For a principal type pseudodifferential operator, we prove that $(\Psi )$ implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker's paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from $\epsilon +3/2$ for any $\epsilon >0$ (Dencker's most recent result) to 3/2 (the present paper). It is already known that $(\Psi )$ does not imply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss $>1$.