<p>Actually, even if in the textbooks in most cases the marginal product is always positive, <strong>it is not unusual to have a production function with negative marginal product</strong>, that is a <strong>total product function  that, beyond a certain level, decreases with an increase of the input under consideration,</strong> say labor.</p>
<p>This kind of production function is a <strong>short period production function, where some input is fixed.</strong></p>
<p>The graphs of the production function and of the marginal product, in this case,  usually look as in the following picture, where the relationship among Total, Marginal  and Average product function  is shown:</p>
<p><a href="https://i.sstatic.net/LikFj.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LikFj.png" alt="enter image description here" /></a></p>
<p>As you can see, beyond the level <span class="math-container">$L=24$</span>, the total production decreases and the marginal product becomes negative.</p>
<p>As an example of a microeconomic textbook where a similar production function is illustrated, you  can see for instance <a href="https://www.amazon.it/Robert-Microeconomics-Behavior-Eighth-Hardcover/dp/B004NHRR0W/ref=sr_1_3?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;crid=23RE30PLWWDFN&amp;dib=eyJ2IjoiMSJ9.-YY_PS8UAaIPQIOdrN1Rgm9AR149RlPbKNuGO-2sfssxn7jVbDy8CBrKOt2kgfsY7PA8MMVGb6fVHvVt7wYHG0nbbaeOBX1RDk9w98j8AuD3iSmJxPfEsra9bNnyVzZkr3bCPTEWQkuPhB1qFyeU1g.Z3DpAQYRQD2g89K0DV8YaRmUmjjDQHj3MTOBIvSdiqg&amp;dib_tag=se&amp;keywords=frank+microeconomics&amp;qid=1705164779&amp;sprefix=frank+microeconomics%2Caps%2C244&amp;sr=8-3&amp;ufe=app_do%3Aamzn1.fos.9d4f9b77-768c-4a4e-94ad-33674c20ab35" rel="nofollow noreferrer">Frank, <em>Microeconomcs and Behavior</em></a>.<span class="math-container">$^1$</span>
But if you search on Google there are plenty of <a href="https://www.google.com/search?q=production+function+negative+marginal+productivity&amp;tbm=isch&amp;ved=2ahUKEwjBlI7F3dqDAxXl8AIHHW8tAccQ2-cCegQIABAA&amp;oq=production+function+negative+marginal+productivity&amp;gs_lcp=CgNpbWcQDDoECCMQJzoHCCMQ6gIQJzoKCAAQgAQQigUQQzoFCAAQgAQ6DggAEIAEEIoFELEDEIMBOggIABCABBCxAzoQCAAQgAQQigUQQxCxAxCDAToECAAQHjoHCAAQgAQQEzoICAAQCBAeEBM6BggAEAgQHlDaDVjLigFg2qEBaAFwAHgAgAHFAYgBtCqSAQUyNy4yNJgBAKABAaoBC2d3cy13aXotaW1nsAEKwAEB&amp;sclient=img&amp;ei=DbOiZYHaKeXhi-gP79qEuAw&amp;bih=1067&amp;biw=2048&amp;client=firefox-b-e#imgrc=e4hzqNgvp4Z2iM" rel="nofollow noreferrer">images of production function with negative marginal product</a></p>
<p>A reason why this can happen is the <strong>usual argument of the  <em>flower pot</em>,</strong> that is an example of scarcity of an input in the short run, that  illustrates the <em>law of diminishing returns</em>: it is not possible to grow the world’s food supply in a flower pot.</p>
<p>With an input, land,  fixed at a low level (the flower pot)  the increase of other inputs, say labor, rapidly cease to have a positive effect on the production of wheat: the efficiency of the production will decrease quickly (decreasing marginal product) and we can imagine a situation in which the production stops to grow and a level at which  too many workers on the land can damage, instead of increasing, the production.</p>
<p>The reason why most textbooks present the positive part of marginal product only are formal reason, as pointed out  in the answer by @1muflon1, and also economic reasons:   a rational manager, as the wage is positive,  will never decide to employ the input, labor, beyond the level <span class="math-container">$L=24$</span>  in the picture.</p>
<p>So, it is reasonable to consider the part of the production function with positive marginal product only.<span class="math-container">$^2$</span></p>
<hr />
<p><span class="math-container">$^1$</span> <sup>  Unfortunately, I have  the Italian edition of Frank's book only, so I can't cite the pages of the English edition.  For the Italian edition, see Frank, <em>Microeconomia</em>, Fourth Edition, Mac Graw-Hill, 2006, pp. 265-266.</sup></p>
<p><span class="math-container">$^2$</span> <sup> See <em>Ibid</em>., pp. 262-263. </sup></p>

