David Miliband has joined forces with his party leader brother Ed behind Labour plans to deliver a "living wage" of well over £7.20 an hour – rising to more than £8.30 in London – for millions of workers in both the public and the private sectors.

The Miliband brothers, whose relationship has been tense since Ed narrowly defeated David in the 2010 leadership contest, are working closely together on how to make the living wage – as opposed to the lower minimum wage – the new norm and a core economic policy for Labour at the next election.

In a sign that relations are thawing with his brother and with the left of the party, David today reveals radical ideas on the living wage that are being considered by the party in a joint article for the Observer with Dave Prentis, general secretary of the public service union Unison. The former foreign secretary, who has declined an open offer to join the shadow cabinet, says that financial incentives could be offered to local authorities and other employers that require their private contractors to pay the living wage, as a condition of them getting work.

Reflecting studies going on in Labour's policy review, he and Prentis say that local authorities could be rewarded with some of the savings accrued from reductions in the bill for in-work benefits.

As a string of Labour-run local authorities prepare to be recognised as official living wage employers, Miliband and Prentis hail the minimum wage delivered by New Labour in 1998 as a "great success".

But defining a living wage as a rate of pay that allows workers to provide themselves and their families with the "basis of a decent life", they insist that more must be done to lift people out of in-work poverty and address the crisis of falling living standards in low to middle-income families.

"Recent reports have highlighted the growing gap between rich and poor, including an important warning from the Resolution Foundation that by 2020 families will be no better off than they were in 2000," they say. "This problem needs to be mended if we are to make progress as a nation – fair pay for a fair day's work has to be part of that basic deal."

They say that, rather than destroying jobs, paying a living wage above the minimum acts as a boost to the local economies. "There are clear and proven benefits to local communities and businesses, too – people have more money to spend and they spend it locally.

"Big companies such as KPMG and many public sector employers such as the Greater London Authority, where mayor Boris Johnson has carried on the work of Ken Livingstone, the devolved administrations and many large local government and civil service employers are paying the living wage."

The current minimum wage is £6.19 for people aged 21 and over and £4.98 for those aged to 18 to 20.

Labour believes that, rather than imposing the living wage in both the private and public sector, employers should be encouraged to adopt the policy voluntarily after seeing the economic and employment benefits. As well as examining how financial incentives through procurement policies could spread the living wage, Labour's policy review is also looking at the possibility of setting up "Living Wage Zones" in which employers could be offered transitional help as they move from paying the minimum to a living wage.

Another idea being considered is to oblige large employers to make public the proportion of employees on their books who are paid at a rate below the living wage.

The independent Resolution Foundation's Commission on Living Standards last week suggested that the law be changed so that all listed companies on the stock exchange have to make clear how many people are paid less than the living wage.

Both Ed and David Miliband put the living wage at the heart of their campaigns to lead the Labour party in 2010. Before the last election David Cameron also expressed backing, saying it was a "good and attractive idea".

On a visit to Islington in north London on Monday to discuss how Labour councils across Britain have succeeded in implementing the living wage, Ed Miliband is expected to describe the living wage as an idea "whose time has come". "The next step is to help more people, including workers in the private sector, have the dignity of earning a living wage. This is one way we can begin building a One Nation economy where prosperity is fairly shared, because it is only by coming together that we can succeed as a country.

"We're looking at a series of ideas as part of Labour's policy review. It's not about making spending commitments, it's about learning from our experience in local government and listening to businesses who tell us how paying the living wage can be the right thing for their business."

The Institute of Fiscal Studies, in a report two years ago, said that, for every person moved on to the living wage, the saving to the Treasury would be about £1,000 a year. This week the living wage campaign will announce new rates.

The list of Labour councils already paying a living wage includes Birmingham, Oxford, Preston and the London boroughs of Hounslow, Lambeth, Camden, Islington, Lewisham, Southwark and Hackney.