Thrust may or may not be reduced during the climb. Typically at 1,000' above field elevation (which can vary with the airport, the aircraft, and the operator, as well as the conditions or circumstance of flight (such as an engine failure, for example), climb thrust is introduced.
Climb thrust may be a power reduction, or it may be a power increase, depending on what was used for takeoff power. Sometimes we see it as a thrust reduction, but sometimes we have a thrust increase, too...sometimes followed again by a thrust reduction while passing through 10,000'.
We use two different types of departure profiles, both basically noise abatement departure profiles, when all engines are working. One involves making the climb thrust setting at a thousand feet, and accelerating to retract flaps also beginning at that altitude. The other departure involves climb thrust set at a thousand feet, with a climb continued at our V2 speed until 3,000' above field elevation. At that point our vertical speed is reduced to either five hundred feet per minute or a thousand feet per minute, and the aircraft is allowed to accelerate while gradually retracting flaps. We then accelerate to 250 knots, or the minimum airspeed for no flaps if it's higher than 250 knots (typically about 280 for us).
We then continue up to 10,000', where we reduce our rate of climb again in order to accelerate to our enroute climb airspeed. At this time, depending on the flight, we may call for a reduced climb thrust setting, or we may continue with the climb thrust we've already got set.