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 will 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, though the distinction is immaterial. 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 to 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. Consumption is act of hoarding and the depreciated fraction of the original hoard is that which has been consumed. Using the ratio for depreciation obtains:

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

Considering that future-goods are invested capital and that unconsumed-goods are present goods obtains:

interest-rate = present-goods-price / invested-goods-price
depreciation-rate = present-goods-price / hoarded-goods-price

One should be careful in the use of equations in economic theory. Math is an abstraction and its use can introduce assumptions. In this case future goods have been related to present goods under the assumption that one is traded for the other, as in the interest relation. This however is the case. One's hoard of capital is an opportunity to invest, and one's investment is an opportunity to consume. The existence of one is an actual trade for the other. By not trading one for the other one is expressing that no increase in utility is obtained from doing so.

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 money price of the amount of the good.

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

Substituting and rearranging obtains:

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

Given that all prices are money prices, next consider the implications of liquidating the hoard at any time. Money does not depreciate as do goods. A pure money depreciates only by the opportunity cost of not investing it. This cost is measured in interest, which is the time preference for economic goods. Money represents the opportunity to trade for a good. As such its depreciation is determined by the preference for goods. If one erroneously includes the opportunity cost of money as depreciation in determination of interest, the irrational result is that hoarded and invested goods are always in the same amount. Time preference is interest and therefore interest cannot determine time preference.

money-interest-rate = (hoarded / invested) * money-interest-rate
hoarded = invested

However in the case of actual money the sum of demurrage and seigniorage is the depreciation rate. Notice that depreciation only affects the hoard as investment represents future goods, which do not depreciate until they become present goods.

money-interest-rate = (hoarded / invested) * money-depreciation-rate

Therefore, in the theoretical case of a pure money, the base interest rate is the ratio of hoarded to invested capital.

base-interest-rate = hoarded / invested

In the theoretical case of actual money, the interest rate varies directly with the base interest rate, by the factor of money depreciation.

money-interest-rate = base-interest-rate * money-depreciation-rate

In actuality, the interest rate varies directly with the base interest rate, by the factor of goods (and money) depreciation.

interest-rate = base-interest-rate * depreciation-rate

Notice that the interest rate is the true savings relation, between consumption savings and investment savings. This ratio of hoard to investment is the fraction of each person's capital reserve in the context of fractional reserve lending. Both are reflections of time preference.