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Cost includes both expired and deferred cost. Expired cost is the cost that has been already incurred, while deferred cost is one which has been incurred but its economic benefit is not received, prepaid expense or expenditure on research and development are some of the examples of deferred cost.
An expense includes only expired cost which is used up in earning revenues in a company’s main operations. In other words it is a cost with a matching economic benefit during a particular period. Example of it includes expenses such as advertising, rent raw material cost etc….
Cost does not necessarily mean an expense, but an expense is part of the cost.