Published on 2 Feb 2013

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Chapter 10 Analytical Problems Answers

1. At the socially efficient level of emissions (E*=50), the total payments are $12000

(the TAC of $ 4500 plus compensation to salmon fishery of $7500). To show that

total payments are minimized, consider increasing emissions to 51. The firm saves

TAC of $295.5 but expects to pay additional compensation of $300 equal to the

increase in TD to the fishery. The total payments therefore increase by $4.5. As the

MD is greater than the TAC, for any increase in emissions above E*, the TAC saved

must be less than the compensation paid and so total payments must increase. Going

the other way, if emissions are reduced to 49, the additional TAC incurred is $305,

while the damages and hence compensation saved is $297. Hence total payments

increase by $8. As the MD is less than the TAC for any level of emissions below E*,

the TAC incurred must be greater than the compensation saved and hence total

payments must increase. It then follows that the only level of emissions for which

total payments cannot be decreased is E*.

This question provides an opportunity for more mathematically inclined students to

practice their calculus. Integrating, we have

Total payments (TP)= 800E-5E2 + 32000-3E2

To minimize total payments, the first order necessary condition is:

dTP/dE=800-10E-6E=0 (MAC = marginal compensation), giving E*= 50.

The second order sufficient condition for a local minimum is satisfied as

d2 TP/dE2= -16 < 0. Hence E* = 50 is the level of emissions that minimizes total

payments by the firm.

2. Total surplus to society (where society is represented by the two firms) is found by

equating MAC to MD. 8E=800-10E gives E*= 44.44 tonnes. MAC(44.4) = $355.55.

If the fishery initially has the property rights to a clean environment, the net gains

from trading are (800×44.4)/2 = $17,777.6. If the chemical factory initially has the

property rights to pollute, the net gains from trading are (35.55×640)/2=$11,377.76.

In the text book example where the MD curve was less steep (MD= 6E), the gains

from trade were $7200 if the chemical firm had initial property rights to pollute and

$20,000 if the fishery had initial property rights to a clean environment. In both cases

the gains from trade are larger if initially the fishery has property rights to a clean

environment. This reflects the fact that if the fishery initially has the rights, then the

initial allocation is further from the socially efficient allocation than if the chemical

plant initially has the property rights. The social surplus gains have a higher variance

in the text book case because the marginal damage curve is flatter, so that total

damages are lower for a given level of emissions. The socially efficient level of

emissions falls. Starting with property rights for a clean environment, in the text

book example the fishery has less to loose in damages per unit emissions by trading

property rights, although the socially efficient level of emissions has risen. The

fishery suffers damages of $7500 in textbook example and $7901 with steeper MD

for a net loss in surplus of $401 in text book example. The chemical plant has more

to gain as the socially efficient level of emissions is higher. On net, the gains are

higher than the losses relative to the case with the steeper MD curve. In the case