Respuesta :
Answer:
[tex]k = (\frac{0.1}{0.2})^{3/2}=0.354[/tex]
In the steady state level of Country A, the capital per worker is 0.354
[tex]k = (\frac{0.3}{0.2})^{3/2}=1.837[/tex]
In th steady state level of country B, the capital per worker is 1.387.
Now we can find the steady level income per worker like this:
Country A
[tex]y=k^{1/3}= (0.354)^{1/3}=0.707[/tex]
Country B:
[tex]y=k^{1/3}= (1.837)^{1/3}=1.225[/tex]
And if we want to find the consumption per worker we can apply this formula:
[tex] c= (1-s) y[/tex]
Country A
[tex]c=(1-0.1)0.707=0.636[/tex]
Country B
[tex]c=(1-0.3)1.225=0.858[/tex]
Explanation:
We assume that we have a common production function for country A and B given by:
[tex]Y=F(K,L)= K^{1/3}L^{2/3}[/tex]
And we can find the per worker production function [tex]y=f(X)[/tex] like this:
[tex]\frac{Y}{L}= \frac{K^{1/3}L^{2/3}}{L}=K^{1/3}L^{-1/3}= (\frac{K}{L})^{1/3}[/tex]
And we can express the function just in terms of a constant like this:
[tex]y=k^{1/3}[/tex]
From the info given by the problem we have:
Depressciation [tex]\delta =0.2[/tex]
Savings for A [tex]S_A = 0.1[/tex]
Savings for B [tex]S_B = 0.3[/tex]
And we assume that [tex]y=k^{1/3}[/tex]
Let's begin finding the steady state level of capital per worker, we need to satisfy the following condition:
[tex]s f(x) = \delta k[/tex]
The reason is because. The growth of capital per worker [tex]\Delta k =[/tex] is given by investment per worker [tex]sf(k)[/tex] minus the depreciation per worker [tex]\deta k[/tex] , and we have then [tex]\delta k= sf(k)-\delta k[/tex] and if is steady then k=0.
[tex]S_A f(x)= \delta k[/tex]
[tex]0.1 k^{1/3}=0.2 k[/tex]
[tex]0.1 =0.2 k^{2/3}[/tex]
[tex]k = (\frac{0.1}{0.2})^{3/2}=0.354[/tex]
In the steady state level of Country A, the capital per worker is 0.354. For country B we can do a similar procedure like this:
[tex]s f(x) = \delta k[/tex]
[tex]S_B f(x)= \delta k[/tex]
[tex]0.3 k^{1/3}=0.2 k[/tex]
[tex]0.3 =0.2 k^{2/3}[/tex]
[tex]k = (\frac{0.3}{0.2})^{3/2}=1.837[/tex]
In the steady state level of country B, the capital per worker is 1.387.
Now we can find the steady level income per worker like this:
Country A
[tex]y=k^{1/3}= (0.354)^{1/3}=0.707[/tex]
Country B:
[tex]y=k^{1/3}= (1.837)^{1/3}=1.225[/tex]
And if we want to find the consumption per worker we can apply this formula:
[tex] c= (1-s) y[/tex]
Country A
[tex]c=(1-0.1)0.707=0.636[/tex]
Country B
[tex]c=(1-0.3)1.225=0.858[/tex]