CC BY-SA 4.0
2024-01-13T17:41:58.150Z
My question is:  if it is reasonable to only consider the positive MP portion of the production function, why do we allow negative marginal utility in consumer theory?
AcadEconInd
CC BY-SA 4.0
2024-01-13T17:57:35.090Z
Because it is equally reasonable also to allow the negative part, it reasonable to neglect it but it is also reasonable to consider it. It is a possible type of production function with reasonable caracteristic. The consideration of the positive part only can come when we consider the process of maximization of the producer, but in the theory of production is perfectly reasonable the negative part. On the contrary, it can be more  'complete' as illustrated in the flower pot example. And what I said is that it is not true that the textbooks consider the positive part only.
BakerStreet
CC BY-SA 4.0
2024-01-13T18:03:29.073Z
In consumer theory we can allow the existence of bads, instead of goods, so we can have negative utility, and in the theory of production we can think of a 'bad' , inefficient, use of the inputs.
BakerStreet
CC BY-SA 4.0
2024-01-13T18:29:48.020Z
It is ‘reasonable’ to consider only the positive part here means ‘we can, it is convenient for simplification’, it does not mean ‘it is wrong, unreasonable, to consider also the negative part’, it is pefectly justified.
In the theory of the consumer too we can dispose of the negative utility of the ‘bads’, setting Good =Absence of the bad, so we have only goods, for convenience.
BakerStreet
CC BY-SA 4.0
2024-01-13T17:25:55.080Z
2024-01-13T17:25:55.080Z
BakerStreet
<p>Negative marginal product is allowed theoretically. In fact law of diminishing marginal returns requires for marginal product to be negative at some point.</p>
<p>Most textbooks work with well-behaved production functions because it makes complex problems easier to solve.</p>
<p>For example, if you use Cobb-Douglass function that does not have negative marginal returns, you are virtually always guaranteed that your fundamental equation will reduce to the so called Bernoulli differential equation, and that is one of the few types of differential equations we know how to solve explicitly using simple analytic methods. Proving uniqueness and stability of the equilibrium is also extremely simple.</p>
<p>Hence the reason why you don’t see negative marginal product in textbooks is that it’s simply too difficult for average Economics student (even at graduate level) to solve economic problems without well-behaved production function. Moreover, such problems can’t often be solved explicitly, and it can be difficult to prove the equilibrium you found is unique and stable.</p>
<p>This is why most student oriented literature will use very limited pool of production functions. Mostly you will see just Cobb-Douglass with few exceptions.</p>

CC BY-SA 4.0
2024-01-13T15:36:22.103Z
I think he means production functions with strictly negative marginal producs for each positive level of the input considered
Matteo Bulgarelli
CC BY-SA 4.0
2024-01-13T17:16:12.553Z
@1muflon1 But the mathematical structure of utility and production functions is same. If we can solve consumers problem with x-y, why is it difficult to solve a producers problem with k-l?
AcadEconInd
CC BY-SA 4.0
2024-01-13T17:26:15.130Z
@AcadEconInd it wouldn’t be difficult but k-l production function make 0 sense. Then you just use 0 labor input so that’s not theoretically interesting or realistic. The kind of production function that makes sense is one where production increases up to some point in both inputs and then declines at some point.
1muflon1
CC BY-SA 4.0
2024-01-13T17:43:10.620Z
I get that. But similarly, x-y utility function also should not make sense. Why do we still allow its possibility?
AcadEconInd
CC BY-SA 4.0
2024-01-13T21:45:16.347Z
@AcadEconInd utility x-y could make sense. You could argue x is good and y is bad (eg pollution), next you can have budget constraint where you pay p for acquiring x and pay q for *removing* y and it could still be interesting problem. I don’t believe any textbook has just simple problem where you have x-y st px + qy = m because that would also be uninteresting trivial problem as it would just reduce to max U(c) st px=m. However, when it comes to production it does not make sense to say that any amount standard factor of production (labor, capital, land) has always declining marginal product
1muflon1
CC BY-SA 4.0
2024-01-13T21:57:56.623Z
For example to have realistic production function with even 1 factor (say capital) that has declining marginal productivity you would need something like $F= A(-K^3+K^2)$ assuming K is normalized to be between [0,1], you might manage to create maybe some slightly simpler quadratic production function (eg $F = A(-K^2+K)$) with these properties but generally you cannot have a linear production function where marginal product increase and then decreases. These kind of production functions would already increase significantly complexity even outside growth theory
1muflon1
CC BY-SA 4.0
2024-01-13T15:17:36.083Z
2024-01-13T15:17:36.083Z
1muflon1
<p>In microeconomics, we usually 'allow' utility functions with negative partial derivatives, indicating a 'bad' commodity, such as <span class="math-container">$u(x,y)=x^2-y$</span>. Naturally, a utility-maximising consumer with a usual budget constraint <span class="math-container">$px+qy=m$</span> will choose <span class="math-container">$y=0$</span> at the optimal. I understand these do not represent well-behaved preferences, but we still allow them in theoretical analysis.</p>
<p>However, we usually don't see production functions with negative marginal products, such as <span class="math-container">$f(k,l)=k-l$</span>. Most textbooks impose the positive marginal product condition on production functions.</p>
<p>Why this difference?</p>

