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Optional: Five Calculus Proofs of Marginal Principles Used in AEM 2500, 2010
It appears that several students are keen to understand the mathematics behind some of our
“Marginal” solutions used in AEM 2500.
Below I provide some very simply calculus-based
demonstrations of some of the major MB = MC type results we use.
THESE ARE OPTIONAL AND
NOT REQUIRED FOR THE COURSE.
1.
The First Equimarginal Principle: Society maximizes net economic surplus at the
quantity (Q) at which Marginal Benefits (MB) = Marginal Costs (MC).
Society’s objective is to maximize Net Economic Surplus (NES) = Total Benefits (TB)
minus Total Costs (TC). Both TB and TC are functions of Q, and will be written as TB(Q)
and TC(Q) respectively – I will use this type of notation throughout this handout.
The
solution is to set up the objective function, take the derivative with respect to Q, and set
this derivative equal to zero (which identifies the maximum).
±²³
ொ
ܶܤሺܳሻ െ ܶܥሺܳሻ
Set first derivative equal to zero
ௗ்ሺொሻ
ௗொ
െ
ௗ்ሺொሻ
ௗொ
ൌ Ͳ
Rearrange and solve
ܯܤ ൌ
ௗ்ሺொሻ
ௗொ
ൌ
ௗ்ሺொሻ
ௗொ
ൌ ܯܥ
Graphically, this point is indicted where the marginal benefits (
demand
) and marginal
costs (
supply
) curve cross, with the grey shaded area depicting the maximum net
benefits (NES) in a market for Q.
This basic principle can be translated into MB and MC
curves for pollution, pollution abatement etc.
Objective function:
$
MC
MB
Quantity

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2.
For a linear (inverse) demand curve of the form P = b – mQ, the marginal revenue
(MR) curve has the same intercept (b) but twice the slope (-2m).

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