Everyone who had less than $100k would be richer.
In the limit you could give everyone a trillion dollars, and we'd all be equal.
This is a thought experiment. My point is that wealth is real and meaningful and can be adjusted.
In the medium term, you have to ask yourself who is able to capture this money. Not for sure the homeless that has no means to start with. Probably will be captured by the people that control directly or indirectly the manufacturing means. So the rich will be the one capturing most of the helicopter money.
The reason is that money is not capital. Money by itself does not produce value. One of the most effective ways to increase social mobility is by investing in education because allows people to acquire social capital that they could invest to generate real capital and then money.
Bill Gates doesn't live in a $40B house.
After he gets a million dollars he will have money to buy a $30 loaf of bread.
But the truth is: money is just a tool, a way to get something.
In your example, there's no point in giving a million dollars to the homeless, it would make much more sense to make the $3 loaf cost 0,3 cents. Now everybody can buy it.
But if you just drop money in the lap of somebody that doesn't know how to produce value and change this value for money, those who produce the real value and owns the means to produce it will still making more money from those who can't produce.
To solve that: take the homeless, provide basic care and opportunities for him start to produce value. Then the money will come.
The world doesn't need most of the people to work. Today what you actually consume is produced by <20% of people working. Most products are produced by machines and not labor.
The homeless don't need to work, because society can provide for them even if they don't.
Actually, I guess, the real losers would be middle class...
The thing is that the dollar as no value by itself, its value derives from what it could buy. If the economic activity increases generating more value, the value of the dollar will increase. And that's not just theory, that's what happened in the last 20 years were a large number of population in developing countries were raised out of poverty. Check the Milanovic chart for seeing it [1].
The second problem is that it is not necessary that a society that has an unequal wealth distribution is also unjust. Suppose to have a society where everybody between 20 and 60 years old works for 1K a month. Before 20 years old the could use debt to leave, after 60 years old to their death (let's say 80 years old) they live with the savings. You would agree that this society is just, everybody makes the same amount of money. And yet there is inequality in wealth distribution. Youngs just before turning 20 will have a huge debt, people turning 60 will be, conversely, in the top 1%.
This example could sound naïve, replace it with student loan and pension funds and you will see that is pretty much the status of US economy.
[1] http://blogs.ft.com/off-message/files/2013/10/Change-in-real...