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 $y &gt; x$, and let us add an amount $\delta$ to the income of both persons. Then we should have that:
$$
SL(x + \delta) - SL(x) &gt; SL(y + \delta) - SL(\delta).
$$
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:
$$
\frac{d SL(x)}{dx} &gt; \frac{d SL(y)}{dy}.
$$
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