MLE of a shifted exponential threshold

You observe $X_1, \dots, X_n$ i.i.d. from the shifted exponential density $f(x) = e^{-(x - \theta)}$ for $x \ge \theta$ (and $0$ below $\theta$), where the threshold $\theta$ is unknown.

Find the MLE of $\theta$, show it is biased, and correct it.