Spatial convergence of real wages in Russian cities

Abstract

This paper provides an empirical test of spatial wage convergence in Russian cities. Using geo-coded data covering 997 Russian cities and towns from 1996 to 2013, I show that real city wages (i) converge over time and (ii) are significantly affected by the initial levels of real wages in neighboring cities. I also find that cities of the Far North, where a special wage policy is implemented, were converging more slowly than the rest of the country. I find a significant negative impact of regional subsidies on real wages in cities outside the Far North and that the effect of extractive industries on real wage has become weaker. These results are robust to the radius of spatial interaction, and my conclusions hold if remote settlements are not taken into account.

Appendices

Appendix 1: empirical papers on regional income convergence using spatial Russian data

See Table 5.

Table 5 Literature on Russian regional income convergence using spatial data

Appendix 2: Proof of Proposition 1

To prove part (i), we use the chain rule to obtain

$$\begin{aligned} \frac{\mathrm {d}\mathbf {y}}{\mathrm {d}t}= \frac{\partial \mathbf {y}}{\partial \varvec{\lambda }}\cdot \frac{\mathrm {d}\varvec{\lambda }}{\mathrm {d}t}. \end{aligned}$$

Combining this with (2) yields

$$\begin{aligned} \frac{\mathrm {d}\mathbf {y}}{\mathrm {d}t}= \mathbf {H}(\varvec{\lambda },\mathbf {D}), \end{aligned}$$

(9)

where

$$\begin{aligned} \mathbf {H}(\varvec{\lambda },\mathbf {D}) \equiv \frac{\partial \mathbf {y}}{\partial \varvec{\lambda }} \cdot \mathbf {F}(\varvec{\lambda },\mathbf {D})\cdot \mathbf{V}(\varvec{\lambda },\mathbf {D}). \end{aligned}$$

As discussed above, the mapping $\mathbf {y}(\varvec{\lambda },\mathbf {D})$ is invertible, so we can define

$$\begin{aligned} \mathbb {G}(\mathbf {y},\mathbf {D})\equiv (\mathbf {H}\circ \mathbf {y}^{-1})(\mathbf {y},\mathbf {D}), \end{aligned}$$

where $\mathbf {y}^{-1}(\cdot )$ is the mapping inverse to $\mathbf {y}(\varvec{\lambda },\mathbf {D})$ for each given distance matrix $\mathbf {D}$, while $\circ$ denotes composition of mappings. Doing so and setting

$$\begin{aligned} g_{i}(y_{i},\mathbf {y}_{-i},\mathbf {D})\equiv \ln G_{i}(y_{i},\mathbf {y}_{-i},\mathbf {D}), \end{aligned}$$

we find that (9) becomes (4). This completes the proof of part (i).

To prove part (ii), observe first that uniqueness of the steady state follows immediately from the necessity and sufficiency of real wage equalization for a spatial configuration to be a long-run equilibrium (see the discussion after equation (3). To show stability, let $\mathcal {U}\in \Delta _{n-1}$ be an open neighborhood of $\varvec{\lambda }^{*}$, such that any solution $\varvec{\lambda }(t)$ to (4) starting in $\mathcal {U}$ converges asymptotically to $\varvec{\lambda }^{*}$. Set $\mathcal {V}\equiv \mathbf {y}(\mathcal {U})$. Clearly, $\mathcal {V}$ is an open neighborhood of $\mathbf {y}^{*}$. Furthermore, by definition of $\mathcal {V}$ for any solution $\mathbf {y}(t)$ to (4) such that $\mathbf {y}(0)\in \mathcal {V}$ there exists a solution $\varvec{\lambda }(t)$ to (2) such that $\varvec{\lambda }(0)\in \mathcal {U}$. Because $\varvec{\lambda }(t)$ asymptotically converges to $\varvec{\lambda }^{*})$, and because $\mathbf {y}(\varvec{\lambda },\mathbf {D})$ is continuous, it must be that $\mathbf {y}(t)$ asymptotically converges to $\mathbf {y}^{*}$. This means $\mathbf {y}^{*}$ is a stable steady state. This completes the proof of part (ii).

