StudySpace: Economics in Development, 6th Edition

Worked Example: The Valuation and Management of Copper Resources in Zambia

The date is 1995 and Zambia’s Nchanga open-pit copper mine is widely regarded as having only five years worth of exploitable ore. Suppose that the production plan entails extracting 100,000 metric tons of copper each year for five years. Assume that the state-owned mining company expects world copper prices to be stable at $2,500 per metric ton (after adjusting for inflation). The cost of mining, smelting, refining, and delivering copper, including a normal return on capital, is a steady $1,900 per metric ton. Therefore, the resource rent is $600 per metric ton.

The economic value of the copper deposit is the present value of expected resource rents from the deposit. If the discount rate is 12 percent, then the resource rent in each year t must be multiplied by 1/(1.12)^t to obtain the present value, where t = 0 for 1995. Under the stated conditions, the copper deposit is worth $242.2 million in 1995, as shown in the following table. (Check the right-hand column to make sure you understand how to calculate present values.)

Gross revenue in 1995 is $250 million (100,000 metric tons × $2,500 per metric ton). To generate this revenue, Zambia has to deplete its natural capital by $60 million, which is the resource rent on the tonnage produced in 1995. Unless Zambia reinvests $60 million of the gross income, the total capital stock will fall and the income stream will be unsustainable. As explained in the text, standard national accounting methods neglect such resource depletion. Thus, the national accounts for 1995 overstate the net product from the Nchanga mine by $60 million.

You may have noticed that something is wrong with this scenario: It violates the maximizing rule for exploiting a natural resource over time. To maximize the present value of the resource rents, extraction should be managed so that the marginal net benefit (MNB) for any year t is equal to the discounted value of the MNB for year t + 1. In algebraic terms, MNB_t = MNB_t+1 /(1 + r). In the example, a ton of copper produced in 1995 generates a marginal resource rent of $600, whereas the discounted value of a marginal ton produced in 1996 is $600/1.12 = $536. Zambia could gain by shifting production at the margin from 1996 back to 1995. A similar adjustment would pay off in later pairs of years.

For example, let copper production rise to 140,000 tons in 1995, matched by reductions of 10,000 tons in years 1996 through 1999. Suppose that this reallocation does not affect price or cost. Although total extraction remains unchanged, the present value of the resource rents rises by more than $5 million, to $247.9 million (see Exercise 1). To be more realistic, large changes in the extraction rate could affect costs and perhaps world market prices. Also, the presence of fixed costs sets a floor on the feasible level of production in any year. So there is a limit to the gains from such rescheduling. Nonetheless, the stable-output production plan shown above would be far from optimal.

Exercises

1. It is your turn to analyze the copper resources at Zambia’s Nchanga mine. As in the Worked Example, the date is 1995. The state-owned mining company expects the world price of copper to remain at $2,500 per metric ton. Company economists anticipate that the cost of production, inclusive of a normal return on capital, will remain at $1,900 per ton.

a.

(i) Calculate the present value of resource rents, on the basis of the schedule of copper production shown in Table 20–1. Using a 12 percent discount rate, fill in all of the blanks in the table.

Click here for Table 20-1

(ii) What is the economic value of the natural resource deposit at Nchanga mine?

$ million.

What is value of resource depletion from production in 1995?

$ million.

To what extent will the standard national accounts overstate the net product in 1995 from the Nchanga mine?

$ million.

(iii) If the discount rate rises to 15 percent, then the economic value of the natural resource deposit to $ million.

(iv) What is the economic logic of valuing the resource less when the discount rate rises? (An answer that simply refers back to the formula is insufficient.)

b. With the extraction pattern as in Table 20–1, the company economists revise their projection of copper prices. They now foresee real copper prices rising by 10 percent per year, while costs remain stable at $1,900 per metric ton.

(i) Under this scenario the price of copper will be

$2,500 per metric ton in 1995

$ in 1996

$ in 1997

$ in 1998

$ in 1999

(ii) The real resource rent per ton (equal to price minus cost) will rise from $600 in 1995 to

$ in 1996

$ in 1997

$ in 1998

$ in 1999

(iii) With the 12 percent discount rate, what is the economic value of the natural capital lying underground at Nchanga now? (Don’t forget to multiply the resource rent per ton by the number of tons per year.)

