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 =