According to Business Dictionary, a constant-cost industry is an industry in which the ratio of units produced to the production cost per unit is constant, even when demand increases. This industry also keeps input prices the same.
According to AmosWEB, in a perfectly competitive industry, the demand and supply curves offset each other. An initial increase in demand causes equilibrium price and quantity to rise, but with free entry and exit, new firms enter the industry as they are induced by higher retail prices. This increase of firms leads to an increase in supply with a lower equilibrium price but a higher quantity. Yet, the supply curve shifts to the original price level of the old equilibrium price, thereby maintaining the same cost in the long run but having higher output.