The experience in Europe is that there is a certain amount of cross-border shopping when there are countries that have very significantly different rates of tax. But it's usually a very limited phenomenon, because the difference in value-added tax rates between neighbouring countries in Europe is usually fairly small, and therefore you would need to be planning to purchase a great deal for it to finance any significant sort of trip. I think, in fact, the greatest incidence of cross-border shopping is not with respect to differences in value-added tax rates, but differences in excise duty rates.
On things such as alcohol and tobacco, there's considerable cross-border shopping between, for example, the United Kingdom and France that is driven by the differences in excise duty. The VAT difference would go the other way, but I don't think French people go to England because of the 2% difference in the VAT rate.