Question 2. (20 points)
The production function reveals the relationship between national output (Y) and the combination of input factors, including technology (A), capital (K), and labor input (L). With the above notations, try to answer the following questions.
(1) In a country, the production function of its national output is Y = F(K, L) = A * K^(3/4) * L^(1/2). Does its production function have constant returns to scale, increasing returns to scale, or decreasing returns to scale? And why? (show your work)
(2) In another country, without considering the technology level (A = 1), its production function can be expressed as Y = F(K, L) = K^(3/4) * L^(1/4). We have a saving rate (s) equal to 40% and a capital depreciation rate (δ) of 20%. Given the information here, how much will be its steady-state capital per labor (k) in this country? Note: k = K/L and output per labor y = f(k).
(3) Following the question above, this country's saving rate (s) increases to 45% and the capital depreciation rate (δ) drops to 15%. How much is its steady-state capital per labor (k) now?
(4) Again, for the second country, its saving rate (s) is 60% and the capital depreciation rate (δ) equals 15%. Also, we have information about its population growth rate (n) and technological progress (g). Here n = 5% and g = 4%. Then how much will be consumption per labor (c) in this country at the steady state.