Solving Constrained Optimization Problem with Two-Period Model of Human Capital

Question

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.

Answer 1

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.