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.

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Notes

  1. 1.

    The meta-analysis of Abreu et al. (2005) finds that across 48 studies the average convergence rate is 4.3 %, much higher than Barro’s 2 %. In part, this finding is due to smaller samples. In part, it is due to the use of fixed effect estimation, which raises the convergence coefficient. Given our findings, a cross-country convergence rate above 2 % only deepens the puzzle of why regions don’t converge faster than countries.

  2. 2.

    Different limits to mobility have different consequences for welfare. In the case of non-tradability of certain locally produced goods, such as housing, perfect mobility of labor would suffice to equalize the living standards of workers across regions (as differences in price levels would offset nominal income differences). Barriers to mobility of labor would in contrast entail differences in the living standards of workers across regions. In our analysis, we try to directly measure living costs as well as potential regulatory barriers to mobility and look at migration of productive factors.

  3. 3.

    One can view Eq. (3) as resulting from a two-period OLG structure in which the young are endowed with raw labor and invest its remuneration into physical and human capital whose return they consume when old.

  4. 4.

    In this one-good model, there is no trade in goods across regions, but in a multi-goods model of Hecksher–Ohlin type, imperfect capital mobility would be isomorphic to imperfect trade in goods.

  5. 5.

    We view this random shock as stemming from a transitory (multiplicative) shock to regional productivity \(A_i \).

  6. 6.

    See the Online Appendix 4 for a list of sample countries and years with DHS data.

  7. 7.

    Specifically, we compute the deflator as \((HC_{i,j,t} /HC_{j,t} )^{0.3}\), where \(HC_{i,j,t} \) is the housing cost in region \(i\) of country \(j\) on period \(t\) and \(HC_{j,t} \) is the average cost of housing in country \(j\) and period \(t\).

  8. 8.

    In column [7], the coefficient on years of schooling is 0.0056 while the coefficient on regional GDP per capita is 0.0227. This implies that one extra year of schooling increases steady-state GDP per capita by about 24 % (=0.0056/0.0227). To interpret the implication of this coefficient in terms of mincerian returns to schooling, take our production function where per capita output is \(y=Ah^{\alpha }\). Given that \(h\) combines human and physical capital, but we do not have data on the latter, assume that physical capital is a linear function of human capital, namely \(K=zH\). Then, given the formula for \(h\) laid out in Sect. 2 and the mincerian equation \(H=e^{\mu S}\), we can approximate \(y\approx A(1+z^{\alpha \frac{\theta }{\theta +\gamma }})e^{\alpha \mu \bar{S}}\), where \(\bar{S}\) is average years of schooling. This formula implies that \(dlny=\alpha \mu d\bar{S} \). To match the regression estimate, coefficients should be such that \(\alpha \mu =0.24\). Given that \(\alpha \) is close to one, the country-wide mincerian return \(\mu \) should be about 0.25, which is the ballpark of the values accounted for in Gennaioli et al. (2013) by using managerial human capital. The same calculation implies that in our preferred specification in column (3) the mincerian return is close to 10% .

  9. 9.

    A 1 % difference in convergence rates has a substantial impact on the length of time to converge. For example, per capita GDP in the poorest region in the median country in our sample is 40 % below the country mean. Closing a 40 % gap with the steady state level of income would take 25 years at a 2 % convergence rate but only 17 years at a 3 % convergence rate.

  10. 10.

    We also tried: (1) an index of the regulation of capital flows from Abiad et al. (2008), (2) an index of the regulation of the banking from Abiad et al. (2008), (3) an index of capital controls from Schindler (2009), and (4) the number of months of severance payments for a worker with 9 years of tenure on the job from Aleksynska and Schindler (2011).

  11. 11.

    Formally, the regressions in Panel A of Table 7 do not include the term \(\alpha \cdot \beta \cdot d_c \cdot lny_t \) appearing in Eq. (8). The reason is that national and regional incomes are strongly correlated. Thus, having a set of interactions between national income and country-level determinants of the speed of convergence creates multicollinearity problems. Nevertheless, the results on interactions are qualitatively similar if we add national income as a control (Table 7B).

  12. 12.

    Results are qualitatively similar for the index of capital controls (i.e. 7.40 percentage points faster growth when GDP per capita is 20 % below the steady state and the index of capital controls is one standard deviations above its average) and the index of banking regulation (i.e. 14.0 percentage point faster growth when GDP per capita is 20 % below the steady state and the index of banking regulation is one standard deviations above its average).

  13. 13.

    We also explored the convergence of the standard deviation of GDP per capita (“sigma convergence”). To that end, we computed the change in the within-country standard deviation of regional GDP per capita between the first and last cross-section of each country and regressed it on the following country-level variables: (1) the (log) initial GDP per capita, (2) the growth of GDP per capita, (3) initial years of schooling, (4) change in schooling, (5) government consumption as a percent of GDP, and (6) government transfers and subsidies as a percent of total government expenditure. In unreported univariate OLS regressions, years of schooling—with a positive coefficient—is the only significant regressor.

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Correspondence to Andrei Shleifer.

Additional information

We are grateful to Jan Luksic for outstanding research assistance, to Antonio Spilimbergo for sharing the structural reform data set, and to Robert Barro, Peter Ganong, and Simon Jaeger for extremely helpful comments. Gennaioli thanks the European Research Council for financial support through the Grant ERC-GA n. 241114. Shleifer acknowledges financial support from the Kauffman Foundation.

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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

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Gennaioli, N., La Porta, R., Lopez De Silanes, F. et al. Growth in regions. J Econ Growth 19, 259–309 (2014). https://doi.org/10.1007/s10887-014-9105-9

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Keywords

  • Convergence
  • Mobility barriers
  • Human capital

JEL Classification

  • O43
  • O47
  • R11