<p>The problem is that you are ignoring the division <span class="math-container">$ \frac {0} {0} $</span>, which is in <span class="math-container">$ \frac {\partial L} {\partial v} $</span>. Before looking at the solution, and seeing that indeed when <span class="math-container">$ \theta_ {2} = 0 \ \Rightarrow \ v ^ {*} = 0 $</span>, I want to note that the first constraint <span class="math-container">$ c_ {1} = w - (1 -v) \theta_ {1} h_ {1} ^ {a1} $</span> can be more realistic and logical. It is easy to see that <span class="math-container">$ \frac {\partial c_ {1}} {\partial v}&gt; 0 $</span> in the restriction, which implies that by saving to invest in human capital, your income increases in the first period. This implies that there is no trade-off between investing human capital and consumption (if the interest rate were less than one it would reduce the problem a little, but it would still be inherently wrong). And second <span class="math-container">$ c_ {1} = w - (1-v) \theta_ {1} h_ {1} ^ {a1}$</span> does not allow saving wealth, only income. A more reasonable restriction would be this <span class="math-container">$ c_ {1} = (1-v) (w - \theta_ {1} h_ {1} ^ {a1}) $</span>
\
I'm going to ignore the second observation and go on to answer your question, I just thought it relevant to point it out. We are going to work with this restriction <span class="math-container">$ c_ {1} = w + (1-v) \theta_ {1} h_ {1} ^ {a1} $</span>. This does not affect <span class="math-container">$ \frac {\partial L} {\partial c_ {1}}, \frac {\partial L} {\partial c_ {2}} $</span>, but it does <span class="math-container">$ \frac {\partial L} {\partial v} $</span>. The third first order condition would be:</p>
<p><span class="math-container">\begin{align}
\frac {\partial L} {\partial v} = \lambda_ {1} \theta_ {1} h_ {1} ^ a - - \theta_ {2} h_ {1} ^ a) (1 + v) ^ {1-a} a \lambda_ {2} \beta = 0 \\
\frac {\partial L} {\partial v} = \lambda_ {1} \theta_ {1} - \theta_ {2} (1 + v) ^ {1-a} a \lambda_ {2} \beta = 0
\end{align}</span></p>
<p>We derive the euler equation by introducing the constraints at <span class="math-container">$ \frac {\partial L} {\partial v} = 0 $</span>:</p>
<p><span class="math-container">\begin{align}
\frac {\theta_ {1}} {w- (1 + v) h_ {1} ^ {a}} = \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {\theta_ {2} h_ {1} ^ {a} (1 + v) ^ {a} }
\end{align}</span></p>
<p>It seems that you can safely say that <span class="math-container">$ \frac {\theta_ {2}} {\theta_ {2}} = 1 \ \forall \theta_ {2} $</span>, but this is not true when <span class="math-container">$ \theta_ {2} = 0 $</span>, but this is not the case because when this occurs <span class="math-container">$ \frac {\partial L} {\partial v} = 0 \ \forall c_ {1}, c_ {2}, v $</span>. This is made clear by multiplying <span class="math-container">$ \frac {\partial L} {\partial v} = 0 $</span> by <span class="math-container">$ \theta_ {2} $</span>:</p>
<p><span class="math-container">\begin{align}
\frac {\partial L} {\partial v} = \frac {\theta_ {1}} {w- (1 + v) h_ {1} ^ {a}} + \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {\theta_ {2} h_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = \frac {\theta_ {1} \theta_ {2}} {w- (1 + v) h_ {1} ^ {a}} + \frac {\beta \theta_ {2} (1 + v) ^ {1-a} a} {h_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = 0 \ \forall c_ {1}, c_ {2}, v
\end{align}</span></p>
<p>Therefore the solution of the euler's equation for <span class="math-container">$ v $</span> is valid if only if <span class="math-container">$ \theta_ {2} \neq 0 $</span>. What is the optimal value of <span class="math-container">$ v $</span> if <span class="math-container">$ \theta_ {2} = 0 $</span>? Since we have three unknowns and <span class="math-container">$ v $</span> only appears in 2 of them, we cannot derive a solution by substitution. You have to see what the effect of v is on the utility function. To achieve this, we obtain the optimal values of <span class="math-container">$ c_ {1} $</span> and <span class="math-container">$ c_ {2} $</span> by solving for the euler equation (without substituting the constraints and we obtain the following:</p>
<p><span class="math-container">\begin{align}
c_ {1} ^ {*} = \frac {\theta_ {1}} {(1 + v) a} \\
c_ {2} ^ {*} = \frac {\theta_ {2}} {\theta_ {1}} (w (1 + v) ^ {1-a} - (1 + v) ^ {- a} h_ {1} ^ {a} \theta_ {1} = 0
\end{align}</span></p>
