Emissions trading markets have been touted as the most efficient mechanism to achieve environmental goals at least cost. Whether in the form of voluntary markets or in a mandatory framework like in the first phase of the European Union (EU) Emission Trading Scheme (ETS), the regulator sets a cap on the emissions which can occur without penalty, and provides emissions allowances accordingly. The recipients are free to use these emission certificates to cover their emissions, or to sell them to the firms which are expected to emit more than what they can cover with their original allocations. As observed in most existing programs, cap-and-trade systems can fail to reach their emission targets as too generous an allocation of pollution permits serves as a disincentive for emissions reductions and deflates pollution prices. Moreover, the implementation of the first phase of the EU-ETS has been widely criticized on one more sensitive account: providing significant (some went as far as calling them obscene) windfall profits for power producers. Here we weight on this debate with the results of a rigorous quantitative modeling undertaking, providing insight into what went wrong in the first phase of the EU-ETS, and proposing alternative reduction schemes with provable advantages. Using market equilibrium models and numerical tools, we demonstrate that properly designed market-based pollution reduction mechanisms can reach pre-assigned emissions targets at low reduction cost and windfall profits, while being flexible enough to promote clean technologies. In the present article, we illustrate our claims with the results of a hypothetical cap-and-trade scheme for the Japanese electricity market.