Economy
& Energy 
Capital Productivity in Brazil in the XX Century 
Annex 1: Brazil and the Additional Protocol of the IAEA

Article: Evaluation of Capital Productivity in Brazil in the XX Century
Aumara
Feu Introduction Capital productivity (inverse of capital/product ratio) is a factor of vital importance for understanding the reasons of the Brazilian per capita growth stagnation in the last two decades. This theme, capital productivity in Brazil, was object of detailed analysis in the PhD thesis of Aumara Feu (2004) published in the No 42 issue of e&e (January/February 2004). Incidentally, it should be emphasized that the level and behavior of the capital productivity varies according to the methodology adopted for the construction of the Brazilian capital stock series. The e&e Projetar_e macroeconomic analysis program, Aumara Feu (2004), Hofman (1992, 2000), Morandi (1998) and Bacha and Bonnelli (2001) present different estimates about this stock, considering that until 2003 there has been no official series relative to this variable. The lack of historical data for a considerable period concerning the gross formation of fixed capital at constant prices also increases the possible estimation mistakes in the calculation of capital stocks and consequently in the determination of capital productivity. The publication “Estatísticas do Século XX” (XX Century Statistics) (IBGE (2003) has data about the gross and net capital stock as well as data about gross formation of fixed capital (FBKF) divided by type of good and by institutional sector. The FBKF is divided by three types of goods: residential construction, nonresidential construction and machines and equipment from 1901 (or 1908) to 2000. From 1947 on the data is subdivided in a forth group of goods, namely others, which in the present report is grouped in the machines and equipment item. In what concerns the division by institutional sector, the FBKF is subdivided in two groups: public administration and families and enterprises . The availability of data about the gross formation of fixed capital for the first half of the century permits to improve the calculation of the capital stock and consequently of the capital productivity and permits to evaluate its behavior (from 1920 on) with an acceptable degree of confidence by making hypothesis about the stock of investment at the end of the XIX century. The objective of the present report is to reconstruct the capital stock series and the capital/product ratio (K/Y), inverse of capital productivity, according to the new (1993) IBGE data. It aims as well to compare the estimates of the present study, according to different methodologies, with those supplied by IBGE, verifying the significance of varying the depreciation according to the capital age and the contribution by type of goods and by institutional sector to the determination of the capital/product ratio behavior. For this purpose, Section I describes the methodology used in the calculation of capital stock, Section II evaluates the results of capital stock according to these different methodologies and compares it with the stock supplied by IBGE (2003), Section III shows the depreciation due to the use of these methodologies, Section IV calculates the series of the capital/product ratio resulting from this stock and finally Section V shows the contribution by type of good and by institutional sector to the total K/Y ratio behavior. Section I  Methodology for calculating the capital stock The different methods for calculating the capital stock are described in Aumara Feu (2004). Among them one could point out: a) method of constant depreciation rate on the stock , present in most growth models, exogenous or endogenous ones, where the initial capital is determined and this stock is depreciated at a constant geometric rate. It should be emphasized that whenever one has available initial stock by type of good, one can use different depreciation rates according to the type of good; b) method of permanent stock ^{ [1] } (MPE), also denominated perpetual stock  sum of past net investments. Incidentally, net investment means gross investment without the corresponding depreciation, determined according to the depreciation function used and the time of life of the type of good. In the first method, the depreciation rate on the capital stock is constant and exogenous. On the other hand, in the second method the rate on the capital is endogenous and varies according to the history of the investment. That is, in the latter the depreciation rate on the investment is fixed and since the stock is a result of the past net investments sum, the depreciation rate on the stock is changed as the proportion of investment relative to the gross domestic product varies along time, modifying the capital stock ^{ [2]}. The first method is simpler to be computed and does not need information about past investments. However, it is more susceptible to errors in the initial stock determination and can underestimate or overestimate the stock by neglecting which changes in the capital age affect the capital erosion. The significance of this last error will depend on the magnitude of investment rate alteration in the considered period. In order to verify alterations in the stock due to the use of these methods and according to new data concerning gross formation of fixed capital supplied by IBGE(2003) ^{ [3]}, we have calculated the stock in the present report by three different ways: 1. MPE with linear depreciation with gap  using, according to Aumara (2004), a 10% gap of the capital time of life, assumed to be equal to that supplied by OECD (1999) by type of good. 2. MPE with linear depreciation  using time of life parameters supplied by IBGE (2003). 3. Method with constant geometric depreciation rate on the stock  using a depreciation rate “equivalent” ^{ [4]} to the time of life supplied by IBGE (2003), applied to the capital stock of the previous year. In the first way the times of life used are 48 years for construction and 19 for machines and equipment, with a gap of 2 years for the start of depreciation for machines and equipment and 5 years for construction. Therefore, since it is assumed that the investment is aggregated to the capital stock in the year following its integration, an investment in year zero becomes effective in year one and starts, if the gap is 5 years, to be scraped in year six. In the two last ways, the estimated useful life is 50 years in residential construction, 40 years in nonresidential construction and 20 years for machines and equipment. In these ways, the standard gap is one year. That is, the investment is integrated in year zero, becomes productive in year one and starts to be depreciated in year two Taking into account these times of life, the calculation of capital stock is given: a) in the permanent stock method by : (1) where the capital (K) in t is given by the sum of the invested goods (I), still in the scraping process, less the depreciation of these goods, according to its time of life. That is, given the time of life (v) by type of capital (i) and the gap period (m), exogenous to the model, one calculates the depreciation rate regarding the capital good (di) as (1/(vimi))^{[5]}.
