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

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 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

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 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 QQ+ N 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

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

1 st 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 is Willingness - 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 green Moving inwards to Q from Q* Avoid pink costs (under mc) Give up green plus pink willingness This 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 is negative. Better off making Q*

Pollution Let MC f be the marginal costs incurred by the firm Let MC p be the marginal costs caused by pollution and not paid by the firm MC = MC p + MC f – previous example MC p 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 (MC f ) Can get back to social welfare max with either a tax or a restriction on quantity

Set Up MC f MC p MC MC f + MC p = MC. Arrows are same size and show that distance between MC and MC f is just MC p qpqp Before regulation supply is MC f and demand is D, so output is q p. p D

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

Maximize W - All costs MC f MC p MC qsqs Supply, MC, equals demand at q s Profit - pollution costs = p q p - area under MC = W - all costs To expand output to q p one incurs a social loss of the red area: area under MC and above demand We assume FC = 0 for convenience p D qpqp

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 MC f MC p MC qsqs {Maximum W - all costs} less {W - all costs from producing “competitive” output} = Deadweight Loss We assume FC = 0 for convenience p D qpqp

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 q s MC f MC p MC qsqs Tax T=MC-MC f at q s : Makes demand to firm D -1 (q) - T which is red line, D shifted down by T. Firm now produces at MC f (q s ) = D -1 (q s ) - T D $/unit units T

Firms Prefer Controls to Taxes MC f MC p MC Unreg. Q qsqs Before regulation profits are red and pink areas When regulation reduces Q Profits are the pink plus green areas. Tax T=MC-MC f at q s : Q is still q s, green area is tax take and only pink remains 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 qtqt qeqe Going from q e to q t Loss in willingness = Gain from less costs = DWL =