<p>I kept searching for another hour in this Stack Exchange and it happens that a similar dynamic optimization have been already explained in this other topic <a href="https://economics.stackexchange.com/questions/4589/understanding-subscripts-in-first-order-conditions-of-dynamic-optimization-probl?rq=1">Understanding subscripts in first order conditions of dynamic optimization problems</a> .</p>
<p>It completely cleared my doubt. If someone gets here because of this same question, go there for a solution.</p>

CC BY-SA 4.0
2022-11-28T21:07:32.683Z
2022-11-28T21:07:32.683Z
PGabriel96
<p>First of all, I'd like to say sorry if I couldn't be more specific in the title, I really tried to synthesize the core of my doubt.</p>
<p>I was reading <strong>The Econometric Analysis of Calibrated Macroeconomic Models</strong>, by K. Kim and A.R. Pagan (present in the book <strong>Handbook of Applied Econometrics, Volume 1: Macroeconomics</strong>) and, in order of constructing some steady state variables, they introduce us to an expected utility function (1) and to a production function (2), which is our constraint for the maximization problem of (1):</p>
<p><span class="math-container">$E_t\left\{\sum_{j = 0}^\infty \beta^j U(C_t, (1 - L_t)) \right\} = E_t \left\{ \sum \beta^j [(1  - \sigma)^{-1} C_t^{\theta(1 - \sigma)} ( 1 - L_t)^{(1 - \theta)(1 - \sigma)}] \right\}$</span> <span class="math-container">$(1)$</span></p>
<p><span class="math-container">$Y_t = A_t K_t^\alpha L_t^{1 - \alpha} = A_t F(K_t, L_t) $</span> <span class="math-container">$(2)$</span></p>
<p>It is introduced as well a law of motion of capital (3):</p>
<p><span class="math-container">$K_{t+1} = (1 - \delta) K_t + I_t $</span> <span class="math-container">$(3)$</span></p>
<p>Then, we can formulate the lagrangian:</p>
<p><span class="math-container">$ \mathcal{L} = E_t \sum \beta^j \left\{ [(1  - \sigma)^{-1} C_t^{\theta(1 - \sigma)} ( 1 - L_t)^{(1 - \theta)(1 - \sigma)}] + \lambda_t [ A_tF(K_t, L_t) - C_t - K_{t+1} + (1 - \delta)K_t - G_t] \right\}$</span> <span class="math-container">$(4)$</span></p>
<p>The lagrangian above is used to choose the First Order Conditions (FOCs) for <span class="math-container">$C_t$</span>, <span class="math-container">$L_t$</span>, <span class="math-container">$K_{t+1}$</span> and <span class="math-container">$\lambda_t$</span>.</p>
<p>I will skip to <span class="math-container">$K_{t+1}$</span>'s FOC because it is exactly where my doubt lies. Its FOC is presented by:</p>
<p><span class="math-container">$E_t \left\{ \beta \lambda_{t + 1}[ {A_{t+1} F_K + (1 - \delta)]} - \lambda_t \right\} = 0 $</span> <span class="math-container">$ (5) $</span></p>
<p>I tried using the rationale of shifting every time dependent variable one time period ahead and then calculating the FOC, but my calculation still doesn't match this result in <span class="math-container">$ (5) $</span>.</p>
<p>If it is some trivial arithmetic or calculus step that I am missing, I'd already be glad if someone could point me where I could find something similar in detail so I could figure it out on my own.</p>
<p>Thank you for your attention.</p>

<macroeconomics><mathematical-economics><lagrangian>
Differentiating over multiple time horizons to get FOCs
PGabriel96
# Differentiating over multiple time horizons to get FOCs

(Asked by: username_0 on 2022-11-28)

First of all, I'd like to say sorry if I couldn't be more specific in the title, I really tried to synthesize the core of my doubt.

I was reading **The Econometric Analysis of Calibrated Macroeconomic Models**, by K. Kim and A.R. Pagan (present in the book **Handbook of Applied Econometrics, Volume 1: Macroeconomics**) and, in order of constructing some steady state variables, they introduce us to an expected utility function (1) and to a production function (2), which is our constraint for the maximization problem of (1):

$E\_t\left\{\sum\_{j = 0}^\infty \beta^j U(C\_t, (1 - L\_t)) \right\} = E\_t \left\{ \sum \beta^j [(1 - \sigma)^{-1} C\_t^{\theta(1 - \sigma)} ( 1 - L\_t)^{(1 - \theta)(1 - \sigma)}] \right\}$ $(1)$

$Y\_t = A\_t K\_t^\alpha L\_t^{1 - \alpha} = A\_t F(K\_t, L\_t) $ $(2)$

It is introduced as well a law of motion of capital (3):

$K\_{t+1} = (1 - \delta) K\_t + I\_t $ $(3)$

Then, we can formulate the lagrangian:

$ \mathcal{L} = E\_t \sum \beta^j \left\{ [(1 - \sigma)^{-1} C\_t^{\theta(1 - \sigma)} ( 1 - L\_t)^{(1 - \theta)(1 - \sigma)}] + \lambda\_t [ A\_tF(K\_t, L\_t) - C\_t - K\_{t+1} + (1 - \delta)K\_t - G\_t] \right\}$ $(4)$

The lagrangian above is used to choose the First Order Conditions (FOCs) for $C\_t$, $L\_t$, $K\_{t+1}$ and $\lambda\_t$.

I will skip to $K\_{t+1}$'s FOC because it is exactly where my doubt lies. Its FOC is presented by:

$E\_t \left\{ \beta \lambda\_{t + 1}[ {A\_{t+1} F\_K + (1 - \delta)]} - \lambda\_t \right\} = 0 $ $ (5) $

I tried using the rationale of shifting every time dependent variable one time period ahead and then calculating the FOC, but my calculation still doesn't match this result in $ (5) $.

If it is some trivial arithmetic or calculus step that I am missing, I'd already be glad if someone could point me where I could find something similar in detail so I could figure it out on my own.

Thank you for your attention.

**Tags:** macroeconomics, mathematical-economics, lagrangian

---

## 1 answers

### Answer by username_0 on 2022-11-28. Score: 1 | Accepted answer

I kept searching for another hour in this Stack Exchange and it happens that a similar dynamic optimization have been already explained in this other topic [Understanding subscripts in first order conditions of dynamic optimization problems](https://economics.stackexchange.com/questions/4589/understanding-subscripts-in-first-order-conditions-of-dynamic-optimization-probl?rq=1) .

It completely cleared my doubt. If someone gets here because of this same question, go there for a solution.


---

