Question

5. Suppose that there is an industry with two firms (1 and 2) with marginal abatement cost curves as follows MAC1 1,000 2E1 MAC2 500 E2 E1 and E2 represent the respective emissions of each firm in tonnes [3] (a) Solve for initial emissions for each fim and rewrite both of the MAC equations as functions of abatement [6] (b) Derive the Industry MAC curve for this industry . Briefly explain each of the steps you take to find the industry curve 3] (c) Calculate the total abatement cost by firm and for the industry as a whole if the regulatory authority specifies that the industry abates 300 tonmes of emissions by having each firm abate one-half of this target (so each firm must abate 150 tonnes) [2] (d) Is this allocation of abatement cost-effective? Briefly explain why or why not. [3] (e) Briefly state what the cost-effective distribution of an industry abatement target of 300 tomes would be between the two firms. Explain your answer [3] (f) Compute the total abatement cost for each firm and the industry as a whole under the cost-effective allocation of 300 tonnes of industry abatement as stated in part (e). Briefly explain how and why this differs from the industry TAC obtained in part (c)?

Answer 1

a) For rewriting both of the MAC equations as functions of abatement

2E₁ =1000- MAC₁

Dividing throughout by 2

E₁ =500- 1/2MAC₁

and also E₂ =500- MAC₂

Now if there is no abatement ,both E₁ and E₂ will be the highest at 500

b) For finding the industry MAC curve,we have to combine the MAC curves to find an overall marginal abatement cost curve such that the Marginal cost of abatement of a unit of emission is the least at any particular level of emission. For abatement to happen most efficiently,we will get MAC₁=MAC₂ at each level of abatement (it can be explained as if one was lower than the other, we will keep abating from that firm until its MAC increases to the level of the other and so on,eg. if E₁ was reduced to 499 first,we will have MAC₁=2 and MAC₂=0,now since second is lower we will have to decrease emission from that until it arrives equal to E₂=498 where MAC₂ becomes equal to 2)

So MAC₁=MAC₂ gives 1000-2E₁=500-E₂

E₂=2E₁-500

so that E=E_industry=E₁+E₂=3E₁-500

E₁=(E+500)/3

putting this in MAC₁ we get MAC₁= 2/3(1000-E)

MAC₁=MAC₂=MAC_industry=2/3(1000-E)

You can intuitively see this as when MAC₁=MAC₂=20 at E₁=490 and E₂=480,you will have MAC=2/3(1000-490-480)=2/3*30=20 i.e. cost of abating one more unit of emission is 20 for each firm and so the industry as a whole at this level of emissions

c) For firm 1 to abate 150 tonnes, we need to find summation of MAC₁ from E=500 to E=350

TAC=0+2+4+6....300=2(0+1+2+3+4...150)=150*151=22,650

For firm 2 to abate 150 tonnies, we need to find summation of MAC₂ from E=500 to E=350

TAC=0+1+2+3.....150=150*151/2=11,325

TAC for the industry as a whole=22650+11325=33,975

Note: You can also find this by finding the areas under MAC curves and doing definite integration with limits as TAC is just the area under MAC curve

d) This allocation of abatement is not cost-effective because one firm has had to bear more of the Cost of abatement. It would have been more efficient if each unit of emission had been abated at the least marginal cost possible from whichever firm it was possible.

e) For cost-effective abatement, we get that the Marginal cost of abatement should be equal for both the firms at each level of abatement

Thus as above, from equating MAC₁=MAC₂ we get E₂=2E₁-500, also such that E₁+E₂=3E₁-500=(1000-300)=700 (since 300 tonnes have to be abated)

so E₁=1200/3=400 and E₂=2*400-500=300

so this abates 100 from E₁ and 200 from E₂ which minimizes the MAC at each level of emission i.e. abating 2 from E₂ for every 1 from E₁

f)For firm 1 TAC will be sum of MAC from E₁=500 to 400

=0+2+4...200=2(0+1+2+....100) =100*101=10100

For firm 2 TAC will be sum of MAC from E₂=500 to 300

=0+1+2+3+4....199+200= 200*201/2=20,100

TAC for the industry will thus be 10100+20100=30200 which is lower than the 33,975 since this split of abatement between the firms is cost-effective unlike the previous one and the MAC at each level is the least possible unlike in the previous one

Hope it helps,do ask if you have any doubts