Find degressive cost function that goes through two points

Question

I need to find the root function (degressive cost function) that goes through the points $$ P_{1} = \left[ \begin{array}{c} 50 \\ 79 \end{array} \right], \qquad P_{2} = \left[ \begin{array}{c} 5000 \\ 3200 \end{array} \right] $$ where the function is (I believe) something like $$ f(x) = \frac{x^{c}}{a} $$

Does anyone know how to find the function? Thank you very much

Answer 1

Let's say you have $f(x_1)=y_1$ and $f(x_2)=y_2$ (where $x_1,x_2,y_1,y_2$ are known), and that $f(x)=\frac{x^c}{a}$. To determine $c$ and $a$, observe that $$\frac{y_2}{y_1}=\frac{f(x_2)}{f(x_1)}=\left(\frac{x_2}{x_1}\right)^c.$$ Taking the natural logarithm and dividing yields $$c=\frac{\log y_2-\log y_1}{\log x_2-\log x_1}.$$ So now $c$ is known, and to find $a$ just rearrange the definition of $f$ to isolate it: $$a=\frac{x_1^c}{f(x_1)}=\frac{x_1^c}{y_1}.$$