In earlier chapters I have ignored, so far as possible, two of the major complications of economics (and life)--time and uncertainty. While I have described production as occurring over time ("widgets per hour"), I have also assumed an unchanging world in which each hour is like the last. In relaxing those assumptions, I will first--in this chapter--describe how the picture must be altered to allow for a changing but still certain world, a world in which people never make mistakes, since they know exactly what is going to happen. In Chapter 13, I will describe some of the effects of further changing the picture to fit the uncertain world in which we live.
The simplest way to introduce time into the picture is by recognizing that a good is when as well as what. An apple today and an apple tomorrow are two different goods, as any hungry child will tell you. Not only is there a price for apples today in terms of oranges today, there is also a price for apples today in terms of apples next year. If I trade 100 apples today for 110 next year, I am receiving an "apple interest rate" of 10 percent, since giving you goods now in exchange for goods in the future is the same thing as loaning you goods in exchange for the goods plus interest in the future. The interest rate (in apples for a one-year loan) is the price of apples today measured in apples a year from now (110/100 = 1.10) minus one. 1.10 -1.00 = .10 = 10%.
The price of apples today in terms of apples a year from now--the rate at which apples today exchange for apples a year from now--is determined in the same way as other prices. If you want to consume fewer apples now and more in the future, you sell apples now in exchange for apples in the future, contributing to the supply of the former and the demand for the latter. If you want to consume now and pay later, you sell future apples in exchange for present ones, contributing to the supply of future apples and the demand for present apples. There is some price--some apple interest rate--at which quantity supplied (of current apples to be exchanged for future apples) equals quantity demanded (of current apples to be exchanged for future apples). That is the market interest rate.
So far, I have assumed that all loans are consumption loans; people buy present apples with future apples (borrow) in order to eat the apples. Another reason is to plant them. Suppose you can take 10,000 apples, remove the seeds from some of them, and trade what is left for the labor of workers who will plant the seeds, water the baby apple trees when they come up, pull weeds, and eventually pick 11,000 apples from your new orchard. If you can do all this in a year (very fast-growing apple trees), you will produce 11,000 future apples using 10,000 present apples as input.
If the apple interest rate is below 10 percent, this is a profitable investment. You borrow 10,000 apples, plant them, pay back 10,000 plus interest a year from now, and have some left over. By doing so, you provide an additional demand for present apples and supply of future apples, which must be included in the total demand and supply determining the apple interest rate. By buying present apples (borrowing apples now), "investing" them, and paying with future apples, you drive up the price of present apples in terms of future apples--the apple interest rate.
As long as the apple interest rate is below 10 percent, planting orchards is profitable. More and more orchards are planted. Each one increases the demand for present apples and the supply of future apples, driving the interest rate (the price of present apples measured in future apples) up. So if you, and everyone else, can convert 10 present apples into 11 future apples by planting an orchard, the apple interest rate cannot stay below 10 percent.
One way of producing future goods from present goods is by planting apple trees; another way is to put the present goods somewhere safe and wait. For goods without significant storage costs (gold bars--provided nobody knows you have them), you can produce one unit of the future good from one unit of the present good, so the interest rate for such goods cannot be less than zero. You would never give 10 ounces of gold in exchange for 9 a year from now, since you could always hide your 10 ounces and have 10 ounces a year from now. That is not true for perishable goods (tomatoes) or for goods that are expensive to store (gold bars--if everyone knows you have them). For such goods, negative interest rates are possible.
Apple Interest Rate = Orange Interest Rate
So far, I have talked only about apple interest rates, leaving open the possibility that there may be a different interest rate for every good. Whether this happens depends on what happens to relative prices (the price of apples now in terms of oranges now, for instance) over time. If the relative prices of all goods stay the same over time (so it always costs the same number of oranges to buy an apple, or a cookie, or a car, or anything else), then all goods must have the same interest rate. If the relative price of one good measured in another is changing, on the other hand, then the two goods will have different interest rates.
To see why this is true, imagine that the apple interest rate is 10 percent and that an apple always trades for two oranges. What is the orange interest rate?
Suppose you have 200 oranges now and want oranges a year from now. You trade your 200 oranges for 100 apples. You then trade your 100 apples today for 110 apples a year from now--lend them out for a year at an apple interest rate of 10 percent. Finally, you trade 110 apples a year from now for 220 oranges. You have, indirectly, exchanged 200 present oranges for 220 future oranges.
The sequence of transactions is shown in Figure 12-la. The solid arrows show the actual transactions; the dashed arrow shows the overall effect. The rates at which the exchanges occur are shown in in parentheses.
If you (and everyone else) can trade present oranges for future oranges indirectly at an interest rate of 10 percent (1 present orange for 1.1 future oranges), nobody will be willing to trade a present orange for less than 1.1 future oranges, so the orange interest rate cannot be less than 10 percent. If you reverse the arrows on Figure 12-la and run the cycle backward (borrow 100 apples and trade them for 200 oranges, then pay the debt a year from now with 110 apples that you get by trading 220 oranges for them), you convert future oranges into present oranges at an interest rate of 10 percent (1.1 future oranges for 1 present orange). If you (and anyone else) can get present oranges in this way at a cost of 1.1 future oranges each, nobody will pay more than 1.1 future oranges for a present orange, so the orange interest rate cannot be more than 10 percent.
Since the orange interest rate cannot be more or less than 10 percent, it must be 10 percent. So the orange interest rate and the apple interest rate are the same. Precisely the same argument applies for any other goods. If the relative prices of two goods stay the same over time, they must have the same interest rate; if the relative prices of all goods stay the same over time, all goods must have the same interest rate. This is what economists call the real interest rate, in contrast to the nominal interest rate--the rate at which you can exchange dollars today for dollars a year from now--the interest rate reported in the daily paper.
Figure 12-1. Arbitrage over time--converting
present oranges into future oranges, using apples as
intermediates. Solid arrows show actual transactions; dashed
arrows show the net effect.
Orange Interest Rate
We now know what happens if relative prices stay the same over time. What happens if they do not? Suppose that the price of apples measured in oranges is falling; an apple buys 2 oranges today, but a year from now, 1 apple will exchange for only 1.5 oranges. Running through the same example (present oranges to present apples to future apples to future oranges), you find that you have traded 200 present oranges for 165 future oranges, for an orange interest rate of about -17.5 percent. The cycle is shown on Figure 12-lb. Just as on Figure 12-1a, you can run through the cycle in either direction, lending or borrowing oranges at an interest rate of about - 17.5 percent.
