Question

Joe’s coffee house operates under the production function Q =2√? + ?^2/3, where L is the number of worker hours and K is the number of coffee machine hours.

a. Clearly show what type of returns to scale is exhibited.

b. Does labor exhibit diminishing marginal returns when capital is fixed? Explain.

Answer #1

Q = F(k, L) = 2√? + ?^2/3

Let L = K = 2

then Q = 2√2 + 2^2/3

Q = 2.83 + 1.6

Q = 4.3

Now, we double the input L = K = 4

Q = 2√4 + 4^2/3

Q = 4 + 2.53

Q = 6.53

We saw that as we double the inputs L and K, The output is less than doubled. So, the production function exhibits Diminising Returs to Scale DRS

b)

MPL = dQ/dL = (2 / 2√?) = 1/√?

d(MPL)/dL = - (1/2√?*L)

Hence, the second-order derivative of MPL is negative which tells us that there are diminishing marginal returns to labor.

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