Consider the solow model a firm output according to

Question

Consider the Solow model. A firm output according to Cobb-Douglas production function Y_t = K^θ(Z_tN_t)^1-θ, where K_t is physical capital stock, and Z_tN_t is units of effective worker. Labor N_t grows at a rate of g_n and efficiency of labor Z_tgrows at a rate g_z. The household consumes a fixed fraction of its income each period, equal to (1-s). It invests the other fraction of its income, s, in new capital (that is, I_tt), with capital accumulating according to: K_t+1 = (1-δ) +I_t.

• 1. Now define per effective worker variables as c_t= C_t/(Z_tN_t), i_t = I_t/(Z_tN_t), k_t = K_t/(Z_tN_t), and y_t = Y_t/(Z_tN_t). Rewrite the capital accumulation equation in terms of the redefined variable
• 2. Derive an analytic expression for the steady state per effective worker capital stock, k*
• 3. What will be the growth rate of K_t in the steady state in which k_t is constant at k*. Show the steps in getting your result/answer.
• 4. Define the Golden Rule saving rate as the saving rate which maximizes steady state consumption per effective worker, ct= Ct/(ZtNt). For the given Cobb-Douglas production function, find an expression for this Golden Rule saving rate.