Question

For each of the following production functions, determine whether returns to scale are decreasing, constant, or increasing,

a. Q = 2K + 3L + KL

b. Q = 20K^0.6 L^0.5

c. Q = 100 + 3K + 2L

d. Q = 5K^0.459 L^0.541

e. Q = 5K^0.37 L^0.56

f. Q = K/L

Answer 1

Basically, the returns to scale refers to how much output changes given a proportional change in all inputs, where all the inputs change by the same factor.

To obtain the next output, we shall multiply all our functions by a factor z

a. Q = 2K + 3L + KL

$Let 𝑄_{0}=𝐹(𝐾,𝐿)=2𝐾+3𝐿+𝐾𝐿$

multiply it by factor 𝑧 and call it 𝑄1

$𝑄1=𝐹(𝑧𝐾,𝑧𝐿)=2(𝑧𝐾)+3(𝑧𝐿)+(𝑧𝐾)(𝑧𝐿)=𝑧(2𝐾+3𝐿+𝑧𝐾𝐿)$

$𝑧(2𝐾+3𝐿+𝑧𝐾𝐿)>𝑧(2𝐾+3𝐿+𝐾𝐿)$

$2𝐾+3𝐿+𝑧𝐾𝐿>2𝐾+3𝐿+𝐾𝐿$

$𝑧𝐾𝐿>𝐾𝐿$

$𝑧>1.$

This indicates increasing returns to scale. 

b. Q = 20K^0.6 L^0.5

$𝑄1=𝐹(𝑧𝐾,𝑧𝐿)=20(𝑧𝐾)^{0.6}(𝑧𝐿)^{0.5}=𝑧^{0.6}𝑧^{0.5}20𝐾^{0.6}𝐿^{0.5}=𝑧^{1.1}𝑄_{0 }$

$𝐹(𝑧𝐾,𝑧𝐿)> 𝑧𝐹(𝐾,𝐿).$

This function expresses increasing returns to scale

c. Q = 100 + 3K + 2L

$𝑄1=𝐹(𝑧𝐾,𝑧𝐿)=100+3(𝑧𝐾)+2(𝑧𝐿)$

In this case;

$𝐹(𝑧𝐾,𝑧𝐿)<𝑧𝐹(𝐾,𝐿)$

$100+3(𝑧𝐾)+2(𝑧𝐿)<100𝑧+3(𝑧𝐾)+2(𝑧𝐿)$  This production function represents decreasing returns to scale.

d. Q = 5K^0.459 L^0.541

Let $𝑄_{0}=𝐹(𝐾,𝐿)=5𝐾^{𝑎}𝐿^{𝑏}$ , where 𝑎+𝑏=1, be the initial production function, then after multiplying it by factor 𝑧 we obtain: 

$𝑄_{1}=𝐹(𝑧𝐾,𝑧𝐿)=5(𝑧𝐾)^𝑎(𝑧𝐿)^𝑏=𝑧^𝑎𝑧^𝑏5𝐾^𝑎𝐿^𝑏=𝑧^{𝑎+𝑏}𝑄_{0}=𝑧𝑄_{0}$

$𝐹(𝑧𝐾,𝑧𝐿)=𝑧𝐹(𝐾,𝐿).$