Recall our assumption of rationality? Yes, by rationality we mean people’s preference relations are complete and transitive. In order to talk about demand, we now need two more (sorry, guys) assumptions, namely desirability and convexity.
Desirability simply means you prefer more to less. It is represented by the property of monotonicity that means if I ask you to choose between 3.1 apples and 3 apples given the same constraints, you opt for the former. Wait a minute, you say. What if it is not an apple, but something bad, like trash? Well, simply modify the offer statement: 3.1-unit reduction of trash and 3-unit reduction of trash.1
Convexity (of preference relation) means your willingness to give up a unit of a particular good in order to get another unit of different good in exchange given your constraints is increasing the more you have the former and the less the latter. (Note: our definition of convexity in the consumption and budget set still hold). This is called diminishing marginal rate of substitution.
So far we have been talking about preference. How do we really analyze it? We usually use a tool called utility function. This is simply a means to express how you would respond when facing a set of goods given the prices and your income. In order for us to represent preference relation with a utility function, we need (oh, shoot!) to assume continuity. It says, if you prefer 1 apple to 1 orange, 2 apples to 2 oranges, you can’t suddenly, out of blue, prefer 3 apples to 3 oranges.2
How do we put the utility function into use, then? By solving a maximization problem. That is, we suppose an individual is trying to maximize his satisfaction (i.e. utility) given his choice set and budget set. By maximizing we mean, he will use up all his income to consume the goods of interest (saving can be a form of a good; I see your eyebrows rising). We would continue on this.
1 But hold on, you say. You’re craving for ice cream. I give you one cone, you ask two. I give you two and offer a third. You start to look less eager, but you still take it. I offer a fourth; you give me a no-thanks-I’m-fine. This is called diminishing marginal utility. It is not contradictory to the monotonicity assumption of the preference relation. You can still prefer more to less, but the additional satisfaction the ‘more’ gives you is becoming less and less as the quantity grows. Four cones of ice cream are too much already for you; you prefer three. Remember that we have constraints that limit preference? Yes, one of the physical constraints is quantity that might be bound to taste (or well, your stomach capacity). In this case we can say that your set of ice cream is limited to three. But don’t take this anecdote very seriously; rather, we usually go around such problem with a weaker restriction called local non-satiation – you’re never satisfied, 'locally'. Meaning, you can still prefer 2.999999999999999-unit of apples to 3 apples and at the same time, prefer 3.000000000000001-unit of apples to 3 apples. But let’s not dwell into this technical necessity. We’re safe for now.
2 Again, do not take the numbers too seriously. It is the order that matters.