the logo after 1 em space; see diagram B), or under the logo the right of the logo (centered vertically along the height of (the space between the logo and the words "THE WORLD the logo after 1 em space; see diagram B), or under the logo BANK" should be the same as the cap height of the words, (the space between the logo and the words "THE WORLD aligned flush left; see diagram C). When the logo is placed 46268 BANK" should be the same as the cap height of the words, on aalignedorflush dark black background, the logo should reverse to left; see diagram C). When the logo is placed whiteon(see diagram D). a dark or black background, the logo should reverse to PRMED Knowledge Brief white (see diagram D). 2. The words "THE WORLD BANK" should be set in ALL Growth andlogoIncome Convergence IncomeCAPS, Convergence 2. The words "THE WORLD BANK" should be set in ALL Univers Bold. The size of the type in relation to the CAPS, Univers Bold. The size of the type in relation to the should remain constant. Always use the art provided in Jesús Crespo Cuaresma an electronic file or in CRC. logo should remain constant. Always use the art provided in Jes´us Crespo Cuaresma Department of Economics, University of Innsbruck, Austria an electronic file or in CRC. Department of Economics, University of Innsbruck, Austria jesus.crespo-cuaresma@uibk.ac.at jesus.crespo-cuaresma@uibk.ac.at Placement of Logo Placement of Logo All World Bank books must display the World Bank logo on All World Bank books must display the World Bank logo on The Solow model of economic growth (Solow, 1956, Swan, 1956) concludes that poorer countries will TheSolowmodelofeconomicgrowth(Solow,1956,Swan,1956)concludesthatpoorercountrieswill the front and back covers, the spine, and the title page. the front and back covers, the spine, and the title page. tend to grow faster than richer ones--providedcover countries share the same production function, tend to grow faster than richer onesprovidedcover countries share the same production function, that that savings rate and population growth, and labour-augmenting technology grows at the same rate in labour-augmenting technologybegrows atattheeither rate in all countries. The existence of income convergence hascover, beenshould savings rate and population growth,FrontFrontthe allexogenousgrowthmodelversusendogenousgrowthmodelsthatdonotnecessarilyconcludeontheof countries. The existence of income convergence has thus beenusually taken tosameatest On the frontfront andOn cover, the logologo thusthe should be placed at either the usuallyplaced taken to be athe be testof lowerlowerupper left left corner and should be accompaniedby the or or upper corner and should be accompanied by the exogenous growth model versus endogenous growth models--that do not necessarily conclude on the existenceofconvergenceinincomepercapitaamongeconomies. Herewedescribedierentconcepts words "THE WORLD BANK." Placement of the World Bank words "THE WORLD BANK." Placement of the World Bank existence of convergence in income per capita among economies. Here we describe different concepts ofconvergenceusedintheempiricalliteratureoneconomicgrowthandsummarizetheresultsofthis logo logo blockthe front cover should bebe as follows:the outside block on on the front cover should as follows: the outside of convergence used in the empirical literature edgeeconomic growth and summarize the results of this literature. literature. edge onthe logo (the outer box) should be between 2.25 picas of of the logo (the outer box) should be between 2.25 picas (3/8 inch) and 3.75 picas (5/8 inch) from the trim. The logo (3/8 inch) and 3.75 picas (5/8 inch) from the trim. The logo should be placed equidistant from both trim and spine. See should be placed equidistant from both trim and spine. See samples on next page. samples on next page. The Solow model and incomeThe The Solow model and income convergence The World Bank logo is the only logo to appear on front covers convergence World Bank logo is the only logo to appear on front covers and spines of publications published by EXTOP. Any exception Total output (Yt) is assumed to depend on physicalCobb-Douglas),productioncopublishers) Total output (Yt) is assumed to depend on physicalneeds and spines of publications published by EXTOP. Any exception augmenting) technology (At) accordingAdditionala augmenting) technology (At) accordingAdditional to this guidelinelogos (totbe labouror by the publisher. to this guideline needs to be approved by the publisher. capital (Ktapproved input (Lt) and (labour- ), labour returnsreturns on all inputs, to scale on all inputs, to atoCobb-Douglas production function with costant capital(forKcosponsors input (Lt) and (labour- logos (for cosponsors or copublishers) appear at the function with costant appear at the bottom of the back cover, along with the World Bank logo. to scale bottom of the back