Consider the following production function: Q=(L+K)1/2

(a) What is the Marginal Product of Labor (MPL)?
(b) What is the Marginal Product of Capital (MPK)? Are they diminishing?
(c) What is the Average Product of Labor (APL)?
(d) What is the Average Product of Capital (MPK)?
(e) What is the TRSL, K ? Is the absolute value of TRSL, K diminishing in L or K?Are there constant, decreasing, or increasing returns to scale?

(a) MPL = 1/2

(b) MPK = 1/2

(c) APL = 1/2+ K/2L

(d) APK = = L/2K+ 1/2

(e) TRSL,K = 1

Explanation:

Q = (L+K)1/2 = L/2 + K/2 ……………………………………………………. (1)

Where Q is the total output, L is Labour and K is capital

(a) What is the Marginal Product of Labor (MPL)?

To get MPL, differentiate equation (1) with respect to L as follows:

MPL = dQ/dL = 1/2  

(b) What is the Marginal Product of Capital (MPK)?  

To get MPK, differentiate equation with respect to K as follows:

MPK = dQ/dL = 1/2  

Both MPL and MPK are diminishing since each of them is 1/2 which less than one.

(c) What is the Average Product of Labor (APL)?

To calculate APL, divide equation (1) by L as follows:

APL = Q/L = (L/2)/L + (K/2)/L

                 = 1/2+ K/2L

(d) What is the Average Product of Capital (MPK)?

To calculate APK, divide equation (1) by K as follows:

APL = Q/K = (L/2)/K + (K/2)/K

                 = L/2K+ 1/2

(e) What is the TRSL,K ?

TRSL,K implies Technical Rate of Subsitution of L for K = MPK/MPL

TRSL,K = MPK/MPL

Since both MPK and MPL are each equal to 1/2 as calculated under (a) and (b) above, we substitute for them in the TRSL,K as follows:

TRSL,K = MPK/MPL = (1/2)/(1/2) = 1

(e.1.) Is the absolute value of TRSL,K diminishing in L or K?

No, it neither diminishing in L nor K since the answer is equal to 1.

(e.2.) Are there constant, decreasing, or increasing returns to scale?

Since the TRSL,K is equal to 1, it implies there is a constant return to scale.

I wish you the best.

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