assignment paper and solution problem set 1
Suppose the inverse supply and demand functions in the borogove market
are given bythe followingfunctions:
Demand: P = 150 â 6Q + 0.02Q2for 0 < Q < 25 Supply: P = 50 + 0.75Q + 0.02Q2for 0 < Q <25 where Q is the units of borogove produced or consumed per year, and P isdollars/unit. Suppose also that the production of borogove emits pollution. The externality of the pollution associated with each additional borogove has rising marginal damages suchthat: MED =0.12Q with Q defined as before. Assume that the rate of emissions per unit of borogove is fixed (or invariable). a. What level of borogove will the unregulated market produce? What will be the priceof borogove in the unregulatedmarket? b. What is the socially optimal level of borogoveproduction? c. What is the potential net social gain from correcting the pollution externality and producing borogove at the optimallevel? d. Suppose the legislature passes a regulation that shifts borogove production to the socially optimal level. What is the net present value of correcting the externality over the first twenty years of the program? Use a social discount rate of5%. Problem2 Consider a case where there are three firms contributing to the pollution of the Cabernet airshed. Their demand functions for the right to pollute are given asfollows: Firm A: Q = 64 â P Firm B: Q = 32 â 1/3P Firm C: Q = 30 â2/3P where Q is the amount of pollution in tons per year and P is the willingness to pay in dollars perton. a. Draw the individual and aggregate demand functions for the right to pollute in the Cabernet airshed (Hint: Note that the aggregate is not a simple summation. If this problem gives you trouble, you may want to review the tutorial on aggregating cost and demand curves that is on the class website). Emissions Demand Curves for Individual Firms and IndustryAggregate b. What is the total amount of unabated air pollution in thisairshed? c. Now draw the marginal abatement cost (MAC)curve. d. Suppose that the Cabernet environmental agency determines that the MED function for emissions is MED = 50 for the entire range of emissions. Draw the marginal abatement benefit (MAB) curve onto the MACcurve e. What is the efficient level of abatement in this case? The efficient level ofemissions? f. What is the totalcost of abatement to reach the efficient level ofemissions? Aggregate Marginal Abatement Cost Curve And Marginal Abatement Benefit Curve with Total AbatementCost g. What is the allocatively efficient level of abatement for each firm? What is the total cost of abatement to eachfirm? Problem3 Adapted from problem 5, p111 intext The Great Fracking Natural Gas company has just established a new well in the ranching community of Grassville, USA. It turns out that some of the chemicals used in theFracking process are contaminating the groundwater. The total damage to ranchers due to livestock losses from drinking the water is 50p2, where p is the level of pollution in this case.The total damage to families who must travel to town and purchase clean water each monthis 300p2. The marginal savings to Great Fracking for being allowed to pollute is 4000 â40p. a. Find the aggregate (including both types of consumers) marginal damage for the public bad. b. Graph the marginal savings and aggregate marginal damage curves with pollution on the horizontalaxis. c. How much will Great Fracking pollute in the absence of any regulation or bargaining? What is this societyâs optimal level ofpollution? d. Starting from the uncontrolled level of pollution calculated in part (c), find the marginal willingness to pay for pollution abatement, A, for each consumer class (abatement reduction in pollution; zero abatement would be associated with the uncontrolled level of pollution). Find the aggregate marginal willingness to pay for abatement. e. Again starting from the uncontrolled level of pollution, what is the firmâs marginal cost of pollution abatement? What is the optimal level ofA? Are the problems of optimal provision of the public bad (pollution) and the public good (abatement) equivalent? Explain why or whynot. Problem4 Consider two adjacent businesses: a music studio and a tutoring center. Rock music from the studio can be heard in the tutoring center, causing concentration difficulties for its patrons. The table below shows the total value of each business at various levels of noise. Assume the owners of the two organizations can bargain costlessly. a) If the music is legal, what is the outcome of the situation, in terms of the level of noise? What is the potential range of payments that might be made and by whom? b) If the music is illegal (restricted to 10), but enforcement is only in response to complaints, how does your answer in part (a)change? Problem5 Problem 2, p114 intext Consider a pollution problem involving a paper mill located on a river and acommercial salmon fishery operating on the same river. The fishery can operate at one of two locations: upstream (above the mill) or downstream (in the polluted part of theriver). Pollution lowers profits for the fishery: without pollution, profits are $300 upstreamand $500 downstream; with pollution, profits are $200 upstream and $100 downstream. The mill earns $500 in profit, and the technology exists for it to build a treatment plant at the site that completely eliminates the pollution, but at a cost of $200. There are two possible assignments of property rights: (i) the fishery has the right to a clean river and (ii) themill has the right to pollute theriver. a. What is the efficient outcome (the maximum of total jointprofit)? b. What are the outcomes under the two different property rights regimes, when there is no possibility ofbargaining? c. How does your answer to (b) change when the two firms can bargaincostlessly? Problem6 Problem 6, p115 intext A beekeeper and a farmer with an apple orchard are neighbors. This is convenient for the orchard owner since the bees pollinate the apple trees: one beehive pollinates one acre of orchard. Unfortunately, there are not enough bees next door to pollinate the whole orchard and pollination costs are $10 per acre. The beekeeper has total costsof TC = H2+ 10H + 10 and marginal costs MC = 10 + 2H where H is the number of hives. Each hive yields $20 worth ofhoney. a. How many hives would the beekeeper maintain if operating independently of the farmer? b. What is the socially efficient number ofhives? c. In the absence of transaction costs, what outcomes do you expect to arise from bargaining between the beekeeper and