Communication is occurring between two satellites. The transmission obeys the free space law. However, the signal is too weak. The vendor offers two options: The vendor can use a higher frequency that is twice the current frequency or can double the effective area of both antennas. Which one of the options will offer more received power? Or if both will offer the same improvement, how much improvement in the received power will be obtained from the best option (all other factors remaining equal)?

From the free space law (more specifically from the Friis equation), we know that the received power is proportional to the gains of each antenna and inversely proportional to the square of the distance between them:P_{rec} / P_{trans} = G_{trans} G_{rec} (lambda / {4 pi R})^2 Where P_{rec} and P_{trans} are the power at receiver and transmitter respectively, G are the gains, lambda is the wavelength, and R is the distance.Gain in turn is given by effective area and wavelength as follows:G = {4pi A}/{lambda^2} Where A is the effective area.Thus, if we double the area of each transmitter, we will increase both gains by 2, and the overall power received by 4. But if we double the frequency, we will cut the wavelength in half, which means we would reduce received power to 1/4 if gain remained the same; but in fact if effective area is held constant, gain will increase by a factor of 4, precisely canceling this effect out. Thus, the power received will be equal regardless of what frequency we choose.The best option is therefore to increase the effective area, increasing received power by a factor of 4. Increasing frequency would not help.
http://people.seas.harvard.edu/%7Ejones/es151/prop_models/propagation.html

http://www.antenna-theory.com/basics/friis.php

https://en.wikiversity.org/wiki/Friis_Transmission_Equation