The Welfare Theorem & The Environment © 1998 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

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

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) plus 1 times MC(2) $/unit MC MC(2) tall 1 3 Q 2 1 wide

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

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

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

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*

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*

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 qp 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

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

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

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 {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

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 units qs

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

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.