<p>But this implies that <span class="math-container">$ c_ {2} ^ {*} $</span> is a corner solution, therefore the constraint <span class="math-container">$ c_{2} = \theta_ {2} h^ {a} $</span> is not binding, so which is not valid. I'm not going to go into much detail why, but you can learn more in Chapter 18 of Simon and Blume's  <em><strong>mathematics for economists</strong></em>  book. The idea is that if  <span class="math-container">$c_ {2} $</span> is 0 the constraint of this variable cannot be fulfilled, it remains as an inequality and is irrelevant to the optimization problem. So it becomes an optimization problem in 2 veritable <span class="math-container">$ v, c_ {1} $</span> but how <span class="math-container">$ \frac {\partial c_ {2} ^ {*}} {\partial v} &lt;0 $</span>. So the optimal level of <span class="math-container">$ v $</span> is 0? Without any restriction on the value of <span class="math-container">$ v $</span> the solution is <span class="math-container">$ v = - \infty $</span> !!!. With restriction its optimal value is equal to 0. Note: this is only true if you make a modification to the utility function, otherwise the optimization problem is not defined. See explanation in &quot;edit&quot;.</p>
<h1>Edit</h1>
<p>I made some corrections to what I had previously written. On that <span class="math-container">$ v $</span> does not depend on <span class="math-container">$ \theta_ {2} $</span>, it does; only in a similar way to what happens when you use quasilinear utility functions. In these types of functions, the good that appears linearly in the utility function does not depend on income if you derive first-order conditions. The problem with this is that the first order conditions are only valid, in this case, when the quantities consumed of both goods are positive. This comes in any intermediate microeconomics book.
It turns out this is general. The first order conditions of an optimization problem in which it is not explicitly specified that the values of the endogenous variables must be greater than or equal to 0. This implies that there will also be parameter values that make the optimal solution let 0 be the endogenous variables. In this case, <span class="math-container">$ \theta_ {2} $</span>.</p>
<p>The problem is that when <span class="math-container">$ \theta_ {2} $</span> equals 0, the objective function is undefined <span class="math-container">$ \theta_ {2} = 0 \ \Rightarrow \ c_ {2} = 0 \ \Rightarrow \ log (c_ {2 }) = - \infty $</span>, so the value of <span class="math-container">$ theta_ {2} = 0 $</span> is not possible. But with slight modifications like changing in the utility function <span class="math-container">$ log (c_ {2}) $</span> for <span class="math-container">$ log (c_ {2} +1) $</span>, the problem is defined and by substituting the constraint of <span class="math-container">$ c_ {2} $</span> in the utility function, it becomes independent of <span class="math-container">$ c_ {2} $</span> and the optimization problem becomes 2 variables and since <span class="math-container">$ v $</span> is only a cost, the optimal solution is <span class="math-container">$ 0 $</span> with restriction and <span class="math-container">$ - \infty $</span> Without restrictions. So when <span class="math-container">$ \theta_{2} = 0 \ \Rightarrow \ v^{*} = 0$</span>, and when <span class="math-container">$ \theta_ {2} \neq 
 0$</span> the optimal value of <span class="math-container">$v$</span> is a constant that comes out of the problem of optimization. So there is a dependency, but it breaks when <span class="math-container">$ \theta_ {2}&gt; 0 $</span>.</p>
<p>Now why is there no dependency when <span class="math-container">$ \theta_ {2}&gt; 0 $</span> ?. As @Bertrand commented, this could be changed if there were disutility of accumulation of human capital in the utility function. But it is not the only way, you can also allow to save wealth to invest in human capital, this will create the expected dependence for <span class="math-container">$\theta_{2}&gt;0$</span>. I think that change is very reasonable; why the agent would be not allowed to save wealth to invest in human capital?</p>
<p>Regarding what they told you about what happens when <span class="math-container">$ w = 0 $</span>, again you will see that this puts more restrictions on the parameters, in fact it has to be true that <span class="math-container">$ a \beta = 1 $</span>, otherwise the optimization problem is not definite.</p>

