Machine Learning Analysis of the Impact of Increasing the Minimum Wage on Income Inequality in Spain from 2001 to 2021

The Spanish Tax Administration Agency (Spanish: Agencia Estatal de Administración Tributaria, AEAT) is a public institution. It is attached to the Ministry of Economy and Finance through the former State Secretariat for Finance and Budget. It handles the application of the tax system under the constitutional principle that everyone must contribute to supporting public expenditure according to their economic capacity. It prepares the annual tax collection reports, which provide information on the amount and annual evolution of the tax revenues managed by the AEAT. It offers a database in Excel format called ”Distribucion salarios” in the supplementary material. This database collects information on employees in Spain who receive income from a company or entity required to provide a list of those receiving such income. This database does not include information on households that pay wages to employees in the household, nor does it reflect the reported wages of the self-employed.

Payments are defined as income, in cash or kind, paid by the reporting unit (company or institution) in the form of annual income. Employees are included in the database even if they worked for only one day. If an employee has worked for more than one company or entity, the amount shown is the sum of all payments made to that employee by the different companies from which he/she received his/her salary. The information is completely anonymous and is presented in intervals of $200, from $0 to $80,000. The last interval is open, with no maximum value specified. A total of 400 annual salary levels are presented. Each cell represents the value corresponding to each bracket by year (2001-2021). Three variables are analyzed: number of employees, salary, and withholding tax.

This paper analyzes other macroeconomic data regularly published by the National Statistics Institute (INE), such as; MNW, Consumer Price Index, Unemployment Index, Gross Domestic Product, and Public Debt. In no case are the data deflated, as they are all nominal.

The formula used to calculate the Gini index is:

To calculate the Gini index for a year, the number of registered employees in that year is needed. The database provides this information in brackets. The number of employees with a gross annual income equal to the interval, the total gross income, and the total income tax withheld are provided for each interval (range of $200). The mean is considered to be the distribution value that makes the variance zero since the probability distribution of each ith interval is considered to be identically distributed. This results in a vector for each interval composed of j values that are identical and equal to the mean of that interval.

Each year vector is a union of all intervals vectors of this year.

To analyze the relationships between macroeconomic variables, graph theory was used. A graph is a collection of nodes (also called vertices) connected by edges (undirected)Newman [2018]. The pattern of interactions between the nodes (individuals or entities) can be captured through the graph structure. The purpose of graph (or network) analysis is the study of relationships between individuals to discover knowledge about global and local structures.

In this paper, the nodes of the graph are defined as all macroeconomic variables, and the edges are defined as moderate or strong correlations between them. The linear correlation between two nodes is represented by $corr(i,j)$, and the Spearman correlation is defined as moderate or strong if $corr(i,j)\geq 0.5$ Suchowski [2001] in case of direct correlation. An $edge(i,j)$ is defined if $abs(corr(i,j)\geq 0.5$.

Detecting communities in networks is one of the most popular topics in modern network science. Communities, or clusters, are typically groups of nodes that are more likely to be connected than to members of other groups, although other patterns are possible. There are no universal protocols, neither for defining a community itself nor for other crucial issues such as validating algorithms and comparing their performance.

The Louvain method hierarchically performs a greedy optimization Blondel et al. [2008], assigning each vertex to the community of its neighbors that yields the highest number, and creating a smaller weighted network whose vertices are the clusters found previously Fortunato and Hric [2016]. Partitions found on this super-network hence consist of clusters including the ones found earlier, and represent a higher hierarchical level of clustering. Software used: Gephi v 0.10 Jacomy et al. [2014].

The statistical model is assumed to be

where $\mu\ N(0,\mathrm{\Sigma})$. The ordinary least squares for independent identically distributed errors (MSE). R- square formula is,

The Durbin-Watson (DW) statistic is a test for autocorrelation in the residuals from a statistical model or regression analysis. Values from 0 to less than 2 indicate positive autocorrelation and values from 2 to 4 indicate negative autocorrelation.

Durbin-Watson test statistics d is given as,

where $N$ is the number of observations and $e_{i}$ is the residual for each observation $(i)$. The software used is statsmodels packages for PythonSeabold and Perktold [2010].

The Random Forest Regressor (RFR) is an ensemble learning model. It combines the predictions of multiple models to produce more accurate results than a single model Breiman [2001]. A decision tree (DT) is a simple model that predicts the outcome by performing a partition based on the predictor (input variable) that provides the greatest reduction in mean squared error (MSE).

Where $y$ and $\hat{y}$ are the measured and predicted values of the samples in a node, respectively, and n is the number of samples in a node. A node that cannot branch further due to a non-decreasing MSE is called a leaf node, and the average of the samples in that node becomes a candidate for prediction. When unseen data is entered into the final DT model, the data moves according to predetermined branching criteria. The value of the leaf node where the data finally arrives is used as the predicted value of the DT. Scitik-Learn is de machine learning software used in PythonPedregosa et al. [2012].

