[This post is not as clear as I want it to be. Sorry.]
Most discussions of long run secular stagnation, and short run liquidity traps, ignore land. They shouldn't.
If a central bank runs out of other options to increase aggregate demand, it could always use helicopter money. Or it could buy land.
Is it possible that a central bank that was allowed to buy land would ever need to use helicopter money to prevent aggregate demand falling below target? Would it ever be desirable for central banks to issue helicopter money rather than buying land?
Let's look first at the long run, then at the short run.
Long run secular stagnation.
If land lasts forever, and is in fixed supply, it is a savings vehicle exactly like Samuelsonian "money", except that it pays rents. Samuelsonian "money" pays a rate of return, in the form of capital gains, equal to the growth rate of GDP. If expected nominal rents are a strictly positive ratio of NGDP, the infinite horizon nominal rate of return on holding land will exceed the NGDP growth rate.
A sufficient condition to rule out secular stagnation (a permanent liquidity trap) is that the central bank targets a strictly positive NGDP growth rate, and that money pays 0% nominal interest. This ensures that the nominal rate of return on holding land is strictly positive, and hence exceeds the zero nominal rate of return on holding money. In the long run, land dominates money in rate of return. A permanently binding Zero Lower Bound is ruled out.
If you want to avoid long run secular stagnation, make sure that the central bank targets a strictly positive NGDP level path growth rate, and don't pay interest on central bank money.
What about the short run?
So let's assume the central bank is sensible, and targets a strictly positive NGDP level path growth rate, and does not pay interest on the money it issues.
There is nothing to prevent a temporary increase in desired saving causing the natural rate of interest to go temporarily very negative. For example, think of a world with no investment opportunities, and where the consumption good cannot be stored. If everyone loses their appetite for consumption this year, the real interest rate must fall enough to persuade them to consume this year rather than next year, when their appetites return.
In a barter economy with land and no money, people would want to hold land, even though they expected a negative rate of return (the capital losses would exceed the rents). In a monetary economy, with NGDP targeting, the real rate of return on holding money would be the expected growth rate in RGDP minus the target growth rate in NGDP.
For a big enough temporary increase in desired saving, the one-period rate of return on holding money could dominate the one period natural rate of return on holding land.
In such a case, a central bank that targeted NGDP by doing land market operations would run out of land to buy. It would need to issue helicopter money to prevent NGDP falling below target.
And when the temporary increase in desired saving ends, and the economy returns to normal, the central bank would need to use the vacuum-cleaner to reverse the helicopter money, to prevent NGDP rising above target. And it would need to use open market sales of land to reverse the open market purchases of land, to prevent NGDP rising above target.
In this case, the central bank would be buying land expecting to sell it again with a capital loss exceeding the rents. It would be expecting to earn negative profits on its land market operations.
What is the difference between buying-then-selling land, and helicopter-then-vacuum-cleaner money?
When the central bank buys land, it at least gets something in exchange for the money it issues. It sells the land at a loss, but it at least gets something for the land it sells, plus the rents. Its loss will be less than 100%. When the central bank does helicopter money, it gets nothing in exchange for the money it issues. It makes a 100% loss. And this matters when the central bank needs to reverse the operations, and sell land, or do a vacuum cleaner, to prevent NGDP rising above target.
Buying land results in a smaller loss to the central bank than helicopter money, and means the subsequent vacuum cleaner operation can be smaller. Buying land means a smaller subsequent increase in future taxes than helicopter money.
If taxes are lump sum, this does not matter. But if taxes are not lump sum, and are distorting taxes, this does matter. With distorting taxes, land market operations mean smaller fluctuations in tax rates than helicopter-then-vacuum-cleaner operations. Tax-smoothing considerations means that buying land beats helicopter money as a way to increase aggregate demand.
Of course, I can't think of a single case in monetary history where a central bank has run out of land to buy. Or even run out of real capital assets to buy. Or even run out of financial assets to buy.
But yes, the need for helicopter money to increase aggregate demand is theoretically possible, in the short run. Because it is theoretically possible for the central bank to run out of assets to buy.