The home secretary, Theresa May, has defended her plan to expel international students from Britain after graduation, despite criticism from Sir James Dyson that the policy is a short-term vote winner that will damage the economy in the long term.

When May was pressed in the Commons by Conservative and Labour MPs on her response to Dyson, she said something had to be done about the situation in which 121,000 students from outside Europe came to study in Britain each year but only 51,000 left.

May told MPs that if nothing was done, there would be an estimated 600,000 non-EU overseas students coming to Britain each year by the 2020s.

Dyson, one of Britain’s most prominent inventors and business leaders, writing in the Guardian on Monday, criticised May’s manifesto pledge to require overseas students to leave the country and apply for a new visa if they wanted to work in Britain as “a short-term vote winner that leads to long-term economic decline”.

His criticism echoed concerns about growing shortages of qualified scientists and engineers in Britain from science and industry leaders.

May’s plan also came under fire from fellow Conservative MPs as well as suffering repeated attacks from the Labour benches.

Sir Peter Luff, the Conservative MP for Mid Worcestershire, asked for assurances that “efforts to curb immigration won’t harm our higher education system or prevent British industries accessing the skills they can only find internationally as a result of any new restrictions on visas for graduates of British universities?”

A second Conservative MP, James Gray, also demanded to know the impact of the plan on businesses such as Dyson – based in his North Wiltshire constituency – which relied on scientists and engineers from overseas. “How would you answer my constituent Sir James Dyson, who argues that if your latest remarks about automatically sending all students home on completion of their studies were to be taken literally then that would have dire consequences for businesses like his who rely on engineers, scientists and others from overseas?”

The shadow home secretary, Yvette Cooper, also attacked the plan to expel overseas graduates, demanding to know if May was proud of targeting postgraduates: “Does she believe that overseas graduates should all have to return home before they can apply for a highly skilled job in British science or the NHS? Yes or no.”

Cooper suggested the plan had been “cooked up” by May’s special advisers and joked that it could be the reason why they had been banned from the Conservative candidates’ list.

But May defended her plan, saying her overseas student migration policy was based on attracting the “brightest and best” from around the world and that student visa applications had gone up in the year to September 2014.

“But we also have to recognise that the latest surveys also show that 121,000 students came in from overseas while only 51,000 left in that year and that by the 2020s we will see 600,000 overseas students each year in this country,” she said.