In a world without trade, what would happen to the cost that American consumers would have to pay for Logitech products?
This is kind of a difficult question to answer because of the way that it is phrased. There are many different kinds of trade, and you would have to know which ones were being eliminated. After all, buying things at the store is simply trading money for products. If you eliminated all trade, you wouldn't be able to get the speakers because noone would want your money. Noone would want to make the speakers because they wouldn't want to trade their time for cash. So you see, without all forms of trade the question isn't how much a person would pay for Logitech's products, it's "how would a person even get Logitech's products?"
But I am going to assume you mean something more limited than that. I am going to assume you mean "global trade between countries." In this case, people would still accept money for labor and money for the products they produced, just not products produced somewhere other than their home country. In this case, Americans would have to pay a lot more for Logitech products than they do now. Logitech would not have access to inexpensive foreign production facilities to assemble their items and would have to rely on parts manufactured and assembled in the US. There may not be great supplies of all the raw materials required here, and so their might be a premium for certain components (because you can't just import, say, the oil required to make the plastic housings.) This would drive prices up considerably.
The good thing about this would be that products would become more "dear" to people and our society would have to switch to a more self-sufficient one. We would have to keep things longer and pay more attention to the recycling of our resources. Of course, you'd never get a good French wine or Persian rug, but hey, we would always have the Big Mac and Coca Cola!
"World without trade" is a major change from the existing situation in the world. If it was possible to make this change, its impact will be disastrous not only for America but for the entire world. In such major changes it is difficult to predict or estimate the prices of the product of one company like Logitech in one market like America. For all that we know, Logitech may be one of the thousands (or even tens of thousands) of the companies that will be forced to close down, if world trade was to close down completely.
World trade refers to buying and selling across national borders. World trade adds to efficiency and productivity of all the countries through the mechanism of division of labour working at most macro levels. It enable a country to produce and export what it can manufacture most economically, and in return import other things from countries that find it advantageous to export them. In this way, each country is able to get a total package of goods and services that is better than what it will be able to produce by itself.
A world without trade will make every country in the world a poorer country, and reduce the average income and prosperity of people in every country. This means that the total production as well as demand of every type of product, including Logitech products, will decline. The actual price of product will also depend on other factors like rate of inflation. But the price of most of the products in real terms will definitely tend to rise because of lower efficiency and higher costs.