You are considering how many hours per day (out of 24) to spend at work. Every h

Question

You are considering how many hours per day (out of 24) to spend at work. Every hour you spend at work, you get paid a wage of $60. However, this is also an hour you have to spend away from your family. Your marginal value of first hour you spend with your family every day at $240, the second is $230, the third is $220, and so on.

Describe what the marginal cost and marginal benefit of working are.

Calculate how many hours you should work. [CAREFUL! $240 is the marginal value of the first hour you spend with your family – it’s what you give up for the 24th hour you spend working, not what you give up for the 1st hour – what’s the value of giving up the 1st hour?]

Explain why we can think of a marginal value (what you get from spending time with your family) as a marginal cost (cost of working). You will want to use the term “opportunity cost.”

Explanation / Answer

Solution:

According to the question, marginal benefit of working, i.e,the amount of benefit you receive by working for an additional hour = wage received per hour = $60 (constant over all hours of working)

Also marginal value you derive by spending time with your family is the marginal cost of working. Note that for an additional hour you spend with your family, you receive lower marginal value. Thus, the marginal value decreases with hours spend with your family. The hours not spent with your family are spent working, so we can also say that marginal cost increases with the number of hours worked.

So, when you spend 24th hour working, and giving up that 24th hour with your family (first hour with your family), marginal cost (MC) incurred = 240

Similarly, when you spend 23rd hr working, giving up 2nd hour with your family, MC incurred = 230. So following this pattern we can observe that working for nth hour is 10*n

So if I denote work hours by W and family hours by F, function for marginal cost of working = 10*W

Value of giving up the first hour with your family, that is first hour you spend working = 10*1 = 10

You should work till the marginal benefit of working equals marginal cost of working

That is, till the point 60 = 10*L

So we get L = 6. You should work for 6 hours.

Why marginal value of family time could be thought of as marginal cost of working: since you spend all your time either with family or working, if you are doing any one thing, you are giving up on the other. When you spend an hour working, you give up your time, that one hour, on next best thing you could do which was spend that hour with family. Since by spending that hour with family, you would have derived say X amount of marginal value, which you otherwise don't since you decide to spend that working. Hence this is a cost you incurr by working. So the opportunity cost of working for that hour = X which is marginal value of family time for that hour. This way we can say marginal value/benefit of family time is same as marginal cost of working.

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