An interest rate is a pure number, r. One dollar lent for one time period at rate r is repaid with (1+r) dollars. More generally, $M are repaid with $M(1+r)t , t periods later.
Present value: The present value of $M, t periods in the future is the amount of money in period zero that need be invested at interest to yield $M in period t.
PV(M,t,r) = M(1+r)-t.
The present value of M per year forever is M(1+r)-1 + + M(1+r)-n + = M/r
Real World Rates
Nominal interest rates are the interest rates that people pay and receive. They are the rates one can find in the daily paper. For example, 30-year treasury securities have an interest rate of 6.18%; 1 year treasuries, 5.10%. Corporate bonds pay more. These securities are promises to pay a fixed interest rate, regardless of the amount of inflation or earnings of the firm.
Common stocks are also assets and allow money to be transferred from one period to the next. Their value ultimately depends upon the earnings of the firm. When held from one year to the next, they change price (called a capital gain or loss) and pay dividends. Over long periods of time, the realized gain (dividends plus capital gains) on common stocks is about 11%.
Real rates of interest are the nominal rate less the inflation rate. The ex-post (meaning realized not expected for the future) real rate for treasuries is on the order of 3%.
Long term investments usually pay more than short term investments because of greater risk (though it is hard to provide empirical verification of this statement). Long term rates can also be viewed as the expected value of a series of short term investments. Thus, when the long rate is below the short rate, one possibility is that iinvestors expect the short rate to fall.
Risky investments, like stocks, pay more than riskless investments, like treasuries.
A Little Investment Theory
Output, y, is made from inputs, one of which we call capital, K, and the rest of which we don't care about. Capital could be machines or it could be natural capital, like clean air. The latter case is much more interesting, so we won't describe it. Moreover, we will assume that capital never wears out or depreciates. Write y = f(K, ) for the production function. The (discrete approximation to the) marginal product of capital is MPK = f(K+1, ) - f(K, ). With the zero depreciation assumption and a price for output of P, an additional unit of capital earns P MPK in every future period. Now let the cost of the unit of capital be c. (This is actually a complicated number, because the tax law and depreciation have considerable interaction, but here just call it c.) If the c dollars were borrowed at rate r, then r c dollars would be owed every year (the interest on the c dollars). Now, by our usual arguments r c = P MPK, or investors would want to invest less or more. Therefore, r = P MPK / c, and this is true for every business.
Should government use the same rate as business? Diamond and Mirrlees answer yes. What might make government special is that government might try to alter income distribution by its investments. In their analysis Diamond-Mirrlees allow the government to impose commodity taxes, that is taxes on prices of the goods consumers buy. These taxes are sufficient to reallocate income, so the government does not need to favor projects, like development of arid regions, in order to reallocate income. There are other answers to this question, including the iconoclastic statement by one of my colleagues, that interest rates are simply a policy variable and have no normative content. A more persistent complaint is that savings are too low because future generations are somehow undervalued. One can make (although not me) an argument that the proper social discount rate is zero.
Making an Economy Grow faster
Since output depends upon capital and labor, making either one larger results in a larger economy. It now seems clear that there is little ability to expand the labor force: men and unmarried women are already working, and only married, non working women can be lured into the labor force with higher wages or lower taxes. There appears to be scope for more capital formation, however. Capital does not appear to be greatly mobile between countries, so more savings in the US would not leak out to the rest of the world. More savings should be achievable with higher (real) after tax interest rates. Thus lower taxes should make for more capital. The degree to which this is true is yet to be discovered, however this is the lower taxes leads to growth position and is the intellectual justification for lower capital gains taxes.