Chapter 5 CONVERGENCE IN THE NEOCLASSICAL MODEL

Transcription

1 Chapter 5 CONVERGENCE IN THE NEOCLASSICAL MODEL Introduction The concept of economic growth has been examined thus far by looking at the performance of a single economy. In particular, we have analyzed the level and the growth rate of per capita income without any reference to similar variables in other economies. However, an important aspect of growth theory involves the analysis of variables in relative terms. For example, it is interesting to examine the path of per capita income with reference to a benchmark level, which is either given exogenously or determined by domestic economic policy. In broad terms, the concept of convergence of a variable is defined as the path of the variable towards a specific value. In the case of per capita income mentioned earlier, this value can be either constant (e.g. a specific income level expressed in constant prices) or change over time (e.g. the USA or the average EU income). Below, we examine several particular interpretations of this general definition. Convergence across countries This form of convergence occurs when countries attain a common reference value in terms of any economic variable, like the income level, the growth rate, the level of consumption, the standard of living, the interest rate, or any other real or nominal variable. This interpretation is particularly helpful in growth theory as it is used in the analysis of differences between the income levels recorded across countries. For example, Figure 5.1 compares the growth patterns of the USA and Greece since Despite the trend for convergence exhibited by these two developed countries, the real income in the USA is twice as big as the level in Greece. In addition, the differences between income levels persist over time.

2 94 P.Kalaitzidakis S.Kalyvitis Figure 5.1. Per capita GDP in the USA and Greece (in constant dollar prices) CHANGE TO EU INCOME Ελλάδα Η.Π.Α Source: World Bank, World Development Indicators (2001). It useful to note that, according to this general definition, convergence between two countries does not necessarily imply that the less-advanced economy should always experience a rise in income growth. The reduction in the gap can be equally well driven by a falloff in the income of the richer country, whereas the income of the poorer country remains constant or experiences a relatively slower decline. Hence, in economic policy terms, achieving convergence implies setting a target for the specific variable (say income) that is relatively higher than the level in the economy under discussion. Convergence across regions Examining convergence across different regions in a country requires again a point of reference (usually the income level). We can find out, for example, whether there is a significant difference between the income levels of Epirus area and Athens and if this gap is likely to persist or wear off over time. Note again that a declining gap is not necessarily equivalent to improvement in absolute terms. At national level, the aggregate income may decrease (compared to the GNP of another country), but at the same time, different regions of the country can exhibit convergence since the high income of the richer region (e.g. Athens) may decline more than the income of laggards. In conclusion, convergence across regions may also occur in the absence of growth when the richer region (e.g. the center ) does worse than the poorer one (e.g. the periphery ).

3 Economic Growth: Theory and Policy 95 Convergence across groups of countries The analysis of convergence across groups of countries requires a sample of countries that share an identical level of technological progress as well as similar economic, social, or even ethnic characteristics. Actually, it will be shown below that economies with different structural parameters may exhibit differential growth patterns despite being governed by common economic policies. Hence, our analysis should involve similar economies since it is irrelevant to examine completely different economies (e.g. a European vs. an African country). In this sense, economies with similar characteristics (like those comprising the EU) is for example an interesting case to study (see Box 5.1). Nevertheless, we should not rule out a comparative analysis across groups of countries (e.g. between European and African economies), which is also likely to yield interesting results concerning growth and income inequality at the global level. To conclude this introduction, there are three broad dimensions in the concept of convergence in growth theory and policy: (i) domestic (i.e. convergence within a country), (ii) national (i.e. convergence across countries) and (iii) global (i.e. convergence across groups of countries). To grasp the meaning of convergence, we will examine below both its theoretical background and the related stylized facts. As a first step, we will extend the neoclassical growth model in order to introduce several theoretical convergence issues and analyze its predictions regarding growth and convergence. Next, we will use available data on growth and income in order to investigate various ways of examining the associated empirical hypotheses of the model regarding convergence. Convergence in the Solow-Swan model To address the questions about convergence, we will reconsider the simplest version of the neoclassical model without technological progress. As we will see, this model yields the central predictions on convergence used by economists. Let us assume an economy i with a Cobb-Douglas production function: Y = AK i a 1 a i Li (5.1)

