Question

in equilibrium, what is the relationship between MRS and marginal utilities

Answer 1

Marginal utility is the added utility (benefit) of some small amount extra of some good. For instance, how much happier you are if you get an extra apple, or how much happiness you loose if you have to give up your apple today (negative amount extra).

The marginal rate of substitution measures how easy it is to replace one good with another while not changing your utility level at the margin. So if you like an extra banana only half as much as you would like an extra apple (assumption), i could take 1 apple from you and give you 2 bananas (or the other way around) and you would be as happy before as after the trade. Note that technically this means that the marginale rate of substitution between apples and bananas is 2 in this example, it is the ratio of the marginal utility of apples and the marginal utility of bananas.

For this ratio to be part of a price equilibrium on the market, the ratios of prices of apples and bananas should also be 2. If the price of an apple would be lower than twice the price of bananas, you could sell 2 bananas, buy 1 apple and be equally happy with your fruit, but have more money.