Introduction

The most important property of the neoclassic model of growth is to determine the existence of conditional convergence (^{Barro and Sala-i-Martin, 2009}). This concept is applied when the growth rate of an economy is related to the distance between the level of product of this economy and its own stationary state, and should not be confused with the absolute convergence that indicates that poor economies tend to grow faster than the rich ones. The two concepts are identical only when one group of economies tends to converge toward the same stationary state.

^{Esposti (2015)} formulates and estimates a model where the forces of convergence and divergence are combined to generate a convergence process in Italian regions. The theoretical and methodological implications indicate that the reduction of regional disparities in agricultural productivity is an objective that makes sectorial (agricultural) policies acceptable and desirable from a perspective of regional development, and that they may have as main objective promoting the convergence of productivity growth, favoring the public nature of the best agricultural technologies.

In México, ^{Rodríguez and Sánchez (2002)}, ^{Fuentes and Mendoza (2003)}, and ^{Chiquiar (2005)} point to a process of divergence after the process of commercial openness, while ^{Valdivia (2007)} and ^{Gómez et al. (2010)} indicate one of convergence. However, there are only a couple of studies that include criteria of participation of the agricultural and livestock sector to evaluate the per capita GDP evolution (^{Saavedra and Rello, 2013}; ^{Asuad et al., 2007}).

In 1986, México entered the General Agreement on Tariffs and Trade (GATT), and in 1994 the North American Free Trade Agreement (NAFTA) was signed. The theoretical framework of these instruments of trade liberalization is in the postulates of comparative advantage and establishes that the countries must be specialized in those economic activities where they have their most abundant factor. In North America, México should be specialized in work-intensive activities and the other countries in capital-intensive activities. Within this theoretical context there will be sectors that are more favored and other less favored, but, in general, a per capita profit should be exhibited in the population of the countries that participate in this international trade.

During the first 10 years of NAFTA, México’s agricultural and livestock foreign trade had noticeable growth (^{Rosenzweig, 2005}). The primary GDP showed an increase of 1.9% annual average, with higher dynamism in the fruit and vegetable subsector. In basic grains the greatest achievement is centered in tax transferences, and in the livestock subsector in the greater access to imported fodders. The agricultural and livestock sector has a higher concentration in production and trade, and paid agricultural employment increased due to its higher productivity, although the absolute agricultural one continues its structural downward trend.

According to ^{Saavedra and Rello (2013)}, in the first 10 years of NAFTA the Mexican fruit and vegetable exports to the USA increased, but the standardization of domestic prices with international ones has generated pressure on the farmers, whose production costs increased due to the reduction of government subsidies and to the insufficiency of the institutions of support to agricultural and livestock producers.

According to ^{Saavedra and Rello (2013)} and ^{Escalante and Catalán (2008)}, the NAFTA balance seems rather negative for the majority of rural producers: wheat, rice, sorghum, maize and powdered milk imports grew significantly, counterbalancing the positive effects of the increase in exports; the agricultural and livestock trade balance has been loss-making almost every year since 1995 and agriculture has ceased to be a source that generates foreign currency, function that it had covered satisfactorily for several decades.

The objective of this study was to evaluate regional convergence in México based on two criteria of participation in the agricultural and livestock sector: 1) evolution of the primary sector per capita GDP and, 2) the per capita GDP per state based on an agricultural and livestock classification of the states by level of participation of the GDP and by level of employment in each one.

The hypothesis attempted to prove that: 1) It is not possible to identify a process of divergence in the study period; and 2) The speed of the convergence is not constant in time.

In this regard, ^{Asued et al. (2007)} maintain that “The openness of the Mexican economy and NAFTA have fostered regional divergence in the states that make up the agricultural and livestock, manufacturing and services regions where the economic concentration in the regions of greatest disparity reinforces the disparity”. In turn, ^{Germán and Escobedo (2011)} suggest that commercial openness has favored to a greater extent the regions with comparative advantages in investment and trade, while ^{Gómez-Zaldivar (2012)} found a generalized convergence up to the 1980s and since that date a relative dispersion.