cover, along with the World Bank logo. where (0,1). Labour input and technology are assumed to grow at constant rates n and g, Yt =YKt (AtLt) t = Kt (At1L-t)1 , - , where (0,1). Labour input and technology are assumed to grow at constant rates n and g, respectively. Physical capital is accumulated through savings (with a constant savings rate s) respectively. Physical capital is accumulated through savings (with a constant savings rate s) and depreciates at a constant rate , and depreciates at a constant rate , dKt dKt dt (1) (1) We can write (1) in terms of eective labour as dt = K t = sYt - Kt. = K t = sYt - Kt. We can write (1) in terms of effective labour as kt = syt - (n + + g)kt, (2) where kt = Kt/(AtLt) and y =tYt/(AttLt) = kt . The tsteady state level of capital per unit of eective labour (k) can be found by setting kt = 0, which leads to k = sy - (n + + g)k , (2) where kt = Kt/(AtLt) and y = Yt/(AtLt) = kt . The steady state level of capital per unit of effective labour (k) can be found by setting kt = 0, which leads to s(k) = (n + + g)k. (3) Graphically, the equilibrium level ofk is given by theintersection point of the investment per s(k) = (n + + g)k . (3) unit of eective labour curve, s(k), with the break-even investment line, [(n++g)k], as shown Graphically, the equilibrium level of k is given by the intersection point of the investment per in Figure 1.1 Countries with levels of capital per unit of eective labour below k (see k1 in unit of Figure labour curve, s(k), with the break-even investment line, [(n++g)k], as shown effective1) present positive growth in the stock of capital per unit of eective labour (see (2)), in Figure 1. Countries with levels of capital per unit of effective labour below k (see k1 in Fig- while1countries to the right of k will tend to decrease their stock of capital per unit of eective ure 1) present positive growth in the stock of capital per unit of effective labour (see (2)), while labour. countriesLog-linearizingofaround tend to decrease their stock of capital per unit of effective labour. to the right k willthe steady state level of income per unit of eective labour, Log-linearizing around the steady statedlevel=ofincome per unit of effective labour, dln(yt) t [ln(y) - ln(yt)], (4) 1Thebreak-even investment line represents the investment needed to avoid the capital stock from falling. dln(yt) dt = [ln(y) - ln(yt)], (4) 1 1The break-even investment line represents the investment needed to avoid the capital stock from falling. Figure 1: The steady state in the Solow model y,sy (n+ +g)k y=f(k) y* dk/dt sf(k) dk/dt k 1 k* k 2 k which implies that the growth rate of income per unit of eective labour (and thus income per capita) is related to the distance to the steady state level of income. This means that the Solow model concludes that (after controlling for those factors that determine dierences in the steady state level of income per capita) poorer countries should grow at higher rates that richer countries. Unconditional and conditional -convergence A natural empirical test of income convergence from (4) is based on regressing the growth rate of income per capita on initial income levels for a cross-section of countries or regions (see Barro and Sala-i-Martin, 1992). A negative (positive) correlation between these two variables indicates the existence of so-called unconditional -convergence (-divergence). Figure 2 presents the corresponding scatterplot (growth rates of GDP per capita versus initial GDP per capita in the period 1970-2000) for all countries in the world for which Penn World Table data are available and the same scatterplot for Spanish provinces in the period 1980-2005 (source: Cambridge Econometrics). As can be seen from the scatterplots, considering more homogeneous groups of economic units (which are more likely to be succesfully modelled through the theoretical setting put forward above), the empirical relevance of unconditional convergence appears more evident. Figure 2: Income growth versus initial incomeWhole world and Spanish provinces .08 Spanish provinces .04 .06 1970-2000 .03 .04 1980-2005 GDPpc, of .02 .02 income, rate .00 rate .01 growth -.02 growth .00 Annual -.04 Annual -.01 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 log(GDPpc), 1970 Income level, 1980 The Solow model predicts income convergence across countries which share the same production function, investment rate, population growth, depreciation rate and common growth rate of 2 technology. In order to account for such dierences, we can control for these (and potentially other) variables in the -convergence regression. If a negative partial correlation between initial income and subsequent income growth appears after controlling for other covariates, conditional -convergence is said to exist. -convergence and divergence Intuitively, convergence takes place if the dispersion of income across