Respuesta :
Answer:
The value of "[tex]\bold{q_1=2.5}[/tex]".
Explanation:
Given value:
[tex]U= Max \ q_1^{0.8} \ q_2^{0.2}\\\\[/tex]
Differentiate the above equation with respect of [tex]q_1[/tex], which will give [tex]MUq_1[/tex] as follows:
[tex]MUq_1= q_2^{0.2}(\frac{0.2}{q_1^{0.8}})\\\\[/tex]
[tex]=0.2(\frac{ q_2^{0.2}}{q_1^{0.8}})[/tex]
Differentiate the equation with respect of [tex]q_2[/tex], which will give [tex]MUq_2[/tex] as follows:
[tex]MUq_2= q_1^{0.8}(\frac{0.8}{q_1^{0.8}})\\\\[/tex]
[tex]=0.8(\frac{ q_1^{0.8}}{q_2^{0.2}})}{}[/tex]
for balancing the equation
[tex]\frac{MUq_1}{P_1}=\frac{MUq_2}{P_2}\\\\\frac{MUq_1}{MUq_2}=\frac{P_1}{P_2}\\\\[/tex]
[tex]\frac{0.2(\frac{ q_2^{0.2}}{q_1^{0.8}})} {0.8(\frac{ q_1^{0.8}}{q_2^{0.2}})}}= \frac{8}{4}\\\\\frac{(\frac{ q_2^{0.2}}{q_1^{0.8}})} {4(\frac{ q_1^{0.8}}{q_2^{0.2}})}}= \frac{2}{1}\\\\\frac{(\frac{ q_2^{0.2}}{q_1^{0.8}})} {(\frac{ q_1^{0.8}}{q_2^{0.2}})}}= 8\\\\\frac{q_2}{q_1}=8\\\\q_2=8q_1\\\\[/tex]
Calculate the value of [tex]q_1[/tex] and [tex]q_2[/tex] as follows:
[tex]100 =p_1q_1+P_2q_2\\\\100= 8q_1+4(8q_1)\\\\100=8q_1+32q_1\\\\100=40q_1\\\\q_1=\frac{100}{40}\\\\q_1=2.5[/tex]
[tex]q_2=8q_1\\\\\therefore q_1=2.5\\\\q_2=8\times 2.5\\\\q_2=20.0\\\\q_2=20[/tex]
