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Secondary benefits

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Lecture on secondary benefits and climate policy. Will be in next edition of the Climate Economics textbook.

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Secondary benefits

  1. 1. 𝐵 = 𝛼𝐴 + 𝜒𝐺 Two criterion emissions, G and A, measured as emission reductions. Benefits, B, of emission reduction: Emission controls, R and S, affect both: 𝐺 = 𝜋𝑆 + 𝜌𝑅, 𝜌 ≫ 𝜋 𝐴 = 𝜎𝑆 + 𝜏𝑅, 𝜎 ≫ 𝜏 Costs, C, of emission reduction: 𝐶 = 0.5𝜅𝑆2 + 0.5𝜆𝑅2 Rework benefits: 𝐵 = 𝛼 𝜎𝑆 + 𝜏𝑅 + 𝜒 𝜋𝑆 + 𝜌𝑅 = 𝑆 𝛼𝜎 + 𝜒𝜋 + 𝑅 𝛼𝜏 + 𝜒𝜌
  2. 2. Benefits, B, of emission reduction: Costs, C, of emission reduction: 𝐶 = 0.5𝜅𝑆2 + 0.5𝜆𝑅2 𝐵 = 𝛼 𝜎𝑆 + 𝜏𝑅 + 𝜒 𝜋𝑆 + 𝜌𝑅 = 𝑆 𝛼𝜎 + 𝜒𝜋 + 𝑅 𝛼𝜏 + 𝜒𝜌 Optimal control for R: 𝜕𝐵 𝜕𝑅 = 𝛼𝜏 + 𝜒𝜌 = 𝜆𝑅 = 𝜕𝐶 𝜕𝑅 ⇒ 𝑅′ = 𝛼𝜏 + 𝜒𝜌 𝜆 The secondary benefit, 𝜏>0, increases the optimal rate of abatement. 𝜕𝐵 𝜕𝑆 = 𝛼𝜎 + 𝜒𝜋 = 𝜅𝑅 = 𝜕𝐶 𝜕𝑆 ⇒ 𝑆′ = 𝛼𝜎 + 𝜒𝜋 𝜅
  3. 3. 𝐵 = 𝛼√𝐴 + 𝜒𝐺 Now make the secondary benefits less than linear. Benefits, B, of emission reduction: Emission controls, R and S, affect both: 𝐺 = 𝜋𝑆 + 𝜌𝑅, 𝜌 ≫ 𝜋 𝐴 = 𝜎𝑆 + 𝜏𝑅, 𝜎 ≫ 𝜏 Costs, C, of emission reduction: 𝐶 = 0.5𝜅𝑆2 + 0.5𝜆𝑅2 Rework benefits: 𝐵 = 𝛼√ 𝜎𝑆 + 𝜏𝑅 + 𝜒 𝜋𝑆 + 𝜌𝑅
  4. 4. Benefits, B, of emission reduction: Costs, C, of emission reduction: 𝐶 = 0.5𝜅𝑆2 + 0.5𝜆𝑅2 Optimal control for R: 𝜕𝐵 𝜕𝑅 = 𝛼𝜏 2√ 𝜎𝑆 + 𝜏𝑅 + 𝜒𝜌 = 𝜆𝑅 = 𝜕𝐶 𝜕𝑅 ⇒ 𝑅′ = 𝛼𝜏 2𝜆√ 𝜎𝑆′ + 𝜏𝑅′ + 𝜒𝜌 𝜆 The secondary benefit, 𝜏>0, increases the optimal rate of abatement R, but that increase falls with abatement of the other criterion emission S. 𝐵 = 𝛼√ 𝜎𝑆 + 𝜏𝑅 + 𝜒 𝜋𝑆 + 𝜌𝑅
