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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?

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    $\begingroup$ 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 $\endgroup$
    – Oliver Queen
    Commented Jan 4 at 15:11
  • $\begingroup$ @Giskard this question can be closed. Your answer is more than enough! $\endgroup$
    – Oliver Queen
    Commented Jan 21 at 12:13
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    $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Jan 21 at 18:00

2 Answers 2

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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.

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  • $\begingroup$ I was thinking something as a Cobb-Douglas $q = A \cdot (y)^{\alpha}$, but I am not sure...to make my example more specific... $\endgroup$
    – Oliver Queen
    Commented Jan 4 at 15:13
  • $\begingroup$ Not sure what you mean by this function. If $y = 0$ then no cost seems to be sunk? $\endgroup$
    – Giskard
    Commented Jan 4 at 15:15
  • $\begingroup$ maybe it's totally wrong...let me think about your answer $\endgroup$
    – Oliver Queen
    Commented Jan 4 at 15:16
  • $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Jan 4 at 15:26
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    $\begingroup$ 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. $\endgroup$
    – Oliver Queen
    Commented Jan 4 at 23:27
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Something like $F(K,L) = zK^{\alpha}L^{\beta}$ if $p\times (zK^{\alpha}L^{\beta}) - wL - rK \geq S$ and $-M$ otherwise, with $M,S > 0$

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  • $\begingroup$ This is a discontinuous profit function; you can easily make 0 profit here with $K=L=0$, so where is the sunk cost? $\endgroup$
    – Giskard
    Commented Jan 13 at 6:54
  • $\begingroup$ 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. $\endgroup$
    – Matteo Bulgarelli
    Commented Jan 13 at 9:02
  • $\begingroup$ 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"? $\endgroup$
    – Giskard
    Commented Jan 13 at 9:05
  • $\begingroup$ I mean, why not? OP asked for a production function that exhibited sunk costs, I posted one... $\endgroup$
    – Matteo Bulgarelli
    Commented Jan 13 at 9:08
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    $\begingroup$ 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. $\endgroup$
    – Matteo Bulgarelli
    Commented Jan 13 at 9:16

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