(Population growth but no technology growth) Consider an economy that 12 is described by the production function Y = K^1/3L^1/ 3. Moreover the depreciation rate of capital is (delta) o = 0.05 and the population growth rate is n = 0.05 (there is no technology growth)
(a) What is the per-worker production function, that is y = Y/L ? What is the marginal product of capital, that is dy/dk?
b) If the saving rate is s= 0.4, find the steady state level of capital per worker k*. What is the steady state level of consumption per worker c*? What is the steady state level of investment per worker i*? What is the steady state level of output per worker y*?
(c) Find the golden rule level of capital per worker k*g, i.e. the steady state level of capital per worker that yields the highest steady state consumption per worker. Find also the golden rule consumption per worker level c*g that corresponds to this capital per worker and compare it to the one in question b. Is cg higher than c*?
(d) Find the savings rate sg that guarantees that we will reach the golden rule capital per worker k*g
(e) If the government decides to implement policies that change the savings rate from c = 0.4 to sg, what will be the immediate impact of this policy change on c, i, and y? What will be the impact in the long run? (Provide numbers for both the immediate impact and the long run impact)
(f) What is the growth rate of GDP per worker in the long run? The growth rate of aggregate GDP?