To prove part (iii), we log-linearize the system (4) in the vicinity of $\mathbf {y}^{*}$. This yields

$$\begin{aligned} \frac{\mathrm {d}\ln y_{i}}{\mathrm {d}t}= \sum _{j =1}^{n}b_{ij}(\mathbf {D})(\ln y_{j}-\ln y^{*}), \end{aligned}$$

(10)

where $b_{ij}$ are defined by

$$\begin{aligned} b_{ij}(\mathbf {D})=g_{i}(\mathbf {y}^{*})\cdot \left. \frac{\partial \ln g_{i}}{\partial \ln y_{j}}\right| _{\mathbf {y}=\mathbf {y}^{*}}. \end{aligned}$$

Solving (10), we obtain

$$\begin{aligned} \ln \mathbf {y}(t) - \ln \mathbf {y}(0) = [\mathbf {I}-\exp (t\cdot \mathbf {B}(\mathbf {D}))]\cdot [\ln \mathbf {y}^{*} - \ln \mathbf {y}(0)], \end{aligned}$$

(11)

where

$$\begin{aligned} \mathbf {B}(\mathbf {D})\equiv (b_{ij}(\mathbf {D}))_{i,j=1,\ldots ,n}, \end{aligned}$$

while $\exp [t\cdot \mathbf {B}(\mathbf {D})]$ is the matrix exponential defined as follows:

$$\begin{aligned} \exp [t\cdot \mathbf {B}(\mathbf {D})]\equiv \mathbf {I} + t\cdot \mathbf {B}(\mathbf {D}) +\frac{t^{2}}{2!}\cdot [\mathbf {B}(\mathbf {D})]^{2} + \cdots \end{aligned}$$

Fix some moment of time $T>0$ and set:

$$\begin{aligned} \mathbf {W}(\mathbf {D})\equiv \mathbf {I}-\exp [T\cdot \mathbf {B}(\mathbf {D})],\quad \varvec{\alpha }\equiv \mathbf {W}(\mathbf {D})\ln \mathbf {y}^{*}. \end{aligned}$$

Denote the diagonal entries $w_{ii}(\mathbf {D})$ of the matrix $\mathbf {W}(\mathbf {D})$ by $\beta _{i}$. Using this notation and stating (11) in coordinate form yields (5). This completes the proof of part (iii) and proves the whole proposition.

Appendix 3: descriptive statistics

See Fig. 6 and Table 6.

Fig. 6

Means of real city wages. Number of cities in parentheses

Table 6 Correlations between real city wages by regions and real GRP per capita

Appendix 4: panel unit root test results

See Tables 7 and 8.

Table 7 HT panel unit root test results

Table 8 IPS panel unit root test results

Appendix 5: Bayesian model estimation

Consider an SDM

$$\begin{aligned} \mathbf {y}=\mathbf {X}\beta +\rho \mathbf {W}\mathbf {y}+\theta \mathbf {W}\mathbf {X}+\varepsilon \end{aligned}$$

(12)

here $\mathbf {X}$ is an $n\times k$ matrix of explanatory variables, $\beta$ is a $k\times 1$ parameter vector, $\mathbf {W}$ is a $n\times n$ spatial matrix, $\rho$ and $\theta$ are spatial scalar parameters, $\varepsilon$ is an $n\times 1$ disturbance vector, $\varepsilon \sim N(0,\sigma ^{2}\mathbf {I}_{n})$, here $\mathbf {I}_{n}$ is the identity matrix of size n. An intercept parameter in (12) is omitted for simplicity.

Following LeSage (1997), LeSage and Parent (2007), I use Metropolis within Gibbs sampling procedure (Metropolis–Hastings sampling for the parameter $\rho$ and Gibbs sampling from the normal distribution for the parameters $\beta ,\theta$ and normal inverse gamma distributions for $\sigma ^{2}$).

I carry out 100,000 draws of the Metropolis within Gibbs sampling procedure, excluding the first 10,000 draws, to produce posterior estimates for each model in Tables 2 and 3. Estimates are mainly based on a spatial toolbox developed by LeSage (2001) for MATLAB software. I run Raftery–Lewis diagnostic tests for each parameter of the chain, in order to detect convergence to the stationary distribution and to provide a way of bounding the variance of estimates of quantiles of functions of parameters. In the Raftery-Lewis test, q is a quantile of the quantity of interest (e.g., 0.05), r is a level of precision desired (e.g., $\pm \,0.025$), and s is a probability associated with r (e.g., 0.90). The above-mentioned number of draws of the Metropolis within Gibbs sampling procedure is sufficient for $q=0.001$, $r=0.0005$, and $s=0.999$ for all the estimated models. Significance levels provided in Tables 2 and 3 are derived from simple descriptive statistics of the Markov chain iterations for each sample.