Consider a pollution accumulation problem with two pollutants where E₁ and E₂ denote emissions and S₁ and S₂ the stocks of pollution for the two pollutants. The abatement cost functions of the two pollutants are given by C_i(E_i)=Y_i[E̅_i-E_i]²/2 while the damage function is given by:

(a) Set up the dynamic optimality conditions.

(b) Determine the steady state.

(c) Carry out the comparative statics with respect to r and .

(d) Determine a general closed form solution for the optimal emission and co-state variable paths, and for the pollution stocks.
Assume now the parameters are:d₁ = 0 d₂ = 0.1γ₁ = γ₂ = 2 E̅₁ = E̅₂ = 10 r = 0.06

(e) Consider first the fully symmetric case with α₁ =α₂ = 1 β₁ = β₂ = 0.1 Simulate the dynamics of the system. Draw the time paths for optimal emissions co-state variables and the pollution stocks. Using a suitable software (Mathematica, Mathlab or the like), plot a 3D-picture of the optimal emission (co-state variable) path on the S₁/S₂ plane using different initial values for the pollution stocks.

(f) Repeat (d) for asymmetric α^, s by choosing α₁ = 1, α₂ = 1.1

(g) Repeat (d) for asymmetric β^, s by choosing β₁ = 0.9 β₂ = 1.1

(h) To simulate the system with more realistic values choose a more flexible abatement cost function according to C_i(E_i)=Y_il[E̅_i- E_i]+Y₁₂[E̅_i-E_i]²/2, i=CO₂, CH₄. For the parameters choose d₁ = -0.352 d₂ =4-10^{-13 γ} _co2,1 = -3.483, γ_CH4,1= -0832 γ_co,2 =6.08.10^-9 γ_CH4,2 = 7.87-10^-6 E̅₁= 30.10 ⁹ ,E̅₂ = 112.10⁶  (measured in tons of CO₂ equivalents) and finally r= 0.02 Determine the optimal paths, the steady states, and the shadow price ratio between the optimal price for CO₂ and CH₄. How does that ratio vary over time?

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D(S, S)=d [a,S + aS]+d[a,S + aS F