Negative differential resistance (NDR) was recently observed in carbon nanotube junctions just before breaking and hypothesized to arise from the formation of monatomic carbon wires in the junction. Motivated by these results, a first-principles scattering-state approach, based on density functional theory, is used to study the transport properties of carbon chains covalently connecting metallic carbon nanotube leads at finite bias. The I- V characteristics of short carbon chains are predicted to exhibit even-odd behavior, and NDR is found for both even and odd chain junctions in our calculations.