The Problem With The "Induced Demand" Theory of New Housing

I was arguing with an acquaintance about New York's sky-high rents, and he made an interesting argument: he suggested that new luxury housing actually makes prices higher, by making the city more desirable to the wealthy and thus encouraging them to bid up housing prices.  In other words, the law of supply and demand doesn't reduce housing prices: supply just increases demand rather than reducing prices.  

If this argument was right, a high-cost city that allowed no new buildings would stop being a high-cost city.  This experiment seems to have been tried in San Francisco.  In no year since 2000 has the city allowed more than 1 or 2 new residential building permits per 10,000 people, far lower even than New York City (let alone such production powerhouses as Houston).  How'd that work out? According to City Data, housing prices are higher now than they were before 2008.