It may have occurred to you that this is not the first time you have seen this kind of argument. We used exactly the same procedure in Chapter 4 to show that if you knew the price of all goods in terms of one good, you could deduce the price of any good in terms of any other good. The process that we have illustrated here in Figures 12-la and 12-lb was described there in a somewhat less complicated form; it is called arbitrage. All we have done is to expand the argument to apply to goods that are labeled by when they are as well as by what they are.
The real interest rate is the rate at which you can convert goods now into goods next year--whether by trading goods for goods directly or by converting goods into money, converting money this year into money next year (lending it out), and converting money next year into goods next year. The nominal interest rate is the rate at which you can convert dollars this year into dollars next year. If the real interest rate is 10 percent, that means that if instead of consuming ten apples, or oranges, or automobiles you save them and lend them out, you will be paid back 11 next year. If the nominal interest rate is 10 percent and you lend out $10, you will get $11 back.
Real and nominal interest rates are equal if, as we shall assume throughout most of this chapter, the price of goods measured in money is not changing--the inflation rate is zero. If there is a positive inflation rate, the nominal rate is higher than the real rate; you get more future dollars for present dollars than future goods for present goods, but future dollars buy fewer goods than present dollars. We consume apples and oranges and automobiles, not dollars, so it is the real, not the nominal, interest rate that is relevant to the decision of whether to spend now or save for the future.
In most economies, relative prices change over time, so the apple interest rate and the automobile interest rate are likely to be different. The real interest rate is then a weighted average of the interest rates for different goods, with the weighting based on how much of each good the individual consumes. This leads to further complications, since the amount of each good consumed is also changing over time.
You have 6 oranges, 3 apples, and an ice cream cone. If markets exist for oranges, apples, and cones, and if, as we generally assume, the costs of arranging to buy and sell goods are negligible, you can transform that bundle of goods into any other bundle with the same total price--by selling what you have and buying what you want. You are therefore indifferent between any two such bundles--not in the sense of being equally willing to consume each but in the sense of being able to transform one into the other by market transactions. So one useful way of summing up what you have is by calculating what it is worth; this makes it possible to compare (for purposes of buying and selling but not of consuming) very disparate bundles. I do not like diamonds and do like ice cream cones, but I would rather have a one-carat diamond than an ice cream cone--even Baskin-Robbins's Pralines and Cream. In this sense, $110 worth of anything is preferred to $100 worth of anything else.
The same method can be used to evaluate bundles across time. Suppose I am offered two employment contracts: one consists of $40,000/year for ten years, the other of $31,000 the first year and a $2,000 raise for each of the next nine. Each contract is a bundle of ten different goods. The different goods are "money this year," "money next year," and so on. How can I compare them?
I can compare them by using the price of "money this year" in "money next year" (or "money two years from now" or . . .) to find a single market value for the bundle, just as I do for a bundle of different goods at the same time. Suppose the interest rate, at which I can either borrow or lend, is 10 percent. In that case, I can convert $1,000 this year into $1,100 next year (by lending) or $1,100 next year into $1,000 this year (by borrowing). In the first case, I lend out the $1,000 (losing the use of it now) and get $1,100 ($1,000 plus $100 in interest) a year from now; in the second case, I borrow $1,000 this year, and pay back the principal plus $100 in interest with the $1,100 I will have next year.
The difference between money this year and money next year is not the same as the difference between how much money will buy this year and how much it will buy next year. Even if we expect prices to stay the same, most of us would rather have a dollar now than a dollar in the future, just as most of us would rather have an apple now than an identical apple in the future. So even when there is no inflation, the nominal interest rate is usually positive--if you give up a dollar this year, you can get more than a dollar next year in exchange.
The present value of a series of payments is their total value measured in terms of money in a single year. Suppose, in the example I gave, I want the present value in year 1 of the series of payments associated with the first employment contract. Forty thousand dollars at the beginning of year 1 is worth $40,000 in Year 1, so the present value of the first term is easy. Forty thousand dollars in Year 2 can be converted into (1/1.1) x $40,000 in Year 1; if I borrowed that sum in Year 1, I could exactly pay it off with my Year 2 income. Forty thousand dollars in Year 3 is equivalent to (1/1.1) x (1/1.1) x $40,000 in Year 1, and so on. The third column of Table 12-1 shows the present values of the payments in the first series. Adding them up we find that the present value of the first series of payments is $270,362. That is the sum I could borrow in Year 1 and exactly pay off with the entire 10-year stream of payments.
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I can, in the same way, calculate the present value of the second series of payments (Table 12-1, column 5). It turns out that it is smaller. This implies that the first stream of income could, by appropriate borrowing and lending, be converted into the second with something left over. So the first stream of income is unambiguously preferable to the second, just as a bundle of goods worth $100 is unambiguously superior to a bundle worth $90, since one can sell the former, buy the latter, and have $10 left.
How would I convert the first stream of payments into the second? The answer is shown on columns 6 and 7 of the table. I would save (and lend out) $9,000 of my first year's salary (leaving me with $31,000 to spend, just as in the second stream), $7,000 of the second year's, $5,000 of the third, $3,000 of the fourth, $1,000 of the fifth. At that point I would have accumulated $25,000 plus interest. In the sixth year, I would pay myself $1,000 from my savings, in the seventh $3,000, and so on, for a total of $25,000. Column 7 of Table 12-1 shows, for each year, the accumulated savings, including interest. At the end, I would have had the same amount to spend each year as with the second employment contract and would have $24,688 left over.
So far I have been describing present value in words with numerical examples. Translated into algebra, we have the following formula:
PV(t)=
Here PV(t) is the present value at time t of an n
year stream of payments; yi is the payment in year i and
r is the market interest rate. 
is the mathematical symbol for "sum"; in general:
with the sum being over as many different xi's as there are values of i.
If we wish to evaluate the present value as of the beginning of the stream of payments, we have t=0,
PV(0)=
Present value calculations provide a way of evaluating any project, employment contract, or the like that can be described as a stream of payments, positive (revenue) or negative (cost), through time. If the present value of a stream is positive, then it is worth having if you do not have to give up something else, such as an alternative job with a higher positive present value, in order to get it. If it is negative, it is not worth having. If you must choose between two, the one with the higher present value is preferable.
