4.3  Age-efficiency and age-price schedules, economic depreciation, and the user cost of capital

When forming an aggregate quantity index, prices are used to weight the different quantities. Hall (1968) showed that the rental price of capital (user cost of capital) is the relevant price when aggregating different types of capital For some assets the rental price for different asset vintages is observable because there is an active rental market (for example, residential and non-residential buildings). However, for other assets, rental markets do not exist or are very thin (for example, rental markets may not exist for certain types of specialised machinery). In this situation firms purchase capital assets and pay an implicit rental for their use. Although the user cost of capital is not directly observable, it can be imputed using information on the price of a new asset, the rate of economic depreciation, and asset price inflation.

The assumption of perfect competition implies the price of a -vintage asset in period is equal to the discounted stream of future rentals , that is:

where is the discount rate and the time the asset’s life ends.

Noting that, equation (11) can be rewritten as:

If equation (12) states that the price of a new asset at the beginning of period is equal to the discounted value of the rental for period plus the discounted price of the one period old asset at the beginning of .

When measuring the user cost of capital it is common to incorporate asset price inflation. When asset price inflation is incorporated the price of a -vintage asset in period is equal to the price of the -vintage asset in period multiplied by one plus that rate of asset price inflation:

where is the asset price rate of inflation.

Substituting the similar expression for from equation (13) into equation (12) and solving for the user cost of capital yields:

Equation is the basic user cost formula. The first term of equation (14) is the finance cost associated with purchasing the asset (or the opportunity cost from investing the funds used to purchase the asset elsewhere). The second term is the loss in the value of the asset due to ageing and represents economic depreciation. The final term represents the capital gains or losses associated with owning the asset.

Equation is often expressed in rate form as follows:

where is the depreciation rate from an -vintage asset in period .[14]

In equation (15)the price of a new asset () and ex post asset inflation () are observable. The discount rate () can be obtained from financial markets or an ex post internal rate of return can be computed using the approach suggested by Jorgenson and Griliches (1967). The latter approach involves equating capital income with the product of the user cost of capital and the productive capital stock and then solving for the discount rate.

To calculate the economic depreciation rate () information is used on the age-efficiency schedule that is used to calculate the productive capital stock. This recognises the fact that:

One cannot select an efficiency pattern independently of the depreciation pattern and maintain the assumption of competitive equilibrium at the same time. And, one cannot arbitrarily select a depreciation pattern independently from the observed pattern of vintage asset prices (suggesting a strategy for measuring depreciation and efficiency).

Hulten 1990:129

To calculate economic depreciation using the age-efficiency schedule it is first necessary to calculate the age-price schedule from the age-efficiency schedule. The age-price schedule () gives the relative value of a -vintage asset to the value of a new asset. Economic depreciation for a particular asset is then calculated by tracing the loss in value of an investment which is derived by multiplying the initial investment by the relevant value from the age-price schedule. The age-price schedule is usually normalised so that .

The link between the age-efficiency schedule and age-price profile can be seen as follows. In a competitive market the ratio of the -vintage rental price to the -vintage rental price asset is equal to the relative efficiency of the -vintage asset to the -vintage asset:

(16)    

Substituting equation (16) into equation (11) yields:

(17)    

Hence the vintage asset price sequence is a function of the age-efficiency profile.

Assuming the user cost of capital grows at a constant nominal rate (that is, for ), then equation (17) can be rewritten as:

(18)    

Consider a new asset purchased at period for and its subsequent sequence of vintage prices. The rental price in period is found by solving equation (17) for (which is possible since is observed). The subsequent sequence of vintage prices is also calculated by using equation (17) and the assumption that the rental price grows at a constant rate.

Values of the age-price schedule are found by taking the price of the new asset to the price of the -vintage asset:

(19)    

where is the price of the new asset in period .