In this paper, we analyze the closed-loop performance of a recently introduced economic model predictive control (MPC) scheme with self-tuning terminal cost. To this end, we propose to use a generalized terminal region constraint instead of a generalized terminal equality constraint within the repeatedly solved optimization problem, which allows us to obtain improved closed-loop asymptotic average performance bounds. In particular, these bounds can be obtained a priori. We discuss how the necessary parameters for the generalized terminal region setting can be calculated, and we illustrate our findings with two numerical examples.