The Gini index of the gross wages received by all workers in the years under study is analyzed and called Gross-Gini. A similar analysis called the net Gini, is carried out on net income (gross salary minus withholding tax). It shows the effect of income tax progressivity as measured by the difference between the two indexes.Table 1 shows the annual results of the database “Distribucion salarios” provided by AEAT. The nominal increase in average annual gross earnings was like the annual increase in minimum wage. Over the twenty years of the study, they increased by €7,500 and €7,300, respectively. The annual minimum wage percentile is calculated concerning annual gross income. In most years, more than 30% of workers did not earn the equivalent of the annual minimum wage. This is due to a significant level of underemployment. Tourism in Spain is seasonal and accounts for more than 13% of the total employed Cabrer-Borras and Rico [2021]. Seasonality in the agricultural sector in Spain accounts for 5% of employment Molinero-Gerbeau et al. [2021]. Over the period under review, the progressivity of income tax has remained stable. Income inequality is favored by this progressivity. The difference between the calculated Gini indices indicates the reduction in income inequality brought about by the progressivity of the tax.

In the following analysis of the dataset, we reduce the intervals from 400 to 5. To do this, we will increase the intervals from €200 to €20,000. The average withholding tax per interval and its evolution are analyzed over the study period in Figure 1. The average withholding rate shows a general downward trend that is more pronounced in the lower average wage brackets. Comparing the ratio from 2001 to 2021, the decline is similar in the five three-percentage-point intervals, but the impact on net income is much greater in the range below 20K (from 6.09% in 2001 to 3.64% in 2021) than in the range between 60K and 80K (29.31% in 2001 to 25.85% in 2021).

The average gross annual salary is calculated for the entire population (Mean Gross Salary) and a calculation of the average gross annual salary for the range [$10,000 - $80,000] (Mean Gross Salary Range) is added, i.e., for each year, salaries and employees with gross annual salaries below $10,000 and above $80,000 are not considered. Both lines are highly correlated with each other, as well as with GDP in nominal euros, Figure 2. Two periods of economic growth that significantly increase the average wage are characterized: 2001-2008 and 2015-2021.

Figure 3 provides a visual analysis of the intuitive relationship between continuous increases in the minimum wage, especially in recent years, and unemployment. It is observed that youth unemployment has the highest elasticity for periods of economic crisis. No increase in unemployment among workers over 55 years of age is observed to be associated with an increase in the minimum wage. The National Statistics Institute (INE) periodically publishes unemployment figures. Quarterly figures are used in this case.

The following variables are evaluated.

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  Diff.- The difference between the average gross salary of all employees and the gross salary range [$10,000-$80,000].

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  Employees. - Number of employees by year.

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  GINI. - Net Wage Gini Index.

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  TAXES. - Withholding tax.

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  GDP. - Gross Domestic Product.

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  UnempRate. - Unemployment Rate.

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  NMW. - National Minimum Wage.

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  Mean Salary. - Mean Gross wage in Range.

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  DEBT. - National Debt.

The size of the nodes depends on their degree. The higher the degree, the more the variable is related to the rest. The linear correlation between two nodes $\left(i,j\right)$ is represented by $corr\left(i,j\right)$. An $edge\left(i,j\right)$ is defined if $abs\left(corr\left(i,j\right)\right)\geq\ 0.5$. The linear Spearman correlation of each edge is given by its value. The color is defined by the two communities found. The interpretation of the graph shows that the variables of the same color have a stronger relationship with each other. Taxes and GDP have a crucial importance in a relationship with both community variables. GINI is only related to its community variables. The results are shown in Figure 4:

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  Modularity: 0,156

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  Number of Communities: 2

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  Average Degree: 4,889

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  Maximum Nodes Degree: 6 (GPD, TAXES).

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  Minimum Nodes Degree: 3 (GINI).

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  Maximum linear correlation ¿ 0.9: (NMW - DEBT), (NMW - Mean Salary), (DEBT - Mean Salary).

Diff is the only node that has multiple negative correlations (Employees, GDP, and TAXES). It is strongly negatively correlated with the number of employees ($\rho=-0.89$). The NMW is strongly positively correlated with Mean Salary ($\rho=0.96$).

It is convenient to detail the results through an analytical analysis after a visual analysis of the study. Three types of regressions are analyzed: a multivariate linear regression, a regression model based on machine learning algorithms, and a time series regression model.

The Mean Salary was calculated considering all employees in all bands of the analyzed database. The bands below €10,000 and above €80,000 have been excluded from the calculation of the Mean Salary Range. The first band corresponds to employees who did not work the calendar year and distorts the average. Similarly, the band above €80,000 is open and includes remarkably high salaries, which also distorts the average. For this reason, the variable ’average salary range’ corresponds to the average of employees whose gross annual salary is between [10K-80K]. There has been a shift in the proportions of employees by rank, particularly since 2016, coinciding with more substantial increases in the minimum wage. Over the years, the [20K-40K] range has increased from 5.85% (12.51% of gross income) of the population in 2001 to 15.61% (20.96% of gross income) in 2021. This is because underemployment (workers who want to work more hours and whose contracts do not cover 40 hours a week or all months of the year) exceeds 15% of total employment (Brecha de género en el empleo por tipo de empleo y periodo, n.d.). This analysis is shown in Figure 6.

Spearman’s linear correlation value between the DIFF variable and GINI was $\rho=\ 0.75$ shown in Fig 4. DIFF has been found to be the most influential in predicting the Gini Index score.

The evolution of the percentile of the minimum wage over the period under study is analyzed and presented in Figure 7, together with other macroeconomic variables. A percentile above 40 is noticeable. This means that 40% of the wages recorded in the database are paid below the Minimum Wage. The evolution of the percentile is related to an increasing minimum wage and to the evolution of the average gross wage.