<p>Concepts: <br>
<strong>Production function</strong> <br>
Measured in units of output, depends on inputs. It has no information on prices or costs, so it cannot directly include fixed/sunk costs. The only way it can imply the existence of sunk costs is by saying that some amount of inputs has to be spent/has already been spent. This means restricting the range. E.g.; <span class="math-container">$f(y) = y$</span>, <span class="math-container">$y&gt;2$</span>.</p>
<p><strong>Cost function</strong> <br>
Can be calculated from the production function and input prices. If the range of the production function is restricted, at least some amount of the inputs has to be spent, then the cost function will include fixed/sunk costs. By definition, fixed costs are the amount of money you pay when you want to produce zero units of output. E.g.;  <span class="math-container">$$
C(q,w) = \left\{\begin{array}{cc}
2w \text{ if } q \leq 2, \\
qw \text{ if } q &gt; 2.
\end{array}\right.
$$</span></p>
<hr />
<p>One can also restrict the range of feasible inputs using your production set notation:</p>
<p><span class="math-container">$$\mathcal{Y}=\{(-y, q)|\text{ such that $q\leq f(y)$ and $y\geq y_s$}\}$$</span></p>
<p>where <span class="math-container">$y_s&gt;0$</span> represents the quantity of the input already commited to production (sunk).</p>
<p>If the function <span class="math-container">$f$</span> is concave, the production set will be convex, though I am not sure what purpose this serves.</p>

CC BY-SA 4.0
2024-01-04T15:13:56.203Z
I was thinking something as a Cobb-Douglas $q = A  \cdot (y)^{\alpha}$, but I am not sure...to make my example more specific...
Oliver Queen
CC BY-SA 4.0
2024-01-04T15:15:08.930Z
Not sure what you mean by this function. If $y = 0$ then no cost seems to be sunk?
Giskard
CC BY-SA 4.0
2024-01-04T15:16:35.953Z
maybe it's totally wrong...let me think about your answer
Oliver Queen
CC BY-SA 4.0
2024-01-04T15:26:06.173Z
So if you have the long run production function $f(y_1,y_2) = y_1^{\alpha_1} y_2^{\alpha_2}$ then you commit $y_s$ amount of the first input, you have sunk costs. Not sure if this answers your question, I am not quite sure what you want.
Giskard
CC BY-SA 4.0
2024-01-04T15:32:11.247Z
A single input, single output production function, which means that we have $f(y) = y^\alpha$ where $\alpha<1$. If this is now the convexity is satisfied and the non-increasing returns to scale are violated this production function involves sunk costs...?
Oliver Queen
CC BY-SA 4.0
2024-01-04T15:37:45.567Z
No? How do *you* define sunk costs?
Giskard
CC BY-SA 4.0
2024-01-04T15:38:46.960Z
Also, with the function you specified it is not possible to violate non-increasing returns to scale.
Giskard
CC BY-SA 4.0
2024-01-04T15:40:58.810Z
Sunk costs is the  amount money that has already been spent and cannot be recovered. I want ro propose a production function with sunk costs. So checking Mas-Colell, Whinston and Green, a production function that involves sunk costs has two properties as we have already discussed...I hope I am clear now
Oliver Queen
CC BY-SA 4.0
2024-01-04T15:54:29.677Z
ok with $f(y)=y^{\alpha}$ for any $t\in [0,1]$ we have that $f(ty)=tf(y)$ so the non increasing returns to scale is for sure not violated....just saw that. But if $f(y)=c+y^{\alpha}$, with $0<\alpha<1$, $c$ denotes a constant, then for sure the concavity of $f$ is satisfied and so the production set is convex.
Oliver Queen
CC BY-SA 4.0
2024-01-04T19:37:01.257Z
"*So checking Mas-Colell, Whinston and Green, a production function that involves sunk costs has two properties as we have already discussed...I hope I am clear now*" Sorry man, but no, you are not clear. Please refer exact passage of MWG so I can read this for myself.
Giskard
CC BY-SA 4.0
2024-01-04T23:27:34.447Z
In chapter $5$ of Mas-Colell, Whinston and Green, check in page $131$, the figures for the sunk costs, $5.B.3.(a)$ and $5.B.3.(b)$. It is easier to check $5.B.3.(a)$ to understand that non-increasing returns to scale is violated (or sunk consts do NOT violate non-decreasing returns to scale), and convexity is satisfied. Also, inaction is NOT possible when firms face sunk costs (i.e., $0\notin \mathcal{Y}$). Based on the plots and the properties that are written in the book this is something that you can easily figure out.
Oliver Queen
CC BY-SA 4.0
2024-01-04T15:09:50.757Z
2024-01-04T19:45:43.040Z
2024-01-04T19:45:43.040Z
Giskard
<p>Something like <span class="math-container">$F(K,L) = zK^{\alpha}L^{\beta}$</span> if <span class="math-container">$p\times (zK^{\alpha}L^{\beta}) - wL - rK \geq S$</span> and <span class="math-container">$-M$</span> otherwise, with <span class="math-container">$M,S &gt; 0$</span></p>