The
quantity of years in which the depreciation should include (2) Therefore, according to equations 1 and 2, since the capital stock is a function of the historical investments in MPE, the depreciation rate is endogenous, depending on the time of life (v) considered, the capital composition and the investment variation along time. b) in the method with constant depreciation rate on stock:
where the depreciation rate on the stock of the year is exogenous. Since IBGE (2003) assumes different times by type of goods, the depreciation rate will be different, in spite of being constant, according to the stock of type of good. On the other hand, the depreciation rate of the total stock will be the division between the sum of the corresponding depreciation by type of stock ( ), by the sum of stocks ( ). Therefore, also in the third way, the depreciation rate on the total stock will be a function of its composition. In what regards the depreciation rate on the stock, it was estimated taking into account the growth rate of investments and it is denominated "equivalent" depreciation rate. The Annex 2 shows that the "equivalent" depreciation rate for a good with (v) time of life whose investment increased in the considered period with a rate g is: ……………..(3) where c is a constant such that In the present report, the "equivalent" rates considered are shown in Table 1 where the times of life and the growth rate of investments in the period (1901/2000 for construction and 1908/2000 for machines and equipment) are shown as well. Table 1  “Equivalent” rates by type of good
Section II  Stock Evaluation In this section are presented the results of the estimates of capital stock in the three forms described in the previous section and these results are compared with those supplied by IBGE (2003). It should be remembered that in the method of permanent stock with linear depreciation and in the method with constant geometric depreciation rates on the capital stock, the rates were estimated considering times of life per type quite the same as those used by IBGE (2003). Therefore, it is expected to get similar results. Capital stocks from 1901to 2000 were estimated in the three forms, considering, according to what is described below, that the series is reliable (with little significant errors) from 1920 on. In the MPE case it was necessary to extrapolate the investments in the years before those when data were available for a time equivalent to the life per type of good. For example, investments of residential goods from 1850 on were estimated in order to obtain the stock of this type of good in 1901. In order to estimate the previous investments it was considered a constant growth rate of investments equal to the average verified in the first half of the XX century for the set of investments. The participation in the investments was taken as the average in the available years before the National Accounts (1908 to 1946). It should be remembered that the influence of data estimation concerning past investments decreases rapidly. These data represent 13% and 0.6% of the stock, respectively, in 1920 and 1940. As a consequence, the capital stock evaluation (and the capital/product (K/Y) ratio) in these years is not influenced by the choice of the previous values. Furthermore, an iterative method, described below, was used that permits obtaining estimates of the initial capital stock from the behavior of the K/Y curve. The two methodologies led to similar results for the first years of the series and are practically coincident from 1920 on. The iterative method for estimating the initial stock starts when one preliminary K/Y ratio is determined for the year zero . When this value is multiplied by the product of year zero one gets the preliminary capital stock for this year . In this way, according to the considered depreciation rate, the capital stock series is built according to equation 2. That is, the capital stock in year one is given by:
Therefore, repeating the procedure above successively for the following years, one gets the estimate of the capital stock and the K/Y behavior. When applying the method, the initial K/Y value is chosen in an iterative way minimizing the variation of the K/Y initial values and they remain approximately constant. The total values of the stock are presented in the annex tables A1.1, A1.2 and A1.3 for the three methodologies. In Table 2 are shown the results for the total stocks in million reais of 1999, calculated in the present report and those supplied by IBGE (2003).
