  We analyze an ideal gas like models of a trading market. We propose a new fit
for the money distribution in the fixed or uniform saving market. For the
marketwith quenched random saving factors for its agents we show that the
steady state income ($m$) distribution $P(m)$ in the model has a power law tail
with Pareto index $\nu$ exactly equal to unity, confirming the earlier
numerical studies on this model. We analyze the distribution of mutual money
difference and also develop a master equation for the time development of
$P(m)$. Precise solutions are then obtained in some special cases.

Physics and Society;Statistical Mechanics;Statistical Finance