Much of my childhood was spent in south London's Battersea park, and a lot of my adulthood too, first with my four children in Battersea's adventure playground, and now with my grandchildren. The place inhabits the memories of generations, both the well-heeled and those from the surrounding tower blocks, clambering, sliding, swooping on ropes and tyres, a bit rough, a bit risky, cool enough for older ones with nowhere else to go. The wickedness of Wandsworth council now charging £2.50 per child to play there defies belief. What next, pay for the air we breathe?

But Wandsworth, boasting the lowest council taxes, has always pioneered Conservative thinking. What Wandsworth does now other Tory councils are sure to follow. Charging children to use the playground, it says, delivers "best value for money for local taxpayers" – or for Tory voters finding the playground more congenial with the riff-raff barred.

These Wandsworth zealots give the game away. Cameron's genius has been to disguise the social injustice of his government's policies with the soothing unction of caring words that Margaret Thatcher could never bring herself to utter: she was no dissembler. So far, pious talk from David Cameron, Nick Clegg and Iain Duncan Smith about social mobility, fairness, cutting poverty and helping the lowest paid has beguiled the public.

But truth will out – and the Institute for Fiscal Studies is the great truth-teller. As the OECD warned last month, the IFS estimates that government policies will push another 300,000 children into poverty – and this despite Clegg's tiny help of lifting some low-paid people out of tax. Economists at the Department of Work and Pensions must have told Duncan Smith the effect his plans would have but he still affects that curious air of injured saintliness when confronted with the facts.

Meanwhile this week's annual Office for National Statistics income data give the final verdict on the Labour era. Tony Blair and Gordon Brown may have missed their target, but they did cut child poverty by a quarter to the lowest rate in 25 years. Labour did even better for the old, with fewer poor pensioners than for 50 years.

The IFS predictions are no surprise. Unemployment will cause most poverty; then consider the effect of cuts falling on the same families over and over again, with inflation at 5%, child tax credits cut and child benefit frozen. Council tax credit is devolved and no longer ring-fenced; so too is the social fund for emergency help, which councils like Wandsworth can set at any level. Housing benefit caps will force families to move away from family support. The Children's Society estimates universal credit will cut childcare credits not by 10% but by 20%, forcing mothers out of work. The education maintenance allowance means families lose £30 a week for teenagers. Switching benefits, uprating from RPI to CPI, means a loss of 10%, every decade, for ever.

Duncan Smith naturally ignored ONS figures showing Labour's success with poverty, focusing instead on widening inequality on Labour's watch – but his effrontery takes some beating: "This gap between the richest and poorest has accelerated over the last five years despite an astonishing £150bn injected into tax credits alone. The end result has been to make benefit dependency and worklessness inherent to the UK way of life, with the middle- and low-income earners picking up the bill."

In fact the IFS said Labour had succeeded in slowing the growth in inequality with those derided tax credits. The IFS graphs show the great difference between a Labour and a Tory government: Thatcher turbo-charged inequality, Labour pulled it back; but now expect another hyper-rise.

Do enough people care? That is Labour's dilemma, never sure if the 70% who are doing OK or very well are concerned about the 30% non-home owners left behind. Day after day, the government and its supportive press pump out the nonsense that money doesn't matter: poverty is due to lack of aspiration, "immobility" or bad parenting. In the same breath it cuts support for those remedies: youth work, careers advice, teen pregnancy and Sure Start.

Polls show voters are conflicted, depending on what question is asked. They are torn between anger at scroungers, pity for the deserving, fear of being ripped off and a desire to live in a society that treats the less fortunate kindly.

People are disgusted by soaring wealth at the top that's so well-disguised the IFS says it's impossible to measure the true incomes of the top 1%. Anger and compassion vie with each other in most people. The art of left of centre politics is to arouse that natural generosity of spirit and indignation at injustice while reassuring people that their good nature is not abused by idlers or cheats.

Most people never have been Conservative, being unwilling to connect with the nastiness exemplified by Wandsworth. But Conservatism can use its media dominance to spread cynicism. Meanwhile the progressive majority with strong social justice instincts is increasingly and disastrously dispersed among Labour, Lib Dem, nationalist or green parties – partly because Labour seemed to lack authentic, infectious social conviction. Just enough waverers were persuaded by Cameron's display of emotional intelligence to believe he cares about the downtrodden. But this week's figures show the public will soon be confronted with just how bogus he is.

There are no mysteries about poverty, neither its causes nor its cures – which are more jobs, more money, more education and more Sure Start. What works has been studied by researchers for a hundred years. We know it all, and yet we grow closer in inequality to America and further from the rest of Europe. The government's astonishing trajectory of cuts means that, according to Professors Peter Taylor-Gooby and Gerry Stoker, by 2013 public spending will be a lower proportion of GDP in Britain than in the US.

The only great mystery is how to construct a politics where people trust that remedies will work and that wealth can be more fairly shared to the benefit of all. That requires a Labour party that itself believes it wholeheartedly, before it can start persuading others.