thefarmer? d. How high would total transaction costs have to be to erase all gains frombargaining? Problem7 Adapted problem 1, p132 intext Assume an economy of two firms and two consumers. The two firms pollute. Firm one has a marginal savings function of MS1(e) = 18 â 2ewhere eis the quantity of emissions from the firm. Firm two has a marginal savings function of MS2(e) = 12 â e. Each of the two consumers has marginal damage MD(e) = 0.5e, where ein this case is the total amount of emissions the consumer is exposedto. a. Graph the firm-level and aggregate marginal savingsfunctions. b. Graph the aggregate marginal damagefunction. c. What is the optimal level of pollution, the appropriate Pigouvian fee, and emissions from eachfirm? Problem8 Assume that there are two firms each emitting 40 units of pollutants into the environment for a total of 80 units in their region. The government sets an aggregate abatement standard of 30 units. The polluters' total abatement cost functions are asfollows: Polluter 1: TAC1= 450 + 4.6(A1)2+ 0.20(A1)3Polluter 2: TAC2 = 370 + 10.2(A2)2 +0.20(A2)3 where Ai= the amount of abatement by polluter I and costs are indollars. a. Suppose the government allocates the abatement responsibility equally such that each polluter must abate 15 units of pollution. Graphically illustrate this allocation and analytically assess the cost implications (i.e., use the above formulas in amathematical analysis). Your analysis should include the total cost to each firm, the marginal costto each firm, and the total cost tosociety. b. Now, assume that the government institutes an emission fee of $900 per unit of pollution. How many units of pollution would each polluter abate? Is the $900 feea cost-effective strategy for meeting the standard?Explain. c. What allocation of abatement responsibility will reach the abatement goal of 50 units at the lowest social cost? What will be the MAC for each firm and for society under this optimalallocation? Problem9 Suppose the state is trying to decide how many miles of a very scenic river it should preserve. There are 10,000 people in the city, each of whom has an identical inverse demand function given by p= 36 â 0.08q, where qis the number of miles preserved and p is the per-mile price he or she is willing to pay for qmiles of preservedriver. a. If the marginal cost (dollars per mile) of preservation is p= 2500 + 8q2, how manymiles would be preserved in an efficient allocation? Round to three decimalplaces. b. What would be the marginal value of saving the last mile up to the efficientallocation? c. What would be the total costs ofpreservation? Total cost is the area under the marginal cost curve up to the quantity produced (or in this case preserved). On the graph below, this is Area C. So to find this, integrate the marginal cost function between 0 and the efficient allocation, 31.777miles. The total cost function is thus 2500q + (8/3)q3. This gives us a total cost of d. How large would the net benefitsbe Problem10 The marginal private cost of the production of stooges is described by the following equation, where Q is the quantity of stoogesproduces. MPC = 8 +0.8Q Unfortunately, the production of stooges has a detrimental impact on the environment, estimated by the following damage function: MD =1.2Q The private market demand for the production of stooges is described by the following equation: MPB = 80 â4Q The government decides to set a Pigouvian tax to account for the environmental damages caused by stooges. What is the optimal Pigouvian tax? What is the expected government revenue generated from thistax? Problem11 Suppose the individual demands for wetlands (a pure public good) by the four citizens in a town are givenby Q1 = 10 - P Q2= 5 - 0.1P Q3= 7 â 0.2P Q4 = 12 -0.5P where Qiis the quantity of wetland demanded by individual i, p is individual iâs price(or willingness to pay) forwetlands. a. Diagram the individual demand curves and the societal demand curve forwetlands. b. Suppose there is a constant marginal cost of 65 to supply wetlands. What is thesocially optimal level ofwetlands? Problem12 Suppose that a chemical manufacturing plant is releasing nitrogen oxides into the air, and these emissions are associated with health and ecological damages. Economists have estimated the following marginal costs and benefits for the chemical market, where Q is monthly output in thousands of pounds and P is price perpound. Marginal Social Benefit [MSB]=50-0.4Q Marginal Private Cost [MPC]=2+0.4Q Marginal Environmental Cost[MEC]=0.2Q Your job isto: a) Find the competitive equilibrium quantity (Q0) and price(P0). b) Find the efficient equilibrium quantity (Q1) and price(P1). c) Draw a graph of this market, showing the MSB, MPC, MSC curves, Q0, Q1, P0, andP1. d) Find the dollar value of a Pigouvian tax that would achieve the efficient equilibrium. Problem13 Suppose there is a firm in the town of Oakley that pollutes the water of the townâs lake. The damages to the town from the pollution, and the cost to the firm of cleaning up the pollution, are shown in the followingfigure. a. What does the Coase Theorem tell us to expect if an alienable right to prevent pollution is given to thetown? b. What level of pollution will emerge from a Coasian bargain? Provide a numerical answer. c. What is the range of payments the firm might make to the town following a mutually agreeable bargain? Provide a numericalanswer. d. How does your answer change if the initial property right in pollution is given to the firm? Provide a numericalanswer. Problem14 An unregulated industry produces zappas and sells them in a competitive market. The inverse supply function for the firm in this marketis: P = 6 +2Q where P is the price per unit of zappa produced (dollars/zappa) and Q is the number of zappas produced per year. The elasticity of supply (Es) is 2.125. The elasticity of demand (Ed) is -4.25. The equilibrium price in the unregulated market is 34/3dollars/zappa. a. What is the function describing the demand in the unregulatedmarket? b. What is the consumer and producer surplus in the unregulatedmarket? Now suppose the production of zappas produces emissions that cause people to freak out. To address the emissions problem, suppose the government sets a regulation thatincreases the cost of production of zappas by 2dollars/zappa. c. What is the new equilibrium price and quantity? What assumption(s) did you make when determining thesevalues? d. Ignoring the benefits associated with the emission reduction, what is the change in consumer, producer, and socialsurplus?