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I heard that profit of all firms in long-term becomes zero. But I am not convinced of why it is like that. Can anyone explain this mathematically?

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    You'd have to give us more mathematical definition of the model and the problem for this to qualify as a math problem suitable for this site.2012-12-12
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    At least the average profit should be zero.2012-12-12
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    It doesn't sound like a claim that would be true under even unrealistically simplified macroeconomic assumptions (such as a finite money supply). It's easy to imagine a toy economy where all businesses continually make profits and pay out dividends; the income of each business comes from private individuals buying services from it with money they were paid in dividends from their owning other businesses. In such a world, a single unit of money can keep generating profits again and again as it circulates.2012-12-12
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    I suspect the reason is that any company making a profit will necessarily engender competition, until the profits are whittled to zero.2012-12-12
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    @HenningMakholm: Your argument holds water in terms of accounting profit. I suspect what OP concerns is economic profit.2012-12-16

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This is a question in economics rather than mathematics. Nevertheless, it's an interesting observation which has a more general analogue in the field of complex dynamic systems. Namely, if a system is rich enough to be ever developing and to spawn identifiable subsytems that are forced to compete, then in the long run the identity of almost every such subsytem will be transient. This applies to life at the level of individuals and species, and to social organizations (e.g. companies) in the case of the human species. It notably doesn't apply in astronomy, because astronomical bodies orbit each other or recede from each other: they are not "forced to compete". Complex systems are in principle formalizable in mathematics, but there is no canonical way to do this. Thus it seems unlikely that a rigorous explanation in mathematical terms is on the horizon.