4 96 P.Kalaitzidakis S.Kalyvitis Box 5.1. European Union policies on real convergence An officially declared policy goal of the European Union is cohesion across member states. This goal is supported by the implementation of social and economic measures aiming at real convergence (i.e. convergence of income and productivity across countries). A relevant question is therefore whether convergence between richer and poorer members has indeed taken place and, if this is the case, what is the degree of convergence between the developed and the less developed member states. The next figure plots the evolution of income levels in Greece, Ireland, Spain and Portugal, which are the poorest EU members, against the average EU level over the period of the 1990s. EU average GDP versus GDP in the less-developed EU economies (ΕΕ-15=100) EE Ireland Spain Greece Portugal Source: European Commission (1999). Obviously, Ireland is the only economy that has attained real convergence, whereas the other member states remain far from the EU average GDP. To support the goal of convergence, the European Union has launched a set of policies -termed Structural Policies- which aim at eliminating income disparities in a sustainable manner. The main financial instruments used to achieve these goals are the Structural Funds and the Cohesion Fund. Financial transfers are allocated for the development of infrastructure in vital sectors, like telecommunications, transports and energy, for the improvement of human capital, and for the support of various structural measures.

5 Economic Growth: Theory and Policy 97 As was shown in previous chapters, the following relation gives the steady-state capital-labor ratio in this economy: 1 1 α sa k i = δ (5.2) n + Consequently, the expression for the steady-state income per capita is: 1 1 α a 1 α s y i = A δ (5.3) n + Finally, the relation a y i = Ak i yields the growth rate g y of country i as: g = ag (5.4) yi k i Recall that the main predictions of the neoclassical model without technological progress are that at the steady state g y = g k = 0 for any economy, and that the capital-labor ratio k increases at a diminishing rate while approaching the steady-state point k (provided of course that the initial capital stock k ( 0) < k ). Using these results, we can now state the following proposition concerning convergence across economies. Proposition 5.1. Consider two economies i=1,2 that have the same production function given by equation (5.1) and the same values of the parameters s, n, δ. In the long-run equilibrium, these economies will exhibit the same level of income per capita, given by equation (5.3). The conclusion above is one of the most powerful predictions in the theory of exogenous economic growth. According to Proposition (5.1), structurally similar economies will converge in the long run to the same income level. Recall now that along transition to the steady state, the dynamics of the growth rate of k can be expressed algebraically as follows: (1 a) gk = sak ( n + δ ) (5.5)

6 98 P.Kalaitzidakis S.Kalyvitis where the subscript i is omitted for simplicity. Equations (5.4) and (5.5) imply then that: (1 a) g y = asak a( n + δ ) (5.6) g y Equation (5.6) yields that < 0, meaning that the growth rate is k inversely proportional to the capital-labor ratio along the economy s transition to the steady-state. Therefore, the lower the initial capital-labor ratio, the higher the growth rate will be. We can now sum up these results in the following Proposition, which is known as the absolute convergence hypothesis. Proposition 5.2. ( absolute convergence hypothesis) In the neoclassical model of exogenous growth, the economy tends to grow faster for a low level of initial capital-labor ratio compared to the case of a higher capital-labor ratio. Proposition 5.2 reveals again the basic mechanism that drives economies during their transition to the long-run equilibrium. In particular, if capital exhibits diminishing returns (α<1), an economy with lower capitallabor ratio exhibits a higher marginal product of capital and thus, grows faster compared to a similar economy with a higher capital-labor ratio. Hence, the differences across countries will tend to fade out over time, with per capita income and its growth rate gradually converging until reaching an identical long-run equilibrium level for both countries, respectively. The hypothesis of absolute convergence provides the main building block for empirical tests on the fit of the model when confronted with data on groups of economies. According to this hypothesis, if all economies have similar characteristics described by equations (5.1) to (5.6), the poor ( rich ) countries with low (high) initial capital k 0, and, thus, low (high) initial income y 0, will display a higher (lower) growth rate. That is, in practice these two variables are expected to be negatively correlated. Later on in this chapter we will examine whether this hypothesis is supported by empirical observations. At the current stage, we will attempt to answer the following question: is it always true that two economies with different levels of initial income will exhibit different (and inversely related) growth rates? In other words, is it possible for a typical African country with low capital and income levels to grow faster than Switzerland?

