Showing public returns to be greater than private returns mathematically Take this headline 

"OECD figures show public benefits more than individuals from tertiary education." 

How would I present this mathematically, to show that public returns are greater than private returns?
This would be true if there are positive externalities from tertiary education. That is, the rest of the society also derives some positive benefit from tertiary education. For example, if the benefit to an individual getting tertiary education is \$250,000 (in incremental lifetime earnings and other nonpecuniary benefits) and the benefit to society is another \$50,000, then public benefits exceed private benefits. This would be the case for example, if education lead to a more civic society that others value. Opposite will be true with negative externalities (incremental earnings of educated come at the expense of lower earnings for someone less educated).

The decision of whether tertiary education is social welfare enhancing or not depends not just on externalities but also on cost of providing that education. In the above example with \$250,000 private benefit and \$50,000 externalities, tertiary education is welfare enhancing if the cost of education is \$275,000 but nor if the cost is \$325,000.

If modeling theoretically, suppose value of A is VA, and value of B (everyone other than A) is VB when A is not educated. Next, suppose value of A is VA', and value of B (everyone other than A) is VB' when A is educated and everything else is same. Then, the headline is equivalent to the condition VB' > VB.

Empirically, I think the headline is not surprising and is most likely true. To test it, you need an exogenous change in education. For example, what happens to peers/neighbors/cities when the fraction of education changes exogenously. Maybe change in immigration laws. Or if a natural disaster closes schools, compare the effect to the effect in other places with similar disaster but no effect on education. The problem is that most changes are not completely exogenous.
This would be true if there are positive externalities from tertiary education. That is, the rest of the society also derives some positive benefit from tertiary education. For example, if the benefit to an individual getting tertiary education is \$250,000 (in incremental lifetime earnings and other nonpecuniary benefits) and the benefit to society is another \$50,000, then public benefits exceed private benefits. This would be the case for example, if education lead to a more civic society that others value. Opposite will be true with negative externalities (incremental earnings of educated come at the expense of lower earnings for someone less educated).

The decision of whether tertiary education is social welfare enhancing or not depends not just on externalities but also on cost of providing that education. In the above example with \$250,000 private benefit and \$50,000 externalities, tertiary education is welfare enhancing if the cost of education is \$275,000 but nor if the cost is \$325,000.

If modeling theoretically, suppose value of A is VA, and value of B (everyone other than A) is VB when A is not educated. Next, suppose value of A is VA', and value of B (everyone other than A) is VB' when A is educated and everything else is same. Then, the headline is equivalent to the condition VB' > VB.

Empirically, I think the headline is not surprising and is most likely true. To test it, you need an exogenous change in education. For example, what happens to peers/neighbors/cities when the fraction of education changes exogenously. Maybe change in immigration laws. Or if a natural disaster closes schools, compare the effect to the effect in other places with similar disaster but no effect on education. The problem is that most changes are not completely exogenous.