Approximating income inequality dynamics given incomplete information: an upturned Markov chain model

Abstract

This article aims to understand mobility within income distribution in cases where there is incomplete information about how individuals transit between income distribution brackets. Understanding these transitions is crucial for evaluating and designing economic policies that affect the population in the long run. For this reason, we propose a methodology that may assist decision-makers to improve policies related to poverty reduction. We start by assuming that the income distribution bracket a person holds exclusively depends on the previous generation’s income bracket, i.e. it has the memoryless property. Therefore, our model resembles a Markov chain model with a steady state distribution that describes the distribution of the income brackets in the long run, and a transition matrix that describes the transitions between income distribution brackets from generation to generation. In contrast to a Markov chain, we assume a given steady state, in order to analyze the space of consistent transition matrices that could generate the steady state proposed. Additionally, we used the joint distribution simulation algorithm developed by Montiel and Bickel (Decis Anal 9:329–347, https://doi.org/10.1287/deca.1120.0252, 2012) to analyze the transition matrix, which allows us to understand the effects of partial information. We test the model with official data from the National Institute of Statistics and Geography and the Social Mobility Survey in Mexico.

Appendix

We derive to its dual form OP (14) from the Lagrangian’s canonical form, where $\mathbf { \alpha }, \mathbf { \mu } \ge 0$.

$$- \sum\limits_{{i,j}} {(a_{{ij}} + d_{{ij}} )} \cdot \ln (a_{{ij}} + d_{{ij}} ) + \overrightarrow {\lambda } (b - Bd) + \overrightarrow {\beta } ( - Cd) + \overrightarrow {\alpha } (a + d) + \overrightarrow {\mu } [1 - (a + d)].{\text{ }}$$

(20)

Deriving with respect to $d_{ij}$ we get Eq. (21). Where $B^c_{ij}$ and $C^c_{ij}$ are the columns of matrices B and C indexed by i, j.

$$\frac{{ - (a_{{ij}} + d_{{ij}} )}}{{(a_{{ij}} + d_{{ij}} )}} - \ln (a_{{ij}} + d_{{ij}} ) - \overrightarrow {\lambda } B_{{ij}}^{c} - \overrightarrow {\beta } C_{{ij}}^{c} + \alpha _{{ij}} - \mu _{{ij}} = 0.,$$

(21)

$$\ln (a_{{ij}} + d_{{ij}} ) = - 1 - \overrightarrow {\lambda } B_{{ij}}^{c} - \overrightarrow {\beta } C_{{ij}}^{c} + \alpha _{{ij}} - \mu _{{ij}} .,$$

(22)

$$a_{{ij}} + d_{{ij}} = e^{{ - 1 - \overrightarrow {\lambda } B_{{ij}}^{c} - \overrightarrow {\beta } C_{{ij}}^{c} + \alpha _{{ij}} - \mu _{{ij}} }} .$$

(23)

Defining $\varphi _{{i,j}} = - 1 - \overrightarrow {\lambda } B_{{ij}}^{c} - \overrightarrow {\beta } C_{{ij}}^{c} + \alpha _{{ij}} - \mu _{{ij}}$, and substituting Eqs. (23) in (20) we get:

$$- \sum\limits_{{i,j}} {e^{{\varphi _{{i,j}} }} } \cdot \varphi _{{i,j}} + \overrightarrow {\lambda } \cdot (b - Bd) + \overrightarrow {\beta } \cdot ( - Cd) + \sum\limits_{{i,j}} {\alpha _{{ij}} } \cdot e^{{\varphi _{{i,j}} }} + \sum\limits_{{i,j}} {\mu _{{ij}} } \cdot (1 - e^{{\varphi _{{i,j}} }} )$$

(24)

Reducing terms we get:

$$\sum\limits_{{i,j}} {(e^{{\varphi _{{ij}} }} + \mu _{{ij}} )} + \sum\limits_{{i,j}} {\overrightarrow {\lambda } } B_{{ij}}^{c} e^{{\varphi _{{ij}} }} + \sum\limits_{{i,j}} {\overrightarrow {\beta } } C_{{ij}}^{c} e^{{\varphi _{{ij}} }} + \overrightarrow {\lambda } b - \overrightarrow {\lambda } Bd - \overrightarrow {\beta } Cd.$$

(25)

Substituting Eq. (23) and reducing terms we get:

$$\sum\limits_{{i,j}} {(e^{{\varphi _{{ij}} }} + \mu _{{ij}} )} + \overrightarrow {\lambda } b + \overrightarrow {\lambda } Ba + \overrightarrow {\beta } Ca.$$

(26)

Where the vector a is the vectorized version of matrix A. Assuming A is a transition matrix, which rows sum to one, we have $Ca={{\textbf {1}}}$, and $\overrightarrow {\lambda } Ca = \overrightarrow {\lambda } {\mathbf{1}}$. Additionally, we have $\overrightarrow {\lambda } (Ba + b) = \overrightarrow {\lambda } (\pi A + \pi (I - A)) = \overrightarrow {\lambda } \pi$, where I is the identity matrix. Substituting we get:

$$\sum\limits_{{i,j}} {(e^{{\varphi _{{ij}} }} + \mu _{{ij}} )} + \overrightarrow {\lambda } \pi + \overrightarrow {\beta } {\mathbf{1}}.$$

(27)

From Eq. (27) we can formulate the dual problem as:

$$\begin{aligned} \mathop {{\text{Minimize}}}\limits_{{}} & \sum\limits_{{i,j}} \quad (e^{{ - 1 - {\mathbf{\lambda }}B_{{ij}}^{c} - {\mathbf{\beta }}C_{{ij}}^{c} + \alpha _{{ij}} - \mu _{{ij}} }} + \mu _{{ij}} ) + \overrightarrow {\lambda } \pi + \overrightarrow {\beta } {\mathbf{1}} \\ {\text{subject to:}} & \\ & \alpha ,\mu \ge 0,\lambda ,\;\beta \quad {\text{ free}} \\ \end{aligned}$$

(28)