I am trying to solve the optimal policy problem in a New Keynesian DSGE with capital accumulation. Given that capital at time t is predetermined (standard law of motion of capital), after using the linear quadratic approach, I get an objective function with inflation, output and investments at the current period and capital at the current period but also with a lag.
Please notice, that the last one is not an independent of policy term for two reasons: 1) current investments are affected from k(-1), 2)In the objective function I get also two product terms where lagged capital is multiplied with current output, so it is dependent of policy for sure.
Any suggestions on what to do? I have thought the obvious, that is, solve for k(-1) from the law of motion and then substitute with k and investm. Is there any other way to solve this issue, or do you agree to go with the substitution?
Thank you a lot in advance.