𝐵=𝛼 𝐴+𝜒 𝐺

Two criterion emissions, G and A, measured as emission reductions.Benefits, B, of emission reduction:

Emission controls, R and S, affect both:𝐺=𝜋 𝑆+𝜌 𝑅 , 𝜌≫𝜋𝐴=𝜎𝑆+𝜏 𝑅 ,𝜎≫𝜏Costs, C, of emission reduction:

𝐶=0.5𝜅𝑆2+0.5𝜆𝑅2

Rework benefits:𝐵=𝛼 (𝜎 𝑆+𝜏 𝑅 )+𝜒 (𝜋𝑆+𝜌 𝑅 )=𝑆 (𝛼𝜎+𝜒𝜋 )+𝑅 (𝛼𝜏+𝜒𝜌 )

Benefits, B, of emission reduction:

Costs, C, of emission reduction:𝐶=0.5 𝜅𝑆2+0.5 𝜆𝑅2

𝐵=𝛼 (𝜎 𝑆+𝜏 𝑅 )+𝜒 (𝜋𝑆+𝜌 𝑅 )=𝑆 (𝛼𝜎+𝜒𝜋 )+𝑅 (𝛼𝜏+𝜒𝜌 )

Optimal control for R:

𝜕𝐵𝜕𝑅=𝛼𝜏+𝜒𝜌=𝜆𝑅=

𝜕𝐶𝜕𝑅 ⇒𝑅 ′=𝛼𝜏+𝜒𝜌

𝜆

The secondary benefit, >0, increases the optimal rate of abatement.

𝜕𝐵𝜕𝑆=𝛼𝜎+𝜒𝜋=𝜅 𝑅=

𝜕𝐶𝜕𝑆 ⇒𝑆 ′=𝛼𝜎+𝜒𝜋

𝜅

𝐵=𝛼√ 𝐴+𝜒 𝐺

Now make the secondary benefits less than linear.Benefits, B, of emission reduction:

Emission controls, R and S, affect both:

Costs, C, of emission reduction:

𝐶=0.5𝜅𝑆2+0.5𝜆𝑅2

Rework benefits:𝐵=𝛼√ (𝜎𝑆+𝜏 𝑅 )+𝜒 (𝜋 𝑆+𝜌 𝑅 )

Benefits, B, of emission reduction:

Costs, C, of emission reduction:𝐶=0.5 𝜅𝑆2+0.5 𝜆𝑅2

Optimal control for R:

The secondary benefit, >0, increases the optimal rate of abatement R, butthat increase falls with abatement of the other criterion emission S.

𝐵=𝛼√ (𝜎𝑆+𝜏 𝑅 )+𝜒 (𝜋 𝑆+𝜌 𝑅 )

Optimal control:

𝑅 ′= 𝛼𝜏2 𝜆√ (𝜎𝑆 ′+𝜏 𝑅 ′ ) +

𝜒𝜌𝜆

Without the primary benefit, =0, the optimal rate of abatement R is small, and abatement of the other criterion emission S increases.

𝑆 ′= 𝛼𝜎2𝜅√ (𝜎𝑆 ′+𝜏 𝑅 ′ ) +

𝜒𝜋𝜅

If climate change is a hoax, =0:

𝑅= { } over {2 √ left ( 𝛼𝜏 𝜆 𝜎 𝑆 +𝜏 𝑅 right )} ≪ 𝑅𝑆= { } over {2 √ left ( 𝛼𝜎 𝜅 𝜎 𝑆 +𝜏 𝑅 right )} < 𝑆

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:

Costs, C, of emission reduction:

𝐶=0.5𝜅𝑆2+0.5𝜆𝑅2

Rework benefits:𝐵=𝛼 ln (𝜎𝑆+𝜏 𝑅)+𝜒 (𝜋 𝑆+𝜌 𝑅 )

Benefits, B, of emission reduction:

Costs, C, of emission reduction:𝐶=0.5 𝜅𝑆2+0.5 𝜆𝑅2

Optimal control for R:

𝐵=𝛼 ln (𝜎𝑆+𝜏 𝑅 )+𝜒 (𝜋 𝑆+𝜌 𝑅 )

Optimal control for R:

𝑅 ′=𝜏𝜒𝜌−𝜎𝑆 𝜆−√(𝜎𝑆𝜆−𝜏𝜒𝜌 )2+4 𝜏𝜆(𝛼𝜏+𝜎𝜒𝜌𝑆)−2 (𝛼𝜏+𝜎𝜒𝜌 𝑆)

The secondary benefit, >0, increases the optimal rate of abatement R, butthat increase falls with abatement of the other criterion emission S. ???

𝑅′=−𝜎𝑆 𝜆−√ (𝜎𝑆 𝜆 )2

−2 (𝜎𝜒𝜌𝑆 )= 𝜆𝜒𝜌

No secondary benefit, 0

No abatement of the other criterion emission S

𝑅′=𝜏𝜒𝜌−√ (𝜏𝜒𝜌 )2+4𝜏2 𝜆𝛼

−2𝛼𝜏 =𝜒𝜌−√ (𝜒𝜌 )2+4 𝜆𝛼

−2𝛼

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.