This was quite an interesting if controversial paper. Its basically explained by the following chart.
As the authors state:
The flow chart illustrates the transition to a vicious cycle of population aging and a decline in public education. In the early phases of industrialization, compulsory schooling regulated child labor and promoted technological progress, thereby discouraging fertility. A major education reform was eventually triggered by the rise in the young's demand for skill acquisition. However, the increased population share of the old enlarged their political power to curtail the government budget for education. The resulting positive response of private education induced further fertility decline.
The authors review the Japanese, USA, British, French, German situations with respect to economic development, ageing, public education, child labour, poverty, etc. The basic premise is given above. I am not sure that this has been thought before but this requires much thought. Actually, what else can one do? Public education is required. Its important to educate society as the economic development happens to move from labour and agriculture intensive sector to intellectual property / knowledge sector employment. Increased automation will anyway force people to work on higher value add jobs or get demoted out of the labour market.
But running the statistics on the overall education spend on the OECD countries here, I am not sure that the level of educational spending is falling. I think this needs to be reviewed a bit more