7 Economic Growth: Theory and Policy 99 The answer is of course no. The reason is that the hypothesis of absolute convergence applies exclusively to economies with identical structural parameters, which are will display the same level of per capita income in the long run. So, for example, if the population growth rate n is higher in a country than in another one, then according to equations (5.3) and (5.6), the former will exhibit lower steady-state income and growth rate along transition to equilibrium than the latter. We can therefore conclude that, according to our model, convergence should only be anticipated across structurally similar economies (i.e. with identical parameter values). 10 This new insight introduces the concept of convergence known as conditional convergence. Proposition 5.3. ( conditional convergence hypothesis) In the neoclassical model of exogenous growth, economies tend to converge faster to their own steady state the further away they are from it. According to Proposition 5.3, simply looking at the initial equilibrium point and the growth rate cannot yield accurate predictions about convergence. We should first allow for possible differences between the countries steady states. So, an economy is likely to exhibit both a high initial income and fast growth simply because its steady-state income is simultaneously high and relatively far from its current income. For instance, we know that large values of the technology constant A result in high steadystate capital-labor ratio and income (see equation 5.3). A straightforward implication is that a simple comparison with an economy having the same initial capital stock and income level without looking at parameter A, would lead to inaccurate conclusions regarding its convergence pattern. In the following sections, we will use Proposition 5.3 in order to introduce the theoretical background for empirical tests on the convergence hypotheses. Obviously, while conducting these tests we should allow for the differential determinants of the steady state (which in our neoclassical setting are given by parameters Α, α, n, s, δ) and then examine the transition towards this point. 10 It is worth noting that according to the theory two economies with identical parameters will achieve the same steady state. However, the opposite does not hold: as can be readily seen in the model, two economies may coincidentally exhibit the same steady-state income level despite having different parameter values. Therefore, two economies sharing the steady-state income level cannot be viewed as similar without further knowledge of their structural charactheristics.

8 100 P.Kalaitzidakis S.Kalyvitis The theoretical background of tests on convergence As we have seen so far, to establish the presence of convergence we have to test the specific hypothesis according to which there is an inverse relation between the growth rate and the initial income of a country. Algebraically, this hypothesis can be expressed as follows: g = f [log(, )] y y i i, t, t + T t (5.7) where g y = log( yi t T / yi t )/ T i, t, t + T, +, is the average growth rate of economy i during the period from t to t+t, and log( y i,t ) denotes the logarithm of income of economy i in period t. 11 For the Cobb-Douglas production function, the capital growth rate, which determines the income growth rate, is given by equation (5.5). The linear approximation of this equation around the steady-state yields: 12 g k d log( K) k = (1 a)( n + δ ) log( ) (5.8) dt k y k Using equation (5.4) and the relationship log( ) = a log( ), we can also y k approximate the income growth rate by: 11 The use of logarithms will allow us to express the growth rates as first differences of the logarithm of income. 12 The capital growth rate can be written as: g k d log( k) (1 a) = = sak dt ( n + δ ) and its linear approximation around the steady state given by equation (5.2) takes the following form: (1 a ) a 2 d log( k) k g k sa( k ) ( n + δ ) (1 a) A( k ) log( ) dk k Substituting above the expression for k yields relation (5.8).

