Question

Determine whether the following production functions exhibit constant, increasing, or decreasing returns to scale. L, K, and H are inputs and Q is the output in each production function. Initially, set each input = 100 and determine the output. Then increase each input by 2% and determine the corresponding output to see if constant, increasing, or decreasing returns to scale occur.                            

(a) Q = 0.5L + 2K + 40H                                                                                                                                                              

(b) Q = 3L + 10K + 500                                                                                                                                                       

(c) Q = 4L + 6K + 8KL

Answer 1

(a) Q = 0.5L + 2K + 40H  

Initially L = 100 , K = 100 , H = 100

Q = 0.5(100) + 2(100) + 40(100)

= 4250

Now each input increases by 2% which means  

L = 102 , K = 102 , H = 102  

Q = 0.5(102) + 2(102) + 40(102)

= 4335

% change in output = (4335 - 4250)/4250

= 85/4250  

= 0.02 or  

= 2%

Since 2% increase in each input causes 2% increase in output which means production function exhibits constant returns to scale.

(b)

Q = 3L + 10K + 500

= 3(100) + 10(100) + 500

= 1800

Now each input increases by 2% which means  

L = 102 , K = 102  

Q = 3(102) + 10(102) + 500

= 1826  

% change in output = (1826 - 1800)/1800

= 26/1800

= 0.014  

= 1.4%  

2% increase in inputs lead to 1.4% increase in output which less than the increase in input thus, production function exhibits decreasing returns to scale.

(c) Q = 4L + 6K + 8KL  

= 4(100) + 6(100) + 8(100)(100)

= 81000

Now each input increases by 2% which means  

L = 102 , K = 102

Q = 4(102) + 6(102) + 8(102)(102)

= 84252

% change in output = (84252 - 81000)/81000

= 3252/81000

= 0.04014 or  

= 4.014%  

In this case increase in output , 4.014% is higher than increase in input,2% therefore production function exhibits increasing returns to scale.