Similar to the previous exercise, consider a pollution accumulation problem with two pollutants. But now the damage functions are additively separable such that D _i(S _i) =d _i, S _i ² while abatement costs are interdependent with C(E ₁, E ₂)=Y ₁[E̅ ₁ - E ₁] ² / 2 +γ ₂ [E̅ ₂– E ₂ ] ²/2 + w [E̅ ₁–E₁][E̅₂- E₂]. If W>0 we say the two pollutants are substitutes in abatement, while if W i

(a) Set up the dynamic optimality conditions.

(b) Determine the steady state.

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

(d) Determine a general closed form solution for the optimal emission, the co-state variable paths and for the pollution stocks.
Assume now the parameters are given as follows: D₁ = D₂ = 1 D₂ = 0.1 γ₁ = γ₂ = 2 E̅₁= E̅₂ = 10 r = 0.06 β₁ = β₂ = 0.1 w = 1 and alternatively w =-1

(e) Simulate the dynamics of the system. Draw the time paths for optimal emissions, the co-state variables and for the pollution stocks. Using 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.

Data from previous exercise

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  while the damage function is given by

Transcribed Image Text:

C (E ) = y; [ - E ]/2