CC BY-SA 4.0
2020-06-27T08:16:37.040Z
Thank you for your answer! As per your first comment on $c_1$, I indeed make a typo. It was meant to be $c_1 = w + (1 -\nu)\theta_1 h_1^a$, that is, consumption in the first period is equals to initial wealth and income in the first period. I've edited the question accordingly. As for your answer on what happens when $\theta_2 = 0$, that was also insightful. However, my last sentence on $\theta_2 = 0$ was more hypothetical. My question really is, why does optimal $\nu*$ does not involve $\theta_2$ when $\theta_2$ is not equals to 0?
Hosea
CC BY-SA 4.0
2020-06-27T19:01:39.840Z
The short answer is that the optimization problem is not defined if $ \theta_ {2} = 0 $, but you can make it defined; in that case, $ v $ will be equal to 0. and there will be a dependency when $ \theta_ {2} $ changes from 0 to some positive value. See the edition for more details, and to see how to make this dependency different from 0 $\forall \theta_{2}$.
juarez_Samuel
CC BY-SA 4.0
2020-07-01T13:42:33.870Z
Thank you for your advise! Will try out different specification for accumulation of human capital or savings. I think both options you have mentioned are reasonable.
Hosea
CC BY-SA 4.0
2020-06-26T22:54:37.593Z
2020-06-28T15:09:24.087Z
2020-06-28T15:09:24.087Z
juarez_Samuel
<p>I'm trying to solve a constrained optimization problem in human capital model. The objective function is</p>
<p><span class="math-container">$\max_{c_1,c_2,\nu} U = u(c_1) + \beta u(c_2)$</span>,</p>
<p>subjected to</p>
<p><span class="math-container">$c_1 = w +(1-\nu)\theta_1 h_1^a$</span> and <span class="math-container">$c_2 =  \theta_2 h_2^a$</span>.</p>
<p>The list below summarizes the variables used:</p>
<ul>
<li><span class="math-container">$c_t$</span> is the consumption in period <span class="math-container">$t$</span>. Note that <span class="math-container">$u(c) = log(c)$</span>.</li>
<li><span class="math-container">$\theta_t$</span> is the wage rate in period <span class="math-container">$t$</span>.</li>
<li><span class="math-container">$\nu$</span> is time spent in the first period to accumulate human capital. <span class="math-container">$\nu$</span> is normalized to be between [0,1], and <span class="math-container">$(1-\nu)$</span> is time spent working in first period.</li>
<li><span class="math-container">$h_t$</span> is the human capital in period <span class="math-container">$t$</span>. Note that <span class="math-container">$h_2 = h_1(1+\nu)$</span>.</li>
<li><span class="math-container">$a$</span> is the innate ability. Together, <span class="math-container">$\theta_t h_t^a$</span> represents income in period <span class="math-container">$t$</span>.</li>
<li><span class="math-container">$w$</span> is the initial wealth.</li>
</ul>
<p>So, given <span class="math-container">$(w,a,h_1)$</span>, individuals choose optimal <span class="math-container">$\nu$</span> in the first period that determines consumption in both first and second period. <span class="math-container">$\theta$</span> is exogenous variable. Now solving this optimization problem using the method of Lagrange equation:</p>
<p><span class="math-container">$L = u(c_1) + \beta u(c_2) - \lambda_1(c_1 - w -(1-\nu)\theta_1 h_1^a) - \lambda_2(c_2 - \theta_2 h_2^a)$</span>.</p>
<p>Solving for <span class="math-container">$\dfrac{\partial L}{\partial c_1} = \dfrac{\partial L}{\partial c_2} = 0$</span> gives the following two equation:</p>
<p><span class="math-container">$c_1 = \dfrac{1}{\lambda_1}$</span> and <span class="math-container">$c_2 = \dfrac{\beta}{\lambda_2}$</span>.</p>
<p>Now solving for <span class="math-container">$\dfrac{\partial L}{\partial \nu} = 0$</span>:</p>
<p><span class="math-container">$\dfrac{\partial L}{\partial \nu} = -\lambda_1 \theta_1 h_1^a + \lambda_2 \theta_2 a h_1^a(1+\nu)^{a-1}$</span> = 0.</p>
<p>Substituting <span class="math-container">$\lambda_1$</span> and <span class="math-container">$\lambda_2$</span>, we get:</p>
<p><span class="math-container">$\dfrac{\theta_1 h_1^a}{c_1} = \dfrac{\theta_2 a \beta h_1^a(1+\nu)^{a-1}}{c_2}$</span>.</p>
<p>Substituting the equality constraints and solving for <span class="math-container">$\nu$</span>, we get:</p>
<p><span class="math-container">$\dfrac{\theta_1 h_1^a}{w+(1-\nu)\theta_1 h_1^a} = \dfrac{\theta_2 a \beta h_1^a(1+\nu)^{a-1}}{\theta_2 h_1^a (1+\nu)^a}$</span></p>
<p><span class="math-container">$\dfrac{\theta_1 h_1^a}{w+(1-\nu)\theta_1 h_1^a} = \dfrac{a \beta}{1+\nu}$</span></p>
<p><span class="math-container">$\nu = \dfrac{a \beta w + a \beta \theta_1 h_1^a - \theta_1 h_1^a}{(\theta_1 h_1^a)(1+a \beta)}$</span>.</p>
<p>What I do not understand is why <span class="math-container">$\theta_2$</span> does not play a role in determining the optimal <span class="math-container">$\nu$</span>. Logically, individuals invest in human capital in the first period, by foregoing possible income in the first period, to earn more income in the second period. However, even if <span class="math-container">$\lim_{\theta_2 \to 0}$</span>, this solution will still recommend individuals to invest in human capital in the first period, by exactly <span class="math-container">$\dfrac{a \beta w + a \beta \theta_1 h_1^a - \theta_1 h_1^a}{(\theta_1 h_1^a)(1+a \beta)}$</span> much.</p>