Materials and Method

The slow growth model by ^{Solow (1956)} stems from a Cobb-Douglas type production function: Y=AK ^α L ^1-α where Y is the domestic product. A represents the technology level, which is considered greater than zero and a is a constant that takes on values higher than zero, but lower than the unit. L represents the amount of work. Expressed intensively, the following is obtained: Y=AK ^α, which represents the impact of the variation in the accumulation of capital throughout time.

When divided by L, (^{Barro and Sala-i-Martin, 2009: 30}), the prior equation becomes:

In order to convert equation (1) into a non-linear differential where the product does not depend solely on k, the right side of the equation is transformed into per capita terms and is substituted, leaving the equation in the following terms:

where δ: level of capital depreciation, n: population growth rate, s: level of savings in terms of efficiency per worker, g: rate of technical progress and f(x): the production function.

In the neoclassic growth model, the properties of ^{Ramsey’s (1928)} model are the ones that determine the dynamics of the accumulation variables of capital and savings.

Equation (2) considers a Cobb-Douglas type technology where the dynamic process towards the stationary state of the per capita product is given by: . Under the assumption that this function presents constant yields at a scale and that a constant fraction of savings (s) is invested in the production, it can be rewritten as: y^=Ak^a

This expression implies that k*(Δk=0) converges towards a stationary state defined by the Cobb-Douglas type production function.

or its equivalent expression

Substituting this expression in the production function, there is:

where g and δ are constant, ln A₀ = α+ε_i; s is the mean investment rate, n is the mean growth rate of the population, and L is the population.

According to ^{Domenech and Domínguez (2013)}, equation (4) is represented in time t; that is, only data of transversal cut are considered for a series of countries or regions in a specific period and it is considered that these countries or regions are found in their stationary state in time t.

Solow’s model can represent the speed at which an economy converges to its stationary state when different periods of time are analyzed through:

where γ _i=(n+g+δ)(1-α-β) and, therefore, longitudinal information is incorporated.

However, this model fails to fulfill the hypothesis that the expected value of the mean is equal to zero, so that the estimators would not be consistent. One way of generating consistency in the estimators using information of transversal cut is by rewriting (5):

And substituting recursively under the assumption that ln y^* _i remains constant:

This expression can be transformed when the variables are used in per capita terms:

Finally, to evaluate empirically the hypothesis of absolute convergence through regional data the following univariate regression is determined (^{Barro and Sala-i-Martin, 2009}):

This equation is estimated by non-linear least squares. In this expression it is important to highlight two things: first, it is assumed that there are only observations for two moments in time, moment 0 and T; therefore, the average growth rate of the per capita income of economy i is evaluated only in the interval 0 and T. Second, the equation must use a set of data where the various economies converge toward similar stationary states, since despite the existence of differences in technology, preferences and institutions between regions or municipalities, it is likely that these are lower in the regions that share a central government.

With the aim of including fictitious regional variables and in particular the effect of free trade in the agricultural and livestock regions of México, it is possible to specify (9) for their empirical estimation in linear terms, as:

Regressions 9 and 10 become linear by specifying the logarithms of the variables and, therefore, they use minimum ordinary least squares for their estimation. In this equation, β1 is equal to [1-e^λT/T] and represents the convergence parameter. Under these conditions if g is negative, there is a convergence process between regions and if it is positive this process will not take place. Variable D represents two fictitious regional variables that correspond to: a) the very high and high agricultural and livestock region; and b) the low and very low agricultural and livestock region; and variable S is the weighing of sector j on the per capita GDP of the state i at the moment t-T. The yjt variable represents the national average of the per capita income of sector j at moment t. In this case in particular, this structural variable refers to the participation of the primary sector in the total income of each state and indicates how much a state would grow if its agricultural and livestock sector would grow at the average growth rate of the national agricultural and livestock sector.

^{Asuad et al. (2007)} classify the states in México in function of the participation per state in the agricultural and livestock GDP and the agricultural and livestock employment per state (Table 1).

