§6. The Illustration of Cutting a Forest

To vary the illustration from intensive agriculture to forestry, let us apply the option selection idea to cutting a forest. Let us consider as the first option the cutting of the forest at the end of nine years, when the income stream consists of the single-item, the production of 900 cords of wood (or $900 if wood is $1 a cord).63

The second option is holding the forest for another Edition: current; Page: [162] year of growth and cutting it at the end of ten years, to receive an income item of 1000 cords (or $1000, assuming an unchanging price of $1 a cord). The two alternatives may be put in precisely the same tabular form as the one previously employed for the case of forestry and farming as follows:

TABLE 8 Optional Incomes from Forest
	10-Year Plan	9-Year Plan	Difference in Favor of 10-Year Plan
1st year	$000	$000
2nd year	000	000
.........	...	...
9th year	000	900	-900
10th year	1000	000	+1000

The last column shows that the ten-year plan, compared with the nine-year plan, involves a cost of 900 in the ninth year, but involves a return of 1000 in the tenth year. The rate of the return (100) over the cost (900) would thus be a little over 11 per cent. If the rate of interest in the market is 5 per cent, it would evidently pay to wait, that is, to postpone the cutting to the tenth year.

The next option would be to cut in the eleventh year, which, as compared with the previous or ten-year plan, would, let us say, cost 1000 in the tenth year and return 1050 in the eleventh year—in other words, give a rate of return over cost of 5 per cent. Evidently, then, it would be a matter of indifference whether the forest were cut in the tenth or eleventh year, inasmuch as the rate of return over cost would be exactly equal to the rate of interest.

Similar reasoning might show that the choice of the next option, that of cutting the forest in the twelfth year, Edition: current; Page: [163] would yield a return of say 21/1050 or 2 per cent. Inasmuch as 2 per cent is less than the rate of interest, this alternative would be rejected. Thus, equilibrium is found where the rate of return on cost equals 5 per cent, the rate of interest.

The case may be illustrated by Chart 16. Let AB represent the number of cords of wood on an acre of growing trees; let A' B' represent the amount of wood which may be expected at the end of five years; let A'' B'' represent what may be expected in ten years and so on for successive years until the forest reaches its maximum growth, MN, at the end of AM years. The percentage-slope, or rate of ascent,64 of the curve BN at any point, therefore, Edition: current; Page: [164] represents the rate of growth, at any time, of the forest. The value at present (at the point of time A) of the forest, in terms of cords of wood, will be represented, not by the height AB, but in a different manner, as follows: If from B' the discount curve65B' C' be drawn, the ordinate of which, at any time, will represent the discounted values of A' B' at that time, then AC' will represent the present value of A' B', i.e., of the amount of the wood if cut in five years. Similarly, AC'' will represent the present value of A'' B'', the wood if cut in ten years. We draw in like manner a number of discount curves until one is found, tT, which is tangent to the curve BN.At will then be the correct value of the young forest, and D will represent the time at which it should be cut. Clearly, At is quite different from AB, the amount of wood at the present time, and also from DT, the amount of wood at the time of cutting. At is the maximum present value out of all possible choices as to the time of cutting. If the forest is for some reason to be cut at once, its value will be only AB; if it is to be cut at A' its present value will be AC' if at A'', it will be AC''; if at D, it will be At.At is the maximum, for if the forest were cut at any other point of time on either side of T the discount curve passing through that point would evidently lie below the curve tT.

At the time A, then, the wood in the forest is only AB but, assuming proper foresting, the value of the forest in terms of wood is At; the rate of growth of the forest is the percentage-slope of BN at B, but the rate of interest is the percentage-slope (the same at all points) of tT.

At the point of tangency alone, namely T, are the rate of growth and rate of interest (both in terms of wood) Edition: current; Page: [165] identical, and to that extent at least there is truth in the thesis that the rate of interest is the rate of growth. This, however, is not the average rate of growth but the rate of growth at the time of cutting. This is the element of truth in the organic productivity theory of Henry George and Alexander Del Mar. These writers based their theories of interest on the productivity of those particular kinds of capital which reproduce themselves, and reached the conclusion that, in the last analysis, the rate of interest consists in the "average rate of growth of animals and plants."66.

Evidently the theory would be substantially correct if "average" were replaced by "marginal." The example of cutting the forest illustrates the simplest theoretical case of marginal productivity as a true basis of the rate of interest.

But that this element of truth is insufficient of itself to afford a complete determination of the rate of interest is evident when we consider that the point at which the forest is to be cut itself depends, among other causes, upon the rate of interest! If the interest rate rises, the discount curves employed become steeper and the point of tangency T moves toward the left, that is, the forest will be cut earlier.

In no case, of course, is the time of cutting the time of maximum stumpage. To wait for that time would eat up too much interest. The theories of Del Mar and Henry George thus constitute a special case under the opportunity principles.

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