The story begins with a competitive industry consisting of many firms. As usual we will call it the "widget" industry. Economists often use the word "widget" meaning "some small good or service, not specific." (This came from a 1950's musical comedy, "How to Succeed in Business Without Really Trying.") You won't be far off if you think of it as a small manufactured good.
We'll need to make a few simplifying assumptions. Assumption: Each firm operates under constant costs in the long run. We recall that with constant costs in the long run, each firm's long run average cost is a horizontal line. We need two more facts about constant costs in the long run, and one more assumption.
Fact: with constant long run costs, the firm's long run marginal cost is also a horizontal line, identical with the LRAC curve.
Assumption: the firm cost curves remain the same as the number of firms in the industry increases.
Fact: when the assumption is true, the long run average cost, marginal cost, and supply curve of the industry are also the same horizontal line.
With these assumptions, the competitive industry's supply curve is a horizontal line. The discussion would be a little more complex in general, allowing for more complicated supply curves, but the overall results would be the same.
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