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?
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$\begingroup$ Why would quantities always be the same? 2 apples = 100ml of juice. 20 apples = 1L of juice? $\endgroup$– BrsGSep 22, 2022 at 16:26
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$\begingroup$ @BrsG That is an assumption. Suppose a relationship exists where two variables are the same. $\endgroup$– GiskardSep 22, 2022 at 20:16
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$\begingroup$ @Giskard: I see, quantities=variables. This threw me off. $\endgroup$– BrsGSep 23, 2022 at 13:52
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2 Answers
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Consider how you have defined the problem.
There is a function defined as $$f: \mathbb {R} \rightarrow \mathbb {R}$$
$$\;\;\;\;\;\;\;\;\;\;\;\;y=f(I)=I\qquad (1)$$
Look at the domain and the codomain: they are both $\mathbb {R}$, that is, according to the definition of function, $f$ assigns one (and only one) real number to each real number.
Then $f$, according to $(1)$, is defined as the function that assigns to each value in $\mathbb {R}$, that is a real number, of the independent variable $I$ that real number itself.
This kind of function, of a set $A$ to itself, that assigns to each element of the domain the element itself is called the identity, or the identity function.
In your case, $f$ is the identity function between real numbers, that is the equality is between real numbers.
In your economic problem, where domain and codomain are $\mathbb{R}$, these real numbers are some kind of measure, expressed through real numbers, of input and output.
So, the equality has nothing to do with the nature of the objects involved, be they inputs, outputs, oranges or apples. The objects are not necessarily the same, even if their measures, as expressed through real numbers, can be the same.
You can see:
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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.
