Savings Relation

Time preference is the Catallactic assumption of human preference for present goods over future goods. The economic rate of interest is the direct reflection of time preference. It can be shown that this preference is also reflected in the reserve ratio. In other words, the rate of capital reserve against credit is the interest rate.

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

Rothbard's last statement is not literally accurate, as a discount rate is not the same as a ratio. Note that a present good is priced higher than a future good, so in the example 100oz of gold (present good) trades for 105oz delivered in the future (future good).

Interest rate as discount rate:

present-goods-price = future-goods-price * (1 + interest-rate)
interest-rate = present-goods-price / future-goods-price - 1 
interest-rate = 105oz / 100oz - 1 = 5%

Interest rate as price ratio:

interest-rate = present-goods-price / future-goods-price
interest-rate = 105oz / 100oz = 105%

As shown in Depreciation Principle, no actual consumption occurs in the trade of a product from producer and consumer. A product is only consumed to the extent that it depreciates. This is evident in the fact that a product can be resold at present price, recovering the portion not depreciated. Similarly, any remaining portion can be consumed in the future, offsetting the present price of purchasing more of the same. So consumption is the depreciated fraction of the original hoard. Using the ratio for depreciation obtains:

depreciation-rate = unconsumed-goods-price / hoarded-goods-price

Given that all prices are represented in the same money, goods-price can be dropped from both relations with the understanding that each name represents the price of the amount of the good.

interest-rate = present / future
depreciation-rate = unconsumed / hoarded

Considering that future-goods is investment and that unconsumed-goods are present obtains:

interest-rate = present / investment
depreciation-rate = present / hoarded

Substituting and rearranging obtains:

interest-rate = (hoarded * depreciation-rate) / investment
interest-rate = (hoarded / investment) * depreciation-rate