First, presumably the idle equipment in the south would be for sale.
True.
Second the investment even for new equipment would be repayed fast with increasing prices.
Maybe. The price at the grain elevator paid to farmers and the price paid by the ultimate consumer of that grain have little to do with each other, at least in the US. Eventually the producer price should rise if other potential factors aren't introduced: confiscation, price limits, & etc.
Third, governments will subsidize it.
This is tricky. Long term agricultural policy in the US is set to favor the large ag trans-nats: Cargill, ADM, IBP, Monsanto, & their ilk. Short term ag policy (2 years or less) is established by a barely controlled riot.
I hope you are right shown to be correct but I wouldn't count on it.
Fourth, irrigation is practicable IF the water table is sufficient
Irrigation, in the US, is losing effectiveness. The various aquifers are nearing depletion in the short grass prairie and salt accumulation in the vital top 1/2 inch of the topsoil in California is lowering productivity - to put it mildly - in the Central Valley and Sacramento River areas.
No the bigger problem is that the soil in vast expanses of the North has no nutrients suitable for crop production.
Absolutely true. And a whole 'nuther topic and one that I cannot claim to have any great expertise. What I do know is the inadvertent introduction of the humble, common, European earthworm was the best thing you Euros did for the US. The 'castings' (i.e., worm shit) are extremely rich in plant nutrition and their burrowing turns the topsoil over bringing trace minerals up from the subsoil.
Only two genera of Lumbricid earthworms are indigenous to North America whereas introduced genera have invaded areas where earthworms did not formerly exist, especially in the north. Here forest development relies on a large amount of undecayed leaf matter. Where worms decompose that leaf layer, the ecology may shift making the habitat unsurvivable for certain species of trees, ferns and wildflowers. Currently there is no economically feasible method for controlling earthworms in forests, besides preventing introductions. Earthworms normally spread slowly, but can be widely introduced by human activities such as construction earthmoving, or by fishermen releasing bait, or by plantings from other areas. Soils which have been invaded by earthworms can be recognized by an absence of palatable leaf litter. For example, in a sugar maple - white ash - beech - northern red oak association, only the beech and oak leaves will be seen on the forest floor (except during autumn leaf-fall), as earthworms quickly devour maple and ash leaves. Basswood, dogwood, elm, poplar and tuliptree also produce palatable foliage.
Soils which have been invaded by earthworms can be recognized by an absence of palatable leaf litter. For example, in a sugar maple - white ash - beech - northern red oak association, only the beech and oak leaves will be seen on the forest floor (except during autumn leaf-fall), as earthworms quickly devour maple and ash leaves. Basswood, dogwood, elm, poplar and tuliptree also produce palatable foliage.
I would just like to point that your view appears to be very US centric -- not that this is a problem -- but I had in mind other areas like Greenland and Siberia as well.
Aquifer depletion is a significant problem and that is why I put the 'IF' in my statement. I presumed (without knowing) that undepleted aquifers can be found in the North and therefore provide an additional 50 years of reliable water supply for irrigation. Orthodoxy is not a religion.
In general, underground water sources in the US have been discovered and are undergoing depletion. In specific - to my knowledge - there is only one major source that hasn't been developed west of the Mississippi in the US. That source is brackish thus requiring a good deal of energy to allow it to be used. Outside the US? Again, I don't really know but I suspect they have been found and are being depleted (thinking of the Aral Sea) as well wherever agriculture is currently, or has been, practiced.
you are the media you consume.