(Sky News) The tax is a government health drive aimed at encouraging people to buy less foods containing saturated fats.
In practice this means an increase in the price of a pack of butter by 2.20 kroner (25p) to more than 18 kroner (£2.07).
Christian Jensen, who runs a supermarket in Copenhagen, said: "It has been a chaotic week with a lot of empty shelves. People have been filling their freezers.
"But actually I do not think the tax will make that much difference. If people want to buy a cake, they will buy it. But right now they are saving money."
The new tax affects all foods that contain saturated fats, from butter and dairy products and meats to oils and pre-cooked foods.
Denmark's Confederation of Industries (DI) says the tax will be a bureaucratic nightmare for producers and outlets.
A DI spokeswoman said: "The way that this has been put together is an administrative nightmare, and I doubt whether it will give better health."
"As far as we have been able to determine, Denmark is the first country in the world to introduce a fat tax" but we know that other countries are following us closely and have their own plans, she added.