  Consider an exchange mechanism which accepts diversified offers of various
commodities and redistributes everything it receives. We impose certain
conditions of fairness and convenience on such a mechanism and show that it
admits unique prices, which equalize the value of offers and returns for each
individual.
  We next define the complexity of a mechanism in terms of certain integers
$\tau_{ij},\pi_{ij}$ and $k_{i}$ that represent the time required to exchange
$i$ for $j$, the difficulty in determining the exchange ratio, and the
dimension of the message space. We show that there are a finite number of
minimally complex mechanisms, in each of which all trade is conducted through
markets for commodity pairs.
  Finally we consider minimal mechanisms with smallest worst-case complexities
$\tau=\max\tau_{ij}$ and $\pi=\max\pi_{ij}$. For $m>3$ commodities, there are
precisely three such mechanisms, one of which has a distinguished commodity --
the money -- that serves as the sole medium of exchange. As $m\rightarrow
\infty$ the money mechanism is the only one with bounded $\left( \pi
,\tau\right) $.
