Time in Economic Theory - Regional Science Matters - page 55

The present value (PV) of stopping the maturing by storage at time T is
determined as:

e rT Maximize PV ¼ VT ðÞ ð 5 : 1 Þ

;

The necessary condition of optimality of storing time is:

dV = V ¼ r ð 5 : 2 Þ

;

Optimal economic duration of the process thus implies that the storing should be
stopped when:
The rate of growth of value equals the rate of interest .
This rule of thumb is not a special dynamic case, only relevant for point input,
point output decision problems. The following dynamic optimization model shows
that the value maximization condition also holds for harvesting sequences over time
of some renewable biological resource (e.g. wine in some plantation, trees in a
natural forest, or fish in the sea) as shown by the following model.
Maximize harvesting income ¼ R 0 1 p u(t)x(t)

¼ Maximize harvesting income

e rt dt;
Subject to the growth condition:

2

dx = dt ¼ ax ðÞ bx ðÞ u ðÞ x ðÞ

u(t) ¼ the rate of harvesting at time t

x(t) the stock of the biological resource at time t

We assume that the price is kept at a constant level.
Maximizing the Hamiltonian

¼

;

e rt

2

H ¼ pu t ðÞ xt ðÞ λ ax t ðÞ bx t ðÞ ut ðÞ xt ðÞ ð 5 : 3 Þ

leads to an optimal rate of harvesting at each instance of time.
One of the conditions of optimality requires that the rate of harvesting is
determined when

d λ=λ ¼ r ð 5 : 4 Þ

Again we get the rule of thumb of optimal harvesting when:
The rate of growth of value should equal the rate of interest at the value
maximizing rate of harvesting.
The Marxian labor theory of value was finally proved to be wrong. Not only is
land (i.e. natural resources needed) an indispensable fundamental factor of produc-
tion beside labor, as had already been shown by Johann Heinrich von Th¨nen
( 1826 , 1960 ). Also the flow of time itself had now been shown to be a crucial
variable in the determination of the dynamics of capital value.