Sorry, but the fact is CAIs don't give you more power AND they use more fuel because the ECU will compensate. They are just a waste of money but the sound good.
Cold air intakes reduce the temperature of the air going to the engine. Many also reduce the restriction and allow more air to be made available to the engine at higher rpms. They usually only increase performance which can result in lower fuel efficiency. Most tuners who want better fuel economy use cold air intakes in conjunction with a programmable ignition chip to defeat the ecu from overriding the cars' new air to fuel ratio.
simple fact - they don't work on injection systems in fact they cause the ecu to run a cold start program which enriches the fuel and wastes gas, if something as cheap and simple as an air filter increased power and gave better gas mileage don't you think the manufacturers would all fit them even as an option, also on lean burn engines the colder denser air would make the mixture even more lean leading to burnt plugs and valves, the ecu/maf/temperature sensor are all mapped together upsetting the balance would cause performance to deteriorate, the injection system squirts a measured amount of gas at each cycle so more or colder air would make no difference, unlike a carburetor which supplies gas continuously