countries is reduced over time. This concept of convergence is known as -convergence. Figure 3 presents the evolution of the standard deviation of income per capita across world countries and European regions (source: Cambridge Econometrics). The results in Figure 3 give evidence of -divergence across Figure 3: Dispersion of income per capitaWhole world and European regions Standard deviation, log(GDPpc) Standard deviation, log(GDPpc), European regions 1.16 .37 1.14 .36 1.12 .35 .34 1.10 .33 1.08 .32 1.06 .31 1.04 .30 1.02 .29 1970 1975 1980 1985 1990 1995 2000 1980 1985 1990 1995 2000 countries at the world level but convergence within more homogeneous economic areas (in this case, Europe). The statistical signicance of changes in the dispersion of income can be evaluated using the test put forward by Carree and Klomp (1997). It can be easily shown that convergence implies convergence, but the opposite does not necessarily apply (see for example Furceri, 2005). The dynamics of the world distribution of income The convergence analyses presented above (and most of the studies existing on the dynamics of income at the world level) take countries as the natural unit of analysis. The results concerning convergence/divergence across countries do not necessarily imply convergence/divergence across individuals at the world level. A rst hint at the dierences appearing from both approaches can be obtained from weighted cross-country -convergence regressions using population as a weight. The unweighted parameter estimate corresponding to the data presented in Figure 2 for the whole world is 0.002 (standard deviation = 0.002), while the weighted estimate is -0.012 (standard deviation = 0.001), which indicates convergence once that we take into account the size of each country in terms of population. Sala-i-Martin (2006) reconstructs the dynamics of world income across individuals by matching macroeconomic estimates of income per capita (at purchasing power parity) with estimates of income dispersion across individuals within countries. Notwithstanding the degree of uncertainty implied by the fact that within-country dispersion estimates are not available for all countries and need to be projected from neighbouring countries, Sala-i-Martin (2006) nds evidence of income convergence for individuals in the period 1970-2000. Sala-i-Martin's (2006) approach is not without criticism. Milanovic (2003) critizises the approach heavily and pinpoints several reasons 3 why Sala-i-Martin's (2006) study has some drawbacks which tend to exagerate the decrease in global income inequality. In particular, among other issues, a) several nations where within-country income inequality has risen in the period under study (and where data are available!) are omitted of the analysis, b) the data on the distribution of income across households is treated as if they reected income distribution across individuals. Nonlinearities and club convergence Azariadis and Drazen (1990) present a theoretical model where heterogeneity in the marginal productivity of capital across levels of the capital stock leads to multiple equilibria corresponding to steady states at dierent income levels. This result implies that depending on the initial level of income, countries may converge to dierent equilibria and thus may get stuck at relatively low levels of GDP per capita, corresponding to a so-called poverty trap. Empirically, the existence of club convergence can be tested by estimating piecewise-linear models where the initial level of GDP per capita determines the parameters corresponding to the other covariates in the regression equation. Durlauf and Johnson (1996) present empirical evidence of this type of nonlinearities in cross-country growth regressions. References [1] Azariadis, C. and A. Drazen (1990), Threshold externalities in economic development. The Quarterly Journal of Economics, 105, 501526. [2] Barro, R. J. and X. Sala-i-Martin (1992), Convergence. Journal of Political Economy, 100, 223251. [3] Carree, M. A. and L. Klomp (1997), Testing the convergence hypothesis: A comment. Review of Economics and Statistics, 79, 683-686. [4] Durlauf, S. and P. Johnson (1995), Multiple regimes and cross country growth behaviour. Journal of Applied Econometrics, 10, 36584. [5] Furceri, D. (2005), and -convergence: A mathematical relation of causality. Economics Letters,89, 212215. [6] Milanovic, M. (2003), The Ricardian Vice: Why Sala-i-Martins calculations of world in- come inequality are wrong. HEW 0305003, EconWPA. [7] Sala-i-Martin, X. (2006), The world distribution of income: Falling poverty and ... conver- gence, period. The Quarterly Journal of Economics, 121, 351397. [8] Solow, R.M. (1956), A contribution to the theory of economic growth. TheQuarterlyJournal of Economics, 70, 65-94. [9] Swan, T. (1956), Economic growth and capital accumulation, EconomicRecord, 32, 334361. 4