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In many basic macroeconomics textbooks a Cobb-Douglas production function with constant returns to scale is used to express the output of the economy as a function of labor and capital: $Y=AK^aN^{1-a}$.

Question: Given that a is normally assumed to be between 0.3 and 0.5, that the current output of the US economy is around 22T\$/year, and that $N$ (number of worked hours in a year) could be around 0.3TH/year, is there any kind of consensus on what are the values of $K$ and $A$ (at least the current values for the US economy)?

If I assume $a=0.5$, $A=5.4$ and $K=54$, then $Y=A\sqrt{KN}=21.7$, is that an OK choice?

I am puzzled to why I can't seem to find actual values for these parameters. If a book or a paper says that the GDP is expressed with this function, shouldn't it also at least give a realistic example with a range and units for the involved parameters? Maybe I am just looking in the wrong places?

Any thought? Thank you in advance

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    $\begingroup$ I'm not an expert in this area, but as far as I understand, $Y$, $K$, and $N$ are data, $a$ is estimated from the data, and $A$ is the residuals. For example, your model is $\ln Y = \ln A + a\ln K + b\ln N$ [if you impose the restriction that $b=1-a$, the model becomes $\ln (Y/N) = \ln A + a\ln (K/N)$], where $\ln A = \mu + \varepsilon$. You can just calculate $A$ as $Y/(K^a N^{1-a})$ from $Y, K, N$ and $a$. $\endgroup$
    – chan1142
    Mar 21 at 4:25
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K is just a stock of capital. This Fed data show that the stock of US capital is almost 70m at constant prices for the most recent year (2019). But there are other ways of measuring it. You can google papers that estimate multifactor productivity to see what measures for K and N people are using (not everyone uses labor hours too).

A is multifactor productivity. This OECD data show that using 2015 as index year the multifactor productivity in the USA in 2019 was 102. A cannot be measured directly. People estimate it using K and L and using time series/panel regressions. That is why this will be in datasets expressed just as a growth rate or index, we can't see what A is only how it changes.

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