Growth in regions

Abstract

We use a newly assembled sample of 1,528 regions from 83 countries to compare the speed of per capita income convergence within and across countries. Regional growth is shaped by similar factors as national growth, such as geography and human capital. Regional convergence rate is about 2 % per year, comparable to that between countries. Regional convergence is faster in richer countries, and countries with better capital markets. A calibration of a neoclassical growth model suggests that significant barriers to factor mobility within countries are needed to account for the evidence.

Appendices

Appendix 1

Proof of Proposition 1

At time t+1, the employment of capital in region $i$ is equal to $h_{i,t+1} =v_{t+1} \cdot \left( {\hat{h} _{i,t+1} } \right) ^{\tau }\left( {\hat{A}_i \cdot h_{t+1} } \right) ^{1-\tau }$. By replacing in this expression the capital endowment $\hat{h} _{i,t+1} \equiv sy_{i,t} =sA_i h_{i,t}^\alpha ,$ and the aggregate capital shock $h_{t+1} =s\int A_i h_{i,t}^\alpha di$, we obtain that the growth of employed capital in region $i$ is equal to:

$$\begin{aligned} \frac{h_{i,t+1} }{h_{i,t} }=v_{t+1} \cdot h_{i,t}^{\alpha \tau -1} \cdot \left( {sA_i } \right) ^{\tau }\left( {\hat{A}_i \cdot s\cdot \int A_i h_{i,t}^\alpha di} \right) ^{1-\tau }, \end{aligned}$$

which is Eq. (5) in the text.$\square$

A steady state in the economy is a configuration of regional employment $\left( {h_i^*} \right) _i$ and an entailed aggregate capital employment $h^{*}=s\mathop \smallint \nolimits ^ A_i \left( {h_i^*} \right) ^{\alpha }di$ such that the steady state capital $h_r^*$ in any region $r$ is:

$$\begin{aligned} 1=v^{*}\cdot \left( {h_r^*} \right) ^{\alpha \tau -1}\cdot \left( {sA_r } \right) ^{\tau }\left( {\hat{A}_r \cdot s\cdot \int A_i \left( {h_i^*} \right) ^{\alpha }di} \right) ^{1-\tau }, \end{aligned}$$

where $v^{*}$ is the normalization factor in the steady state. This can be rewritten as:

$$\begin{aligned} \left( {h_r^*} \right) ^{1-\alpha \tau }=A_r^{\tau -\left( {1-\alpha } \right) (1-\tau )} \cdot v^{*}\cdot s\cdot \left( {\frac{\int A_i \left( {h_i^*} \right) ^{\alpha }di}{\int A_i^{1-\alpha } di}} \right) ^{1-\tau }. \end{aligned}$$

(14)

There is always an equilibrium in which $h_i^*=0$ for all regions $i$. Once we rule out this possibility, the equilibrium is interior and unique. In fact, Eq. (14) can be written as $h_r^*=A_r^{\frac{\tau -\left( {1-\alpha } \right) (1-\tau )}{1-\alpha \tau }} \cdot C$, where $C$ is a positive constant which takes the same value for all depending on the entire profile of regional capital employment levels. Because the capital employed in a region does not affect (has a negligible impact on) the aggregate constant $C$, there is a unique value of $h_r^*$ fulfilling the condition. By plugging the value of $h_r^*$ into the expressions for $v^{*}$ and $\mathop \int A_i \left( {h_i^*} \right) ^{\alpha }di$, one can find that for $\alpha <1$ and $\tau <1$, there is a unique value of $C$ that is consistent with equilibrium.

Finally, given the fact that $y_{i,t+1} /y_{i,t} =\left( {h_{i,t+1} /h_{i,t} } \right) ^{\alpha }$, the economy approaches the interior steady state according to Eq. (5) in the text.

