Keywords

1 Introduction

The crossing waterway transport is carried out in all Brazilian macro-regions. The main characteristic of this transport is the interconnection of road stretches, usually in a short journey, integrating road systems or also as an element of urban/metropolitan transport. Nevertheless, this type of transport, for thousands of people, is the main or the only option for crossing a body of water. This essentiality reflects the amount paid for the service. Often, it is not modest, due to the lack of a logistical alternative. In this sense, the present work seeks to present a model for forecasting the price of the crossing transport service, especially in the crossing of vehicles. However, before entering the proposed model, it is necessary to understand, in general, the territorial process that explains the number of crossings in Brazil, a movement intensified from the second half of the 20th century (Fig. 1).

Fig. 1.
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Crossing navigation: Manaus - Careiro route in Amazon river

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2 Historical Context

Brazil has undergone major social, economic and territorial transformations in the last decades of 20th century. Can be listed the construction of a new capital; the agricultural and urban frontier expansion; increased of the industrial sector in the Gross Domestic Product (GDP) and; the intensification of the urbanization process.

Regarding the territorial occupation process, I highlight that in the period between 1950 and 2000, the Brazilian population jumped from 51 to 169 million inhabitants; now 73% of the population is urban, compared to 31% in the middle of the last century (Table 1).

Table 1. Brazil - number of people living in urban and rural areas (1950–2021Footnote

Population Estimation available: https://www.ibge.gov.br/estatisticas/sociais/populacao/9103-estimativas-de-populacao.html?=&t=resultados.

).
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In the same decades, there was an increase in municipal units throughout the national territory and, consequently, an increase in the number of cities. In 1950 there were 1,889 municipal units. Currently, the country has 5,570 municipalities (Table 2).

Table 2. Number of municipalities (1950–2021Footnote

Numbers from the Population Estimation Research, available from: https://www.ibge.gov.br/estatisticas/sociais/populacao/9103-estimativas-de-populacao.html?=&t=resultados.

).
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These numbers emphasize the territorial transformation that took place in the last century, with the outlining of new development vectors. As already pointed out by several authors, including Steinberger (1999), Becker (2005) and Queiroz (2007), it is a process of internalization of productive activities, especially agriculture, which generated poles of population attraction for the regions previously considered as “demographic voids”, mainly in the North and Midwest of Brazil.

This process was called “March to the West”, initiated in the Getúlio Vargas government (1930–1945). The creation of the new capital in the Central Plateau of the country, called Brasília, intensified this process. In parallel, with the change in the political axis, there was the construction of several road sections connecting the capital to the other regions of the country, a fact that accelerated the process of expansion of economic and urban borders, demonstrated in a certain way in the data above and in the maps below.

This movement was not always consistent with the occupation of land for agricultural exploitation. The interconnection of the new territorial divisions in the different regions of the country allowed for a greater participation of these in global markets, even with evident inefficiencies and gaps in infrastructure. The crossing lines, responsible for interconnecting road sections without bridges, are partially explained by this context (Figs. 2 and 3).

Fig. 2.
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Brazilian roadways development: 1960–2010

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Currently, Brazil has 90 crossing transport lines under the regulatory authority of the National Waterway Transport Agency – ANTAQ (Fig. 4). Approximately 300 shipping companies authorized by ANTAQ operates the federal lines. Almost 50% of them have a tax framework for individual microentrepreneurs (MEI). The MEI are exempted from submitting bookkeeping, annual survey of balance sheet and economic result. They are also not obliged to issue an invoice when providing the service to final consumer. Such tax benefits have resulted in great informality in the sector.

Fig. 3.
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Brazilian multimodal network (IBGE)

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Fig. 4.
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Brazilian crossing navigation lines.

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3 Service Affordability Price

After understand de crossing historical context, it should be highlight that only in 2015 the transport was sign up as a fundamental right in the Brazilian Federal Constitution (CF). The text compiled from art. 6 of the CF says: “These are social rights: education, health, food, work, housing, transport, leisure, security, social security, maternity and childhood protection, assistance to the destitute, in the form of this Constitution”.

The Brazilian Waterway Transport Regulatory FrameworkFootnote 3 establishes that the State can crossing transport service under the authorization regime. Subsequently, the Law determines freedom prices in free and competitive markets.

The LawFootnote 4 also establishes that the provision of transport services must be carried out in an appropriate manner, satisfying the conditions of regularity, efficiency, safety, topicality, generality, courtesy in the provision of the service, and affordability prices.

Public passenger transport has functional characteristics that differentiate it from the private services regime. Public services are directly linked to the concept of essentiality in order to meet the appropriate service attribute that satisfies the price affordability requirementFootnote 5. In practice, most crossing lines fit the public service characteristics.

In this context, the challenge is to establish practical parameters for the attribute of reasonable prices, considering the essentiality of the service to the community and the guarantee of freedom of prices and tariff services provided for by Law.

4 Monitoring Regulated Prices

In most of the crossing transport markets in inland navigation can be find a few or only one company operating on the lines authorized by ANTAQ. In 2020, more than 65% of the lines in operation had only one authorized shipping company. In some cases, technical and operational line’s characteristics do not allow a significant number of operators. In other cases, there are no companies interested in competing for service’s provision.

The ANTAQ does not yet have a methodology for measuring the affordability price. The topic will be developed in the 2022–2024 Regulatory Agenda. Until the study is completed, this work proposes a linear regression model to predict prices. The model can be useful both in the investigation of current authorizations and for new crossing line authorizations.

In short, it is proposed to compare the intended price with that practiced in the transport market, based only on the length of the crossing line. If the target price is within the model’s prediction range, then the hypothesis that it is statistically similar to the market price is accepted. If it is below or above, the regulatory agency may request a detailed explanation of the proposed pricing.

