# Why decreasing co-state variable (over time) is Compulsory?

I have a short question.

In the continuous time infinite horizon optimal control maximization problem Like

$$\max \int^{\infty}_0 u(c_t)e^{-pt}dt$$

I use current hamiltonian function and I obtain FOCs and solve them.

In this case, co-state variable must be decreasing? That’s $$\frac{\dot{q_t}}{q_t}<0$$ is this negativity compulsory ? Why?

• Hi: I'm not certain that this is the reason but check out "transversality condition". That condition might be why. – mark leeds Dec 7 '19 at 17:31
• Maybe consider the other first-order condition relating the time change of the costate variable to the derivative of the Hamiltonian with respect to the state variable. Either way, it would be easier to possibly clarify your question when you also include the solution to the problem – Ali Dec 7 '19 at 23:17