CC BY-SA 4.0
2020-06-27T10:12:38.643Z
What are your choice variables? From the statement of your maximization problem it looks like that only $c_1$ and $c_2$ are your choice variables. In this case however, with two constraints, there is no degree of freedom left for adjusting anything. You may mention right at the beginning that $h_1, h_2$ are choice variables (if they are). If this is the case you can substitute the constraints into the objective function and solve the new unconstrained maximization problem.
Bertrand
CC BY-SA 4.0
2020-06-27T10:43:19.577Z
The choice variables are $c_1,c_2, \nu$. However, $c_1,c_2$ is determined by exogenous variables and $\nu$, so really the problem is solving for optimal $\nu$.
Hosea
CC BY-SA 4.0
2020-06-27T15:53:00.517Z
Note that for $w=0$ your optimal $\nu$ does not depend on $\theta_1$ either. Your result is due to your timewise separable utility function, and absence of disutility of labor or of human capital formation.
Bertrand
<optimization><human-capital>
Solving Constrained Optimization Problem with Two-Period Model of Human Capital
Hosea
# Solving Constrained Optimization Problem with Two-Period Model of Human Capital

(Asked by: username_0 on 2020-06-23)

I'm trying to solve a constrained optimization problem in human capital model. The objective function is

$\max\_{c\_1,c\_2,\nu} U = u(c\_1) + \beta u(c\_2)$,

subjected to

$c\_1 = w +(1-\nu)\theta\_1 h\_1^a$ and $c\_2 = \theta\_2 h\_2^a$.

