The Laffer curve merely describes the relationship between tax rate and tax revenue. It assumes that workers are most motivated to make money when the rate is 0%. It also assumes that as the rate increases, their motivation decreases. These two assumptions are the basis of the shape of the graph.
It would therefore be understood, based on the two aforementioned assumptions, that at a 0% rate, the total income of the society is of large magnitude. But the graph neglects to depict this.
All in all, I think the graph neglects to depict three thrings:
- The workers' income and how it changes as rates increase,
- the behavior of the workers as rates increase, (this is similar to but different than the first neglect), and
- the basis of the Laffer curve shape.
So, I decided to first graph the behavior of the workers as the rate increases, which would then affect the amount of revenue the government took in (which, when graphed, is the Laffer curve), and then, just for fun, I included the disposable income. The x axis is the tax rate and the y axis is money amount-- in the United States, dollars. All y values are arbitrary.
You could see the graph here: https://www.desmos.com/calculator/7iex7au6ey
Because the behavior of the workers is arbitrary, any depiction of their behavior would be sufficient for my purposes. Therefore, if you click on the link, you can see two different situations graphed, each situation for a different type of behavior. (For simplicity, I disabled the graphs for the second situation).
Situation one: The orange curve is the total income of the economy before taxation, the purple curve is the tax revenue and the black curve is the disposable income.
Situation two (not enabled): The green curve is the total income of the economy before taxation, the blue curve is the tax revenue and the red curve is the disposable income.
The total income curves are shaped like they are because as the tax rate increases, society works less, (or is less motivated to make the same amount of money), and as a whole, has less total income. The tax revenue curves: Because, number one, as tax rates increase, workers are motivated to make less money, and, number two, the rate at which they make less money is increasing, the value of tax revenue does not increase at a constant rate. In fact, after a certain rate (changing depending on the behavior of the workers) tax revenues start to decrease. 80% of 60 dollars is more than 90% of 43 dollars. The tax revenue curves are, in essence, the Laffer curve. And lastly, the disposable income curves are simply,
total income - tax revenue
I have therefore:
- depicted the workers' income and how it changes as rates increase,
- depicted the behavior of the workers as rates increase, and
- I have proven that a bell like shape used to depict the relationship between tax rates and tax revenues is called for.
Now, if one would argue that the Laffer curve should not be bell shaped, and rather should have constant slopes, we could deploy my graphs to show the absurdity of that assumption. The only society that can support a Laffer curve with constant slopes would be in a society where those taxed don't lose motivation to make money. This was, I believe, shown to be an impossibility during Communism in the USSR. Because it is a given that society becomes less and less motivated to make money the more they are taxed, using may graphs we can show that the Laffer curve must have changing slopes.
I graphed a society with indifference to the tax rates using my equations. It can be found this link:
I was playing around with my graphs, creating different situations each depicting different worker behaviors and I noticed I was unable to choose a behavioral situation, (the orange curve in first situation above and green one in the second), where the tax revenue curve (the purple curve in first situation above and blue one in the second) would be a perfect bell shape like Laffer depicted in his curves. So, I decided I would work backwards. I would first draw the tax revenue curve and derive the formula for the associated behavioral curve. This requires simple algebra.
- let j(x) = behavioral curve
- let a(x) = a perfect bell shape = (1/50)(-(x-50)^2) + 50
- let a(x) = money collected at x% rate of total income-- j(x) = (1/100)(x) (j(x))
- a(x) = (1/100)(x) (j(x))
- (100)(a(x)) = (x)(j(x)) (100*a(x))/(x) = j(x)
- ---substitute the equation for the perfect a(x)---
- (100*((1/50) (-(x-50)^2) + 50))/(x) = j(x)
When graphed, j(x), the behavioral curve, has a constant slope. (You could see it here: https://www.desmos.com/calculator/tn3qjflkwj) We said before, "Because, number one, as tax rates increase, workers are motivated to make less money, and, number two, the rate at which they make less money is increasing, the value of tax revenue does not increase at a constant rate."
I assumed that the rate at which people are discouraged is not constant. Clearly, if the Laffer curve is a perfect bell shape, I was wrong. But was I wrong? Would it make sense to say that a worker is discouraged the same amount between rates of 5% and 35% as they are between 50% and 80%? Doesn't it make more sense to say the rate at which they are discouraged is changing? If we assume yes, can we disprove Laffer via using may three graphs-- namely, you can't use a perfect bell curve because it leads a constant rate of discouragement?
My questions are:
- Are my graphs accurate?
- And, if they are, are there preexisting graphs that describe the relationship between workers' income, workers' behavior and tax revenue? or is the Laffer graph the only one of the sort?
- And most importantly, doesn't it make more sense to say that the rate at which people are discouraged is not constant?