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Using R, assume the population income for N= {1.. 7} looks like $X_1$= c(1,3,5,7,7,19,21). Use the Lc() function from the "ineq" package to plot the corresponding Lorenz curve $L_{X_1}(p_1)$.

Next, rerank $n_2$ and $n_6$ so that $X_2$= c(1,18,5,7,7,4,21) while mean is same ($\mu = 9$). Plot the Lorenz curve $L_{X_2}(p_2)$ using Lc().

Next, rerank $n_4$ and $n_5$ so that $X_3$= c(1,18,5,6,8,4,21) while mean is same. Plot the Lorenz curve $L_{X_3}(p_3)$ using Lc(). Note that this redistribution violates strict horizontal equity.

Questions:

  • Using the Lorenz curves above, may I say that all strictly increasing, concave social welfare functions will prefer $L_{X_2}(p_2)$ and $L_{X_3}(p_3)$ as dominant over $L_{X_1}(p_1)$?
  • Can we discuss the "equalizing effect" of the redstribution even though reranking has caused $p_1$ to change?

I ask because Lambert (2001, 37-42) treats the total tax ratio > 0 and the mean post-tax income < mean pre-tax income, and shows that the "concentration curve for post-tax income with respect to pre-tax income" is the same as the Lorenz curve if there is no reranking. In my example, however, the total tax ratio = 0, the mean post-tax income = mean pre-tax income, and there is reranking. I think my confusion relates to the "concentration curve" as an intermediary to our real goal, the comparison of Lorenz curves.

All clarification appreciated.

Lambert. 2001. The Distribution and Redistribution of Income.

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