$ million.

c. Given the new projection of rising copper prices (from part b), let’s see if the production plan in Table 20–1 is optimal.

(i) The resource rent earned on copper in 1995 is $ per ton.

(ii) With the discount rate still at 12 percent, the discounted value of the resource rent per ton of copper in 1996 is $ per ton.

(iii) If prices and costs would remain as stated, would Zambia benefit from producing more in 1995 and less in 1996 or vice versa? Explain.

(iv) The present production plan for 1995 and 1996 would be optimal if the company expected copper prices to rise to $ per metric ton in 1996. (Hint: apply the maximizing rule and remember that MNB is the difference between price and cost.)

d. Return to the original price and cost estimates from the opening paragraph. The government is thinking of selling the mine to Monzanium Monopoly Corporation (MMC). MMC does not share the government’s projections.

(i) In particular, MMC believes that costs would drop to $1,500 per metric ton if the mine were privately owned and managed. The resource rent then would be $ per ton, each year.

(ii) For simplicity, assume that production plan remains as shown in Table 20–1 and that the discount rate is still 12 percent. What is the present value of the resource rents expected by MMC over the five-year period?

$ million.

(Of course, an optimal production plan would allow MMC to do even better than this.)

(iii) The government negotiators insist that the mine will not be sold for less than its own valuation (from part a) plus $100 million. Will MMC accept this deal? At this price, how will the government of Zambia and MMC split the resource rents arising from the privatized operation of the mine? Do both parties come out ahead? Explain.

2. The optimal harvest of a renewable resource is defined by equating marginal benefits and costs. In Madagascar, most families rely on firewood or charcoal made from firewood, for cooking and keeping warm in chilly weather. Firewood, of course, is a renewable resource since limbs and trees grow back. But the woodlands are a common resource with open access.

a. Curve TV in Figure 20–1 shows how the sustainable total value of the annual harvest, in francs, varies as a function of the effort, E, in person-days, to gather firewood. (When firewood does not pass through the market, the value is imputed.)

FIGURE 20–1

(i) Up to point A the TV curve rises at a decreasing rate. The sustainable harvest value rises in smaller and smaller increments as successive units of effort are devoted to gathering firewood. What ecological and economic factors explain the declining payoff per unit of extra effort?

(ii) The TV curve reaches a peak at point A. Additional effort reduces the annual harvest value. Explain how this is possible.

(iii) At point B the sustainable harvest abruptly drops to zero. What is the environmental meaning of this outcome?

b. For simplicity, suppose that firewood is gathered exclusively by children, who otherwise have no gainful work to do. Their parents attribute zero cost to each unit of effort

(i) Mark on the horizontal axis the level of effort that produces the maximum sustainable net benefit, which is defined as the difference between the annual harvest value and the harvest cost (zero here). Label this point E1.

(ii) In terms of effort level, how does the maximum net benefit compare with the maximum total harvest value? Explain.

c. Now allow for a fixed cost per unit of effort. In Figure 20–1, draw a straight line from the origin through point C; label this line TC, for total cost of gathering firewood.

(i) Mark on the horizontal axis the level of effort that produces the maximum sustainable net benefit. Label this point E2.

(ii) In terms of effort level, how does the maximum net benefit in this case compare with the maximum total harvest value? Explain.

d. With open access to the woodlands, individual family decisions determine the actual amount of harvest effort. Assume here that effort costs remain as in part c.

(i) With the effort level at E2, individual families have an incentive to gather more firewood. How can this be when the curves show that the marginal cost exceeds the marginal value?

(ii) Even when effort is at E1, some families still have an incentive to gather more firewood. How can this be when the marginal value is negative beyond E1?

(iii) In the absence of irrational behavior, can individual decisions lead to point B, where the sustainable harvest falls to zero? Explain.

(iv) In general, will the free market produce the efficient level of firewood harvest in Madagascar? Explain.

