¿Es factible disminuir a la mitad la pobreza en Mexico?
| Autor | Ch |
| Cargo | Report |
Halving Poverty in Mexico
Introduction
Poverty goals are key indicators to evaluate the advancement of development. For instance, in September 2000 the world leaders of the United Nations adopted a document known as the Millennium Declaration, which explicitly set an ambitious agenda for international development. It includes a series of goals known as the millennium development goals (MDG). (1) The first of them establishes that countries should reduce by half 1) the proportion of people living below $1 a day, and 2) the proportion of people who suffer from hunger, by 2015, taking the level observed in 1990 as a reference point. Since then, several approaches have been suggested and implemented to study the feasibility of that goal, for example Besley and Burgess (2003), Deaton (2003), and Chen and Ravallion (2004).
This paper develops and applies a simple methodology to estimate two parameters of interest for the analysis of poverty goals: the required time and the minimum necessary growth rate to meet a poverty goal under several growth and distribution scenarios. The methodology has several advantages. First, it can be applied to practically all poverty measures used in applied work. Second, the parameters of interest can be estimated on a case by case basis instead of estimating a cross-country regression. As noticed by Bourguignon (2002), this approach is more appropriate since both the development and the inequality levels of a country do affect the growth elasticity of poverty reduction. Third, the parameters can be estimated from aggregate data. These parameters are estimated for rural and urban Mexico in the case in which poverty goals imply halving both incidence and intensity of poverty in the middle-run. (2) To this end, we use not only the one dollar a day poverty line, but also the official food poverty lines and the two dollar a day poverty line, to take into account country-specifics.
The article is organized as follows: section I describes the methodology. Section II illustrates it using data from Mexico. Finally, section III summarizes the main results and conclusions.
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Methodology
Let [F.sub.t](y) be the cumulative income distribution, and [y.sub.t] (p) the p quantile of that distribution at time t. We focus on poverty measures that can be fully characterized in a general form as follows:
[P.sub.t] = P([[mu].sub.t],z,[L.sub.t]), (1)
where [[mu].sub.t] is the mean income, z the poverty line and [L.sub.t] the Lorenz curve.
As a special case for this class of measures, we have the family of additively separable poverty measures, which can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [pi](.) is the poverty evaluation function and [H.sub.t] the proportion of people whose incomes are below the poverty line, z. (3) For instance, the Foster-Greer-Thorbecke (1984) family of poverty measures is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
Let P * be a poverty goal. The needed mean income, [mu] *, to meet this poverty goal for a given income distribution, L, and an exogenous poverty line, z, is defined as:
[mu] * = inf{[mu]:P([mu],z,L) [less than or equal to] P *}. (4)
Therefore, the time taken to meet the poverty goal, P *, given ,[mu], L, z, and an annual per capita growth rate, [bar.g], can be written as follows:
t([bar.g]) = ln(([mu] */[mu]).sup.1/ln(1+[bar.g]]) (5)
Analogously, the m;n;mum necessary growth rate of per capita income, g ([bar.t]), to meet the poverty goal, P *, in [bar.t] years, holding both the income distribution and the poverty line constant is:
g([bar.t]) = [([mu] */[mu]).sup.1/[bar.t]] - 1. (6)
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1. Incorporating Inequality into the Analysis
Although most of the poverty changes are explained by growth in average incomes, Kraay (2006) shows that changes in income inequality may play an important role in meeting poverty goals in the medium to long-run, particularly in very unequal societies. Nevertheless, incorporating changes in inequality into the analysis creates a further dilemma, given that the income distribution can change in an infinite number of ways.
To handle this problem, we use the lognormal distribution to approximate the distribution of income. This is a standard parameterization in applied work, because it fits the data very well and is tractable (López and Servén, 2006).
Exploiting the one to one mapping that arises under lognormality between the Lorenz curve and the Gini coefficient, G, and using the fact that [L.sub.t] (p) = [PHI]([[PHI].sup.-1](p)-[[sigma].sub.t]) -see Aitchison and Brown (1966)-, it can be shown from (2) that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where [[phi].sub.t] (p) = [phi]([[PHI].sup.-1](p)-[[sigma].sub.t] / [phi]([[PHI].sup.-1](p)) (8)
and [[sigma].sub.t] = [square root of 2[[PHI].sup.-1](1+[G.sub.t]/2), (9)
where [PHI](*) and [phi](*) are, respectively, the cumulative distribution function and the probability density function for the standard normal. Particularly, the Headcount ratio can be reformulated as:
[P.sub.0t] = [PHI](ln(z/[mu].sub.t]/[[sigma].sub.t]+[[sigma].sub.t]/2]). (10)
It can be easily shown that [[partial derivative]P.sub.0]/[partial derivative][mu]
[P.sub.0]([mu],z,G) = [P.sup.*]. (11)
From this equation we can estimate the parameters of interest, t([bar.g]) and g([bar.t]), for counterfactual income distributions.
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