Using the idea of opportunity cost introduced in an earlier chapter, we can reduce the previous paragraph to one simple sentence: "Choose any alternative that has a positive present value." If taking one job means not taking another, then not getting what you would have earned in the second job is the (opportunity) cost of taking the first and should be included in the present value calculation. If the result is still positive, then the present value of the income stream is higher for the first job than for the second, so you should take it.
One interesting present value is the present value of $1/year forever, which turns out to be $1 divided by the interest rate. To see why this is so, note that if you lend out $10 at 10 percent interest (10 equals 1/.10), you can collect $1/year forever. Just collect the interest and keep reinvesting the $10. We shall use this fact shortly.
In the previous 11 chapters we have analyzed the economics of an unchanging world, where every year is exactly the same as the year before. In that context, a question such as "If we sell widgets, will we make a profit?" reduces to the question "Will we make a profit this year?" Since every year is the same, if you make a profit this year you will make a profit every other year as well. In the real world, things are not so simple; a firm may choose to take a loss for several years in order to get profits in the future.
By using present values, we reduce the more complicated problem of choice in a changing world to the simpler problem that we have already solved. A firm trying to decide whether to produce widgets converts all of its future gains and losses into present values and adds them up. If the sum is positive (a net profit), it ought to produce; if the sum is negative (a net loss), it ought not to. Similar calculations can be made by a firm deciding how much to produce, what mix of inputs to use, and so forth. It compares the alternatives in terms of the present value of all gains and losses and chooses the one for which it is highest.
Suppose, to take a particularly simple example,that a firm is considering an investment (a factory, a piece of land, a research project) that lasts forever and produces the same return each year. If the present value of the annual profit made possible by the investment is greater than the cost of the investment, it is worth making; otherwise it is not. As we just saw, the present value of a permanent income stream of $X is $X/r, where r is the market interest rate. So if an investment of $1,000,000 yields an annual return of more than (r) x $1,000,000, it is worth making. If, in other words, the investment pays more than the going interest rate, it is an attractive one.
The calculation is more complicated if you are investing in something that will eventually wear out; in that case, the investment must pay at least the interest rate plus its own replacement cost to be worth making. The corresponding present value calculation is to compare the present value of the stream of income generated by the investment ($X per year for as many years as the machine lasts) with the initial expense plus the present value of any future expenses (maintanance, for example); if the present value of the payments is larger than the expense (the net present value is positive), the investment is worth making.
Redoing the previous 11 chapters in these terms would make this a very long chapter indeed, so I will restrict myself to working out the logic of one particularly interesting case.
Consider a depletable resource, say petroleum. There is a certain amount of it in the ground; when it has all been pumped up, there will never be any more. Firms that own oil wells must decide how to allocate their production over time in order to maximize profits. What will be the result?
Assume, for simplicity, that it costs nothing to produce oil; if you own an oil well containing 1,000,000 barrels of oil, your problem is simply to decide when to sell how much. Further assume that the oil wells are owned by many firms, each with only a few wells, so that each firm is a price taker.
What the firm takes is not a single price but a pattern of prices over time--P1 at the beginning of the first year, P2 at the beginning of the second year, P3 at the beginning of the third year, and so on. Since we are considering a world with change but no uncertainty, at the beginning of the first year everyone already knows what the entire pattern of prices over time is going to be. Suppose that the market interest rate is 10 percent, the first year's price (P1) is $10.00/barrel, and the second year's price (P2) is $12.00/barrel. A firm that sells some of its oil at the beginning of the first year gets a present value (measured in Year 1) of $10.00/barrel. A firm that sells oil a year later gets a present value (again measured in Year 1 so that we can compare the two) of $12.00/ 1.1 = $10.91/barrel. Under those circumstances, all firms would prefer to sell their oil in the second year. If they hold money for a year, they get 10 percent; if they hold oil for a year, they get 20 percent.
But if no oil were offered for sale in the first year, the price would be much more than $10.00/barrel. The price structure I have just described--$10.00/barrel in Year 1, $12.00/barrel in Year 2, and an interest rate of 10 percent--is inconsistent with rational behavior. If it existed, it would make people behave in a way such that it could not exist.
The only way to avoid such inconsistencies is for the pattern of prices over time to be such that P2 is exactly 1.1 times P1, so that the present value a firm gets by selling a barrel of oil is the same whether it sells it in the first year or the second. If it sells it in the first year, it gets $10.00; if it sells it in the second, it gets $11.00. The same argument applies to all future years. The price of oil must go up, year by year, at the interest rate.
You may find this way of describing what "must" happen confusingly abstract. The alternative is to try to describe the process by which an equilibrium set of prices is reached. In doing so, we will ignore the fact that in the perfectly predictable world we are assuming, everyone knows everything in the first minute of the first year, so equilibrium establishes itself immediately.
Imagine, then, that firms are considering a pattern of oil sales that does not lead to the pattern of prices I have described (price rising at the interest rate). A firm notices that it does better by selling its oil in Year 2 than it would by selling it in Year 1 and investing the money. So the firm changes its plans, transferring any production it had planned to make in Year 1 to Year 2. The result is to drive down the price in Year 2 and drive up the price in Year 1, moving both prices towards the pattern I have described. If the present value of the Year 2 price is still greater than the present value of the Year 1 price, another firm changes its plans. The process continues until the Year 2 price is equal to the Year 1 price times 1 + r. The same argument applies for the relation between the Year 2 price and the Year 3 price, the Year 3 price and the Year 4 price, and so on.
Suppose you knew the demand curve for petroleum, the total amount that now exists, and the interest rate. How would you calculate the price? The easiest way is to work backward. The intersection of the demand curve with the vertical axis tells you the highest price petroleum can sell for--the price it will sell for the day the last well goes dry. Call that price Pmax and that year Tdry. A year earlier, in year Tdry -1, the price must be lower by a factor of 1/(1 + r), two years earlier it must be lower by a factor of 1/(1 + r)2 and so forth.
How do we find the date of Tdry? Since we know the price of petroleum in year Tdry -1 and the demand curve, we know how much was consumed that year. The same applies for year Tdry -2 and for each earlier year. Add up consumption year by year, starting at Tdry (quantity demanded = 0) and working back. When the total quantity consumed adds up to the total that now exists, you have reached the present. Since you now know how many years separate the year that we run out of oil from the present year, you know when we will run out. The calculations are shown in Figure 12-2a, where price is calculated from the demand curve, and Figure 12-2b, where quantity is added up year by year. The figures assume an initial quantity of oil of a billion barrels.
