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, and it is shown here that the inverse capital ratio ("savings ratio") is the same quantity.

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.

Rothbard: Man Economy and State

Both time discount and interest rate can be represented as either rate or ratio. Ratios are used here to simplify presentation, though their numeric values may be less intuitive. Examples assume simple interest with a single compounding period. In Rothbard's example above, 100oz of gold (present good) trades for 105oz delivered in the future (future good).

interest-rate = future-value / present-value - 1
interest-rate = 105oz / 100oz - 1 = 5%

discount-rate = 1 - present-value / future-value
discount-rate = 1 - 100oz / 105oz = ~4.8%

interest-ratio = future-value / present-value
interest-ratio = 105oz / 100oz = 105%

discount-ratio = present-value / future-value
discount-ratio = 100oz / 105oz = ~95.2%

As shown in Depreciation Principle, no actual consumption occurs in the trade of a product from producer to consumer. A product is only consumed to the extent that it depreciates. This is evident in that a product can be resold at present price, recovering the monetary value of the unconsumed portion. Similarly, any unconsumed portion can be consumed in the future, offsetting the present price of purchasing more of the same. Consumption is act of hoarding, and the depreciated fraction of the original hoard has been consumed.

Goods that remain unconsumed are future goods relative to the original hoard of goods. Future goods cannot depreciate until they exist, at which point they are present goods. The discount ratio is used to obtain present goods depreciation as follows.

discount-ratio = present-value / future-value
unconsumed-value = hoarded-value * depreciation-ratio

A hoard is the opportunity to invest, and an investment is the opportunity to consume. One is traded for the other until no further increase in utility is obtained from doing so. The interest ratio is used to obtain the investment return for this trade.

interest-ratio = future-value / present-value
invested-value = present-value * interest-ratio

Rearranging and substituting unconsumed-value for present-value obtains the relation between depreciating hoarded goods and non-depreciating invested goods. The capital ratio is the ratio of hoarded (reserved) to invested capital, the inverse of interest.

interest-ratio = invested-value / (hoarded-value * depreciation-ratio)
capital-ratio = (hoarded-value * depreciation-ratio) / invested-value

Converting to rate relations obtains:

interest-rate = invested-value / (hoarded-value * depreciation-ratio) - 1
interest-rate = (1 / capital-ratio) - 1

The interest ratio is the inverse capital ratio. Each is the same reflection of time preference. The interest rate is each person's savings rate, allocated between investment and consumption.