# ECON 260 Lecture Notes - Maxima And Minima

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Published on 2 Feb 2013
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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
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