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## 13.4 Review and Practice

### Summary

Time is the complicating factor when we analyze capital and natural resources. Because current choices affect the future stocks of both resources, we must take those future consequences into account. And because a payment in the future is worth less than an equal payment today, we need to convert the dollar value of future consequences to present value. We determine the present value of a future payment by dividing the amount of that payment by (1 + r)n, where r is the interest rate and n is the number of years until the payment will occur. The present value of a given future value is smaller at higher values of n and at higher interest rates.

Interest rates are determined in the market for loanable funds. The demand for loanable funds is derived from the demand for capital. At lower interest rates, the quantity of capital demanded increases. This, in turn, leads to an increase in the demand for loanable funds. In the aggregate, the supply curve of loanable funds is likely to be upward-sloping.

We assume that firms determine whether to acquire an additional unit of capital by (NPV) of the asset. When NPV equals zero, the present value of capital’s marginal revenue product equals the present value of its marginal factor cost. The demand curve for capital shows the quantity of capital demanded at each interest rate. Among the factors that shift the demand curve for capital are changes in expectations, new technology, change in demands for goods and services, and change in relative factor prices.

Markets for natural resources are distinguished according to whether the resources are exhaustible or renewable. Owners of natural resources have an incentive to consider future as well as present demands for these resources. Land, when it has a vertical supply curve, generates a return that consists entirely of rent. In general, economic rent is return to a resource in excess of the minimum price necessary to make that resource available.

### Concept Problems

1. The charging of interest rates is often viewed with contempt. Do interest rates serve any useful purpose?
2. How does an increase in interest rates affect the present value of a future payment?
3. How does an increase in the size of a future payment affect the present value of the future payment?
4. Two payments of \$1,000 are to be made. One of them will be paid one year from today and the other will be paid two years from today. Which has the greater present value? Why?
5. The essay on the viatical settlements industry suggests that investors pay only 80% of the face value of a life insurance policy that is expected to be paid off in six months. Why? Would it not be fairer if investors paid the full value?
6. How would each of the following events affect the demand curve for capital?

1. A prospective cut in taxes imposed on business firms
2. A reduction in the price of labor
3. An improvement in technology that increases capital’s marginal product
4. An increase in interest rates
7. If developed and made practical, fusion technology would allow the production of virtually unlimited quantities of cheap, pollution-free energy. Some scientists predict that the technology for fusion will be developed within the next few decades. How does an expectation that fusion will be developed affect the market for oil today?
8. Is the rent paid for an apartment economic rent? Explain.
9. Film director Brett Ratner (Rush Hour, After the Sunset, and others) commented to a New York Times (November 13, 2004, p. A19) reporter that, “If he weren’t a director, Mr. Ratner said he would surely be taking orders at McDonald’s.” How much economic rent is Mr. Ratner likely earning?
10. Suppose you own a ranch, and that commercial and residential development start to take place around your ranch. How will this affect the value of your property? What will happen to the quantity of land? What kind of return will you earn?
11. Explain why higher interest rates tend to increase the current use of natural resources.

### Numerical Problems

Use the tables below to answer Problems 1–5. The first table gives the present value of \$1 at the end of different time periods, given different interest rates. For example, at an interest rate of 10%, the present value of \$1 to be paid in 20 years is \$0.149. At 10% interest, the present value of \$1,000 to be paid in 20 years equals \$1,000 times 0.149, or \$149. The second table gives the present value of a stream of payments of \$1 to be made at the end of each period for a given number of periods. For example, at 10% interest, the present value of a series of \$1 payments, made at the end of each year for the next 10 years, is \$6.145. Using that same interest rate, the present value of a series of 10 payments of \$1,000 each is \$1,000 times 6.145, or \$6,145.

Table 13.3 Present Value of \$1 to Be Received at the End of a Given Number of Periods

Percent Interest
Period 2 4 6 8 10 12 14 16 18 20
1 0.980 0.962 0.943 0.926 0.909 0.893 0.877 0.862 0.847 0.833
2 0.961 0.925 0.890 0.857 0.826 0.797 0.769 0.743 0.718 0.694
3 0.942 0.889 0.840 0.794 0.751 0.712 0.675 0.641 0.609 0.579
4 0.924 0.855 0.792 0.735 0.683 0.636 0.592 0.552 0.515 0.442
5 0.906 0.822 0.747 0.681 0.621 0.567 0.519 0.476 0.437 0.402
10 0.820 0.676 0.558 0.463 0.386 0.322 0.270 0.227 0.191 0.162
15 0.743 0.555 0.417 0.315 0.239 0.183 0.140 0.180 0.084 0.065
20 0.673 0.456 0.312 0.215 0.149 0.104 0.073 0.051 0.037 0.026
25 0.610 0.375 0.233 0.146 0.092 0.059 0.038 0.024 0.016 0.010
40 0.453 0.208 0.097 0.046 0.022 0.011 0.005 0.003 0.001 0.001
50 0.372 0.141 0.054 0.021 0.009 0.003 0.001 0.001 0 0

