Capital Input Cost

Abstract

Unlike labour productivity change, the measurement of total factor productivity change (or difference) crucially depends on the measurement and decomposition of capital input cost. This chapter discusses in detail the basics of its measurement and shows that one can dispense with the usual neo-classical assumptions. Various models are reviewed, as well as the role played by the ‘rate of return’.

Appendices

Appendix A: Decompositions of Time-Series Depreciation

Time-series depreciation of an asset of type i and age j over period t is, according to expression (3.6), defined as $P_{i,j-0.5}^{t^-} - P_{i,j+0.5}^{t^+}$, which is the (nominal) value change of the asset between beginning and end of the period. This value change combines the effect of the progress of time, from t ⁻ to t ⁺, with the effect of ageing, from j − 0.5 to j + 0.5. Since value change is here measured as a difference, a natural decomposition of time-series depreciation into these two effects is

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {P_{i,j-0.5}^{t^-} - P_{i,j+0.5}^{t^+} =} \\ & &\displaystyle \qquad (1/2)\left[(P_{i,j-0.5}^{t^-} - P_{i,j-0.5}^{t^+}) + (P_{i,j+0.5}^{t^-} - P_{i,j+0.5}^{t^+})\right] + \\ & &\displaystyle \qquad (1/2)\left[(P_{i,j-0.5}^{t^-} - P_{i,j+0.5}^{t^-}) + (P_{i,j-0.5}^{t^+} - P_{i,j+0.5}^{t^+})\right]. {} \end{array} \end{aligned} $$

(3.79)

This decomposition is symmetric; it has the same structure as the decomposition of a value difference into Bennet indicators. The first term at the right-hand side of the equality sign measures the effect of the progress of time on an asset of unchanged age; this is called revaluation. The revaluation, as measured here, is the arithmetic mean of the revaluation of a j − 0.5 periods old asset and a j + 0.5 periods old asset, and may be said to hold for a j periods old asset.

The second term c oncerns the effect of ageing, which is measured by the price difference of two, otherwise identical, assets that differ precisely one period in age. This is called Hicksian or cross-sectional depreciation. The arithmetic mean is taken of cross-sectional depreciation at beginning and end of the period, and, hence, may be said to hold at the midpoint of period t.

The ratio-type counterpart of expression (3.79) is a decomposition into two Fisher-type indices,

$$\displaystyle \begin{aligned} \frac{P_{i,j+0.5}^{t^+}}{P_{i,j-0.5}^{t^-}} = \left[\frac{P_{i,j-0.5}^{t^+}}{P_{i,j-0.5}^{t^-}}\frac{P_{i,j+0.5}^{t^+}}{P_{i,j+0.5}^{t^-}}\right]^{1/2} \left[\frac{P_{i,j+0.5}^{t^+}}{P_{i,j-0.5}^{t^+}}\frac{P_{i,j+0.5}^{t^-}}{P_{i,j-0.5}^{t^-}}\right]^{1/2}. \end{aligned} $$

(3.80)

The first term at the right-hand side of the equality sign measures revaluation. The second ter m measures cross-sectional depreciation. As one sees, revaluation depends on age, and cross-sectional depreciation depends on time. In the usual model, these two dependencies are assumed away. Revaluation is approximated by $P_{i}^{t^+}/P_{i}^{t^-}$, the price change of a new asset of type i from beginning to end of period t. Cross-sectional depreciation is approximated by 1 − δ _ij, where δ _ij is the (positive) percentage of annual depreciation applying to an asset of type i and age j. The specific formulation highlights the fact that ageing usually diminishes the value of an asset.

Under these two assumptions, the basic time-series depreciation model for an asset of type i and age j, over period t, is given by

$$\displaystyle \begin{aligned} \frac{P_{i,j+0.5}^{t^+}}{P_{i,j-0.5}^{t^-}} = \frac{P_{i}^{t^+}}{P_{i}^{t^-}}(1 - \delta_{ij}) \: \:(j=1,\ldots ,J_i). \end{aligned} $$

(3.81)

For assets that are acquired at the midpoint of period t it is useful to distinguish between new and used assets. Over the second half of period t, the model reads

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{P_{i,0.5}^{t^+}}{P_{i,0}^{t}} & = &\displaystyle \frac{P_{i}^{t^+}}{P_{i}^{t}}(1 - \delta_{i0}) \\ \frac{P_{i,j+0.5}^{t^+}}{P_{i,j}^{t}} & = &\displaystyle \frac{P_{i}^{t^+}}{P_{i}^{t}}(1 - \delta_{ij}/2) \: \:(j=1,\ldots ,J_i), \end{array} \end{aligned} $$

(3.82)

where (1 − δ _ij∕2) serves as an approximation to (1 − δ _ij)^1∕2. The percentage of annual depreciation, δ _ij, ideally comes from an empirically estimated age-price profile for asset-type i. Under a geometric profile one specifies δ _i0 = δ _i∕2 and δ _ij = δ _i (j = 1, …, J _i).

Replacing $P_{i,j+0.5}^{t^+}$ by $\mathcal {E}^{t^-}P_{i,j+0.5}^{t^+}$, $P_{i,0.5}^{t^+}$ by $\mathcal {E}^{t}P_{i,0.5}^{t^+}$, and $P_{i,j+0.5}^{t^+}$ by $\mathcal {E}^{t}P_{i,j+0.5}^{t^+}$ delivers expressions for anticipated time-series depreciation.