3. Indonesia needs to expand electricity production. The country has abundant reserves of relatively inexpensive coal. However, coal burning creates pollution in the form of soot and ash, as well as nitrogen and sulfur oxides, which contribute to smog and acid rain. These costs are externalities because they are borne by parties other than those engaged in the sale and purchase of coal.

Figure 20–2 summarizes conditions in the market for coal. Coal is available in virtually unlimited amounts for $35 per ton; this is shown by the supply curve. Since the supply curve is horizontal, the private marginal cost of coal equals the average cost per ton. Each ton of coal burned, however, imposes $10 worth of external costs on society; this is shown by the curve labeled MEC, for marginal external cost. The competitive equilibrium, which disregards the external cost, is at point A.

FIGURE 20-2

a.

(i) In Figure 20–2, draw the social marginal cost curve by adding together vertically the supply curve and the MEC curve. Label it SMC. This curve combines the private and external costs of using coal.

(ii) Label the intersection of the SMC curve and the demand curve point B. Label the corresponding output QB.

(iii) At the competitive market equilibrium (point A), buyers value the last units of coal at $ per ton. Private supply costs are $ per ton. External costs are $ per ton. On balance, the last units of coal yield a net loss to society of $ per ton. Because of the external diseconomy, the market does not achieve an efficient level of coal production.

(iv) If output is at QB, what is the net benefit or cost to society from the last units of coal produced?

(v) At the efficient output level QB is there any pollution? Explain.

b. To achieve the efficient outcome, where social marginal cost equals the marginal benefit to coal users, the government can directly regulate coal transactions. Or it can impose a tax on coal—a green tax. This shifts the market supply curve upward at each point by the amount of the tax per ton.

(i) A tax of $ per ton will shift the supply curve for coal just enough to achieve a market equilibrium at the efficient output level QB.

(ii) In what sense does the green tax internalize the external cost?

c. A technological change allows the power company to eliminate pollution entirely at a cost of $5 per ton of coal. Thus, adding $5 per ton to the private cost would reduce the external cost to zero.

(i) Now that this pollution-abatement technology is available, is QB still the efficient output level? Explain.

(ii) With the green tax defined above, does the power company have an incentive to adopt the new technology? Explain.

(iii) Suppose the tax is restructured as a charge per unit of pollution rather than per ton of coal. This entails levying a $10 tax on the volume of pollutants produced initially per ton of coal, to achieve the outcome at point B. With this new tax structure, does the power company have an incentive to adopt the new pollution-control technology? Explain.

d. The exercise has been concerned with how government can correct a market failure created by external diseconomies. But government policies often make market outcomes worse. For each situation below, explain how an external diseconomy arises and state whether the government policy moves the market closer to or further from the efficient outcome.

(i) Cattle ranchers clear land for grazing in Brazil’s Amazon basin; the government has subsidized the ranching operations through tax breaks and low-interest loans.

(ii) Cotton growers in Uzbekistan divert large volumes of water from the Amu Darya River for irrigation; the government has supplied the irrigation water at a heavily subsidized price.

(iii) The government of Nigeria increases the price of gasoline, which has been fixed at far below the market equilibrium level.

4. Exercise 3 assumed that the marginal external cost of pollution and the marginal abatement cost both were constant. Letting these costs vary allows us to examine the concept of optimal pollution and the rationale for marketable permits. Consider the case of water pollution from palm-oil mills in Malaysia.

a. External cost rises with the amount of effluent discharged into the rivers; in fact, the external cost rises at an increasing rate. Suppose that the marginal external cost (MEC, measured in Ringgit) is related to the discharge of effluent (DIS, in tons per palm-oil mill per year) according to the equation MEC = 10 + 4DIS.

(i) Draw the line showing this relationship in Figure 20–3; label it MEC.

(ii) The prevailing level of discharge is 100 tons per palm-oil mill per year. The marginal external cost imposed by the last ton of discharge is M$ . (M$ is the symbol for the Malaysian Ringgit.)

