A small rise in the minimum wage in Britain’s main city regions would encourage employers to deploy workers more productively and help boost local economies by more than £1bn, says a thinktank.

According to the study, by the Smith Institute, employers need the spur of a higher minimum wage to shake them out of a spiral of low productivity and low growth that depresses company revenues and has trapped about 2 million workers on the current minimum wage of £7.38 an hour, or £7.83 for those aged 25 years or older.

The leftwing thinktank said the plan – which would be voluntary – would involve increasing rates to the voluntary living wage, which is run by the Living Wage Foundation and recommends employers pay at least £10.20 an hour in London and £8.75 in the rest of the country.

Employers including the drug-maker GlaxoSmithKline, Unilever, the retailer Lush and the accountants KPMG, have already adopted the voluntary wage rates with their own employees. They also make a commitment to force suppliers and clients to pay the higher minimum wage.

More than half a million workers would secure an average annual pay rise of more than £1,700, and UK city regions would benefit from a £560m boost that could top £1bn if the government handed much of the extra tax generated to city mayors and the newly created combined authorities.

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The institute, which is named after the former Labour leader John Smith, said a “local living wage dividend” could increase to £1.1bn, when taking into account economic multipliers such as increased local spending by low-paid workers and local authority spending to improve the infrastructure of the region.

The Sheffield city region, which is run by the Labour mayor, Dan Jarvis, would see the average annual salary of low-paid workers rise by just over £1,000. Taxes receipts from the wage boost would add £13m to the Treasury’s coffers.

Jarvis said: “It is concerning that Sheffield city region has the highest percentage of employees earning below the voluntary living wage of all the regions considered in the report. Ensuring that people are paid a proper wage which meets the cost of living is vital for residents and good for the economy.”

The report, which was funded by the Living Wage Foundation, highlights the role that leading local public and private-sector employers such as universities, hospitals, football clubs and city airports, can play in providing leadership on the living wage.

While most jobs created in the last two years have been full-time and the numbers of zero-hour contract jobs have fallen slightly, wage rates have risen only slowly. Britain’s productivity also remains well below France and Germany, where workers can achieve the same output as UK workers in two-thirds of the time.

The report calls on metro mayors and local authorities to work with these key “anchor institutions” to drive living wage take-up in their towns, cities and regions, and to integrate the living wage into their economic development strategies.

Paul Hunter, the deputy director of the Smith Institute, said: “Big employers often like to talk about the positive role they play in their local community. One way that they can go beyond the warm words is to pay their staff the living wage and demand their suppliers do the same.

“This is not just about good corporate citizenship. Evidence shows workers paid fairly are more productive. And, as our research shows, the living wage can also provide a boost for the local economy on which established employers are dependent.”