If Standard of Living is narrowly defined as the ability to buy more stuff, I would say that this is generally correct. Of all of the principles, this one is really the most subjective and could be described another way.
If a benevolent dictator knew no more about economics other than this principle what would happen? China put a steel furnace in every backyard, if the plan had worked and China had successfully produced more steel than any nation on earth - would that have increased the Chinese standard of living?
Of course not, the standard of living depends on producing quality goods and services that people actually want. And even that alone isn't enough. The principle doesn't state it explicitly, but there seems to be an implicit suggestion that producing more stuff will raise the standard of living. Even that isn't true - standard of living is a very subjective term.
Say that there are two countries. Country A is centrally controlled - they somehow figured out to match and even exceed the productivity of the typically laissez faire market, producing 25% more 'stuff' than Country B. However, Country A also controls every aspect of private life as well, decide which stuff they can buy, how they use it, who they can associate with, when they travel and to where, what their job will be.
On paper, it appears that Country A is wealthier than Country B, but is their standard of living really any better?
I understand what Mankiw is getting at - the more successful the market, the better off its people tend to be. What would be a better way to state this principle?
Pathetic attempt to explain off hard statistics that undermine your worldview.
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