What is the rule relating the ratio of marginal utility to prices of two goods at the optimal choice? explain why, if this rule does not hold, the choice cannot be utility-maximizing.

The optimality condition in the consumer's choice between two goods is the following

The ratio of the marginal utilities is called the Marginal Rate of Substitution between goods x and y, and it arises from the preferences of the consumer towards the two goods. The MRS establishes the rate at which this consumer can give up a certain quantity of good x in exchange for another amount good y, while maintaining the same level of utility (it establishes the units of good x that would satisfy the consumer if he had to reject a certain number of units of y, given his personal preferences).

When MRS is equaled to the price ratio, it allows to calculate the bundle of goods x and y, that provides the maximum utility to this consumer given the market prices of x and y, which means that the solution provides the highest-utility bundle that he can afford.

If the rule does not hold, the choice might be either suboptimal or impossible. When the result is suboptimal, the consumer acquires a bundle that he can afford but that is not the one yielding the maximum utility. If the result obtained is impossible, the consumer cannot afford the bundle, even tough it yields very high utility to him.

this is a pretty broad question and can be a pretty lengthy answer. since 1900, women have gained a lot more rights and have become active in the workforce. with women being in the workforce, men play a larger role in family life. many family tasks are shared between men and women, whereas, women used to be the sole family caretakers.

that should get you started at least.