### On interest rates and fixed contracts

**Matthew Martin**8/19/2013 05:23:00 PM

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Suppose that, rather than choosing how much to spend this period, consumers are faced instead with, say, a 30 year contract that delivers a stream of goods and services consumed annually for a fixed annual fee. Also, pretend there's no inflation or deflation, so this is all in real terms. Obviously, whether you choose to buy the contract depends on its price. But to caculate the price of an intertemporal contract, you need to calculate "real present value"--an accounting concept that asks "given the interest I can earn on savings, how much do I need to set aside right now to be able to afford this contract's payments in each of the next 30 years?"

Suppose you can put your money in a savings account at your bank at an interest rate $r$. So, if you put one dollar into the savings account this year, you'll earn $r$ in interest, so that you have $1+r$ in your savings next year. Following this logic, if your annual payment is $x$, that means you need $\frac{x}{1+r}$ in your savings account this year in order to have $x$ in there next year. And you need $\frac{x}{\left(1+r\right)}$ in there to have $x$ in there two years from now.

That's how real present value calculations work. So if your contract requires a payment of $x$ every year for 30 years, and you can save at an interest rate $r$, then the real present cost of that contract--the amount of money you need to have right now to pay all future payments for it--is given by $$y=\sum_{t=0}^{30}{\frac{x}{\left(1+r\right)^t}}$$ where $y$ is the "real present value" of the cost of the contract. The higher $y$ is, the more expensive the contract is, and the lower $y$ is, the cheaper the contract is.

Now consider a situation where the interest rate $r$ declines, but $x$ remains unchanged. Since $r$ appears in the denominator of each term of the series above (and all terms are positive), that means that lowering $r$ actually increases $y$.

Therefore, by increasing the price of this contract, a lower interest rate actually increases the cost of consumption. Higher cost means less consumption--the lower interest rate is contractionary! Granted, in reality the interest rate is not locked in for 30 years, so in reality we'd need to amend the model to allow for future interest rates to return to long-run rates after a short-term decrease in interest rates. But even in that more complicated formula, a temporary decrease in the interest rate still raises the real present value of the contract price, though by less than in this example.

Does this result lend credibility to the claim that the Fed keeping interest rates too low is actually suppressing aggregate demand through income effects? In theory, no. The reason is that we're looking specifically at a kind of consumption that has investment-like components--it is an investment vehicle that pays out goods and services as dividends in each of the next 30 years. Low rates discourage investment, and thus they do discourage these kinds of contracts. But people don't just take that money and hide it under a matteress--they might as well invest it in that case--but rather go out and spend it on immediate consumption. That is, lowering interest rates shifts consumption away from future periods and into today. Instead of offering a contract that costs $x$ each year for 30 years, the firm might offer a contract that front-loads more consumption, costing $x'>x$ in the first period and $x''<x$ in future periods. The error above is in failing to recognize that contracts are endogenous.