Why is human capital per worker multiplied by wage per worker?

Question

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.

Answer 1

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.