The Hungarian mathematician George Polya had a rule for dealing with anything that looks intractable: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” This applies in politics too. One of the hardest problems Labour faces is how to rebuild a reputation for economic competence when the Conservatives have successfully painted the opposition as unrepentant budget wastrels. To concede the point feels like surrender on enemy terms: accepting a mendacious account of what caused the financial crisis; signing up to cruel cuts. Yet to dispute the point looks like denial of political reality – blaming the voters instead of listening to them.

So, following Polya’s maxim, how does Labour find a solvable problem within the unsolvable one? The trick is to take the Tories (momentarily) out of the equation. Posit super-benign circumstances in which there is no blame for the great crash and even George Osborne says all fiscal paths are equally valid. In this fantasy land the left would still have to confront failings of the Labour state that cannot simply be cast as inadequate funding or corruption by private enterprise.

There is a vast political space between monolithic state control and a privatised market free-for-all

The period 1997-2010 saw much-needed investment in public services, but the social and economic return on that capital dwindled over time. When the question of whether Labour spent too much is removed, the question of how effectively the money was being spent remains.

Tony Blair and Gordon Brown fostered a culture in which the obvious political remedy to every social ill was a task-force in the cabinet office, a tsar and a juicy budget. Some of that money was wisely used but inevitably some was misspent. Ministers rarely displayed indignation, or even curiosity, on the public’s behalf about the portion that might be wasted.

This was bad for the left because it eroded confidence in centrally administered tax-and-spend as a natural expression of solidarity in the collective national interest. It fertilised soil in which the right could later sow suspicion of public spending as confiscation from the virtuous for handouts to the indolent.

Labour’s tendency to measure social improvement in money allocations was also intellectually stultifying. Cash is a palliative treatment for the symptoms of social dysfunction, and compensation for people who find themselves on the losing side of globalisation. It doesn’t always equip them to cope without ongoing state support, which might not be doing them as many favours as ministers hope. Housing benefit, for example, became a Treasury subsidy to landlords, supporting high rents and passing only notionally through the hands of tenants in need of better homes. Tax credits as consolation for low pay became an anaesthetic that numbed the Treasury where it should have felt the need to do something about stagnating incomes.

Such questions are lost in debates about the size of the state measured by public spending as a proportion of national income. Osborne’s plans close in on 36%, a postwar low, and the shrinkage will feel more severe in many areas because expensive items such as pensions and health are protected. But Labour cannot simply campaign for a higher number. Its task is to describe something better, not always bigger. It must advertise a state that pays for itself by preventing costly social problems down the road and by nurturing a happier, more secure, healthier society that yields a more productive economy.

Some thought has already gone into what this might entail, much of it by New Labour types. It is a myth on the left that “Blairites” are wedded to neoliberal dogmas and laissez-faire economics. It was Peter Mandelson in 2008 who called for a new “industrial activism” led by a “more capable, strategic state” in response to the financial crisis. That analysis evolved into a growth strategy developed by Andrew Adonis. His work was studied in Osborne’s Treasury but neglected in Ed Miliband’s election campaign.

In 2010 James Purnell was arguing that New Labour had been “too hands-off with the market and too hands-on with the state” and that the two elements were linked. Market failure creates social breakdown that drives demand for public spending. Miliband embraced only the first half of Purnell’s argument; he was never excited by state reform that would have balanced the message. That expressed a failure of imagination, not a lack of resources.

Jon Cruddas, who led Miliband’s policy review, was not squeamish when talking about dysfunction in the centralised model of public services. He consistently talked about relationships, community, sharing and reciprocity as the basis for a more effective social policy. In June 2014, the IPPR thinktank published The Condition of Britain – a 280-page tome packed with policy ideas for a more equal and prosperous society, based on the premise that taxpayers’ money is a finite resource best spent in ways that sustain public consent: a more contributory element to the welfare system; preferring investment in social infrastructure (such as Sure Start centres) to cash benefits; trusting local authorities to choose their priorities.

To develop that line of thinking, Labour has to rediscover two concepts that some on the left find challenging. First, differential regional outcomes needn’t be denounced as wicked “postcode lotteries”. They might reflect the discovery of a practice that works and can be shared. Second, government doesn’t have to own the whole supply chain of social intervention. It can commission charities, co-operatives, social enterprises and, yes, the private sector to find solutions to public policy problems. There is a vast political space between the poles of monolithic state control and a privatised, outsourced market free-for-all.

From that place Labour then challenges the Tories. The opposition must sound confident that it has a recipe for delivering better services, consistent with the left’s historic ambitions for equality and social justice, yet baked in with an understanding that budgets are limited and public consent is fragile. The contest is then no longer framed in terms of a wasteful Labour state against a thrifty Tory one, but as a modern, innovative, inclusive Labour state, supporting a resilient and productive society, as opposed to a divisive, neglected, mean Tory one that ends up costing us all more in the long run.

Imagining the smarter state is not hard. It is, in Polya’s method, the problem Labour can solve. Fed back into the current political equation it then also helps solve the harder problem of regaining public trust as a potential government.