The combination of Brexit, Covid, and an immigration policy that favours skilled (in practice higher paid) workers has created severe shortages of some lower paid workers. We are seeing that today among lorry drivers, crop pickers, food processors, butchers and other occupations. There are two, seemingly compelling, arguments for why this might be a good thing.
The first is straight supply and demand: a worker shortage means employers raise wages to attract workers. The second is that shortages increase productivity, as firms seek labour saving machines that will reduce their demand for labour. If, like me, you think reducing low pay and increasing low productivity are good things, what is there not to like? Few should begrudge the wages in some of the industries discussed by John Harris or Sarah O’Connor rising.
There are many caveats to these arguments. The most obvious caveat is that it would be far better to have much more modest shortages, which would have the same effect on productivity and wages but which wouldn’t disrupt the wider economy. It should be possible to allow some immigration on a short term basis, without diminishing employers’ knowledge that wages will have to be higher. So far the government has refused to do this. The result is significant damage to the economy as a whole. The rest of this post is about those more modest shortages.
Another caveat is that some of these workers, like those working for the NHS or in social care, are paid for by the state, and it is highly unlikely that their pay will be allowed to rise under this government. A consequence will be permanent shortages unless the government adjusts its immigration rules to allow these workers in.
A third caveat is if workers are working in industries that produce products that can be easily produced abroad, another response to worker shortages is to take production overseas. Production may no longer be able to compete with overseas production if domestic wages rise. But this caveat obviously does not apply to goods that cannot be traded, like social care or, after Brexit, lorry drivers.
A fourth point to never forget is that there are clear losers from ending immigration - immigrants themselves. They are prepared to undertake the considerable costs of working in a foreign country because they are much better off financially from doing so. If you think in global terms rather than from the point of view of natives alone, there are large global benefits to be gained from immigration. One economist argued that barriers to immigration were Trillion Dollar bills on the sidewalk.
The fifth caveat, which is often ignored when talking about immigration, is that we are ignoring one side of what has happened. Immigrants leaving has created labour shortages in many industries, but it has also reduced the overall level of aggregate demand.  We cannot have one without the other. As a result, there is going to be less employment in all the industries unskilled immigrants didn’t work in. Not enough unemployment to fill all the jobs immigrants used to do, but enough to question what the macroeconomic impact of reduced unskilled immigration will be.
This is a crucial question we need to answer before we can accept Larry Elliot’s claim that Brexit has solved a problem (low wages in some industries) created by free movement. We need to look at the economy as a whole, not just workers in a particular industry.
To do this I want to suggest a thought experiment. Imagine a country where (for simplicity) there is no foreign trade. Jobs are either ‘good’ (high wage because they are relatively productive) or bad (low wage that are relatively unproductive). All the good jobs are taken by native workers, while all the low paid jobs are filled by immigrant labour. Although the jobs done by immigrants were badly paid, they still want to do them because they are much better than wages in the countries immigrants came from. Goods produced in the country are consumed by both native and immigrant workers, and both groups have similar preferences.
Now suppose this country sent all the immigrants home. Suddenly the demand for good jobs would fall (because immigrants were not buying them), and there would be many vacancies for bad jobs. Filling the vacancies comes partly from the unemployment caused by high wage workers losing their jobs, but also an increase in the wage of low skilled workers relative to high skilled workers.
Now consider who is better off. Former immigrants are worse off because they have been thrown out of jobs they wanted to do. How about native workers? Whereas before all had been working in high wage jobs, now some of the native workers are working in low paid jobs (although not as low as they once were). As you would expect high paid jobs to be better than low paid jobs, it would seem that no one in the economy is actually better off.
How is that possible? The economy has shrunk because immigrants have left, but the mix of goods produced is unchanged. Without immigrants, some native workers have to be low paid. From native workers’ point of view, some have lost good jobs and are doing worse jobs. Average income per head is unchanged, but whereas before native workers were in the better part because immigrants were in the lower part, now some native workers are in the lower part.
This was the point of an earlier post, which argued that restricting ‘low skilled’ (aka low paid) immigration would mean more native born workers doing lower paid jobs. True, they might not be as low paid as they were when immigrants did them, but I doubt if anyone would call them ‘good jobs’. 
What about low paid jobs getting higher pay stimulating productivity? Again from a macro perspective we need to look at the higher paid jobs, where wages have fallen, reducing the incentive to innovate. I have for some time argued that the best way of increasing productivity is to run the economy hot. Giles Wilkes argues the same in the context of an interesting discussion of worker shortages. In general creating shortages is definitely an inferior way of inspiring productivity growth, because excess demand is a stronger motivator of new investment than higher costs. One of the problems of worker shortages is that it may lead to inflation, which encourages fiscal or monetary policy makers to deflate the economy.
Of course the UK is not like my thought experiment in many ways. In particular, native born workers work alongside many immigrants in low paid jobs. They clearly gain from restricting immigration. But this doesn’t change the result from the thought experiment that on average the position of native workers will get worse. This is a very similar situation to free trade agreements, which often increase overall welfare but where some workers lose out. 
Considerations like this mean that economists tend to think low skilled immigration is a good thing from a purely economic point of view. Focusing on those who gain from reduced immigration is misleading, because it ignores those who lose out. Now there may be good social or political reasons to limit immigration as a whole, but if you want to improve the position of low paid workers a better policy is to increase the minimum wage, or in some industries enforce restrictions in hours worked and ensure better pay for working unsocial hours (bring back and extend the agricultural wages board).
Labour shortages among workers deemed to be unskilled by this government are damaging to the majority of workers in the UK and damaging to those overseas who are prevented from filling the shortages by immigration rules. It will improve the relative position in the income distribution of those working in the industries immigrants have left, but there seem to be better ways of achieving redistribution that do not reduce average incomes at the same time. If you still think the redistribution is worth it, remember you have also made the global distribution of income more unequal.
 Expansionary fiscal or monetary policy are good at dealing with temporary demand shocks, but not permanent ones.
 A simple numerical example may help. Suppose the economy is initially made up of 10 million immigrant workers, and 90 million native workers. Native workers are on average paid 10 units a week, although that is a distribution where some will be getting much less. The immigrants get paid on average 5 units a week (also a distribution). Total national income is equal to 10*5 + 90*10 = 950 units, where 10 is the average income for native workers.
The 10 million immigrants now leave. Immediately aggregate demand falls by 50 units. The demand for native jobs is now 900/950th of its previous level, which makes just over 5% of native workers redundant (just over 5 million). But because preferences are the same and assuming no economies of scale, our new economy must have just under 9 million native workers moving over into the jobs immigrants used to do. So relative wages have to move until we attract just under 4 million native worker to do the work immigrants used to do. These assumptions imply that, once relative prices have adjusted, the new economy is just a scaled down version of the old one, where total income is 950*9/10=855 and average native income has fallen from 10 to 9.5.
 Another thought experiment can illustrate this. Suppose wages are low in a particular industry not because of immigration, but because of foreign competition from low wage economies. Let’s say the industry was textile production. Would we think it was a good idea to create shortages of textiles domestically by restricting imports of textiles? That would lead to more domestic production and therefore almost certainly higher wages and prices in the domestic textile industry. But equally, more people would work in the textile industry, and less in other industries which were more competitive internationally. Productivity in the textile industry might improve, but in the economy as a whole it might fall because workers had been diverted to a relatively low productivity industry.
Economic policy in most countries for decades has been to do the opposite of this thought experiment, by reducing barriers on trade. By specialising in producing the goods we are relatively good at producing, we increase productivity in those industries. Not everyone in the economy gains from this (e.g. domestic textile workers), but average incomes rise.