Economy & Energy
Year VIII -No 44:
June-July 2004  
ISSN 1518-2932

seta.gif (5908 bytes)No 44 Em Português Support:






e&e No 44


Main Page

Capital Productivity:
A further Limitation for the Brazilian Growth

Simplified Methodology for Estimating the Evolution of Capital Productivity

 Nuclear Inspections and Non Proliferation:

Nuclear inspections in Brazil and the French Public Pissoir

The Non-Proliferation Nuclear Treaty NPT.

e&e links



Text for Discussion

Simplified Methodology for Estimating the Evolution of Capital Productivity



The application of a fixed depreciation rate to the capital stock of the preceding year is used in different economic growth theoretical models. When applied to real data for obtaining the capital stock, this procedure implies that the historical investments are ignored. Errors up to 20% relative to the depreciation obtained from real investment data were observed. However, this simplification – besides its theoretical easiness – provides a quick method for evaluating the capital stock and consequently the capital productivity.

As previously demonstrated[1], it is possible to estimate the behavior of the capital productivity variable by applying a fixed depreciation rate to the capital stock of the previous year and adding the investments made on the previous year. In this methodology it is important the choice of the initial stock value and of the depreciation rate adequate for the time of life involved and its relative participation. In order to estimate this depreciation rate it is also necessary to have or to estimate the growth rate of investments in the previous years.

In the present paper (1) we describe a simplified methodology for obtaining the evolution of the capital productivity, (2) we evaluate the influence of varying the parameters used for determining the capital productivity values and (3) we apply the methodology to six countries and compare the Brazilian results with those obtained using the linear depreciation methodology.

In procedure (3) the lack of information regarding investments in the period before that for which information is available is compensated by assuming a regular behavior of the capital/product ratio at the beginning of the period when the parameter is being evaluated. The equivalent depreciation rate adopted is considered to be the same in the studied countries.

 Description of the Methodology

The simplified methodology for evaluating the capital productivity evolution consists of the following steps:

  1. A depreciation rate is chosen considering the expected life of the goods and the investment growth rate in the past. An approximation is to consider for the previous period the same growth rate as that of the GDP for which generally longer series are available. The equivalent depreciation rate d  is given by 

      , t is the investment growth rate and v is the life value (in years) of the depreciated goods
  1. An initial value of capital product ratio p0 is chosen.
  2. The stock in year zero K0 is estimated using the GDP value Y0 available
    K0 = Y0 . ρ0 
  3. Temporary values for  K1 = K0 + I0 - d . K0 are obtained.
  4. In an analogous way successive values of Ki and ρi are obtained
  5. The value of ρ0 is iteratively chosen so that the behavior of ρ in the first years is the expected one (slightly upward).

This procedure can be adopted both for the set of goods and the categories separately. In the case of set of goods it is necessary to choose a depreciation rate adequate for the proportion of machines and equipment and construction in the investment.

We present the results regarding Brazil with the goods divided into residential construction, non-residential construction, machines and equipment and others by the linear depreciation methodology and compare the results obtained with those of the simplified methodology and with grouped investments.

The initial parameters (capital/product ratio in year zero and depreciation rate) for the simplified methodology were the same obtained using the linear depreciation[2].

The simplified methodology was applied to the set of goods and two tests were made: the first one applied to the whole series (from 1908 to 2003) and the second one to the 1950-2003 period.

In the first set of data (1908 to 2003), the initial K/Y  ρ0=1,35 and the depreciation rate d=4.12% annually were used. The value of ρ0 is the same for the initial year and the depreciation rate corresponds to the average of the period.

For the second set of data (1950 to 2003) the same procedure was used considering ρ0=1,66 and d=3,82% annually.

The result is shown in Figure 1 and presents good agreement between the simplified method and the linear depreciation.


Figure 2: Comparison between the simplified method, applied to two different period, and the linear depreciation method.

In Figure 2 it is shown the effect of varying +20% and –20% the initial values of K/l and d.


Figure 2: Influence of initial parameters variation on the behavior of the capital/product ratio using the simplified method.

