In principle, the mark-up of historical dollars in the plant account to current dollars is a process of following the procedure shown in Figure 5.6 for each layer, or vintage, of plant. This requires knowledge of the age of each layer, which is typically unavailable to investors. A useful mathematical approximation can be derived by partitioning the plant account into layers based on: (a) gross plant life (L) and (b) a constant real asset growth rate ( g) over the historical years covering the plant life. With these layers specified, the GDP Deflators are applied consistent with the procedure of Figure 5.6. CAPX is the capital expenditure outlay, or layer. Other symbols are self-evident. The derivation below constructs the plant in current-dollar layers and then derives the original-cost (historical-dollar) layers. A mark-up factor is the ratio of the total current-dollar gross plant to the total historical-dollar gross plant.

Consider a gross plant amount in current dollars where current reflects a particular year, such as 1993. The variables shown below would be stated in 1993 dollars.

Start with the identity,

Assuming a constant, real historic growth rate ( g) in building up the plant account, then:

Since RETIREMENT is CAPX made L years ago:

After substitution, the beginning identity becomes the equation:

In this equation, CAPX can be solved for:

END PLANT can be decomposed into a summation of past capital expenditures as follows:

With the assumption of a constant, real historical asset growth rate ( g) and...