Question

Q=180√LK^0.8

L=5

K=10

1).Construct marginal product of labour and marginal product of capital

2If L=5, K=10 Measure average product of labour and average product of capital

3.return to scale production function

4.if K=5, L=10 calculate marginal rate of products.

5.law of diminishing marginal returns of labour and law of diminishing marginal returns of capital

6.how about the cross partial effect of this production function

Answer 1

$Q=180\sqrt{L}K^{0.8}\\L=5\space K=10$

1.

$MPL=\frac{dQ}{dL}=\frac{180}{2\sqrt L}K^{0.8}=\frac{90}{\sqrt L}K^{0.8}$

$MPK=\frac{dQ}{dK}=180\times0.8\sqrt{L}\space K^{-0.2}\\MPK=144\sqrt{L}K^{-0.2}$

2.

$Q=180\sqrt{L}K^{0.8}$

$\\APL=\frac{Q}{L}=\frac{180\sqrt{L}K^{0.8}}{L}=\frac{180}{\sqrt{L}}K^{0.8}$

$\\APL=\frac{180(10)^{0.8}}{5}=80.49845(10)^{0.8}$

$\\APL=507.9$

$\\APK=\frac{Q}{K}=\frac{180\sqrt{L}K^{0.8}}{K}$

$\\APK=\frac{180\sqrt{L}}{K^{0.2}}=\frac{180\sqrt{5}}{10^{0.2}}=\frac{402.49}{10^{0.2}}$

$\\\\APK=\frac{402.49}{1.5848932}=253.9554$

3.

$Q(K,L)=180\sqrt{L}K^{0.8}$

Scaling both factors of production by t>t

$Q(tK,tL)=180\sqrt{tL}(tK)^{0.8}\\$

$=180(t)^{\frac{1}{2}}\sqrt{L}(t)^{0.8}(K)^{0.8}\\=180\sqrt{L}(K)^{0.8}(t)^{0.8}\\=180\sqrt{L}(K)^{0.8}(t)^{0.5+0.8}\\=(t)^{1.3}180\sqrt{L}(K)^{0.8}$

$Q(tK,tL)=Q(K,L)(t)^{1.3}$

$(t)^{1.3}>(t)$ implies increasing returns to scale

4.

$MRTS=\frac{MPL}{MPK}=\frac{90}{\sqrt{L}}(K)^{0.8}\times\frac{1}{144\sqrt{L}(K)^{-0.2}}$

$=\frac{90}{144} \frac{K^{0.8}(K)^{0.2}}{L}$

$=\frac{90}{144}\frac{K}{L}$

$=\frac{10}{16}\frac{K}{L}$

$MRTS=\frac{5}{8}\frac{K}{L}$

5.

Cross partial effects

$MPL=\frac{90}{\sqrt{L}}(K)^{0.8}$

$\frac{dMPL}{dK}=90\frac{(0.8)}{\sqrt{L}}(K)^{-0.2}\\=\frac{72(K)^{-0.2}}{\sqrt{L}}>0$

$MPK=144\sqrt{L}(K)^{-0.2}$

$\frac{dMPK}{dL}=\frac{144}{2\sqrt{L}}(K)^{-0.2}=\frac{72}{\sqrt{L}}(K)^{-0.2}>0$