Given (1) and (2), is it possible to show the existence of a Bellman equation (3), using Bellman's Principle of Optimality?
- $$\ max \Sigma\beta^s U(C_t)$$
Subject to the following resource constraint:
- $$C_t + K_{t+1}= F(K_F, E_F,S_t)$$
Where: $$E_t = F_E(K_E,E_2,R_t, R_{t+1})$$
$$S_t = \Sigma(1-d)E_t$$
$$R_{t+1}=R_t -E_t$$
And the Bellman equation is given:
- $$V_t (K_t,R_t,E_t)= max_{C_1,K_2,E_1,E_t} {U(C_t)+\beta V_{t+1}(K_{t+1},R_{t+1},E_{t+1}) + \lambda_2 (F_2 (K_2,E_t-E_1)-E_t)}$$
This problem is taken from eq. (27) and eq. (39) of the following textbook: https://are.berkeley.edu/~traeger/pdf/KarpTraegerDraft.pdf.
The specified parameters in the model are:
- Consumption $C_t$
- Capital $K_t$
- The energy sector $E_t$
- The carbon stock $S_t$
- The finite resource $R_t$
Would genuinely appreciate being able to see how this is conducted on this type of model.