Answer in Microeconomics for jocelyn bantono #159742

If the production function of the firm is Q = 50K ^0.5L^0.5   

 a) how will you derive the marginal product?

 b) what is the marginal value product with respect to L and with respect to K?

 c) how will the marginal rate of technical substitution look like for this production function?

a) We can find the marginal product by partially differentiating the production function “Q” with respect to “L”:

“MP_L=dfrac{partial Q}{partial L}.”

b) The marginal value product with respect to “L”:

“MP_L=dfrac{partial}{partial L}(50K^{0.5}L^{0.5})=dfrac{25K^{0.5}}{L^{0.5}}.”

The marginal value product with respect to “K”:

“MP_K=dfrac{partial}{partial K}(50K^{0.5}L^{0.5})=dfrac{25L^{0.5}}{K^{0.5}}.”

c) By the definition of the marginal rate of technical substitution, we have:

“MRTS_L^K=dfrac{dL}{dK}=dfrac{MP_L}{MP_K},”“MRTS_L^K=dfrac{dfrac{25K^{0.5}}{L^{0.5}}}{dfrac{25L^{0.5}}{K^{0.5}}}=dfrac{K}{L}.”