ORIGINAL: Yogi the Great
A better term would be value subtracted.
Some plans could suggest this tax be added at each step along the way. manufactuer/grower sells to assembler/packager (tax added) who sells to wholesale distributer (tax added) who sells to retail distributer (tax added) who sells to consumer (tax added).
A way for ever expanding government to raise enough money for their programs as tax is produced/multiplied at each step along the way instead of one time tax when purchasing. However VAT is set up, the end result like most taxes is higher cost to consumers of the products subject to the tax.
Well in Europe the impact is of course that the cost to the end consumer is higher (i've never seen a tax that makes it cheaper) but the "each step of the way"is not how it works (though I realise you are positing a scenario how it could work in some jurisdiction).
Until the end consumer, each company (registered for VAT) can recover the VAT it suffers on its purchases, but then charges VAT on to its customer. the amount of the tax increases as it moves on the chain because as each company processes it ("adds value") the base cost upon which the tax % is applied increases. So the final consumer (who does not sell on) has to eat the tax. the tax is not cumulative, but increases as suppliers "add value".
And for the progressive bit, at least in the UK the tax leaves out certain "non luxuries". To quote from wikipedia:
The third largest source of government revenues is value added tax (VAT), charged at 20% on supplies of goods and services. It is therefore a tax on consumer expenditure.
Certain goods and services are exempt from VAT, and others are subject to VAT at a lower rate of 5% (the reduced rate, such as domestic gas supplies) or 0% ("zero-rated", such as most food and children's clothing).[25] Exemptions are intended to relieve the tax burden on essentials while placing the full tax on luxuries, but disputes based on fine distinctions arise, such as the notorious "Jaffa Cake Case" which hinged on whether Jaffa Cakes were classed as (zero-rated) cakes—as was eventually decided—or (fully-taxed) chocolate-covered biscuits. Until 2001, VAT was charged at the full rate on sanitary towels.
I particularly like that the UK treats books as essentials, and so no VAT. Sadly the Dutch do charge (albeit at a reduced rate) when I import here from Amazon UK.