Near-full employment should mean pay rises - but cheap imported labour helps keep it low.
She misses a point. Immigration is not necessary to hold wages down. In the HOS model, free trade in goods alone is sufficient to make British wages converge towards third-world levels. This is the factor price equalization theorem, familiar to any economics student.
The thinking here is simple. Imagine we move from a world of restricted trade to freer trade - say, because India and China join the global economy. The UK will then import more goods which require unskilled labour, such as textiles. Demand for unskilled labour in the UK will therefore fall - and so will wages.
If HOS is right, then, immigration is a red herring. Trade alone holds wages down. Which means that a policy of restricting immigration - even if it's feasible - will do nothing to raise wages of the unskilled. It will merely reduce foreigners' liberty, with no offsetting benefits.
The question is how far the HOS model actually describes reality. An early paper by Adrian Wood (Jstor only, I'm afraid) claimed that it did. This was controversial at the time. But since then, we've seen the emergence of offshoring, which allows us to import low-wage serivces as well as goods, and the boom in Chinese and Indian exports. So maybe HOS is more true now than 10 years ago.
If so, Polly shouldn't be moaning about immigration, but about all international trade.
There's more: Tim gives Polly a fuller fisking.