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