Question

1.The Golden Rule in a Solow Model without a Cobb-Douglas Production Function Suppose that the per-worker production function is: 4k tk +3 where yt = Yt/L and kt = Kt/L A.Does this production function exhibit diminishing marginal product of capital? Illustrate and explain. Note that you can use calculus, but you can also create a table. Note that AKt+1- Akt+1 and: B.Suppose that the savings rate in this economy is 36 percent (s- 0.36) and the depreciation rate is 6 percent (d - 0.06). Mathematically calculate and graphically illustrate the steady-state levels of capital per worker (k*), output per worker (y"), consumption per worker (c*), and investment per worker (i') C.Intuitive describe what is meant by the Golden Rule steady-state? D.Suppose this economy wants to achieve the Golden Rule steady-state equibrium, should the savings rate be increased, decreased, or remain the same? Note that I am not asking you to compute the exact savings rate that will achieve the Golden Rule, but I am simply asking whether the current savings rate of 36 percent is too low, too high, or just right to achieve the Golden Rule. There are many ways to do this problem and most of them do not involve calculus

Answer 1

y_t = 4kt/k_t + 3 A MPK = partial differential y_t/partial differential k_t Let u = 4k_t nu = k_t + 3 MPK = u^1 nu - uv^1/v^2 u^1 = 4 nu ^1 = 1 MPK = 4(k_t + 3) -4 k_t(1) /(k_t + 3) ^2 = 4k_t + 12-4k_t/(k_t) ^2 + 6k_t + 9 = MPK = 12/kt^2 + 6k_t + 9 as k_t increases, MPK decreases this implies that this production function exhibit diminishing marginal product capital B) s = 0. 36 d = 0. 06 At steady state: Ak = 0 delta k = sy-sk 0 = (0. 36) (u k_t/R_t + 3) -(0. 06) k_t 0. 36/0. 06 = k_t/(4k_t/R_t + 3) 6 = (k_t + 3/4R_t) k_t 6 = k_t + 3/4 24 = k_t + 3 k_t = 21 steddy state level of capital per workers y = 4k_t/k_t + 3 = 4(21) /21 + 3 = 84/24 y^* = 3. 5 steddy state level of output per worker i = sy = (0. 36) (3. 5) i = 1. 26 steddy state level of investment per worker e = y-i = (3. 5) -1. 26) C^* = 2. 24 consumption per worker at steady state level C) -Golden rule steady state it is the steady state value of k that maximises the consumption per worker At golden rule/MPK = d 12/k^2t + 6k_t + 9 = 0. 06 12/0. 06 = k^2t + 6k_t + 9 = 200 k^2t + 6k_t-191 = 0 by solving this, we get (k_t = 11. 14) which is become the steady state level D) if the economy wants to achieve the golden rule steady state level then the polimaker should reduce the savings rate. to reach the golden rule steady state