# How income inequality can affect GDP

Since income inequality is a big issue in the United States and the upcoming election, I thought it would be instructive to look at how income inequality may affect the total US income (GDP) in a very simple model. This will only be a linear model in the sense that I will model domestic spending given some income without demanding self-consistency so spending equals income. However, I think the same qualitative results will hold. Let us suppose that the a person’s spending as a function of income $I$ is given by $s(I)$. Now let $\rho(I)$ be the distribution of income (i.e probability density function).  The national income is then $I_{total} = N \int s(I) \rho(I) dI$, where $N$ is the US population. We can write this as $I_{total} = N\bar{s}(I)$, where the bar denotes expectation value or average.  So income inequality is measured by how wide $\rho(I)$ is. A perfectly equal society would have $\rho(I) =\delta(I-\bar{I})$. Now let’s first suppose that $s(I)$ is linear so your spending is exactly proportional to your income. In this case, $I_{total}=Ns(\bar{I})$, for any distribution and thus income inequality does not affect GDP. You will always get a GDP equal to spending at the US average. Now suppose that spending is super-linear or convex so you spend more as you earn faster than you earn it. This assumes that you save less the richer you are. In this case, by Jensen’s inequality $I_{total}>Ns(\bar{I})$, and the total GDP is bigger than if everyone spent at the average.  In this case, income inequality would actually increase the pie. Now, finally in the case where the spending function is concave, you spend less as you earn more, or save more as you earn more then $I_{total}, and thus the total spending is less than spending at the average. In this case, income inequality would reduce the GDP. So depending on how spending changes with income, income inequality could decrease or increase the size of the pie. My guess is that spending functions are concave so a little more equality could improve the economy.

## 6 thoughts on “How income inequality can affect GDP”

1. Rick G says:

This is a one-time gain, though. Growth in GDP is usually thought to be determined by investment, which is equal to saving, which is complementary to spending, i.e. I-s(I). And then you get precisely the opposite result, which is that if s(I) is concave, savings and investment are increased by inequality, and so GDP growth is increased by inequality in that model.

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3. @Rick I would say if you wanted to predict the long term effects you would need a fully nonlinear macro model so this is not applicable. Certainly a sudden increase in savings would cause a recession as in 2008. If you follow the Solow model, long term growth is governed by technological change while short term growth is governed both by labor force growth and capital growth. One could argue that more equality would allow widen the pool of possible innovators and thus increase the possibility of technological change. But in the short run, a redistributive tax could spur short term growth.

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4. Rick G says:

Carson, I agree that long term growth is governed by technological change, but that change is driven more by savings and investment than by consumption. I agree that there are many non-linear long run effects to consider. But even in the medium run, the fed determines the path of NGDP, and I don’t see the mechanism by which flattening the income distribution raises RGDP in the short run. Are there lots of idle factories or people seeking work who aren’t finding it? Not in 2016, so it’s hard to see where the extra production would come from.

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5. @rick Given that real incomes have been stuck for most people for the past decade or more, I would say that even though unemployment is under 5%, there are still people on the sidelines and in part time jobs that could be more productive.

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