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Solutions to Problem Set 2

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York University
ECON 3340
Andrea Podhorsky

Chapter 12 #1 (a) + (b) See Figure below. Notice the aggregate marginal savings curve. Its derivation and an explanation follow in part (c). (c) It is straightforward to calculate aggregate marginal damage. Since pollution is non - rival, aggregate dama ge is the vertical sum of the individual marginal damage functions: MD =MD I+MD =e2 e=2e Industry aggr egate marginal savings is slightly more complicated. How much is saved between the two firms for a given level of total emissions? This will require a horizontal summation of marginal savings curves, and the answer to this question depends on how emissions are distributed among the firms. Efficiency would suggest that this distribution take place as to maximize total savings. The necessary condition for thi s is the equimarginal principle: marginal savings should be equated across firms to maximize savings from pollution (minimize total costs of abatement). This is imposed here in deriving the aggregate marginal savings function: el= 5 –MS 1 2e 2=8 -MS2 then e 24 -(1/2) MS 2 Equimarginal principle: MS1 = MS 2 = MS e +e = e = 5 -MS+4 - (1/2)MS l 2 T eT=9-(3/2)MS then MS =6- (2/3) e T This is the effective aggregate level marginal savings function when emissions are distributed cost effectively. The optimal level of emissions is found where this marginal savings function equals aggregate marginal damage: 6- (2/3) e=2e then e*=2.25 The Pigouvian fee that supports this optimal level of emissions is equal to the level of marginal savings (and marginal damage) at the opt imum: fee* =6-2/3(2.25) =2(2.25) =4.5 The last part of the solution is to determine the individual firms' levels of emissions at the optimum. These can be found by setting the original marginal savings functions equal to the Pigouvian fee: 5-e =4.5 then e = 0.5 1 1 1 8-2e 24.5 then e =27/4 Verify that these sum to the optimal total emissions: e +1 =224+7/4=2.25 #2. Consider the market for electricity. Suppose demand (in megawatt hours) is given by Q=50- P and that the marginal private cost of generating electricity is $10 per megawatt hour (P is in the same units). Suppose further that smoke is generated in the production of electricity in direct proportion to the amount of electricity generated. The health damage from the smoke is $15 per megawatt hour generated. a) Suppose  the  electricity  is  produced  by  an  unregulated  monopolist.  What  price  will   be  charged,  and  how  much  electricity  will  be  produced?   b) In  part  (a),  what  is  the  consumer  surplus  from  the  electricity  generation?  What  is   the  net  surplus,  taking  into  account  the  pollution  damage?   (a) Depicted is a monopolist with linear demand and constant marginal private constant marginal external cost. If this firm is unregulat ed, it would maximize profit by choosing a quantity at which marginal revenue is equal to marginal private cost. For this problem: P=50-Q= AND MR=50-2Q MPC = 10 Profit max 50 -2Q=10 then Q`=20, and P' =50-20=30 (b) Refer to the Figure. Ignoring the externality for t he moment, the consumer surplus associated with this market is represented by the shaded area under the demand curve, and above the price: CS = 1/2(50 - 30)20 = 200 Damages from pollution resulting from the generation of electricity are represented by the stippled area: D= 15*20=300 Net consumer surplus from t his market is therefore negative: Net surplus = 200 - 300 = -100 We see that there is nothing that constrains a market from having a negative total effect on consumers. How can this be, if consumers have the option to not participate in a free market? The answer is that consumer surplus from the market itself [part (a)] is optional for any one consumer: they can purchase electricity or not. But the damages from pollution are not optional for any one consumer: if everyone else purchases electricity, I su ffer from the pollution 2 whether I buy electricity or not. The private benefit from purchasing electricity is greater than the private cost of purchasing electricity. This outcome results because of the difference between the private costs of purchasing ele ctricity (payment plus marginal pollution damage to one consumer) and marginal social costs of purchasing electricity (payment plus marginal pollution damage to all consumers). This is the nature of externalities. Chapter 14, #3 a) The damage function given in the problem is written: D(S) = 0.01S Where S is grams of sulfates deposited. (b) Given in the problem is the transfer equation: S = E E 3 E D Where E Eepresents tons of sulfur emitted in England, and EDtons of sulfur fitted in Denmark. The transfer coefficients areEa =l andDa =3 (d) Let E and qDrepresent emissions reductions in England and Denmark, respectively. The goal at this point is to achieve qE+ q D 12 At minimum emissions control costs between the two countries. We know that this is accomplished where the marginal control costs are equal in the two countries: q D 2 q E The two equations in the two unknowns (q Dnd q )Eimply qD=8 and q =4E (e) Now we still want to reduce total emissions between the two countries by 12 tons, but to do so in a way that accounts for the different transfer coefficients. The logic is that reductions in Denmark are more valuable to the Swedes than reductions in England because Denmark is so much closer. This would suggest equalizing the marginal costs of reducing sulfate deposits in Sweden. As discussed in the text, this is the marginal co
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