Differing Inflation Rates for Separate Income Groups; Literature?
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).