The restaurant may or may not distribute the percentage increase in costs evenly across all items being sold. Some items may not receive a markup for marketing purposes. Therefore the dollar menu items could remain even if those items are sold at a net loss. However, the usual trick to avoid losing money is to reduce portion sizes.
There aren't that many places selling items at $1 each though (mostly breakfast items) and they are analogous to free bread that is offered by more upscale restaurants. The $1 items are usually small portions, low quality, and require very little prep time. The businesses use the $1 items to lure in customers so that they can sale other items that are more profitable. Even if you only buy the $1 item on this visit, the restaurant owners know there is the opportunity to be entice the customers to eat something that will generate more profit in the future.
Speaking from past experience when there were increases in the minimum wage, I noted that the prices of hamburgers/sandwiches normally went up about 50 cents when the minimum wage increased. A soft drink would cost a quarter more. The price of fries/chips would also increase by a quarter. The typical burger, fries and drink meal would go up by $1.
I think that the question is whether the price increase needed to support a $1/hour in the minimum wage is closer to 5 cents per item or 50 cents per item. I suspect that it will lie at the higher end of the scale based upon past experience. I realize that I should have been clearer that the 50 cent increase was applicable to the main entree and that other items will see smaller increases.
= new reply since forum marked as read