T-shirts and Calculators: Did You Know? Sales Tax vs. Value Added Tax

We are all familiar with sales tax.  We pay it on our groceries, in some states on clothing items, and on basically anything else you would find in a retail location.  However, did you know there is one tax that is very common in Europe and Russia and was also recently adopted in India? It has accountants scratching their heads, as it is not straightforward in any way- it is called Value Added Tax (VAT).  VAT is levied on a product at every point of its production where the value was added vs sales tax, which we pay at the final sale and is levied on the total value.
Here is an example how it works:
Let’s say for the ease of calculation that the VAT is 10%. If a manufacturer sells cotton for $10, he will collect additional $1 from the buyer to pay the VAT. The buyer will take the cotton and make material and sell it for $20, he will be in turn collecting $2 in VAT from the buyer who purchases the material (out of which $1 will go to the government, and the other $1 will be reimbursement of his VAT paid earlier).  The second buyer will then take the material and turn into clothing and sell it for $30. He will be collecting $3 from his customers, and only $1 will go to the government. The rest is his reimbursement is for the taxes paid previously.
As you can see, the VAT has a  pretty complex structure to account for.  Without the proper documentation, it will be difficult to maintain and be calculated correctly. Some businessmen (women) may consider this a more rational approach, as the tax burden does not all fall on the final customer. In the modern world the process of production for even the simplest of things, often goes through way more then 3 steps and becomes increasingly complicated with each buyer/seller involved.
What do you think?  Should we adopt this method in the United States, or continue on with the traditional method of sales tax?