<p>Your problem is
<span class="math-container">$$
\max_{C_t} \sum_{t = 0}^\infty \ln(C_t) \text{ s.t. } \begin{cases} H_t = H_{1,t} + H_{2,t},\\
H_{t+1} = B H_{2,t},\\
C_t = X^\alpha H_{1,t}^{1-\alpha},\\
H_{2,0} \text{ fixed}
\end{cases}
$$</span>
First, we can substitute out <span class="math-container">$C_t$</span>, <span class="math-container">$H_{1,t}$</span> and <span class="math-container">$H_{2,t}$</span> to get:
<span class="math-container">$$
\max_{H_{2,t}} \sum_{t = 0}^\infty \ln(X^\alpha(H_t - H_{2,t})^{1-\alpha}) \text{ s.t. } \begin{cases}
H_{t+1} = B H_{2,t},\\
H_{2,t} \in [0, H_t],\\
H_{0} \text{ fixed}
\end{cases}
$$</span>
This gives the Bellman equation:
<span class="math-container">$$
v(H_t) =\max_{H_{2,t} \in [0, H_t]}\left\{\ln(X^\alpha (H_t - H_{2,t})^{1 - \alpha}) + \beta v(B H_{2,t})\right\}
$$</span>
Here <span class="math-container">$H_t$</span> is the state variable and <span class="math-container">$H_{2,t}$</span> is the decision variable.</p>
<p>The first order condition gives:
<span class="math-container">$$
\begin{align*}
&amp;-\frac{1 - \alpha}{H_t - H_{2,t}} + \beta v'(B H_{2,t}) B = 0,\\
\leftrightarrow &amp;\beta B v'(H_{t+1}) = \frac{1 - \alpha}{H_t - H_{2,t}}.
\end{align*}
$$</span>
The envelope condition gives,
<span class="math-container">$$
v'(H_t) = \frac{1 - \alpha}{H_t - H_{2,t}} \to v'(H_{t+1}) = \frac{1 - \alpha}{H_{t+1} - H_{2,t+1}}
$$</span>
As such,
<span class="math-container">$$
\frac{1 - \alpha}{H_t - H_{2,t}} = \beta B \frac{1 - \alpha}{H_{t+1}- H_{2,t+1}}
$$</span></p>
<p>So,
<span class="math-container">$$
\begin{align*}
&amp;H_{t+1} - H_{2,t+1} = \beta B (H_t - H_{2,t}),\\
\leftrightarrow &amp;B H_{2,t} - H_{2,t+1} = \beta B (B H_{2,t-1} - H_{2,t}),\\
\leftrightarrow &amp;H_{2,t+1} = (1 + \beta) B H_{2,t} - \beta B^2 H_{2,t-1}.
\end{align*}
$$</span>
We can also express everything int terms of <span class="math-container">$H_{1,t}$</span> to get,
<span class="math-container">$$
\begin{align*}
&amp;\frac{1 - \alpha}{H_t - H_{2,t}} = \beta B \frac{1 - \alpha}{H_{t+1}- H_{2,t+1}},\\
\leftrightarrow &amp; \frac{1 - \alpha}{H_{1,t}} = \beta B \frac{1- \alpha}{H_{1,t+1}},\\
\leftrightarrow &amp; H_{1,t+1} = \beta B H_{1,t}.
\end{align*}
$$</span></p>

