# Why does HDI use log-transformed data for income index?

For calculation of HDI, the Income Index used is $$\mathrm{I_{income} = \dfrac{ln(Income) - ln(min Income)}{ln(max Income) - ln(min Income)}}$$. Why is the log transformed income used here? World Bank has answered that it is to emphasize the diminishing returns to the standard of living but I don't get how log-transform can reflect diminishing returns.

I know that log-transform is generally used to reduce skewness in the data and a log-transformed independent variable does reflect diminishing returns in a linear regression model but for calculating HDI, we take geometric mean and don't perform multiple linear regression.

Decreasing returns to standard of living means that if you add a certain amount $$\delta$$ to the income of a low income person, then the increase in her standard of living is bigger than if you would give the same amount to a high income person.

Let $$SL(x)$$ be the standard of living for a person with income $$x$$.

Take a person with income $$x$$ and a person with income $$y > x$$, and let us add an amount $$\delta$$ to the income of both persons. Then we should have that: $$SL(x + \delta) - SL(x) > SL(y + \delta) - SL(\delta).$$ The left hand side is the increase in the standard of living for the poor person. The right hand side is the increase in the standard of living for a rich person.

Dividing by $$\delta$$ and taking the limit for $$\delta \to 0$$ gives: $$\frac{d SL(x)}{dx} > \frac{d SL(y)}{dy}.$$ The left hand side is the marginal return to the standard of living for a person with income $$x$$ and the right hand side is the marginal return to the standard of living for person with the higher income $$y$$.

Decreasing marginal returns means that this marginal return decreases when income increases. So it is lower for the higher value $$y$$ compared to the lower value $$x$$.

A function with a decreasing slope is called a concave function. So what you want in the end is a function that is

• Increasing, i.e. $$SL(x)$$ increases with $$x$$
• concave, i.e. $$dSL(x)/dx$$ decreases with $$x$$.

One (popular) function that satisfies these two conditions is the natural log function. Note that $$d \ln(x)/dx = 1/x$$ which indeed decreases with $$x$$.

The function that the world bank uses: $$SL(x) = \frac{\ln(x) - \ln(x_{min})}{\ln(x_{max}) - \ln(x_{min})},$$ is just a linear transformation that makes sure that:

• $$SL(x_{min}) = 0$$
• $$SL(x_{max}) = 1$$.

So the function is:

• increasing
• concave
• between 0 and 1
• Why $\ln$ though? Seems like for any increasing concave $f$, the transformation $T(x) = \frac{f(x) - f(x_{min})}{f(x_{max}) - f(x_{min})}$ would have the same properties. Why not use $\sqrt{ \ \ }$, $\sqrt[5]{ \ \ }$ or $\ln \ln$ instead? – Giskard May 14 at 14:34
• @Giskard, You probably have to ask the Word Bank for an answer -- as they don't offer more info on their website ;-). A "statistical reason could be that income is usually log-normally distributed. As such, the computed $SL$ will be (truncated) normally distribution. This makes it probably easier to use as inputs for statistical analysis. – tdm May 14 at 14:43
• Yeah, I was surprised your answer focused so heavily on concavity and did not mention the distribution of income! Or perhaps I missed it? – Giskard May 14 at 14:45