Calculating the price of a depletable resource
by working backward from the date at which it is exhausted.
Tdry is the year the last oil well runs dry. Working
backward from Tdry, price falls at the interest rate, as
shown on Figure 12-2a. Figure 12-2b shows annual and cumulative
consumption; when cumulative consumption reaches the total amount now
existing, we have reached the present.
As you may have realized, I have simplified the problem somewhat by assuming implicitly that everything happens at the beginning of each year, so that quantity consumed depends only on price at that instant. One could solve the problem a little more precisely by letting price rise and quantity demanded fall continuously through the year. Doing that would involve either using calculus or making the geometric calculations even more complicated than they are, which is why I did not do so. I have also assumed that D, the demand curve for oil, is stable over time--quantity demanded changes as a result of the changing price but not as a result of changing automobile technology, population, weather, and so forth. That assumption could also be dropped, but again at the cost of making the calculations more complicated.
In solving the problem, I assumed that the demand curve intersects the vertical axis--that there is some maximum price people will pay for petroleum. An alternative assumption is that as quantity goes to zero, price goes to infinity--people keep buying less and less petroleum at a higher and higher price per gallon. The calculation in that case is more complicated, since it involves an infinite sum (of smaller and smaller quantities of petroleum), but the logic of the problem is essentially the same.
It is interesting to ask how our solution would change if we changed one of the variables. Suppose, for instance, that oil producers have adjusted to an interest rate (present and anticipated) of 5 percent. Some unexpected event raises the interest rate (now and forever after) to 10 percent. What happens to current and future prices and consumption of oil?
You should be able to work out the answer for yourself. At the old rate of production (before the change), oil prices were rising at 5 percent a year; an oil producer got the same return holding oil as holding money. With that pattern of production after the change, producers find that it is more profitable to produce a gallon of oil this year and invest the money, ending up next year with this year's price plus ten percent, than to hold the oil and produce it next year, ending up with this year's price plus five percent. Oil producers alter their plans, shifting production to earlier years. As they do so, the price falls in the early years, since more oil is being produced then, and rises in the later years. When equilibrium has been reestablished, current production is higher and the current price lower than before the change, but the price is rising faster than it was before--10 percent a year instead of 5 percent. After adjusting to the higher interest rate, we are using oil faster than before and will run out sooner--as you can easily prove by starting at Tdry and working back to the present at the higher interest rate.
In discussing a competitive industry in Chapter 9, I pointed out that the structure of the industry was exactly that which would be chosen by a dictatorial administrator ordered to produce the same quantity at the lowest possible cost. A similar statement can be made about a competitive industry producing a depletable resource. The interest rate represents the rate at which goods this year can be converted into goods next year--as shown in the example of the apple orchard earlier in this chapter. The price of petroleum in any year is equal to its marginal value, for reasons explained back in Chapter 4. If the price of petroleum next year is less than 1 + r times the price this year (if, say, P1 = $10.00 and P2 = $10.50), that means that the marginal barrel this year goes to someone with a marginal value for it of $10 and the marginal barrel next year goes to someone with a marginal value of $10.50/barrel. By choosing to produce a barrel next year that we could have produced this year, we are choosing a value of $10.50 next year instead of a value of $10 this year. But if the interest rate is 10 percent, someone who gives up $10 this year can convert it into $11 next year, so it is wasteful to give up $10 this year in exchange for only $10.50 next year. Following out this argument, a wise and benevolent administrator will allocate a depletable resource so as to make its price rise at the interest rate. As long as he does not do so, there is some way of reallocating production over time that produces a net gain.
This is only a sketch of an argument that cannot be made precisely until after the discussion of economic efficiency in Chapters 15 and 16. If you find the explanation confusing, you may want to come back to it after reading those chapters.
Before I leave the subject of depletable resources, several more points are worth making. The first is that the analysis I have given depends on the assumption that the owners of the resource have secure property rights--that they can confidently expect that petroleum they do not sell this year will still be theirs to sell next year.
Suppose that is not true; suppose, for example, that anyone who owns an oil well this year has a 10 percent chance of being expropriated next year. In that case, the same analysis implies that the price of petroleum will increase each year by a factor of 1.1 x (1 + r). Owners of oil wells will sell petroleum next year instead of this year only if the price is enough higher to compensate them both for the interest they lose by not selling the oil until next year and for the chance that when next year arrives, the oil will no longer belong to them.
Most oil, at present, belongs to governments; most of those governments are at least somewhat unstable. The present rulers of Saudi Arabia, for example, would be foolish to base their plans on the assumption that they will still rule Saudi Arabia ten years from now--especially with the fate of the Shah of Iran still recent history. They should be, and doubtless are, aware that money in Switzerland is a more secure form of property than oil in Saudi Arabia.
The effects of insecure property rights are not limited to distant sheiks. The American government may be stable, but its economic policies are not; the imposition of special taxes (such as the windfall profits tax) on oil companies is, in effect, a partial expropriation. If oil companies expect such taxes to increase, it is in their interest to produce oil now instead of saving it for the future--or, to put the conclusion more precisely, it is in their interest to produce more now and less in the future than they would if they did not expect such taxes to increase.
One implication of this argument is that the price of oil at present may be too low! If most of it belongs to people with insecure property rights, they have an incentive to produce more now (driving the present price down) and less in the future (driving the future price up) than if property rights were secure. If, as I claim, the solution under secure property rights is in some sense optimal ("efficient" in the sense to be defined in Chapter 15), then insecure property rights create a less desirable outcome. Initially oil prices are too low and consumption too high; later prices are too high and consumption too low. The allocation of the resource over time is inefficient; too much is consumed now and too little saved for later.
It may occur to some readers to ask whether the price of oil has been increasing at the interest rate over, say, the last fifty or a hundred years. The answer is no. From about 1930 to about 1970, the real price of oil--the price allowing for inflation--fell substantially. The OPEC boycott brought the real price most of the way back up to where it had been in 1930, but events since have brought it back down to about what it was before the boycott--far below where it would be if it had been rising at the interest rate from 1930 to the present.