Table
2: Comparison, in the different methodologies, of total stock of fixed
Source: (1) present report, (2) CD annex to XX Century Statistics (IBGE) IBGE (2003) presents gross and net stocks data, estimated respectively by the geometric rate and by the MPE, considering the same time of life for the two stocks. The results show values for the gross and net stocks significantly different. According to the definition of OECD (2001), the gross stock is larger because in its calculation the gap occurs only in the last year of the capital life (depreciation with sudden death). Since the IBGE publication does not details the methodology used in calculating the capital stock and since the analysis of the capital stock capacity of generating one unit of product (ACHO QUE A FRASE ESTÁ INCOMPLETA), using the net stock, the present report is restricted to comparing the net stocks calculated according to different methodologies.
Figure
1: Capital stock estimated by IBGE (2003) and in the present report The times of life in MPE with linear depreciation with gap are: 48 for construction and 19 for machines and equipment, with gap of 5 and 2 years respectively. In the other cases they are 50 years for residential constructions, 40 years for nonresidential constructions and 20 years for machines and equipment. It can be observed that the net capital stock based on Aumara Feu’s parameters (2004)  that will be considered as reference for the present report, presents values similar to those of IBGE (2003)^{[6]}. Section III  Depreciation Rate in the three Methodologies As previously mentioned, the depreciation rate is obtained in the permanent stock method through equations (1) and (2) and it is a function of the past investments and the considered time of life. On the other hand, in the method with constant depreciation rate on the stock of the previous year, the “equivalent” depreciation rate (equation (3)) is a function of the historical investment rates growth per type of good. In what follows the values obtained using the MPE are compared with the “equivalent rates” for the different types of goods ^{ [7]}. The values of the annual growth of investment were obtained from the exponential adjustment and grouped as public and private goods. For example, the adjustment of data concerning machines and equipment for determining the annual growth was carried out for the set of goods and the “equivalent rate” used is the same for public and private goods. In Figures 2 and 3 the values of the depreciation rate per type of good and the depreciation on the total stock obtained by the MPE and by the “equivalent “ rate are presented
Figure 2: Annual total depreciation rates and depreciation rates per type of good, calculated by MPE and the corresponding “equivalent” depreciation rates (constant rate on the stock that reproduces, approximately, the depreciation calculated using the MPE). It can be observed that even though there is good agreement between the capital stock series calculated either by the MPE method or by the constant “equivalent” rate (Figure 1), the annual depreciation rates vary significantly along the period (Figure 2), depending on the method used. This variation in the depreciation rate by type of capital is due to alterations in the average age of the capital stock caused by variations in the investment rate along time. The “equivalent” total depreciation varies according to the capital composition whereas the total depreciation rate calculated by MPE varies according to the capital composition and age. Therefore, the MPE rate is smaller when the investment rate increases, decreasing the capital age and the annual gap. The coefficients of depreciation rates variation by the MPE relative to the “equivalent” depreciation rate are 3.4% 7.4%, 9.1% and 6.9% respectively for residential construction, nonresidential construction, machines and equipment and total. This demonstrates that investments in machines and equipment and in nonresidential construction goods vary considerably along time, affecting the depreciation rate of these type of goods. For the machine and equipment item, for example, it is observed that at the start of the 1980s, due to large investments in this type of good in the previous decade, it was necessary to replace 7% of this stock in order to maintain the same production capacity. On the other hand, in the mid 1990s, due to the investment decrease in the previous decade, it was necessary to replace 10% of this stock in order to keep constant the production capacity. Furthermore, when one compares the total depreciation rate calculated by the two methods, it is observed that the “equivalent” depreciation rate, due to modifications in the stock composition per type of good, is smaller than the depreciation rate by MPE. In the latter, the variations are due not only to alterations in the composition but mainly, as indicated by Aumara (2004), to alterations in the average stock age. So, according to what was described: when using the constant rate method on the stock one cannot detect variations in the depreciation rate due to alterations in the capital age and estimate alterations in the quantity of investment necessary to replace the capital gap. Section IV  Capital Productivity or its Inverse: the K/Y Ratio In the present section are presented the K/Y ratios (Figure 3) calculated according to the stock division shown in Section II (Figure 1) for the GDP in reais values of 1999 As expected, due