Risk-sharing within Brazil and South America

Abstract

Our findings suggest that risk-sharing is 24.6% higher within Brazilian federal states than between a sample of South American countries. This “border effect” occurs irrespective of the geographical distance between regions which, in turn, decreases risk-sharing (by 2.3% per thousand kilometers). We report that the variance of state disposable income is between 73.1 and 78.5% lower than the variance of gross state product in Brazil. Our results show that fiscal federalism promotes risk-sharing by reducing the volatility of disposable income. The tax-transfers system is progressive as income persistently flows from rich to poorer Brazilian states. We conclude that the benefits from increasing international integration within South America will be higher than in the intranational case.

Appendices

Appendix A Revision of the income forecasts for the VAR(1) case

This appendix demonstrates Proposition 1. The individual and permanent income growth joint process are given by the system (5), which is rewritten here for convenience:

$$\begin{aligned} \begin{pmatrix} \Delta y_{at}\\ \Delta y_{jt} \end{pmatrix} \equiv \begin{pmatrix} A_{j}^{11} &{} A_{j}^{12} \\ A_{j}^{21} &{} A_{j}^{22} \end{pmatrix} \begin{pmatrix} \Delta y_{at-1}\\ \Delta y_{jt-1} \end{pmatrix} + \begin{pmatrix} v_{1jt}\\ v_{2jt} \end{pmatrix}. \end{aligned}$$

which can be written in matrix form as: $\mathbf {\Delta Y}_{jt} = {\textbf{A}}_{j} \mathbf {\Delta Y}_{j t-1} + {\textbf{V}}_{jt}$, where ${\textbf{Y}}_{jt} = \begin{pmatrix} y_{at}\\ y_{jt} \end{pmatrix}$, $\mathbf {\Delta Y}_{jt} = \begin{pmatrix} \Delta y_{at}\\ \Delta y_{jt} \end{pmatrix}$, ${\textbf{A}}_{j} = \begin{pmatrix} A_{j}^{11} &{} A_{j}^{12} \\ A_{j}^{21} &{} A_{j}^{22} \end{pmatrix}$ and ${\textbf{V}}_{jt} = \begin{pmatrix} v_{1jt}\\ v_{2jt} \end{pmatrix}$.

The VAR(1) process can also be written in a $VMA(\infty )$ form: $\mathbf {\Delta Y}_{jt} = \sum _{i=0}^{\infty } {\textbf{A}}_{j}^{i} {\textbf{V}}_{t-i}$ (see Martin et al. 2013, chapter 13). Rewriting the equation above s periods forward, we have $\mathbf {\Delta Y}_{jt+s} = \sum _{i=0}^{\infty } {\textbf{A}}_{j}^{i} {\textbf{V}}_{t+s-i}$, which is equivalent to ${\textbf{Y}}_{jt+s} = {\textbf{Y}}_{jt+s-1} + \sum _{i=0}^{\infty } {\textbf{A}}_{j}^{i} {\textbf{V}}_{t+s-i}$. Taking expectations at time t and $t-1$, we have:

$$\begin{aligned} \mathbb {E}_{t}{\textbf{Y}}_{jt+s} = {\textbf{Y}}_{jt+s-1} + \sum _{i=0}^{\infty } {\textbf{A}}_{j}^{s+i} {{\textbf {V}}}_{t-i} , \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}_{t-1}{\textbf{Y}}_{jt+s} = {\textbf{Y}}_{jt+s-1} + \sum _{i=1}^{\infty } {\textbf{A}}_{j}^{s+i} {\textbf{V}}_{t-i} . \end{aligned}$$

We finally write

$$\begin{aligned} \mathbb {E}_{t}{\textbf{Y}}_{jt+s} - \mathbb {E}_{t-1}{\textbf{Y}}_{jt+s} = {\textbf{A}}_{j}^{s} {\textbf{V}}_{t}. \end{aligned}$$

(17)

Recall that the update on permanent income is written as

$$\begin{aligned} \Delta yp_{at} = (1 - \beta )\sum _{s=0}^{\infty }\beta ^{s} (\mathbb {E}_{t}y_{at+s}-\mathbb {E}_{t-1}y_{at+s}), \end{aligned}$$

and

$$\begin{aligned} \Delta yp_{jt} = (1 - \beta )\sum _{s=0}^{\infty }\beta ^{s} (\mathbb {E}_{t}y_{jt+s}-\mathbb {E}_{t-1}y_{jt+s}), \end{aligned}$$

for both the aggregate and regional cases, respectively. Hence, we can write

$$\begin{aligned} \mathbf {\Delta Yp}_{jt} = (1- \beta ) \sum _{s=0}^{\infty }\beta ^{s}(\mathbb {E}_{t}{\textbf{Y}}_{jt+s} - \mathbb {E}_{t-1}{\textbf{Y}}_{jt+s}), \end{aligned}$$

where $\mathbf {{\Delta Yp}}_{jt} = \begin{pmatrix} \Delta yp_{at}\\ \Delta yp_{jt} \end{pmatrix}$. Using Eq. (17):

$$\begin{aligned} \mathbf {\Delta Yp}_{jt} = (1- \beta ) \sum _{s=0}^{\infty }(\beta {\textbf{A}}_{j})^{s} {\textbf{V}}_{t}. \end{aligned}$$

