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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!

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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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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!

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