Spatial Aspects of Convergence Evidence from Russian Regions
2.2 Theoretical framework – the Solow Growth Model
The benchmark theoretical framework for convergence and growth is the neoclassical Solow-Swan exogenous growth model (Solow 1956; Swan, 1956). In this paragraph I will briefly summarize the results of the Solow model and then discuss the issue of non-independent observations to introduce the spatial econometrics models. This brief explanation is based on Anderson and Thiesse (2004, Part III) and Romer (2006).
Consider an economy where aggregate physical capital Kt and labor Lt are used to produce an homogenous good.
At denotes labor-augmenting technical change or the effectiveness of labor which grows at an exogenous rate g. Capital depreciates at an exogenous rate δ. The saving rate s is exogenously defined as the fraction of output to be used as an input in t+1 and the rest is used for consumption c. The production function takes the form
Yt = F(Kt, Lt At) (1)
with constant returns to scale and diminishing marginal products for K and L, which depends only on the K/L ratio.
Defining quantities per unit of effective labor, with y* = Y/(Lt At) and k* = K/(Lt At), the production function simplifies to
y* = f(k*) (2)
The fundamental difference equation of the Solow model is given by
kt+1* - kt* ≤ sf(kt*) – (δ + n + g)kt* (3)
where n is the growth rate of population and ct*=(1-s) f(k*).
From (3) we can see that the rate of change of the capital stock per unit of effective labor is the difference between two terms: sf(kt*), which is the actual investment per unit of effective labor, and (δ + n + g)kt*, which is the breakeven investment, i.e. the amount of investment necessary to keep k at its existing level.
The steady-state is the point where physical capital per effective unit of labour k* is constant.
It corresponds to a stationary point of equation (3) so that the equation becomes
0 = sf(kt*) – (δ + n + g)kt* (4) The amount of investment, sf(k), is employed to replenish the K/L ratio given depreciation δ, population growth rate n and technological progress g.
At k*, investments sf(kt*) equal effective depreciation (δ + n + g)kt* and k remains constant over time.
The behavior of aggregate variables in steady-state can be summed up as follows:
- effective labor AL grows at rate g+n
- capital grows at g+n because when k is constant, K=ALk*=AL
- the assumption of constant returns to scale implies that the aggregate output grows at rate g+n.
In steady-state, all variables grow at constant rates and the rate of growth of output per capita is determined only by the rate of growth of technological progress g.
The Solow model proves that given any initial level k0, the economy will convergence to the steady-state with a monotonic transition:
- if k0 < k*, the growth rate of capital per unit of effective labor is positive and decreases gradually
- if k0 > k*, the growth rate is negative and increases gradually.
The model implies conditional convergence: countries which are far from the steady-state will grow faster than those which are close to the steady-state.
Steady-state capital, productivity and income are determined by A, δ, g, s and n.
Assuming that countries have the same technology, differences in income and productivity should be explained by cross-country differences in s and n.
The first study to provide an empirical specification for the Solow model was Baumol (1986) with the cross-section regression
log (yt)– log(yo) = a + b* log(yo) (5)
If the coefficient b is negative, there is proof of convergence. Barro (1991) and Barro and Sala-i-Martin (1992) define the concepts of absolute β convergence and conditional β convergence founded on the results of the Solow model. Poor countries
could grow faster than rich countries only if they shared the same steady-states, that is a situation of absolute convergence. If countries instead are characterized by different steady-states, then the concept of conditional convergence applies as it is necessary to condition the different steady-states themselves.
The empirical approach received two main critics through the years: first, the problem of omitted variables which is typical of cross-sectional studies; second, a critics specific to application in regional studies, where observations may not be considered as independent from each other. The use of panel data methods helps to solve, at least partially, these issues (Islam 1995; Arbia, Basile, Piras 2005) as, with regards to differences across regions, they allow to control for unobserved heterogeneity through regional fixed effects αi, constant over time, and time effects ct
ln[yt+k,i /yt,i] = αi + β*lnyt,i + ct + εt,i (6) where i (i=1…N) denotes the observations, and t (t=1…T) denotes time periods.
The dependent variable is the log of the annual growth rate of per-capita GRP and αi are the time-invariant regional fixed effects.
The literature on spatial econometrics explicitly relaxes the assumption of independent observations i, (i=1…N), which is a strong one when dealing with regions of the same country, and controls for spatial dependence (spatial autocorrelation) and/or spatial heterogeneity in the same panel-data environment. Applications are numerous, both with cross-sectional and panel data. See for example Elhorst (2001, 2003), Ertur, Le Gallo, Baumont (2006), Buccellato (2007), Ledyaeva (2009) and Kholodilin, Oshepkov, Siliverstov (2009) for applications to the Russian Federation.
The model for spatial dependence with panel data dates back to Anselin (1988) and Arbia (1989). Spatial dependence can be incorporated in the panel regression model seen above in two ways, either creating a spatial autoregressive model SAR or a spatial error model SEM.
In the SAR model, spatial dependence enters the model as an additional regressor in the form of a spatially lagged dependent variable Wy where W is the weight matrix as defined in section 2.1. This specification is used to verify the presence of spatial interactions between the regions when it is likely that yi is influenced by y from the
neighbouring regions. The panel regression model (6) becomes a fixed-effect spatial lag model with the form
ln[yt+k,i /yt,i] = αi + ρ
∑
= N
j wij
1
ln[yt+k,i /yt,i] + β*lnyt,i + ct + εt,i (7)
where ρ is the spatial autocorrelation coefficient which provides evidence of positive spatial autocorrelation if positive and significant.
In the SEM model, spatial dependence enters the model in the error term εt,i. This specification is used to correct the influence of spatial dependence which takes the form of a nuisance. It simply assumes that the errors are spatially correlated and controls for it.
Therefore the model (6) takes the following form
ln[yt+k,i /yt,i] = αi + β*lnyt,i + ct + εt,i
(8) with εt,i = λ
∑
= N
j wij
1
εt,i + ηι
where λ is the spatial correlation coefficient and ηι are assumed to be normally distributed with zero mean and known variance.
Both models are estimated by using maximum likelihood with the R package splm.
3. Empirical analysis