清华大学环境与资源经济学作业答案.pdf

1 Tsinghua University Prof. Jing Cao Department of Economics Spring 2014 Problem Set 1 ENVIRONMENTAL AND RESOURCE ECONOMICS Problem Set #1 Answer Key Questions: 1. SHORT ANSWER QUESTIONS (40 points; 8 questions; 5 points each) a) Define static efficiency and show what it is using a graph. (Be sure to label all of the important features of the graph.) Answer: An allocation of resources is said to satisfy the static efficiency criterion if the net benefit from the use of those resources is maximized by that allocation. The net benefit is the excess of benefits over costs resulting from that allocation. Net benefits are maximized when the marginal benefit is equal to the marginal cost (i.e. at the allocation defined by q* and P*). b) Define opportunity cost. Answer: For environmental services, the opportunity cost is the net benefit foregone because the resources providing the services can no longer be used in their next most beneficial use. Price Quantity Marginal Cost Demand Curve P* q* Net Benefit 2 c) What is the present value of 1,000 yuan which is to be paid to you two years from today if the discount rate is 10%. What would the present value be if the discount rate were increased to 15%. (Please show your calculations.) Answer: yuanTrPV45.826 %)101 ( 1000 )2;10;1000( 2 yuanTrPV14.756 %)151 ( 1000 )2;15;1000( 2 d) Can there be positive externalities? If so, give an example. Yes. Some examples include: The pollination of an apple orchard by a bee keepers bees. Basic research, produced by a government laboratory, but used by others. A beautiful garden maintained by one person, but also enjoyed by others. e) If an externality is reflected in the price of a good, does it cause a market failure? What is the name for this type of externality? No, because the external effect is internalized in the market price. This is called a pecuniary externality. f) What is the difference between a “common property” resource and an “open access” resource? Common property resources are those which are owned in common rather than privately. Entitlements to use common property resources may be formal, protected by specific legal rules, or they may be informal, protected by tradition or custom. Open access resources can be exploited on a first-come, first-serve basis, because no individual or group has the legal power to restrict access. In the presence of sufficient demand, unrestricted access may cause resources to be overexploited. g) What is the “free rider” problem? As described above, the free rider problem occurs because of the nonexcludability characteristic of public goods. The same amount of the good will be available whether or not an individual pays for it, so there is an incentive to free ride. h) What is the Coase Theorem? When might this type of solution to environmental problems be impossible? The Coase Theorem says that as long as negotiation costs are negligible and the affected consumers can negotiate freely with each other (i.e. when the number of parties is small), a court could allocate an entitlement to either party and an efficient allocation would 3 result. However, this type of solution will usually not work when the number of parties is large, transaction costs are high, complete information is not available to both parties, there are large wealth effects, or when some parties have a potential to act strategically. 2. Assume that the inverse demand function for the petroleum reserve in East China Sea isqP5 . 010, where Pis the price of the resource and q is the quantity. Further assume that the total cost function for the extraction of the oil resources is qTC2. (25 points) a. How much of the resource would be supplied in a static efficient allocation? What would be the size of the net benefit (in yuan)? (5 points) Answer: Since the extraction of the oil reserve is qTC2, now take the derivative, we can derive marginal cost function: 2 dq dTC MC To get the static efficient allocation, marginal benefit, which is the price, should be equal to the marginal cost: 16* 85 . 0 05 . 08 25 . 010 q q q MCqP The net benefit is equal to the area under the demand curve and above the marginal cost curve. Price Quantity Marginal Cost Demand Curve 16 Net Benefit 10 2 20 4 6416)210( 2 1 NB b. Now assume that we are concerned with two (and only two) periods and that the demand and cost conditions will be the same in both periods. (5 points) 11 11 2 5 . 010 qTC qP 22 22 2 5 . 010 qTC qP Further assume a discount rate of 10%. If there were a total of 40 units of the resource to be spread across the two periods, what would be the dynamically efficient allocation (quantity) in each period? Answer: Note that only 16 units of the resource were demanded in the previous part of the problem. So if all of the conditions stay the same in both of the two periods we are dealing with, total consumption would be only 32 units (2 x 16). Since the total amount available for both periods is 40 units, there is no scarcity. Therefore: 16 * 2 * 1 qq c. Now assume that there are only 20 units of the resource to be spread across the two periods. What would be the dynamically efficient allocation in each period? What would be the efficient price in each period? What would be the marginal user cost in each period? (5 points) Answer: First we can specify an equation that defines the present value of the marginal net benefit in each period. For Period 1 this is: 11111 5 . 