Is the value of private schools over-stated or under-stated?

Two recent studies have contradictory findings on the 'value of private schools'.

Analyzing PISA 2011 test scores of 40 countries, Sakellariou of NTU finds that
students of private schools perform better in international assessments compared to students in the public school system. However, when one controls for observable socioeconomic background characteristics at the individual and school level, public school students perform equally well. 
India didn't participate in PISA 2011 but this finding is inline with the at least two other studies in Indian context, finding the same.

Jishnu Das, on the other hand, has a new paper on 'value of private schools', in the context of Punjab, Pakistan. The 'value of private schools' here is the willingness of parents to pay for private school. Das et. al find that 
central determinants of school choice are the distance to school, school fees, and the characteristics of peers. Families are willing to pay on average between 75% and 115% of the average annual private school fee for a 500 meter reduction in distance. In contrast, price elasticities are low: -0.5 for girls and -0.2 for boys. Both distance and price elasticities are consistent with other estimates in the literature, but at odds with a belief among policy makers that school fees deter enrollment and participation in private schooling. 
A voucher policy that reduces the fees of private schools to $0 (from an average annual fee of $13) increases private school enrollment by 7.5 percentage points for girls and 4.2 percentage points for boys.
Essentially, it means that parents desire to put their kids in private schools is more that we we typically tend to think. 

It is also important to note that quality or learning outcomes isn't one of the key determinants of parents in choosing the school, as described above. May be the characteristics of peers is being used as a proxy for the same.

In a different story, Project Syndicate reports that private tutoring has become the de-facto education system in Egypt due to governance failure in public schools. It's not uncommon in such situations for teachers to exploit the grey areas of conflict of interest.
“Things are actually worse since the revolution,” Khalil said while hopping on a minibus to a nearby wealthy residential compound where he gardens. He’s trying to make enough money to hire a private tutor for the exam. Indeed, private tutoring has become the de facto Egyptian education system.
Some teachers have admitted off the record that they teach the bare minimum in class, so that they can profit from the same students in private lessons. According to some estimates, Egyptian families spend more than $1 billion on private tutoring to compensate for poor education – a cost that often amounts to almost a quarter of household income.
One should be positively skeptical of the effect of rules such as "preventing public teachers from running private tuitions" (though it should be a starting point) in such situations because the above condition isn't arising from lack of such rule but due to an underlying governance failure resulting in poor implementation of everything. There are high chances that the new rule banning public school teachers from private tutoring, if framed, will also meet the same fate.

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