EDIT: The responses are all assuming the tax is regressive. Note the POVERTY_LEVEL constant in there. And the tax rate itself could be progressive. 0% under 20k (if that's the poverty level), 15% on the next 30k, 25% on the next 50k, etc.
The real problem is the positive feedback loop where money buys political influence, which then results in laws that favor the already-rich at the expense of everyone else. Taxing consumption won't fix that. Taxing political contributions might fix it, but you can't do that because political contributions (a.k.a. bribes) are now protected under the First Amendment thanks to Citizens United. High marginal income tax rates or a wealth tax (or reversing Citizens United) are the only ways to fix the problem AFAICT.
How come? There are entire categories of consumption that open up with significant wealth? Private aviation, yachts, fine arts are all categories with spending heavily concentrated up top - a poor person is unlikely to own a proportional 1/1000th share of a Gulfstream jet or a Picasso.
Also, most jets are not bought by filthy stinking rich people, they are bought by businesses, mainly airlines, which operate on pretty thin margins. If you raised taxes on jets significantly you would probably put a lot of them out of business. If you tried to raise taxes on jets that were disproportionately bought by filthy stinking rich people they'd just go out and buy different jets to avoid the taxes.
On the other hand, if you're poor then just the bare necessities of life will cost a significant fraction of your income.
The only equitable taxation scheme is a progressive income tax and/or a wealth tax.
See this 2009 IFS report for the nuance in the context of UK VAT - in particular figure 10.1 and 10.2 (http://www.ifs.org.uk/budgets/gb2009/09chap10.pdf).
Often VAT regimes will not tax certain basic good such as bread, eggs, or essential household products, but in the UK most goods including many that you may consider basic have 20% VAT.
Someone being taxed say 20% on their consumption when their typical yearly consumption is $1 million has a much smaller impact to their marginal welfare than someone with a typical expenditure of $10k taxed at 10-15%. If someone has an income and consumption of $10k, then you probably don't really want tax them much or anything at all (you probably want to give them money/job/training/something!).
Provided you offset the increased taxation by consumption with increased services and benefits to the poorest, then you can mitigate most of the worst regressive effects.