The list below summarizes the variables used:

* $c\_t$ is the consumption in period $t$. Note that $u(c) = log(c)$.
* $\theta\_t$ is the wage rate in period $t$.
* $\nu$ is time spent in the first period to accumulate human capital. $\nu$ is normalized to be between [0,1], and $(1-\nu)$ is time spent working in first period.
* $h\_t$ is the human capital in period $t$. Note that $h\_2 = h\_1(1+\nu)$.
* $a$ is the innate ability. Together, $\theta\_t h\_t^a$ represents income in period $t$.
* $w$ is the initial wealth.

So, given $(w,a,h\_1)$, individuals choose optimal $\nu$ in the first period that determines consumption in both first and second period. $\theta$ is exogenous variable. Now solving this optimization problem using the method of Lagrange equation:

$L = u(c\_1) + \beta u(c\_2) - \lambda\_1(c\_1 - w -(1-\nu)\theta\_1 h\_1^a) - \lambda\_2(c\_2 - \theta\_2 h\_2^a)$.

Solving for $\dfrac{\partial L}{\partial c\_1} = \dfrac{\partial L}{\partial c\_2} = 0$ gives the following two equation:

$c\_1 = \dfrac{1}{\lambda\_1}$ and $c\_2 = \dfrac{\beta}{\lambda\_2}$.

Now solving for $\dfrac{\partial L}{\partial \nu} = 0$:

$\dfrac{\partial L}{\partial \nu} = -\lambda\_1 \theta\_1 h\_1^a + \lambda\_2 \theta\_2 a h\_1^a(1+\nu)^{a-1}$ = 0.

Substituting $\lambda\_1$ and $\lambda\_2$, we get:

$\dfrac{\theta\_1 h\_1^a}{c\_1} = \dfrac{\theta\_2 a \beta h\_1^a(1+\nu)^{a-1}}{c\_2}$.

Substituting the equality constraints and solving for $\nu$, we get:

$\dfrac{\theta\_1 h\_1^a}{w+(1-\nu)\theta\_1 h\_1^a} = \dfrac{\theta\_2 a \beta h\_1^a(1+\nu)^{a-1}}{\theta\_2 h\_1^a (1+\nu)^a}$

$\dfrac{\theta\_1 h\_1^a}{w+(1-\nu)\theta\_1 h\_1^a} = \dfrac{a \beta}{1+\nu}$

$\nu = \dfrac{a \beta w + a \beta \theta\_1 h\_1^a - \theta\_1 h\_1^a}{(\theta\_1 h\_1^a)(1+a \beta)}$.

What I do not understand is why $\theta\_2$ does not play a role in determining the optimal $\nu$. Logically, individuals invest in human capital in the first period, by foregoing possible income in the first period, to earn more income in the second period. However, even if $\lim\_{\theta\_2 \to 0}$, this solution will still recommend individuals to invest in human capital in the first period, by exactly $\dfrac{a \beta w + a \beta \theta\_1 h\_1^a - \theta\_1 h\_1^a}{(\theta\_1 h\_1^a)(1+a \beta)}$ much.

**Tags:** optimization, human-capital
> **Comments:**
> **username_1**: What are your choice variables? From the statement of your maximization problem it looks like that only $c_1$ and $c_2$ are your choice variables. In this case however, with two constraints, there is no degree of freedom left for adjusting anything. You may mention right at the beginning that $h_1, h_2$ are choice variables (if they are). If this is the case you can substitute the constraints into the objective function and solve the new unconstrained maximization problem.  
> **username_0**: The choice variables are $c_1,c_2, \nu$. However, $c_1,c_2$ is determined by exogenous variables and $\nu$, so really the problem is solving for optimal $\nu$.  
> **username_1**: Note that for $w=0$ your optimal $\nu$ does not depend on $\theta_1$ either. Your result is due to your timewise separable utility function, and absence of disutility of labor or of human capital formation.  

