Solution:
a.). Derive MRTS = MPLMPK\frac{MP_{L} }{MP_{K}}MPKMPL
Let x = L
y = K
MPL = ∂Q∂L=1x0.5\frac{\partial Q} {\partial L} = \frac{1} {x^{0.5} }∂L∂Q=x0.51
MPK = ∂Q∂K=1y0.5\frac{\partial Q} {\partial K} = \frac{1} {y^{0.5} }∂K∂Q=y0.51
MPLMPK=wr\frac{MP_{L} }{MP_{K}} = \frac{w }{r}MPKMPL=rw
1x0.51y0.5=wr\frac{\frac{1} {x^{0.5} } }{\frac{1} {y^{0.5} } } = \frac{w} {r }y0.51x0.51=rw
y0.5x0.5=wr\frac{y^{0.5}} {x^{0.5} } = \frac{w} {r }x0.5y0.5=rw
x = r2yw2\frac{r^{2}y} {w^{2} }w2r2y
y = w2xr2\frac{w^{2}x} {r^{2} }r2w2x
b.). The cost function of the firm:
TC = rY + wX
TC = r(w2xr2\frac{w^{2}x} {r^{2} }r2w2x) + w(r2yw2\frac{r^{2}y} {w^{2} }w2r2y)
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