  A model of Einstein-Hilbert action subject to the scale transformation is
studied. By introducing a dilaton field as a means of scale transformation a
new action is obtained whose Einstein field equations are consistent with
traceless matter with non-vanishing modified terms together with dynamical
cosmological and gravitational coupling terms. The obtained modified Einstein
equations are neither those in $f(R)$ metric formalism nor the ones in $f({\cal
R})$ Palatini formalism, whereas the modified source terms are {\it formally}
equivalent to those of $f({\cal R})=\frac{1}{2}{\cal R}^2$ gravity in Palatini
formalism. The correspondence between the present model, the modified gravity
theory, and Brans-Dicke theory with $\omega=-\frac{3}{2}$ is explicitly shown,
provided the dilaton field is condensated to its vacuum state.