(iii) The total external cost of pollution (TEC) is related to the level of discharge by the equation TEC = 10DIS + 2DIS². (Note that MEC is the slope of the TEC function at each point.) The total external cost from the prevailing level of discharge is M$ per palm-oil mill.

b. Abatement costs rise with the extent of the clean-up; in fact, abatement costs rise at an increasing rate since easy and cheap measures can reduce pollution up to a point, but further abatement is increasingly expensive. The relationship between the marginal abatement cost (MAC, in Ringgit) and the amount abated (ABT, in tons) is MAC = 10 + 4ABT.

FIGURE 20-3

(i) The minimum level of abatement is none, in which case DIS = 100 and ABT = . The theoretical maximum abatement eliminates all the pollution, in which case DIS = 0 and ABT = . In general, the amount abated is ABT = 100 – DIS.

(ii) In Figure 20–3, draw in and label the line representing MAC. Be careful to draw it so that the minimum value occurs at DIS = 100.

(iii) For the theoretical maximum abatement, the marginal cost of eliminating the last ton of discharge is M$ .

(iv) The total abatement cost is TAC = 10ABT + 2ABT². For the theoretical maximum abatement, the total cost of cleaning up pollution is M$ per palm-oil mill.

(v) Which creates a larger cost to the Malaysian economy: zero abatement or full abatement?

c. Something in between these extremes is to be preferred in the interest of minimizing the overall cost. The rule is to equate MEC and MAC. On the graph, this is the point where the two lines cross.

(i) Identify in Figure 20–3 the optimal level of pollution, as point X on the horizontal axis.

(ii) At point X, the discharge of effluents remains at DIS = tons.

d. Is this truly the optimal outcome? Let’s do some calculations to find out.

(i) At point X the total external cost is TEC = M$ and the total abatement cost is TAC = M$ ; this gives a combined cost to the economy of M$ .

(ii) Suppose the discharge were reduced by 5 tons, compared to point X. This gives DIS = tons and ABT = tons. In this case, TEC = M$ , TAC = M$ , and the combined cost to the economy is M$ . How does this compare with the combined cost at point X?

(iii) What if the discharge were left 5 tons higher than at point X? This gives DIS = tons, TEC = M$ , TAC = M$ , and the combined cost to the economy is M$ .

(iv) If the welfare criterion is to minimize the total cost to society, what do you conclude?

e. Finally, consider the issue of marketable permits. A permit grants the holder the right to discharge 1 ton per year. As awful as this may sound, let’s judge the scheme on its merits.

(i) The government issues X permits to each mill, where X is the optimal discharge level per mill that you identified above. Assuming that the permits can be enforced (which is no harder than enforcing direct controls), how does the total amount of pollution compare with the optimum found above?

(ii) After receiving the initial allotment of permits, you discover a way to clean up effluents at your mill for M$150 per unit of DIS, while at my mill it costs M$210. Explain why there is an incentive now for us to trade permits. Will you sell permits to me, or vice versa?

(iii) What is the effect of our transaction on the total discharge of effluents? What is the effect on total costs to society?

(iv) Does the presence of marketable permits provide a profit incentive for you to seek out less expensive means to reduce pollution to the allowable level? Explain.

5. Net national product (NNP) is defined as gross national product (GNP) less depreciation of what the textbook calls made capital, as distinct from human capital or natural capital. Depreciation is also the difference between gross and net domestic product (GDP and NDP), as well as gross and net investment. Depreciation is an accounting measure of the extent to which assets wear out and need to be replaced to sustain production capacity. It is often measured using the straight-line method, which means that an asset lasting 15 years wears out at a rate of 1/15 per year. This is a simple approximation to the true rate of depreciation.

a. You buy a truck for $30,000 and estimate that it has a useful life of 10 years. Under the straight-line method, the depreciation rate is percent per year and the annual value of depreciation is $ .

b. Suppose that GNP per capita in Bangweulu is $400.

(i) If the capital-output ratio for the economy is 2.0, then the capital stock is $ per capita.

(ii) The capital stock has an estimated average life of 20 years. Using straight-line depreciation, this means that the depreciation rate is percent per year.