<consumer-theory><producer-theory>
Negative marginal utility and negative marginal product
AcadEconInd
# Negative marginal utility and negative marginal product

(Asked by: username_0 on 2024-01-13)

In microeconomics, we usually 'allow' utility functions with negative partial derivatives, indicating a 'bad' commodity, such as $u(x,y)=x^2-y$. Naturally, a utility-maximising consumer with a usual budget constraint $px+qy=m$ will choose $y=0$ at the optimal. I understand these do not represent well-behaved preferences, but we still allow them in theoretical analysis.

However, we usually don't see production functions with negative marginal products, such as $f(k,l)=k-l$. Most textbooks impose the positive marginal product condition on production functions.

Why this difference?

**Tags:** consumer-theory, producer-theory

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## 2 answers

### Answer by username_1 on 2024-01-13. Score: 1

Actually, even if in the textbooks in most cases the marginal product is always positive, **it is not unusual to have a production function with negative marginal product**, that is a **total product function that, beyond a certain level, decreases with an increase of the input under consideration,** say labor.

This kind of production function is a **short period production function, where some input is fixed.**

The graphs of the production function and of the marginal product, in this case, usually look as in the following picture, where the relationship among Total, Marginal and Average product function is shown:

[![enter image description here](https://i.sstatic.net/LikFj.png)](https://i.sstatic.net/LikFj.png)

As you can see, beyond the level $L=24$, the total production decreases and the marginal product becomes negative.

As an example of a microeconomic textbook where a similar production function is illustrated, you can see for instance [Frank, *Microeconomcs and Behavior*](https://www.amazon.it/Robert-Microeconomics-Behavior-Eighth-Hardcover/dp/B004NHRR0W/ref=sr_1_3?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&crid=23RE30PLWWDFN&dib=eyJ2IjoiMSJ9.-YY_PS8UAaIPQIOdrN1Rgm9AR149RlPbKNuGO-2sfssxn7jVbDy8CBrKOt2kgfsY7PA8MMVGb6fVHvVt7wYHG0nbbaeOBX1RDk9w98j8AuD3iSmJxPfEsra9bNnyVzZkr3bCPTEWQkuPhB1qFyeU1g.Z3DpAQYRQD2g89K0DV8YaRmUmjjDQHj3MTOBIvSdiqg&dib_tag=se&keywords=frank+microeconomics&qid=1705164779&sprefix=frank+microeconomics%2Caps%2C244&sr=8-3&ufe=app_do%3Aamzn1.fos.9d4f9b77-768c-4a4e-94ad-33674c20ab35).$^1$
But if you search on Google there are plenty of [images of production function with negative marginal product](https://www.google.com/search?q=production+function+negative+marginal+productivity&tbm=isch&ved=2ahUKEwjBlI7F3dqDAxXl8AIHHW8tAccQ2-cCegQIABAA&oq=production+function+negative+marginal+productivity&gs_lcp=CgNpbWcQDDoECCMQJzoHCCMQ6gIQJzoKCAAQgAQQigUQQzoFCAAQgAQ6DggAEIAEEIoFELEDEIMBOggIABCABBCxAzoQCAAQgAQQigUQQxCxAxCDAToECAAQHjoHCAAQgAQQEzoICAAQCBAeEBM6BggAEAgQHlDaDVjLigFg2qEBaAFwAHgAgAHFAYgBtCqSAQUyNy4yNJgBAKABAaoBC2d3cy13aXotaW1nsAEKwAEB&sclient=img&ei=DbOiZYHaKeXhi-gP79qEuAw&bih=1067&biw=2048&client=firefox-b-e#imgrc=e4hzqNgvp4Z2iM)

A reason why this can happen is the **usual argument of the *flower pot*,** that is an example of scarcity of an input in the short run, that illustrates the *law of diminishing returns*: it is not possible to grow the world’s food supply in a flower pot.