---

## 1 answers

### Answer by username_2 on 2020-06-26. Score: 2 | Accepted answer

The problem is that you are ignoring the division $ \frac {0} {0} $, which is in $ \frac {\partial L} {\partial v} $. Before looking at the solution, and seeing that indeed when $ \theta\_ {2} = 0 \ \Rightarrow \ v ^ {\*} = 0 $, I want to note that the first constraint $ c\_ {1} = w - (1 -v) \theta\_ {1} h\_ {1} ^ {a1} $ can be more realistic and logical. It is easy to see that $ \frac {\partial c\_ {1}} {\partial v}> 0 $ in the restriction, which implies that by saving to invest in human capital, your income increases in the first period. This implies that there is no trade-off between investing human capital and consumption (if the interest rate were less than one it would reduce the problem a little, but it would still be inherently wrong). And second $ c\_ {1} = w - (1-v) \theta\_ {1} h\_ {1} ^ {a1}$ does not allow saving wealth, only income. A more reasonable restriction would be this $ c\_ {1} = (1-v) (w - \theta\_ {1} h\_ {1} ^ {a1}) $
\
I'm going to ignore the second observation and go on to answer your question, I just thought it relevant to point it out. We are going to work with this restriction $ c\_ {1} = w + (1-v) \theta\_ {1} h\_ {1} ^ {a1} $. This does not affect $ \frac {\partial L} {\partial c\_ {1}}, \frac {\partial L} {\partial c\_ {2}} $, but it does $ \frac {\partial L} {\partial v} $. The third first order condition would be:

\begin{align}
\frac {\partial L} {\partial v} = \lambda\_ {1} \theta\_ {1} h\_ {1} ^ a - - \theta\_ {2} h\_ {1} ^ a) (1 + v) ^ {1-a} a \lambda\_ {2} \beta = 0 \\
\frac {\partial L} {\partial v} = \lambda\_ {1} \theta\_ {1} - \theta\_ {2} (1 + v) ^ {1-a} a \lambda\_ {2} \beta = 0
\end{align}

We derive the euler equation by introducing the constraints at $ \frac {\partial L} {\partial v} = 0 $:

\begin{align}
\frac {\theta\_ {1}} {w- (1 + v) h\_ {1} ^ {a}} = \frac {\beta \theta\_ {2} (1 + v) ^ {1-a} a} {\theta\_ {2} h\_ {1} ^ {a} (1 + v) ^ {a} }
\end{align}

It seems that you can safely say that $ \frac {\theta\_ {2}} {\theta\_ {2}} = 1 \ \forall \theta\_ {2} $, but this is not true when $ \theta\_ {2} = 0 $, but this is not the case because when this occurs $ \frac {\partial L} {\partial v} = 0 \ \forall c\_ {1}, c\_ {2}, v $. This is made clear by multiplying $ \frac {\partial L} {\partial v} = 0 $ by $ \theta\_ {2} $:

\begin{align}
\frac {\partial L} {\partial v} = \frac {\theta\_ {1}} {w- (1 + v) h\_ {1} ^ {a}} + \frac {\beta \theta\_ {2} (1 + v) ^ {1-a} a} {\theta\_ {2} h\_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = \frac {\theta\_ {1} \theta\_ {2}} {w- (1 + v) h\_ {1} ^ {a}} + \frac {\beta \theta\_ {2} (1 + v) ^ {1-a} a} {h\_ {1} ^ {a} (1 + v) ^ {a}} = 0 \\
\frac {\partial L} {\partial v} = 0 \ \forall c\_ {1}, c\_ {2}, v
\end{align}

Therefore the solution of the euler's equation for $ v $ is valid if only if $ \theta\_ {2} \neq 0 $. What is the optimal value of $ v $ if $ \theta\_ {2} = 0 $? Since we have three unknowns and $ v $ only appears in 2 of them, we cannot derive a solution by substitution. You have to see what the effect of v is on the utility function. To achieve this, we obtain the optimal values of $ c\_ {1} $ and $ c\_ {2} $ by solving for the euler equation (without substituting the constraints and we obtain the following:

\begin{align}
c\_ {1} ^ {\*} = \frac {\theta\_ {1}} {(1 + v) a} \\
c\_ {2} ^ {\*} = \frac {\theta\_ {2}} {\theta\_ {1}} (w (1 + v) ^ {1-a} - (1 + v) ^ {- a} h\_ {1} ^ {a} \theta\_ {1} = 0
\end{align}

