# interpreting level-log model that has a percentage variable

If I have a level-log model, such as
y= β0+β1log(x), I know the interpretation is "If we increase x by one percent, we expect y to increase by (β1/100) units of y."

So, my questions are

1. If y is a percentage variable, what is the correct interpretation of the resulting regression? Is it still the same but has a unit of percentage instead of unit, so the interpretation becomes "If we increase x by one percent, we expect y to increase by (β1/100) percent of y"? Or we don’t have to divide β1 by 100, so the interpretation becomes "If we increase x by one percent, we expect y to increase by β1 percent of y"?

2. Can I convert my variables (ex: interest rate and inflation rate) to logarithm if it has a unit of percentage? If it can be done, can I change my model into log-log model, although my variable y has a unit of percentage, so the model becomes log(y)= β0+β1 log(x) ? What is the correct interpretation of this model?

Answer for Question 1: If y is a percentage your model will be: $$(\%)y=\beta_0+\beta_1 log(x)$$
$$\frac{d(\%)y}{d log(x)}=\beta_1$$ $$\frac{d(\%)y}{d log(x)}=\frac{d(\%)y}{dx/x}=\beta_1$$ Dividing both sides by 100: $$\frac{d(\%)y}{d log(x)}=\frac{d(\%)y}{100 \times dx/x}=\frac{\beta_1}{100}$$
Your interpretation is: when X increases by 1 percent, y increases by $\beta_1$ percentage points.