Table 13.4 Present Value of \$1 to Be Received at the End of Each Period for a Given Number of Periods

Percent Interest
Period 2 4 6 8 10 12 14 16 18 20
1 0.980 0.962 0.943 0.926 0.909 0.893 0.877 0.862 0.847 0.833
2 1.942 1.886 1.833 1.783 1.736 1.690 1.647 1.605 1.566 1.528
3 2.884 2.775 2.673 2.577 2.487 2.402 2.322 2.246 2.174 2.106
4 3.808 3.630 3.465 3.312 3.170 3.037 2.910 2.798 2.690 2.589
5 4.713 4.452 4.212 3.993 3.791 3.605 3.433 3.274 3.127 2.991
10 8.983 8.111 7.360 6.710 6.145 5.650 5.216 4.833 4.494 4.192
15 12.849 11.718 9.712 8.559 7.606 6.811 6.142 5.575 5.092 4.675
20 16.351 13.590 11.470 9.818 8.514 7.469 6.623 5.929 5.353 4.870
25 19.523 15.622 12.783 10.675 9.077 7.843 6.873 6.097 5.467 4.948
30 22.396 17.292 13.765 11.258 9.427 8.055 7.003 6.177 5.517 4.979
40 27.355 19.793 15.046 11.925 9.779 8.244 7.105 6.233 5.548 4.997
50 31.424 21.482 15.762 12.233 9.915 8.304 7.133 6.246 5.554 4.999
1. Your Uncle Arthur, not to be outdone by Aunt Carmen, offers you a choice. You can have \$10,000 now or \$30,000 in 15 years. If you took the payment now, you could put it in a bond fund or bank account earning 8% interest. Use present value analysis to determine which alternative is better.
2. Remember Carol Stein’s tractor? We saw that at an interest rate of 7%, a decision to purchase the tractor would pay off; its net present value is positive. Suppose the tractor is still expected to yield \$20,000 in net revenue per year for each of the next 5 years and to sell at the end of 5 years for \$22,000; and the purchase price of the tractor still equals \$95,000. Use Tables (a) and (b) to compute the net present value of the tractor at an interest rate of 8%.
3. Mark Jones is thinking about going to college. If he goes, he will earn nothing for the next four years and, in addition, will have to pay tuition and fees totaling \$10,000 per year. He also would not earn the \$25,000 per year he could make by working full time during the next four years. After his four years of college, he expects that his income, both while working and in retirement, will be \$20,000 per year more, over the next 50 years, than it would have been had he not attended college. Should he go to college? Assume that each payment for college and dollar of income earned occur at the end of the years in which they occur. Ignore possible income taxes in making your calculations. Decide whether you should attend college, assuming each of the following interest rates:

1. 2%
2. 4%
3. 6%
4. 8%
4. A new health club has just opened up in your town. Struggling to bring in money now, the club is offering 10-year memberships for a one-time payment now of \$800. You cannot be sure that you will still be in town for the next 10 years, but you expect that you will be. You anticipate that your benefit of belonging to the club will be \$10 per month (think of this as an annual benefit of \$120). Decide whether you should join at each of the following interest rates:

1. 2%
2. 4%
3. 6%
4. 8%
5. You have just purchased a new home. No money was required as a down payment; you will be making payments of \$2,000 per month (think of these as annual payments of \$24,000) for the next 30 years. Determine the present value of your future payments at each of the following interest rates:

1. 2%
2. 4%
3. 6%
4. 8%
6. You own several barrels of wine; over the years, the value of this wine has risen at an average rate of 10% per year. It is expected to continue to rise in value, but at a slower and slower rate. Assuming your goal is to maximize your revenue from the wine, at what point will you sell it?
7. You have been given a coin collection. You have no personal interest in coins; your only interest is to make money from it. You estimate that the current value of the collection is \$10,000. You are told the coins are likely to rise in value over time by 5% per year. What should you do with the collection? On what factors does your answer depend?
8. The Case in Point on the increasing scarcity of oil suggested that the Khurais complex is expected to add 1.2 million barrels to world oil production by 2009. Suppose that world production that year what otherwise be 87 million barrels per day. Assume that the price elasticity of demand for oil is −0.5. By how much would you expect the addition of oil from the Khurais complex to reduce the world oil price?