Sure. But that's not inflation. That's an increase in housing prices. If housing prices rise but rents do not, it doesn't make a difference to consumption - your consumption is either the rent you pay or the imputed rent. If buyers want to bid up the price of housing past what renting it out will pay for, that's their problem.
And at the risk of just reposting this comment on HN all the time. Focusing on central banks as the driver of asset price increases is looking in entirely the wrong direction. There is a third component that pushes both asset prices and central bank policy - the global supply/demand of savings vs investment opportunities. Which is driven mostly by demographics. China's massive working class, and the unprecedented rate at which they are getting wealthier, and their savings rate which is like >10x the average US citizen means there is a huge increase in the global supply of savings. Which bids up asset prices and pushes down yields, as savers compete with each other to buy up the extant profitable and safe opportunities. Central banks are the on the receiving end of this too: over-saving pushes down the natural rate of interest, which means policy rates must be lower to respond (unless you want to condemn some working Americans to unemployment). So long-term interest rates fall, and central banks have less operating space to smooth out the business cycle by moving short-term rates.
And at the risk of just reposting this comment on HN all the time. Focusing on central banks as the driver of asset price increases is looking in entirely the wrong direction. There is a third component that pushes both asset prices and central bank policy - the global supply/demand of savings vs investment opportunities. Which is driven mostly by demographics. China's massive working class, and the unprecedented rate at which they are getting wealthier, and their savings rate which is like >10x the average US citizen means there is a huge increase in the global supply of savings. Which bids up asset prices and pushes down yields, as savers compete with each other to buy up the extant profitable and safe opportunities. Central banks are the on the receiving end of this too: over-saving pushes down the natural rate of interest, which means policy rates must be lower to respond (unless you want to condemn some working Americans to unemployment). So long-term interest rates fall, and central banks have less operating space to smooth out the business cycle by moving short-term rates.