The Realpolitik of Economic Welfare Observations on Democracy, Arrow抯 Impossibility Theorem and Th
The Welfare Theorem & The Environment © 1998, 2011 by Peter Berck.
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Transcript of The Welfare Theorem & The Environment © 1998, 2011 by Peter Berck.
The Welfare Theorem & The Environment
© 1998, 2011 by Peter Berck
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
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
0
10
20
30
40
50
60
70
1 2 3 4 5 6 7
Quantity
Price AREA
Demand
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
q
D
p
q
D
Willingness(Q)
q+n
The willingness to pay for q units is thegreen area while the willingness to payfor q+n units is green and pink. Thereforethe willingness to pay for n extra units isthe pink area
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
MC(2) tall
MC
Q
$/unit
1 2 3
VC(3) is approximately 1 times MC(0) plus 1 times MC(1)plus 1 times MC(2)
1 wide
VC as a function of Q
MC
Quantity
$/unit
Q Q+ N
VC(Q) is the pink area while VC(Q+N) is the gray andthe pink areas. Thus the gray area is the additional costsfrom making N more units when Q have already been made.Note that C(Q+N) - C(Q) = VC(Q+N) - VC(Q) = gray area
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
MC
p
Q
$/un
it
1st Welfare Theorem: Surplus Form
• Competition maximizes the sum of Consumer Surplus and Firm Profit
• Comp. Maximizes Willingness - Cost– willing = surplus + pQ– C(Q)= pQ - profit– so Willing - C(Q) = surplus + profit
Proof by Picture$/
unit
units
MC
D
Q*
The pink quadrilateral is willingness
The grayish area is VC; so the remaining pink triangle isWillingness - VC
A smaller Q?$/
unit
units
MC
D
Q*Q
Decreasing Q results in willingness- VC shrinking to the red area.
As before, at Q* W-VC = triangle That is now the red plus greenMoving inwards to Q from Q*Avoid pink costs (under mc)Give up green plus pink willingnessThis nets to: Green part of triangle Is lost; only red remains
The red area is added VC
Larger Q?$/
unit
units
MC
D
Q* Q
The blue quadrilateral is added willingness,so the remaining red triangle is W - VC and isnegative. Better off making Q*
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
MCf
MCp
MC
MCf + MCp = MC.Arrows are same size and showthat distance between MC andMCf is just MCp
qpBefore regulation supply is MCf anddemand is D, so output is qp.
p
D
Competitive Solution
MCf
MCp
MC
qp
Before regulation supply is MCf anddemand is D, so output is qp. Profit = p qp - area under MCf
Surplus is area under demandand above price.And pollution costs are are under MCp
We assume FC = 0 for convenience
p
D
Maximize W - All costs
MCf
MCp
MC
qs
Supply, MC, equals demandat qs
Profit - pollution costs= p qp - area under MC= W - all costs
To expand output to qp
one incurs a social loss ofthe red area: area underMC and above demand
We assume FC = 0 for convenience
p
D
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
MCf
MCp
MC
qs
{Maximum W - all costs}less{W - all costs fromproducing “competitive” output}=Deadweight Loss
We assume FC = 0 for convenience
p
D
qp
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
MCf
MCp
MC
qs
Tax T=MC-MCf at qs:Makes demand to firm D-1(q) - Twhich is red line, D shifted downby T. Firm now produces atMCf(qs) = D-1(qs) - T
D
$/unit
units
T
Firms Prefer Controls to Taxes
MCf
MCp
MC
Unreg. Qqs
Before regulation profits arered and pink areas
When regulation reduces QProfits are the pink plusgreen areas.
Tax T=MC-MCf at qs: Q is still qs, greenarea is tax take and only pinkremains as profit
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
DWL of Taxes
MC
MC +t
qtqe
Going from qe to qt
Loss in willingness =Gain from less costs =DWL =