CC BY-SA 4.0
2024-01-13T06:54:21.263Z
This is a discontinuous profit function; you can easily make 0 profit here with $K=L=0$, so where is the sunk cost?
Giskard
CC BY-SA 4.0
2024-01-13T09:02:20.977Z
Fine, I updated, now instead of 0 you get $-M$. P.S. the function I posted before was continous and was a production function, not a profit function. This one is not continuous.
Matteo Bulgarelli
CC BY-SA 4.0
2024-01-13T09:05:52.310Z
And why is there a discontinuity? Is that included in some definition of fixed costs that I am not aware of? Also, how is it that whether a fixed cost applies to a production function depends on price? What if this is a state enterprise who produces a service that is not sold? Is it not possible to have a fixed cost in that case? Do you have some definitions that could maybe explain why what you define constitutes a "fixed cost"?
Giskard
CC BY-SA 4.0
2024-01-13T09:08:03.650Z
I mean, why not? OP asked for a production function that exhibited sunk costs, I posted one...
Matteo Bulgarelli
CC BY-SA 4.0
2024-01-13T09:08:57.123Z
I mean you posted a production function, and something that seems pretty random next to it. I don't see why these are fixed costs. Could you maybe explain your thinking and the definitions you started from?
Giskard
CC BY-SA 4.0
2024-01-13T09:11:01.263Z
You can normalize the price to 1 without loss of generality, of course...
Matteo Bulgarelli
CC BY-SA 4.0
2024-01-13T09:16:41.837Z
Sunk cost = fixed entry cost that cannot be recouped if you sell off. So I need a function that even if you don't produce, you still "pay" for it.
Matteo Bulgarelli
CC BY-SA 4.0
2024-01-13T09:18:33.807Z
Btw what you you wrote in the first comment is not true, if you produce 0 you make a loss of $-S$.
Matteo Bulgarelli
CC BY-SA 4.0
2024-01-13T21:02:12.030Z
I am not sure I understand what you mean, can you please quote the not true sentence? Also, why is there a discontinuity in your profit function? Also, if there is no sale, providing the product is a state function, then normalizing $p$ to 1 is meaningless, there is no profit at all; so what do you mean by that?
Giskard
CC BY-SA 4.0
2024-01-15T15:20:21.370Z
My questions are the following
1. is ths $z$ in $F(K,L)$ defined somewhere in your answer? Is it just a real number?
2. Could you write your production set with the exmple of your production function as well or is it difficult to do so?

Than you in advance!
Oliver Queen
CC BY-SA 4.0
2024-01-12T23:37:35.563Z
2024-01-13T09:06:30.053Z
2024-01-13T09:06:30.053Z
Matteo Bulgarelli
<p>I am looking for an example of a single output-single input production function such that involves sunk costs. I have in my mind that a drug - firm that is motivated to make a new drug, the drug has some sunk costs due to the research and development that must be suffered from the firm for every new drug patent, but I do not know how to write this down mathematically speaking. If we assume that the input is <span class="math-container">$y_1$</span> and it is just labour and the output is <span class="math-container">$q$</span>, the production function is denoted as <span class="math-container">$f(y)$</span> and the production set is written as follows</p>
<p><span class="math-container">$$\mathcal{Y}=\{(-y, q)|\text{ such that $q\leq f(y)$ and $y\geq 0$}\}$$</span></p>
<p>What we know about a production function that involves sunk costs is that the presence of sunk costs violates non-increasing returns to scale, namely <span class="math-container">$\forall y\in \mathcal{Y}$</span> and <span class="math-container">$\forall a\in [0,1]$</span>, <span class="math-container">$ay\notin \mathcal{Y}$</span> but convexity is staisfied, namely <span class="math-container">$\forall y, y^{'}\in \mathcal{Y}$</span> and <span class="math-container">$\forall a\in [0,1]$</span>,then <span class="math-container">$ay+(1-a)y^{'}\in\mathcal{Y}$</span>.</p>
<p>So how do I model such a single-output single input production function?</p>
<p>Would a Cobb-dougals where the sum of the exponents <span class="math-container">$\alpha+\beta\leq 1$</span> satisfy such a production function?</p>