to the stock behavior, it is noted that the K/Y ratios do not differ substantially, independent of the methodology used for calculating the stock. It is important to notice that using MPE with linear depreciation and the “equivalent rate”, both considering the same time of life per type of good, both the level and the trend are similar. This fact indicates that the use of the “equivalent rate” on the stock, simplified methodology, does not produce substantial errors in the estimation of the K/Y ratio, provided that the initial stock level has been properly determined In what concerns the series that use the stock calculated by MPE with linear depreciation with gap and the net stock supplied by the IBGE (2003), these are similar in trend and the K/Y ratio behavior of IBGE (2003) is smoother than the ratio calculated by MPE (61.3% and 81.3% variations in the period, respectively). The initial IBGE K/Y ratio is 17% higher than that calculated by MPE and it converged in the 1980s (when depreciation calculated by MPE drops) to the same value and it became 4% higher at the end of the period. The difference between the series is due to the time of life considered and mainly to the depreciation function considered in each methodology. Since we have neither complete information on how IBGE has determined the geometric depreciation rate used in its calculation nor if this rate is used on the stock or on the investment (as in the MPE), we can only infer that the IBGE (2003) methodology may overestimate the initial capital stock considering that this is considerably higher than the value calculated in the present report according to the two methodologies (described in Section II), namely MPE and iterative method. In a general way, the trend of the series, independent of the methodology used in the stock calculation is increasing and indicates a change of plateau that has occurred between 1970 and 1985. One can say that the K/Y rate level would have changed from 2 to 3. That is, at the end of the period it would be necessary 50% more capital than in the mid century in order to generate one unit of product. Figure 3: Comparison of the capital/product ratio using different methodologies. Since the behavior theoretically foreseen for the K/Y ratio is constant in the stationary state, this change of level could indicate that the country has left one stationary state to enter another one in this period. According to Aumara Feu (2004), the country would have undergone a shock in the marginal production of capital in the period that could have been caused by technology incorporation from bordering countries, abundant in labor, during the process of agricultural industrialization and modernization. Finally, it should be emphasized that this change of level is relevant because it increases the quantity of investment necessary to growth and it changes the capital participation in the income as well. According to the Solow model with HarrodDomar technology and CES (ConstantElasticityofSubstitution) function, the capital participation in the product (capital share) varied from 65% to 53%^{[8]}, decreasing along time due to the Capital/Product (K/Y) growth – or decrease of the capital productivity (Y/K). Still according to this model, considering the technological growth and the working population growth as constant and equal to the average values from 1994 to 2002, namely 0.62% and 2.34% respectively, the investment rate in the period, 19.3% of the product, would generate a growth of 3.8% in the 1970s and 2.6% presently. Therefore, the K/Y ratio evolution in the period has decreased the capital participation in the growth accounting as well as the growth possibility for the Brazilian economy. Section V  Capital Productivity per Type of Good and per Institutional Sector in the First Half of the XX Century. The IBGE (2003) data permit also to evaluate the capital stock per type of good: construction, residential and nonresidential, machines and equipment and others ^{ [9]}, as well as per institutional sector: enterprises and families and administration. The stock was calculated by MPE with linear depreciation with gap and by the method with constant depreciation on the stock and it was observed a similar behavior of the series obtained. In what follows the results presented were obtained by MPE. In Figure 4 it is shown the capital stock evolution per type of good between 1920 and 2003. It is interesting (or sorrowful) to observe that the stock of machines and equipment remains practically constant since 1980. Actually, it decreased until 1994 when it decreased 16% relative to 1983, resumed growth since then and in 2002 it reached the level observed in 1983. In terms of stock per capita, machines and equipment decreased 31% in 1994 relative to 1983 and in 2003 the stock was 24% lower than in 1983.
Figure 4: The capital stock graphic per type of goods shows that the stock relative to machines and equipment remained practically stationary since 1983.
In what concerns the participation of the different types of capital stock, it is shown in Figure 5. It is surprising to observe the decreasing trend of machines and equipment participation in the capital stock along the period.
Figure 5: Participation in the stock per type of good The results of the K/Y ratio from 1920 to 2003^{[10]} per type of good (i) are indicated in Figure 6, where the total K/Y ratio is composed of the sum of stocks per type of good divided by the GDP.
