<p>I'm sort of confused what you are asking, but the basic idea behind the transformation is that $y$ is a vector of inputs AND outputs for a firm. Things that are inputs are expressed in negative quantities. So when you are maximizing $\vec{p} \cdot \vec{y}$ for maximal profit, you are paying $p_1 y_1 + \cdots$ for the input(s), and gaining $p_n y_n + \cdots$ for the output(s).</p>

<p>The transformation function simply describes how the input(s) are transformed to make the output(s). The transformation frontier is analogous to the production possibilities frontier, if you are familiar with that, just shifted over to reflect that inputs expressed as negative quantities.</p>

CC BY-SA 3.0
2016-03-01T19:34:07.163Z
2016-03-01T19:34:07.163Z
Kitsune Cavalry
<p>I know that this is an old question, but I thought I'd add an answer in case it's helpful to others with a similar question. My interpretation of the original question was that the question asker was looking for a concrete example of how a production function and a transformation function might be related. So I am providing a simple example. </p>

<p>Suppose we have a firm that produces a single output, <span class="math-container">$ q$</span>, using a Cobb-Douglas production function of the following form: </p>

<p><span class="math-container">$$ q\le f(x_1,x_2)=x_1^\alpha x_2^\beta $$</span></p>

<p>That is, <span class="math-container">$ f(x_1,x_2) $</span> describes the maximum amount of <span class="math-container">$ q$</span> that can be produced by a firm with this technology. So the production possibilities set, <span class="math-container">$ \mathbf Y$</span>, is the set of all vectors <span class="math-container">$ \mathbf y=(-x_1,-x_2,q), \mathbf y\subseteq \mathbb R^3 $</span> (in this case - clearly could be extended to <span class="math-container">$ \mathbb R^n $</span>), such that the inequality above holds. </p>

<p><span class="math-container">$$ \mathbf Y=\{\mathbf y|q\le f(x_1,x_2)\} $$</span></p>

<p>Note that the vector has negative signs on <span class="math-container">$ x_1$</span> and <span class="math-container">$ x_2$</span> because they are <em>inputs</em>, while the output, <span class="math-container">$ q$</span>, is positive. </p>

<p>An equivalent way to represent this is as a <strong>transformation function</strong>, <span class="math-container">$ F: \mathbb R^3 \to \mathbb R$</span>, where </p>

<p><span class="math-container">$$ F(\mathbf y) = q-x_1^\alpha x_2^\beta $$</span></p>

<p>Using this notation, </p>

<p><span class="math-container">$$ \mathbf Y=\{\mathbf y|F(\mathbf y) \le 0 \} $$</span></p>

<p>Where <span class="math-container">$ F(\mathbf y) \le 0 $</span> implies that the first inequality is satisfied. </p>

<p>This is fairly simple to interpret when there is only one output; however the concept of a transformation function is useful because it allows us to think about production technologies that might use multiple inputs and produce multiple outputs. </p>

<p>I hope this is useful! </p>

CC BY-SA 4.0
2023-08-05T20:19:47.790Z
Very nice answer. However, let the inputs be wood and nails, and the output chairs. How could one compute $q-x^\alpha_1x^\beta_2$?
Sergio
CC BY-SA 4.0
2019-06-14T17:06:07.553Z
2019-06-14T17:06:07.553Z
Julia B
<p>In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (p.135):</p>

<p>$\begin{cases}Max\text{ }p*y\\s.t.\text{ }F(y)\leq0\end{cases}$</p>

<p>Where y is production vector and <strong>F(.)</strong> is a <strong>transformation function</strong>. </p>

<p>The only information I have found about this function states that $F:R^L\to R$ and that production set of a firm could be represented as $Y=\{y\in R^L:F(y)\leq0\}$. Also if $F(y)=0$ it is named <strong>transformation frontier</strong> (p.562)</p>

<p>So I am interested in what concrete are these transformation function and transformation frontier. Intuitively it must be related to some feasibility restrictions but I whant to have some full and strickt explanation.</p>

<p>Will be very greatfull for help!</p>

<microeconomics><production-function><optimization><welfare-economics>
Transformation Function
Bogdan
# Transformation Function

(Asked by: username_0 on 2016-03-01)

In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (p.135):

$\begin{cases}Max\text{ }p\*y\\s.t.\text{ }F(y)\leq0\end{cases}$

Where y is production vector and **F(.)** is a **transformation function**.

The only information I have found about this function states that $F:R^L\to R$ and that production set of a firm could be represented as $Y=\{y\in R^L:F(y)\leq0\}$. Also if $F(y)=0$ it is named **transformation frontier** (p.562)

So I am interested in what concrete are these transformation function and transformation frontier. Intuitively it must be related to some feasibility restrictions but I whant to have some full and strickt explanation.

Will be very greatfull for help!

**Tags:** microeconomics, production-function, optimization, welfare-economics


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## 2 answers

### Answer by username_2 on 2019-06-14. Score: 5 | Accepted answer

I know that this is an old question, but I thought I'd add an answer in case it's helpful to others with a similar question. My interpretation of the original question was that the question asker was looking for a concrete example of how a production function and a transformation function might be related. So I am providing a simple example.

Suppose we have a firm that produces a single output, $ q$, using a Cobb-Douglas production function of the following form:

$$ q\le f(x\_1,x\_2)=x\_1^\alpha x\_2^\beta $$

That is, $ f(x\_1,x\_2) $ describes the maximum amount of $ q$ that can be produced by a firm with this technology. So the production possibilities set, $ \mathbf Y$, is the set of all vectors $ \mathbf y=(-x\_1,-x\_2,q), \mathbf y\subseteq \mathbb R^3 $ (in this case - clearly could be extended to $ \mathbb R^n $), such that the inequality above holds.

$$ \mathbf Y=\{\mathbf y|q\le f(x\_1,x\_2)\} $$

Note that the vector has negative signs on $ x\_1$ and $ x\_2$ because they are *inputs*, while the output, $ q$, is positive.

An equivalent way to represent this is as a **transformation function**, $ F: \mathbb R^3 \to \mathbb R$, where

$$ F(\mathbf y) = q-x\_1^\alpha x\_2^\beta $$

Using this notation,

$$ \mathbf Y=\{\mathbf y|F(\mathbf y) \le 0 \} $$

Where $ F(\mathbf y) \le 0 $ implies that the first inequality is satisfied.

This is fairly simple to interpret when there is only one output; however the concept of a transformation function is useful because it allows us to think about production technologies that might use multiple inputs and produce multiple outputs.

I hope this is useful!

> **Comments:**
> **username_3**: Very nice answer. However, let the inputs be wood and nails, and the output chairs. How could one compute $q-x^\alpha_1x^\beta_2$?  

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### Answer by username_1 on 2016-03-01. Score: 2

I'm sort of confused what you are asking, but the basic idea behind the transformation is that $y$ is a vector of inputs AND outputs for a firm. Things that are inputs are expressed in negative quantities. So when you are maximizing $\vec{p} \cdot \vec{y}$ for maximal profit, you are paying $p\_1 y\_1 + \cdots$ for the input(s), and gaining $p\_n y\_n + \cdots$ for the output(s).

The transformation function simply describes how the input(s) are transformed to make the output(s). The transformation frontier is analogous to the production possibilities frontier, if you are familiar with that, just shifted over to reflect that inputs expressed as negative quantities.


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