1. Consider a model economy with a production function
Y = K^{0.2}(EL)^{0.8},
where K is capital stock, L is labor input, and Y is output. The savings rate (s), which is defined as s = S/Y (where S is aggregate savings), is a constant. The aggregate savings finance aggregate investment (thus I_{t} = S_{t}). The population growth rate (n), growth rate of labor efficiency level (g), and depreciation rate of capital (d) are all constants.
(a) Show that this production function indicates constant return to scale.
(b) Show that this production function indicates decreasing marginal product of labor (MPL).
(c) Define capital per efficiency unit worker (k=K/EL) and output per efficiency unit worker (y=Y/EL). Express y as a function of k.
(d) Find steady state levels of k and y (k^{*} and y^{*}). Note that steady state is defined as a state where k does not change over time. Thus, the economy is in steady state at period t if and only if we have k_{t+1} = k_{t} (= k^{*}).
(e) Suppose there are two countries, the developed North (N) and the developing South (S). The North has 48% savings rate (s=0.48) and 0% population growth rate (n=0). The South has 9% savings rate (s=0.09) and 6% population growth rate (n=0.06). Both share the growth rate of efficiency level of 1% (g=0.01) and depreciation rate of 2% (d=0.02). What are the steady state level of y in the North and the South (y_{N}^{*} and y_{S}^{*})?