CC BY-SA 4.0
2024-01-04T15:11:10.040Z
Single input, sinlge output production function that invorves sunk costs. The properties of the sunk consts are two, the following 

$1.$ non-increasing returns to scale must be violated

$2.$ and the convexity assumption must be satisfied
Oliver Queen
CC BY-SA 4.0
2024-01-21T12:13:42.093Z
@Giskard this question can be closed. Your answer is more than enough!
Oliver Queen
CC BY-SA 4.0
2024-01-21T18:00:18.093Z
Hi Oliver! Thank you! However on SEs we only close questions that are somehow 'invalid'. Good questions remain up so that future visitors with the same or similar problems can find and learn from them.
Giskard
<microeconomics><mathematical-economics><production-function>
Could you give an example of production function such that involves sunk costs?
Oliver Queen
# Could you give an example of production function such that involves sunk costs?

(Asked by: username_0 on 2024-01-04)

I am looking for an example of a single output-single input production function such that involves sunk costs. I have in my mind that a drug - firm that is motivated to make a new drug, the drug has some sunk costs due to the research and development that must be suffered from the firm for every new drug patent, but I do not know how to write this down mathematically speaking. If we assume that the input is $y\_1$ and it is just labour and the output is $q$, the production function is denoted as $f(y)$ and the production set is written as follows

$$\mathcal{Y}=\{(-y, q)|\text{ such that $q\leq f(y)$ and $y\geq 0$}\}$$

What we know about a production function that involves sunk costs is that the presence of sunk costs violates non-increasing returns to scale, namely $\forall y\in \mathcal{Y}$ and $\forall a\in [0,1]$, $ay\notin \mathcal{Y}$ but convexity is staisfied, namely $\forall y, y^{'}\in \mathcal{Y}$ and $\forall a\in [0,1]$,then $ay+(1-a)y^{'}\in\mathcal{Y}$.

So how do I model such a single-output single input production function?

Would a Cobb-dougals where the sum of the exponents $\alpha+\beta\leq 1$ satisfy such a production function?

**Tags:** microeconomics, mathematical-economics, production-function
> **Comments:**
> **username_0**: Single input, sinlge output production function that invorves sunk costs. The properties of the sunk consts are two, the following   $1.$ non-increasing returns to scale must be violated  $2.$ and the convexity assumption must be satisfied  
> **username_0**: @Giskard this question can be closed. Your answer is more than enough!  
> **username_1**: Hi Oliver! Thank you! However on SEs we only close questions that are somehow 'invalid'. Good questions remain up so that future visitors with the same or similar problems can find and learn from them.  

---

## 2 answers

### Answer by username_1 on 2024-01-04. Score: 3 | Accepted answer

Concepts:   
**Production function**   
Measured in units of output, depends on inputs. It has no information on prices or costs, so it cannot directly include fixed/sunk costs. The only way it can imply the existence of sunk costs is by saying that some amount of inputs has to be spent/has already been spent. This means restricting the range. E.g.; $f(y) = y$, $y>2$.

**Cost function**   
Can be calculated from the production function and input prices. If the range of the production function is restricted, at least some amount of the inputs has to be spent, then the cost function will include fixed/sunk costs. By definition, fixed costs are the amount of money you pay when you want to produce zero units of output. E.g.; $$
C(q,w) = \left\{\begin{array}{cc}
2w \text{ if } q \leq 2, \\
qw \text{ if } q > 2.
\end{array}\right.
$$

---

One can also restrict the range of feasible inputs using your production set notation:

$$\mathcal{Y}=\{(-y, q)|\text{ such that $q\leq f(y)$ and $y\geq y\_s$}\}$$

where $y\_s>0$ represents the quantity of the input already commited to production (sunk).

If the function $f$ is concave, the production set will be convex, though I am not sure what purpose this serves.

