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I have the following optimal control problem

$$\max_{c_t} \int^{\infty}_0 e^{-p_it}\ln(c_t(i))dt$$

subject to $$\dot{w_t}(i)=rw_t(i) -n_ic_t(i)$$ $$w_0(i)=w_0>0$$

I have some wealthy and infinitely-lived dynasties indexed by i.

$n_i\in (n_{low}, n_{high})>0$ is number of members.

$p_i\in (p_{low}, p_{high})>0$ is discount rate.

I have 4 types dynasties:

Case-1: p is low, n is low

Case-2: p is high, n is low

Case-3: n is high, p is low

Case-4: n and p is high.

$w_t(i)>0$ is wealth stock of dynasty i.

Suppose that, in each dynasty, every member consumes the same amount, and there is one individual in each dynasty who takes decisions concerning consumption per capita, $c_t(i)$.

And note that $r>p_{high}>0$


I have solved this control problem, and I have obtained the following solutions in summary:

Current value Hamiltonian:

$$H=\ln(c_t(i))+q_t[rw_t(i) -n_ic_t(i)]$$

$$\frac{\dot{q_t}}{q_t}=-\frac{\dot{c_t(i)}}{n_ic_t(i)} $$

And

$$\frac{\dot{q_t}}{q_t}=n_i(p_i-r)$$

Euler equation is:

$$\frac{\dot{c_t(i)}}{c_t(i)}=n_i(r-p_i)$$

So, $$c_t(i)=c_0e^{-n_i(r-p_i)}$$

$$\frac{\dot{w_t(i)}}{w_t(i)}=r-n_i\frac{c_t(i)}{w_t(i)}$$

$$q_t=q_0e^{-(p_i-r)t}$$


So far I can solve the problem. However, I am stack with the economic interpretation of the result. which type of a dynasty would I like to be a member of? Which one is reasonable? How can explain this with economic intuition? Please help me at this point.

Thanks a lot.

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  • 1
    $\begingroup$ Why not solve the differential equation for wealth stock? This can easily show you which type of dynasty you would like to be a member off given the difference in the fixed parameters $\endgroup$ – user20105 Dec 7 '19 at 23:23

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