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In Economics, a concept called "Opportunity costs" plays a very important role. This refers to what one could earn if the same resources were put to another use and conversely what is being lost when the same resources are not being utilized for that.
Normal profits are the "opportunity cost of using entrepreneurial abilities." A business owner could use her abilities for a pre-existing company and receive payments for it. By deciding to set up her own venture she is foregoing that. As this is not an accounting cost it is usually overlooked. Similarly a business as a whole could make a profit by producing an alternative product. That would be its opportunity cost.
Normal profits are the total opportunity cost of a business. This is always larger than the accounting profits. Abnormal profits are profits earned in addition to the opportunity costs of the business. If a business considers all the alternatives it has and finds itself making a negative normal profit with any of them, it would ideally shift to producing the alternative.
In economic terms, a normal profit is the profit over and above a normal profit. A normal profit is also defined as zero economic profit. This is when the firm's revenues exactly equal its explicit and its implicit costs.
Explicit costs are the costs that a firm actually pays out. This includes things like salaries paid to its employees and the costs of the materialis it uses.
Implicit costs are the opportunity costs incurred by the owners of a company. This is the value of the other things they could do with their resources instead of using them for the business of this particular firm. For example, if you have a job making $50,000 per year and you quit it to start your own company, you have implicit costs of $50,000 because you could be making that much if you weren't running your own company.
When a firm's revenues exactly match its explict and implicit costs, it makes a normal profit. Anything above that can be called "economic profit" or "abnormal profit."
In the last sentence of justaguide's answer, should "making a negative normal profit" be "making a negative abnormal profit" instead?
In the last sentence of pohnpei397's answer, would it be better to replace "Anything above that can be called" with "The difference from that is called" because abnormal profit can be negative?
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