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Who would suffer most from UK / EU trade restrictions?

A so called "Hard Brexit" would damage both the UK and the rest of the EU. However the proportionate impact on the UK would be far greater.


18 October 2016

Since the EU Referendum result, many opponents of Britain's membership of the European Union have become increasingly vocal about the need for what is referred to as a "Hard Brexit." That is a departure from the EU which also entails departure from the EU's "single market".

In my opinion such advocates give insufficient weight to the economic implications to the UK from leaving the single market. Accordingly I wrote a short piece "Ultra-hard Brexit – a mathematical perspective" for the Conservative Home website. You can read it below.

Mohammed Amin: Ultra-hard Brexit – a mathematical perspective


Reducing trade between the UK and the rest of the EU (“rEU”) would be bad for both. This should be obvious to anyone who has ever opened an economics textbook.

However, I have become dismayed by the number of UK commentators who contend that rEU has more to lose from trade restrictions than does the UK.

Mathematical thinking

When thinking about an issue, it often helps to consider the most extreme cases. For example, consider a function of the form y=f(x).

Does f(x) cross the X axis? In other words, is there a value of x, say x’, for which f(x’)=0? For complicated functions, finding such a value, x’, can be extremely difficult, perhaps even impossible. However, you don’t need to do that to answer the question. You only need to be able to prove three things:

  1. f(x) is continuous. This means that vanishingly small changes in x only give rise to vanishingly small changes in y.
  2. For some value of x, no matter how large (say x= 1 trillion), f(x) is greater than zero.
  3. For some value of x, no matter how small (say x= minus 1 trillion), f(x) is less than zero.

The logical consequence of the above three intermediate propositions is that there must be some value of x, x’, for which f(x’)=0 even if one cannot calculate the value of x’.


As Brexit becomes more extreme, the UK suffers more (in relation to the size of the UK economy and the per capita welfare of its population) than does rEU.


The economic impact of Brexit is essentially a continuous variable. If Brexit comprises a tiny change to the application of some minor EU rule, with nothing else changing, obviously the economic impact is also tiny. The greater the degree of Brexit hardness, the greater the economic impact.

Let us consider the hardest possible form of Brexit. While there are no foreseeable circumstances in which this would happen, this would be a Brexit where after implementation:

To mix metaphors, one could say that there was an Iron Curtain down the English Channel. Actually the above restrictions are much fiercer than those of the Cold War.

In this scenario, obviously both the UK and rEU would suffer greatly. The key question is who suffers more in relation to their economy or on a per capita basis.

One obvious point is that most UK citizens would need to find new holiday destinations! Unless it was going to be “staycations” all the time, holidays would have to be taken in North or South America, Africa or Asia after taking a somewhat circuitous flight route. While Norway and Iceland are not within rEU, their position in the EEA would presumably require them to exclude UK tourists in the same manner as rEU.

Obviously, the GDP of both parties would decline from losing the ability to export to each other. However, the rEU countries would continue to trade within their residual internal market which would have shrunk from 508 million consumers to about 443 million due to the loss of the UK. From a UK perspective, its internal market would have shrunk from 508 million to 65 million.

The 2015 GDP of the entire EU was about €14.6 trillion, and that of the UK about €2.6 trillion, so rEU’s GDP was about €12 trillion. The Office for National Statistics reports that in 2015 the UK’s exports to rEU were about £223 billion (€254 billion at an FX rate of 1.14) while rEU’s exports to the UK were about £291 billion (€ 332 billion at the 1.14 FX rate).

The direct impact of losing all of these exports would reduce rEU’s GDP by about 2.8 per cent (332×100/12,000) and reduce the UK’s GDP by about 9.7 per cent (254×100/2600).

Quod erat demonstrandum.

The above are all primary effects of ultra-hard Brexit as defined. In practice there would be secondary effects. For example if Chinese tourists were unable to travel between the UK and rEU, I would expect them to make rEU the choice for their first visit to this part of the world, since rEU in aggregate has far more tourist attractions than does the UK.

Policy implications

A deal is an agreement reached between two parties after negotiation. What makes deals possible is that the two parties value differently the items that they are going to exchange.

The fundamental negotiating goal of the UK should be to agree a form of Brexit that causes as little damage to the UK economy as possible, and inconveniences UK citizens as little as possible. Conversely, the obvious negotiating goal of rEU is to reach an agreement with the UK that preserves the integrity of rEU and does not encourage any other countries to seek an apparently more attractive life outside the EU. How to handle the negotiations is obviously a challenge for our country’s negotiating team.


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