<p>No, it would not be correct even by a stretch.  Just because there is some superficial similarity, or just because plots of profit function and Laffer curve resemble each other that does not mean they are special cases of each other. For example, <a href="http://www.sci-news.com/astronomy/human-brain-cosmic-web-similarities-09066.html" rel="nofollow noreferrer">the cosmic web structure</a> looks like structure of neurons in our brains, that does not mean our brains are special cases of cosmic web.</p>
<ol>
<li>In economics government is traditionally not considered having profit function. Government has its own social welfare function, similarly as individuals have utility function not profit function. We usually talk about profit functions only in context of firms (e.g. see Varian Microeconomic Analysis or MWG Microeconomic Theory). Taxonomy and semantics aside, while there are some parallels between these concepts we often impose different assumptions on properties, shapes or behavior of these functions so they are not really special cases of each other, even if some undergraduate social-welfare/utility/profit maximizations problems might superficially look very similar.</li>
<li>Even if we would ignore 1. Laffer curve gives you relationship between tax revenue and tax rate. That is not even superficially related to profit, since profit is by definition revenue minus cost (<span class="math-container">$\pi = PQ-C(Q)$</span>). A more close analogue to Laffer curve, would be the relationship between price and revenue, but even here you have to squint your eyes and force the analogy. Firms do not set prices as shares of your income. Even if they did, firm-customer relationship is fundamentally different from taxpayer-government one. So even though there you could at least find some parallels it would not be possible to defend argument that they are special cases of each other.</li>
</ol>

CC BY-SA 4.0
2021-07-14T19:13:27.733Z
"*Firms do not set prices as shares of your income.*" A markup $(P/MC(Q) - 1)$ analogy would work though!
Giskard
CC BY-SA 4.0
2021-07-14T19:21:26.460Z
@Giskard I am not sure about that, market power is not bounded, markup can go to infinity, also how does it relate to shares of personal income?
1muflon1
CC BY-SA 4.0
2021-07-14T19:26:47.847Z
It does not, it is not the exact same situation, it is just a better analogy than the one you proposed.
Giskard
CC BY-SA 4.0
2021-07-14T19:27:56.880Z
More generally I wonder if answering unclear questions based on a guess of what the question is rather than asking for clarification is a good idea. I have seen several instances of this on the site during the last two weeks.
Giskard
CC BY-SA 4.0
2021-07-14T19:28:53.553Z
@Giskard well I don't think this is clear cut case, but I see your point
1muflon1
CC BY-SA 4.0
2021-07-14T20:45:27.793Z
Thank you for your answer. Especially the point 1 about the function of the government. However, for point 2, don't you think since the government has non-profit function, then its revenue becomes equivalent to profit? Also, from the perspective of tax, tax rate is something that generates that revenue as much as quantity of supply is what brings the money. So, "tax rate" for tax revenue must be what "quantity of supply" is for profit.
Turkhan Badalov
CC BY-SA 4.0
2021-07-14T20:50:28.847Z
@TurkhanBadalov 1. no, even government has both revenues and costs, but we do not call it profit since government revenue is fundamentally different from firm one in that government does not sell you anything, it takes from you what it wants. If you would want some sort of government-profit analogue it would be government budget balance which is gov. revenue - gov. spending (i.e. cost) and this is not what Laffer curve describes. 2. No there is absolutely no workable analogy between quantity of supply and tax rate, they are not even remotely similar. if anything tax rate is closer to price
1muflon1
CC BY-SA 4.0
2021-07-14T20:53:09.550Z
but even comparing it to price is still quite a large stretch since it takes share of your income (I guess only for taxation in form of levy or some fee would the analogy be apt - but that is not taxation that laffer curve describes)
1muflon1
CC BY-SA 4.0
2021-07-14T21:00:56.990Z
@1muflon1, why are you saying that there is no workable analogy? Given that we haven't reached maximum yet: when the business wants to earn more, they increase the quantity of supply; when the government wants to earn more, they increase the tax rate. Isn't this sufficient analogy?
Turkhan Badalov
CC BY-SA 4.0
2021-07-14T21:09:08.363Z
@TurkhanBadalov no. 1. First that statement is simply generally not true, it depends on market structure. 2. See all the explanations above, just because there is some maximization problem there does not mean one is special case of the other. The main similarity is that both can be set up as some class of maximization problem, you can do that for various unrelated things. In addition, profit generally might not even have single maxima but multiple ones whereas laffer curve should have just single maximum. There are so many other differences as well
1muflon1
CC BY-SA 4.0
2021-07-14T21:39:38.283Z
@1muflon1, thank you for your reply. It appears mathematically they are not really that different since both are about maximizing concave-down function as it was mentioned by Michael above. If I may ask, how do you know that Laffer curve has only 1 maximum while profit curve can have several maxima? Although the last one is usually depicted as the concave-down parabola as well
Turkhan Badalov
CC BY-SA 4.0
2021-07-14T22:08:51.597Z
@TurkhanBadalov 1. Mathematically optimization is optimization but that’s like saying mathematically there is no difference between 1 human and 1 roll of toilet paper just because in both cases integer 1 is involved. 2. because the disincentive effect of tax should be monotonically increasing, there are some issues such as the fact that government cannot tax leisure directly, but in practice generally they are. When it comes to profit function a firm might have one local maximum operating with current plant and another one if it builds new plant etc because of fixed costs and so on
1muflon1
CC BY-SA 4.0
2021-07-14T22:19:40.687Z
@TurkhanBadalov to put it most bluntly, laffer curve has general functional form of $R=t(1-t)A$ where A is the tax base and t tax rate, whereas profit functions are varied, in fact profit function $\pi(p,q,c)$ does not even needs to be concave, convex or neither convex or concave (like in the case of natural monopoly). Eg $\pi=pq-c$ is valid profit function. Note you should also understand what special case means, it means that one model is more specific version of some another $\pi=pq-c$ is special case of profit function, some concave maximization from physics is not, neither Laffer is
1muflon1
CC BY-SA 4.0
2021-07-14T19:02:31.250Z
2021-07-14T19:02:31.250Z
1muflon1
<p>I have discovered that Profit Maximization and Laffer curves convey similar ideas. Basically, down-side parabola.</p>
<p>Would it be correct to say that Laffer curve is nothing but application of the profit maximization curve by the government? If not, how do they differ?</p>

