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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.

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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
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    $\begingroup$ 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? $\endgroup$
    – Giskard
    May 14 at 14:34
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    $\begingroup$ @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. $\endgroup$
    – tdm
    May 14 at 14:43
  • $\begingroup$ Yeah, I was surprised your answer focused so heavily on concavity and did not mention the distribution of income! Or perhaps I missed it? $\endgroup$
    – Giskard
    May 14 at 14:45

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