This is a question in economics where we create a model for GDP per capita. GDP per capita can be caculated in such formula where you take GDP/population. This formula (GDP/population) can be broken down in a way where you see what affects the GDP per capita formula "in the real world". I'll attach a picture down below so you understand it.
I have another formula where someone has changed this into something I can't understand really. So I was wondering if someone could explain this image below and why the changes were made. I'll explain more below the second picture.
Y = GDP
B = Population
H = Hours worked
N = Employed workers
B15-74 = Population which age bracket is eligible to work
I can't understand why you change the whole formula in this way as delta in the numerator and the regular expression in the denominator. The way this is explained in the question, I can write below with another attachment
. This image to the left is suppose to describe the "relative change in hours worked per employee". So does this mean that the whole equation is the relative change in BNP per capita. But what does this mean "relative change" I want the mathematical explanation of why they would change the whole equation this way and remove the multiplication and instead add addition between the fractions. And also why they would change the fractions in such way, as delta in the numerator and the regular expression in the denominator.
The equation for $\frac YB$ is the same as the equation for $\frac{GDP}{POP}$ except for renaming the terms. You should be able to see that these equations are identities in that all the intermediate terms cancel. Mathematically we could replace any of these terms with an unrelated variable, like replacing Employment with the number of flowers blooming in my yard. As long as we do it in both the numerator and denominator it is fine. They have chosen the intermediate terms to try to make the ratios meaningful. They explain some about why they chose those variables in the text below the first equation. $\frac {GDP}{Hours\ worked}$ is the average production per hour worked. We can compare different countries to see if one is more efficient than another this way. Incidentally, the $\frac YL$ they mention at the end would be $\frac YN$ in the terms of the second entry.
$\frac {\Delta (something)}{something}$ represents the fractional change in $something$. If you have $1000$ cars and buy $50$ more we would say the fractional change in the number of cars you have is $\frac {\Delta \ cars}{cars}=\frac {1050-1000}{1000}=0.05=5\%$. The last equation reflects a general result that if you have a quantity that is the product of a number of variables, the fractional change is (approximately) the sum of the fractional changes of the variables. The proof goes through logarithms and calculus. If you have $A=BC$ we can take the logarithm of each side, then take the derivative and get $$A=BC\\ \log A=\log B + \log C\\ \frac {\Delta A}A=\frac {\Delta B}B+\frac {\Delta C}C$$ Your equation does the same with four terms.