4  Physical capital stock and the user cost of capital

Measuring capital and multifactor productivity requires measures of physical capital inputs. As the flow of physical capital services is not directly observable, productivity analysts usually assume the flow of capital services is proportional to the capital stock. Ideally the capital stock measure should be formed taking into account the loss in productive capacity of capital assets that occurs over time. Subsection 4.1 discusses measurement of productive capital stocks, where stocks are constructed from past investments and where the loss in the productive capacity of capital assets is taken into account. Subsection 4.2 illustrates the construction of the productive capital stock using a simple numerical example. Finally, subsections 4.3 and 4.4 discuss and illustrate measurement of rental prices and how these are linked to economic depreciation and the loss in productive capacity of capital assets. Readers interested in further details on measuring physical capital stocks and rental prices should consult, for example, Hulten (1990), Hulten and Wykoff (1995), and Diewert and Lawrence (2000).

4.1  Productive capital stock

Productive capital stocks endeavour to measure the total productive capacity of different types of capital assets in existence at a point in time. Suppose information on investments in a particular asset type is available for period to period for and is denoted by the vector . Furthermore assume that the productive capacity of an asset in period that is now periods old (that is, the -vintage asset) is given by:

(9)    

where denotes the relative productive capacity of a -vintage asset to the productive capacity of a new asset. The series is known as the age-efficiency schedule and is usually normalised so that . The age-efficiency schedule shows the decline in the productive capacity of an asset over its economic life.[11]

Three commonly used age-efficiency patterns are the linear, ‘one-hoss-shay’ and geometric age-efficiency schedules. The linear age-efficiency schedule assumes that the productive capacity of an asset depreciates linearly over the asset’s economic life. The ‘one-hoss-shay’ efficiency pattern assumes the productive capacity of an asset remains constant over its economic life but then falls to zero when the asset’s economic life ends.[12] The geometric age-efficiency pattern assumes the productive capacity of an asset declines at a constant rate.

Given real investment data, the functional form of the age-efficiency schedule, and assuming past vintages of a particular asset can be aggregated, the productive capital stock for a particular asset type in period is calculated as follows:

(10)    

Equation (10) is known as the perpetual inventory model of the productive capital stock.[13]

4.2  Numerical example of the productive capital stock

Table 6 presents hypothetical data on the (new) price of purchasing a capital asset () and the nominal investment in the asset (). The volume of investment in the asset () is found by dividing the nominal investment series by the price series (that is, dividing column (2) by column (1)). The age-efficiency schedule, which is displayed in column (4), assumes the life of the asset is five years and the loss in productive capacity is linear (that is, the age-efficiency schedule () is linear).

Table 6 – Productive capital stock

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
	1.0	100.0	100.0	1.00	100.0						100.0
	1.1	120.0	109.1	0.80	80.0	109.1					189.1
	1.3	150.0	115.4	0.60	60.0	87.3	115.4				262.7
	1.5	160.0	106.7	0.40	40.0	65.5	92.3	106.7			304.4
	1.6	180.0	112.5	0.20	20.0	43.6	69.2	85.3	112.5		330.7
	1.7	195.0	114.7	0.00	0.0	21.8	46.2	64.0	90.0	114.7	336.7

To calculate the productive capital stock for the asset it is first necessary to account for the diminished productivity capacity of each investment over time. Columns (5) to (10) present the productive capacity of the asset for each investment. These series are calculated by multiplying the initial investment by the relevant value from the age-efficiency schedule. For example, to find the productive capacity of an investment in the asset in that was made in , the initial investment () of 100 is multiplied by 0.4, the value of the age-efficiency schedule for a three period old asset (). Likewise, the productive capacity of an investment in the asset in that was made in (69.2) is found by multiplying 115.4 () by 0.6, the value of the age-efficiency schedule for a two period old asset ().

Finally, the productive capital stock of the asset is calculated using equation by adding the efficiency adjusted investments for each period. The resulting productive capital stock for the asset is reported in column (11).