The Welfare Theorem & The
Environment
© 1998, 2011 by Peter Berck
Outline
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Surplus as measure of consumer satisfaction
VC as area under MC
Competition maximizes Surplus plus Profit
Not true with “externality:” Pollution
Use of Tax to reach optimality
Use of Regulation to reach optimality
Willingness to Pay
• Willingness to pay is area under demand.
– demand price P(Q) is amount willing to pay for
next unit
– So total willing to pay for Q units is P(1) + P(2) +
...+ P(Q)
• lower riemann sum and an approximation
• the area under the demand curve between 0 and Q
units, which is the integral of demand, is (total)
willingness to pay
Calculating Total Willingness
70
60
50
Price
40
AREA
Demand
30
20
10
0
1
2
3
4
Quantity
5
6
7
Consumer Surplus
• Consumer surplus is willingness to pay less
amount paid
• Amount paid is P Q
Consumer surplus is willingness to pay
less amount paid
• Willingness is pink + green. Surplus is just
the pink
p
D
q
Willingness(Q)
The willingness to pay for q units is the
green area while the willingness to pay
for q+n units is green and pink. Therefore
the willingness to pay for n extra units is
the pink area
p
D
q
q+n
Approximating VC from MC
• MC(Q) is C(Q+1) - C(Q)
– C(1) = MC(0) + C(0) = MC(0) + FC
– C(2) = C(1) + MC(1) = MC(0) + MC(1) + FC
– C(Q) = MC(0)+…+MC(Q-1) + FC
• VC(Q) = MC(0) + …+ MC(Q-1)
VC is area under MC
VC(3) is approximately 1 times MC(0)
plus 1 times MC(1)
MC
plus 1 times MC(2)
$/unit
MC(2) tall
1
2
3
1 wide
Q
VC as a function of Q
VC(Q) is the pink area while VC(Q+N) is the gray and
the pink areas. Thus the gray area is the additional costs
from making N more units when Q have already been made.
Note that C(Q+N) - C(Q) = VC(Q+N) - VC(Q) = gray area
$/unit
MC
Q
Q+ N
Quantity
Cost and Profit
$/unit
• VC(Q) is MC(0) + MC(1) + ...+ MC(Q-1)
• profit: p =pQ - VC(Q) - FC
• p + FC = Green + Black - Black = Green
MC
p
Q
1st Welfare Theorem:
Surplus Form
• Competition maximizes the sum of Consumer
Surplus and Firm Profit
• Comp. Maximizes Willingness - Cost
– willing = surplus + pQ
– C(Q)= pQ - profit
– so Willing - C(Q) = surplus + profit
Proof by Picture
$/unit
The pink quadrilateral is willingness
The grayish area is VC;
so the remaining pink triangle is
Willingness - VC
MC
D
Q*
units
A smaller Q?
Decreasing Q results in willingness
- VC shrinking to the red area.
$/unit
As before, at Q* W-VC = triangle
That is now the red plus green
Moving inwards to Q from Q*
Avoid pink costs (under mc)MC
Give up green plus pink willingness
This nets to: Green part of triangle
Is lost; only red remains
D
Q
Q*
units
$/unit
Larger Q?
The red area is added VC
The blue quadrilateral is added willingness,
so the remaining red triangle is W - VC and is
negative. Better off making Q*
MC
D
Q*
Q
units
Pollution
• Let MCf be the marginal costs incurred by the
firm
• Let MCp be the marginal costs caused by
pollution and not paid by the firm
• MC = MCp + MCf
– previous example MCp could be a constant t
MC of Pollution
• Health related costs: Asthma, cancer from
diesel exhaust, cancer from haloethanes in
water…
• Destruction of buildings from acid rain.
Includes Parthenon
• Acid rain destruction of lakes
Social Welfare
• Max Willingness to Pay less ALL costs
maximizes welfare
• Economic system maximizes willingness less
firm’s costs (MCf)
• Can get back to social welfare max with either
a tax or a restriction on quantity
Set Up
D
MC
MCf + MCp = MC.
Arrows are same size and show
that distance between MC and
MCf is just MCp
MCf
p
Before regulation supply is MCf and
demand is D, so output is qp.
MCp
qp
Competitive Solution
Before regulation supply is MCf and
demand is D, so output is qp.
Profit = p qp - area under MCf
Surplus is area under demand
and above price.
And pollution costs are are
under MCp
D
MC
MCf
p
MCp
qp
We assume FC = 0 for convenience
Maximize W - All costs
D
Supply, MC, equals demand
at qs
Profit - pollution costs
= p qp - area under MC
= W - all costs
MC
p
To expand output to qp
one incurs a social loss of
the red area: area under
MC and above demand
We assume FC = 0 for convenience
MCf
MCp
qs qp
Dead Weight Loss
• 1. Find the socially right output. Find its
Willingness – Costs
• 2. Find any other output. Find its Willingness
– Costs
• 3. DWL = (W-C)right-(W-C)wrong
Deadweight Loss of Pollution
D
MC
{Maximum W - all costs}
less
p
{W - all costs from
producing “competitive” output}
=
Deadweight Loss
MCf
MCp
qs qp
We assume FC = 0 for convenience
Actual Policies
• Air, Water, Toxics, etc are nearly all in terms of
standards (quantity like controls) rather than
in terms of pollution fees
• Is this a surprise?
A tax can achieve qs
T
D
MC
$/unit
Tax T=MC-MCf at qs:
Makes demand to firm D-1(q) - T
which is red line, D shifted down
by T. Firm now produces at
MCf(qs) = D-1(qs) - T
MCf
MCp
qs
units
Firms Prefer Controls to Taxes
Before regulation profits are
red and pink areas
MC
Tax T=MC-MCf at
qs: Q is still qs, green
area is tax take and only pink
remains as profit
When regulation reduces Q
Profits are the pink plus
green areas.
MCf
MCp
qs
Unreg. Q
DWL of taxation
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A tax results in too low an output.
Find the DWL.
(First find the no-tax-first-best equilibrium)
No find the with tax quantity
Now find the triangle
DWL of Taxes
MC +t
Going from qe to qt
Loss in willingness =
Gain from less costs =
DWL =
MC
qt qe