Regional Input-Output Matrices and an Application to Analyze a Manufacturing Export Shock in Mexico

The basic approaches to estimate a RIOM with indirect methods are invariably based on a national IOM. Once the latter has been obtained, its elements are transformed in accordance with the chosen methodology, as well as on different criteria related to the distinctive features of the region for which a RIOM is constructed.

The basic procedure to construct a national IOM starts by assuming that the gross output of an economy with “n” sectors in one period can be represented as follows²:

where x is a (n x 1) vector of gross outputs; Z is a (n x n) matrix of (intermediate) sales to other productive sectors (with each of its elements given by z_ij); f is a (n x 1) vector of the final demands; and “i” represents a column vector of 1’s of dimension (1 x n).

In the input-output approach, the fundamental assumption is that the flow of goods and services of any given sector “i” demanded by sector “j” (i.e, the z_ij elements of the Z matrix) depends exclusively on the total production of “j” (x_j), where this relation is expressed as follows:

Based on this definition, notice that:

where a_ij is a coefficient that captures, for sector “j”, a fixed relation between the level of production of “j” and the level of input “i” used to obtain the referred production. These coefficients are called “fixed technical coefficients”, implying that all productive sectors have Leontief production functions and, therefore, that all productive sectors have constant returns to scale.³ Then (1) can be expressed as:

Finally, solving for x:

where L = (I - A)^-1.

L is known as the “Leontief inverse matrix”, or “matrix of total requirements”, which its elements depend on a_ij. This matrix allows us to identify the impact of exogenous shocks on gross output by means of the so-called multiplier effects, and in which we are interested for the purposes of impact analysis. These multiplier effects are classified into direct (the effect on the economic sector that receives the exogenous shock), and indirect (the effect generated by the affected sector on other sectors of the economy it interacts with); while their sum is known as total multiplier.⁴ The intuition behind these multipliers is that when a sector experiences, for instance, a positive exogenous shock, it leads to greater productive activity in the same sector (a direct effect), as a result there is higher demand for intermediate inputs from other sectors of the economy involved in the productive process (an indirect effect), and so on. This process continues in a way that the economy´s production grows more compared to the initial impact. This generates greater value added, and more employment in the economy.⁵

To obtain the coefficients a_ij and, therefore, the IOM in Mexico´s case, we rely on INEGI’s IOMs estimates of 2012. The result generated from the RIOMs for Mexico, based on such IOM, is presented in the following section.

The main goal of this paper is to construct RIOMs for Mexico. The literature on the subject indicates that the construction of a RIOM can be carried out using “direct methods”, that is, methods that require statistical data obtained from surveys, just like a national IOM is constructed.⁶

Alternatively, RIOMs can also be generated using “synthetic approaches”, that is, using indirect and semi-direct methods (also known as hybrid methods), which help transform information available at the aggregate level to information at the regional level.⁷ The essence of these methods lies in adjusting the elements of an IOM to obtain components of a RIOM. Thus, in all cases there is an IOM as a starting point.

A practical problem with indirect methods, however, is that in order to convert technical coefficients from the national to the regional level, there is a set of different alternatives, which depend on the application of the so-called Location Quotients (LQ). A LQ is an analytical statistic that measures a region’s industrial specialization relative to the nation, which is computed as an industry’s share of a regional total for some economic variable divided by the industry’s share of the value for the same statistic at the national level and for which there are various alternative methodologies to compute them.⁸

Given these alternatives, different studies have focused on evaluating the performance of the different LQs to construct RIOMs. Such evaluations compared the RIOMs estimated with “direct” methods against those obtained with “indirect” methods, and concluded that the best performance was observed using the Flegg method (FLQ).⁹ Therefore, the Flegg approach will be used in this paper to estimate the regional technical coefficients requited to obtain the RIOM.¹⁰

It should be stressed that the only economic series required for the estimation of Flegg’s coefficients (FLQij) is states´ GDP (which is the one used to calculate regional GDP) and national GDP per sector, which in Mexico is provided by INEGI. The following section describes in detail how the rest of the necessary components to construct a RIOM are obtained.