CC BY-SA 4.0
2021-07-14T18:34:37.417Z
"*I have discovered that Profit Maximization and Laffer curves convey similar ideas. Basically, down-side parabola.*" is excellent, if intended as satire. Would you agree that a rainbow also conveys a similar idea?
Giskard
CC BY-SA 4.0
2021-07-14T18:37:10.607Z
Kidding aside, it is not quite clear what you are asking. Surely you do not mean a geometric shape, so similar idea in what sense? If the government is trying to maximize tax revenue, then both curves have to do with maximization, but the fact that a tax maximization problem and a profit maximization problem both involve maximization is probably not very surprising.
Giskard
CC BY-SA 4.0
2021-07-14T20:25:41.000Z
@Giskard, I would suppose that same geometric shape  already says that the nature of both relationships is the same. For instance, even when discussing time complexity of algorithms, we don't really care what is the exact growth function but we rarher say whether this is quadratic, cubic, etc. While I don't intend to connect time complexity with economics, I want to say the fact that both relationshops have the same curve should tell us about their same intrinsic nature. Shouldn't it?
Turkhan Badalov
CC BY-SA 4.0
2021-07-14T21:27:55.640Z
I guess the connection is maximizing a strictly concave function that admits both positive a negative derivatives somewhere without any additional constraints.
Michael Greinecker
CC BY-SA 4.0
2021-07-14T21:32:32.833Z
@Michael Greinecker, exactly how I am seeing it. Thank you for clarifying more precisely!
Turkhan Badalov
CC BY-SA 4.0
2021-07-15T03:25:42.220Z
@MichaelGreinecker But that is just a textbook example of the curves, actual strict concavity is not guaranteed?
Giskard
CC BY-SA 4.0
2021-07-15T05:42:42.223Z
@Giskard In the case of profit maximization, there are familiar conditions giving that. In the case of the Laffer curve, it was at least Arthur Laffer's view that this is the actual shape when he published his theory on a napkin.
Michael Greinecker
CC BY-SA 4.0
2021-07-15T05:46:11.557Z
@MichaelGreinecker Yes, I am familiar :) I am just pointing out that the connection is very strenuous - basically there is a connection between the very basic idealized textbook depictions of these curves. Hardly a connection in concepts. Also, see [this comment](https://economics.stackexchange.com/questions/45845/isnt-laffer-curve-a-special-case-of-profit-maximization-curve-if-not-the-same?noredirect=1#comment74816_45846).
Giskard
CC BY-SA 4.0
2021-07-15T13:27:36.257Z
@user253751, are you sure that the govnernment wants to minimize well-being? Maybe you wanted to say maximize?
Turkhan Badalov
CC BY-SA 4.0
2021-07-15T15:03:00.367Z
Governments don't want to maximize taxes. They are supposed to maximize well-being. If not, then you live in a corrupt oppressive regime. The Laffer curve is only relevant when someone claims their country is on the wrong side of it, which it almost always is not.
Criticize SE actions means ban
<profit-maximization>
Isn't Laffer curve a special case of Profit Maximization curve if not the same?
Turkhan Badalov
# Isn't Laffer curve a special case of Profit Maximization curve if not the same?