There are at least three possible explanations for the apparent divergence between theory and fact. The first is that the economic theory of depletable resources is wrong. The second is that the theory is logically correct but that one of its assumptions--a predictable world--does not apply. If, for example, each year people overestimated future demands and/or underestimated future supplies, future prices would consistently turn out lower than expected and price would fail to rise over time at the interest rate. Economists are generally skeptical of such an explanation because it requires not merely mistakes but consistent mistakes; one would expect that after a decade or two of overestimating future oil prices, people would learn to do better--especially people who own oil wells.
The third, and most interesting, explanation of the observed pattern of prices is that oil is not a depletable resource! If this seems like an odd idea, consider that the world has been "about to run out of oil" for most of the past century; for most of that time, proven reserves have been equal to between 10 and 20 years of production.
I started my analysis of a depletable resource by assuming that there were no production costs, so that the price of the resource was entirely due to the limited quantity. Suppose I had not made that assumption. How would the existence of production costs affect the conclusion?
Assume that production costs can be predicted with certainty. In that case, we can repeat our previous analysis, simply substituting "price minus production cost" for price. Price minus production cost is what the owner of an oil well ultimately gets by selling his oil. If it rises faster than the interest rate, all producers are better off holding their oil for future production; if it rises more slowly than the interest rate, all producers are better off selling everything immediately--or at least as fast as they can get the oil out of the ground without raising production cost substantially in the process. In equilibrium, price minus production cost must rise at the interest rate--provided the owners of oil wells have secure property rights.
So one explanation of what has actually happened to oil prices is that most of the price is production cost--where that includes not only the cost of pumping the oil but also the cost of finding it. If production cost in that sense has been falling over time, then price could be falling as well--even if price net of production cost was rising.
In the previous discussion, we were considering a pure depletable resource--a resource whose price was entirely determined by its limited supply. Consider, at the other extreme, a resource of which only a limited amount exists but for which production costs are substantial and for which that "limited amount" is very large compared to the quantity demanded at a price sufficient to cover the cost of production. The amount is so large that technology, law, and political institutions will have changed beyond recognition long before the supply is exhausted.
Under those circumstances, saving the good now in order to sell it when supplies run short is not a very attractive idea--before that happens we may have stopped using it, the owner may have been expropriated, or the world may have ended. Changes in its price over time will be almost entirely determined by changes in production cost. The good is, strictly speaking, depletable, but that fact has no significant effect on its price. The pattern of oil prices over the past ninety years or so suggests that that may well be how the market views petroleum.
If so, then the insecure property rights discussed earlier imply almost exactly the opposite of what they implied before. If the price of oil is determined by the cost of finding and producing it, then insecure property rights make the price of oil higher, not lower, than it would otherwise be. If someone who invests in finding and drilling an oil well has a 50 percent chance of having his well expropriated as soon as it starts producing, his return if he does keep the well must be at least twice his costs in order for him to be willing to make the investment. His return depends on the price he sells the oil for, so the price of oil will be higher in a world of insecure property rights. The same condition that makes the present price of a depletable resource (more precisely, a resource whose price is mostly due to its limited total quantity rather than to its cost of production) lower makes the present price of a resource whose price is mostly due to cost of production higher!
If an individual can buy apples for $0.50 apiece, he will adjust his consumption of apples until the marginal utility of an apple to him, the utility he gets from consuming one more apple, is the same as the marginal utility to him of $0.50--or, in other words, until the marginal value of an apple is $0.50. That is the argument by which we demonstrated, back in Chapter 4, that price equals marginal value--P = MV.
An interest rate is also a price. If the apple interest rate is 10 percent, the price of an apple this year is 1.1 apples next year. So I will adjust my consumption of apples in both years until the increased utility I get from consuming one more apple this year is the same as the increased utility I get from consuming 1.1 additional apples next year.
"On a list of the differences between Lily and me it would be near the top that I park so I won't have to back out when I leave and she doesn't."--Archie Goodwin
Why would an apple next year give me less utility than an apple this year? There are two major reasons. The first is impatience. Most of us, given the choice between the same pleasure now or in the future, would prefer to have it now. If so, then in comparing alternative patterns of pleasure over time--alternative utility streams--we will discount utility just as we discount income. If, in making a choice today, I am indifferent between a 100-utile pleasure now or a 105-utile pleasure next year, I may be said to have an internal discount rate (for utility) of 5 percent.
My internal discount rate--my impatience--is a characteristic of my tastes; it describes my preferences between pleasures now and pleasures in the future, just as my utility function describes my preferences between apples and oranges. The market interest rate depends not merely on my tastes but on the tastes and productive abilities of everyone else as well. There is no particular reason why the two rates should be equal. Impatience may explain why the value to me of an apple now is greater than the value to me (now) of an apple a year from now, but it does not explain why the ratio of the two is exactly equal to the market interest rate.
The equality between price and marginal value is not a characteristic of the consumer's tastes; it is a result of his rational behavior in deciding how much of what good to consume. This is true when the choice is between the same good at different times just as much as when it is between different goods at the same time. I will use Figure 12-3 to show how this works for a consumer whose internal discount rate is zero in a world where the market interest rate is 10 percent .
Suppose the consumer whose marginal utility curve is shown on Figure 12-3 consumed the same number of apples each year--say 1,000 (he likes apples). The marginal utility of apples would be 10-utiles per apple, as shown on the figure. Since he consumes the same number of apples each year, he receives the same pleasure from consuming one more apple whichever year he consumes it. Since his internal discount rate is zero, he is indifferent between equal pleasures now and in the future. So he is indifferent between consuming an apple this year, next year, or in any future year.
The apple interest rate is assumed to be 10 percent, so he can trade 10 apples this year for 11 apples next year. Giving up 10 apples this year costs him about 100 utiles; consuming 11 more apples next year gives him about 110--the numbers are approximate because marginal utility will change a little over the range of quantities from 990 apples per year to 1,011 apples per year. He is indifferent between utiles this year and utiles next year, so losing 100 of the former and gaining 110 of the latter is a net gain. The consumer revises his consumption plans; instead of consuming 1,000 apples each year, he decides to consume 990 this year (point B on Figure 12-3) and 1,011 next year (point D).