The regional intermediate consumption of sector “j” that stems from sector “i” ( $Z_{ij}^R$ ) is obtained based on the following definition:

where $X_j^R$ is the regional gross output of good “j” and $a_{ij}^R$ is the regional technical coefficient.

Out of these two components, only $a_{ij\mathrm{ }}^R$ is available, which compels us to estimate $X_j^R$ . To do so, let us keep in mind that the regional gross output of sector “j” ( $X_j^R$ ) is defined as:

where $V_j^R$ is the regional gross value added; ${ZT}_j^R$ is the total regional demand for intermediate goods; $M_j^R$ are regional imports; and $T_j^R$ are payments of regional taxes.

Given that there is no information at the state level for $M_j^R\mathrm{ }$ and $T_j^R$ , they are estimated, as a result, as follows:

where it is assumed, on the one hand, that the national and regional average propensity to import are also equal ( $m_j^N=m_j^R$ ); and, on the other hand, that the effective national and regional tax rates are equal ( $t_j^N=t_j^R$ ). These two last assumptions are used, given that there is only information available on $m_j^N\mathrm{ }$ and $t_j^N$ .

Based on the above definitions, expression (7) becomes¹³:

Finally, solving for $X_j^R$ :

Thus, the gross output of sector “j” in region “R” ( $X_j^R$ ) can be estimated with the formula referred above, as it contains the numerator $V_j^R$ , that is, the value of the regional GDP of economic sector “j”, and for which there is information; and in the denominator there is a series of parameters that are regional technical coefficients ( $a_{ij}^R$ ), the region’s average propensity to import ( $m_j^R$ ) and the regional effective tax rate ( $t_j^R$ ), for which there are also estimates. When the components of regional gross output ( $X_j^R$ ) and regional technical coefficients ${(a}_{ij}^R)$ are available, it is possible to estimate the following for each sector “j”:

Meanwhile, each of these components indicates the consumption of each sector “j” of the different sectors that provide inputs. Using this formula, we form the region’s intermediate consumption matrix, which is one of the components of a RIOM. The sum of all elements yields total intermediate consumption of sector “j” of regional origin:

As mentioned in Section II, the Flegg method assumes that the techniques of national and regional production are the same. Under this assumption, at the regional level each sector “j” would be consuming intermediate goods with a value of:

Still, given that the Flegg method is applied to adjust regional production technique differences from what the region is supposed to consume, if it maintains national production techniques, minus what it, in fact, consumes given the adjusted production techniques that are obtained using the Flegg method, the total intermediate consumption of sector “j” stemming from the rest of the states ( ${ZRE}_j$ ) equals:

The RIOM is complemented by estimating, at the regional level, the components of the final demand (C^R, I^R, G^R, EXP^R, M^R)¹⁴, as well as the components of the payment industry: value added (V^R), taxes ( $T_j^R$ ) and imports ( $M_j^R$ ); along with the disaggregation of value added, that is taxes ( ${TSPNS}_j^R$ ), wages ( ${REM}_j^R$ ) and the payment of capital ( ${EBO}_j^R$ ).

As regarding the components of regional final demand, for the case of private consumption and the regional government spending (C^R and G^R, respectively), the national values of these variables are multiplied by the share of regional population in the national total; while regional investment (I^R), that equals the gross capital formation plus the change in inventories, is obtained from multiplying the value of national investment by the share of regional GDP in the national total.¹⁵ For the case of EXP^R, state exports’ data released by INEGI were used in order to obtain the corresponding regional values.

With respect to the components of the payment industry, M^R was calculated as the average ratio of output to imports at the national level and the regional gross output, while $T_j^R$ , which corresponds to taxes on goods, is estimated by applying the tax rate of the goods at the national level ( $t_j^N$ ) to the regional gross output ( $X_j^R$ ). Consequently, the item of regional wages ( ${REM}_j^R$ ) is estimated multiplying the national figures by the share of regional wages in national wages (wage earners, self-employed, employers, and non-paid employees); while the tax on production $\left({TSPNS}_j^R\right)$ is obtained by multiplying the payroll tax (ISN) by the sectorial share of wages in the region. The capital payment ( ${EBO}_j^R$ ) is obtained as the (residual) difference between V^R and the rest of above-mentioned components. It should be noted that, insofar as the rest of components of V^R have errors in the measurement, the analysis of the capital payment can fail to generate reliable results.