  5. 5. Optimal control: 𝑅′ = 𝛼𝜏 2𝜆√ 𝜎𝑆′ + 𝜏𝑅′ + 𝜒𝜌 𝜆 Without the primary benefit, 𝜒=0, the optimal rate of abatement R is small, and abatement of the other criterion emission S increases. 𝑆′ = 𝛼𝜎 2𝜅√ 𝜎𝑆′ + 𝜏𝑅′ + 𝜒𝜋 𝜅 If climate change is a hoax, 𝜒=0: 𝑅" = 𝛼𝜏 2𝜆√ 𝜎𝑆" + 𝜏𝑅" ≪ 𝑅′ 𝑆" = 𝛼𝜎 2𝜅√ 𝜎𝑆" + 𝜏𝑅" < 𝑆′
  6. 6. 𝐵 = 𝛼 ln 𝐴 + 𝜒𝐺 Now make the secondary benefits less than linear. Benefits, B, of emission reduction: Emission controls, R and S, affect both: 𝐺 = 𝜋𝑆 + 𝜌𝑅, 𝜌 ≫ 𝜋 𝐴 = 𝜎𝑆 + 𝜏𝑅, 𝜎 ≫ 𝜏 Costs, C, of emission reduction: 𝐶 = 0.5𝜅𝑆2 + 0.5𝜆𝑅2 Rework benefits: 𝐵 = 𝛼 ln(𝜎𝑆 + 𝜏𝑅) + 𝜒 𝜋𝑆 + 𝜌𝑅
  7. 7. Benefits, B, of emission reduction: Costs, C, of emission reduction: 𝐶 = 0.5𝜅𝑆2 + 0.5𝜆𝑅2 Optimal control for R: 𝜕𝐵 𝜕𝑅 = 𝛼𝜏 𝜎𝑆 + 𝜏𝑅 + 𝜒𝜌 = 𝜆𝑅 = 𝜕𝐶 𝜕𝑅 ⇒ 𝛼𝜏 + 𝜎𝜒𝜌𝑆 = 𝜎𝑆𝜆 − 𝜏𝜒𝜌 𝑅 + 𝜏𝜆𝑅2 ⇒ 𝑅′ = 𝜏𝜒𝜌 − 𝜎𝑆𝜆 ± (𝜎𝑆𝜆 − 𝜏𝜒𝜌)2+4𝜏𝜆(𝛼𝜏 + 𝜎𝜒𝜌𝑆) −2(𝛼𝜏 + 𝜎𝜒𝜌𝑆) 𝐵 = 𝛼 ln 𝜎𝑆 + 𝜏𝑅 + 𝜒 𝜋𝑆 + 𝜌𝑅
  8. 8. Optimal control for R: 𝑅′ = 𝜏𝜒𝜌 − 𝜎𝑆𝜆 − (𝜎𝑆𝜆 − 𝜏𝜒𝜌)2+4𝜏𝜆(𝛼𝜏 + 𝜎𝜒𝜌𝑆) −2(𝛼𝜏 + 𝜎𝜒𝜌𝑆) The secondary benefit, 𝜏>0, increases the optimal rate of abatement R, but that increase falls with abatement of the other criterion emission S. ??? 𝑅′ = −𝜎𝑆𝜆 − 𝜎𝑆𝜆 2 −2 𝜎𝜒𝜌𝑆 = 𝜆 𝜒𝜌 No secondary benefit, 𝜏 =0 No abatement of the other criterion emission S 𝑅′ = 𝜏𝜒𝜌 − 𝜏𝜒𝜌 2 + 4𝜏2 𝜆𝛼 −2𝛼𝜏 = 𝜒𝜌 − 𝜒𝜌 2 + 4𝜆𝛼 −2𝛼
  9. 9. Secondary benefits • If greenhouse gas emission reduction helps solving other problems (air pollution, energy security), then we should do more of it • However, greenhouse gas emission reduction would be a clumsy way to solve other problems and other problems do not justify a lot of greenhouse gas emission reduction • Other problems justify a lot of other problem solving.

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