At first, you could consider simply putting weights on GDP per capita and the GINI coefficient for your measure of social welfare ($W$).

$$W = Y^\alpha (1-G)^{1-\alpha}$$

So social welfare depends both on income inequality and GDP. A few problems arise with this naive model.

• What is the correct weight to put on $\alpha$?
• Perfect forced income inequality as well as perfect forced income equality hurts the economy, and this sort of measure would not take this into account.
• As GDP increase, one might intuitively think that income inequality increases, since usually GDP increases with large scale technological shocks, which have high variance over time, and it increases slowly with labor productivity, which has very low variance over time in comparison.

So what are some ways of solving these issues?

1.) You could try to guess what politicans treat $\alpha$ as by looking at the GINI and GDP per capita over time and find the $\alpha$ that maximizes discounted welfare over that time period, though that makes the assumption that on average, competing/bargaining for constituent interests maximizes social welfare.

2.) The GINI coefficient is measured as $\frac{A}{A+B}$, relative to the Lorenz curve, so you could put weights on A and B as well to show that perfect equality/perfect inequality is not preferable, but then that is a whole other subjective animal.

3.) For the last point, I don't have any particular recommendations of dealing with it, except maybe accounting for lag values.

That would be my general guess for approaching this kind of question anyhow. Take it for what you will.

At first, you could consider simply putting weights on GDP per capita and the GINI coefficient for your measure of social welfare ($W$).

$$W = Y^\alpha (1-G)^{1-\alpha}$$

So social welfare depends both on income inequality and GDP. A few problems arise with this naive model.

• What is the correct weight to put on $\alpha$?
• Perfect forced income inequality as well as perfect forced income equality hurts the economy, and this sort of measure would not take this into account.
• As GDP increase, one might intuitively think that income inequality increases, since usually GDP increases with large scale technological shocks, which have high variance over time, and it increases slowly with labor productivity, which has very low variance over time in comparison.

So what are some ways of solving these issues?

1.) You could try to guess what politicans treat $\alpha$ as by looking at the GINI and GDP per capita over time and find the $\alpha$ that maximizes discounted welfare over that time period, though that makes the assumption that on average, competing/bargaining for constituent interests maximizes social welfare.

2.) The GINI coefficient is measured as $\frac{A}{A+B}$, relative to the Lorenz curve, so you could put weights on A and B as well to show that perfect equality/perfect inequality is not preferable, but then that is a whole other subjective animal. The idea is that even long term equality or inequality can affect long term GDP growth, even if current GDP looks okay.

3.) For the last point, I don't have any particular recommendations of dealing with it, except maybe accounting for lag values.

That would be my general guess for approaching this kind of question anyhow. Take it for what you will.