Question #249979
Let the production function of a firm is given as
1
Expert's answer
2021-10-11T16:46:17-0400

Solution:

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

Let x = L

      y = K

MPL = QL=1x0.5\frac{\partial Q} {\partial L} = \frac{1} {x^{0.5} }


MPK = QK=1y0.5\frac{\partial Q} {\partial K} = \frac{1} {y^{0.5} }


MPLMPK=wr\frac{MP_{L} }{MP_{K}} = \frac{w }{r}


1x0.51y0.5=wr\frac{\frac{1} {x^{0.5} } }{\frac{1} {y^{0.5} } } = \frac{w} {r }


y0.5x0.5=wr\frac{y^{0.5}} {x^{0.5} } = \frac{w} {r }


x = r2yw2\frac{r^{2}y} {w^{2} }


y = w2xr2\frac{w^{2}x} {r^{2} }

 

b.). The cost function of the firm:

TC = rY + wX

TC = r(w2xr2\frac{w^{2}x} {r^{2} }) + w(r2yw2\frac{r^{2}y} {w^{2} })

TC = w2xr+\frac{w^{2}x} {r} + r2yw\frac{r^{2}y} {w}

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