CC BY-SA 4.0
2024-12-31T10:30:18.877Z
2024-12-31T10:30:18.877Z
tdm
<p>We have the following Lucas Model setup: a representative consumer:</p>
<p><span class="math-container">$$\sum_{t=0}^\infty\beta^t\log(C_t)$$</span></p>
<p>Production function that uses <span class="math-container">$X$</span>, a fixed amount of land and <span class="math-container">$H_{1,t}$</span>, human capital:</p>
<p><span class="math-container">$$C_t=X^{\alpha}H_{1,t}^{1-\alpha}$$</span></p>
<p>The production function for human capital is given by:</p>
<p><span class="math-container">$$H_{t+1}=BH_{2,t}$$</span></p>
<p>And total human capital is distributed between production and human capital accumulation:</p>
<p><span class="math-container">$$H_t=H_{1,t}+H_{2,t}$$</span></p>
<p>I woud like to do 2 things: (1) formulate the Bellman equation, and (2) find FOCs and Envelope Conditions.</p>
<p>My difficulties stem from the fact that I am accustomed to seeing physical capital <span class="math-container">$K_t$</span> in these problems. As it stands now, I set up my Bellman equation recursively:</p>
<p><span class="math-container">$$v(H)=\max\Big\{\log(C_t)+\beta v(H')\Big\}$$</span></p>
<p>I then plugged in for <span class="math-container">$C_t$</span>, <span class="math-container">$H'$</span>, and <span class="math-container">$H_1$</span> to get my Bellman equation:</p>
<p><span class="math-container">$$v(H)=\max_{H_2}\Big\{\alpha\log(X)+(1-\alpha)\log(H-H_2)+\beta v(BH_2)\Big\}$$</span></p>
<p>I am reasonably confident that this is correct, but it leads to some quirks when I look at FOCs and ECs. I do not believe we max with respect to <span class="math-container">$X$</span>, since it's just some fixed amount of land.</p>
<p>Now part (2). The envelope condition just refers to the partial of the Bellman equation with respect to <span class="math-container">$H$</span> in this case, so that should be:</p>
<p><span class="math-container">$$\frac{\partial v(H)}{\partial H} = \frac{1-\alpha}{H-H_2}=\frac{1-\alpha}{H_1}$$</span></p>
<p>The FOC is where I'm getting stuck. I need to differentiate with respect to <span class="math-container">$H'$</span>, so:</p>
<p><span class="math-container">$$0 = \beta B\frac{\partial v(H')}{\partial H'} + \frac{1-\alpha}{H_1}$$</span></p>
<p>I am not convinced this is quite right. Furthermore, I'm a bit confused as to the role of <span class="math-container">$X$</span> in all of these equations. I either have to ignore it as I am, or I need to integrate it into my FOCs and ECs, which doesn't seem right to me given that it's a constant (although I had considered this at first, given that it was analogous to <span class="math-container">$K$</span> like I've seen it in the model before).</p>
<p>Any suggestions would be sincerely appreciated. Thank you for your time.</p>

<optimization><bellman-equations>
Lucas Human Capital Model with Land
scoopfaze
# Lucas Human Capital Model with Land

(Asked by: username_0 on 2024-12-31)

We have the following Lucas Model setup: a representative consumer:

$$\sum\_{t=0}^\infty\beta^t\log(C\_t)$$

Production function that uses $X$, a fixed amount of land and $H\_{1,t}$, human capital:

$$C\_t=X^{\alpha}H\_{1,t}^{1-\alpha}$$

The production function for human capital is given by:

$$H\_{t+1}=BH\_{2,t}$$

And total human capital is distributed between production and human capital accumulation:

$$H\_t=H\_{1,t}+H\_{2,t}$$

I woud like to do 2 things: (1) formulate the Bellman equation, and (2) find FOCs and Envelope Conditions.

My difficulties stem from the fact that I am accustomed to seeing physical capital $K\_t$ in these problems. As it stands now, I set up my Bellman equation recursively:

$$v(H)=\max\Big\{\log(C\_t)+\beta v(H')\Big\}$$

I then plugged in for $C\_t$, $H'$, and $H\_1$ to get my Bellman equation:

$$v(H)=\max\_{H\_2}\Big\{\alpha\log(X)+(1-\alpha)\log(H-H\_2)+\beta v(BH\_2)\Big\}$$

I am reasonably confident that this is correct, but it leads to some quirks when I look at FOCs and ECs. I do not believe we max with respect to $X$, since it's just some fixed amount of land.

