"The conclusion, then, is that algorithmic trading (at least in the time period studied, in which stocks were generally rising) does improve market efficiency in the sense of higher liquidity and better price discovery."
But is it real liquidity or the illusion of liquidity?
Liquidity is, more or less, money ready to be invested.
The question of whether sophisticated strategies really provide this is complex, like the strategies themselves. If you want background, I think Doug Noland's Credit Bubble Bulletin has done a good job of addressing these questions over the years.
At the same time, I think we can see simple things. The big question isn't day-to-day-liquidity but liquidity-when-you-need it. By that measure, when we look at recent and older wild-swings in the market and especially the "flash crash", it seems fairly evident that the spectrum of "sophisticated strategies" don't provide liquidity-when-you-need-it and that is increasingly a problem.
The rest of your analysis is nonsense.
The existence of money to be invested is what makes a stock markets liquid...
Money to be invested is the necessary ingredient of a "liquid market". And a liquid market is a complex thing to measure. A market can easily seem liquid if lots of shares trade. But if it's the same shares over and over again on a day-to-day basis and if any time a large block appears, the price goes way down, then the market has an illusion of liquidity rather than real liquidity.
While we do control for share price levels and volatility in our empirical work, it remains an open question whether algorithmic trading and algorithmic liquidity supply are equally beneficial in more turbulent or declining markets.
A nominal transaction tax would eliminate most of the ultra-high frequency stuff and could be put to better use. Volumes would drop and spreads would widen, but it wouldn't really matter up to some reasonable amount.
Buyers pay the offer, sellers the ask. If you want to save the spread you can leave a resting order and wait for someone to take you out but you have no guarantee of a fill (or at least a timely one).
Individual traders who are value-oriented or technical traders, as well as mutual funds etc, gain money on "buy low, sell high". Of course mutual funds and the like do so on a very long term basis. In any case, they pay the spread. So if the spread is narrower then they make more money in average.
Example: if the Apple's quote is 95/105 (buy/sell) today and 105/115 tomorrow, an invidual trader buying and selling would just break even (buy at 105 and sell at 105) even though Apple's notional fair value has risen. But if there are more market makers, the quotes would be 99/101 and 109/111. Then our individual trader would make 109 - 99 = 10 in the same market conditions.
You're right that a wider spread wouldn't matter much to an individual trader, but that's mainly because their main cost is trading fees (paid to the broker) anyway. For mutual funds, though, a 1 basis-point change in the spread means a lot of money.
Let's say on average it was $0.01 per dollar. As soon as someone entered the market with some microstructure expertise and strategy, everyone but the microstructure guy starts making (say) $0.005 half the time and losing $0.006 the other half.
Reducing some of the randomness and providing more consistency is certainly worth something; you can imagine an example where it was +/- $0.50 per dollar reduced to +$.005, - $.006, but note (assuming the same dollar volume) that the HFT guy would still be making the same amount of money as in the previous example, where he was providing much less value. I.e. the cost that HFT commands isn't strongly tied to the value it provides. This is without even delving into the latency-arbitrage arms race cluster fuck.
No it wouldn't. Unless you make some really odd assumptions (like the traders are monkeys), you could still have tons of limit orders and NO executions. That's what happens to iliquid stocks.