Education Spending Ratio Oscar Tabarez, the current football manager of Uruguay National Team, have urged, after his good results, his governement to raise, as promised, the Education Spending on GDP Ratio to 6%. 

It is a very good resolution indeed and it makes me looking how my country, France, is ranked. 
Someone, could help to understand why France (i'm french) has a low level in comparison with Mexico, Brazil or Argentina and is one of the last country of OECD.

source : https://data.oecd.org/eduresource/education-spending.htm

Are we more efficient with Public Expenditure ? 

It is large issue but any comments will help me.
My work focuses on the Economics of Education, although from a more micro perspective. I can honestly say it's a complicated question and there are many people working on optimal education spending.  

First, a bit of simple math. Let $EdExp$ be education expenditure, $GDP$ be Gross Domestic Product, $E_{students}$ be enrolled students, $S_{pop}$ be school-age population, $N_{pop}$ be non-school age population. One can decompose Education Expenditure by GDP ratio in this way:

$$ \frac{EdExp}{GDP} \equiv \frac{E_{students}}{S_{pop}} \times \frac{S_{pop}}{(S_{pop}+N_{pop})} \times (\frac{EdExp}{E_{students}} \times \frac{(S_{pop}+N_{pop})}{GDP})$$

Note this isn't a "made-up formula", it's an accounting identity. This is true in the US, China and in a tribe of herdsmen. I'll translate its terms to english. $\frac{EdExp}{GDP}$ is Education Spending to GDP ratio, which Tabarez thinks should be 6%. $\frac{E_{students}}{S_{pop}}$ is the enrollment rate, the fraction of people who are school-aged and attend school. $\frac{S_{pop}}{(S_{pop}+N_{pop})}$ is the dependency ratio, the fraction of the population that is school aged. Lastly, $\frac{EdExp}{E_{students}} \times \frac{(S_{pop}+N_{pop})}{GDP}$ is education expenditure per student divided by GDP per capita. This is the proportion of average income that is spent on the average student.

Countries like France have high GDP per capita and low dependency ratios, which means they can sustain high expenditure per student without having to compromise a large share of GDP on education. Uruguay and Brazil, on the other hand, have high dependency ratios and somewhat low GDP per capita. Enrollment rate isn't very high too. So, in order for these countries to sustain high expenditure per student (which is what ultimately counts), they need to reserve a larger share of GDP for education.

That's one part, the one having to do with demography and aggregate spending. I'd like to talk about the one some of my research is concerned with, spending efficiency. Uruguay is not an extreme case as Brazil, but it shares many problems with its northern neighbor.

Out of the ~6.5% of GDP that Brazil spends in education, only about 1% goes to public fundamental/elementary schools. About 1% goes to public high schools and the rest goes to public universities. This is just terrible. You can't spend 2% of GDP on dozens of millions of students and then 4% on 1 million of them and expect the first system to have reasonable quality. Then, we have the institutional problems. In most states, schools aren't allowed to pay bonuses to better teachers and many poor municipalities can't even hire trained teachers because of low pay, depending on self-taught locals to teach. Access to federal resources such as financing for new materials and data to better understand your school's demands is somewhat easy but mostly untrained, unqualified school managers don't know what to do with these resources. 

In the cities and school that do have somewhat qualified personel and more money, public tenure rules in Brazil make it impossible to fire low productivity teachers and managers and even promote the good ones. Leading to overall lack of incentives. 

On one hand, spending matters. It's hard to imagine schools giving good education if there aren't blackbords, chairs and basic equipment in it. On the other hand, poor institutional design and lack of incentives can make tons of money go to waste. 

As a rule of thumb, anything in the 4-5% education spending to GDP ratio is reasonable. More than that means mostly resources going to waste if there aren't serious attempts to create better incentives and improve school management.