Answer in Microeconomics for caleb #159815

1.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?

2. if Qd = a/b – cP and Qs = d/e + fP, in market equilibrium (Qd = Qs) what is P, Qd and Qs?

1)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}.”

2) Equating “Q_d” and “Q_s”, we get:

“dfrac{a}{b}-cP=dfrac{d}{e}+fP,”“P=dfrac{ae-bd}{be(c+f)}.”

Substitunig “P” into the “Q_d” and “Q_s”, we get:

“Q_d=dfrac{a}{b}-cdfrac{(ae-bd)}{be(c+f)},”“Q_s=dfrac{d}{e}+fdfrac{(ae-bd)}{be(c+f)}.”