> **Comments:**
> **username_0**: I was thinking something as a Cobb-Douglas $q = A  \cdot (y)^{\alpha}$, but I am not sure...to make my example more specific...  
> **username_1**: Not sure what you mean by this function. If $y = 0$ then no cost seems to be sunk?  
> **username_0**: maybe it's totally wrong...let me think about your answer  
> **username_1**: So if you have the long run production function $f(y_1,y_2) = y_1^{\alpha_1} y_2^{\alpha_2}$ then you commit $y_s$ amount of the first input, you have sunk costs. Not sure if this answers your question, I am not quite sure what you want.  
> **username_0**: A single input, single output production function, which means that we have $f(y) = y^\alpha$ where $\alpha<1$. If this is now the convexity is satisfied and the non-increasing returns to scale are violated this production function involves sunk costs...?  
> **username_1**: No? How do *you* define sunk costs?  
> **username_1**: Also, with the function you specified it is not possible to violate non-increasing returns to scale.  
> **username_0**: Sunk costs is the  amount money that has already been spent and cannot be recovered. I want ro propose a production function with sunk costs. So checking Mas-Colell, Whinston and Green, a production function that involves sunk costs has two properties as we have already discussed...I hope I am clear now  
> **username_0**: ok with $f(y)=y^{\alpha}$ for any $t\in [0,1]$ we have that $f(ty)=tf(y)$ so the non increasing returns to scale is for sure not violated....just saw that. But if $f(y)=c+y^{\alpha}$, with $0<\alpha<1$, $c$ denotes a constant, then for sure the concavity of $f$ is satisfied and so the production set is convex.  
> **username_1**: "*So checking Mas-Colell, Whinston and Green, a production function that involves sunk costs has two properties as we have already discussed...I hope I am clear now*" Sorry man, but no, you are not clear. Please refer exact passage of MWG so I can read this for myself.  
> **username_0**: In chapter $5$ of Mas-Colell, Whinston and Green, check in page $131$, the figures for the sunk costs, $5.B.3.(a)$ and $5.B.3.(b)$. It is easier to check $5.B.3.(a)$ to understand that non-increasing returns to scale is violated (or sunk consts do NOT violate non-decreasing returns to scale), and convexity is satisfied. Also, inaction is NOT possible when firms face sunk costs (i.e., $0\notin \mathcal{Y}$). Based on the plots and the properties that are written in the book this is something that you can easily figure out.  

---

### Answer by username_2 on 2024-01-12. Score: 0

Something like $F(K,L) = zK^{\alpha}L^{\beta}$ if $p\times (zK^{\alpha}L^{\beta}) - wL - rK \geq S$ and $-M$ otherwise, with $M,S > 0$

> **Comments:**
> **username_1**: This is a discontinuous profit function; you can easily make 0 profit here with $K=L=0$, so where is the sunk cost?  
> **username_2**: Fine, I updated, now instead of 0 you get $-M$. P.S. the function I posted before was continous and was a production function, not a profit function. This one is not continuous.  
> **username_1**: And why is there a discontinuity? Is that included in some definition of fixed costs that I am not aware of? Also, how is it that whether a fixed cost applies to a production function depends on price? What if this is a state enterprise who produces a service that is not sold? Is it not possible to have a fixed cost in that case? Do you have some definitions that could maybe explain why what you define constitutes a "fixed cost"?  
> **username_2**: I mean, why not? OP asked for a production function that exhibited sunk costs, I posted one...  
> **username_1**: I mean you posted a production function, and something that seems pretty random next to it. I don't see why these are fixed costs. Could you maybe explain your thinking and the definitions you started from?  
> **username_2**: You can normalize the price to 1 without loss of generality, of course...  
> **username_2**: Sunk cost = fixed entry cost that cannot be recouped if you sell off. So I need a function that even if you don't produce, you still "pay" for it.  
> **username_2**: Btw what you you wrote in the first comment is not true, if you produce 0 you make a loss of $-S$.  
> **username_1**: I am not sure I understand what you mean, can you please quote the not true sentence? Also, why is there a discontinuity in your profit function? Also, if there is no sale, providing the product is a state function, then normalizing $p$ to 1 is meaningless, there is no profit at all; so what do you mean by that?  
> **username_0**: My questions are the following 1. is ths $z$ in $F(K,L)$ defined somewhere in your answer? Is it just a real number? 2. Could you write your production set with the exmple of your production function as well or is it difficult to do so?  Than you in advance!  

---

