Answer:
[tex](0.5 \times 8q_2)+q_2=100\\\\5q_2=100\\\\q_2=20[/tex]
since [tex]q_2 = 20[/tex]
[tex]q_1 = 8*20\\\\q_1=160[/tex]
Explanation:
U(q₁ q₂)
[tex]q_1^{0.8}q_2^{0.2}\\\\P_1= \$0.5 \ P_2=\$1 \ Y=100[/tex]
Budget law can be given by
[tex]P_1q_1+P_2q_2=Y\\\\0.5q_1+q_2=100[/tex]
Lagrangian function can be given by
[tex]L=q_1^{0.8}q_2^{0.2}+ \lambda (100-0.5q_1-q_2)[/tex]
First order condition csn be given by
[tex]\frac{dL}{dq} =0.8q_1^{-0.2}q_2^{0.2}-0.5 \lambda=0\\\\0.5 \lambda=0.8q_1^{-0.2}q_2^{0.2}---(i)[/tex]
[tex]\frac{dL}{dq} =0.2q_1^{0.8}q_2^{-0.8}- \lambda=0\\\\ \lambda=0.2q_1^{0.8}q_2^{-0.8}---(ii)[/tex]
[tex]\frac{dL}{d \lambda} =100-0.5q_1-q_2=0\\\\0.5q_1+q_2=100---(iii)[/tex]
From eqn (i) and eqn (ii) we have
[tex]\frac{0.5 \lambda}{\lambda} =\frac{0.8q_1^{-0.2}q_2^{0.2}}{0.2q_1^{0.8}q_2^{-0.8}} \\\\0.5=\frac{4q_2}{q_1}\\\\q_1=8q_2}[/tex]
Putting [tex]q_1=8q_2[/tex] in euqtion (iii) we have
[tex](0.5 \times 8q_2)+q_2=100\\\\5q_2=100\\\\q_2=20[/tex]
since [tex]q_2 = 20[/tex]
[tex]q_1 = 8*20\\\\q_1=160[/tex]
