1- The instantaneous utility function is given byu(q)=q-(1/2)q where q is consumption of a good (control variable) and is a positive constant. However, during production of this good a pollutant releases into the environment. Accumulation of this pollutant over time damages the environment. Suppose that we can model the instantaneous environmental damages by a linear function of the form D(S) =bS, where b is a positive constant and S is the accumulation of the pollution (state variable). Meaning at every point of time the welfare is given by the utility net of the damages, i.e w(q,S)=q-(1/2)q-bS

Pollution accumulation process is given by S'=q.

Suppose that the discount rate is given by r and we want to maximize the discounted net utility (utility minus environmental damage) over a planning horizon of 10 years. The initial stock of the pollution is S(0) = 5 and to avoid a catastrophic climate change to happen, S(10) shouldn't pass the critical value of 200 units.

a) Write down the optimal control problem and the Hamiltonian that maximizes the present value of the total welfare over this 10 years.

b) Write down the optimality conditions.

c) Find the optimal trajectories of consumption (q), pollution stock (S) and shadow price ( or ).