  We study the tropical version of the contraction morphism $\mathcal{T}$
between moduli spaces of stable and pseudostable curves. By promoting
$\mathcal{T}$ to a logarithmic morphism, we obtain a piecewise linear function
between the generalized cone complexes parameterizing tropical stable and
pseudostable curves. The ray corresponding to the contracted divisor $\delta_1$
is not contracted to the cone point but mapped onto a ray of
$\mathcal{M}_{g,n}^{{\rm trop}, {\rm ps}}$, with a slope reflecting the
geometry of the desingularization of a plane cusp. We explore in detail the
situation of $g=1$, where the tautological geometry of both spaces is fully
described by piecewise polynomial functions on the tropical moduli spaces.
