Positive errors with mean $\theta$: the slope is still unbiased, the intercept absorbs $\theta$

Consider $Y_i=\alpha+\beta X_i+\varepsilon_i$ with $\varepsilon_i\sim f(\varepsilon)=\frac1{\Gamma(\theta)}e^{-\varepsilon}\varepsilon^{\theta-1}$, $\varepsilon>0$. Then which statement is true?