(iii) The annual value of depreciation is $ per capita. So net national product (NNP) per capita is $ . NNP is percent of GNP. This is fairly typical.

(iv) To produce this output, the economy digs up mineral resources equivalent in value to 6 percent of GDP, and soil loss from agriculture is estimated at 4 percent of GDP. Deducting this depletion of natural capital, the adjusted net national product (ANNP) is $ per capita.

(v) To maintain its capital stock and ensure the sustainability of the economy’s capacity to generate income, total investment in Bangweulu must be no less than percent of GNP.

c. Economist Robert Repetto and associates estimated resource depletion and degradation in Indonesia during the early 1980s. Table 20 –2 shows their results, with line 3 added to account for depreciation of made capital (assuming conservatively that this depreciation amounts to 5 percent of GDP).

Click here for Table 20-2

(i) Calculate the growth rate of GDP and fill in line 2 in the table.

(ii) Subtract depreciation from GDP to get net domestic product (NDP), and fill in line 3.

(iii) Find the sum of lines 5a, 5b, and 5c to obtain total depletion of natural capital, and fill in line 5d.

(iv) Deduct the depletion of natural capital from NDP to get adjusted net domestic product (ANDP); fill in line 6.

(v) Then calculate the growth rate of ANDP and fill in line 7.

(vi) How does the growth rate of ANDP relate to the growth rate of GDP? Explain.

d. Look at Indonesia’s investment performance, before and after taking into account depreciation and depletion.

(i) In Table 20–2, deduct depreciation from gross domestic investment (GDI) to get net domestic investment (NDI). Then deduct depletion of natural capital to get the adjusted net domestic investment (ANDI). Fill in lines 9 and 10.

(ii) To gain perspective, compute GDI and ANDI as percentages of GDP and ANDP, respectively. Fill in lines 11 and 12.

(iii) Compare the stories told by lines 11 and 12. What do you conclude about investment performance in Indonesia, relative to gross domestic product?

(Note: Keep in mind that the exercise has used an arbitrary basis for determining the depreciation of made capital.)

6. Optional: How does a debt-for-nature swap work? In 1987 Bolivia had just conquered hyperinflation, but one legacy of previous mismanagement was an external debt of $5.8 billion, or $875 per capita (compared to per capita GNP of $530). Because of Bolivia’s poverty and the austerity required to curb inflation, the government could not afford the luxury of funding national park services. Enter an international conservation organization (ICO) that is interested in preserving Bolivian forests. The ICO sees room for a deal.

a. Since the country was virtually insolvent, international creditors were selling Bolivian debt obligations for about 15 cents on the dollar.

(i) Therefore, the ICO can purchase $20 million of Bolivian debt from an overseas bank for $ million.

(ii) The $20 million debt obligation bears an interest rate of 9 percent, so Bolivia owes $ million per year in interest payments to whomever holds the debt paper. (If a payment is missed, the interest charge still accrues as additional debt which Bolivia would have to deal with in the future.)

(iii) At the prevailing exchange rate of 2.05 Bolivianos per dollar, the government’s annual interest charge for this debt obligation equals B million.

b. The ICO approaches the government of Bolivia with the following deal.

(i) We will buy your $20 million debt obligation, feed the legal documents through a shredder, recycle the scraps of paper, and forgive the debt. This will save you $ million per year, or B million, in interest charges alone.

(ii) In return, we ask you to commit B2 million per year to pay for management of a designated national park. Compared to the interest charges on the debt, you save B million per year.

(iii) By shredding your debt obligation, we also reduce the debt principal that you owe by the equivalent of B million—not to mention future legal fees for negotiating with the bank on possible debt relief. Of course, the deal also leaves you with a well-managed national park for future generations to enjoy.

c. This is a debt-for-nature swap. Why does the ICO prefer the swap over simply giving an equal dollar amount to Bolivia to finance management of the national park?

(Hint: The yield on dollar investments is less than the interest rate on Bolivian debt, which the ICO purchases at a deep discount. Second hint: Local responsibility.)

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