Figure
6: Evolution of the capital/product ratio , total and per type of good The capital/product ratio shown in Figure 6 indicates the existence of a plateau around 1.6 until the end of the Second World War and two other plateaus previously mentioned of an approximated value of 2 until the start of the 1970s and 3 from the 1980s on. It should be remembered that the afterwar period as well as after the first petroleum shock (1973) corresponded to abundance of external resources for buying capital goods due respectively to war credits and availability of petrodollars. It should be noticed that extending the start of the series to 1920 is important because it makes more clear the change of plateau between mid 1970s and mid 1980s. Figure 6 also shows the evolution of contributions per type of good: machines and equipment (+ others) and construction. The contribution from the first group of goods has oscillated around a plateau close to 0.5. In the crisis period in the thirties and in the II World War period there occurred a drop of contributions from machines and equipment that was recovered in the postwar period. With the economic growth in the 1970s the contribution of the machines and equipment relative to the GDP increased, confirming the larger participation of investment in machines and equipment in the growth process, demonstrated by De Long e Summers (1991, 1993) in a cross section analysis. On the other hand, from 1983 on the machines and equipment stock relative to the GDP started to decline and in this period occurred the progressive deindustrialization of the Brazilian economy. In Figure 7, where it is shown the capital/product ratio divided by residential and nonresidential construction and the others item is separated, one can observe the largest contribution, namely that of nonresidential construction, to the growing behavior of the total K/Y ratio. It should also be noted the smaller contribution of the machines and equipment item (without the others item) that incidentally is below the value it had in the seventies. The residential construction contribution behaves as expected, that is, it increased in periods of low growth since the necessity of residences is less elastic relative to variations in economic growth.
Figure 7: Contribution to the Capital/Product ratio per type of good Finally it was calculated the contribution per type of institutional sector to the capital/product ratio, shown in Figure 8.
Figure 8: Contribution to the K/Y ratio per institutional sector. It can be observed that the most of the K/Y ratio increase (loss of capital productivity) occurred in the institutional sector: enterprises and family. More exactly, between 1973 and 1984 there was an increase of practically 50% in the capital/product ratio due to the enterprise and family aggregate. It should be observed that the governmentowned enterprises are included in this institutional sector. Incidentally, in the National Accounts chapter (Eustáquio Reis et allii) of the IBGE publication, it is shown that the participation of the governmentowned enterprises has increased from 1973 on. That is, even though the public sector, in its direct administration, has not been responsible for the capital productivity decrease the same is not true for the enterprises under the control of the State. In a later study, it would be interesting to verify capital stock data where the contribution of the stateowned enterprises is explicit. Conclusions The estimates analyzed in the present report about the Brazilian capital stock, according to new data supplied by IBGE (2003) are coincident in what concerns the indication of a substantial capital productivity decrease in mid seventies and mid eighties. This decrease corresponds to a K/Y ratio growth which changed from level 2 to level 3^{[11]}, and this seems to indicate that the Brazilian economy changed from one stationary state to another one. According to Aumara Feu (2004) this change coincides with a shock on the marginal capital productivity, observed in 1970s. It should be pointed out that with this capital productivity decrease, according to calculations based on the Solow model with CES production function and HarrodNeutra technology, an investment rate of 19.3% of the product could generate, ceteris paribus, 3.8% growth in the 1970s whereas in the fist decade of the XX century the expected growth is 2.6%. Actually, the K/Y ratio values from 1920 to 2003 suggest two changes of series level in the period. In a schematic way, one can think of three plateaus: the first one from the start of the postwar period (1947), the second one between 1955 and 1973 and the third one from 1983 until the last calculated year (2003). In approximated numbers one could have 1.6, 2.0 and 3.0 for the plateaus or 0.63, 0.50 and 0.33 for the capital productivity. In what concerns the methodologies used, it should be pointed out that the IBGE (2003) initial stock in 1950 is significantly higher than the value calculated in the present report according to two methodologies, iterative process and permanent stock method as well as according to depreciation functions: linear, linear with gap and the one resulting from a constant rate on the stock per type of good. It should also be emphasized that depreciation in the permanent stock method has varied in the period from 3.4% to 4.3%, according to the capital age with a 7% variation coefficient on the constant rate. This result shows that in spite of the fact that the different results of the stock are