Problem
In this exercise we extend the Solow-Romer model to include population growth. The total number of workers Lt grows at a constant growth rate ¯ n = 0.01 (i.e., 1% per year). A constant fraction l of the workers in each period work on producing new ideas and the rest of the workers work on producing output, i.e., LAt = l*Lt, LYt = (1- l)Lt. New ideas are produced according to ?At = z *At^ß *L^? *At. (Notice the di?erence between this speci?cation and the one we studied in class!) Let ß = 0.8 and ? = 0.3. Output is produced as usual according to the Cobb-Douglas production function Yt = At * Kt^(1/3) * Lyt^(2/3) .
(a) Find the constant growth rate of At. (Hint: divide both sides of the previous equation by At to and At / At = z* At^(ß-1) * Lt^? /(At). For the growth rate of At to be constant, At^(ß-1)* LAt^? must be constant.)
(b) Explain why with ß = 1 the stock of ideas At could not grow at a constant rate.
(c) Consider a balanced growth path along which output and capital grow at the same constant rate gY = gK. What is the growth rate of output?
(d) Suppose the savings rate is 20% and depreciation is 4%. What is the capital-output ratio?
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.