We investigate the electronic structures of (X = Li, K, Cs) by using Tran and Blaha's modified Becke and Johnson exchange potential. Calculated energy gaps are substantially better than previous first-principles results with respect to experimental values. The substantial improvement is achieved because the conduction bands are correctly calculated with the new exchange potential. The approach should be applicable to other similar materials. The elastic properties of (X = Li, K, Cs) are also studied in detail with the generalized gradient approximation such as bulk modulus, shear modulus, Young's modulus, Poisson's ratio, sound velocities, and Debye temperature.