NEW YORK (Reuters) - United Airlines said on Thursday its first-quarter capacity would shrink, but revenue per passenger would rise, as it trims its flight schedule to cut costs.

The No. 2 U.S. airline, owned by UAL Corp (UAUA.O: Quote, Profile, Research), said earlier this week that it will shrink its fleet by up to 4 percent this year to combat the skyrocketing cost of jet fuel.

In a regulatory filing on Thursday, United said first-quarter consolidated traffic -- measured as the number of miles it flies paying passengers -- is expected to fall between 2.75 percent and 3.25 percent.

Its consolidated capacity -- the number of miles it is selling seats for -- is expected to fall 0.1 percent.

Airline capacity cuts are expected, as carriers battle high fuel prices and weakening demand, particularly in the United States.

Earlier this week, rival Delta Air Lines Inc (DAL.N: Quote, Profile, Research) announced a plan to cut jobs and flying capacity, while Northwest Airlines Corp (NWA.N: Quote, Profile, Research) said it was evaluating possible capacity cuts.

United said it expected first-quarter consolidated passenger unit revenue -- which measures revenue per passenger -- to increase between 8.5 percent and 9.5 percent over the same quarter last year, helped by the capacity contraction.

However, it said operating cost per available seat mile, excluding fuel and special items, would increase 3 percent to 3.5 percent for the first quarter of 2008.

In a message to employees on Tuesday, United Airlines Chief Executive Glenn Tilton said the airline aims to eliminate 15 to 20 of its older, less fuel-efficient, narrow-body planes. United's fleet currently has 460 aircraft.