Savings Relation

Time preference is the catallactic assumption of human preference for present goods over future goods. It is well established that time preference is reflected as the interest rate. According to Murray Rothbard, in his Man, Economy and State:

The level of the pure rate of interest is determined by the market for the exchange of present goods against future goods, a market which we shall see permeates many parts of the economic system. [...] Thus, if, on the time market, 100 ounces of gold exchange for the prospect of obtaining 105 ounces of gold one year from now, then the rate of interest is approximately 5 percent per annum. This is the time-discount rate of future to present money. [...] The pure rate of interest will then be the going rate of time discount, the ratio of the price of present goods to that of future goods.

Murray Rothbard: Man Economy and State

However, it is individual capital ratio that determines the interest rate. The interest ratio is that of future to present good price. It is the market price premium required to compensate an owner for time without his good – or the price of time. As with all prices, it is determined entirely by individual preferences, in its case time preference, expressed as individual trades.

The time preference of an individual can be represented as the ratio of the price of his hoard to that of his lending. Together these amounts are his savings. In trading a fraction of his hoard for its future value, one expresses that its future amount is worth more to him than presently. Conversely, by not doing so one expresses the opposite valuation.

A hoard is the opportunity to invest (lend) and an investment is the opportunity to consume. One is traded for the other until no further increase in value is obtained from doing so. By investing, one values the future amount more than the present amount not invested. By not investing, one values the present amount not invested more than the future amount. If this were not the case there would be a lower or higher level of investment respectively. This valuation, manifested as a trade, is the expression of one’s time preference.

Perhaps more fallacies have been committed in discussions concerning the interest rate than in the treatment of any other aspect of economics. It took a long while for the crucial importance of time preference in the determination of the pure rate of interest to be realized in economics; it took even longer for economists to realize that time preference is the only determining factor. Reluctance to accept a monistic causal interpretation has plagued economics to this day.

The individual does not control the market interest rate. The individual controls his capital ratio given the market interest rate. The capital ratio is how individual time preference is expressed. The interest rate is how those preferences are priced by the market.

The following vertical bar chart provides an example of an individual’s savings.

           Individual Capital Ratio (time preference)
          |
          |
  21 [
  20 [ H  |
  19 [ HH |
  ---[----+--------------------- Market Interest (time price)
U 18 [ HHH|TTTTTTTTTTTTTTTTT
N 17 [ HHH|PTTTTTTTTTTTTTTTT
I 16 [ HHH|PPTTTTTTTTTTTTTTT
T 15 [ HHH|PPPTTTTTTTTTTTTTT
  14 [ HHH|PPPPTTTTTTTTTTTTT
O 13 [ HHH|PPPPPTTTTTTTTTTTT
R 12 [ HHH|PPPPPPTTTTTTTTTTT
D 11 [ HHH|PPPPPPPTTTTTTTTTT
I 10 [ HHH|PPPPPPPPTTTTTTTTT
N  9 [ HHH|PPPPPPPPPTTTTTTTT
A  8 [ HHH|PPPPPPPPPPTTTTTTT
L  7 [ HHH|PPPPPPPPPPPTTTTTT
   6 [ HHH|PPPPPPPPPPPPTTTTT
V  5 [ HHH|PPPPPPPPPPPPPTTTT
A  4 [ HHH|PPPPPPPPPPPPPPTTT
L  3 [ HHH|PPPPPPPPPPPPPPPTT
U  2 [ HHH|PPPPPPPPPPPPPPPPT
E  1 [ HHH|PPPPPPPPPPPPPPPPP
     =======================
       abc|defghijklmnopqrst
UNIT BY DESCENDING MARGINAL VALUE

Each ordinal increment represents a marginal value increment. Symbols H, P, and T represent Hoard, Present and Time value increments respectively. Hoarded value is the present value of a unit not lent. Present value is that of a lent unit had it not been lent. Time value is the expected net value (principal + interest) of the lent unit over a period of time at the market interest rate for that period.

Each vertical bar on the horizontal axis represents one monetary unit, yet each unit has a different marginal value to the owner, as a consequence of marginal utility. This value is expressed on the vertical axis as bar height. One should not conflate value with price. The value of each owned unit increases as the hoard decreases, and therefore the net value of the same interest rate (price for money, decreases as the hoard decreases, until it becomes negative (where no more is lent).

The individual’s time preference is demonstrated by his valuation between marginal units “c” (the next unit to potentially be lent) and “d” (the last unit lent). The former’s present value is higher than can be offset by its potential time value, so it is not lent. The latter’s present value is not, so it is lent. If the market interest rate rises such that the increase in return on lending “c” exceeds the cardinal value increment of “b” (i.e. chart cell “b19”) then “c” will be lent. If the market interest rate falls such that the decrease in return on “d” exceeds “c18” then the loan of “d” will be liquidated.

Total savings is 20 units (units “a” through “t”). Total hoarding is 3 units (“a” through “c”). Total lending is 17 units (“d” through “t”). The individual’s capital ratio is therefore 3/17 (~17.65%), represented on the chart as a vertical line between units “c” and “d”. The opportunity cost of the hoard is 3 units x the market interest rate. The return on the lending is 17 units x the market interest rate.

