Density income and Total Income.

Question

Suppose that the density income function is $f(r) = a*exp(r^2)$ with $r \in(0; 10)$ (hundred thousand euros) and $a=exp(10)-1$. Assume that the total number of people in this economy is 1 million. Compute the total income in the interval [5; 7] (total amount of money in the economy from people earning between $r = 5$ and $r = 7$ (hundred thousands euros).

Answer 1

If I ask Alpha to integrate the density income function, I get a huge number-almost $3 \cdot 10^{46}$ That should give the total number of people. It also seems strange to have almost all of the people at the top of the income spectrum. The total income would be the integral of $r\cdot f(r)$ over the range of income, so it looks like $N_0$ is the total population and $y*$ is the average income (it would be good to define these). Then the total income in the band from $r=5$ to $r=7$ would be $\int_5^7 ar\exp(r^2)dr$, but the normalization is off. To fix the normalization, we would multiply by $\frac {N_0}{\int_0^{10} ar\exp(r^2)dr}$ to get $\frac {N_0 \int_5^7 ar\exp(r^2)dr}{\int_0^{10} ar\exp(r^2)dr}$. You shouldn't divide by the average income-check the units.