Considering the fact that Rome and Carthage were large trading partners and Rome destroyed it. Do we have any evidence that suggests Rome went through an economic downturn?
Is a recession the way we think of one even possible back then? Or would it just be a famine?
Not quite. In some respects, quite the opposite. The Romans systematically looted all of their conquests and brought the "spoils of war" back home. They now had total control over Mediterranean shipping as well as direct control over the grain supply from N. Africa.
However, there were negative economic consequences, including a glut in the slave market (always happened after a major victory) and grain market. Most importantly, many of the troops returning home had lost their farms due to the length of time they were on campaign. This caused a huge social crisis.