1. The Welfare Theorem & The Environment
© 1998 by Peter Berck

2. Outline 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

3. 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

4. Calculating Total Willingness

5. Consumer Surplus
Consumer surplus is willingness to pay less amount paid
Amount paid is P Q

6. Consumer surplus is willingness to pay less amount paid
Willingness is pink + green. Surplus is just the pink p q D

7. 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+n q

8. 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)

9. VC is area under MC VC(3) is approximately 1 times MC(0)
plus 1 times MC(1)
plus 1 times MC(2)
$/unit MC
MC(2) tall 1 3 Q 2
1 wide

10. 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
Quantity Q
Q+ N

11. Cost and Profit VC(Q) is MC(0) + MC(1) + ...+ MC(Q-1)
profit: p =pQ - VC(Q) - FC
p + FC = Green + Black - Black = Green
$/unit MC p Q

12. 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

13. Proof by Picture The pink quadrilateral is willingness
The grayish area is VC;
so the remaining pink triangle is
Willingness - VC
$/unit MC D
units Q*

14. A smaller Q? Decreasing Q results in willingness
shrinking to the red area.
VC(Q) is just the blue so;
remaining red is W - VC;
if Q* made then red+ pink
is W - VC; shouldn’t
have output less than Q*
$/unit MC D
units Q Q*

15. 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*
$/unit MC D Q
units Q*

16. 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

17. 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

18. 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

19. Set Up D MC MCf + MCp = MC. Arrows are same size and showqp
Before regulation supply is MCf and
demand is D, so output is qp. D MC
MCf + MCp = MC.
Arrows are same size and show
that distance between MC and
MCf is just MCp
MCf p
MCp

20. Competitive Solution D Before regulation supply is MCf and
demand is D, so output is qp. MC
Profit = p qp - area under MCf
Surplus is area under demand
and above price.
And pollution costs are are
under MCp
MCf p
MCp qp
We assume FC = 0 for convenience

21. Maximize W - All costs D Supply, MC, equals demand at qs MC
Profit - pollution costs
= p qp - area under MC
= W - all costs
To expand output to qp
one incurs a social loss of
the red area: area under
MC and above demand p
MCf
MCp qs qp
We assume FC = 0 for convenience

22. 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

23. Deadweight Loss of PollutionMC
{Maximum W - all costs}
less
{W - all costs from
producing “competitive” output} =
Deadweight Loss p
MCf
MCp qs qp
We assume FC = 0 for convenience

24. 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?

25. 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
units qs

26. 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
MCf
When regulation reduces Q
Profits are the pink plus
green areas.
MCp
Unreg. Q qs

27. DWL of taxation 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
After Max gets done with monopoly, find the DWL of monopoly.