5 Methodology

A random sample of 42 lines of vehicle crossings was selected from the federal crossing line’s database (Table 3). The table of prices charged in December 2021 was checked for each line. Considering the variety of vehicles listed in the operators’ price list, the passenger car was selected as standard for the modelling.

At least one ANTAQ authorized shipping company was operating the crossing lines selected. The typical sample vessel has a steel hull, 12 m of beam, 50 m length and 505 deadweight tons, in average. The type of engine was not considered. The operational schedule programmed foresaw at least five daily trips. Finely, Google Earth was used to estimate the average length (in kilometers) of each crossing line.

In a preliminary approach, the scatter plot of price per kilometer by line extension indicated a logarithmic pattern. Under the assumption of association between variables, the Linear Regression method was adopted to assess the effect of an explanatory variable (extension) on the response variable (price per kilometer). The method also allows forecasting values for the response variable.

The linear regression model expressed by Johnson (2007, p. 362) was represented by Eq. (1):

$$Y=Z*B+ \varepsilon , \qquad E\left(\varepsilon \right)=0 \; and \; Cov\left(\varepsilon \right)={\sigma }^{2}I$$
(1)

where: Y is dependent variable (or response variable); Z is the matrix of the independent variable (or predictor variable); B is the matrix of unknown parameters to be estimated by the n independent observations; \(\varepsilon\) is the random residual that assumes zero mean, constant variance and zero covariance (uncorrelated).

When the residuals have a normal distribution, the prediction interval (IP) for a future observation can be written as:

$$IP\left({Y}_{f};\gamma \right)= {\widehat{y}}_{f}\pm {t}_{\gamma }{S}_{e}\sqrt{1+\frac{1}{n}+\frac{{({x}_{f}-\overline{x })}^{2}}{\sum {({x}_{i}-\overline{x })}^{2}}}$$
(2)

Considering the complexity of the calculations involved, the entire process of model selection, statistics calculation and model checking was supported by the Statistica software.

Table 3. Sample selected of Brazilian vehicle crossing lines.
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6 The Linear Regression Model

It was selected the model that met all the assumptions of linear regression at a significance level of 5%. The equation of the adjusted regression model for crossing a car was y = 3.2672 − 0.8827 * x, where, y = Ln (unit price of car in R$/km) and x = Ln (line length in km). The R2 statistic, that measures the variability explained by the model, presented a value equivalent to 94.96% (Table 4). Both regression coefficient, the intercept and the explanatory variable, had a significance level lower than 0.05 (Table 5). The following tables presents the summary of the analysis of variance.

Table 4. Summary ANOVA statistics
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Table 5. Regression summary statistics
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Considering the selected sample, it was concluded that the length of the line influence significantly the price charged for crossing a car, explaining about 95% of its variability. The model makes it possible to make predictions about the market crossing price. The Table 6 presents as example the forecast interval (with 95% of confidence) generated by the model for crossing lines with 100 m, 500 m, 1 km and 1.5 km in length.

Table 6. Forecast interval for some crossing lines extension.
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In a crossing line of 500 m in length, it is expected that the cost of crossing a car varies from R$ 15.13 to R$ 38.67. If the practiced (or intended) price is outside this range, then the Agency may analyze the case seeking for additional information to identify dissonant market practices.

6.1 Model Checking

If the model is valid, each residual is an estimate of the error, which is assumed to be normal random variable with mean zero and variance S2 (Johnso 2007, p. 381).

The normality of residuals examined graphically showed residuals that tend to align with the quantiles of a theoretical normal distribution (Fig. 5). Although some asymmetry is observed, the hypothesis of normality was accepted.

Still in relation to the analysis of the residuals of the regression model, their independence was tested. The Durbin-Watson statistic (d) that measures the correlation between each residue and the residue corresponding to the immediately previous observation indicated the value of 1.975, confirming the hypothesis of independent residues (Table 7).

Table 7. Durbin-Watson test independent residuals
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Fig. 5.
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Normal probability plot of residuals from Eq. (1).

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The homoscedasticity (or constant variance) of the residuals examined by graphic visual analysis (Bussa 2003) indicated residuals scattered randomly around zero. In this case, the hypothesis of constant variance was accepted (Fig.6).

Fig. 6.
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Predicted vs residual scores.

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7 Application

The crossing price forecast model was applied to another 8 (eight) crossing lines regulated by ANTAQ. They prices charged were outside the model’s forecast range. Three of them were below the lower limit and the last five were above the upper limit of the forecast range. In other words, these line are statistically significant different from the model market (Table 8).

This result indicates that the regulatory agency should get additional information to identify why the price in these lines are asymmetry to the modeled market. The following table presents the crossing lines taken from the model.

The posterior analysis identified that in the case of the crossing line Anapu (PA)/Vitória do Xingu (PA) - Directive BR-230 (Belo Monte) the operator practices a discount policy for cars to the detriment of large vehicles. For the other crossings, the ongoing analysis requested operational and economic-financial information from the companies.

Table 8. Forecasting statistics for eight crossing lines applied.
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8 Conclusions

The linear regression model selected for forecasting the transport market price is statistically valid. For the crossing lines surveyed, there is a strong correlation between the price of a car crossing and the length of the line, so this information was enough to generate price market estimates.

The application of the regression model in other lines in operation identified eight cases in which the practiced price is outside the forecast range. In one of these cases, it was identified a practice of discounts for automobiles. For the other cases, the ongoing analysis requested additional information from the authorized companies.

The statistical method applied in this work does not replace the study of price formation for regulatory purposes. In order to curb practices harmful to competition and the abuse of economic power, the regulatory Agency may use methods of price determination such as price-cap, yardstick competition, revenue-cap, or also use the legal framework of competition law.