Once the aforementioned operations are carried out, we obtain the necessary information to construct RIOMs for the Mexican economy, along with estimates of value added at the regional level. ¹⁶ Since there are four regions (Northern, North-Central, Central and Southern) according to the regionalization proposed in the “Reporte Sobre las Economías Regionales” published by Banco de México (see Figure 1), there are four matrices, one for each region.

The exercise will consider, for simplicity, a shock on total manufacturing exports of USD 10,000 million, which represents 3.5 percent of total manufacturing exports from Mexico to the US in 2015. The regional distribution of the shock, in dollar terms, will be derived by multiplying the share of each region’s manufacturing exports in total manufacturing exports, by the previously mentioned USD 10,000 million shock. Since official data indicate that the shares of regional manufacturing exports in total manufacturing exports are 61.3, 22.5, 13.0, and 3.2 percent for the Northern, Central, North-Central and the Southern regions, the regional shocks in dollar terms will amount to USD 6,130 million; USD 2,245 million; USD 1,300 million and USD 320 million, respectively (Table 1a).¹⁷

With an estimate of the four regional shocks at hand, the next step consists of distributing them within each region across the 12 manufacturing subsectors considered, to obtain the calculation of the regional input-output matrices. In each region, the shock is distributed in terms of the share of the subsectors’ exports in total manufacturing exports, and is shown in Table 1b. Thus, we have four regional distributions of external shocks since the shares of regional manufacturing exports in total manufacturing exports differ across regions. This table is quite revealing as it shows that the impacts in the Northern, North-Central and Central regions concentrate in Machinery, Computer, Electrical and Electronic Product, and Transportation Equipment Manufacturing (subsectors 333-336), with shares of 76.6, 75.0 and 71.5 percent, respectively. In the South, on the other hand, the main subsectors to absorb the shock are Chemical (45.1 percent), Primary Metal (17.3 percent), and Food Manufacturing (13.8 percent).

Figure 2 provides the shares of state manufacturing exports in national manufacturing exports, reinforcing the view of why external shocks may affect regions differently. As it can be seen, the North is the region which concentrates the largest shares of state manufacturing exports concerning national exports, and hence, is the most sensitive to external shocks. It is then followed by the Central, North-Central and Southern regions.

Following the methodology outlined in subsections II and III, we now proceed to estimate direct and indirect effects on regional gross output, value added and employment of the shocks mentioned above. It is important to mention at this point that we will work with input-output matrices in which the “manufacturing sector” is divided in the 12 subsectors already shown in Tables 1a and 1b. Thus, in our analysis, the direct multiplier for the whole manufacturing sector within each region will be derived by adding the direct effects obtained across the 12 regional manufacturing subsectors. Clearly, given the complementarities among manufacturing subsectors, “indirect effects” are also expected to emerge among them. Hence, the total indirect multiplier will be obtained by adding the indirect effects from the manufacturing subsectors, to the indirect effects stemming from subsectors other than manufacturing. Tables 2-4 present these results, with gross output and value added expressed in MXN 2012 million; and employment expressed in number of workers.¹⁸

One feature that immediately stands out in Tables 2-4 has to do with large differences of total effects across regions. In particular, notice that the impacts of the external shock on gross output, value added and employment in the North are, by far, the largest. This result, however, should not be considered a surprise as it was previously mentioned that this region concentrates 61.3 percent of total manufacturing exports, followed by the Central region with a distant 22.5 percent, the North-Central region with 13.0 percent, and, lastly, the Southern region, with a meager 3.2 percent of total manufacturing exports. Since the differences mentioned above imply that absolute impacts are very heterogeneous across regions, comparisons of absolute direct and indirect effects are of little value. Hence, in what follows we will be referring, first of all, to relative measures of the direct and indirect effects within each region (obtained by dividing the direct and indirect effects by the regional total effect); and, second, to the measures of changes at the regional level in the relevant variables (gross output, value added, and employment) which result from the shock, expressed as fractions of the 2012 absolute value of the corresponding variable.