Now part (2). The envelope condition just refers to the partial of the Bellman equation with respect to $H$ in this case, so that should be:

$$\frac{\partial v(H)}{\partial H} = \frac{1-\alpha}{H-H\_2}=\frac{1-\alpha}{H\_1}$$

The FOC is where I'm getting stuck. I need to differentiate with respect to $H'$, so:

$$0 = \beta B\frac{\partial v(H')}{\partial H'} + \frac{1-\alpha}{H\_1}$$

I am not convinced this is quite right. Furthermore, I'm a bit confused as to the role of $X$ in all of these equations. I either have to ignore it as I am, or I need to integrate it into my FOCs and ECs, which doesn't seem right to me given that it's a constant (although I had considered this at first, given that it was analogous to $K$ like I've seen it in the model before).

Any suggestions would be sincerely appreciated. Thank you for your time.

**Tags:** optimization, bellman-equations


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## 1 answers

### Answer by username_1 on 2024-12-31. Score: 2 | Accepted answer

Your problem is
$$
\max\_{C\_t} \sum\_{t = 0}^\infty \ln(C\_t) \text{ s.t. } \begin{cases} H\_t = H\_{1,t} + H\_{2,t},\\
H\_{t+1} = B H\_{2,t},\\
C\_t = X^\alpha H\_{1,t}^{1-\alpha},\\
H\_{2,0} \text{ fixed}
\end{cases}
$$
First, we can substitute out $C\_t$, $H\_{1,t}$ and $H\_{2,t}$ to get:
$$
\max\_{H\_{2,t}} \sum\_{t = 0}^\infty \ln(X^\alpha(H\_t - H\_{2,t})^{1-\alpha}) \text{ s.t. } \begin{cases}
H\_{t+1} = B H\_{2,t},\\
H\_{2,t} \in [0, H\_t],\\
H\_{0} \text{ fixed}
\end{cases}
$$
This gives the Bellman equation:
$$
v(H\_t) =\max\_{H\_{2,t} \in [0, H\_t]}\left\{\ln(X^\alpha (H\_t - H\_{2,t})^{1 - \alpha}) + \beta v(B H\_{2,t})\right\}
$$
Here $H\_t$ is the state variable and $H\_{2,t}$ is the decision variable.

The first order condition gives:
$$
\begin{align\*}
&-\frac{1 - \alpha}{H\_t - H\_{2,t}} + \beta v'(B H\_{2,t}) B = 0,\\
\leftrightarrow &\beta B v'(H\_{t+1}) = \frac{1 - \alpha}{H\_t - H\_{2,t}}.
\end{align\*}
$$
The envelope condition gives,
$$
v'(H\_t) = \frac{1 - \alpha}{H\_t - H\_{2,t}} \to v'(H\_{t+1}) = \frac{1 - \alpha}{H\_{t+1} - H\_{2,t+1}}
$$
As such,
$$
\frac{1 - \alpha}{H\_t - H\_{2,t}} = \beta B \frac{1 - \alpha}{H\_{t+1}- H\_{2,t+1}}
$$

So,
$$
\begin{align\*}
&H\_{t+1} - H\_{2,t+1} = \beta B (H\_t - H\_{2,t}),\\
\leftrightarrow &B H\_{2,t} - H\_{2,t+1} = \beta B (B H\_{2,t-1} - H\_{2,t}),\\
\leftrightarrow &H\_{2,t+1} = (1 + \beta) B H\_{2,t} - \beta B^2 H\_{2,t-1}.
\end{align\*}
$$
We can also express everything int terms of $H\_{1,t}$ to get,
$$
\begin{align\*}
&\frac{1 - \alpha}{H\_t - H\_{2,t}} = \beta B \frac{1 - \alpha}{H\_{t+1}- H\_{2,t+1}},\\
\leftrightarrow & \frac{1 - \alpha}{H\_{1,t}} = \beta B \frac{1- \alpha}{H\_{1,t+1}},\\
\leftrightarrow & H\_{1,t+1} = \beta B H\_{1,t}.
\end{align\*}
$$


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