Intertemporal optimization: When to use the Hamiltonian vs Lagrangian

Question

Assume a producer wishes to maximize the net present value, choosing optimal quantities of K and L. variables are time dependent. y is the production, p is the price of y. K is capital, r is the price of capital, L is labor, w is wage.

$$ \max_{K(t), L(t)} \pi(t)= \int_{t=0}^T e^{-\rho.t}. (p(t) . y(t) - [r(t).K(t) + w(t).L(t)]).dt $$ Subject to the constraint: $$ S.t: y(t) = K(t)^a . L(t)^b $$

I'm trying to find the quantities of Y, K and L to maximize the net present value of profit, but I don't know what to use to solve this optimization problem, the lagrangian or the Hamiltonian? I don't know what to use because all the variables are expressed in time $t$, not $t-1$ or $t+1$.

Answer 1

I think you can exclude Hamiltonian, if you do Ramsey model with assumption that : " no costs of adjustment of capital in time ⇒ the problem of maximizing the present value of profit reduces to the problem of maximizing profit in each period"

meaning that firms do not choose that if today they will increase some variable tmrw they will have it less (contrary to Housholds utility fce) thus the profit lifetime maximisation can be reduced to the one time maximisation, therefore Hamiltonian is not necesarry (as it is used for intratemporal problems). Therefore I would use only Lagrangian. However Im not sure about this term in fce e^(-ro * t) as we use it not in the firms profit but in the Houshold utility fce where one uses Hamiltonian as the e^(-ro * t) is used for prefering consumtion now rather than tmrw. So maybe check the assumption I've mentioned above and also the e^(-ro * t).

Answer 2

Discretizing $t$ we have

$$ \max_{K(t), L(t)} \pi(t)= \int_{t=0}^T e^{-\rho.t}. (p(t) . y(t) - [r(t).K(t) + w(t).L(t)]).dt\approx \sum_{j=0}^{j=n}e^{-\rho \delta j}\left(p(\delta j)K^a(\delta j)L^b(\delta j)-r(\delta j)K(\delta j)-w(\delta j)L(\delta j)\right) $$

then calling

$$ O(K_j,L_j) = \sum_{j=0}^{j=n}c_j\left(p_jK^a_jL^b_j-r_jK_j-w_jL_j\right) $$

the stationary points are at the solutions for

$$ \cases{ O_{K_j}= a p_j K^{a-1}_jL^b_j-r_j=0\\ O_{L_j}= b p_j K^a L^{b-1}_j-w_j=0 } $$

now assuming $K_j>0, L_j>0$ we have

$$ \cases{ (a-1)\ln K_j + b \ln L_j = \ln\left(\frac {r_j}{a p_j}\right)\\ a \ln K_j+(b-1)\ln L_j = \ln\left(\frac {w_j}{b p_j}\right) } $$

thus obtaining expressions to $K^*_j, L^*_j$ optimals.

$$ \cases{ K_j^* = \left(\frac{r_j}{a p_j}\right)^{\frac{1-b}{a+b-1}}\left(\frac{w_j}{b p_j}\right)^{\frac{b}{a+b-1}}\\ L_j^* = \left(\frac{r_j}{a p_j}\right)^{\frac{b}{a+b-1}}\left(\frac{w_j}{b p_j}\right)^{\frac{a-1}{a+b-1}} } $$