Appendix B: Geometric Profiles

Consider the fundamental asset price (equilibrium) equation in expression (3.29). To avoid notational clutter it is assumed that there is no tax. It is further assumed that the future interest rate is constant; that is, r ^t+j = r (j = 0, …, J _i). Then

$$\displaystyle \begin{aligned} P_{i,j-0.5}^{t^-} = \frac{u_{ij}^{t}}{1+r} + \frac{u_{i,j+1}^{t+1}}{(1+r)^{2}} + \ldots + \frac{u_{i,J_{i}}^{t+J_{i}-j}}{(1+r)^{J_{i}-j+1}}. \end{aligned} $$

(3.83)

A geometric age-efficiency profile¹⁹ means that $u_{ij}^{t}/u_{i1}^{t} = \phi _{ij} = (1 - \delta _i)^{j-1}$ (i = 1, …, I;j = 1, …, J _i). Equivalently, $u_{i,j+1}^{t}/u_{ij}^{t} = 1 - \delta _i$. By substituting this into expression (3.83) one obtains

$$\displaystyle \begin{aligned} P_{i,j-0.5}^{t^-} = \frac{u_{ij}^{t}}{1+r} + \frac{u_{ij}^{t+1}(1-\delta_i)}{(1+r)^{2}} + \ldots + \frac{u_{ij}^{t+J_{i}-j}(1-\delta_i)^{J_{i}-j}}{(1+r)^{J_{i}-j+1}}. \end{aligned} $$

(3.84)

For a one-period older asset of the same type one then has, similarly but with one term less,

$$\displaystyle \begin{aligned} P_{i,j+0.5}^{t^-} = \frac{u_{i,j+1}^{t}}{1+r} + \frac{u_{i,j+1}^{t+1}(1-\delta_i)}{(1+r)^{2}} + \ldots + \frac{u_{i,j+1}^{t+J_{i}-j-1}(1-\delta_i)^{J_{i}-j-1}}{(1+r)^{J_{i}-j}}. \end{aligned} $$

(3.85)

Using again the geometric-profile assumption, expression (3.85) can be written as

$$\displaystyle \begin{aligned} P_{i,j+0.5}^{t^-} = \frac{u_{ij}^{t}(1-\delta_i)}{1+r} + \frac{u_{ij}^{t+1}(1-\delta_i)^2}{(1+r)^{2}} + \ldots + \frac{u_{ij}^{t+J_{i}-j-1}(1-\delta_i)^{J_{i}-j}}{(1+r)^{J_{i}-j}}, \end{aligned} $$

(3.86)

or

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} P_{i,j+0.5}^{t^-} {=} (1-\delta_i)\left( \frac{u_{ij}^{t}}{1+r} + \frac{u_{ij}^{t+1}(1-\delta_i)}{(1+r)^{2}} + \ldots + \frac{u_{ij}^{t+J_{i}-j-1}(1-\delta_i)^{J_{i}-j-1}}{(1+r)^{J_{i}-j}}\right).\\ \end{array} \end{aligned} $$

(3.87)

The part between brackets appears to be identical to $P_{i,j-0.5}^{t^-}$ minus its last term, so that

$$\displaystyle \begin{aligned} P_{i,j+0.5}^{t^-} = (1-\delta_i)\left(P_{i,j-0.5}^{t^-} - \frac{u_{i,J_{i}}^{t+J_{i}-j}}{(1+r)^{J_{i}-j+1}}\right). \end{aligned} $$

(3.88)

Now $u_{i,J_{i}}^{t+J_{i}-j}$ is the user cost of an asset, of type i that is j years old in period t, in the final year of its (economic) existence. If J _i →∞ then this user cost will expectedly tend to 0. The denominator $(1+r)^{J_{i}-j+1}$ will then tend to ∞. Together it is reasonable to expect that

$$\displaystyle \begin{aligned} P_{i,j+0.5}^{t^-} \approx (1-\delta_i)P_{i,j-0.5}^{t^-}, \end{aligned} $$

(3.89)

which means that the age-price profile of the asset is approximately geometric.

Reversely, consider expression (3.25) without the tax component:

$$\displaystyle \begin{aligned} u_{ij}^{t} = (1+r^{t})P_{i,j-0.5}^{t^-} - P_{i,j+0.5}^{t^+} \: (j=1,\ldots ,J_{i}). \end{aligned} $$

(3.90)

For a one-period older asset of the same type then

$$\displaystyle \begin{aligned} u_{i,j+1}^{t} = (1+r^{t})P_{i,j+0.5}^{t^-} - P_{i,j+1.5}^{t^+}. \end{aligned} $$

(3.91)

Assume now a geometric age-price profile; that is, $P_{ij}^{t}/P_{i0}^{t} = \varphi _{ij} = (1 - \delta _i)^{j}$ (i = 1, …, I;j = 1, …, J _i). Equivalently, $P_{i,j+1}^{t}/P_{ij}^{t} = 1 - \delta _i$. Substituting this into expression (3.91) delivers

$$\displaystyle \begin{aligned} u_{i,j+1}^{t} = (1+r^{t})(1 - \delta_i)P_{i,j-0.5}^{t^-} - (1 - \delta_i)P_{i,j+0.5}^{t^+} = (1 - \delta_i)u_{ij}^{t}, \end{aligned} $$

(3.92)

which means that the age-efficiency profile of the asset is geometric.