I have some questions about Macroeconomics I want some profe

I have some questions about Macroeconomics. I want some professional tutors help me solve them.Please give me detailed solutions. Thank you!

Solution

Given that

The law of motion of physical capital per unit of effective labor:

                        k(t) = K(t)/A(t)L(t)

The Differential equation:                           

                        k(t) = sy(t) – (n+g+) k(t)

            Where: s = Saving rate

                        n = Population growth rate

                        g = Growth rate of technology, and

                        = Depreciation rate of capital

The output per unit of effective labor is:

                        y(t) = Y(t)/ A(t)L(t)

1 .If Production function is: y(t) = K(t) (A(t) L (t) ^1-for 0 < < 1 and

Show that:       y(t) = k(t)

Discrete time indexed by subscript t;       where: t = 0, 1, 2....

We can consider the given Cobb-Douglas production function like as follows:

                        Y_t = F(K_t, L_tE_t) = K _t (L_tE_t)^1-

            Then    y_t = Y_t/L_tE_t

                        K _t (L_tE_t)^1-

                Y_t/L_tE_t = (K_t/ L_tE_t)

                         y_t = k_t

2. k* denote the steady state level of physical capital. The level of capital such that k(t) = k* for every t and is derived as the solution to k (t) = 0

How does the stock of physical capital k_t change over time?

If we can solve for the dynamics of k_t:

Capital gain: Investment, i_t = sf(k_t) = sk_t                  Where: i_{t =} I_t/L_tE_t

Capital Loss: Depreciated capita plus loss due to population growth and technological change (n + g + )k_t

Law of motion k* (discrete time) is:

                        k* = 0; i_t - (n + g + )k_t              

            If,        k* < 0; i_t < (n + g + )k_t         

            If,        k* > 0; i_t > (n + g + )k_t

Steady-state of the saving rate (s) is:

                        s = (n + g + )(k_t/y_t).