You're going to have to explain this one to me. Why would productivity increase (I assume when you say it will change you mean increase) as the result of having fewer people available to fill each unskilled vacancy?
Its not so simple. Skilled immigration tends to increase productivity, and consequently wages, as it is easier for companies to fill positions like you say.
The challenges come when you have a large labour supply, and particularly when you have a large supply on low skilled labour.
When company a company wants to scale up or become more efficient, it has a choice between investing in skills of existing staff; getting new technology; other productivity improvements such as process overhaul; or hiring more staff to pick up the slack.
A large labour supply means that this equation is more often slanted towards hiring a new person.
This has the effect of companies delaying investments that would improve productivity.
This effect is particularly acute when talking about low skilled labour, as the labour is cheaper in general, and the technology investments tends to be very expensive in comparison with the cost of staff.
This means that a large amount of immigration, and unskilled immigration in particular, depresses productivity growth and wages. This suppression is not uniform, it effects those on low incomes much more than those on higher incomes - who may in fact still be experiencing the positive effects of immigration.
This paper analyses the effect immigration has on the wages of native workers. Unlike most previous work, we estimate wage effects along the distribution of native wages. We derive a flexible empirical strategy that does not rely on pre-allocating immigrants to particular skill groups. In our empirical analysis, we demonstrate that immigrants downgrade considerably upon arrival. As for the effects on native wages, we find a pattern of effects whereby immigration depresses wages below the 20th percentile of the wage distribution but leads to slight wage increases in the upper part of the wage distribution. This pattern mirrors the evidence on the location of immigrants in the wage distribution. We suggest that possible explanations for the overall slightly positive effect on native wages, besides standard immigration surplus arguments, could involve deviations of immigrant remuneration from contribution to production either because of initial mismatch or immigrant downgrading.
http://restud.oxfordjournals.org/content/80/1/145.abstract
Taking a macro view of wages doesn't tell the whole picture.