How to optimize this dynamical system? Needing guidelines

I'm trying to solve a growth model, where the author indicates is a dynamical system. I want to ask if someone would help me with some guidelines of how to optimize this, I've been trying to solve it throw lagrangian but, as it is a dynamical system is not the right method. The model is a growth model where individuals choice consumption, number of children $$(n_t)$$ ,quality of them in terms of education $$(e_{t+1})$$ and intergenerational transfers $$(s_{t+1})$$. Also it follows that human capital $$(h_{t+1})$$ follows the function $$h_{t+1}=h(e_{t+1})$$.

The optimization problem is to maximize the utility (3) subject to (5) knowing that $$h_{t+1}=h(e_{t+1})$$ This is a project where I'm learning on the way and by myself without any guidance, so I will be very thankfull for any help.

Here I leave the functions. The paper is 'Fertility clubs and economic growth' of Ahituv and Moav (2005) and you could find the model in 'Cheap children and the persistence of poverty' Moav (2001) It does not specify anything about the time of maximization, I tried to solve it as an OLG too but I did not arrive to any result of showed up. Also it does not specify about any particular t, so I guess I should get a time path.

Thanks!!

• I think you can have better answers if you specify: i) The problem to solve explicitly. (Max, Min. Which function?). ii) And also it seems there's lack of some sort of law of motion for physical and human capital, that is roughly speaking an equation that contains, $k_{t+1}$ and $k_{t}$, and the same for $H_t$. Anyways, if it's useful to say, the Lagrangian method works as well in dynamic optimization, and I think this could be one of those cases. – nrivera Jul 12 '20 at 0:13
• Hi! Thanks for your answer. The problem consists on maximice (3) subject to (5) and knowing that ht+1=h(et+1) which is a condition you can set directly into the utility and budget constraint. There is no any law of motion specified in the model. – RLF Jul 12 '20 at 2:56
• @Roberto maximize (3)? For a random $t$? I don't think that is it. Perhaps discounted sum of utilities? – Giskard Jul 12 '20 at 8:25