Figure 12-3 A consumer's marginal utility for
apples. The consumer follows a pattern of consumption over time
for which the present value (discounted at the discount rate for
utility) of the marginal utility of an apple consumed next year
equals 1/(1 + r) times the marginal utility of an apple
consumed this year, where r is the apple interest rate. If the
consumer's discount rate for utility equals the apple interest rate,
he consumes the same number of apples each year (point C).
Since he is now going to consume fewer apples in the first year than he had initially planned, their marginal utility will be higher. Since he is going to consume more apples in the second year, their marginal utility will be lower. As you can see from the figure, the marginal utility of apples in Year 1 (990 apples per year) is about 10.25 utiles per apple; the marginal utility in Year 2 (1,110 apples per year) is about 9.75 utiles per apple. Even with the revised plan, 11 apples next year are still worth more, in utiles, than 10 apples this year. So long as this is true--so long as the marginal utility of apples in Year 1 is not at least 10 percent greater than in Year 2--the consumer can improve his situation by revising his consumption plan and transferring consumption from Year 1 to Year 2. If the marginal utility of apples in Year 2 were more than 10 percent greater than in Year 1, he could improve his situation by revising in the opposite direction--transferring consumption from Year 2 to Year 1. He will achieve his optimal plan only when the amount he consumes in each year is such that the ratio of the marginal utility of apples in Year 1 to the marginal utility of apples in Year 2 is 1.1--equal to the price of apples in Year 1 measured in apples in Year 2. This, of course, is our old friend the equimarginal principle:
.
The ratio of prices, measured in some common unit (say dollars in Year 1), is equal to the price of Year 1 apples measured in Year 2 apples. The solution is shown on Figure 12-3 as points A (first year) and E (second year). The marginal utilities are about 10.5 and 9.5, so their ratio is about 1.1.
The analysis can easily be generalized to more than two years. We have shown that the ratio of marginal utilities between Year 1 and Year 2 must be 1.1. The same argument applies between Year 2 and Year 3, Year 3 and Year 4, and so forth. Consumption of apples rises, year by year, in such a way as to make the marginal utility of apples fall by 10 percent per year.
So far, we have done our calculations on the assumption that the consumer has an internal discount rate of zero--he is indifferent between identical pleasures now and in the future. Now let us consider a consumer who has an internal discount rate of 5 percent. He too will adjust his consumption plans until he is indifferent between an additional 10 apples this year and an additional 11 apples next year. Since he regards 10 utiles this year as equivalent to 10.5 next year, his optimal consumption plan will be one for which 11 apples next year produce 5 percent more utility next year than 10 apples this year produce this year. His consumption for this year will be point B on Figure 12-3; his consumption for next year will be point D.
Finally, consider a consumer whose internal discount rate is 10 percent. His consumption will be point C this year and point C next year. Each year he consumes 1,000 apples and receives a marginal utility of 10 utiles per apple. Since he regards 10 utiles this year as equivalent to 11 next year, he can benefit neither by increasing his present consumption (10 apples more this year, 11 fewer next year) nor by decreasing it.
We have now answered the question that started this discussion. The marginal value of a future apple, measured in present apples, will be the same as the price of a future apple, measured in present apples; MV = P. Hence the internal discount rate for apples will be equal to the apple interest rate--and similarly, the internal discount rate for dollars will be equal to the dollar interest rate. The internal discount rate for apples is the sum of the internal discount rate for utility and the rate at which the marginal utility of apples declines with time. The rational consumer will adjust his consumption plans until that sum equals the apple interest rate.
We have also provided a way of looking at another question of considerable interest--what determines how much people save or borrow. The individual consumer has a flow of income, an internal discount rate, a utility function, and an interest rate at which he can borrow or lend. His objective is, by appropriate borrowing and lending, to arrange his consumption over his lifetime in such a way as to maximize the present value of his utility. That is simply a more formal statement of what we have just been doing with apples--rearranging consumption wherever doing so gets us more utility, discounted at the individual's internal discount rate back to time zero, than it costs us.
Consider an individual--a professional baseball player, say--who will have a very high income in the early part of his life and a much lower one later on. If he spends his money as it comes in, he will consume $100,000 a year at age twenty-five and only $20,000 a year at age fifty. The utility from the last $1000 of $100,000 is probably much less than the utility from adding $1000 plus twenty-five years of accumulated interest to $20,000. So he reduces his expenditure in the early years, invests part of his income, and has the money to use later, when it will be more useful to him.
The same argument applies in reverse to someone with the opposite income pattern--a medical student, say, who has very little income through about age twenty-five but expects to be doing very well at age fifty. He adjusts in the opposite direction, transferring consumption from the later years to the earlier years by borrowing.
We can describe the behavior of such individuals a good deal more precisely by applying the results of the previous section to dollars instead of apples. Assuming, as usual, rational individuals and a predictable world, each consumer will borrow and lend in such a way as to make his internal discount rate for dollars equal to the market interest rate--the rate at which he can borrow or lend dollars.
This implies that the individual's marginal utility for income, discounted back to the present at his internal discount rate for utility, should fall year by year at the interest rate. A dollar spent now will give him the same discounted utility as a dollar plus interest spent next year; the increase in the money exactly balances the fall in its discounted marginal utility. If that were not the case, he could make himself better off by increasing or decreasing his savings. We are again back with the equimarginal principle, applied over time instead of space.
Budget line diagrams for intertemporal
choice. B1 on Figure 12-4a shows a consumer with
$40,000 in Year 1 or $50,000 in Year 2. B2 on Figure 12-4b
shows a consumer with $30,000 each year. On both figures,the interest
rate is 25 percent.
Figure 12-4 shows how the allocation of consumption over time can be analyzed using budget lines and indifference curves. Consider someone who has $40,000 and is deciding how to divide his consumption between this year and next year. If he spends it all now, he will have $40,000 of consumption this year and none next. If he consumes nothing this year, he can lend it all out at an interest rate of 25 percent and consume $50,000 next year. The budget line B1 on Figure 12-4a shows the alternatives available to him. He maximizes his utility (at X1) by spending $16,000 this year and lending out the other $24,000. Next year he gets back $24,000 in principal plus $6,000 in interest, for a total of $30,000, all of which he consumes. His demand for loans, at an interest rate of 25 percent, is -$24,000. The minus sign means that he is making loans--he is a supplier, hence he is "demanding" a negative quantity..
