Question

(Growth Accounting) Suppose that the representitive firm's production function is Cobb- Douglas: (a) Show that the growth rate of total output can be decomposited as (Hint: You may take logarithm to equation (4) on both sides, then use chain rule to take derivative with repect to time t) (b) Suppose in the year 2016, compared to 2015 the total output increases by 3%, and the technology keeps the same (no change in A). Also the labor force inceases by 0.05% compared to 2015. What is the growth rate of total capital compared to 2015? (c) Suppose between 2018 and 2017, the difference between the total output's growth rate and the total labor force's growth rate is 1%, and the difference beween the total output's growth rate and the total capital's growth rate is 2%. Then what is the growth rate of total factor productivity (A) between 2018 and 2017?

Answer 1

Y = F(k, N^d) = z K^0.5 (N^d)^0.5 y rightarrow output, k rightarrow capital stock, N^d rightarrow labor input z rightarrow total factor productivity. z = 6, k = 4, wage rate is 3. Let us assume the price of is �p�. y = z k^0.5 (N^d)^0.5 y = (6) (4)^0.5 (N^d)^0.5 y = (6) (2) (N^d)^0.5 y = 12 (N^d)^0.5 partial differential y/partial differential n^d = MP_L = (12) (0.5) (N^d)^0.5 MP_L = 6/(N^d)^0.5 Marginal revenue of product: The firm�s product demand curve. \r\n Marginal revenue product of labor = (Marginal product of labor) (Price) MRP_L = (MP_L) (P) For optional labor, MRP_L should be equal to wage rate. i.e, MRP_L = w MP_L middot P = w 6/(N^d)^0.5 middot P = 3 (N^d)^0.5 = (6/3) P (N^d)^0.5 = 2 p Sugaring both sides N^d = 4 P^2 optimal labor. profit = Total Revenue - Total cost profit = (P) (y) - (WN^d + rk). Profit = (P) (Z k^0.5 N^d^0.5)