Answer to Question #240426 in Macroeconomics for Ana

Question

5

. Check

whether the following functions satisfy the following properties:

i) Increasing i

n both inputs.

ii) Diminishing marginal products.

iii) Constant returns to scale.

iii) Complementarity between capital and labor.

a) Y=K^0.5^N0.7

b) Y= 2K+3N

c)Y = K^0.3N^0.7

d) Y= 2N+2

Use the fact that given a function of the form Y=K^αN^β marginal products are given by

MPN=βK^αN^β−1, MPK=αK^α−1N^β

Answer 1

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

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

Hence it is increasing in both inputs

ii)

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

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)

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

Hence it is increasing in both inputs

ii)

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

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)

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

Hence it is increasing in both inputs

ii)

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

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)

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

Hence it is increasing in both inputs

ii)

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

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