Deriving "y" from Aggregate Production Function?

Question

How do I get the income per capita function from the aggregate prod. function? I need "y" to continue further into my problem..

Here is the info I have...

Assume that the prod. function is Y= K (TO THE 1/2 POWER) L (TO THE 1/2 POWER)

s= .2

depreciation= 0.05

economy starts off w/ k=4

Also...how do you solve for the steady state of y and c?

Answer 1

Y=(KL)^(1/2)

Y/L=(KL)^(1/2) / L = (KL)^(1/2) / (L^(1/2))^2

y=(K/L)^(1/2)

K/L=k

y=√k

Steady-state:

sy=δk

0.2√k=0.05k

4=√k

k=16

y=√k=4

c=y-δk = √k - δk = 4 - 0.05*16 = 3.2

Current level: k=4

Steady-state: k=16

Conclusion: output per worker will increase because per worker capital will increase from 4→16, to increase steady-state output per worker it is required:

♦ change production function;

OR

♦ change depreciation (switch to another type of capital)

OR

♦ change saving rate

Golden rule for given production function:

y-δk → MAX

y' - (δk)' = 0

y' = (δk)'

1/2√k=δ

√k=1/2*0.05 = 1/0.1=10

k*=100

y*=√k* = 10

δk=sy

0.05k=s*√k

0.05√k=s

s*=10*0.05=0.5

c*=y* - δk*=√k* - 0.05k* = 10-5=5

Answer 2

there is two points in your question which are not connected.

.- Deriving y means to find dy/dL or dy/dK (calculus notation)

.- Income per capita can be given by y/P (arithmetical notation) where P is population.

for some economists y<>Y for others y=Y; depends of notation you are using....please precise notation...