But this implies that $ c\_ {2} ^ {\*} $ is a corner solution, therefore the constraint $ c\_{2} = \theta\_ {2} h^ {a} $ is not binding, so which is not valid. I'm not going to go into much detail why, but you can learn more in Chapter 18 of Simon and Blume's ***mathematics for economists*** book. The idea is that if $c\_ {2} $ is 0 the constraint of this variable cannot be fulfilled, it remains as an inequality and is irrelevant to the optimization problem. So it becomes an optimization problem in 2 veritable $ v, c\_ {1} $ but how $ \frac {\partial c\_ {2} ^ {\*}} {\partial v} <0 $. So the optimal level of $ v $ is 0? Without any restriction on the value of $ v $ the solution is $ v = - \infty $ !!!. With restriction its optimal value is equal to 0. Note: this is only true if you make a modification to the utility function, otherwise the optimization problem is not defined. See explanation in "edit".

# Edit

I made some corrections to what I had previously written. On that $ v $ does not depend on $ \theta\_ {2} $, it does; only in a similar way to what happens when you use quasilinear utility functions. In these types of functions, the good that appears linearly in the utility function does not depend on income if you derive first-order conditions. The problem with this is that the first order conditions are only valid, in this case, when the quantities consumed of both goods are positive. This comes in any intermediate microeconomics book.
It turns out this is general. The first order conditions of an optimization problem in which it is not explicitly specified that the values of the endogenous variables must be greater than or equal to 0. This implies that there will also be parameter values that make the optimal solution let 0 be the endogenous variables. In this case, $ \theta\_ {2} $.

The problem is that when $ \theta\_ {2} $ equals 0, the objective function is undefined $ \theta\_ {2} = 0 \ \Rightarrow \ c\_ {2} = 0 \ \Rightarrow \ log (c\_ {2 }) = - \infty $, so the value of $ theta\_ {2} = 0 $ is not possible. But with slight modifications like changing in the utility function $ log (c\_ {2}) $ for $ log (c\_ {2} +1) $, the problem is defined and by substituting the constraint of $ c\_ {2} $ in the utility function, it becomes independent of $ c\_ {2} $ and the optimization problem becomes 2 variables and since $ v $ is only a cost, the optimal solution is $ 0 $ with restriction and $ - \infty $ Without restrictions. So when $ \theta\_{2} = 0 \ \Rightarrow \ v^{\*} = 0$, and when $ \theta\_ {2} \neq
0$ the optimal value of $v$ is a constant that comes out of the problem of optimization. So there is a dependency, but it breaks when $ \theta\_ {2}> 0 $.

Now why is there no dependency when $ \theta\_ {2}> 0 $ ?. As @Bertrand commented, this could be changed if there were disutility of accumulation of human capital in the utility function. But it is not the only way, you can also allow to save wealth to invest in human capital, this will create the expected dependence for $\theta\_{2}>0$. I think that change is very reasonable; why the agent would be not allowed to save wealth to invest in human capital?

Regarding what they told you about what happens when $ w = 0 $, again you will see that this puts more restrictions on the parameters, in fact it has to be true that $ a \beta = 1 $, otherwise the optimization problem is not definite.

> **Comments:**
> **username_0**: Thank you for your answer! As per your first comment on $c_1$, I indeed make a typo. It was meant to be $c_1 = w + (1 -\nu)\theta_1 h_1^a$, that is, consumption in the first period is equals to initial wealth and income in the first period. I've edited the question accordingly. As for your answer on what happens when $\theta_2 = 0$, that was also insightful. However, my last sentence on $\theta_2 = 0$ was more hypothetical. My question really is, why does optimal $\nu*$ does not involve $\theta_2$ when $\theta_2$ is not equals to 0?  
> **username_2**: The short answer is that the optimization problem is not defined if $ \theta_ {2} = 0 $, but you can make it defined; in that case, $ v $ will be equal to 0. and there will be a dependency when $ \theta_ {2} $ changes from 0 to some positive value. See the edition for more details, and to see how to make this dependency different from 0 $\forall \theta_{2}$.  
> **username_0**: Thank you for your advise! Will try out different specification for accumulation of human capital or savings. I think both options you have mentioned are reasonable.  

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