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The production set has a simple meaning: It is the set of all production vectors that are feasible to a firm.

The production function also has a simple meaning: It gives the output quantity for a given vector of input quantities (for a firm that produces only one output. )

But what is the production transformation function? It is defined as the function $F(y)$ such that the production set is $\{y: F(y)\leq 0 \}$. Essentially, it is a way to encode the production set.

But is there also a meaningful interpretation of this function itself, rather than merely as a mathematical device to define the production set in a way that's analytically tractable?

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It is, as you say, a mathematical device to define the production set in a way that is analytically tractable

I have seen one description as thinking of it as a measure of the amount of technical progress now required to make the combination $y$ a feasible one, so

  • if $F(y)=0$ then that combination is just feasible with current technology,
  • if $F(y)\lt 0$ that combination is feasible with less than current technology,
  • if $F(y)\gt 0$ that combination requires more than current technology enables and so is currently not feasible

though this looks more like handwaving and rationalisation rather than something actually meaningful

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One reason a direct interpretation may be difficult is that the function is seemingly without unit. That is if for a function $F()$ the inequality $F(y) \leq 0$ yields the set of feasible productions then the function $G() = 2F()$ works as well. The same is true for any other scalar multiplication or any monotonic transformation of $F$ that leaves the $F(y) = 0$ level curve in place.

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