9 Economic Growth: Theory and Policy 101 g y y (1 a)( n + δ ) log( ) (5.9) y The general form of the above equation is: y g y = β log( ) (5.10) y where β=(1-α)(n+δ). It can be easily seen that convergence is determined by coefficient β. In particular, for β>0, if the initial income is higher (lower) than the steady-state income, the economy will grow faster (more slowly). This form of convergence is known in the literature as β-convergence. The latter also introduces the concept of convergence speed, which has already been discussed in Chapter 3. In particular, according to equation (5.8), coefficient β shows how fast the economies approach their long-run equilibrium: higher (lower) β implies faster (lower) convergence. It is obvious that coefficient β can shed light in the analysis of economic growth: if β is large and convergence is fast, the economy is expected to approach its long-run equilibrium in a relatively short time period and, therefore, we can use steady-state analysis to examine the growth pattern. However, in the case of a low β, the economy will be far from its long-run equilibrium and its growth path can be better approximated by the transitional dynamics. Now, solving equation (5.10) yields: βt βt log( y ) = (1 e ) log( y) + e log( y0 ) (5.11) t Income at any time t depends on the constant value of the initial income y 0 and of the steady-state income y. In addition, for higher values of β, income y t will be closer from its steady-state level. At t=t, equation (5.11) becomes: y log( y T T 0 ) βt (1 e ) y = log( ) (5.12) T y 0 Equation (5.12) is a simplified form of equation (5.7) and shows that the average growth rate after T periods is inversely related to the levels of the initial and the steady-state income. Therefore, just as in the case of the

10 102 P.Kalaitzidakis S.Kalyvitis absolute and conditional convergence, β-convergence depends not only on initial income level, but also on its steady-state level. We can now state the following Proposition about the different forms of β-convergence. Proposition 5.4. (absolute and conditional β-convergence) In the neoclassical model of exogenous growth, absolute β-convergence is measured in terms of initial income of the economy, whereas conditional β-convergence is given by the deviation of initial income from the its steadystate level. Just as in the case of absolute and conditional convergence discussed earlier, the above forms of β-convergence have similar properties. Therefore, in order to establish whether the conditions supporting convergence are satisfied, we should look at the steady states of the economies. We can now examine how the above results apply to a collection of data from different economies, i.e. when cross section data are used. In particular, we raise the following question: if each of these economies tends to converge to its steady-state income level, will then convergence emerge at the global level, i.e. will the inequality of income distribution be worldwide diminished? At first glance, there should be a positive correlation between the two forms of convergence: for a small income gap across economies worldwide, individual economies should tend to converge to equilibrium income per capita. But does the opposite hold, namely does β-convergence lead to small income dispersion? This time the answer is no and the reason is that this income gap can stay constant (or increase) despite the presence of β- convergence. For example, a poor country may grow much faster than a rich one so that the income of the former may turn out to exceed the level of the latter and thus the income gap will increases while there is β-convergence in both countries under consideration This phenomenon is known in the scientific literature as Galton s Fallacy. Back in the 19 th century, Sir Francis Galton observed that children having fathers that were taller than the average tended to have heights close to the average of the population heights and not close to the average of their family members heights (which was higher than the population average). This tendency of heights to converge to the average level (called mean reversion) was regarded by Galton as a proof of the fact that the dispersion of all individuals heights should diminish over time. However, this hypothesis does not hold in reality. For more details on the relation between Galton s Fallacy and the empirical studies on growth and convergence, see Quah (1993) and Bliss (1999).

