Does the price of food being more expensive mean you are going to eat less and be skinny?
No doubt it is a contributing factor.
prices are higher too because quality is higher, because food standards are fairly strict.
For example, eggs are more expensive than in the US. That is because the law demands higher animal welfare standards. In turn, we don't use chemicals to disinfect eggs before packing them. this in turn keeps the protective layer on the outside of the egg intact. This in turn means that eggs can be stored at room temperature for lengthy periods, and we don't have any concern about eating raw egg in e.g. our deserts without having to buy pasteurized eggwhite etc. Raw egg is used quite a lot.
One of the arguments in the Brexit negotiations is the fear that the UK will drop standards and import chlorinated chicken from the US. The problem isn't so much the chlorination itself, which is done in the EU with e.g. lettuce. The problem is that US chicken relies on chlorination to keep it edible until consumption, whereas EU chicken relies on the animals being in good health from the moment they hatch to the moment they land on your plate.
So obviously, we eat less. When we go eat in the diner at Charlie's place, I cannot get over the fact that my royal breakfast costs less than 10$. In Belgium that would be a 18 euro breakfast at least. Financial reality dictates that we limit portion sizes.
The other thing is that by law, country of original and every single additive needs to be on the label, which in turn makes company be careful with additives. So there's a lot less things in our food than might have adverse effects.