My mathematical knowledge is pretty limited so I apologies in advance for the trivial question.
I know that for an equation of the form: $Y = aX + bZ$ The amount of Y "contributed" by X is a and the amount contributed by Z is b.
I want to be able to have the same understanding of my parameters for:
$ Y = X^a * Z^b$
The reason being that I have a model which I want to approximate using an equation of this form and it is important for me to understand what proportion of the output is can be attributed to each parameter.
In the first example above : $Y = aX + bZ$, say Y is the amount of ill people in a population, X the average amount of meat eaten by the population and Z the average temperature the population lives in, it is nice to know that, if a < b trying to reduce the temperature in buildings is likely to reduce the amount of ill people more than if the same efforts are deployed in reducing the amount of meat eaten.
The actual regression of the model is given by:
$\ln(Y) = 3.72+0.66\ln(X_1)+0.14\ln(X_2)-0.35\ln(X_3)+0.03\ln(X_4)-0.01\ln(X_5)+0.04\ln(X_6)+0.78\ln(X_7)$
$Y = 41.26*X_1^{0.66}*X_2^{0.14}*X_3^{-0.35}*X_4^{0.03}*X_5^{-0.01}*X_6^{0.04}*X_7^{0.78}$
Well, something similar obtains, but it relates to proportional changes, since you are using logs. So if you double X, the proportional change in Y (that is, $\Delta Y/Y)$ is $2^a.$ And if the exponents sum to one, then doubling all inputs doubles output.