Vince Cable put himself on a collision course with his new ministerial colleague yesterday as he firmly ruled out Tory calls to make it easier for employers to sack staff and defended Liberal Democrat plans for a "wealth tax".

He sought to assert his authority as Business Secretary over his Tory ministers after David Cameron appointed his close ally Michael Fallon to the department in last week's reshuffle – with an instruction to act as a champion for industry.

Mr Cable set himself in opposition to Mr Fallon over the Tories' support for "no-fault dismissal", under which small companies would be allowed to sack staff without explanation. He made clear he was in charge of employment policy and added: "There is job insecurity: we don't want to add to it."

In a newspaper interview yesterday, Mr Fallon criticised the Liberal Democrats' proposal for extra tax on the wealthy. But the Business Secretary told The Andrew Marr Show there was still massive inequality in Britain and said: "Wealthy people could contribute more."

Taking another swipe at his Coalition partners, Mr Cable criticised the bureaucracy surrounding visas for foreign students. Having previously attacked the Home Secretary Theresa May's bid to drive down immigration to "tens of thousands" a year, he said: "There's an enormous amount of red tape around the whole immigration system – permits for workers, visas for people coming from China, students."

With senior Tories now calling for the Coalition to change its policy and support a third runway at Heathrow, Mr Cable, whose South-west London constituency of Twickenham would be affected by such a move, was adamant this will not happen. "There's an absolute political commitment not to expand Heathrow," he said.

Despite the weekend's disagreements, Coalition ministers will seek to bury their differences today as they announce more than 3,000 business regulations are to be scrapped and hundreds of thousands of firms spared health and safety inspections.

Mr Cable and Mr Fallon will put on a united front as they promise a "red-tape blitz" and to protect companies from "compensation culture" claims.

They will announce health and safety inspections will be limited from next year to employers operating in higher-risk areas such as construction and agriculture and to those that have poor safety records. Tomorrow, Mr Cable will set out the Government's business strategy for coming years and the sectors, including the aerospace, automotive and creative industries, that ministers will prioritise for support.