How do capacity constraints affect the slope of the Phillips curve? In particular, do changes in output affect prices by more when firms face more strongly diminishing returns? We investigate this question in a New Keynesian framework. Intuitively, one may conjecture that the answer is yes. With more strongly diminishing returns, marginal costs increase more when production increases, also raising prices by more. The Phillips curve, the relationship between output and inflation, should therefore be steeper with more strongly diminishing returns. But this intuition is incomplete in models with sticky prices ` a la Calvo. In the standard New Keynesian model with Calvo pricing frictions, more strongly diminishing returns actually flatten the Phillips curve, given the standard parameterisation of the model. The reason is the reluctance of firms who alter their prices to let them get too far out of line with the prices set by firms that do not adjust; more strongly diminishing returns make it advantageous for such firms to raise their prices by less. We explain how the Phillips curve might be amended to avoid this implausible feature, for example, by using Rotemberg pricing or introducing wage stickiness.