With an input, land, fixed at a low level (the flower pot) the increase of other inputs, say labor, rapidly cease to have a positive effect on the production of wheat: the efficiency of the production will decrease quickly (decreasing marginal product) and we can imagine a situation in which the production stops to grow and a level at which too many workers on the land can damage, instead of increasing, the production.

The reason why most textbooks present the positive part of marginal product only are formal reason, as pointed out in the answer by @1muflon1, and also economic reasons: a rational manager, as the wage is positive, will never decide to employ the input, labor, beyond the level $L=24$ in the picture.

So, it is reasonable to consider the part of the production function with positive marginal product only.$^2$

---

$^1$  Unfortunately, I have the Italian edition of Frank's book only, so I can't cite the pages of the English edition. For the Italian edition, see Frank, *Microeconomia*, Fourth Edition, Mac Graw-Hill, 2006, pp. 265-266.

$^2$  See *Ibid*., pp. 262-263.

> **Comments:**
> **username_0**: My question is:  if it is reasonable to only consider the positive MP portion of the production function, why do we allow negative marginal utility in consumer theory?  
> **username_1**: Because it is equally reasonable also to allow the negative part, it reasonable to neglect it but it is also reasonable to consider it. It is a possible type of production function with reasonable caracteristic. The consideration of the positive part only can come when we consider the process of maximization of the producer, but in the theory of production is perfectly reasonable the negative part. On the contrary, it can be more  'complete' as illustrated in the flower pot example. And what I said is that it is not true that the textbooks consider the positive part only.  
> **username_1**: In consumer theory we can allow the existence of bads, instead of goods, so we can have negative utility, and in the theory of production we can think of a 'bad' , inefficient, use of the inputs.  
> **username_1**: It is ‘reasonable’ to consider only the positive part here means ‘we can, it is convenient for simplification’, it does not mean ‘it is wrong, unreasonable, to consider also the negative part’, it is pefectly justified. In the theory of the consumer too we can dispose of the negative utility of the ‘bads’, setting Good =Absence of the bad, so we have only goods, for convenience.  

---

### Answer by username_2 on 2024-01-13. Score: 0

Negative marginal product is allowed theoretically. In fact law of diminishing marginal returns requires for marginal product to be negative at some point.

Most textbooks work with well-behaved production functions because it makes complex problems easier to solve.

For example, if you use Cobb-Douglass function that does not have negative marginal returns, you are virtually always guaranteed that your fundamental equation will reduce to the so called Bernoulli differential equation, and that is one of the few types of differential equations we know how to solve explicitly using simple analytic methods. Proving uniqueness and stability of the equilibrium is also extremely simple.

Hence the reason why you don’t see negative marginal product in textbooks is that it’s simply too difficult for average Economics student (even at graduate level) to solve economic problems without well-behaved production function. Moreover, such problems can’t often be solved explicitly, and it can be difficult to prove the equilibrium you found is unique and stable.

This is why most student oriented literature will use very limited pool of production functions. Mostly you will see just Cobb-Douglass with few exceptions.

> **Comments:**
> **username_3**: I think he means production functions with strictly negative marginal producs for each positive level of the input considered  
> **username_0**: @1muflon1 But the mathematical structure of utility and production functions is same. If we can solve consumers problem with x-y, why is it difficult to solve a producers problem with k-l?  
> **username_2**: @AcadEconInd it wouldn’t be difficult but k-l production function make 0 sense. Then you just use 0 labor input so that’s not theoretically interesting or realistic. The kind of production function that makes sense is one where production increases up to some point in both inputs and then declines at some point.  
> **username_0**: I get that. But similarly, x-y utility function also should not make sense. Why do we still allow its possibility?  
> **username_2**: @AcadEconInd utility x-y could make sense. You could argue x is good and y is bad (eg pollution), next you can have budget constraint where you pay p for acquiring x and pay q for *removing* y and it could still be interesting problem. I don’t believe any textbook has just simple problem where you have x-y st px + qy = m because that would also be uninteresting trivial problem as it would just reduce to max U(c) st px=m. However, when it comes to production it does not make sense to say that any amount standard factor of production (labor, capital, land) has always declining marginal product  
> **username_2**: For example to have realistic production function with even 1 factor (say capital) that has declining marginal productivity you would need something like $F= A(-K^3+K^2)$ assuming K is normalized to be between [0,1], you might manage to create maybe some slightly simpler quadratic production function (eg $F = A(-K^2+K)$) with these properties but generally you cannot have a linear production function where marginal product increase and then decreases. These kind of production functions would already increase significantly complexity even outside growth theory  

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