Answer to Question #246131 in Macroeconomics for MINI

Question #246131
Consider a dynamic economy withgk= 3%,gh−i= 2%gh−ni= 1%,Q=K0.5H0.2L0.3population growth = 1.5%, population growth with immigration = 2.5%.Calculate the errors for ignoring human-capital contribution of immigrants in the calculationof real-wage series for 40 periods.
1
Expert's answer
2021-10-04T10:18:46-0400

a)"Y=K^{0.5}N0.7"

(I)"\\frac{\\delta Y}{\\delta K}=K^{-0.5}N0.7>0\\space as\\space K,N>0\\\\\n\n\\frac{\\delta Y}{\\delta N}=0.7K^{0.5}N^{-0.3}>0\\space as\\space K,N>0"

Hence it is increasing in both inputs

ii)

"MP_K=\\frac{\\delta Y}{\\delta K}=0.5K^{-0.5}N0.7\\\\\\frac{\\delta MP_K}{\\delta K}=-(0.5)(0.5)K^{-1.5}N^{0.7}<0\\\\\n\n\n\n\nMP_N=\\frac{\\delta Y}{\\delta N}=0.7K^{0.5}N^{-0.3}\\\\\\frac{\\delta MP_N}{\\delta N}=-(0.7)(0.3)K^{0.5}N^{-1.3}<0\\\\\nAs\\space \n\n\n\\frac{\\delta MP_K}{\\delta K}<0 \\space and \\space \\frac{\\delta MP_N}{\\delta N}"

It satisfies diminishing marginal products

iii)

"Y=K^0.5N0.7"

Doubling K and N we get

"Y(2K,2N)=(2K)^{0.5}(2N)0.7\\\\=(2)^{1.2}K^{0.5}N0.7"

As"\\space N^{1.2}" =Doubling K and N more than double Y. Therefore it does not satisfy constant returns to scale.

iv)

This is a cobb-douglas production function hence there is no complementarity between capital and labor.


b)"Y=2K+3N"

(I)

"\\\\\n\\frac{\\delta Y}{\\delta K}=MP_K=2>0\\space as\\space and \\\\\\frac{\\delta Y}{\\delta N}=MP_N=3>0"

Hence it is increasing in both inputs

ii)

"\\frac{\\delta MP_K}{\\delta K}=-(0.3)(0.7)K^{-1.7}N^{0.7}<0\\\\\n\n\n\n\n\n \\frac{\\delta MP_N}{\\delta N}=-(-0.7)(0.3)K^{0.3}N^{-1.3}<0"

It satisfies diminishing marginal products

iii)

"Y=2K+3N"

Doubling K and N we get

"Y(2K,2N)=2(2K)+3(2N)\\\\=2[2K+3N]=2Y"

As doubling K and N doubles Y. Therefore it satisfies constant returns to scale.

iv)

Now, slope "\\frac{MP_K}{MP_N}=\\frac{2}{3}=constant"

Hence both inputs are substitutes of each other.


C)"Y=K^{0.5}N^{0.7}"

(I)

"\\frac{\\delta Y}{\\delta K}=MP_K=0.3K^{-0.7}N^{0.7}>0\\\\\n\\frac{\\delta Y}{\\delta N}=MP_N=0.7K^{0.3}N^{-0.3}>0"

Hence it is increasing in both inputs

ii)

"MP_K=\\frac{\\delta Y}{\\delta K}=0.5K^{-0.5}N0.7\\\\\\frac{\\delta MP_K}{\\delta K}=-(0.5)(0.5)K^{-1.5}N^{0.7}<0\\\\\n\n\n\n\nMP_N=\\frac{\\delta Y}{\\delta N}=0.7K^{0.5}N^{-0.3}\\\\\\frac{\\delta MP_N}{\\delta N}=-(0.7)(0.3)K^{0.5}N^{-1.3}<0\\\\\nAs\\space \n\n\n\\frac{\\delta MP_K}{\\delta K}<0 \\space and \\space \\frac{\\delta MP_N}{\\delta N}"

It satisfies diminishing marginal products

iii)

"Y=K^{0.3}N^{0.7}"

Doubling K and N we get

"Y(2K,2N)=(2K)^{0.3}(2N)0.7\\\\=2K^{0.3}N^{0.7}\\\\=2Y"

Doubling K and N doubles Y. Therefore it satisfies constant returns to scale.

iv)

This is a cobb-douglas production function hence there is no complementarity between capital and labor.


D)"Y=2N+2"

(I)

"\\\\\n\\frac{\\delta Y}{\\delta K}=MP_K=0=0\\space as\\space and \\\\\\frac{\\delta Y}{\\delta N}=MP_N=2>0"

Hence it is increasing in both inputs

ii)

"\\frac{\\delta MP_K}{\\delta K}=0\\\\\n\n\n\n\n\n \\frac{\\delta MP_N}{\\delta N}=>0"

It does not satisfy diminishing marginal products

iii)

"Y=2N+2"

Doubling K and N we get

"Y(2N)=2(2N)+2"

Therefore it does not satisfy constant returns to scale.

iv)

There is no complementarity between capital and labor


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS