I need a lot of help on this. Not sure where to start on any of the questions. S

Question

I need a lot of help on this. Not sure where to start on any of the questions.

Suppose the economy's production function is given by: Suppose further that the saving rate in the economy is fixed at s and that the number of workers grows at the rate g N in every period. In addition, the capital stock of the economy depreciated at rate delta in every period. Write down output per worker and investment per worker as functions of capital per worker. What is the level of investment per worker which is required in this economy in each period to maintain a given level of capital per worker? Draw a diagram with capital per worker on the x-and output per worker on the y-axis. In your diagram, graph the three functions you derived in part a and b., i.e. output per worker, investment per worker and required investment per worker to hold the capital stock constant. Denote the steady state of the economy by (k/N)* and (Y/N)*.If z = xayb, then gz agx + bgy where gz denotes the growth rate of z and so on. Suppose that the growth rate of workers doubles. How will this change your graph in part (c)? What happens to steady state capital per worker and output per worker? Using two diagrams along the lines of Figure 11-7 in the textbook, graph the dynamic effects of the increase in the growth rate of workers on output per worker and on the growth rate of output per worker.

Explanation / Answer

a) output per worker (y) = Y / N = K^ ( alpha) N^ (- alpha) ; = k ^ ( alpha)

note "K" is capital , "k" is capital per worker ;

"Y" is output , "y" is output per worker

Investment per worker = k_t - ( 1- depreciation)* (1+gn)^-1 * k_t-1

= k_t - ( 1- depreciation- gn) * k_t-1

b) investment required = k^ ( alpha) - (1- depreciation - gn) *k^alpha

=> l^(alpha) - k^ ( alpha) - (1- depreciation - gn) *k^alpha  

= > investement required = ( gn + depreciation)*k^alpha

c) we know Investment per worker = s* y_t =   k_t - ( 1- depreciation - gn) * k_t-1;

hence k_t = s* k_t ^ ( alpha) +( 1- depreciation - gn) * k_t-1;

this is the equation for movement of capital stock ;

at steady state change in per capita stock is zero, hence k_t = k_t-1 ;

then we get k = (s/ (depreciation +gn)) ^(1 / (1- alpha) )

so steady state per capita capital stock is = (s/ (depreciation +gn)) ^(1 / (1- alpha) )

at steady state per capita output is = (s/ (depreciation+gn) ) ^(alpha / (1- alpha) )

at steady state per capita investment is= s*(s / (depreciation +gn) ) ^(1 / (1- alpha) )

d) at steady state all will grow at 0 ( zero ) growt rate;

as growth rate of steady state variable is zero.

e) now steady state ouptut will be :

steady state per capita capital stock is = (s/ (depreciation +2*gn)) ^(1 / (1- alpha) )

at steady state per capita output is = (s/ (depreciation+2*gn) ) ^(alpha / (1- alpha) )

at steady state per capita investment is= s*(s / (depreciation +2*gn) ) ^(1 / (1- alpha) )

Graph will be those by solow model .

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