Much of the discussion of the recent British vote to leave the European Union takes it for granted that the result will be less free trade for the U.K. While that is possible, so is the opposite result. Britain can still negotiate a free trade agreement with the EU, as several non EU countries have done, assuming both sides want it. And leaving the EU leaves Britain free to negotiate free trade agreements with other countries, most obviously the Commonwealth.
The critical issues are the positions of the U.K. government and its potential trading partners, including the EU. Many who voted for Brexit were motivated by a desire to reduce trade and/or immigration, but not all. The winning coalition seems to have included both protectionists and free traders. The free traders who voted for Brexit plus the free traders who voted against it might well add up to a majority.
For those who supported free trade, the objection to the EU was the rest of the package, in particular extensive regulation. Many people take it for granted that if you have free trade such regulation is needed to prevent countries from cheating, regulating the national market in ways that favor their producers. That argument assumes that national governments want to cheat on free trade. It treats a free trade agreement as a deal where each country gives up something it values, its own trade restrictions, in exchange for the other country doing the same.
Much talk about trade views it that way. Politically speaking that view is correct, since trade restrictions are a way in which politicians can benefit well organized producer groups in exchange for their political support. Economically speaking, however, that view is false. The gain from protecting U.K. manufacturers from foreign competition comes at the cost of their customers and U.K. export industries.
Unilateral free trade, the policy of England in the 19th century and Hong Kong in the 20th, produces a net benefit for the inhabitants of the country that adopts it, quite aside from any benefits to their trading partners. From the standpoint of the welfare of the citizens rather than their rulers, the usual trade negotiation consists of each side offering to stop shooting itself in the foot in exchange for the other side doing the same. If governments engaged in trade negotiations were trying to maximize the welfare of their inhabitants, there would be no need for either tariffs or agreements on regulation, since there would be no incentive for the governments to use regulation to cheat on trade agreements.
If supporters of free trade in the U.K. and potential trading partners are sufficiently numerous and sufficiently well informed, Brexit should lead to an increase in free trade. If they are numerous but poorly informed, believe that the benefit comes from a trading partner abandoning its restrictions, it still might lead to an increase. We will have to wait and see.