Anexo 2 Taxa de depreciação do estoque de capital equivalente a um tempo de depreciação linear N.

The investment depreciation rate is defined as the stock fraction in year t that is depreciated in the following year. The equivalent depreciation rate would be the rate that reproduces the depreciation resulting from a set of investments in v years that are linearly depreciated.

Figure A2.1: In the first graphic, investment values of 100 units with a time of life (v) of 10 years and in the second one, values of the surviving investment and stock according to the age of the goods and the total stock.

As an example, let's consider an annual constant investment I of 100 units, linearly depreciated along 10 years as shown in Figure A2.1. The stock would be stabilized in 550 units and the equilibrium depreciation would be 100/550=18.2%. In the stock composition there would be 10 units correponding to t years backwards (depreciated 10/10=100% ), 20 units of t-1 years backwards depreciated 10/20=50% and so on. Finally, the stock of the previous year would be 100 units that would not be depreciated yet.

In this example the stock would be the sum of the arithmetic progression (10+20+ 30+ ..........+100) with t=10 terms and would be equal to 550. In the generic case for t years of life and constant investment I one would have an arithmetic progression with common difference equal to I/t and the stock K would be

K=I/t.(1+2+.....t)=I/t*(t.(t+1)/2)=I.(t+1)/2

the depreciation in equilibrium would be equal to the investment (D=I) and the depreciation would be:

As will be verified, since there is no stationary state when investment varies along the time likewise there is no exact equivalent rate. Then the rate to be considered depends on the investments' history.

The capital stock K in an year t would be obtained from the investment of the previous year It-1 and from the depreciation for the year t (Dt).

K_t = K_t-1 + I_t-1 – D_{t
(1)}

We are assuming that the investment is incorporated to the stock in the year after it is effectuated and that the depreciation function is linear. As a consequence, the depreciation is equal to the average investments in the the years previous to that of the investment. Therefore one will have for the generic year t

(2)

This is so because we are supposing that the investment made in year t-2 becomes productive (it is incorporated to the stock) in the following year t-1 and starts to be depreciated in year t.

To simplify things, let us consider the investment made from zero year on.

It should be remembered that the stock before the zero year, according to the adopted function, will be extinguished in the year v+1. This means that the stock of the zero year will be totally depreciated in v+1.That is, the sum of the investments input (investments) less the sum of the investments output (depreciation), from zero year on, will form the stock in year v+1.

Table A2.1 shows the investments, depreciation and stock that occurred in the period.

As a hypothesis concerning the investment, let's consider that it has grown at j annual rate.

That is

(3)

where

The table below shows the evolution of the investment, depreciation and stock calculation along time, according to equations (1), (2) and (3).

Year	Investment	Investment depreciation in the period	Accumulated capital stock due to investments in the period
0		0	0
1
2
3
.......
v
v+1	since we have	Since only the last v years influence depreciation
v+m

According to the table above, dividing the depreciation by the stock, one gets the depreciation , for any year where , that is equivalent to a linear depreciation during v years, considering the annual investment growth rate j. The equivalent depreciation rate is given by:

that is, is constant in time for fixed values of v (time of life) and c (related to the investment rate growth j)

In Table A2.2 we show the equivalent depreciation rates for different lifes and different annual investment growth rates.

Table A2.2

Life / Investment growth	0%	1%	2%	3%	4%	7%
10 years	18,18%	17,91%	17,66%	17,41%	17,17%	16,52%
19 years	10,00%	9,71%	9,44%	9,19%	8,96%	8,35%
20 years	9,52%	9,23%	8,96%	8,71%	8,48%	7,88%
40 years	4,88%	4,58%	4,33%	4,11%	3,92%	3,50%
48years	4,08%	3,79%	3,54%	3,33%	3,16%	2,80%
50 years	3,92%	3,63%	3,38%	3,18%	3,01%	2,67%

In the present applications we have considered the "equivalent" rates found in Table A2.3 where we also indicate the time of life and the investment growth rates per type of good in the period when the data is available (1901/2000 for construction and 1908/2000 for machines and equipment). We have considered investment growth rates obtained by exponential adjustment of IBGE (2003) data.

Table A2.3

	Residential Construction	Non-residential Construction	Machines and Equipment
Life	50	40	20
Annual Investment Growth	6,6%	6,3%	5,3%
"Equivalent" Depreciation Rate	2,7%	3,6%	8,2%

Figure A2.2 shows the depreciation rates obtained using the linear function and the above expression for the “equivalent” rate

Figure A2.2: Equivalent depreciation rates and depreciation rate varying according to the capital stock age (Method of Permanent Stock) for different types of goods

and for the aggregated value (total).

Graphic Edition/Edição Gráfica:
MAK
Editoração Eletrônica

Revised/Revisado:
Sunday, 28 August 2005.

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