The wiretap channel is now a fundamental model for information-theoretic security. After introduced by Wyner, Csiszár and Körner have generalized this model by adding an auxiliary random variable. Recently, Han, Endo and Sasaki have derived the exponents to evaluate the performance of wiretap channels with cost constraints on input variable plus such an auxiliary random variable. Although the constraints on two variables were expected to provide larger-valued (or tighter) exponents, some non-trivial theoretical problems had been left open. In this paper, we investigate these open problems, especially concerning the concavity property of the exponents. Furthermore, we compare the exponents derived by Han et al. with the counterparts derived by Gallager to reveal that the former approach has a significantly wider applicability in contrast with the latter one.