11 Economic Growth: Theory and Policy 103 Now, what are the implications in terms of the relationships developed previously? In the simplest case of successive time periods (Τ=1) between t- 1 and t, equation (5.12) can be rewritten as follows: log( y / y 1 ) (1 e ) log( y) (1 e ) log( y 1 ) t β β t = t β t t + c log( y ) = e log( y 1 ) (5.13) where β c = (1 e ) log y. Income dispersion is then given by the variance: 14 2β = t Var[log( y )] e Var[log( y 1 )] t (5.14) Defining the variance as Var[log( y t 2 t )] σ and substituting above implies: 2 2β 2 = t 1 σ t e σ (5.15) Hence, in order for income dispersion to be reduced intertemporally we must have that β>0 (β-convergence), in which case: 2 2 t < σ t 1 σ (5.16) Inequality (5.16) is known in the growth literature as σ-convergence and states the reduction of income dispersion (captured by its variance) over time. As can be readily seen by equation (5.15), a necessary condition for σ- convergence to hold is that β>0. On the other hand, the reverse is not true, 2 2 i.e. we may get σ t > σ t 1 even when β>0, provided that the initial value of the variance is lower than the one prevailing in the steady state (see Barro and Sala-I-Martin, 1995, chapter 11). In turn, the following Proposition can be stated on the relationship between β-convergence and σ-convergence. Proposition 5.5. (relation between β-convergence and σ-convergence) In the neoclassical model of exogenous growth β-convergence is a necessary but insufficient condition for σ-convergence. 14 The variance of a variable Χ around its mean Ε(Χ) is given by the formula Var(X)=E(X- E(X)) 2 and satisfies the following property: Var(kX)=k 2 Var(X) for k=constant.

12 104 P.Kalaitzidakis S.Kalyvitis As we will see, this theoretical relationship is of little use in the empirical context, as the two forms of convergence usually coexist. However, for reasons of strictness we should clarifying once again the exact meanings of the two concepts: β-convergence characterizes the path of the economy s income towards its steady-state level, whereas σ-convergence refers to the intertemporal evolution of global dispersion of income. Hence, both concepts are crucial in the analysis of convergence to the steady-state income per capita. So far we have presented the most important theoretical issues on convergence. As we will see, these concepts also support the empirical tests of convergence hypotheses. In the next section we will look at empirical measures of convergence and discuss the results of many empirical studies attempting to determine different measures of convergence. Convergence tests: empirical studies During the last decades, and especially after the publication of Solow s exogenous growth model, applied macroeconomists turned their attention to empirical studies testing convergence across countries. The following reasons account for the increased interest in this kind of tests: The model of exogenous growth yielded the first specific and, more importantly, measurable results concerning the convergence process. Apart from its importance for economics, empirical measurements of convergence also shed further light upon the social and political aspects of the issue of global income distribution. A larger variety of statistical data became easily available through several economics data banks maintained consistently by various economists and research teams. Improvement in theoretical econometric methods involving, among other things, endogeneity and heterogeneity issues, along with the development of user-friendly computer-based estimation offered more accurate answers to many quantitative issues involving the nature of data on economic growth. A straightforward test regarding the presence of convergence can be carried out by plotting countries income per capita over a long time horizon. To gain a first picture of the long-run performance in various economies we plot in the figure below the log of income per capita in three selected economies (USA, UK, Portugal) between 1870 and These economies are chosen based on data availability and on their substantial disparities in the level of income.

13 Economic Growth: Theory and Policy 105 Figure 5.2. Log of per capita GDP in 3 countries, Kingdom Portugal USA United Source: Easterly and Rebelo (1993). Examining the figure above we can draw the following conclusions: - Despite starting from the same income level, the two more developed economies (USA and UK) diverged significantly during this period. Obviously, if this path persists, per capita GDP levels in these countries will continue to diverge in absolute terms. Therefore, a question that arises naturally is whether one of these countries diverges along an unstable transition path during the period under consideration (implying a rejection of the Solow model) or if their income levels simply converge to a new long-run equilibrium level, which is not the same for both countries. - Income in Portugal, which is the less developed country, displays a similar evolution. However, there is a large gap between Portugal s per capita income and that in the two developed countries. Moreover, this difference persists over time. The relevant question in this case is therefore whether less developed economies can possibly converge to the developed ones, or if less developed countries have a different (lower) long-run equilibrium income. However, we should always keep in mind that the simple graphical illustration of per capita income suffers from several shortcomings: The conclusions drawn above concern only certain developed economies for which long-run data is available. No predictions can be drawn for the majority of developed economies.

