In this paper, we initiate our investigation of log canonical models for
(
M
¯
g
,
α
δ
)
(\overline {\mathcal {M}}_g,\alpha \delta )
as we decrease
α
\alpha
from 1 to 0. We prove that for the first critical value
α
=
9
/
11
\alpha = 9/11
, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that
α
=
7
/
10
\alpha = 7/10
is the next critical value, i.e., the log canonical model stays the same in the interval
(
7
/
10
,
9
/
11
]
(7/10, 9/11]
. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.