# The number of relative prices without money

So recently I am reading a lot of monetary literature and this comes up when they talk about the unit of account function of money:

$$(n / 2)(n-1)$$

Can we prove this or where does it come from and what is the intuition here?

– EconJohn
Jul 4, 2023 at 1:11
• Bofinger, P., 2001. Monetary policy: goals, institutions, strategies, and instruments. and Issing, O., 2011. Einführung in die Geldtheorie. for example mention this formula. Jul 4, 2023 at 11:48
• I think more usually in mathematics it is written as n(n-1)/2 Jul 24, 2023 at 12:00

The formula you wrote comes from the mathematical concept of combination.

In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter: you take a certain number of objects from a set, without caring of the order of the selection.

The number of combinations of $$n$$ things taken $$k$$ at a time, without repetition (each element is taken only once), can be indicated as $$C^n_k$$, and it is equal to the so-called binomial coefficient, indicated as $$\binom{n}{k}$$:

$$C^n_k=\binom{n}{k}\equiv \frac {n!}{k!(n-k)!}$$

In our case, of an exchange economy with $$n$$ goods, the number of markets (the exchange spots of two goods) is given by the combinations of $$n$$ objects taken two at a time, that is:

$$\;$$

$$C^n_2= \binom{n}{2}=\frac {n!}{2!(n-2)!}= \frac {n(n-1)(n-2)(n-3)... 2\cdot 1}{2(n-2)(n-3)... 2 \cdot 1}=\frac {n(n-1)}{2}\qquad (1)$$ $$\;$$

If you consider relative prices regardless of the order of the goods, that is, given two goods $$A$$ and $$B$$, you consider $$\frac{p_A}{p_B}$$ and $$\frac{p_B}{p_A}$$ to be the same price, the number of relative prices is also given by the formula $$(1)$$ ($$p_A$$ and $$p_B$$ are the nominal prices of $$A$$ and $$B$$).

$$\;$$

You can see:

https://en.wikipedia.org/wiki/Combination

https://en.wikipedia.org/wiki/Binomial_coefficient

• You are welcome! Jul 4, 2023 at 12:46
• I do have one more question: If we consider an exchange economy with $n$ goods and $one$ universal used medium of exchange in which every other good is priced can we formulate it also with the binomial coefficient (what would $k$ be in this case)? Intuitively I get that it is $n-1$ because every other good is priced in that specific good beside the good it self. Jul 4, 2023 at 13:13
• I agree with you, if you are saying that the only prices you consider are $\frac{p_i}{p_U}$, $i=1,...,n-1$ (the goods except the universal medium), where $p_U$ is the price (arbitrary) of the 'universal medium of exchange'. Jul 4, 2023 at 13:30
• This true for $n>2$. Indeed, for $n\neq 1$ we have, (dividing both sides by $n-1$), $\frac {n(n-1)}{2}>n-1 \iff n/2>1 \iff n>2$ Jul 5, 2023 at 16:09
• Oh now its obvious for me. Thank you very much! I think this is all. Have a nice day :) Jul 5, 2023 at 19:27