Production Over Time

Abstract

Time is a key input in all kinds of production. However, the subject of using time for production is more relevant in some industries than in others. The distinguishing feature of agriculture and other biologically based industries compared to other industries is the fact that production often takes quite a long time. Just think of the production of wood from beech trees, which first reach maturity after 50–100 years of growth!

Appendix

14.1.1 Interest Calculation

With an interest rate of i, an amount of a MU at the beginning of year 1 (time t ₀) is equivalent to an amount of a(1 + i)ⁿ MU at the end of year n. Similarly, an amount of b MU at the end of year n is equivalent to an amount of b(1 + i)⁻ⁿ at the beginning of year 1.

If, during n years, there is a series of net payments (these can be negative or positive) of b ₀, …, b _n−1 at the beginning of each year t = 1, …, n then the accumulated sum of this series of net payments is equivalent to $\sum \limits_{{t = 0}}^{n - 1} {{b_t}{{(1 + i)}^{{(n - t)}}}}$at the end of year n. This amount is referred to as the future value (FV) which is defined as:

$$ FV = \sum\limits_{{t = 0}}^{{n - 1}} {{b_t}{{(1 + i)}^{{(n - t)}}}} $$

(A.1)

The future value FV can be converted to a present value (NPV (Net Present Value)) by calculating the amount that would be equivalent to FV at the beginning of year 1. This is done by multiplying FV by (1 + i)⁻ⁿ which produces:

$$ NPV = \Bigg(\sum\limits_{{t = 0}}^{{n - 1}} {{b_t}{{(1 + i)}^{{(n - t)}}}} \Bigg){(1 + i)^{{ - n}}} = \sum\limits_{{t = 0}}^{{n - 1}} {{b_t}{{(1 + i)}^{{ - t}}}} $$

(A.2)

A one-time amount of A (e.g. an investment at the beginning of year 1) can be converted into an equivalent series of equally large amounts over n years by multiplying the one-time amount by the annuity factor $a_{{i\left| n \right.}}^{{ - 1}}$, where$a_{{i\left| n \right.}}^{{ - 1}}$ is defined by:

$$ a_{{i\left| n \right.}}^{{ - 1}} \equiv \frac{i}{{1 - {{(1 + i)}^{{ - n}}}}} $$

(A.3)

An amount of b calculated as:

$$ b = Aa_{{i\left| n \right.}}^{{ - 1}} = A\frac{i}{{1 - {{(1 + i)}^{{ - n}}}}} $$

(A.4)

can, thus, be interpreted as an amount which, if paid out at the end of the year t over n years, the series of payments would be equivalent to the one-time amount A at the beginning of year 1.

The formula (A.4) can also be used to “even out” a payment series of b ₀, …, b _n−1, paid out at the beginning of each of a total of n years to an equivalent series of equally large amounts of b, paid out at the end of each of the n years. This is done by multiplying NPV in (A.2) with the annuity factor $a_{{i\left| n \right.}}^{{ - 1}}$ which produces:

$$ b = a_{{i\left| n \right.}}^{{ - 1}}NPV = a_{{i\left| n \right.}}^{{ - 1}}\sum\limits_{{t = 0}}^{{n - 1}} {{b_t}{{(1 + i)}^{{ - t}}}} = \frac{i}{{1 - {{(1 + i)}^{{ - n}}}}}\sum\limits_{{t = 0}}^{{n - 1}} {{b_t}{{(1 + i)}^{{ - t}}}} $$

(A.5)

The previous formulas can be used to show that a series of payouts of amount c at the end of each year of a total of n years, is equivalent to a one-time amount (NPV), paid out at the beginning of year 1, calculated as:

$$ NPV = c{a_{{n\left| i \right.}}} = c\frac{{1 - {{\left( {1 + i} \right)}^{{ - n}}}}}{i} $$

(A.6)