Answer to Question #159815 in Microeconomics for caleb

Question #159815

1.If the production function of the firm is Q = 50K 0.5L0.5   

 

 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
Expert's answer
2021-01-29T20:03:34-0500

1)a) We can find the marginal product by partially differentiating the production function QQ with respect to LL:


MPL=QL.MP_L=\dfrac{\partial Q}{\partial L}.

b) The marginal value product with respect to LL:


MPL=L(50K0.5L0.5)=25K0.5L0.5.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 KK:


MPK=K(50K0.5L0.5)=25L0.5K0.5.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:


MRTSLK=dLdK=MPLMPK,MRTS_L^K=\dfrac{dL}{dK}=\dfrac{MP_L}{MP_K},MRTSLK=25K0.5L0.525L0.5K0.5=KL.MRTS_L^K=\dfrac{\dfrac{25K^{0.5}}{L^{0.5}}}{\dfrac{25L^{0.5}}{K^{0.5}}}=\dfrac{K}{L}.

2) Equating QdQ_d and QsQ_s, we get:


abcP=de+fP,\dfrac{a}{b}-cP=\dfrac{d}{e}+fP,P=aebdbe(c+f).P=\dfrac{ae-bd}{be(c+f)}.

Substitunig PP into the QdQ_d and QsQ_s, we get:


Qd=abc(aebd)be(c+f),Q_d=\dfrac{a}{b}-c\dfrac{(ae-bd)}{be(c+f)},Qs=de+f(aebd)be(c+f).Q_s=\dfrac{d}{e}+f\dfrac{(ae-bd)}{be(c+f)}.

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