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A growing body of empirical evidence suggests that the dynamics of wealthwithin a population tends to be non-ergodic, even after rescaling theindividual wealth with the population average. Despite these discoveries, theway in which non-ergodicity manifests itself in models of economic interactionsremains an open issue. Here, we shed valuable insight on these properties bystudying the non-ergodicity of the population average wealth in a simple modelfor wealth dynamics in a growing and reallocating economy called Reallocatinggeometric Brownian motion (RGBM). When the effective wealth reallocation in theeconomy is from the poor to the rich, the model allows for the existence ofnegative wealth within the population. We show that then, in RGBM ergodicitybreaks as the difference between the time-average and the ensemble growth rateof the average wealth in the population. In particular, the ensemble averagewealth grows exponentially whereas the time-average growth rate isnon-existent. Moreover, we find that the system is characterized with acritical self-averaging time period. Before this time period, the ensembleaverage is a fair approximation for the population average wealth. Afterwards,the non-ergodicity forces the population average to oscillate between positiveand negative values since then the magnitude of this observable is determinedby the most extreme wealth values in the population. This implies that thedynamics of the population average is an unstable phenomenon in a non-ergodiceconomy. We use this result to argue that one should be cautious wheninterpreting economic well-being measures that are based on the populationaverage wealth in non-ergodic economies.

Ergodicity breaking in wealth dynamics: The case of reallocating geometric Brownian motion