We have verified that the simplified methodology applied to the capital stock supplies the basis for constructing the capital/product curve in cases where data concerning previous investment of the studied period are not available.

 The determination of the initial K/Y rate (or of the initial stock) and of the time of life depends on the criteria that will be more or less arbitrary, which in its turn depends on the availability of other information about the previous period.

 In the example considered it was possible to have the results from the other procedure for determining the initial parameters. In Figure 2 it is shown the effect of changing 20% the input parameters of the 1950-2003 period in the determination of K/Y. For the final year and for 1960 it is shown in Table 1 the deviation relative to the calculation using these parameters as compared with the values calculated with the average rate and with initial depreciations.

Table 1: Deviation between the values calculated  relative to the value obtained using the parameters (for 1950) ρo=1,66 and d=3,82% 

  Variation of the initial K/Y Variation of the time of life

















It can be observed that the K/Y value at the end of the period is practically unaffected by the 20% variation in the initial value. Even in 1960 (ten years after the initial year), a 20% variation in the capital/product ration results in a difference lower than 5% in the projected K/Y values. In 1970, as can be observed in Figure 2, it is already impossible to distinguish the lines resulting from initial values that differ 40%.

In what concerns the effect of changing the depreciation rate, it is crescent with time. At the final year it reaches –10% or 12% of the K/Y ratio estimated value. It is important to point out that 20% and even 40% variations in the chosen depreciation rate (comparing the lower curve with the upper one) do not induce qualitative interpretation errors as it has occurred with the K/Y ratio.

Application of the simplified methodology to some countries.

The International Monetary Fund publishes economical series of different countries[i]. In studies concerning the influence of capital productivity on the growth process, we are interested in those countries where, like Brazil, there has been significant variations in the capital productivity in the last fifty years. The described method was applied to different countries considering the total investment (without subdivision by type of investment) relative to the GDP. That is, GDP and investment was considered in their nominal value. The GDP deflator were applied to these values.

The annual depreciation rate adopted for all countries was 4% and the initial capital/product ratio was introduced so that the variation of the first years of the series became similar to that of the following years. This procedure, whose limitations were discussed, could be compared in some cases with the results obtained by more elaborate methodologies.

The countries chosen were those whith a significant variation in the capital/product in the period chosen included in the IMF data. It is no coincidence that many of these countries have experienced deep modifications in their productive system. The countries chosen were Japan, South Korea, Italy, Brazil, Chile and India. The K/Y ratio curves are shown in Figure 3.

In what concerns Brazil, it can be observed that the choice of the initial K/Y ratio was made by the criterion of reproducing the behavior of the following years and this has the effect of overestimating the initial value so that the first part of the curve is maintained constant. In the following years – less influenced by the initial choice – the curve is similar to that shown in Figure 2 where the po value is overestimated (capital/product in the initial year). Concerning the absolute values, it should be considered that the Figure 2 values are at constant prices while those of Figure 3 are at current prices. For the year 2000 the K/Y value by the simplified method is 2.8. Using the linear depreciation method for current prices the value is 2.9

Figure 3: Evolution of the capital/product ratio for different countries  using investment data published by the IMF.

In the case of Brazil, one can observe in Figure 4 a comparison between the results calculated with the linear depreciation and with the simplified methodology. The largest difference is due to the choice of the K/Y ratio initial value. It should be pointed out that the depreciation rate chosen (4%) is very close to that corresponding to the linear depreciation method.

Figure 4: Comparison of the results obtained for the capital/product ratio using the simplified and linear depreciation methods.


The simplified method proved to be useful and trustworthy for obtaining an approximation for the capital productivity of countries. Based on the described methodology, the evolution of the capital productivity was calculated for 6 countries. The results are discussed in the present e&e issue under the title "Capital Productivity: a further Limitation to Brazilian Growth".

[2] This choice is only possible when it has been previously used for the non-simplified evaluation; in the following item, the influence of the deviations of these parameters on  the calculated productivity will be estimated.



Graphic Edition/Edição Gráfica:
Editoração Eletrônic

Tuesday, 11 November 2008

Contador de visitas