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Consider the production function [tex]Y = \sqrt{K}\sqrt{N}[/tex].

a) What is the output if both labor and capital are halved?

b) Is this production function characterized by constant returns to scale? Explain.

c) Write the production function as a relation between output per worker and capital per worker. Does this modified production function exhibit constant returns to scale between output per worker and capital per worker? Justify your answer.

Answer :

Consider the production function Y = √K√N.a) The output if both labor and capital are halved

The production function Y = √K√N can be written as Y = K1/2N1/2.

So, if both labor and capital are halved, the new production function would be

Y = (K/2)1/2(N/2)1/2,

which simplifies to

Y = (K1/2N1/2)/2 = Y/2.

So, if labor and capital are halved, the output will be half the original output.

b) Is this production function characterized by constant returns to scale?Yes, the given production function is characterized by constant returns to scale. Let us assume that we multiply labor and capital by a factor of λ. Then the new production function would be

Y = λ(K1/2)(λN)1/2

= λK1/2λN1/2.λK1/2λN1/2

= λ(K1/2N1/2)

= λY.

So, the output also increases by a factor of λ. Therefore, the production function exhibits constant returns to scale

.c) Write the production function as a relation between output per worker and capital per worker The production function can be written as

Y = K1/2N1/2.

Dividing both sides by N, we get

Y/N = K1/2N-1/2.

Now, dividing both sides by K, we get

Y/K1/2N-1/2K = (Y/K1/2N-1/2)/(K1/2N1/2)

= (Y/K)/(KN)1/2

= (Y/K)L1/2,

where L = N/K is the number of workers per unit of capital. So, the production function can be written as

Y/K1/2L1/2 = Y/KL.

The modified production function exhibits constant returns to scale between output per worker and capital per worker because if we multiply both capital and labor by a factor of λ, then

Y/KL = (λY)/(λKL)

= Y/KL. Therefore, the modified production function exhibits constant returns to scale.

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