ABSTRACT
As n -» °°, R* -> 2r. However, the approach is not monotonic increasing: at n — 1, R* = 2r; it then falls to a minimum value, which is always
K = ^[(<2 + <2* ) - ( S + S*)].6
The accounting profit for any year is:
A = L ( Q - S ) ,
so that R *,in this instance, is
2 r ( Q - S ) l(Q + Q * ) - ( S + S*)]
[la]
greater than r, at values of n which are related to r itself. For example, when r = 5 per cent, the minimum value of R* occurs between the tenth and twentieth year; when r = 30 per cent, it occurs between the fifth and tenth year. That is, in the ‘one-hoss shay’ case, the account ant’s measure of the rate of profit will give different answers for two businesses which are alike in every respect except that the machines of one are longer-lived than those of the other. If, for example, the rate of profit is 30 per cent, the accountant’s answer for n = 5 years is 42.1 per cent and for n = 30 years, 53.4 per cent. 2.4 In passing, it could be mentioned that American studies show that some businessmen estimate rates of return as the ratio of accounting profit to the gross (that is, undepreciated) value of assets. The book value of capital when straight-line depreciation is used is half the gross value. If the investment projects concerned are ‘one-hoss shays’ and n is large, the accounting rate of profit would approximately equal r. Expression [la] may therefore provide a theoretical justification for this business practice.9 2.5 The corresponding expressions for cases 2, 3, and 4 - falling; rising; and rising, then falling quasi-rents - are:
2 R* = - n
(1 -& ”) ( ! + r - b )
for case 3 read a (> 1) for b\
( am-l l-fr/(l+r)V” jm-ifMl-fo/(l+r)}" ?* _ 2 I a — 1____ \ 1 — b ) 1 + r —a______ \(1 + r — &)(! +
" 1 -U K 1 + r)jm (b [1 + r))n-m]\ v 1 + r —a \ (1 + r — 6)(1 + r)m I
[Id] 2.6 In case 2, R* -» 0 as n -> °°. There is a value of n where R* = r, as R* is greater than r for n — 2. The rapidity with which this occurs, for certain values of the variables, is shown in Table 4.1. Table 4.1 Values of R *, case 2 (balanced stock, physical capital) (percentages)
2 years 5 years 10 years 20 years 30 years r (non-bracketed figures, 6 = 0.5; bracketed figures, b = 0.9)
5 6.7 [7.4] 3.7 [5.7] 2.0 [4.9] 1.0 [3.8] 0.1 [3.0] 10 13.4 [n.a.] 7.4 [n.a.] 4.0 [n.a.] 2.0 [n.a.] 1.3 [n.a.] 20 n.a. [30.3] n.a. [24.4] n.a. [21.4] n.a. [16.4] n.a. [12.5] 30 40.8 [46.0] 22.5 [37.9] 12.0 [33.5] 6.0 [25.2] 4.0 [18.9]
Would it be too fanciful to suggest that the low levels of rates of profit which Nevin found in British manufacturing may in part be due to a combination of quasi-rent patterns similar to those of case 2 and rather large n’sl 2.7 In case 3, the value of R* is greater than r (and the values of R* for cases 1 and 2) at small values of n , falls slightly as n increases, but then quickly increases, approaching 00 as n -> With a = 1.1, n = 20 years, and r = 30 per cent, R* is already 108.8 per cent; with a = 1.5, the corresponding figure is 796 per cent! (R* is not defined for the case of 1 4-r = a.) Case 4 is a combination of cases 2 and 3. For given values of n, the value of R* lies in between the values of R* of the two previous cases.10 As m -+ n, the case 3 result comes to dominate the expression; as m -> 0, the case 2 result comes to dominate. This is illustrated in Table 4.2 where certain values of R* for cases 2 ,3 and 4 are shown; r = 10 per cent, a = 1.3, b — 0.7, and n = 20 years. Just by a fluke of counterbalancing forces, the accountant’s measure could give the right answer in case 4.
