People claim that if only rent regulations were abolished, the law of supply and demand would result in prices evening out across the spectrum of apartment rents, with the lower ones (rent-regulated) going up and the high ones going down.
The situation in New York, however, is not
X - Y = Z,
but a virtually unlimited demand from people willing to pay X. Thus the equation will never result in Z, a happy middle ground determined by economic rules. Instead of there being X and Y, as there are now, or the hypothetical Z, there would be just X. Market rate as it now stands. For all. You can't have a stable formula when there is a virtually unlimited demand willing to step into the breach.
As more and more housing is built, the supply level rises to meet the demand level, and prices level off and stabilize (and often drop).
Right now, there is an artifically low level of supply vis a vis the demand, because of rent control.
And no, you are probably not going to be paying as low as you would in any other city, but it will be lower overall eventually if rent control is scrapped.
When you have a housing market driven by 20-something year olds trying to get to the top of corporations, law firms and banks and brokerage houses, prices are capped at the total combined salaries of about a half-dozen yuppies sharing a 2 bedroom apartment. It is not a family friendly environment and long term social stability does not result.
You can endlessly describe as socialists or communists those who believe that it is ok to engage in a bit of social engineering in order to encourage family stability.
But the problem is really complex, there is no free-market anyway because of all sorts of other political social and economic factors, and the freemarketeers are as self-deluded as the communists who want everything given to the poor.