<p>For your first question, it is hard to say whether your graphs are valid, as you are more or less assuming random functional forms. More accurate graphs would derive the functions based on utility maximization of the taxpayer.</p>

<p>Furthermore, you are wrong to think the validity of the Laffer curve depends on a perfect bell shape. The Laffer curve simply states that the revenue maximizing tax rate is less than 100%, i.e. there are tax rates where lowering the tax can increase revenue. </p>

<p>The Laffer curve can more accurately be derived without explicit functional forms:</p>

<ol>
<li><p>Tax revenue at a tax rate of 0 is 0.</p></li>
<li><p>At a tax rate of zero, tax revenue must be increasing with the tax. A small marginal tax increase does not discourage the worker from generating income, if we evaluate his first order conditon (optimization problem) at a tax of zero.</p></li>
<li><p>Tax revenue at a tax of 100% or more is also zero. A household that dislikes labor and gets no net income from labor will not work.</p></li>
</ol>

<p>Then (e.g. by the intermediate value theorem) it follows that there is a tax rate that maximizes revenue, because a function that is increasing at the initial point and starts at zero and ends at zero must have a maximum somewhere, since it must start going down at some point.  For the argument of Laffer, this is all you need. Whether the curve is a perfect bell or not or whether it has humps is irrelevant. </p>

<p>Most economists would not dispute the existence of a Laffer curve. The controversy about this curve is about whether we are on the left or on the right of the revenue maximizing tax rate.</p>

CC BY-SA 4.0
2018-05-11T13:52:55.617Z
I understand that the Laffer curve is merely Rolle's Therom for economics. But can't we better explain the reason the Laffer curve is bell like in the first place with the total income curve I graphed? Wouldn't it make more sense to draw the Laffer curve as a function of discouragement?
user3814413
CC BY-SA 4.0
2018-05-11T14:45:25.297Z
It doesn't matter how exactly you draw the Laffer curve. The exact shape has neither academic nor policy applications. Depending on the behavioral assumptions you impose on the taxpayers many different shapes of the Laffer curve become sensible, so it is hard to adequately answer yoru question. The tax discouragement is likely not constant, but it does not matter for the Laffer argument. Furthermore, due to the progressive nature of taxation in reality, it is difficult to draw a proper shape of the curve anyway, since even the tax rate itself, let alone its effects, are not constant.
BB King
CC BY-SA 4.0
2018-05-11T07:22:58.513Z
2018-11-23T09:05:33.693Z
2018-11-23T09:05:33.693Z
BB King
<p>The only thing that the "Laffer Curve" concept asserts is that tax income as a function of the tax rate will have a maximum at a tax rate value lower than 100%, and then tax income will drop, as the tax rate continues to increase.  </p>

<p>No assertion is made about the exact shape of the curve, its curvature, the argmax point, etc. These may perhaps be calibrated per country, period, etc. Or, specific theoretical models may produce their own Laffer Curve.  </p>

<p>The main goal here is to stress the fact that the tax rate affects the behavior of economic agents in many ways, from working less to tax evasion, and that it eventually leads to undesirable consequences if set too high.</p>

CC BY-SA 4.0
2018-05-13T20:37:10.413Z
2018-05-13T20:37:10.413Z
Alecos Papadopoulos
<p>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. </p>

<p>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. </p>

<p>All in all, I think the graph neglects to depict three thrings:</p>

<ol>
<li>The workers' income and how it changes as rates increase,</li>
<li>the behavior of the workers as rates increase, (this is similar to but different than the first neglect), and</li>
<li>the basis of the Laffer curve shape. </li>
</ol>

<p>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. </p>

<p>You could see the graph here:
<a href="https://www.desmos.com/calculator/7iex7au6ey" rel="nofollow noreferrer">https://www.desmos.com/calculator/7iex7au6ey</a></p>

<p>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).</p>

<p>Situation one: The <strong>orange</strong> curve is the total income of the economy before taxation, the <strong>purple</strong> curve is the tax revenue and the <strong>black</strong> curve is the disposable income.</p>

<p>Situation two (not enabled): The <strong>green</strong> curve is the total income of the economy before taxation, the <strong>blue</strong> curve is the tax revenue and the <strong>red</strong> curve is the disposable income.</p>

<p>The <strong>total income curves</strong> 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 <strong>tax revenue curves</strong>: 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 <strong>disposable income curves</strong> are simply,</p>

<blockquote>
  <p>total income - tax revenue</p>
</blockquote>

<p>I have therefore: </p>

<ol>
<li>depicted the workers' income and how it changes as rates increase,</li>
<li>depicted the behavior of the workers as rates increase, and</li>
<li>I have proven that a bell like shape used to depict the relationship between tax rates and tax revenues is called for.</li>
</ol>

<p>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. </p>

<p>I graphed a society with indifference to the tax rates using my equations. It can be found this link: </p>

<p><a href="https://www.desmos.com/calculator/q2uslua7nc" rel="nofollow noreferrer">https://www.desmos.com/calculator/q2uslua7nc</a></p>

