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.
You assume that higher spending causes higher GDP. This is not necessarily true. 

Saving income will increase GDP through investments (unless you're in a Keynesian trap). Think about the most standard growth model, where the future (and steady state) GDP increases strictly in the savings rate.

Hence, whatever increases the savings rate (in your toy model, higher inequality), potentially increases future GDP. This was an argument brought forward in Barro (2000). Version with linear savings rate are in Bertola et al (2006) 

Of course, this is the neoclassical answer. There are many reasons why high inequality decreases growth/GDP which work through political economy models or similar. I'll list some of them despite you not asking explicitly for them, perhaps it's of use for future visitors:


Fiscal Policies: Equality leads to less necessary government redistribution and hence more incentives for investments (Meltzer and Richard, 1981; Corcoran and Evans, 2010; Persson and Tabellini, 1994)
Crime: Inequality reduces opportunity costs for illegal activities that harm GDP (Alesina and Perotti, 1993; Barro 2000)
Imperfect Credit Markets: Fixed investment costs or similar may lead to higher growth in unequal societies (Barro 2000)
Saving Incentives: With "not much to lose", poor people face moral hazard in unequal societies, a mechanism that decreases growth with inequality (Banerjee and Newman, 1991)
Fertility-Education: Equality in human capital leads to smaller fertility rates, and hence lower GDP. The Argument is too long to be summed up here; see Perotti (1996) and Croix and Doepke (2003))