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The difference between these two is that private costs are only one part of overall social costs. Social costs take into account not only private costs, but also the externalities that come as a result of a given economic decision.
Private costs are the costs with which we are all familiar. These are the costs that we actually have to pay when we decide to do something. Let us take the example of owning and driving a car. Clearly, there are many private costs involved here. If I own and drive a car, I have to buy the car. Then I have to pay for the gas I use. I have to maintain the car. I have to buy insurance. I have to spend time in the car being unable to do much else other than driving. These are all private costs because I actually have to pay them and they are specific to me.
Social costs also include the externalities of driving my car. When I drive my car, it contributes to some extent to polluting the atmosphere. I do not directly pay for this, but it affects society. When I drive my car, I contribute to traffic congestion. Again, I do not pay for this, but I have at least a marginal effect on many other people who are also stuck in congestion, wasting their time.
Social costs is a category that includes both the more tangible private costs and the external costs of any given economic activity.
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