Answer to Question #249979 in Microeconomics for Kajal

Question

Let the production function of a firm is given as

Answer 1

Solution:

a.). Derive MRTS = $\frac{MP_{L} }{MP_{K}}$

Let x = L

      y = K

MP_L = $\frac{\partial Q} {\partial L} = \frac{1} {x^{0.5} }$

MP_K = $\frac{\partial Q} {\partial K} = \frac{1} {y^{0.5} }$

$\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

$\frac{\frac{1} {x^{0.5} } }{\frac{1} {y^{0.5} } } = \frac{w} {r }$

$\frac{y^{0.5}} {x^{0.5} } = \frac{w} {r }$

x = $\frac{r^{2}y} {w^{2} }$

y = $\frac{w^{2}x} {r^{2} }$

 

b.). The cost function of the firm:

TC = rY + wX

TC = r($\frac{w^{2}x} {r^{2} }$) + w($\frac{r^{2}y} {w^{2} }$)

TC = $\frac{w^{2}x} {r} +$ $\frac{r^{2}y} {w}$