One of the big questions about universal basic income (UBI) is "how are you going to pay for it?". While there are many good schemes such as increased taxes that can pay for it, there's one thing a lot of people don't consider. Why don't we just print more money?
Because of inflation right? If we printed more money than the value of money goes down and so no extra value has entered the economy, right? Yes, but I'm here to prove to you that even though no value has actually entered into the economy the poor still get richer at the expense of the rich getting poorer. How does this make sense? Lets find out.
To set up our society suppose we have an economy with some amount of total money M and some amount of total people n. We say we want to print some amount of money C to pay exclusively for universal income. We say that before applying universal basic income the total value of the kth person's worth is given by the function Q(k). Notice I used the word "value" here, value is described simply as the fraction of the total amount of money in the economy that person has, not the total amount of actual money that person has. This will become important when we start talking about inflation because after inflation money is worth less so talking about money in raw terms is not very useful, in UBI everyone gets more money. Instead we want to know how much value that person gets and so we have to make that distinction.
When we're printing new money we cause inflation, the value of a single dollar is worth less than it used to be. I am going to call the degree the value of a single dollar decreases our inflation factor and it's given trivially by this equation.
We also want to know how much money each money (not value) someone will get from UBI, if we assume the money is distributed equally the amount of money someone gets can be given as
This means the total value a person will have after UBI is given out is the total amount of money they have (the money they have initially + the UBI payment) multiplied by the new value of the dollar (the inflation factor). Describing this equation:
Factoring out we get
Great but we care about how their value changes not their absolute new value, we can calculate this by simply dividing their new value by their old value to give us.
So lets notice something here, d and j are both constants so the only variable on the RHS is the initial money we started with. The proportion of gain or loss we get is inversely proportional to the amount of money we initially had. The more money we had the more loss we should expect, the less money we have the less loss we should expect. This means that poor people should expect greater benefits from UBI than rich people even when factoring into account inflation, but how good are these benefits really?
To figure this out we should ask ourselves how much money do we need to break even, ie: not gain or lose any value after UBI. If we have more than this amount of money we should expect to lose value and if we have less than this amount of money we should expect to gain value. We can figure this out by setting the LHS side to 1 and solving.
Great! But this form of the equation is a little abstract, let us expand out the inflation factor d and the UBI income per capita j to see something quite interesting...
That's right in order to break even the amount of money you need to start with is the average amount of money in the society. If you have below average income you will GAIN value and if you have above the average you will LOSE value. Therefore printing money favours the poor at the expense of the rich! Woohoo we've solved it right?
"But surely the rich could raise prices, and then make UBI useless!"
Argh those dastardly rich! But are we actually sure that the rich raising prices will disadvantage the poor? Let us investigate, to model this we should introduce a price of a good, we'll call the value of the good G so that the number of goods (A for amount) someone can buy is simply given as
Notice that G here is once again value not money. If we were to call the value of the good after UBI is implemented (and the rich raise prices) G' then the equation:
Does NOT mean the prices of the good hasn't changed, it means the VALUE of the good hasn't changed or that the price of the good increased with inflation exactly. Noting that we will introduce a factor q which is an arbitrary factor that is how much the rich increased the items value after UBI. Giving the new value of our good as:
If q is greater than one then the rich have raised prices unfairly and if q is less than one then the rich have failed to raise prices and the poor are unfairly advantaged. If q is exactly equal to one than the rich have done the right thing and have raised the prices of the good exactly proportional to the amount of inflation. The new number of items a person can buy is now equal to their new value divided by the new value of the good.
Expanding out Q'(k) gives us our final equation
We want to know whether someone can buy MORE goods or LESS goods so we divide the new amount of items someone can buy with the previous amount of items someone can buy
Doing some cancelling we get
Cool! So now we know how much more a specific person can buy but once again we want to know who is advantaged (can buy more) and who is disadvantaged (can buy less). To figure this out we once again set our LHS to 1 to find out what value you need to break even, if you are above that value you will be able to buy less and if you are below that value you will be able to buy more. Solving we get:
Great! Somethings to note here. Lets suppose prices do not change with inflation, what happens here? In this case q is exactly equal to the inflation factor d which means we get a division by 0, that seems broken but it's not it just means there is no solution to this equation which means no matter how much money you have you'll ALWAYS be able to buy more goods. This makes sense because again, everyone gets more money so if the prices don't change to reflect that then everyone's buying power increases.
Ok but what happens if the rich do the right thing and increase their prices with inflation, in this case q just becomes 1 and we get this familiar equation again:
Low and behold that's the exact same equation we got before and once again when we expand d and j we get.
That's right even if prices increase with the resulting inflation the poor still can buy more if they have less than the average income. Why? Because even if the rich get the same value for the goods that they're selling the value still initially changed to favour the poor when UBI was introduced so the poor STILL have more buying power even when you take into account a price increase.
What if the rich raised the prices even more, beyond inflation? First of all, yes this certainly COULD happen but the rich don't need UBI to do this, they can increase prices unfairly right now! The mechanisms that are stopping the rich from raising the prices of goods arbitrarily high right now will still exist even after UBI is implemented. But suppose the rich did raise prices higher than inflation, what would happen?
Well you can plug in some random values of q, any value greater than 1 unfairly favours the rich and any value less than 1 unfairly favours the poor. But for values greater than 1 you'll notice something, there will always be an amount of money that you can have, below which, you will be able to buy more items. When q is greater than 1 the rich do get richer and the middle class gets poorer but the lowest class STILL manages to get richer, this stops becoming true when q is so high that the amount of money you need to have less of is smaller than the price of a single good. Can the rich raise prices so drastically high? Of course, but again whether they could do that isn't a question of UBI, they could raise prices this high with or without UBI and it'd have the same effect, people can buy less items.
Anyway I hope that clears up why printing money to pay for UBI can still have a beneficial effect on society.
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