All Terrain Thinking

A Compendium of things I think are Important

Economics: It's not just whats' in your wallet

Capital and Interest Rates

The link between income and interest rates is very clear for those of you who have money in savings accounts or CDs. As the interest rate rises, the earning power of your money increases. We saw some evidence of this in the mid 1990s as the Federal Reserve (an institution we will talk about in macro) drove down interest rates and groups representing the elderly complained that this was reducing the income of those who were living off interest on their accounts.

Businesses, younger individuals who were borrowing money, and homeowners with mortgages on the other hand, were happy with the reduction in rates since this meant that they would pay less for their funds. For example, if I had a 30 year mortgage of $100,000 and the rate increased from 6.25 to 7.25 percent, the monthly payment would increase from $616 to $682, a yearly increase of more than $800.

Interest rates, the price of borrowing funds, can be a bit confusing at first however because there are so many different rates. If you buy a house you will pay a mortgage rate, but in fact there are many rates depending upon the type of mortgage - how long, how many points (a down payment of sorts, and the variability of the rate. If you do not pay off your credit card balance, then you will pay the credit card rate which will tend to be a bit higher than the rate on a car loan. Corporations and companies can also borrow money by issuing bonds which can have substantially different interest rates. GM will pay less than Local Lonnie's Loops, but more than the US government when it borrows, and all will tend to pay higher rates than those paid by your local town or state. Furthermore, if the government wants to borrow for a long period, 10-30 years, it will pay more than if it wants to borrow money for 6 months or a year.

At this time in our micro course we will not go into a detailed treatment of the factors responsible for this wide array of interest rates. In effect we will be talking about a generic interest rate and we can assume that when we talk about an increase that we would expect to see varying increases across the array of interest rates. [for a more detailed treatment of the determinants of interest rates and you can check out the treatment in the macro component]. For now we will return to our simple supply-demand model where demanders are borrowers and the suppliers are lenders. When you go to the bank to get a mortgage you are demanding funds and when you deposit money into the bank you are a lender [you actually give your money to the bank who will then lend it out - which is why we call banks intermediaries].

We also need to introduce two concepts are often used interchangeably - investment and capital. To see the difference let's return to our homeowner who pays the mortgage each month. We would say this individual is making a monthly investment which will result in the accumulation of physical capital - a house (we will talk about human capital when we talk about labor).

The same would be true for a firm that borrows money (makes an investment) to buy new machinery (physical capital) to be used to produce output to be be sold. The amount someone would be willing to pay for the investment would be dependent upon the amount of output produced by the machine and its price (the marginal revenue product). For example, I suspect you would have no trouble deciding on whether or not to make an investment of $1,000 to buy a machine that could produce $2,000 of 'stuff.'

In the 'real world,' however, things are likely to be a bit more complicated. The machine you buy will be purchased with a loan that will be repaid over a period of time which is acceptable since the machine is durable and will produce 'stuff' over a period of time. For example let's consider the situation faced by Tubby T's which is considering an investment in a machine to design T-shirts. The machine costs $13,000, payable today, and is expected to generate $7,000 in additional profit immediately and $7,000 at the start of next year. Should we make the investment to buy this machine (capital)?

As a first pass, one might simply look at the numbers as presented in the table below which show a net total profit of $500 and be tempted to decide that the investment should be made.

Future Revenues and Costs of Tubby T's New Machine

The problem is we have not yet taken account of the fact that all of the pieces in the transaction are not taking place today.  This causes a problem since we all know that we would not trade $100 today to receive $100 a year from now. There is, however, a rate at which we would be willing to exchange a $ today for a $ one year from today. The key to answering this question, to establishing the appropriate exchange rate, is present value analysis which provides us with a systematic way of converting dollars at different times into dollars today so we can compare them.

In the example above, let's assume interest rates are 9 percent - you can borrow or lend money at 9 percent. This means that if you lend $100 at the beginning of the year, you will have $109 at the end of the year - and if you borrow $100 at the beginning of the year, you will need to pay back $109 at the end of the year [in the 'real' world you will probably have a payment schedule with monthly payments on a loan which complicates the math, but the fundamental structure is the same]. In this example, since the $100 grew to $109 by the end of the year, the present value of the $109 received one year from today would be $100. Stated somewhat differently, $100 is the amount we must invest today at 9 percent so that we will get $109 in one year.

Returning to the problem above, we now must attempt to convert the $7,000 earned a year from now into its present value. To do that we can use the present value formula:

PV = FV/(l+g)^T

• PV = Present Value ( value at beginning of the period)
• FV = Future Value ( value at the end of the period)
• g = Average growth rate per period
• T = Number of time periods

In this problem T=1, FV = $7,000, and g = 9 percent. Plugging these values into the equation we would get the present value to be $6,422. With this figure we can now rethink our investment decision. The 'new' numbers appear in the table below. At a 9 percent interest rate, the present value of the $7,000 next year is $6,422 - we would need to invest $6,422 today to get $7,000 in one year. We are actually paying $6,500 [ $13,500 - $7,000] today to allow us to earn $7,000 in one year when the alternative is to go to the bank and invest $6,422 and have $7,000 at the end of the year - a saving of $78. What we find is that the present value of the machine is less than the current cost which means we should not make the investment.

Present Value of Revenues and Costs of Tubby T's New Machine
9% interest rate

What would happen to our decision if the interest rate changed? What if the interest rate were 7 percent? The results appear in the table below. In this case it appears that the revenue generated by the machine exceeds the costs and we should make the investment. At the lower interest rate, we would need to invest $6,542 today to get $7,000 in one year, but this is more than we would need to pay for the machine-so we should buy it.

Present Value of Revenues and Costs of Tubby T's New Machine
7% interest rate

What are the generalization we can take with us? Most importantly, we can now see the relationship between interest rates and the demand for funds. All other things equal, we can expect that as the interest rate falls, more projects to have a positive net profit and therefore investment should increase. In our simple example, a reduction in interest rates from 9 to 7 percent would change Tubby's mind on the investment decision from a no to a yes. If others acted in similar fashion we would expect the negative relationship which we would translate into the demand curve below - the negative slope indicating the negative relationship between the cost of funds and the number of borrowers (demanders).

The supply side is a bit simpler. Just ask yourself how much you would be willing to lend if you could double your money in one year, if you could turn $100 to $200. For the majority of people we would expect they would be willing to lend more (save more) than they would if they could turn the $200 into $109. Why give up the enjoyment we could derive from the funds today for just a bit more next year? The result would be a positive relationship between interest rates and the supply of money.

If we combine the two curves we have the supply and demand curves below. The market will settle at an interest rate of r* with borrowing / lending equal to M*. As you would expect with any market analysis, the rate of interest will change if there is a change in any of the factors that affect either suppliers (lenders) or demanders (borrowers), but a thorough treatment of this can be found in macro. For now we will accept the fact that anything that increases demand will tend to drive up interest rates, while an increase in supply will tend to drive them down. For example, as the baby boom generation moved into the age bracket where they would be buying homes we could expect this would increase demand for mortgages and this increased demand for borrowing would tend to put upward pressure on interest rates. Similarly, as the boomers move into their 50s and high savings years, we could expect an increase in the supply of funds (lending) which would tend to put downward pressure on rates.

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