Answer to Question #249954 in Microeconomics for Kush Kumar

Question

Problem 5.8. For each of the following production functions, determine whether returns to scale are decreasing, constant, or increasing when capital and labor inputs are increased from K = L = 1 to K = L = 2. a. Q = 25K0.5L0.5 b. Q = 2K + 3L + 4KL c. Q = 100 + 3K + 2L d. Q = 5Ka Lb , where a+b= 

Answer 1

Let's use $F(zK,zL)=AK^aL^b$

a. Q$=25K^{0.5}L^{0.5}$

Let Q$_0=F(K,L)=25K^{0.5}L^{0.5}$ be initial production function then after multiplying by z we get:

$Q_1=F(zK,zL)=25(zK)^{0.5}(zL)^{0.5}=z^{0.5}z^{0.5}25K^{0.5}L^{0.5}=z^{1.0}Q_0$

In this case F $(zK,zL)>zF(K,L)$

This production function represents increasing returns to scale.

b. Q$=2K+3+4KL$

Let $Q_0=F(K,L)=2K+3L+4KL$ be the initial production function. Let us multiply it by factor z and call it Q$_1$

$Q_1=F(zK,zL)=2(zK)+3(zL)+4(zK)(zL) =z(2K+3L+4zKL)$

So $F(zK,zL)>zF(K,L)$

Production function represents increasing returns to scale.

c. Let Q$_0=$ F$($ K,L$)=100K+3K+2L($ initial production function$)$

After multiplying it by factor z:

Q$_1=F(zK,zL)=100+3K+2L$

In this case F$(zK,zL)<$ $zF(K,L)$

$100+3(zK)+2(zL)<100z+3(zK)+2(zL)$

This is a decreasing returns to scale

d. $Q_0=F(K,L)=5K^aL^b$ where $a+b=1$

Let's multiply it by factor z

$Q_1=F(zK,zL)=5(zK)^a(zL)^b=z^az^b5K^aL^b$

$z^{a+b}Q_0=zQ_0$

Here $F(zK,zL)=zF(K,L)$

The production function represents a constant returns state.