# Box: Basic Formulas for the Perpetual Inventory Method

Net stocks are estimated using the perpetual inventory method with geometric depreciation. The method begins with the investment for each year i in type of asset j, that is, $I_ij_$. For current-cost and real-cost valuation, the investment that is used in the calculation is in 1992 dollars; that is, the investment is in current dollars that have been deflated by the price index for that type of asset with a 1992 reference year. For historical-cost valuation, investment in dollars is used in the calculation so that assets are measured using the prices that existed when they were acquired.

The annual geometric rate of depreciation for type of asset j, $\delta _j_$, is related to the declining-balance rate for type of asset j, $R_j_$, (which is the multiple of the comparable straight-line rate) and to the average service life for type of asset j in years, $T_j_$, by:

$\delta _j_ = R_j_/T_j_.$

Consider the contribution of investment during year i in type of asset j to the real-cost net stock at the end of year t, $N_tij_$. New assets are assumed, on average, to be placed in service at midyear, so that depreciation on them is equal to one-half the new investment times the depreciation rate. Therefore, the contribution to the real-cost net stock at the end of year t is given by:

$N_tij_ = I_ij_ \left(1-\delta _j_/2\right)\left(1-\delta _j_\right)^t-i^ ,$

where $t\ge i$.

To calculate the real-cost net stock at the end of year t for asset j, N_tj_, the contributions are summed over all vintages of investment flows for that asset so that:

$N_tj_ = \sum _i=1_^t^ N_tij_.$

The equations used to estimate historical cost stocks are identical to equations (2) and (3) except that the investment flows are expressed at historical cost rather than at real cost.

Current-cost estimates of the net stock of asset j (in dollars), $C_tj_$, are obtained by multiplying the real-cost net stock at the end of year t for asset j by the value at the end of year t of the price index that was used to deflate nominal investment in asset j, $P_tj_$, so that:

$C_tj_ = P_tj_ N_tj_.$

The current-cost net stock of assets at the end of year t, $C_t_$, is estimated as the sum of the stocks given above across all types of assets, so that:

$C_t_ = \sum _j_ C_tj_.$

Depreciation on an asset j during year t, $D_tj_$, equals the net stock of asset j at the end of year t-1 plus investment in asset j during year t less the net stock of asset j at the end of year t, so that:

$D_tj_ = N_t-1,j_ + I_tj_ - N_tj_.$

This equation holds under real-cost and historical-cost valuation.

Current-cost depreciation, $M_tj_$, is calculated by multiplying real-cost depreciation from equation (6) by the average price of asset j during year t, $&overline;P_tj_$, so that:

$M_tj_ = &overline;P_tj_ D_tj_.$

Current-cost depreciation for an aggregate of assets in year t is calculated by summing across the various types of assets, that is, by:

$M_t_ = \sum _j_ M_tj_.$

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