The same figure could just as easily show someone who has nothing this year but will get $50,000 next year. By borrowing against his future income (at 25 percent), he can consume $40,000 in Year 1 and nothing (his debt just cancels his income) in Year 2. Alternatively, he can consume nothing this year and the whole $50,000 next year. B1 shows the alternative patterns of consumption available to him. His alternatives, and therefore his optimum pattern of consumption, are exactly the same as in the previous case. He borrows and consumes $16,000 in Year 1. After paying back $20,000 in principal and interest, he has $30,000 left to consume in Year 2. His demand for loans at 25 percent is +$16,000.
Why does the same figure represent both situations? In both, the consumer has the same wealth--a present value of $40,000 in Year 1. As we showed earlier, if the consumer can freely borrow or lend at the same interest rate, all income streams with the same present value are equivalent.
Figure 12-4b shows a consumer with an income of $30,000 each year. If he consumes his income as it comes in, he will be at point E--his initial endowment. He maximizes his utility by instead saving $6,000 from his first $30,000. His consumption (point X2) is $24,000 in Year 1 and $37,500 in Year 2. His demand for loans at 25 percent is -$6,000.
We could, if we wished, use diagrams like these to derive the demand curve for loans of a single individual, just as we derived a demand curve from indifference curves back in Chapter 3. We would start with an indifference curve map representing the individual's taste for two "goods"--consumption in Year 1 and consumption in Year 2. For a given endowment--a stream of income--we would draw a series of budget lines corresponding to different interest rates. For each budget line we would find the optimal point and calculate how much the individual would borrow or lend in order to get there from his initial endowment. We would end up with a curve showing amount borrowed as a function of the interest rate. Since the paper we draw the curves on only has two dimensions, we would be limited to analyzing behavior in a two-period world. The mathematics could be generalized easily enough to a many-period world, but we could no longer draw the diagrams.
In all of these cases, just as in the indifference curve diagrams of Chapter 3, the optimum occurs where the budget line is tangent to an indifference curve. The slope of the budget line shows the rate (1+r) at which consumption in Year 1 can be exchanged for consumption in Year 2 . The slope of the indifference curve shows the rate (1+d) at which you are just willing to exchange consumption in Year 1 for consumption in Year 2 , where d is your internal discount rate for dollars. At the point of tangency the two slopes are equal, so the internal discount rate for dollars is equal to the market interest rate. We have derived the equimarginal principal over time in exactly the same way we derived it between goods in Chapter 3.
Budget line diagrams for a consumer whose
ability to borrow is limited. Figures 12-51 and 12-5b show a
consumer who cannot borrow; in Figure 12-5b, this has no effect on
his consumption choices. Figure 12-5c shows a consumer who can
borrow, but only at a higher interest rate than he lends at.
Figure 12-5a shows a somewhat different case--a consumer with an endowment F who can lend (again at 25 percent) but cannot borrow--perhaps because nobody trusts him to pay the money back. If he could borrow he would, putting him at point X; since he cannot, the best he can do is to stay at F, spending each year's income that year. Figure 12-5b shows a similar situation, but one in which the consumer's inability to borrow has no effect; even if he could borrow, he wouldn't. Finally, Figure 12-5c shows the more realistic case of an individual (again with endowment F) who can both borrow and lend, but at different rates. He can lend at 10 percent, but must pay 25 percent if he wishes to borrow
So far we have been analyzing the decision to borrow or lend by a single consumer. If we add up the behavior of all consumers, we get the total supply and demand for loans by consumers. The higher the interest rate is, the more consumers shift consumption towards their later years, since a higher interest rate increases the amount you can get later if you give up a dollar now. So a higher interest rate decreases the net demand (demand minus supply) for loans by consumers.
What factors determine the shape of the net demand curve for loans--the relation between the interest rate lenders are paid for the use of their money and the amount consumers choose to lend or borrow? One factor is the pattern of lifetime earnings and of expenditure opportunities. If the number of careers which, like medicine, require lengthy training increases, then so will the demand for loans. We saw this effect on Figure 12-4a, where the same individual changed from net lender to net borrower when his income was shifted from Year 1 to Year 2. If medical technology improves in a way that gives old people new and very valuable ways of spending their money, the utility function for income to the old rises. Individuals choose to spend less of their income when young in order to save it to pay medical bills when they are old. Since they must lend out the money they save in order to get interest on it, the supply of loans increases.
A second factor is the internal discount rate. If some cultural change makes people more concerned about their own (or their children's) future, their savings will go up and their borrowing down. If everyone decides to enjoy life today whatever the consequences, savings go down and borrowing up.
If all lending and borrowing were of this sort, then total borrowing and total saving would have to be equal; you cannot borrow a dollar unless someone else saves it and lends it to you, so net demand for loans (at the equilibrium interest rate) must be zero. In that situation, changes in the net demand schedule end up as changes in the market interest rate. If demand for loans rises and supply falls, the interest rate goes up until quantity demanded and quantity supplied are again equal.
All lending and borrowing is not of this sort. In addition to individuals borrowing or lending in order to adjust their consumption patterns over time, there are also firms borrowing in order to invest--the equivalent, in my earlier example, of using apples to grow apple trees. As we saw earlier, the decision of whether or not to make a particular investment is based on a present value calculation. The higher the market interest rate, the lower the present value of a future return. So if interest rates are high, firms will only invest in projects that have a very high return relative to the initial investment. The lower the interest rate, the larger the number of projects that yield a positive net present value. So the demand curve for loans by firms is downward sloped--quantity of loans demanded increases as the interest rate falls.
Just as we can use indifference curves to analyze intertemporal choice of consumption by individuals, so we could use isoquant curves and isocost lines to analyze the corresponding decision by firms. Just as the firms in Chapter 9 chose the lowest cost bundle of inputs to produce a given quantity of output, so here firms choose the lowest cost combination of Year 1 inputs (purchased by borrowing against Year 2 income) and Year 2 inputs with which to produce Year 2 output. Different interest rates imply different slopes for the isocost lines, different lowest cost combinations of inputs, and thus different amounts of borrowing by firms.
Individuals and firms are not the only participants on the capital market. Governments borrow, both from their citizens and from foreigners, in order to finance present expenditures with future taxes. And capital may flow into (or out of) the country--foreigners may find they get a better return in America than at home and therefore choose to lend money to American consumers, firms, or governments. Individuals, firms, and governments both here and abroad all contribute to the supply and demand curves that determine the market interest rate.
