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We examine the uncapacitated single-item lotsizing problem with backlogging and start-up costs where Wagner–Whitin costs are assumed. We generalize some theoretical results obtained in [5] for the polyhedral description of the convex hull of feasible solutions for models that can be viewed as particular cases of the one treated in this paper (models without start-up costs and models where backlog is not allowed). In the presence of Wagner–Whitin costs (which satisfy p _t+ h ̃ _t⁺−p _t+1⩾0, for 0⩽t⩽n−1, and p _t+1+ h ̃ _t⁻−p _t⩾0 , for 1⩽t⩽n, where p _t, h ̃ _t⁺ and h ̃ _t⁻ are the unit production, storage and backlogging costs; in the case that unit production costs are constant over time, the Wagner–Whitin assumption corresponds to non-negative holding costs and backlogging costs) we present a linear extended formulation with O(n) variables and O(n²) constraints. By projection, we obtain a linear formulation in the original …