When a voltage is applied to a layer of a dielectric elastomer, the layer reduces in thickness and expands in area. A recent experiment has shown that the homogeneous deformation of the layer can be unstable, giving way to an inhomogeneous deformation, such that regions of two kinds coexist in the layer, one being flat and the other wrinkled. To analyze this instability, we construct for a class of model materials, which we call ideal dielectric elastomers, a free-energy function comprising contributions from stretching and polarizing. We show that the free-energy function is typically non-convex, causing the elastomer to undergo a discontinuous transition from a thick state to a thin state. When the two states coexist in the elastomer, a region of the thin state has a large area, and wrinkles when constrained by nearby regions of the thick state. We show that an elastomer described by the Gaussian statistics cannot stabilize the thin state, but a stiffening elastomer near the extension limit can. We further show that the instability can be tuned by the density of cross links and the state of stress.