(Asked by: username_0 on 2021-07-14)

I have discovered that Profit Maximization and Laffer curves convey similar ideas. Basically, down-side parabola.

Would it be correct to say that Laffer curve is nothing but application of the profit maximization curve by the government? If not, how do they differ?

**Tags:** profit-maximization

> **Comments:**
> **username_1**: "*I have discovered that Profit Maximization and Laffer curves convey similar ideas. Basically, down-side parabola.*" is excellent, if intended as satire. Would you agree that a rainbow also conveys a similar idea?  
> **username_1**: Kidding aside, it is not quite clear what you are asking. Surely you do not mean a geometric shape, so similar idea in what sense? If the government is trying to maximize tax revenue, then both curves have to do with maximization, but the fact that a tax maximization problem and a profit maximization problem both involve maximization is probably not very surprising.  
> **username_0**: @username_1, I would suppose that same geometric shape  already says that the nature of both relationships is the same. For instance, even when discussing time complexity of algorithms, we don't really care what is the exact growth function but we rarher say whether this is quadratic, cubic, etc. While I don't intend to connect time complexity with economics, I want to say the fact that both relationshops have the same curve should tell us about their same intrinsic nature. Shouldn't it?  
> **username_2**: I guess the connection is maximizing a strictly concave function that admits both positive a negative derivatives somewhere without any additional constraints.  
> **username_0**: @username_2, exactly how I am seeing it. Thank you for clarifying more precisely!  
> **username_1**: @username_2 But that is just a textbook example of the curves, actual strict concavity is not guaranteed?  
> **username_2**: @username_1 In the case of profit maximization, there are familiar conditions giving that. In the case of the Laffer curve, it was at least Arthur Laffer's view that this is the actual shape when he published his theory on a napkin.  
> **username_1**: @username_2 Yes, I am familiar :) I am just pointing out that the connection is very strenuous - basically there is a connection between the very basic idealized textbook depictions of these curves. Hardly a connection in concepts. Also, see [this comment](https://economics.stackexchange.com/questions/45845/isnt-laffer-curve-a-special-case-of-profit-maximization-curve-if-not-the-same?noredirect=1#comment74816_45846).  
> **username_0**: @user253751, are you sure that the govnernment wants to minimize well-being? Maybe you wanted to say maximize?  
> **username_3**: Governments don't want to maximize taxes. They are supposed to maximize well-being. If not, then you live in a corrupt oppressive regime. The Laffer curve is only relevant when someone claims their country is on the wrong side of it, which it almost always is not.  

---

## 1 answers

### Answer by username_4 on 2021-07-14. Score: 3 | Accepted answer

No, it would not be correct even by a stretch. Just because there is some superficial similarity, or just because plots of profit function and Laffer curve resemble each other that does not mean they are special cases of each other. For example, [the cosmic web structure](http://www.sci-news.com/astronomy/human-brain-cosmic-web-similarities-09066.html) looks like structure of neurons in our brains, that does not mean our brains are special cases of cosmic web.

