There are several questions on the interpretation of coefficients in log-linear models such as
Interpreting regression coefficients of log(y+1) transformed responses
Log linear model interpretation - % Contributions?
Interpretation in log linear regressions with coefficients bigger than 1
but so far I have not found an question that deals with the interpretation of coefficients of explanatory variables presenting shares/percentages. So the model I'm trying to estimate is:
$$\ln (y_s) = \alpha + \beta \frac{\text{number of unemployed}_s}{\text{number of people in the labor force}_s} + \epsilon_s$$
The index $s$ is for states and $\ln(y_s)$ is the natural logarithm of the average income in state $s$. I estimated a beta coefficient of $-3$. The typical interpretation would be that income decreases by $\beta * 100 = -3 * 100$ dollars for a one unit increase in the explanatory variable. But since the explanatory variable here is a share/proportion, would the correct interpretation be one without multiplication of the coefficient by 100? It seems unreasonable that a one unit increase in the share of unemployed to total labor force would decrease average earnings by 300%. A 3% reduction makes more sense but I have yet to understand what would be the correct interpretation.
Thanks a lot in advance and no need to mention the potential endogeneity problems here; I'm really just looking for the interpretation of beta.