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Some additional notes on transfer coefficients and spatial considerations

Some additional notes on transfer coefficients and spatial considerations. Because almost everyone was confused last time!. Notation. E i = Uncontrolled emissions from firm i. e i = Post-control emissions from firm i. A i = E i -e i = Abatement from firm i.

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Some additional notes on transfer coefficients and spatial considerations

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  1. Some additional notes on transfer coefficients and spatial considerations Because almost everyone was confused last time!

  2. Notation • Ei = Uncontrolled emissions from firm i. • ei = Post-control emissions from firm i. • Ai = Ei-ei = Abatement from firm i. • MCi(Ai) = Marginal cost of abatement (typically an increasing function) • MD(p) = Marginal damage from pollution (typically an increasing function) • p = p(e1,e2,…) = pollution as function of emissions • dp/dei = ai = transfer coefficient = proportion of emissions that cause damage.

  3. Optimal amount of pollution • 1 polluter: MC(A) = 2.5*A • Marginal damage: MD = 10 (every unit of pollution causes $10 in damage, so marginal benefit of abatement is $10). • Determine optimal amount of Abatement. • Equivalently, we can determine optimal amount of pollution.

  4. Optimal Abatement $ MC = MB 2.5A = 10 A*=4 MC of abatement 10 MB of abatement A*=4 Abatement

  5. Optimal emissions Recall MC(A) = MC(E-e)=2.5(E-e) If E=10, MB(e) = 25-2.5e $ 25 MC = MB 25-2.5e = 10 e* = 6 This makes sense: A=E-e, 4=10-6 MC of emissions 10 MB of emissions 6 10 Emissions

  6. Add space • Pollution control cost: c(E-e) • Pollution damage cost: D(p(e)) • Total cost to society of emissions e: C(E-e) + D(p(e)) • Regulator’s objective: Mine{C(E-e) + D(p(e))}

  7. Solving this problem • Take derivative, set = 0. • dTC/de = -c’ + dD/dp * dp/de = 0 Effect of 1 unit of emissions on pollution (transfer coefficient, a) Marginal cost of abatement Marginal damage from pollution Optimal amount of emissions must satisfy: MC/a = MD

  8. Add multiple polluters and space • Regulator solves: Min{ei}S ci(Ei-ei) + D(p(e1,e2,…,eN)) dTC/dei = -ci’ + dD/dp * dp/dei Rule for any number of polluters: MCi/ai = MD….so MCi/ai = MCj/aj

  9. Example with 3 firms • All MC the same: MC(Ai) = 2.5Ai • Marginal Damage = 10 (so marginal benefit of abatement = 10). • a1 = 1.0, 2.5A1 = 10, A1* = 4.0 • a2 = .01, 2.5A2/.01 = 10, A2* = .04 • a3 = 10, 2.5A3/10 = 10, A3* = 40 • Firm with greatest influence must abate the most (firm 3)…they will have high marginal abatement cost relative to other two firms.

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