Extraction Path Recently a large cave with about 200.000 tons of guano was rediscovered in north-eastern Zimbabwe (this is truel) Suppose that the government wishes to dig up and sell the guano over a period of three years, and that it faces the following demand functions: In year Qd = 161.000 – 100 P In year Qd = 180.000 – 100 P In year Qd = 190.000 – 100 P The marginal extraction cost is $200/ton, the interest rate is 10%, and there is a fixed, up-front cost (access roads, drying plinths, etc.) of $40 million a How much guano should be mined in each year (the “optimal extraction path”)? b Is this proposed project profitable? Explain, by setting up the cost-benefit analysis c How would the answer to a be changed if the deposit has only 150.000 tons? a Assume that marginal extraction costs are zero Sine user cost = P – MEC we have UC = P here We expect UC to rise with the rate of interest, so here we expect P to rise with the rate of interest, giving P2 = P1 (1 + 0,1) and P3 = P1 (1 + 0,1)(1 + 0,1) We also know that since the resource is to be used up in the three years, Q1 + Q2 + Q3 = 200.000 Substitution gives 161.000 – 100.P1 + 180.000 – 100P2 + 190.000 – 100.P3 = 200.000 Replacing P2 and P3 gives 531.000 – 100P1 – 100P1.(1,1) – 100P1 (1,21) = 200.000 Whish may be solved to give P1 = $1.000 and so Q1 = 61.000 P2 = $1.100 and so Q1 = 61.000 P3 = $1.210 and so Q1 = 61.000 Note: If the MEC = 200, as the problem originally indicated, then we have that (P2 – 200) = 1,1 (P1 – 200) and (P3 – 200) = 1,2 (P1 – 200) Appropriate substitutions yield prices of $1018,73; $1100,60 and $1190,66 in the three years, and quantities sold of 59, 127, 69, 940 and 70,934 respectively [try the exercise!] b A cost-benefit analysis sets out the cash flow, which indicates the inflow, and outflow, of cash in each year These should then be discounted to give the net present value Here is how we would set it up for the mine, in the case where MEC = $200/ton The figures are in millions of dollars Year Year Year Inflows: Sales 60,23 76,98 84,46 Outflows: Investment 40,00 Extraction 11,83 13,99 14,19 Net cash lows 8,40 62,99 70,27 This is clearly a very profitable project, for the revenues cover the costs in year 1! Repeat the computations of part a to give P1 = $1.151,06 and so Q1 = 45.894 P2 = $1.266,16 and so Q1 = 53.384 P3 = $1.392,78 and so Q1 = 50.772 Assuming that the marginal extraction cost is zero here too Coal and External Costs a Suppose that the demand curve for coal is given by Qd = 480 – 2P, the supply curve by Qs = 10p, and that the industry is competitive Graph the demand and supply curves, and find the market price and quantity b Expert estimate that for every ton of coal used, the rest of society has to bear costs of $15/ton (= “marginal external costs”) So the government puts a $15/ton tax coal Draw the new supply curve, and show and find the price which consumers of coal will now have to pay, and the quantity produced c Briefly comment on the pros and cons of using each of the following to deal with pollution caused by using coal: i) Taxing the output of coal ii) Taxing the pollution caused by using coal iii) Regulating/limiting the output of coal-generated pollution directly a See diagram here Qd = 480 – 2P and Qs = 10P In equilibrium demand equals supply, so Qd = Qs so 480 – 2P = 10P, giving P = $40, Q = 400 b When the consumer pays Pd, the producer gets Pd – tax = Pd – 15, which is the “supply price” (Ps) So Qs = 10 (Pd – 15) Set this equal to Qd, and solve to find P = $52,50 and Q = 375 Note that although the tax is $15, consumers pay only $12,50 more than before; suppliers absorb the other $2,50 c Tax coal output Easy to But puts a burden on clean coal as well as dirty (i.e high sulfur) coal And provides no incentive to install pollution control devices Tax pollution Economists prefer this; it provides the right incentive to use cleaner fuel, and/or clean up Problem: how set the tax rate Regulation The U.S relies mainly on this (e.g no-lead gasoline, etc.) Can be effective, but often at high cost No incentive to continually improve Forests a You plant a tree which grows 50% in the first year, 49% in the second year, and so on – i.e it grown 1% less quickly in each succeeding year The real interest rate you face is 6% When should you cut the tree? Explain You are interested in harvesting trees from a 15 square kilometer area of tropical forest Two types of trees are worth cutting; species A trees can be sold for $100/cubic meter and species B for $70/cubic meter There are 2000 stems of species A and 5000 of species B which are potentially worth harvesting; the rest of the forest will be left more or less intact, so this level or harvesting is considered sustainable Your overhead fixed costs are $ 80.000 (to build a logging road, buy some equipment, etc.); it then costs $50 per cubic meter harvested i) What is the maximum amount you would pay for permission to work this concession (i.e what is the rent generated by this forest)? ii) How much tax revenue would the government get if it put a tax of 30% on the selling price of the wood? (Assume you cannot sell the wood for more than $100 and $70 respectively) Explain iii) How much tax revenue would the government get from a 45% tax on the profits of the company? Is this a better or a worse tax than the tax in part ii)? why? a The tree grows 50% in the first year, 49% in the second, and so on It will grow at 6% in year 45, at which time it should be cut; after that your money would grow more quickly in the bank (at 6%) than in the form of the tree (at 5%, 4%, etc.) Algebraically, Tree growth = (51 – year); cut when tree growth = 6%; so (51 – year) = Therefore year = 45 b (i) The number s are as follows: Specie s A B Price Output Revenue Extraction cost Gross profit $100 $70 2.000 5.000 $200.000 $350.000 $100.000 $250.000 Less overhead = net profit $100.000 $100.000 $80.000 $120.000 You would pay a maximum of $120.000 for a permit to cut; this is the resource rent (ii) A tax of 30% on species would leave an after-tax price of $49, which is less than the extraction cost of $50 per cubit meter Thus you would not cut this species The tax would leave an after tax gross profit of $20 (= $100 - $30 $50) on species 2, for a gross profit of $60.000 But this is not enough to cover the overhead costs Thus the firm would shut down, and the government would collect no revenue (iii) A tax of 45% on $120.000 would yield $54.000, and would not discourage cutting the trees This is clearly preferable to the situation in (ii)