Growth Convergence and Regional Inequality in India (1981–2012)

Abstract

There has been significant research effort to study the impact of liberalisation on growth and distribution in India. Using per capita income (PCI) data for the period 1981–82 to 2012–13 (28 regions for the entire period and 31 regions for 2001–2 to 2012–13) at the sub-national level in India we examine the claims of divergence and stratification (twin peak formation) as has been claimed in some of the recent literature. We confirm that there is divergence of PCI. We present the first set of tests of multimodality in the Indian convergence debate using Silverman (J R Stat Soc 43:97–99, 1981; Density estimation for statistics and data analysis. Monographs on statistics and applied probability, Chapman & Hall, London, 1986) procedure. Weighted kernel density plots and multi-modal tests reveal that there is emergence of “multi-modes” in the distribution of PCI, not just twin modes. The spatial pattern of growth reflects an area of stagnation in the eastern-central belt—Bihar, Madhya Pradesh, Uttar Pradesh and Orissa, and in the north eastern part of India—Assam and Manipur and a decline in Mizoram. Sikkim demonstrates fastest growth, whereas Gujarat, Haryana, Kerala, Maharashtra, Punjab, Tamil Nadu (among the big states) and Himachal Pradesh, and Andaman and Nicobar (small state and Union Territories) maintained their position. Karnataka, and Andhra Pradesh (among the southern states), Arunachal Pradesh and Nagaland (among the north eastern states) along with Jammu and Kashmir, Uttarakhand and Chhattisgarh, moved up in the growth ladder. The continuation of growth stagnation in most of the BIMARU states poses a challenge to received theories of growth convergence and raises developmental concerns that the increased play of market forces in the Indian economy have not been able to overcome.

Appendix

Transition Probability Matrix

For the Markov transition matrices we assume that the probability of variable “$\hbox {s}_\mathrm{t}$” taking a particular value depends only on its past value “$\hbox {s}_\mathrm{t-1}$” according to the first-order Markov chain

$$\begin{aligned} \hbox {P}\{\hbox {s}_\mathrm{t}= \hbox {j} \,\vert \, \hbox {s}_\mathrm{t-1}= \hbox {i} \}= \hbox {Pij} \end{aligned}$$

(1A)

where “$\hbox {P}_\mathrm{ij}$” indicates the probability that state “$\hbox {i}$” will be followed by state “$\hbox {j}$” and

$$\begin{aligned} \hbox {P}_\mathrm{i1}+\hbox {P}_\mathrm{i2}+\cdots +\hbox {P}_\mathrm{in}=1 \end{aligned}$$

(2A)

The transition matrix is presented as;

$$\begin{aligned} P= \left[ \begin{array}{llll} {P_{11}}&{} \quad {P_{12}}&{} \quad {\ldots } &{} \quad P_{1}n \\ {P_{21}}&{} \quad {P_{22}}&{} \quad {\ldots } &{} \quad P_{2}n \\ {\ldots .}&{} \quad {\ldots }&{} \quad {\ldots } &{}\quad \ldots \\ {\ldots .}&{} {\ldots }&{}\quad {\ldots } &{}\quad \ldots \\ {Pn_{1}}&{} \quad {Pn_{2}}&{} \quad {\ldots } &{} \quad Pnn \\ \end{array} \right] \end{aligned}$$

(3A)

The transition probability matrix measures in each cell the transition from one state of relative income to the same or another state of relative income. It therefore, measures the probability with which the income level in a country or region rises, falls, or remains unchanged between two periods (Magrini 2007). These probabilities are normalised so that the sum of each row probabilities adds up to 1.

Kernel Density Estimator

A probability density function f(x) of a random variable X is defined as

$$\begin{aligned} f\left( x \right) ={\lim }_{h\rightarrow 0} \frac{1}{2h}P\left( {x-h<X<x+h} \right) \end{aligned}$$

(4A)

For any given “h”, we can estimate $P(x -h<X <x +h)$ by the proportion of the sample falling in the interval $(x -h, x +h)$. Thus a natural estimator $\hat{f}$ of the density is given by choosing a small number “h”, where “$\hbox {n}$” refers to the real observations:

$$\begin{aligned} \hat{f} \left( x \right) =\frac{1}{2hn}P\left[ number~of~X_1 ,\ldots ..,X_n\,falling~in~(x-h,x+h )\right] \end{aligned}$$

(5A)

This is described as the naive estimator. A weight function “w” is defined as:

$$\begin{aligned} w\left( x \right) =\left\{ \begin{array}{ll} \frac{1}{2}, &{} \quad \text {if}\,\left| x \right| <1 \\ 0, &{} \quad \text {otherwise} \\ \end{array}\right. \end{aligned}$$

(6A)

This suggests that the naive estimator can be written as

$$\begin{aligned} \hat{f} \left( x \right) =\frac{1}{n}\mathop {\sum }\limits _{i=i+1}^n \frac{1}{h}w\left( {\frac{x-X_i }{h}} \right) \end{aligned}$$

(7A)

To generalize the naive estimator, the weight function “$\hbox {w}$” is replaced by a kernel function K which satisfies the condition

$$\begin{aligned} \mathop {\smallint }\nolimits _{-\infty }^\infty K\left( x \right) dx=1 \end{aligned}$$

(8A)

Therefore, the kernel estimator with kernel K is defined by

$$\begin{aligned} \hat{f} \left( x \right) =\frac{1}{nh}\mathop {\sum }\limits _{i=1}^n K\left( {\frac{x-X_i }{h}} \right) \end{aligned}$$

(9A)

where “h” is the window width, also called the smoothing parameter or bandwidth.

There are two choices to be made here,

• a) appropriate kernel function, and

• b) the bandwidth “$\hbox {h}$”.

Population weighted Estimators

The sum “$\hbox {K}$” in Eq. (7A) above, is now replaced by the weighted product “wK”. The weighted estimator given below is expected to alter the height of the individual bumps (see Gisbert 2003).

$$\begin{aligned} \hat{f} \left( x \right) =\frac{1}{nh}\mathop {\sum }\limits _{i=1}^n wK\left( {\frac{x-X_i }{h}} \right) \end{aligned}$$

(10A)