If we ever get around to writing the principles of microeconomics book we have in us (along with about 5 other book ideas that we float and never get around to), one of the topics I'm sure we will include is discounting and the time-value of money. Very few principles books of which I am aware touch on discounting (of the six I have in my office--pictured to the right-- only one defines discounting and that is in a 1.5 page appendix).

I believe that one of the most important practical things we can teach students in a first-time economics class is the basics of comparing benefits and costs (in all their forms). Critical to that understanding is understanding the value of money today versus the value of money tomorrow--since costs and benefits often pile up at different rates. To make the point I usually couch the discussion in terms of what you would be willing to give up today to get more stuff tomorow. This can get mathematically complicated fast, but I think the basic points can be made without going into too much math.

Suppose you have $100 today. If you put that money in a savings account that earns 5% interest per year, how much money will you have one year from today? Answer: $105. To see this, just convert the 5% interest rate to the decimal form 0.05 by dividing 5 by 100, multiply by the principal of $100 to find the interest earned in a year ($100 * 0.05 = $5) and add that back to the principal to get $105.

That $105 is called the *future value* of $100 @ 5% annual interest for one-year.

But suppose you save your $100 for longer, say 3 years. How much will the $100 be worth after 3 years @ 5% interest per year? To figure this out we just notice that at the end of year 1 we will have $105 that we are reinvesting for another year at 5% interest. So at the end of year 2 we will have $110.25, which we reinvest again in year 3 to gain another years interest. After year 3 we will have $115.76.

More generally, the future value of $X invested at r% is $X*(1+r/100) after year 1. using the interest factor of (1+r/100), after year 2 we will have ($X*(1+r/100))*(1+r/100), and after year 3 we will have (($X*(1+r/100))*(1+r/100))*(1+r/100). Combining things, we end up with $X*(1+r/100)^3.

Even more generally, the future value of $X invested at r% interest for N years is $X*(1+r/100)^N.

Pretty simple right (I know you've probably glazed over at this point)?

Perhaps of more relevance for benefit-cost analysis is not to ask what today's money would be worth tomorrow, but instead ask what would tomorrow's money be worth today. For example, suppose we have a business idea that is going to cost us $95 today but we will earn $105 one year from now. Is the business idea worth it? It is tough to compare today's dollars to tomorrow's dollars. So we have to figure out a way to make the costs and benefits comparable.

One way we could figure that out is to ask what else we could do with our $95 today rather than invest in our business idea. One idea would be to save the money. If the interest rate is 5%, using our future value formula we know that we would have $95*(1.05)=$99.75 if we put the money in the bank. If we inverst the $95 in our business idea however, that $95 would turn into $105. So it looks like the best decision is to invest in the business idea for $95 and have $105 one year from now.

Another way to think about the same problem is to ask how much the $105 is worth in today's dollars. This is called the present value. The present value of $105 one-year from now can be found by answering the question how much money would we have to invest today ($PV) to get $105 one year from now. Using our future value formula, if we invest $PV today at 5% interest we can get back the future value of $105 next year: $PV*(1.05) = $105. Solving for the present value gives: $PV=$105/(1.05), or after calculating, $PV=$100.

In general, the present value of $Z received after N years with an interest rate of r% is: $Z/((1+r/100)^N).

To summarize:

The future value of $X invested for N years at r% interest is:

$FV=$X*(1+r/100)^N

The present value of $Z received after N years at r% interest is:

$PV=$Z/((1+r/100)^N)

There are a bunch of other present and future value calculations we could talke about, like the future and present value of a stream of payments rather than one time investments, or the value of money continuously compounded, but the relatively simple present and future values of a lump sum give us the inutition wee need to start to ask and answer some very interesting questions, like:

**Why is do we need to get more money in the future to give up money today**? The simple answer, which most of the time is the right answer, is that people are impatient. As would most people, I would rather get something today rather than wait until tomorrow. Some people are more impatient than others, so they go to the bank and try to borrow money (take out a loan). The impatience people express in terms of the value of money today versus teh value of money tomorrow is sometimes called the time-value of money.

**Where do banks get their money to loan to the impatient people**? From the more patient people of course. So the people who want money now pay a premium (called interest) to the people who are willing to forego some of their money today in return for more money tomorrow.

A similar story can be told for businesses. Many businesses need money now in order to run their busnesses (buy equipment, supplies,...). They are willing to pay a premium for to get that money.

**Where do businesses get the money for upfront costs**? They get it from investors who are willing to give up some of their money today in return for more money in the future. This might come through investments in stocks, or through commercial lending, or venture capital, or.... But again, the simple answer is those who need money now are willing to pay a premium to those who are willing to forego money today in return for more money tomorrow.

**Would you rather have $100 today or $105 one year from today**? This is an interesting question because decisions like this made by consumers and producers every day can impact interest rates. If the interest rate is 5%, and most would rather have $100 today rather than $105 one-year from now, then savings rates will be very low. With low savings rates, interest rates will start to rise. The interest rate is just the price of not saving--or put another way, interest is foregone income. So by failing to save, you are giving up the opportunity to earn interest. You could've had $5 more next year, but you chose to spend the $100 today. As more people fail to save, the price of not saving (interest rate) increases.

**How far will interests rise**? As savings decrease, interest rates will rise until you are indifferent between saving the $100 for a year and spending it. If interest rates rise too far, then everyone will stop spending today and simply save their $100. As we get a glut of savings in banks, interest rates start to fall. In the end we get an equilibrium interest rate that represents the trade-offs that people are willing to make between money today and money tomorrow. This is often referred to as the discount rate.

**Is there really only one discount rate?** This is a really tough question and one that goes far beyond what we want to talk about in Env-Econ 101, but we will just leave it at this: some people think there is only one discount rate, some people think there are different discount rates depending on teh type of investment being made. Some people think there are different discount rates depending on what is being valued. Some people think that discount rates are constant over time while others think that discount rates vary depending on how long in the future the cost or benefit happens. In short, there is very little agreement on discount rates.

**So what should we use as the discount rate? **Another tough question. In its primer on Regulatory Impact Analysis, the U.S. Office of Information and Regulatory Affairs states, "To provide an accurate assessment of benefits and costs that occur at different points in time or over different time horizons, an agency should use discounting. Agencies should provide benefit and cost estimates using both 3 percent and 7 percent annual discount rates..." So 5% is a good round number if you only want to calculate things once.

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