Answer to Question #222734 in Economics of Enterprise for shane

Question #222734

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


1
Expert's answer
2021-08-04T09:38:27-0400

Q=180LK0.8L=5 K=10Q=180\sqrt{L}K^{0.8}\\L=5\space K=10


1.

MPL=dQdL=1802LK0.8=90LK0.8MPL=\frac{dQ}{dL}=\frac{180}{2\sqrt L}K^{0.8}=\frac{90}{\sqrt L}K^{0.8}


MPK=dQdK=180×0.8L K0.2MPK=144LK0.2MPK=\frac{dQ}{dK}=180\times0.8\sqrt{L}\space K^{-0.2}\\MPK=144\sqrt{L}K^{-0.2}


2.

Q=180LK0.8Q=180\sqrt{L}K^{0.8}


APL=QL=180LK0.8L=180LK0.8\\APL=\frac{Q}{L}=\frac{180\sqrt{L}K^{0.8}}{L}=\frac{180}{\sqrt{L}}K^{0.8}


APL=180(10)0.85=80.49845(10)0.8\\APL=\frac{180(10)^{0.8}}{5}=80.49845(10)^{0.8}


APL=507.9\\APL=507.9


APK=QK=180LK0.8K\\APK=\frac{Q}{K}=\frac{180\sqrt{L}K^{0.8}}{K}


APK=180LK0.2=1805100.2=402.49100.2\\APK=\frac{180\sqrt{L}}{K^{0.2}}=\frac{180\sqrt{5}}{10^{0.2}}=\frac{402.49}{10^{0.2}}


APK=402.491.5848932=253.9554\\\\APK=\frac{402.49}{1.5848932}=253.9554


3.

Q(K,L)=180LK0.8Q(K,L)=180\sqrt{L}K^{0.8}

Scaling both factors of production by t>t

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


=180(t)12L(t)0.8(K)0.8=180L(K)0.8(t)0.8=180L(K)0.8(t)0.5+0.8=(t)1.3180L(K)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)1.3Q(tK,tL)=Q(K,L)(t)^{1.3}


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

4.

MRTS=MPLMPK=90L(K)0.8×1144L(K)0.2MRTS=\frac{MPL}{MPK}=\frac{90}{\sqrt{L}}(K)^{0.8}\times\frac{1}{144\sqrt{L}(K)^{-0.2}}


=90144K0.8(K)0.2L=\frac{90}{144} \frac{K^{0.8}(K)^{0.2}}{L}


=90144KL=\frac{90}{144}\frac{K}{L}


=1016KL=\frac{10}{16}\frac{K}{L}


MRTS=58KLMRTS=\frac{5}{8}\frac{K}{L}


5.

Cross partial effects

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


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


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


dMPKdL=1442L(K)0.2=72L(K)0.2>0\frac{dMPK}{dL}=\frac{144}{2\sqrt{L}}(K)^{-0.2}=\frac{72}{\sqrt{L}}(K)^{-0.2}>0


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