14 106 P.Kalaitzidakis S.Kalyvitis Usually, the sample of countries analyzed is not chosen irrespective of the likelihood of convergence. The economies for which data can be found are typically those that have converged to a high level of income, which is actually the reason for which statistical data is available. Despite the high quality of available data, cross-country comparisons are not always feasible due to different statistical measurements used by national statistical offices and the often measurement errors, especially during the earlier years of the sample. In addition, despite the fact that the long-time horizon (over 100 years) is relevant for tracing the long-run growth tendency of economies, it may still not be enough to reveal important dimensions of convergence, such as β-convergence and σ-convergence, or the long-run equilibrium income in these economies. These limitations were largely overcome by the publication of the Penn World Tables (a description was given in Chapter 4). The immediate step was to use this data in order to test the basic prediction about convergence of the neoclassical model, namely that initial income is inversely related to the average growth rate or, equivalently, that richer (poorer) economies exhibit lower (higher) growth rates i.e. they display absolute convergence. Following this approach, Figure 5.3 plots per capita income during the 1960s against the average growth rate between 1960 and 1989 for 119 countries. Figure 5.3. Growth rate versus initial per capita income (119 countries) 8 7 Per capita income (in logs), Growth rate, Source: Levine and Renelt (1992).

15 Economic Growth: Theory and Policy 107 As it can be readily seen, there is no sign of correlation between the two variables plotted above. More exactly, a fitted linear trend shows a positive correlation between per capita income and the growth rate implying that richer (poorer) economies tend to have higher (lower) growth rates! This conclusion challenges one of the main predictions of the neoclassical model since it questions the existence of β-convergence, which is also the necessary condition for σ-convergence. But is this indeed what the neoclassical model predicts about convergence? Recall that in the previous section this form of convergence has been termed absolute β-convergence. In fact, the neoclassical model bases its main results on variables like economy s initial income and its own long-run equilibrium level. Therefore, we shouldn t test for convergence economies that are likely to have different long-run equilibrium income. Since there is no reliable method for estimation of a country s steady-state income, a general criterion should be adopted in order to classify economies into homogenous categories. Only after this classification takes place can any convergence test be applied; as expected, this recalls us the prerequisites of conditional β-convergence. Following this idea, Figure 5.4 illustrates the two variables for a sample of 24 OECD countries. This is a group of economies that consistently exhibit high per capita income over many decades. So, we expect that they fall into a category of economies with similar per capita income and also many common features from the standpoint of their structure and behavior during the period analyzed. Figure 5.4. Growth rate versus initial per capita income (24 ΟECD countries) Per capita income (in logs), Growth rate, Source: Levine and Renelt (1992).

16 108 P.Kalaitzidakis S.Kalyvitis Figure 5.4 reveals a completely different image from Figure 5.3. Here, initial income and growth are inversely related; moreover, this inverse relation is statistically significant. 15 Indeed, many empirical studies have supported the accuracy of the figure above on the basis of data that confirm the existence of conditional β-convergence across similar economies. Sala- I-Martin (1996) examined the relation given in equation (5.12) for a sample of 110 countries and found that the coefficient β fluctuates around 0.02, meaning that annual convergence amounts to 2% of the distance between actual and steady-state income. That is, in general economies converge to their long-run equilibrium income, with their growth slowing down as they approach their equilibrium income. However, this does not mean that the long-run per capita income is the same across countries. It should be noticed that the tests mentioned thus far do not apply exclusively to countries. In fact, an extensive literature has been developed around the issue of convergence across regions within a country. Since it is expected that regions display relatively smaller differences in their per-capita income than countries, conditional β-convergence should provide a suited hypothesis to test. Barro and Sala-I-Martin (1995) studied the implications of equation (5.12) in the U.S. states, for which there is available data for the period They estimated that β is not far from 0.02, which is close to the result found by similar studies of regions for other developed countries. Coefficient β also denotes the speed of convergence. In particular, a value of 2% means that it will take 35 years for economies to cover half of the gap between initial and equilibrium income. Thus, it turns out that convergence is a rather slow phenomenon, even when we allow for the determinants of steady-state income. As we have seen earlier, β-convergence is a necessary but insufficient condition for σ-convergence (see Proposition 5.5). A straightforward implication is then that σ-convergence cannot occur across economies worldwide (since there is no sign of β-convergence in these economies), but it is possible to arise across ΟECD countries. Figure 5.5 illustrates graphically the dispersion of real income per capita in both cases. 15 The value of the correlation coefficient between the two variables is 0.68, which is considered high enough for cross-sectional data. Note also that similar conclusions are drawn if we examine clusters of countries with an approximately equal income level (e.g. lessdeveloped economies, countries with common geographical characteristics etc).

