Expectedly, each school has a unique perspective. The Austrian school largely differentiates itself based on the richness of its theory of capital and interest. However, before we discuss the Austrian theory of interest, it might be instructive to examine the nature of interest by invoking as little economics as possible.
Dimensional Analysis of Interest Rates
When posed the question "what is interest?", one common response is that it is the price of money. This is not true. The price of money, like any other good, is the amount of something else that must be traded for it. To see this, consider the converse: what is the price of a tomato? Well, it is the number of dollars that must be given up in exchange for a tomato. Ok, how about the price of a banana? Again, it is the number of dollars that must be given up in exchange for a banana. Because dollars are money the price of all goods is expressed in dollars, whereas the price of dollars is expressed in other goods.
Returning to the tomato, let's say its price is 2 dollars. Price = 2 dollars per tomato. Expressed this way, we find that the price of a tomato has dimensions. Recall from high school physics that all statements about reality have dimensions. Dimensions give meaning beyond the mere quantity. To say I am 6 tall, or 70 tall is meaningless. I should say that I am 6 feet tall, or 70 inches tall. The dimension of my height is length, expressed in feet or inches. Similarly, the price of a tomato has dimensions dollars per tomato. Price = 2 $/tomato. The same is true for bananas. Price = 4 $/banana. In general, the dollar price of X has dimensions $/X. Price = y $/X. Taking the reciprocal, we get: price = 1/y X/$. Price of what? Why, of dollars, of course! By this simple mathematical argument, we determine that the price of dollars is the amount of good X (happens to be 1/y) that must be trade for one dollar. Thus, while the prices of all goods is expressed in dollars, the price of dollars is expressed in terms of other goods. We can further extend this by noting that if the dollar price of tomatoes is 2 and the dollar price of bananas is 4, then the tomato price of bananas -- the number of tomatoes that must be exchanged for one banana -- is 2: 1/2 tomato/$ x 4 $/banana = 2 tomatoes/banana.
So what has this musings about dimensions taught us about interest rates? Well, what are the dimensions of interest rates? Look up any financial source and you will find interest rates quoted as r % per year. Dimensions = 1/year. Thus, without invoking any economics, we can determine that interest rates are the price of time. Economics enters the picture when we begin to question why time has value, and how one determines that value.
Why Time Has Value
Time has value because, all else equal, humans prefer consumption sooner than later. This concept is known as time preference. Historically, the charging of interest, or usury as it was pejoratively referred to, has been much maligned by the church, Marxists, and others. There is no reason for this myopic prejudice. Interest is a natural phenomenon that arises from the voluntary interactions of individuals expressing their subjective preferences.
Robert Murphy discusses this concept in his article Why Do Capitalists Earn Interest Income:
Since no one would be willing to give $10,000 now in exchange for a promise of $1,000 payments for each of the next ten years, it naturally follows that no one would pay $10,000 for our hypothetical tractor. Because of this fact—that present goods are worth more than future goods—the tractor can be purchased for less than $10,000Robert Murphy also has a great article on Bohm-Bawerks critique of the exploitation theory of interest.
Determining The Price of Time
The concept of time preference is intuitive and should resonate with the reader. The harder question to answer is how one puts a price on time. Wikipedia defines interest as "a fee paid on borrowed capital"; the operative term being capital. Capital is real wealth (tangible physical goods), not paper claims to wealth. In essence, the holder of capital through the act of saving is foregoing consumption today for consumption in the future. He demands payment for this because of his time preference. That the borrower acquiesces to his demand is a result of the wealth generation process: the borrower expects, by acquiring capital today, to be able to repay that capital plus interest in the future. It is the interaction of these two phenomena that established the price of time as the marginal efficiency of capital.
Reasoning with money tends to obfuscate the insight so let's consider a barter economy and then see if we cannot extend the results to a monetary economy. In a barter economy, a saver saves real wealth and an entrepreneur demands real wealth for investment. Assume the baker saves 10 loaves of bread and the shoemaker in order to build a shoe-making machine needs 10 loaves of bread, 1 for each day that he works on the machine and cannot sustain himself through other means. (Aside: this is the concept of the subsistence fund which shows that savings and not consumption (as in the Keynesian framework) are the drivers of economic growth.)
The two agree that in return for the 10 loaves today, the shoemaker will return 11 loaves when his machine is done. By agreeing to part with the 10 loaves today that could have been otherwise used to purchase, say, a new suit, or a squash racket, the baker has expressed him time preference. He willingly foregos consumption today for increased consumption in the future. Had his time preference been higher, he would have demanded a higher rate of interest. The shoemaker, on the other hand, agrees to return 11 loaves because he expects his new machine to enhance his ability to produce shoes. Had the productivity increase been smaller, the interest rate he could afford would be lower. As various savers and entrepreneurs interact with each other to coordinate savings and investment, the savers express their time preference and the entrepreneurs are guided by the productivity gains they hope to achieve, which illustrates that the price of time is the marginal efficiency of capital.
Introducing money into the equation changes nothing. In the barter economy, the shoe maker must find a baker willing to save bread and offer him a rate sufficient to satisfy his time preference and within the productivity gains he expects. This is just another form of the double coincidence of wants. To alleviate this problem, we introduce money into the system, but do not change the essence of the interactions.
This is a simple description of how the free market allocates capital towards investment. Through this coordination the price of time is established. It is what Wicksell termed the neutral rate. For a more detailed explanation, see Shostak's article on marginal utility and interest formation. See also natural and neutral interest rates by Roger Garrison.
We have argued that interest rates are the price of time and that they should be established by allowing the market to clear free of manipulation. When the central bank increases the money supply or private banks increase the money supply through fractional reserve banking, they cause distortions to the market for capital. If entrepreneurs have first access to the new money then they can acquire the goods they need without convincing anyone to actually save those goods. Thus, expanding the money supply is nothing but forced savings. Ideally, a free market on a gold standard will set the interest rate, and not a central bank that has no idea what the interest rate should be. What is most strange is that mainstream economists will agree that the government should play no part in setting the price of goods and services like cars and orange juice. Yet, they all insist that the government should control the most important price in the economy, the price of time.