In the optional section of Chapter 6, I showed that a trade deficit is equivalent to a net inflow of capital and argued that whether our current deficit is a good or a bad thing depends on why that inflow is occurring. We are now in a position to state the argument a little more clearly.
A capital inflow occurs because foreign investors can get more for their money here than at home--it reflects relatively high real interest rates. If the reason is, as sometimes asserted, that Americans have become increasingly impatient, unwilling to give up present utility for future utility, then it is a symptom of a change that will ultimately make us poorer--we are consuming our future income and some day the bill will come due. If the reason is that American firms have lots of good investment opportunities and are therefore happy to offer higher rates than Japanese firms, the bill will still come due, but we will have the returns from those investment opportunities to pay it with.
The fact that real interest rates have become relatively high here may reflect a change, not here, but abroad. Perhaps Japanese savers have become less impatient, or wealthier, or Japanese firms have fewer good investment opportunities than before, or the Japanese government has stopped borrowing money from its people. Any of those changes would lower Japanese real interest rates, making America a more attractive place to invest. The result would be a capital inflow and hence a trade deficit.
1. One of my examples of goods that are inexpensive to store was "gold bars--provided nobody knows you have them." Why are gold bars expensive to store if people know you have them?
2. What does your answer to Problem 1 suggest might be one of the factors affecting interest rates?
3. The apple interest rate is zero, the peach interest rate is 10 percent. Currently the price of a peach is 1 apple. What will the price of a peach be next year? The year after?
4. According to the numbers at the bottom of Table 12-1, the first stream of income is worth $270,362, the second is worth $259,892, and their difference, the amount you would have if you received the first while simultaneously paying out the second, is $24,688. The numbers do not appear to add up. What is wrong? Show that the numbers are actually consistent.
5. You have a choice between two jobs. One of them pays you $20,000/year for four years, with the payment coming at the end of each year. The other pays you $19,000/year for four years, plus a "recruitment bonus" of $3,000 at the beginning of the first year. The market interest rate is 10 percent. Which job should you take?
6. Some years ago, a prominent consumer magazine ran an article on how to choose a mortgage. Different ways of borrowing a given amount of money (with or without down payment, short term or long term, etc.) were compared according to the total number of dollars you had to pay out during the term of the mortgage--the fewer dollars the better.
a. What sort of conclusion do you think they reached? According to their criterion, what is the best kind of mortgage?
b. Do you agree with their criterion? Can you describe two mortgages for the same amount of money, one of which results in your paying out more total dollars but is clearly better? If so, do. Assume that your income is sufficient to pay either mortgage.
7. You can build a factory for $1,000,000 that will permit you to manufacture 100,000 widgets per year at a cost of $1/widget (not counting the cost of the factory). The market interest rate is 10 percent.
a. The factory will last forever; you do not expect the price of widgets to change. What is the lowest price for widgets at which the factory is worth building?
b. The factory will last only three years. What is the lowest price for widgets at which the factory is worth building?
8. The government borrows money by selling at auction $1,000 bonds, payable in two years, with no interest payments. The market interest rate is 10 percent.
a. How much will the bonds sell for?
b. Even though the bonds do not "pay interest" (the buyer receives $1,000 when the bonds mature and nothing before that), buyers still end up receiving interest on their investment. Explain.
c. What interest rate are buyers of the bonds actually receiving on their investment? Explain.
d. What will happen if, immediately after the bonds are sold, the market interest rate unexpectedly falls to 5 percent?
(Hint: The bonds must sell, both initially and after the change in interest rates, for a price at which the buyers are indifferent between buying them and investing their money at the market interest rate. If the selling price were higher than that, nobody would buy them; if it were lower, nobody would make any other investment.)
9. A bank offers you the following deal. Deposit $1000 with them and they will give you a pair of binoculars. Five years later, you get your thousand dollars back--but no interest. Assuming that the market interest rate is 5 percent, how much are you really paying for the binoculars?
10. A bank offers you the following deal. Deposit $10,000 with them and they will give you a savings bond worth $2,000. Four years later, they will give you your money back--but no interest. What interest rate are you really getting?
11. The following is a list of prices for wheat futures printed in February; the price for a wheat future is the price you pay now in exchange for delivery of a bushel of wheat at some future date.
March May July September November January $2.40 $2.70 $1.20 $1.50 $1.80 $2.10
a. When is the new crop harvested? Explain.
b. About how much does it cost to store a bushel of wheat for a month? Explain.
For both parts you may, if you wish, assume that the interest rate is zero. (Note: This problem requires some original thinking by you; it has not been done anywhere in the chapter. The correct analysis is, however, similar to the analysis of other problems that are in the chapter.)
12. I can borrow as much as I want at 10 percent; I can lend as much as I want at 8 percent. What can you say about my internal discount rate for money in a year when:
A. I borrow money.
B. I lend money.
13. Is my internal discount rate for utility determined by my tastes, my opportunities, or both? Is my internal discount rate for money determined by my tastes, my opportunities, or both? Explain.
14. I am trying to decide whether to have my wisdom teeth out. I estimate the cost, in time, pain, and money, at 10,000 utiles. The benefit is a reduction in minor dental problems of about 1 utile a day. I expect that I, my teeth, and my dental problems will last forever.
a. I decide not to have my wisdom teeth out. What can you say about my internal discount rate?
b. I decide to have my wisdom teeth out. What can you say about my internal discount rate?
15. What effect would each of the following have on interest rates?
a. A new religion spreads that preaches the virtues of thrift; converts save their money for their old age.
b. A new religion spreads that preaches that the end of the world is imminent.
16. Figure 12-4a shows the same budget line corresponding to two possible situations--one in which income is $40,000 in Year 1 and nothing in Year 2, and one in which it is nothing in the Year 1 and $50,000 in Year 2.
a. Redraw (or xerox) the figure, showing the endowments E1 and E2 corresponding to the two situations.
b. For each situation, calculate the demand curve over a range of interest rates.
c. Give another initial endowment that would also correspond, at an interest rate of 25 percent, to the same budget line.
The analysis of depletable resources in this chapter is not a product of recent concerns with the problem, summarized in phrases (and book titles) such as "limits to growth" and "spaceship earth." It was produced more than fifty years ago by Harold Hotelling. His original article is:
"The economics of exhaustible resources." JPE 39, 137-75.