<p>This is an intertemporal dynamic problem, but the fact that the restriction contains no derivatives allows us to solve it using a static optimization approach. Replacing the constraint into the objective:
<span class="math-container">$$
\pi(t) = \int_0^T \Big[ p(t)K(t)^{a}L(t)^{b} - r(t)K(t) - w(t)L(t) \Big]e^{-\rho t} dt
$$</span></p>
<p>The fact that there are no derivatives in the constraint means that what the agent does in one instant <span class="math-container">$t$</span> of time, does not affect the future. This is why we can simply differentiate with respect to the instruments, <span class="math-container">$L(t)$</span> and <span class="math-container">$K(t)$</span>.</p>
<p><span class="math-container">$$ \frac{d\pi(t)}{dL(t)} = \int_0^T \Big[ bp(t)K(t)^{a}L(t)^{b-1}-w(t) \Big]e^{-\rho t} dt =0 $$</span></p>
<p><span class="math-container">$$ \frac{d\pi(t)}{dK(t)} = \int_0^T \Big[ap(t)K(t)^{a-1}L(t)^{b}-r(t)\Big]e^{-\rho t} dt =0 $$</span></p>
<p>These are the FOC of the problem.</p>

CC BY-SA 4.0
2023-12-31T13:04:54.450Z
2023-12-31T15:14:26.473Z
2023-12-31T15:14:26.473Z
Pekisch
<p>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 function, p is the price of y. K is capital, r is the price of capital, L is labor, w is wage.</p>
<p><span class="math-container">$$ \max_{K(t), L(t)} \pi(t)= \int_{t=0}^T (p(t) . y(t) - [r(t).K(t) + w(t).L(t)]).e^{-\rho.t} .dt
$$</span>
Subject to the constraint:
<span class="math-container">$$ S.t: y(t) = K(t)^a . L(t)^b $$</span></p>
<p>Can I have some guidance on how to solve this optimization problem? Do I simply need to set up the lagrangian? Derive it with respect to K and L? I don't think I need to use the Hamiltonian.</p>

<producer-theory><dynamic-optimization><intertemporal-choice>
Intertemporal profit maximization
Meg
# Intertemporal profit maximization

(Asked by: username_0 on 2023-12-28)

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 function, p is the price of y. K is capital, r is the price of capital, L is labor, w is wage.

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

Can I have some guidance on how to solve this optimization problem? Do I simply need to set up the lagrangian? Derive it with respect to K and L? I don't think I need to use the Hamiltonian.

**Tags:** producer-theory, dynamic-optimization, intertemporal-choice

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## 1 answers

### Answer by username_1 on 2023-12-31. Score: 2

This is an intertemporal dynamic problem, but the fact that the restriction contains no derivatives allows us to solve it using a static optimization approach. Replacing the constraint into the objective:
$$
\pi(t) = \int\_0^T \Big[ p(t)K(t)^{a}L(t)^{b} - r(t)K(t) - w(t)L(t) \Big]e^{-\rho t} dt
$$

The fact that there are no derivatives in the constraint means that what the agent does in one instant $t$ of time, does not affect the future. This is why we can simply differentiate with respect to the instruments, $L(t)$ and $K(t)$.

$$ \frac{d\pi(t)}{dL(t)} = \int\_0^T \Big[ bp(t)K(t)^{a}L(t)^{b-1}-w(t) \Big]e^{-\rho t} dt =0 $$

$$ \frac{d\pi(t)}{dK(t)} = \int\_0^T \Big[ap(t)K(t)^{a-1}L(t)^{b}-r(t)\Big]e^{-\rho t} dt =0 $$

These are the FOC of the problem.


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