## LEARNING OBJECTIVE

- How much of a limited resource should be consumed today, and how much should be saved for future consumption?

For the past 60 years, the world has been “running out of oil.” There are news stories about the end of the reserves being only 10, 15, or 20 years away. The tone of these stories is that, at this time, we will run out of oil completely and prices will be extraordinarily high. Industry studies counter that more oil continues to be found and that the world is in no danger of running out of oil.

If you believe that the world will run out of oil, what should you do? You should buy and hold. That is, if the price of oil in 20 years is going to be $1,000 per barrel, then you can buy oil at $40 per barrel, hold it for 20 years, and sell it at $1,000 per barrel. The rate of return from this behavior is the solution to (1+r) 20 = 1000 40 .

This equation solves for r = 17.46%, which represents a healthy rate of return on investment. This substitution is part of a general conclusion known as the **Ramsey rule**:The solution to this problem is known as Ramsey pricing, after the discoverer Frank Ramsey (1903–1930). For resources in fixed supply, prices rise at the interest rate. With a resource in fixed supply, owners of the resource will sell at the point maximizing the present value of the resource. Even if they do not, others can buy the resource at the low present value of price point, resell at the high present value, and make money.

The Ramsey rule implies that prices of resources in fixed supply rise at the interest rate. An example of the Ramsey rule in action concerns commodities that are temporarily fixed in supply, such as grains after the harvest. During the period between harvests, these products rise in price on average at the interest rate, where the interest rate includes storage and insurance costs, as well as the cost of funds.

Example: Let time be \(\begin{equation}t=0,1, \ldots\end{equation}\), and suppose the demand for a resource in fixed supply has constant elasticity: \(\begin{equation}\mathrm{p}(\mathrm{Q})=\mathrm{a} \mathrm{Q}-1 \varepsilon\end{equation}\). Suppose that there is a total stock R of the resource, and the interest rate is fixed at r. What is the price and consumption of the resource at each time?

Solution: Let Qt represent the quantity consumed at time t. Then the arbitrage condition requires

\begin{equation}\text { a } Q 0-1 \varepsilon(1+r) t=p(Q 0)(1+r) t=p(Q t)=a Q t-1 \varepsilon\end{equation}

Thus, \(\begin{equation}Q t=Q 0(1+r)-t \varepsilon\end{equation}\). Finally, the resource constraint implies

\begin{equation}R=(Q 0+Q 1+Q 2+\ldots)=Q 0(1+(1+r)-\varepsilon+(1+r)-2 \varepsilon+\ldots)=Q 01-(1+r)-\varepsilon\end{equation}

This solves for the initial consumption Q_{0}. Consumption in future periods declines geometrically, thanks to the constant elasticity assumption.

Market arbitrage ensures the availability of the resource in the future and drives up the price to ration the good. The world runs out slowly, and the price of a resource in fixed supply rises on average at the interest rate.

Resources like oil and minerals are ostensibly in fixed supply—there is only so much oil, gold, bauxite, or palladium in the earth. Markets, however, behave as if there is an unlimited supply, and with good reason. People are inventive and find substitutes. England’s wood shortage of 1651 didn’t result in England being cold permanently, nor was England limited to the wood it could grow as a source of heat. Instead, coal was discovered. The shortage of whale oil in the mid-19th century led to the development of oil resources as a replacement. If markets expect that price increases will lead to substitutes, then we rationally should use more today, trusting that technological developments will provide substitutes.Unlike oil and trees, whales were overfished and there was no mechanism for arbitraging them into the future—that is, no mechanism for capturing and saving whales for later use. This problem, known as the tragedy of the commons, results in too much use (Garett Hardin, Science, 1968, Tragedy of the Commons). Trees have also been overcut, most notably on Easter Island. Thus, while some believe that we are running out of oil, most investors are betting that we are not, and that energy will not be very expensive in the future—either because of continued discovery of oil or because of the creation of alternative energy sources. If you disagree, why not invest and take the bet? If you bet on future price increases, that will tend to increase the price today, encouraging conservation today, and increase the supply in the future.