1. In economics government is traditionally not considered having profit function. Government has its own social welfare function, similarly as individuals have utility function not profit function. We usually talk about profit functions only in context of firms (e.g. see Varian Microeconomic Analysis or MWG Microeconomic Theory). Taxonomy and semantics aside, while there are some parallels between these concepts we often impose different assumptions on properties, shapes or behavior of these functions so they are not really special cases of each other, even if some undergraduate social-welfare/utility/profit maximizations problems might superficially look very similar.
2. Even if we would ignore 1. Laffer curve gives you relationship between tax revenue and tax rate. That is not even superficially related to profit, since profit is by definition revenue minus cost ($\pi = PQ-C(Q)$). A more close analogue to Laffer curve, would be the relationship between price and revenue, but even here you have to squint your eyes and force the analogy. Firms do not set prices as shares of your income. Even if they did, firm-customer relationship is fundamentally different from taxpayer-government one. So even though there you could at least find some parallels it would not be possible to defend argument that they are special cases of each other.

> **Comments:**
> **username_1**: "*Firms do not set prices as shares of your income.*" A markup $(P/MC(Q) - 1)$ analogy would work though!  
> **username_4**: @username_1 I am not sure about that, market power is not bounded, markup can go to infinity, also how does it relate to shares of personal income?  
> **username_1**: It does not, it is not the exact same situation, it is just a better analogy than the one you proposed.  
> **username_1**: More generally I wonder if answering unclear questions based on a guess of what the question is rather than asking for clarification is a good idea. I have seen several instances of this on the site during the last two weeks.  
> **username_4**: @username_1 well I don't think this is clear cut case, but I see your point  
> **username_0**: Thank you for your answer. Especially the point 1 about the function of the government. However, for point 2, don't you think since the government has non-profit function, then its revenue becomes equivalent to profit? Also, from the perspective of tax, tax rate is something that generates that revenue as much as quantity of supply is what brings the money. So, "tax rate" for tax revenue must be what "quantity of supply" is for profit.  
> **username_4**: @username_0 1. no, even government has both revenues and costs, but we do not call it profit since government revenue is fundamentally different from firm one in that government does not sell you anything, it takes from you what it wants. If you would want some sort of government-profit analogue it would be government budget balance which is gov. revenue - gov. spending (i.e. cost) and this is not what Laffer curve describes. 2. No there is absolutely no workable analogy between quantity of supply and tax rate, they are not even remotely similar. if anything tax rate is closer to price  
> **username_4**: but even comparing it to price is still quite a large stretch since it takes share of your income (I guess only for taxation in form of levy or some fee would the analogy be apt - but that is not taxation that laffer curve describes)  
> **username_0**: @username_4, why are you saying that there is no workable analogy? Given that we haven't reached maximum yet: when the business wants to earn more, they increase the quantity of supply; when the government wants to earn more, they increase the tax rate. Isn't this sufficient analogy?  
> **username_4**: @username_0 no. 1. First that statement is simply generally not true, it depends on market structure. 2. See all the explanations above, just because there is some maximization problem there does not mean one is special case of the other. The main similarity is that both can be set up as some class of maximization problem, you can do that for various unrelated things. In addition, profit generally might not even have single maxima but multiple ones whereas laffer curve should have just single maximum. There are so many other differences as well  
> **username_0**: @username_4, thank you for your reply. It appears mathematically they are not really that different since both are about maximizing concave-down function as it was mentioned by username_2 above. If I may ask, how do you know that Laffer curve has only 1 maximum while profit curve can have several maxima? Although the last one is usually depicted as the concave-down parabola as well  
> **username_4**: @username_0 1. Mathematically optimization is optimization but that’s like saying mathematically there is no difference between 1 human and 1 roll of toilet paper just because in both cases integer 1 is involved. 2. because the disincentive effect of tax should be monotonically increasing, there are some issues such as the fact that government cannot tax leisure directly, but in practice generally they are. When it comes to profit function a firm might have one local maximum operating with current plant and another one if it builds new plant etc because of fixed costs and so on  
> **username_4**: @username_0 to put it most bluntly, laffer curve has general functional form of $R=t(1-t)A$ where A is the tax base and t tax rate, whereas profit functions are varied, in fact profit function $\pi(p,q,c)$ does not even needs to be concave, convex or neither convex or concave (like in the case of natural monopoly). Eg $\pi=pq-c$ is valid profit function. Note you should also understand what special case means, it means that one model is more specific version of some another $\pi=pq-c$ is special case of profit function, some concave maximization from physics is not, neither Laffer is  

---

