The cost-minimization hypothesis proposes that positive allometry in sexually selected traits can be explained if the proportional energetic maintenance costs of weapons decrease as traits increase in size. Energetic maintenance costs are the costs of maintaining homeostasis. They are slow, persistent energy sinks that are distinct from ephemeral costs of growth. Because some tissues expend more energy on maintenance than others, energetic maintenance costs can be inferred from proportional tissue composition. For example, soft tissues require more energy for maintenance than exoskeleton, so an arthropod claw that is 50% soft tissue and 50% exoskeleton would have higher energetic maintenance costs than one that is 30% soft tissue and 70% exoskeleton. I tested the cost-minimization hypothesis using proportional tissue composition as a proxy for energetic maintenance costs in snapping shrimp ( Alpheus heterochaelis and Alpheus estuariensis ) and fiddler crabs ( Uca pugilator ). As predicted, larger weapons comprised proportionally less soft tissue mass and more exoskeleton mass than smaller weapons. Furthermore, I extended cost-minimization to explain trait exaggeration: individuals might exaggerate traits by investing more mass in exoskeleton. As predicted, exoskeleton mass proportional to weapon mass increased as exaggeration increased. These results support and extend the cost-minimization hypothesis to explain positive allometry and weapon exaggeration.