<p>Coudl you write the entire problem as you read in the book? I don't understand which variables are control variables or state variables. Condition [2] is found in this way</p>
<p><span class="math-container">$\displaystyle-\frac{dH}{dy}=\dot{costate}$</span> (equation of motion of costate variable)</p>
<p>there is also one more condition, which is the following</p>
<p><span class="math-container">$\displaystyle\frac{dH}{dcostate}=\dot{y}$</span> (equation of motion of state variable)</p>

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2022-11-01T10:12:04.733Z
2022-11-01T10:12:04.733Z
Pepus
<p>I have problems with writing the current value Hamiltonian and finding the related necessary and sufficient conditions:</p>
<p>I should max the RHS of : <span class="math-container">$\rho U_t = log (\chi) + \frac{1}{\rho} log(\lambda)+ M(t)log(\lambda) $</span></p>
<p>s.t.(resource constraint): <span class="math-container">$a(\lambda)t + \chi = L$</span> ; and <span class="math-container">$M=\dot{\mu}$</span></p>
<p>According to the book, substitute for <span class="math-container">$\chi$</span> from the resource constraint and solve the problem that has <span class="math-container">$\mu \ge 0$</span> and <span class="math-container">$\lambda \ge0 $</span> as the only constraints</p>
<p><span class="math-container">$\frac{log(\lambda)}{\rho}=\frac{a(\lambda)}{L-a(\lambda)\mu}$</span> <span class="math-container">$~~$</span>[1]</p>
<p><span class="math-container">$\frac{a'(\lambda)\mu}{L-a(\lambda)\mu}=\frac{\mu}{\rho \lambda}$</span> <span class="math-container">$~~$</span>[2]</p>
<p>Where <span class="math-container">$\chi$</span> is the total amount of labor in manufacturing corresponding to the optimal aggregate demand for intermediate goods (since one unit of labor manufactures one unit of intermediate, and we have a unit continuum of industries).
<span class="math-container">$\lambda$</span> is the step-size of the innovation (innovation evolves stochastically and increases in each period by a factor of <span class="math-container">$\lambda &gt; 1$</span>, with probability <span class="math-container">$\mu dt$</span> in a time interval <span class="math-container">$dt$</span>. <span class="math-container">$\mu$</span> is the rate of innovation</p>
<p>M(t) is the expected number of successes in innovation before time <span class="math-container">$\tau$</span> : <span class="math-container">$M(\tau)= \int_{0}^{\tau} \mu(s) d s$</span>.</p>
<p><span class="math-container">$L$</span> is labor and <span class="math-container">$a(\lambda)$</span> is an increasing function.</p>
<p>Sorry for this question but what I learned in class was:</p>
<p>FOC:</p>
<ol>
<li><p>derivative hamiltonian w.r.t. controls variables=0</p>
</li>
<li><p>derivative hamiltonian w.r.t. state variable = d(costate variable)t</p>
</li>
<li><p>transversality condition</p>
</li>
</ol>
<p>How to find [1],[2]?</p>

<macroeconomics><economic-growth>
Current Value Hamiltonian
John M.
# Current Value Hamiltonian

(Asked by: username_0 on 2022-08-23)

I have problems with writing the current value Hamiltonian and finding the related necessary and sufficient conditions:

I should max the RHS of : $\rho U\_t = log (\chi) + \frac{1}{\rho} log(\lambda)+ M(t)log(\lambda) $

s.t.(resource constraint): $a(\lambda)t + \chi = L$ ; and $M=\dot{\mu}$

According to the book, substitute for $\chi$ from the resource constraint and solve the problem that has $\mu \ge 0$ and $\lambda \ge0 $ as the only constraints

$\frac{log(\lambda)}{\rho}=\frac{a(\lambda)}{L-a(\lambda)\mu}$ $~~$[1]

$\frac{a'(\lambda)\mu}{L-a(\lambda)\mu}=\frac{\mu}{\rho \lambda}$ $~~$[2]

Where $\chi$ is the total amount of labor in manufacturing corresponding to the optimal aggregate demand for intermediate goods (since one unit of labor manufactures one unit of intermediate, and we have a unit continuum of industries).
$\lambda$ is the step-size of the innovation (innovation evolves stochastically and increases in each period by a factor of $\lambda > 1$, with probability $\mu dt$ in a time interval $dt$. $\mu$ is the rate of innovation

M(t) is the expected number of successes in innovation before time $\tau$ : $M(\tau)= \int\_{0}^{\tau} \mu(s) d s$.

$L$ is labor and $a(\lambda)$ is an increasing function.

Sorry for this question but what I learned in class was:

FOC:

1. derivative hamiltonian w.r.t. controls variables=0
2. derivative hamiltonian w.r.t. state variable = d(costate variable)t
3. transversality condition

How to find [1],[2]?

**Tags:** macroeconomics, economic-growth

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## 1 answers

### Answer by username_1 on 2022-11-01. Score: 2

Coudl you write the entire problem as you read in the book? I don't understand which variables are control variables or state variables. Condition [2] is found in this way

$\displaystyle-\frac{dH}{dy}=\dot{costate}$ (equation of motion of costate variable)

there is also one more condition, which is the following

$\displaystyle\frac{dH}{dcostate}=\dot{y}$ (equation of motion of state variable)


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