# On the relationship between income distribution and GDP?

I was thinking about the following simple example when I wondered what the theoretical effects wealth equality or inequality may have on GDP:Suppose there is a society with three individuals who have enough money to cater for all their needs, and there is an additional 600 units of disposable income left over.Let us assume that $S_i$ is the disposable income allocated to person $i$ and let us first assume person $i$ will then spend an amount proportional to $\\sqrt{S_i}$ (This assumption is incorrect, but I'm just using it as an example of a non-linear income to expenditure relationship example, which I will compare below with a linear assumption), that is, disposable expenditure would be given by $k\\sqrt{S_i}$ for some constant k.Now if all the excess income was allocated equally among the three members, the total excess expenditure is $k(\\sqrt{200} + \\sqrt{200} + \\sqrt{200}) = 42.43k$.However if all of the excess income is allocated to a single individual, the total excess expenditure is $k\\sqrt{600} = 24.49k$Thus given the assumption that expenditure is proportional to the square root of disposable income, a lower GDP should be observed in economies with increased inequality of disposable income.However if expenditure is proportional to the square of disposable income, then it can easily be shown with a similar argument that wealth distribution will increase the GDP in this simple model.According to modern economics principles, in simple terms how does distribution of disposable income affect GDP?*Note: I'm not interested in the politics behind equality, such as riots and revolutions etc that might result from severe inequality, but merely the financial/economic principles.*

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