Appendix 2: Convergence coefficients for generic values of depreciation and population growth

Our main analysis assumes a zero rate of population growth ($n=0$) and full depreciation ($\delta =1$). Focusing on this case allowed us to obtain an exact closed form for our main estimating equation. We now perform a log-linear approximation to derive convergence coefficients when $n$ and $\delta$ are generic.

In region $i$, the growth of per capita GDP between periods $t$ and $t+1$ is equal to $ln(y_{i,t+1} /y_{i,t}$) which, by the assumed production function, is equal to $\alpha ln\left( {\frac{h_{i,t+1} }{h_{i,t} }} \right) \cong \alpha \left( {\frac{h_{i,t+1} }{h_{i,t} }-1} \right) .$ There is a direct link between a region’s income growth and the growth of the region’s per capita capital employment.

Let us therefore find the law of motion for $h_{i,t}$ for generic values of $n$ and $\delta$. Denote by $\hat{H}_{i,t}$ the capital endowment of region $i$ at time $t$, and by $H_{i,t}$ the same region’s employment of capital. The law of motion for $\hat{H}_{i,t}$ then fulfills:

$$\begin{aligned} \hat{H}_{i,t+1} =sA_i H_{i,t}^\alpha L_{i,t}^{1-\alpha } +\left( {1-\delta } \right) \cdot \hat{H}_{i,t} . \end{aligned}$$

The capital stock next period is equal to undepreciated capital $\left( {1-\delta } \right) \cdot \hat{H}_{i,t}$ plus this period’s savings $sA_i H_{i,t}^\alpha L_{i,t}^{1-\alpha }$. To express the equation in per capita terms, we devide both sides of the above equation by the region’s population $L_{i,t}$ at time $t$ and obtain:

$$\begin{aligned} \frac{\hat{H}_{i,t+1} }{L_{i,t} }&\equiv \frac{\hat{H}_{i,t+1} }{L_{i,t+1} }\cdot \frac{L_{i,t+1} }{L_{i,t} }\equiv \hat{h} _{i,t+1} (1+n)\\&=sAh_{i,t}^\alpha +\left( {1-\delta } \right) \cdot \hat{h} _{i,t} . \end{aligned}$$

The law of motion of the region’s per capita capital endowment can be approximated as:

$$\begin{aligned} \hat{h} _{i,t+1} \cong sA_i h_{i,t}^\alpha +\left( {1-\delta -n} \right) \hat{h} _{i,t}. \end{aligned}$$

(15)

To solve for regional GDP growth, we need to transform the above equation into a law of motion for regional capital employment $h_{i,t}$. To do so, we can exploit our migration equation (4) to write:

$$\begin{aligned} \hat{h} _{i,t} =\left( {h_{i,t} } \right) ^{\frac{1}{\tau }}\cdot \left( {\hat{A}_i \cdot h_t } \right) ^{-\frac{1-\tau }{\tau }}\cdot \left( {v_t } \right) ^{-\frac{1}{\tau }}\cdot \end{aligned}$$

By plugging the above equation into (15) we then obtain, after some algebra, the following equation:

$$\begin{aligned} \frac{h_{i,t+1} }{h_{i,t} }-1\cong \left[ {sA_i h_{i,t}^{\frac{\alpha \tau -1}{\tau }} \cdot \left( {v_t } \right) ^{\frac{1}{\tau }}\cdot \left( {\hat{A}_i \cdot h_t } \right) ^{\frac{1-\tau }{\tau }}+\left( {1-\delta -n} \right) } \right] ^{\tau }\cdot \left[ {\frac{h_{t+1} }{h_t }} \right] ^{1-\tau }\cdot \left[ {\frac{v_t }{v_{t+1} }} \right] -1. \end{aligned}$$

