Eq (4) of Bils and Klenow (2000)

Question

I hope some of you read Economics. I was wondering if anyone can help me in deriving equation (4):

$\ln A_{i}\left( t\right) =\beta\ln h_{i}\left( t\right) +\ln\bar{A}\left( t\right) +\xi_{i}\left( t\right)$,

in the following article:

BILS, M. AND P. J. KLENOW (2000):"Does Schooling Cause Growth?," American Economic Review, 90(5), 1160-1183.

Notation:

$\bar{A}:$ the exogenously growing “world technology frontier”,

$h_{i}(t):$ the average level of human capital in country $i$,

$\beta:$ discount rate,

$A_{i}:$ technology level in country $i$ at time $t$ and

$\xi_{i}:$ some kind of error (not sure)

Thanks

update: I've updated the link - it's from P. Klenow's website.

Answer 1

It essentially says that you should stay in education until the marginal discounted cost of tuition including wages foregone at your skill level at that time is equal to the discounted difference between the wages from marginal skill gains from education and the wages from skill gains from work experience, taking account remaining lifespan in which these wages are earned.

Answer 2

Considering the context within economics and, more specifically, econometrics, it's sort of a log-log equation of relations between the variables you listed.

So, consider that $\bar{A}$ is exogenous. Thus, it is not determined by the equation and inherently considered to be a constant.

$A_i(t)$ corresponds to the level of technology at time $t$ and is the subject this equation seeks to model. To get the non-linear parent distribution from which the motivation of this equation came, consider the transformation of raising $e$ to the power of both sides. Simplification will yield: $$A(t) = \bar{A}(t)\,h(t)^{\beta}$$

Now consider it "intuitively". This equality says that the level of technology in a country at a given time, $t$, is determined by an arbitrary "technology frontier" at time $t$, $\bar{A}(t)$, multiplied by the average level of human capital (education) which is $h_i(t)$ raised to the power of the discount rate, $\beta$.

You can think of the discount rate as the amount of "resources", I guess, devoted to education. Read about the social discount rate for a better idea of what that might be modeling. The equation tells you that the level of technology at a given time is determined by the amount of technology which exists already multiplied by the discounted level of human capital at that time. In ultra-laymen's terms: The new level of technology is determined by the old level of technology magnified by the amount of education people are receiving.

Henry's description seems probably more accurate given the context of the article. I didn't actually read it. I was just speaking off what you posted here. Hopefully my words help you see a little bit more of where they might have gotten the idea for this model.

Edit: The reason they log both sides is so they can develop a $\textbf{linear}$ model to extrapolate $A_i(t)$ for values of $h_i(t)$ within an error of $\xi$ from the linear model. $\xi$ usually follows a distribution $N$~($0, var(\xi)$).