EDUCATION is flush with data comparisons, from the Programme for International Student Assessment (PISA) run by the OECD, a mainly rich-world think-tank, which ranks 15-year-olds in core subjects every three years, to TIMSS and PIRLS, tests of younger pupils’ mathematics, science and reading levels administered by national research institutions. But such pecking orders cannot tell governments how much they should spend on education, or what the money should go on. Two new pieces of research shine light on these questions. An “efficiency index”, published on September 5th and constructed by academics working with GEMS Education Solutions, a consultancy, analyses the impact of spending on outcomes in 30 rich and developing countries. The OECD’s annual Education at a Glance, published on September 9th, looks more broadly at school financing and structures, and how these affect results and progress to university and work. Both offer lessons for governments around the world.

As far as star pupils and stragglers are concerned, the efficiency index resembles other rankings. Finland and South Korea shine (two of PISA’s other high performers, China and Singapore, were omitted because some data were unavailable). Brazil and Indonesia, which do poorly in PISA, are both very inefficient, too.

But what the rest of the pack get for their outlay varies widely (see table). Taking into account teachers’ pay, class sizes and pupils’ PISA scores, the former Soviet-bloc countries, notably the Czech Republic and Hungary, are highly efficient. The Mediterranean countries—Greece, Portugal and Spain—are strikingly not. America’s pricey schools system disappoints in both efficiency and outcomes. Elsewhere in the prosperous Anglosphere, Australia and New Zealand do better in both.

The link between results and teachers’ pay is surprisingly weak. An experienced Finnish teacher earns on average $42,800. But Switzerland pays teachers at the same point in their careers $68,000, and though it gets creditable maths results, coming ninth out of 65 in the most recent PISA assessment in 2012, it does much less well in reading and science. The figure in Germany is $54,000 a year. Its results have improved recently, but at a cost that leaves it a lowly 25th in the efficiency ranking.

Education spending depends not only on what teachers earn, but on how many of them there are—and in many places that number is rising, as rich countries cut class sizes in the hope that children will learn more. Parents, convinced that their children will do better with fewer classmates, are keen on the policy, too. But again, the data provide little support.

Both Korea and Finland have high pupil-teacher ratios (third- and fifth-highest of the 30 countries GEMS studied, respectively). France and Norway, with few pupils per teacher even by rich-world standards, trail in performance rankings, coming 25th and 30th in PISA 2012. Portugal, one of Europe’s laggards, has just half as many pupils per teacher as Finland (partly because the number of teachers did not drop as birth rates fell). Only when classes become truly unwieldy do outcomes seem to suffer: along with Brazil, the other country with a higher pupil-teacher ratio than Korea is Chile, which also has poor results.

Adam Still of GEMS thinks that many of the highest spenders have probably passed “peak efficiency”—the point at which more money brings diminishing returns. Andreas Schleicher, the data-gatherer who oversees PISA, reckons that differences in spending explain less than a fifth of the variation in countries’ outcomes. Such conclusions run counter to the claims of teaching unions, which generally argue that smaller classes and higher pay are essential if outcomes are to improve.

“Quality of teachers has a clearer impact than class size,” says Mr Schleicher. A rich country may decide to pay teachers well in order to get the best graduates into schools or to underscore its esteem for the profession. Giving teachers plenty of support as they enter the classroom, and continuing their training throughout their careers, will be more effective than increasing their numbers—and cheaper.