By noting that the aggregate capital stock grows at the rate $\left( {h_{t+1} /h_t } \right) =s\left( {y_t /h_t } \right) +(1-n-\delta )$, we can rewrite the above law of motion as:

$$\begin{aligned}&\frac{h_{i,t+1} }{h_{i,t} }-1\cong \left[ {sA_i h_{i,t}^{\frac{\alpha \tau -1}{\tau }} \cdot \left( {v_t } \right) ^{\frac{1}{\tau }}\cdot \left( {\hat{A}_i \cdot h_t } \right) ^{\frac{1-\tau }{\tau }}+\left( {1-\delta -n} \right) } \right] ^{\tau }\\&\quad \quad \times \left[ {s\left( {y_t /h_t } \right) +(1-n-\delta )} \right] ^{1-\tau }\cdot \left[ {\frac{v_t }{v_{t+1} }} \right] -1. \end{aligned}$$

A steady state is identified by the condition $h_{i,t+1} =h_{i,t} =h_{i,SS}$ and thus $h_{t+1} =h_t =h_{SS}$. Because in the steady state there is no migration, and the human capital endowment of a region is also equal to its ideal employment level, we also have that $v_{t+1} =v_t =1$. As a result, the steady state is identified by the following conditions:

$$\begin{aligned}&sA_i h_{i,SS}^{\frac{\alpha \tau -1}{\tau }} \cdot \left( {\hat{A}_i \cdot h_{SS} } \right) ^{\frac{1-\tau }{\tau }}=\left( {\delta +n} \right) ,\\&s\left( {y_{SS} /h_{SS} } \right) \equiv s\left( {\int A_i h_{i,SS}^\alpha /h_{SS} } \right) =\left( {n+\delta } \right) . \end{aligned}$$

If we log-linearize with respect to regional employment $h_{i,t}$ and national output $y_t$ the right hand side of the law of motion of $h_{i,t}$ around the steady state above, we find that for any $\tau >0$ we can write the following approximation:

$$\begin{aligned} \frac{h_{i,t+1} }{h_{i,t} }-1\cong -\left( {\delta +n} \right) \cdot \left( {1-\alpha \tau } \right) \cdot \ln \left( {\frac{h_{i,t} }{h_{i,SS} }} \right) +\left( {\delta +n} \right) \cdot \left( {1-\tau } \right) \cdot \ln \left( {\frac{y_t }{y_{SS} }} \right) . \end{aligned}$$

By exploiting the fact that $ln\left( {\frac{y_{i,t+1} }{y_{i,t} }} \right) \cong \left( {\frac{y_{i,t+1} }{y_{i,t} }-1} \right) \cong \alpha \cdot \left( {\frac{h_{i,t+1} }{h_{i,t} }-1} \right)$, we can then write:

$$\begin{aligned} \frac{y_{i,t+1} -y_{i,t} }{y_{i,t} }\cong -\left( {\delta +n} \right) \cdot \left( {1-\alpha \tau } \right) \cdot \ln \left( {\frac{y_{i,t} }{y_{i,SS} }} \right) +\left( {\delta +n} \right) \cdot \alpha \cdot \left( {1-\tau } \right) \cdot \ln \left( {\frac{y_t }{y_{SS} }} \right) . \end{aligned}$$

As a result, the speed of convergence is equal to $\left( {\delta +n} \right) \cdot \left( {1-\alpha \tau } \right)$ and regional growth increases in country level income with coefficient $\left( {\delta +n} \right) \cdot \alpha \cdot \left( {1-\tau } \right)$. These coefficient boil down to those obtained under the exact formulas of our model when $\left( {\delta +n} \right) =1$ (and thus when, as assumed in the model, $\delta =1$ and $n=0)$. When, on the other hand, $\left( {\delta +n} \right) \ne 1$, the mapping between our estimates and the economy’s “deep” parameters will be different, entailing different values for $\alpha$ and $\tau$.

Appendix 3

See the Appendix Table 11.

Table 11 Description of the variables

Appendix 4

See the Appendix Table 12

Table 12 Fertility, land quality, employment in agriculture