Answer to Question #159815 in Microeconomics for caleb

Question

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?

Answer 1

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:

$$Q_d=\dfrac{a}{b}-c\dfrac{(ae-bd)}{be(c+f)},$$ $$Q_s=\dfrac{d}{e}+f\dfrac{(ae-bd)}{be(c+f)}.$$