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I am studying a paper entitled "The Past and Future of Knowledge-Based Growth" by Strulik et al (2013) where the budget constraint is written as:

$$w_{t}h_{t}(1-\tau n_{t})=c_{t}^{1}+s_{t}+n_{t}e_{t},$$

where $w_{t}$ are wages, $h_{t}$ is the level of human capital, $\tau$ is the amount of time to raise a child, $n_{t}$ is the number of children, $c_{t}^{1}$ is consumption, $s_{t}$ are savings, and $e_{t}$ is educational investment. Wages are a function of human capital since human capital is an input into production and therefore determines marginal product (wages).

My question is as follows. Why are wages multiplied by the human capital term, $h_{t}$, in the equation above when just including $w_{t}$ would presumably imply workers are earning the return to educational investment via human capital. I have seen this sort of setup often and trust there's an explanation, however I do not fully understand the economic intuition here.

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  • $\begingroup$ Why you want to have marginal return in a budget constraint? A budget constraint is just what an agent obtains must be less than or equal to what she spends. $\endgroup$ Commented Mar 11, 2022 at 21:56
  • $\begingroup$ I just edited the question to get rid of the word marginal there. But why is it the case that individuals earn the return to their education via the human capital function and the marginal product of some production function that determines wages. Isn't that double-counting in a way? $\endgroup$
    – econogist
    Commented Mar 11, 2022 at 23:10
  • $\begingroup$ No. Wage is on the efficient unit of human capital. $\endgroup$ Commented Mar 11, 2022 at 23:21
  • $\begingroup$ I think I understand. Are you saying that the $h_{t}$ is the human capital per time unit and the marginal product of the production function is the wage that must be paid to each unit of human capital? And therefore wage must be multiplied by the human capital per time unit worked, $w_{t}h_{t}$? $\endgroup$
    – econogist
    Commented Mar 11, 2022 at 23:55
  • $\begingroup$ Yes. Actually this is the reason why it is called human "capital". Firms don't care about how many human capital each worker holds but only the total efficient units of human capital. However, this is not the case if you have a model with different types of workers and they are not perfect substitutive. $\endgroup$ Commented Mar 12, 2022 at 1:20

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This probably goes back to the way human captial is defined. Let $f(\ell)$ be a production function of the firm where $k$ is capital and $\ell$ is the amount of "basic" labor hours.

Now assume that there are two different types of workers, one with low human capital level, which we normalize to $1$ and one with human capital level $h > 1$.

Here human capital is defined in terms of productivity difference. Namely, if the firm hires $\ell_h$ hours from workers having human capital level $h$ then it is is `as if' it would hires $h \times \ell_h$ hours from workers having human capital level $1$. So workers with human captial $h$ are $h$ times more productive.

In other words, if you have an amount $\ell_h$ of human captial $h$ hours and an amount $\ell$ of human captial $1$ hours, it is "as if" the firm buys a total of $h \ell_h + \ell$ hours of human captial 1 hours.

Then total output of the firm is given by $f(\ell + h \ell_h)$. As wage is equal to marginal product, we have that: $$ \begin{align*} &\frac{d f}{d \ell} (\ell + h \ell_h) h = \frac{w_h}{p}.\\ &\frac{d f}{d \ell} (\ell + h \ell_h) = \frac{w}{p} \end{align*} $$ where $p$ is output price, $w_h$ is the wage of a human capital $h$ worker and $w$ the wage of a human capital 1 worker. So: $$ w_h = h w. $$ If your human captial is $h$ then your wage is $h$ times as high.

This framework assumes that workers with different human capital levels are in some way perfect subsitutes of each other (after taking into account productivity difference). This follows from the assumption that $\ell$ and $h \ell_h$ enter additively into the production function. This may, or may not be a realistic assumption.

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