board lot at least).
For these two counteracting reasons, many companies will do splits or
consolidations to keep their prices in or near the two-digit range.

The Dow stocks are under a singular pressure, which I think they are
probably strong enough to resist, but you never know. A company's
weighting in the Dow is proportional to its nominal price. Each time
one of them splits 2:1, its contribution to the index is halved.
Since management might want a high weight (makes it harder to
underperform the index) they might be tempted to postpone a split.

There are some differences in volatility among low-price and high-price
stocks, which may once have been due to rounding of prices to the
nearest 1/16th, which means each move is a bigger move on a 25 cent stock.
However, these days it may be a circular effect of the last point:

Lastly and more importantly, there is a longstanding "emotional"
meaning given to stocks at different price levels. Many serious
analysis books will recommend that you avoid stocks under $20, for example.
This may have a tenuous basis in fact: since some large majority
of stocks went public at a price of at least $10, something trading
at $1 has shown a hint of something bad in their past.

Though most of these effects should have no meaning, it is true
that stocks at certain nominal prices do have lower returns on
average than those at other prices. However, the variation
is so wide that this is of interest only academically, as it isn't
a useful predictor of any given stock.

Have a look at this
Bear in mind that most of the research in this thread is based on
stocks which have been picked by one or more quantitative stock
selection methods which have been shown to outperform in backtesting,
not on a full population of all stocks possible.
The best numbers are in the fifth post in the thread. In summary:
For stocks which were picked using one of these methods, the returns
in the following month tended to be somewhat higher for lower priced
stocks, though those under $5 were either very good or very bad
depending on the specific test. The bottom 30% of stocks selected by
these methods then sorted by price pretty repeatedly outperformed the
other 70% selected in these circumstances. That places the cutoff at
somewhere around $25 per share. ($18 to $26, depending on the test).
Again, this test may not have useful predictive value for individual investments.
The underlying business is what's much more important!

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