Consider an abatement cost function C(e,)=(abe) 2 /2b, where is the amount of private investment to

Consider an abatement cost function C(e,κ)=(a−be−κ)²/2b, where κ is the amount of private investment to reduce abatement costs.
There are J identical firms. The social damage function is D(E)=d·E²/2.

(a) Determine the socially optimal allocation, including the level of investment for each firm. Determine also the socially optimal tax rate and the optimal number of emission permits.

(b) Repeat for the cost function C(e,κ)=a·(e/κ).

(c) Assume now that the firm has joint costs for producing output and abating emissions, given by C(x,e,κ)=(ax−be−κ)²/2b+cx²/2, and there are again J identical firms. The inverse demand for output is P(X)=A−BX, where X=xJ is total output. Repeat part (a).