17 Economic Growth: Theory and Policy 109 Figure 5.5. Intertemporal dispersion of real income per capita worldwide and across ΟECD countries 1,20 1,00 0,80 0,60 0,40 0, World OECD Source: Sala-I-Martin (1996). Figure 5.5 shows that the dispersion of income at global level does not decline, but on the contrary, increases over time. This confirms the finding that rich countries tend to become richer, whereas the relative position of poor countries worsens. In other words, the unequal distribution of global income is intensified at the expense of poor countries. On the other hand, the isolated analysis of OECD countries reveals the existence of convergence (except for the period ). Consequently, income inequalities across OECD countries decrease significantly over time. These findings (i.e. σ- convergence except for ) also hold when regions of developed countries are examined. Most of the available surveys tend to agree that convergence occurs across similar economies at a speed of 2%. But what are the implications of this convergence speed according to the neoclassical model? Recall that relation (5.10) shows that coefficient β is equal to the product between the labor share and the sum of the population growth rate and the depreciation rate. Allowing for β to equal approximately 0.02 and assuming that n and δ 0.01 (i.e. 1% average annual rate of population growth) and 0.05 (i.e. 5% average annual rate of depreciation) respectively, we get that the fraction of capital should be 0.67 (=1-0.33). Note though that a series of empirical studies (see Chapter 4) has found that this fraction equals 0.30, which is much lower than the prediction above The model examined here does not allow for technological progress. If productivity grows at a constant rate, we should include this rate in the sum (n+δ). Assuming that the average productivity growth rate equals 0.02 (2% yearly), the fraction of capital should be even higher and roughly equal to 0.75.

18 110 P.Kalaitzidakis S.Kalyvitis This apparent contradiction arising from initial estimations of coefficient β was resolved by Mankiw, Romer and Weil (1992), whose work was discussed extensively in Chapter 4. Their empirical methodology assumes that aggregate capital incorporates both physical and human components and finds that the fraction of capital approximates 0.7. Hence, the analysis of Mankiw, Romer and Weil does not only that reveal a new dimension of the neoclassical model, but also supports its predictions about convergence. β (1 e T ) Finally, it is worth emphasizing that the coefficient in T equation (5.12) is inversely related to the number of time periods, T. Hence, when we look at the linear relation between income and growth and assume more time periods over which convergence is examined, we should expect to find lower values for this coefficient. The reason is that the economy approaches the long-run equilibrium at a decreasing rate, which affects the average growth rate over the period under discussion. It is clear then that the period size constitutes an important factor of the empirical estimation. 17 Conclusions The issue of convergence is one of the central aspects of the neoclassical model of exogenous growth. It has drawn the interests of economists studying the development of countries and regions and, despite its simplicity, this model remains an important tool in the growth theory. Its central predictions, which have been confirmed by many empirical studies, can be summarized as follows: Each economy converges to its own long-run equilibrium, which is determined by a series of exogenous parameters. Convergence to long-run equilibrium income does not lead automatically to leas income inequality. Economies worldwide do not show a consistent convergence pattern, whereas homogenous groups of countries and their regions tend to converge to their steady-state income. The annual rate of convergence is very low and equals roughly 2%. One para here 17 Note that for Τ approaching infinity (i.e. sample size increases), the coefficient tends to zero, whereas for T approaching zero, (i.e. sample size decreases), the coefficient tends to β.