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I am reading an economics paper which contains a model with households and banks. In the model, banks are owned by households. The authors make this assumption when they discuss households (section 2.2, page 8):

There is a continuum of risk-neutral households denoted by $h \in [0,1]$ ...

When they move to discussing banks, they make a similar assumption (section 2.3, page 9):

There is a continuum of banks denoted by $b \in [0,1]$ ...

I have seen countless papers in economics with such assumptions regarding continuum of agents. Most of the time, I just gloss over this but I don't really understand what these assumptions mean, what they are useful for and why there are there in the first place. It seems like it's something basic and nobody discusses it but I really want to grasp the meaning and purpose of this assumption.

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Economic models often have a finite number of agents. In such models, agents are commonly named using the positive integers (i.e. 'natural numbers'): so the set of agents is $\{1, 2, ..., n\}$.

Other economic models have an infinite number of agents. For example, the number of agents might be assumed to be the same as the number of positive integers. In that case, the set of agents is $\mathbb{N}=\{1, 2, ...\}$. It is said to have a 'cardinality' (or size) of aleph-null.

Finally, some economic models have even more agents. In the examples you mention, each agent is named using a real number in $[0, 1]$; so the set of agents is the set of real numbers in $[0, 1]$. This set is said to have cardinality aleph-one and can be shown to be 'larger' than the set of natural numbers (in the sense of 'larger' defined by Cantor).

Of course, this is not very realistic -- the justification, presumably, is to make the mathematics easier. In particular, it allows for integration (e.g. to find total consumption in the economy) as opposed to summation.

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