# Consider the following Cobb-Douglas function for aggregate economy

Question 1 (Growth Accounting) Consider the following Cobb-Douglas function for aggregate economy. r :AIK’ILI’” (1) where Y is aggregate output, A1 is the level of technology, K is capital stock, and L is total labor hours. I. Express the equation {2) with respect to time. Take log and take derivative with respect to time. Provide an equation of growth
rate of aggregate output. 2. In the economic growth model, A] is also called Solow residual {i.e., TFP). factors (e.g., these are factors affecting technology
according to our deﬁnition of technology} can be in A1? Now consider a model that decompose total labor into quality and quantity of work as follows. r : AIKafringl_a : made—”LP” : Ami—”K”.Ll—a (2)
where L total hour and A2 qualityr of labor hour (e.g., this value increase as overall skill-level of labor improve — one hour of average
labor used to produce 5 output for given capital but now one hour of average labor produce l0 output with the same capital) 3. Express the equation {3) with respect to time. Take log and take derivative with respect to time. Provide an equation of growth
rate of aggregate output. Interpret the equation in growth rate form. r : Alsf’xﬂagLV—”l ‘”2 (3)
where K1 is Capital of AI (more of equipment} or ICT (information-communication technology) related capital and K2 is traditional
capital (e.g.. factor):r or places to produce). 4. Express the equation {4) with respect to time. Take log and take derivative with respect to time. Provide an equation of growth
rate of aggregate output. Interpret the equation in growth rate form.