Quizzed by MPs, the head of the Prosperity UK think-tank denied the annual bill would be £13bn, but acknowledged: “We don’t have a figure.”
Mr Johnson has made so-called “alternative arrangements” to solve the border controversy a key plank of his promise to renegotiate Theresa May’s divorce deal.
On Monday, the Tory leadership favourite insisted there were “abundant” technical solutions, but “no single magic bullet”, to settle the issue.
It would also require the EU opening up computer systems, to prevent VAT fraud, and argues for the Irish Republic to shifts food and animal standards from EU to UK rules – both of which are likely to be opposed.
The report, published on Monday, is seen as crucial to Mr Johnson’s attempts to strike a new divorce deal, taking out the flashpoint Irish ‘backstop’, which would tie the UK to EU customs rules.
Mr Singham admitted “there are costs” to the blueprint and argued there would need to be a “transitional adjustment fun for small traders”.
He defended the absence of a cost, telling the committee: “Our job, in the last four weeks, was to come up with the arrangements that we think will makes sense.”
And he added: “Many of the things we are suggesting here do not impose a significant cost because they are electronic procedures, many of which people are already doing.”
Mr Singham pointed to a “trusted trader” proposal, which would remove the need for many checks, adding: “It’s not just a question of imposed costs, there are a lot of savings.”
But he was warned, by Simon Hart, the committee’s Conservative chairman, that the full report – due next month – would be undermined if it failed to include costings.
“I think it could potentially weaken the efficacy of what you are saying unless there is some pounds, shillings and pence,” Mr Hart said.
The interim report, co-chaired by Conservative MPs Nicky Morgan and Greg Hands, argues there is no need for the kind of futuristic high-tech systems branded “unicorns” by critics.
But Owen Smith, the former Labour shadow Northern Ireland Secretary said the “vague” proposals amounted to no more than “a desperate attempt to make a square peg fit into a round hole in order to try and hold the warring factions of the Conservative Party together”.
He said it was clear that the solutions did not have consent from either side of the border.
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