OK. Here is where the error in the analogy begins.
"I'm going to reduce the cost of your daily dinner by $20.
Lunch for 10 now cost just $80.
The group still wanted to pay their bill the way we pay our taxes, so the first four men were unaffected.
They would still eat for free.
But what about the other six men, the paying customers?
How could
they divide the $20 windfall so everyone would get his fair share??
They realized that $20 divided by six is $3.33.
There is no $20 windfall to "split up". The restaurant owner never gave them $20 to split up. Instead of the restaurant owner giving the diners $20, the cost of the lunch has been reduced, so there is a new lower amount to be proportioned out.
Here's how it should be analyzed. The bill is no longer $100, the bill is now $80. Thus:
The first four guys paid 0% of the bill before, they will pay 0% of the bill now.
The fifth guy paid 1% before ($1.00), he pays 1% now, or $.80.
The sixth guy paid 3% before ($3.00), he pays $2.40 now.
The seventh guy paid 7% before ($7.00), he pays $5.60 now.
The eighth guy paid 12% before ($12.00) he pays $9.60 now.
The ninth guy paid 18% before ($18.00) he pays $14.40 now.
The tenth guy paid 59% before ($59.00) he pays $47.20 now.
This would be the fair way to split up the bill if they wanted to keep all "diner rates" proportionally the same.
Similarly, tax cuts are not money the government gives you to divide up, it's money they no longer take from you. But what if you were to apply the Bush tax cut percentages--rates went down 4.6% for the wealthy and 3% for the next three brackets--to the meal, you would reduce each diner's original percent of the $100 bill by an equivalent amount. Thus:
The richest guy was paying 59% of the bill when the lunch was $100, now he will pay 54.4% of the $80 bill, or $43.52.
The next richest guy was paying 18% of the bill when the lunch was $100, now he will pay 15% of the $80 bill, or $12.00.
The next guy was paying 12%, he will now pay 9% of $80, or $7.20.
...and so on.
The way you've got it described is exactly like the old problem with the 3 guests staying in the hotel and they are charged $30 for the room. But then the manager realizes the room should have cost only $25 so he sends the bellhop up to the room to give each person a dollar and to keep two dollars for his trouble. So now each person has paid $9 each, or $27 total, and the bellhop kept $2, so that's $29. What happened to the other dollar?