Assuming that the matrix $\beta {\textbf{A}}_{j}$ is invertible, the following expression is valid:

$$\begin{aligned} \sum _{s=0}^{\infty }(\beta {\textbf{A}})^{s} = (\textit{I}- \beta {\textbf{A}}_{j})^{-1}, \end{aligned}$$

we then have:

$$\begin{aligned} \mathbf {\Delta Yp}_{jt} = (1- \beta ) (\textit{I}- \beta {\textbf{A}}_{j})^{-1} {\textbf{V}}_{t}, \end{aligned}$$

and by denoting ${\textbf{B}}_{j} = (\textit{I}- \beta {\textbf{A}}_{j})^{-1}$, the last equation can be written as

$$\begin{aligned} \mathbf {\Delta Yp}_{jt} = (1- \beta ) {\textbf{B}}_{j} {\textbf{V}}_{t}, \end{aligned}$$

or, alternatively as

$$\begin{aligned} \begin{pmatrix} \Delta yp_{at}\\ \Delta yp_{jt} \end{pmatrix}= (1- \beta ) \begin{pmatrix} B_{j}^{11} &{} B_{j}^{12} \\ B_{j}^{21} &{} B_{j}^{22} \end{pmatrix} \begin{pmatrix} v_{ajt}\\ v_{jt} \end{pmatrix}. \end{aligned}$$

Appendix B Variance Decomposition of Disposable Income

We provide another way of measuring the role of fiscal federalism on risk-sharing by following Asdrubali et al. (1996). We start by employing a variance decomposition of disposable income using the identity, $Y_{j} = \frac{Y_{j}}{YD_{j}} \frac{YD_{j}}{C_{j}} C_{j}$, which results in

$$\begin{aligned} 1 = \beta _{F} + \beta _{C} + \beta _{\mu }. \end{aligned}$$

(18)

where $Y_{j}$ represents the gross state product of region j, $YD_{j}$ is the corresponding disposable income, which consists of $Y_{j}$ minus taxes plus transfers to region j; $C_{j}$ denotes consumption. Equation (18) thus represents a decomposition of total risk-sharing. The parameter $\beta _F$ stands for the contribution of the tax-transfer system, $\beta _C$ for the financial and other channels, and finally, $\beta _{\mu }$ represents that part of income that is not shared. The sum $\beta _{F} + \beta _{C}$ represents the percentage of income shocks that are shared. Following Mélitz and Zumer (1999), we establish that $\beta _{\mu }$ is predetermined, based on our previous results. We thus consider that $\beta _{\mu }=14.6\%$, as this is our “preferred” estimate for the regional risk-sharing degree (panel estimation, assuming an AR(1) specification for income). Given the hypothesis that the error terms are independent, we can write:

$$\begin{aligned} \beta _{F} = \frac{{{\,\textrm{Cov}\,}}(\Delta y_{j}, \Delta y_{j} - \Delta yd_{j})}{{{\,\textrm{Var}\,}}(\Delta y_{j})}. \end{aligned}$$

The coefficient $\beta _{F}$ is found by estimating the following equation:

$$\begin{aligned} \Delta y_{jt} - \Delta yd_{jt} = v_{F,t} + \beta _{F} \Delta y_{jt} + \epsilon _{F,jt}, \end{aligned}$$

(19)

where $v_{F,t}$ represents a time fixed effect, which allows for an unobservable risk-sharing across regions in a given year; $\epsilon _{F,jt}$is the error term. Variables in lower cases are in natural logarithms. Time subscripts are suppressed for simplicity and j represents the respective region.

Table 14 State-level AR(1)

Regression errors are heteroskedastic in our sample because the variance of gross state product differs substantially between states across time. Hence, we estimate Eq. (19) using two step Generalized Least Squares (GLS), as in Asdrubali et al. (1996) and Sørensen et al. (2007). The database is the same one used to test for fiscal federalism in Sect. 4.1. We find that 13.85% of the shocks are smoothed out by the federal government through taxes and transfers ($\beta _{F}$). As we assumed that 14.60% of the shocks are not smoothed ($\beta _{\mu }$), the smoothing by other channels ($\beta _{C}$) amounts to 71.55%. The finding of a 13.85% reduction in risk via the fiscal channel is in line with our previous estimates.

Estimation results for individual states

This section reports the AR(1) and VAR(1) (Eqs. (4) and (5)) estimation results for each state, in addition to the individual risk-sharing results (Eq. 1) for each income specification.

Table 15 State-level VAR(1)

Table 16 State risk-sharing estimations (Random Walk)

Table 17 State risk-sharing estimations (AR(1))

Table 18 State risk-sharing estimations (VAR(1))

Robustness: estimation results of Eq. (16)

The first row of Table 20 presents the findings using a fixed effect panel model for the Brazilian states and Eq. (16). The second row shows the OLS results for Brazil within South American countries. Table 20 reports the OLS results of (16) estimated for each state individually. The degrees of freedom are low in the state-by-state estimations as there are 17 observations per state. As can be seen, the results provide evidence against full risk-sharing for both regional and international cases. However, the estimated $\tau$ is lower for the states, supporting the conclusion of a higher degree of intranational integration relative to the international one.

Table 19 Risk-sharing standard estimations

Table 20 State risk-sharing standard estimations