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Cobb-Douglas is $Y = AK^{1-\alpha}L^{\alpha}$. What exactly is $L$ in Cobb-Douglas? Is $L$ the number of workers available in a single year? Working hours?

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It can be both. The number of working hours are of course easier to interpret. You can also make the assumption that every employee works some fixed number of hours, say 8 per day (or 220 per month, or 38 a week, etc). In this case if you denote by $n$ the number of workers and $L$ the number of hours worked you have of course $$ n \cdot 8 = L $$ and then $$ A \cdot K^{1-\alpha} \cdot L^{\alpha} = A \cdot K^{1-\alpha} \cdot ( n \cdot 8 ) ^{\alpha} = 8^{\alpha} \cdot A \cdot K^{1-\alpha} \cdot n ^{\alpha} = \left(8^{\alpha} \cdot A\right) \cdot K^{1-\alpha} \cdot n ^{\alpha}. $$ So you can choose $L$ to represent either quantity as long as you take it into consideration when choosing/calibrating $A$.
(Provided that some conditions like $n \cdot 8 = L$ hold or are a reasonable approximation of reality.)

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