# Age-efficiency, age-price and depreciation rate functions

## Age-efficiency functions

14.36 There is a lack of empirical data about the shape of age-efficiency functions and the choice is a matter of judgement. Although capital stock levels are sensitive to the shape of the age-efficiency function, average growth rates are not. (In fact, if real GFCF is held constant over time, the choice would have no impact on the capital stock growth rate, but it would affect the capital stock level.) The ABS has chosen to use hyperbolic functions, the same approach as that used by the US Bureau of Labor Statistics (BLS). In a hyperbolic function, the efficiency of the asset declines by small amounts at first and the rate of decline increases as the asset ages.

14.37 Hyperbolic decline has the form:

\(\large E_t = \frac{M-A_t }{M-bA_t }\)

where \(E_t\) is the efficiency of the asset at time \(t\) (as a ratio of the asset's efficiency when new).

\(M\) is the asset life as per the Winfrey distribution (discussed below)

\(A_t\) is the age of the asset at time \(t\)

\(b\) is the efficiency reduction parameter.

14.38 The efficiency reduction parameter b is set to 0.5 for machinery and equipment, and 0.75 for structures - the same parameter values as used by the US BLS. The higher value for non-dwelling construction redistributes efficiency decline to occur later in the asset's life, relative to machinery and equipment, the efficiency decline of which is distributed more evenly throughout the asset's life. For computer software, b is set to 0.5. For livestock, b is also set to 0.5. Clearly, a more accurate age-efficiency function and age-price function could be assumed by recognising that livestock are immature for a number of years before they begin service as mature animals. However, such improvements compromise model simplicity and the improvements from doing so would be quite small. For mineral exploration b is set to 1, implying that there is no efficiency decline in exploration knowledge. The opposite is the case for artistic originals, where b is set to 0, implying straight-line efficiency decline.

14.39 Graphs below show (i) the main types of age-efficiency functions and (ii) the age-price functions relating to each of the age-efficiency functions. When the hyperbolic functions for each of the possible lives of an asset are weighted together (as per the Winfrey distribution), the resulting average age-efficiency function resembles a logistic function with a point of inflection towards the end of its maximum life.

## Image

### Graph 14.1 Age-efficiency functions

## Description

### Graph 14.1 Age-efficiency functions

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### Graph 14.2 Age-price functions

## Description

### Graph 14.2 Age-price functions

## Age-price functions

14.40 Age-price functions are calculated using average age-efficiency functions and a real discount rate. The age-efficiency function describes the decline in the flow of capital services of an asset as it ages. Using the discount rate, the net present value of future capital services can be readily calculated. For instance, when multiplied by a suitable scalar, the first value of the age-price function represents the present discounted value of the capital services provided by an asset over its entire life. The second value of the age-price function represents the present discounted value of the capital services provided by an asset from the end of its first year until the end of its life. The third value represents the present discounted value of the capital services provided by an asset from the end of its second year until the end of its life, and so on. Age-price functions are normalised and adjusted for mid-year purchase, to allow for some consumption of fixed capital occurring in the first year. The ABS has chosen a real discount rate of 4 per cent, the same as that used by the US BLS and which approximates the average real 10-year Australian bond rate.

14.41 When the net present values of the different assets are aggregated for a particular period, they form the net capital stock for that period.

## Depreciation rate functions

14.42 In real terms, depreciation (or COFC) is the difference between the real economic value of the asset at the beginning of the period and at the end of the period. The depreciation rate function is calculated as the decline in the age-price function between assets of consecutive ages. When multiplied by a suitable scalar, it shows the pattern of real economic depreciation or COFC over an asset's life. Consumption of fixed capital for each vintage of each asset type is then aggregated to form the total consumption of fixed capital for that period. It can also be calculated as GFCF less the net increase in the net capital stock; that is, GFCF less the difference between the net capital stock at the end of the period and at the beginning of the period).