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?
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2$\begingroup$ What are you reading exactly? please clarify this question $\endgroup$– EconJohn ♦Commented Jul 4, 2023 at 1:11
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1$\begingroup$ Bofinger, P., 2001. Monetary policy: goals, institutions, strategies, and instruments. and Issing, O., 2011. Einführung in die Geldtheorie. for example mention this formula. $\endgroup$– BlankerHansCommented Jul 4, 2023 at 11:48
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1 Answer
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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:
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$$C^n_2= \binom{n}{2}=\frac {n!}{2!(n-2)!}= \frac {n(n-1)(n-2)(n-3)... 2\cdot 1}{2(n-2)(n-3)... 2 \cdot 1}=\frac {n(n-1)}{2}\qquad (1)$$ $\;$
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 $B$).
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You can see:
answered Jul 4, 2023 at 12:23
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1$\begingroup$ 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. $\endgroup$ Commented Jul 4, 2023 at 13:13
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1$\begingroup$ 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'. $\endgroup$ Commented Jul 4, 2023 at 13:30
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1$\begingroup$ This true for $n>2$. Indeed, for $n\neq 1$ we have, (dividing both sides by $n-1$), $\frac {n(n-1)}{2}>n-1 \iff n/2>1 \iff n>2$ $\endgroup$ Commented Jul 5, 2023 at 16:09
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1$\begingroup$ Oh now its obvious for me. Thank you very much! I think this is all. Have a nice day :) $\endgroup$ Commented Jul 5, 2023 at 19:27