We study a free boundary problem arising from the modeling of an idealized electrostatically actuated MEMS device. In contrast to existing literature, we consider a three-dimensional device involving a hinged elastic plate. The model couples the electrostatic potential to the displacement of the elastic plate, which is caused by a voltage difference that is applied to the device. The electrostatic potential is harmonic in the free domain between the elastic plate and a rigid ground plate. The elastic plate displacement solves a fourth-order parabolic equation with hinged boundary conditions and a right-hand side proportional to the gradient trace of the electrostatic potential on the elastic plate. We establish local and global well-posedness of the model in dependence of the applied voltage difference and show that only touchdown of the elastic plate on the ground plate can generate a finite time singularity. Next, we consider stationary solutions and prove that such solutions exist for small voltage values and do not exist for large voltage values. To prove the nonexistence result, we show that the fourth-order elliptic operator with hinged boundary conditions satisfies a positivity preserving property and has a positive eigenpair.