Hi artemsolod,

It’s great to hear that that KKT conditions helped : )

As for the KKT conditions wouldn’t those be: \nabla U(\vec{C}) - \sum_{t=0}^T \mu_t \nabla \left(F(K_t,1) + (1-\delta) K_t- C_t - K_{t+1} \right) = \vec{0}

For question 1, in first-order condition, we are not (just) solving derivatives wrt
\mu but wrt each variable + \mu. This is actually one part of the KKT condition (think how this relates to the **Stationarity** condition you listed here).

There appears to be no 3rd option for output to go (like burn it down). Even if there is, we can judge by monotonicity of the utility function that it would be strictly better to consume and, hence, get a (simpler?) problem that has *equality* constraint on capital and a simple dynamic on consumption (i.e. output - investment).

For this question, I am not too sure if I understand your question correctly, so my answer might be slightly off. I guess what you are asking is why we are not using all the budget we have to increase our utility so that we have an equality constraint that has consumption equal to budget (please correct me if. I understand it incorrectly). In the model section section, you can find our setup and see that monotonicity is not the only thing that shapes the utility function. There is also curvature associated with the utility function and production function. It is less obvious that we can use equality to capture the optimality. Using Lagrange multipliers allows for a more general and systematic approach that can handle a variety of constraints, including both equality and inequality constraints, simultaneously.

Without it 4.11 becomes \mu_t\left[(1-\delta)+f'(K_t)\right] - \mu_{t-1}=0

I think it is 4.10? 4.10 is the first-order condition wrt to K_t. To find the first-order condition with respect to K_t, we differentiate the Lagrangian function with respect to K_t, and set the result equal to zero. Note that K_t is dependent on the capital a period before (this is also why \beta here makes sense).

but is this what we want from the modeling prospective? This creates consumption spikes at the end of the period a la “There is no tomorrow”.

First of all, this is a finite state case. The terminal condition K_{T+1}=0 is derived from the first-order necessary condition involving the Lagrange multiplier. It also makes economic sense as it is not optimal to leave any leftover capital that isn’t converted to utility. We want to spend what we have and convert it into utility before the end.

I understand that you might not be happy with the consumption pattern (i.e., spending too much at the end). You can add some other factors into the model, for example, adding a bequest to the next generation and giving the bequest some utility so that it gives some incentives for agents to hold back consumption and preserve some capital at the end. There are other ways you can adjust the model so that it can capture features that you would like to have.

Hope this helps. Very happy to discuss further.