I was designing a 2 quadrant multiplier but the output was always reading a constant output for whatever input. The circuit in question is attached. I would like to get pointers on how to choose the resistor values and also how to correct the circuit so that I can achieve reliable results.

 

It looks like you are applying you signals directly to the bias voltage sources. Unless you have set them otherwise they have zero equivalent impedance. This causes a fight between the signal source and the bias source. This is not a good thing in a simulator and is even worse in the real world.

Try putting resistors between the bias voltage sources and the input it biases and see what happens.