<p>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.</p>

<ol>
<li>let j(x) = behavioral curve </li>
<li>let a(x) = a perfect bell shape = (1/50)(-(x-50)^2) + 50 </li>
<li>let a(x) = money collected at x% rate of total income-- j(x) = (1/100)(x) (j(x))</li>
<li>a(x) = (1/100)(x) (j(x))</li>
<li>(100)(a(x)) = (x)(j(x)) (100*a(x))/(x) = j(x)</li>
<li>---substitute the equation for the perfect a(x)--- </li>
<li>(100*((1/50) (-(x-50)^2) + 50))/(x) = j(x)</li>
</ol>

<p>When graphed, j(x), the behavioral curve, has a constant slope. (You could see it here: <a href="https://www.desmos.com/calculator/tn3qjflkwj" rel="nofollow noreferrer">https://www.desmos.com/calculator/tn3qjflkwj</a>) We said before, "Because, number one, as tax rates increase, workers are motivated to make less money, and, number two, <strong>the rate at which they make less money is increasing</strong>, the value of tax revenue does not increase at a constant rate."</p>

<p>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 <em>more</em> 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?</p>

<p>My questions are:</p>

<ol>
<li>Are my graphs accurate?</li>
<li>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?

<ol start="3">
<li>And most importantly, doesn't it make more sense to say that the rate at which people are discouraged is not constant?</li>
</ol></li>
</ol>

<taxation>
Wasn't Laffer Wrong?
user3814413
# Wasn't Laffer Wrong?

(Asked by: username_0 on 2018-05-11)

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:

1. The workers' income and how it changes as rates increase,
2. the behavior of the workers as rates increase, (this is similar to but different than the first neglect), and
3. 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:

1. depicted the workers' income and how it changes as rates increase,
2. depicted the behavior of the workers as rates increase, and
3. 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:

<https://www.desmos.com/calculator/q2uslua7nc>

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.

1. let j(x) = behavioral curve
2. let a(x) = a perfect bell shape = (1/50)(-(x-50)^2) + 50
3. let a(x) = money collected at x% rate of total income-- j(x) = (1/100)(x) (j(x))
4. a(x) = (1/100)(x) (j(x))
5. (100)(a(x)) = (x)(j(x)) (100\*a(x))/(x) = j(x)
6. ---substitute the equation for the perfect a(x)---
7. (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:

1. Are my graphs accurate?
2. 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?
   3. And most importantly, doesn't it make more sense to say that the rate at which people are discouraged is not constant?

**Tags:** taxation

---

## 2 answers

### Answer by username_1 on 2018-05-11. Score: 5

For your first question, it is hard to say whether your graphs are valid, as you are more or less assuming random functional forms. More accurate graphs would derive the functions based on utility maximization of the taxpayer.

Furthermore, you are wrong to think the validity of the Laffer curve depends on a perfect bell shape. The Laffer curve simply states that the revenue maximizing tax rate is less than 100%, i.e. there are tax rates where lowering the tax can increase revenue.

The Laffer curve can more accurately be derived without explicit functional forms:

1. Tax revenue at a tax rate of 0 is 0.
2. At a tax rate of zero, tax revenue must be increasing with the tax. A small marginal tax increase does not discourage the worker from generating income, if we evaluate his first order conditon (optimization problem) at a tax of zero.
3. Tax revenue at a tax of 100% or more is also zero. A household that dislikes labor and gets no net income from labor will not work.

Then (e.g. by the intermediate value theorem) it follows that there is a tax rate that maximizes revenue, because a function that is increasing at the initial point and starts at zero and ends at zero must have a maximum somewhere, since it must start going down at some point. For the argument of Laffer, this is all you need. Whether the curve is a perfect bell or not or whether it has humps is irrelevant.

Most economists would not dispute the existence of a Laffer curve. The controversy about this curve is about whether we are on the left or on the right of the revenue maximizing tax rate.

> **Comments:**
> **username_0**: I understand that the Laffer curve is merely Rolle's Therom for economics. But can't we better explain the reason the Laffer curve is bell like in the first place with the total income curve I graphed? Wouldn't it make more sense to draw the Laffer curve as a function of discouragement?  
> **username_1**: It doesn't matter how exactly you draw the Laffer curve. The exact shape has neither academic nor policy applications. Depending on the behavioral assumptions you impose on the taxpayers many different shapes of the Laffer curve become sensible, so it is hard to adequately answer yoru question. The tax discouragement is likely not constant, but it does not matter for the Laffer argument. Furthermore, due to the progressive nature of taxation in reality, it is difficult to draw a proper shape of the curve anyway, since even the tax rate itself, let alone its effects, are not constant.  

---

### Answer by username_2 on 2018-05-13. Score: 2

The only thing that the "Laffer Curve" concept asserts is that tax income as a function of the tax rate will have a maximum at a tax rate value lower than 100%, and then tax income will drop, as the tax rate continues to increase.

No assertion is made about the exact shape of the curve, its curvature, the argmax point, etc. These may perhaps be calibrated per country, period, etc. Or, specific theoretical models may produce their own Laffer Curve.

The main goal here is to stress the fact that the tax rate affects the behavior of economic agents in many ways, from working less to tax evasion, and that it eventually leads to undesirable consequences if set too high.


---

