Answer to Question #181366 in Microeconomics for Ishaq Mubarak

Question #181366

Given utility maximization problem U=Q1Q2 subject to 1OQ1 +2Q2=240

a. Derive the lagrange function.

b. Derive the first order condition.

c. Using cramer's rule to find the critical value of q1q2 and lamda



1
Expert's answer
2021-04-15T20:43:07-0400

a) Lagrange function:

Z=Q1Q2+λ(24010Q12Q2)Z = Q1Q2+ λ(240-10Q1-2Q2)

b) first-order conditions:

ZQ1=Q2λ10=0ZQ2=Q1λ2=0Zλ=24010Q12Q2=0.ZQ1 = Q2− λ10 = 0\\ ZQ2 = Q1− λ 2 = 0 \\ Zλ = 240 − 10Q1 −2 Q2 =0.


Zλ=24010Q12Q2=0Z\lambda=240-10Q1-2Q2=0

ZQ1=Q2λ10=0ZQ1=Q2- \lambda10=0

ZQ2=Q1λ2=0ZQ2=Q1-\lambda2=0


c) [01021001210]\begin{bmatrix} 0 & -10 & -2 \\ -10 & 0 &1 \\ -2 & 1 & 0 \end{bmatrix} [λQ1Q2]\begin{bmatrix} \lambda \\ Q1 \\ Q2 \end{bmatrix} = [24000]\begin{bmatrix} -240 \\ 0 \\ 0 \end{bmatrix}


Q1M=2402[10]=12=\frac{240}{2[-10]}= -12


Q2M=2402[2]=60=\frac{240}{2[-2]}=-60


λ=2402[10.2]=6\lambda=\frac{240}{2[-10.-2]}=6




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