Suppose that an economy’s production function is Cobb-Douglas with parameter = 0

Question

Suppose that an economy’s production function is Cobb-Douglas with
parameter = 0.3.

a. What fractions of income do capital and labor receive?
b. Suppose that immigration increases the labor force by 10 percent. What happens to total
output (in percent)? / The rental price of capital? / The real wage?
c. Suppose that a gift of capital from abroad raises the capital stock by 10 percent. What
happens to total output (in percent)? / The rental price of capital? / The real wage?
d. Suppose that a technological advance raises the value of parameter A by 10 percent. What
happens to total output (in percent)? / The rental price of capital? / The real wage?

My teacher gave us this problem without giving us any practice problems of this type at all. I have no idea how to even begin such a problem. Any and all help would be MUCH appreciated!

Explanation / Answer

For "perfect/fair" economic system:

a)
Labor receives 30% of income
Capital receives 70% of income

b)
I've just used practical approach - MS-Excel software for calculations. (Although calculus aproach may be applied too - but it will not give you exact result because it's only true for K0 and L0 and A0)

Y +6.899%
r +6.899%
w -2.8188%

... Y ........ A ... K ..... L ........ R ............ W ............. r ............. w
3.78929 ... 5 ... 2 ... 0.50 ... 1.13679 ... 2.65250 ... 0.56839 ... 5.30501
4.05073 ... 5 ... 2 ... 0.55 ... 1.21522 ... 2.83551 ... 0.60761 ... 5.15547

Y - Income
A - Technology
K - Amount of capital
L - Labor-force
R - Total income of capital R=Yx0.3
W - Total income of labor W=Yx0.7
r - income per unit of capital r=R/K
w - income per unit of labor w=W/L

Y= ((4.05073 / 3.78929) - 1) x 100 = +6.899304492%
A = ((5/5)-1) x 100 = 0%
K = ((2/2)-1) x 100 = 0%
L = ((0.55/0.5)-1) x 100 = +10%
R = ((1.21522/1.13679)-1) x 100 = +6.899304492%
W = ((2.83551/2.65250)-1) x 100 = +6.899304492%
r = ((0.60761/0.56839)-1) x 100 = +6.899304492%
w = ((5.15547/5.30501)-1) x 100 = -2.818814099%

c)
Here is another method for similiar task (this one is more desirable for your teacher):
(1+Y%) = (1+A%) x ((1+K%)^0.3) x ((1+L%)^0.7)
A=0% 1+A = 1
L=0% 1+L = 1
K=+10% 1+A = 1+10% = 1.1
1+Y = 1x (1.1^0.3) x 1 1.029005759
Y 1.029005759 - 1 = 0.029005759 = +2.9%

R - Total income of capital R=Yx0.3
W - Total income of labor W=Yx0.7
r - income per unit of capital r=R/K
w - income per unit of labor w=W/L

You can simplify it to:

Y° = 1
L° = 1
w° = W°/L° = 0.7Y°/L°
w¹ = W¹/L¹ = 0.7Y¹/L¹
1+w% = w¹/w° = (0.7Y¹L°) / (0.7L¹Y°) = (Y¹L°) / (L¹Y°) = Y¹ / L¹
Y¹ = Y° + Y = 1+0.029005759 = 1.029005759
L¹ = L° + L = 1+0% = 1.00
1+w% = 1.029005759 / 1 = 1.029005759
w% = 1.029005759 - 1 +2.9%

Y° = 1
K° = 1
r° = R°/K° = 0.3Y°/K°
r¹ = R¹/K¹ = 0.3Y¹/K¹
1+r% = r¹/r° = (0.3Y¹K°) / (0.3K¹Y°) = (Y¹K°) / (K¹Y°) = Y¹ / K¹
Y¹ = Y° + Y = 1+0.029005759 = 1.029005759
K¹ = K° + K = 1+10% = 1.10
1+r% = 1.029005759 / 1.1 0.93545978
r% = 0.935459781 - 1 -0.06454 -6.454%

Y +2.9%
r -6.45%
w +2.9%

d)
.......... Y ........ A ... K ..... L ........ R ............ W ............. r ............. w
1.. 16.24505 ... 5.0 ... 2 ... 4 ... 4.87351 ... 11.37153 ... 2.43676 ... 2.84288
2.. 17.86955 ... 5.5 ... 2 ... 4 ... 5.36087 ... 12.50869 ... 2.68043 ... 3.12717
% 10.0000 .. 10.0 ... 0 ... 0 .. 10.0000 ... 10.00000 .. 10.00000 . 10.00000


Y = +10%
r = +10%
w = +10%

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