0825 . 010qqMCPR 5 We can see this from the following graph: For the second period, the equation that defines the present value of the marginal net benefit is: 2 222 2 4545.2727. 7 %101 25 . 010 )1 ( )(q q r MCP RPV Note that when q2 = 0, PVR2 = 7.2727. And when PVR2 = 0, 16 4545. 2727. 7 2 q . 7.2727 16 Period 2 q2 PVR2 0 8 16 Period 1 R1 q1 0 6 If we put the net present benefit curves for the two periods on the same graph, we can see that the allocation ( * 2 * 1,q q) that will maximize the total net present benefit (the area under the two curves) will be defined where the two curves intersect. 16 16 Quantity in Period 2 Quantity in Period 1 0 20 20 0 8 7.2727 R1 PVR2 q q 1 2 7 Now we need to calculate the point where the two curves intersect: )(4545.2727. 75 . 08 2211 RPVqqR And in our example, 20 21 qq, so, )20(4545.2727. 75 . 08 11 qq 7147. 92853.1020 ,2853.10 * 2 * 1 q q Now that we have determined the optimal quantities (= the dynamically efficient allocation) to extract in each period, we can calculate the optimal prices. Here we just plug the optimal quantities into the inverse demand functions: 8574. 42853.105 .10 1 P And for the second Period: 1427. 57147. 95 .10 1 P The marginal user cost is defined as: ttt MCPR So for Period 1: 8574. 228574. 4 111 MCPR And for Period 2: 1427. 321427. 5 222 MCPR d. What is the relationship between the marginal user cost in period 1 and period 2? What is the relationship between the marginal user cost and the price in each period? (5 points) Answer: 1 . 18574. 2/1427. 3 1 2 R R =1+0.1 And here we can recall that “.1” (10%) is the rate of interest or the discount rate, r. So the relationship is: )1 ( 12 rRR The marginal user cost (or royalty) rises at the rate of the opportunity cost of capital (or the interest rate or the discount rate). This is known as Hotellings Law or Theorem. Which also implies that: 8 )1 ( 2 1 r R R or )( 21 RPVR (I.e. that the present value of the royalty is the same in both periods.) And as previously discussed, the relationship between the marginal user cost and price in each period is: ttt MCPR e. Now assume the same demand and cost conditions as those given above and that there are again only 20 units of the resource to be spread across the two periods. Now, however, the discount rate has been raised to 15%. What would be the dynamically efficient allocation, price, and marginal user cost in each period? What is the relationship between the marginal user cost in period 1 and period 2? (5 points) Answer: The calculations are the same as in part c. However, now substitute r = 15% for r = 10%. Note that the equation for the present value of the net present benefit is now: 15.1 25 .10 )( 2 2 q RPV So when we equate the two curves: )(4348.9565. 65 .8 2211 RPVqqR The dynamically efficient allocation is: 4188.10 * 1 q and 5812. 9 * 2 q The optimal prices are: 7906. 4 * 1 P and 2094. 5 * 2 P The marginal user costs are: 7906. 2 * 1 R and 2094. 3 * 2 R And the relationship between the marginal user cost in period 1 and period 2 is again: %15115. 17906. 2/2094. 3/ 12 RR or that )1 ( 12 rRR 9 3. Suppose the inverse demand function (= marginal benefit function) for an environmental resource is P = 10 - 1q. Further suppose that the marginal cost function is MC = 1q. It can be shown that the total benefit from using the resource is given by the equation TB10q 1 2 q2 and the total cost is given by the equation TC 1 2 q2. Now plot the curves for the marginal benefit and marginal cost for the range q = 0 to q = 10 on one graph. Then plot the curves for the total benefit and total cost on a second graph. Finally show the relationship between the two graphs. (20 points) The marginal benefits (MB), marginal costs (MC), total benefits (TB), and total costs (TC) are calculated below. The net benefit (NB) can also be calculated as the difference between the total benefits and total costs. q MB = 10 - 1q MC = 1q TB10q 1 2 q2TC 1 2 q2 NB 0 10 0 0 0 0 1 9 1 9.5 .5 9 2 8 2 18 2 16 3 7 3 25.5 4.5 21 4 6 4 32 8 24 5 5 5 37.5 12.5 25 6 4 6 42 18 24 7 3 7 45.5 24.5 21 8 2 8 48 32 16 9 1 9 49.5 40.5 9 10 0 10 50 50 0 The graphs for marginal benefits and costs and total benefits and costs are plotted on the next page: Graphs for marginal benefits and costs and total benefits and costs: 10 4. The country of SEMs primary enemy is the heavily-armed and aggressive country of Gov. SEMs national defense can be provided by a concrete wall built around its borders. The demand for protection from Gov comes from three distinct groups, each with 100 members. The marginal annual values placed on protection by typical individual members of each group are: MV1 = 200 - .1W MV2 = 150 - .2W MV3 = 100 - .5W where W is the height of the concrete wall measured in feet. The cost of SEMs protective wall is $37,000 per foot of height for concrete and labor. Its crummy concrete, so the wall must be rebuilt every year! Suppose the government of SEM provides all national defense. What is the optimal height of the wall? (15 points) National defense