First, it is not obvious that government schooling has positive externalities. Government officials can pursue their own interests in providing education because tax revenue frees them from having to cater to what parents want. In a democracy, their interests are in having a population “educated” to accept the rule of the state. Furthermore, I doubt if government education is highly correlated with crime rates. Does Mankiw cite studies? Finally, I don’t see how having government schooling K-12 results in developing more technology. At best, one could say that advanced technical training in college and graduate school might provide the ground work for it. But, even here, technological advance depends on the innovative character of the person and a freedom of the institutional setting in which he works rather than on training, which must be limited to what is already known.
Second, there are two cases of positive externalities. One is where the person generating the externality would not increase his production regardless of a government subsidy. Obvious, no government subsidy can be justified in this case. Only in the other case where the person would increase production could a subsidy be potentially justified,
Third, in the case where a person would increase production it isn’t obvious that the subsidy couldn’t be provided voluntarily. Even my local NPR station has fund drives and solicits over $100,000 in donations each time. Surely, if people really value the provision of some good, they will be willing to financially support its production. Maybe the government subsidy that NPR receives results in too many public radio programs or public radio programs with little social benefit or public radio programs that are too costly to produce.
Fourth, if the government uses coercion to provide the subsidy, then it creates “forced riders.” Using coercion takes the decision of government officials outside the realm of economic calculation. Therefore, their decision cannot be economizing for society as a whole. Government officials make production decisions on the basis of their own preferences, not by objectively comparing the preferences of others and producing what others find more valuable.
Take a look at Ludwig von Mises on externalities in Human Action:
And Murray Rothbard in Power and Market: