  Recently, the intrinsic magnetic topological insulator MnBi$_2$Te$_4$ has
attracted great attention. It has an out-of-plane antiferromagnetic order,
which is believed to open a sizable energy gap in the surface states. This gap,
however, was not always observable in the latest angle-resolved photoemission
spectroscopy (ARPES) experiments. To address this issue, we analytically derive
an effective model for the two-dimensional (2D) surface states by starting from
a three-dimensional (3D) Hamiltonian for bulk MnBi$_2$Te$_4$ and taking into
account the spatial profile of the bulk magnetization. Our calculations suggest
that the diminished surface gap may be caused by a much smaller and more
localized intralayer ferromagnetic order. In addition, we calculate the spatial
distribution and penetration depth of the surface states, which indicates that
the surface states are mainly embedded in the first two septuple layers from
the terminating surface. From our analytical results, the influence of the bulk
parameters on the surface states can be found explicitly. Furthermore, we
derive a $\bf{k}\cdot \bf{p}$ model for MnBi$_2$Te$_4$ thin films and show the
oscillation of the Chern number between odd and even septuple layers. Our
results will be helpful for the ongoing explorations of the MnBi$_x$Te$_y$
family.