It is important to note that since value is subjective, only the individual’s valuation of the amount of interest is meaningful in this context. The market rate of interest raises his ordinal valuation of the units lent to between “18” and “19”. The chart therefore represents market interest as a horizontal line between those increments.

Only the choice to lend or not lend expresses time preference. Depreciation occurs in what is hoarded, not in what is lent. As shown in the Depreciation Principle, hoarding is consumption. The common perception that a trading from “producer” to “consumer" constitutes consumption is a clear error. One can decrease his rate of depreciation and thereby make his hoard last longer, but to be reflected as time preference one must change his rate of lending.

Notice that, relative to the previous chart, a decrease in the interest rate by the value of the 18th ordinal increment implies that one less unit is lent.

           Individual Capital Ratio (time preference)
           |
           |
  21 [
  20 [ H   |
  19 [ HH  |
  18 [ HHH |TTTTTTTTTTTTTTTT
U ---[-----+-------------------- Market Interest (time price) 
N 17 [ HHHH|TTTTTTTTTTTTTTTT
I 16 [ HHHH|PTTTTTTTTTTTTTTT
T 15 [ HHHH|PPTTTTTTTTTTTTTT
  14 [ HHHH|PPPTTTTTTTTTTTTT
O 13 [ HHHH|PPPPTTTTTTTTTTTT
R 12 [ HHHH|PPPPPTTTTTTTTTTT
D 11 [ HHHH|PPPPPPTTTTTTTTTT
I 10 [ HHHH|PPPPPPPTTTTTTTTT
N  9 [ HHHH|PPPPPPPPTTTTTTTT
A  8 [ HHHH|PPPPPPPPPTTTTTTT
L  7 [ HHHH|PPPPPPPPPPTTTTTT
   6 [ HHHH|PPPPPPPPPPPTTTTT
V  5 [ HHHH|PPPPPPPPPPPPTTTT
A  4 [ HHHH|PPPPPPPPPPPPPTTT
L  3 [ HHHH|PPPPPPPPPPPPPPTT
U  2 [ HHHH|PPPPPPPPPPPPPPPT
E  1 [ HHHH|PPPPPPPPPPPPPPPP
     =======================
       abcd|efghijklmnopqrst
UNIT BY DESCENDING MARGINAL VALUE

This holds at every increment down to the level of interest in which the individual does not lend.

                           Individual Capital Ratio (time preference)
                           |
                           |
  21 [                     |
  20 [ H                   |
  19 [ HH                  |
U 18 [ HHH                 |
N 17 [ HHHH                |
I 16 [ HHHHH               |
T 15 [ HHHHHH              |
  14 [ HHHHHHH             |
O 13 [ HHHHHHHH            |
R 12 [ HHHHHHHHH           |
D 11 [ HHHHHHHHHH          |
I 10 [ HHHHHHHHHHH         |
N  9 [ HHHHHHHHHHHH        |
A  8 [ HHHHHHHHHHHHH       |
L  7 [ HHHHHHHHHHHHHH      |
   6 [ HHHHHHHHHHHHHHH     |
V  5 [ HHHHHHHHHHHHHHHH    |
A  4 [ HHHHHHHHHHHHHHHHH   |
L  3 [ HHHHHHHHHHHHHHHHHH  |
U  2 [ HHHHHHHHHHHHHHHHHHH |
E  1 [ HHHHHHHHHHHHHHHHHHHH|
  ---[---------------------+---- Market Interest (time price) 
     =======================
       abcdefghijklmnopqrst|
UNIT BY DESCENDING MARGINAL VALUE

Similarly this relation holds to the point where the individual lends all of his capital.

       Individual Capital Ratio (time preference)
       |
       |
  ---[-+------------------------ Market Interest (time price) 
  21 [ |TTTTTTTTTTTTTTTTTTTT
  20 [ |PTTTTTTTTTTTTTTTTTTT
  19 [ |PPTTTTTTTTTTTTTTTTTT
U 18 [ |PPPTTTTTTTTTTTTTTTTT
N 17 [ |PPPPTTTTTTTTTTTTTTTT
I 16 [ |PPPPPTTTTTTTTTTTTTTT
T 15 [ |PPPPPPTTTTTTTTTTTTTT
  14 [ |PPPPPPPTTTTTTTTTTTTT
O 13 [ |PPPPPPPPTTTTTTTTTTTT
R 12 [ |PPPPPPPPPTTTTTTTTTTT
D 11 [ |PPPPPPPPPPTTTTTTTTTT
I 10 [ |PPPPPPPPPPPTTTTTTTTT
N  9 [ |PPPPPPPPPPPPTTTTTTTT
A  8 [ |PPPPPPPPPPPPPTTTTTTT
L  7 [ |PPPPPPPPPPPPPPTTTTTT
   6 [ |PPPPPPPPPPPPPPPTTTTT
V  5 [ |PPPPPPPPPPPPPPPPTTTT
A  4 [ |PPPPPPPPPPPPPPPPPTTT
L  3 [ |PPPPPPPPPPPPPPPPPPTT
U  2 [ |PPPPPPPPPPPPPPPPPPPT
E  1 [ |PPPPPPPPPPPPPPPPPPPP
     =======================
       |abcdefghijklmnopqrst
UNIT BY DESCENDING MARGINAL VALUE