Consider for example a production function with one variable.
Let’s say the function is f:R->R y=I, where y is the quantity produced and I is the quantity of some input used for production.
In this situation the quantity of some good produced and the quantity of some input used for production is always the same.
We still would say there are two quantities/variables but what’s the difference between them when they are always the same?

  • $\begingroup$ Why would quantities always be the same? 2 apples = 100ml of juice. 20 apples = 1L of juice? $\endgroup$
    – BrsG
    Sep 22 at 16:26
  • $\begingroup$ @BrsG That is an assumption. Suppose a relationship exists where two variables are the same. $\endgroup$
    – Giskard
    Sep 22 at 20:16
  • $\begingroup$ @Giskard: I see, quantities=variables. This threw me off. $\endgroup$
    – BrsG
    Sep 23 at 13:52

1 Answer 1


A bijection between two things does not mean that these two things are the same.

For each night there is exactly one day, yet night and day are quite objectively different (e.g., in average temperature).

I can always transform 1 push-up's worth of energy (approx. 7kcal, but kcal is also a made up unit of measurement) into a push-up; yet the energy and the push up are IMO obviously quite different, for example the first could be transformed into many things.

Suppose there is an economy with two industries, producing apples and oranges, the production functions of which are $F_A(L) = L$ and $F_O(L) = L$; i.e. in both industries the amount of output is equal to the amount of labor used. Yet clearly, labor and output are different, as it is possible that the 10 units of labor available are distributed so that the outputs in the industries are 8-2 respectively; thus neither output is equal to total labor. It is possible to transform an apple into an orange by reallocating labor, but apples are not oranges.

Bonus: Rational numbers and integer numbers are not the same, though a bijection exists. Admittedly this is more tricky when infinite quantities are involved.


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