The town will be eventually unequal that is inevitable. Wealth inequality does not primarily stem from some primordial unequal wealth distribution.
Wealth, is in economics the value of net assets you own and hence it is formed from saved or invested (macroeconomically they are the same thing) income accumulated over time.
Even if you do one-off redistribution of wealth people:
- have different skills and productive capacity (and other factors) leading to different income.
- have different saving and spending habits.
Which will ultimately lead to unequal wealth distribution.
Of course the difference in income and in spending habits and saving is further derived from other factors. For example, income from some work might be more stochastic (e.g. prospecting), so luck for example also plays a role. Difference in preferences can be due to different genetics and upbringing.
Hence, unless you actively continuously redistribute wealth, wealth inequality is inevitable.
To not have any wealth inequality administrator would have to ensure everyone is literally a copy of everyone else, has the same preferences, buys the same things, has same saving rate and chooses the same investments. Moreover, everyone would have to do the same job because even 2 identical people might be differently productive in 2 different jobs and make sure all incomes are deterministic or somehow ensure everyone has the same luck. This is clearly unrealistic and outside of the control of administrator.