Information about the GDP for each state from 1940 to 2001 was used (^{Germán-Soto and Escobedo, 2011}), and it was extended until 2010 with data from the Economic Information Bank of the National Institute of Geography, Statistics and Information (Instituto Nacional de Geografía, Estadística e Informática, INEGI). In addition, information of the agricultural and livestock GDP per state for 1991 and 2007 from the general agricultural and livestock census 1991 and 2007 was used, and the population from 1940, 1990 and 2010 of the state and municipal database system from INEGI (several years).

The per capita GDP from the primary sector was established with information from the GDP of the primary sector from 1991 and the population reported in 1990 by the population census of that year, and that of 2010 was established with the GDP from the primary sector in 2007 and the population in 2010 by the population census of that year.

Results and Discussion

Table 2 shows the linear estimations of equation (9) and equation (10) for Mexican states, using the State Gross Domestic Product (Producto Interno Bruto Estatal, PIBE) and the regional agricultural classification. The first column identifies the periods of estimation, the second points out the coefficients by ordinary least squares from equation (9), and the third presents the coefficients of the equation (10).

The punctual estimation for β of the total sample of the states included during the period from 1940 to 2010 was -0.0085, with a standard error of 0.0017 and an estimation of R ² of 0.51. The value of β indicates that the speed of convergence between poor and rich states is only 0.85 % during the study period.

The second column points to a value of β of -0.0088 with a standard error of 0.0018 and R ² of 0.53. The similarity between the coefficient estimated in 9 and 10 would indicate that the speed at which the average per capita income converges between the states studied is not substantially different from the speed at which this income converges within each of the regions in which the states were classified.

Considering the period that covers from 1940 to 1990, the value of coefficient β was -0.11 with a standard error value of -0.0019 and R ² of 0.59 without considering fictitious regional variables. When these dummy variables are taken into account the value of β is estimated at -0.11 with a standard error of 0.002 and R ² of 0.61. That is, the speed of convergence does not change substantially, but the magnitude does; during this period evidence is found of absolute convergence between the per capita income of the agricultural and non-agricultural states. The growth rate is estimated at 1.13 % annual.

However, when the period of 1990 to 2010 is analyzed, the results are different. The estimated value of β is 0.0022 with standard error of 0.004 and R ² of 0.012 when equation (9) is estimated. When the regional and structural variables are included (equation 10), the value of β is estimated at -0.006 with standard error of 0.004 and R ² of 0.008. The sign is negative, but the values of the coefficients estimated are not statistically different from zero because of their standard error value.

These results show the existence of a regional convergence process during the period of 1940 to 2010 and in the period of 1940 to 1990. However, this process is not reflected in that of 1990 to 2010. In order to contrast the convergence analysis from 1990 to 2010, equations 9 and 10 were estimated, but using the agricultural and livestock per capita domestic product instead of the state gross domestic product. The variable of agricultural regional classification considered four regions instead of two (high, very high, low, and very low agricultural and livestock participation); in addition, the percentage participation of the state agricultural GDP was used as an agricultural structural variable.

The results are shown in Table 3.

For equation 9 the value of β was 0.0031 with standard error of 0.005 and R ² of 0.012. In equation 10 the value of β was 0.0049 with standard error of 0.006 and R ² of 0.34. Because the coefficients estimated are not statistically different from zero, we cannot generate any conclusion regarding the process of regional convergence/divergence within the range of data and the period of time analyzed.

Conclusions

In this study we explore the relationship between the per capita domestic product and the free trade process in México. Economic convergence was determined in two periods: from 1940 to 1990 and from 1940 to 2010. However, the convergence process becomes slower when the period of 1940 to 2010 is considered. For the period of 1990 to 2010 it is not possible to obtain conclusions about a convergence or divergence process, whether the state per capita gross domestic product is used or the agricultural and livestock per capita gross domestic product is used because the coefficients estimated are not different from zero. However, it is possible to point out that if the magnitude of the free trade process that has taken place in our country until now remains constant, it will not be possible to maintain the hypothesis that the speed of convergence is stable in time for the agricultural regions considered in the study.

The results agree with those by ^{Mas et al. (2005)} for the case of Spanish regions where the existence of a regional convergence process is described; however, they differ from the results by ^{Esposti (2012)} where the existence of a divergence process in Italian regions is described.