Differing Inflation Rates for Separate Income Groups; Literature? So here's a few ideas that have been floating around my head since undergraduate university. It seems that money owned by the highest net wealth owners is always moving around and making a return, but only in markets that don't particularly affect the lower classes (debt and equity trading for example). 

So might there be significantly different rates of inflation or expected rates of inflation for different income groups? As income inequality rises, would money demand also be lower for lower income groups and higher for high income groups, which would create inflation for luxury goods and such? If a central bank cares about population welfare and they know their policy may impact inflation differently for different income groups, couldn't this affect their planning/optimal policy?

I know these are a lot of questions. I am mainly interested in whether there have been studies or theories about any of these sorts of questions before and whether these questions are actually interesting, empirically. Feel free to give me suggestions on how to hone in this post; it's a lot of speculative questions, I know.

Edit: Otherwise, how might you guys approach this problem? How might you guys and gals try to model inflation for different income groups?
Let us first boil this question down to a simple story: Suppose bread prices are very flexible while luxury car prices are very sticky. If the central bank now changes its monetary policy to increase inflation, then bread will become relatively more expensive compared to luxury cars. If the central bank anticipates this, they may prefer a more restrictive monetary policy to avoid redistribution from the bottom to the top.

To analyze this, two literature strands may help you. First, heterogeneous price stickiness has been studied in

Carvalho, Carlos. "Heterogeneity in price stickiness and the real effects of monetary shocks." Frontiers in Macroeconomics 6.3 (2006).

This paper essentially breaks with the standard assumption of identical price stickiness across sectors. Thus, at some point one sector (bread) may experience higher inflation than another sector (luxury cars), since luxury car prices are more sticky and there is an expansionary monetary policy. The framework for welfare analysis in sticky price models is given in:

Woodford, Michael. Optimal monetary policy inertia. No. w7261. National Bureau of Economic Research, 1999.

However, you not only have heterogeneity in price stickiness, but also heterogeneity in consumers. This is the second strand in the literature you should consider, the estimation of demand systems and welfare effects therein. The main reference (and recent Nobel memorial prizewinner) is:

Deaton, Angus, and John Muellbauer. "An almost ideal demand system." The American economic review (1980): 312-326.

If you combine these two analyses, you will have an answer to your question. However, this would require you more or less to write a paper about this (which may very well be worth it).