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.