If we assume that these devices have Star Trek style limitations and are not true infinite free/energy matter, then there is no reason our economies could not continue.
Replicators in Star Trek are not capable of replicating things larger than them (totally reasonable limitation to expect), require large amounts of energy (another reasonable assumption for our replicators), cannot replicate advanced technologies or people, and regularly break down.
Just to name a few major economic industries which will continue on in post replicator society:
Entertainment (especially live entertainment, you can't replicate a whole concert in your bedroom.)
Services (Anything done by people, doctors, lawyers, teachers, massage therapists, etc.)
Construction and heavy manufacturing (If we assume everyone has a microwave sized household replicator they still can't replicate themselves a car, large companies that invest in larger replicators will produce cars, and obviously houses or malls are too big to replicate and will need to be assembled out of replicated components.)
Restaurants, bars, ski resorts, etc. (All are more than just food or a little now and can't be replicated in your kitchen.)
Mining and energy (Exceedingly important for powering replicators.)
Replicator Repair (If it's anything like Star Trek they will break constantly.)
Prostitution (World's oldest profession won't be going anywhere.)
Professional Sports.
etc.
Fact is very little of our economy is actually based on making trinkets which could be easily done by replicators. Instead a great deal of it involves human actions which can' be easily replaced.
Replicators in Star Trek are not capable of replicating things larger than them (totally reasonable limitation to expect), require large amounts of energy (another reasonable assumption for our replicators), cannot replicate advanced technologies or people, and regularly break down.
Just to name a few major economic industries which will continue on in post replicator society:
Entertainment (especially live entertainment, you can't replicate a whole concert in your bedroom.)
Services (Anything done by people, doctors, lawyers, teachers, massage therapists, etc.)
Construction and heavy manufacturing (If we assume everyone has a microwave sized household replicator they still can't replicate themselves a car, large companies that invest in larger replicators will produce cars, and obviously houses or malls are too big to replicate and will need to be assembled out of replicated components.)
Restaurants, bars, ski resorts, etc. (All are more than just food or a little now and can't be replicated in your kitchen.)
Mining and energy (Exceedingly important for powering replicators.)
Replicator Repair (If it's anything like Star Trek they will break constantly.)
Prostitution (World's oldest profession won't be going anywhere.)
Professional Sports.
etc.
Fact is very little of our economy is actually based on making trinkets which could be easily done by replicators. Instead a great deal of it involves human actions which can' be easily replaced.