achieved via Josephson parametric amplifiers (JPA). In the past, these amplifiers exhibited low power added efficiency (PAE), which is roughly the fraction of pump power that is converted to output signal power. This is increasingly relevant because recent attempts to build high saturation power amplifiers achieve this at the cost of very low PAE, which in turn puts a high heat load on the cryostat and limits the number of these devices that a dilution refrigerator can host. Here, we numerically investigate upper bounds on PAE. We focus on a class of parametric amplifiers that consists of a capacitor shunted by a nonlinear inductive block. We first set a benchmark for this class of amplifiers by considering nonlinear blocks described by an arbitrary polynomial current-phase relation. Next, we propose two circuit implementations of the nonlinear block. Finally, we investigate chaining polynomial amplifiers. We find that while amplifiers with higher gain have a lower PAE, regardless of the gain there is considerable room to improve as compared to state of the art devices. For example, for a phase-sensitive amplifier with a power gain of 20 dB, the PAE is ~0.1% for typical JPAs, 5.9% for our simpler circuit JPAs, 34% for our more complex circuit JPAs, 48% for our arbitrary polynomial amplifiers, and at least 95% for our chained amplifiers.