similar, whenever one wants to measure the quantity of investment necessary for replacing the capital erosion and in order to grow, this value can vary significantly according to the method used. The capital productivity decrease observed between the beginning of the seventies and eighties corresponds to the stock increase without the corresponding product growth. This stock increase, if it is decomposed by type of good, was mainly due to contributions from nonresidential constructions. There occurred also a contribution increase from machines and equipment stock at the time, but this increase is reverted from 1983 on. Incidentally, the capital stock of machines and equipment was practically constant in real value in the two last decades, with a decreasing participation in the total stock. The participation decrease of this type of good as a production factor may be associated not only with the productivity increase but also with the Brazilian deindustrialization process in the last decades. Actually, the figures suggest that we have more sheds and less machines. Finally, in what concerns the contribution per institutional sector to the stock increase and to the productivity decrease , the public administration had a participation in what might be called “first shock of capital productivity decrease” but not in the second one. On the other end, the participation of the stateowned enterprises deserves to be analyzed since, according to the IBGE publication, there has been a substantial increase of investment from these enterprises mainly in 1976. That is, the capital productivity decrease is not connected with the public administration but may be linked with the action of the stateowned enterprises. Annex 1: Results for the capital stock in Brazil by three methods Annex 2: Depreciation rate of the equivalent capital stock at a linear depreciation time v ____________________________ . Bibliographic References Bacha, Edmar L. e Regis Bonnelli, ”Crescimento e Produtividade no Brasil: o que nos diz o Registro de Longo Prazo.” Rio de Janeiro: Seminários da Diretoria de Estudos Macroeconômicos do IPEA 52, 2001. De Long, J., Bradford e Lawrence H. Summers, “Equipment Investment and Economic Ghrowth.” Quartely Journal of Economics CVI (1991). ___________, “How Strongly do Developing Economies Benefit from Equipment Investment?.” Journal of Monetary Economics 32 (1993). Ferreira, Pedro C., João V. Issler e Samuel A. Pessoa, “Testing Production Funtions Used in Empirical Growth Studies.” Economic Letters (2003). Feu, Aumara, “Capital Productivity in Brazil from 1950 yo 2002.” Economy&Energy, 42 (January/February 2004). Hofman, André A, "Capital Accumulation in Latin America: A Six Country Comparison for 19501989.” Review of Income and Wealth 38 (1992). ___________, "Standardised Capital Stock Estimates in Latin America: a 195094 update.” Cambridge Journal of Economics 24 (2000). OCDE(2001) Instituto Brasileiro de Geografia e Estatística, “Estatísticas do Século XX.” Rio de Janeiro, 2003. Morandi, Lucilene, “Estoque de Riqueza e a Poupança do Setor Privado no Brasil  1970/95.” Rio de Janeiro: Textos para Discussão do IPEA 572, 1998. [1] Used by Raymond Goldsmith (1951) for the first time, and among other authors, by Hofman (2000), Ferreira, Isler and Pessoa (2003) and Aumara Feu (2004). The application of the method is detailed by the last author. [2] In Aumara Feu (2004), it was observed that the depreciation rate on the capital stock in Brazil varied from 3.4 and 4.3% from 1950 to 2001 due to alterations in the capital stock age and composition. The author shows also that the weight of age variation due to alterations in the investment rate in the country along the period was larger than the weight of variations of the capital stock composition in what concerns modification of the depreciation rate along the time. [3] The values for the years previous to those of IBGE (2003) are in part due to a recomposition work carried out by the Instituto Brasileiro de Economia (Brazilian Institute of Economy)  IBRE/FGV [4] The “equivalent” depreciation rate was calculated in the present report considering that IBGE uses geometric depreciation rates corresponding to the time of life supplied by type of good but does not mention which rates are used. [5] In order to simplify notation the subscript i in the formulas is not shown. [6] Incidentally, the series estimated by the IPEA/IBGE research group along the last years has come close to the series calculated by Aumara Feu (2004). That is, the series published by IBGE (2003) with geometric depreciation is closer to that calculated by Aumara Feu (2004) than the series estimated by Morandi (1998) and Morandi, Zygielszyper and Reis (2000) using MPE with fixed depreciation rate. [7] Per institutional sector: public administration and families and enterprises, the same depreciation rates were applied. [8] The methodology and the data used in calculating the growth accounting according to this model are described in Aumara Feu (2004). [9] The stock for the item others was estimated considering the same time of life of the machines and equipment item that is presented separately or together with this item. [10] For the year 2000 it was used the data supplied by IBGE in the Trimester National Accounts. [11] Approximate values of K/Y Annex 1: Results for the capital stock in Brazil by three methods Annex 2: Depreciation rate of the equivalent capital stock at a linear depreciation time v

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