Question 9
What is the optimal consumption bundle, i.e. the solution to the original problem of maximizing \(log(x_{1})+log(x_{2})\) subject to the budget constraint \(p_{1}x_{1}+p_{2}x_{2}\)?
True \(\left(x_{1},x_{2}\right)=\left(\frac{y}{2\ p_{1}},\frac{y}{2\ p_{2}}\right)\)
False \(\left(x_{1},x_{2}\right)=(\frac{y}{\ p_{1}},\frac{y}{\ p_{2}})\)
Explanation:
Plugging \(\lambda^{*}=\frac{y}{2\ p_{1}p_{2}}\) into \(\left(x_{1}(\lambda),x_{2}(\lambda)\right)=(\lambda p_{2},\lambda p_{1})\) gives
\(\begin{equation}\left(x_{1}(\lambda^{*}),x_{2}(\lambda^{*})\right)=(\frac{y}{2\ p_{1}p_{2}}p_{2},\frac{y}{2\ p_{1}p_{2}}p_{1})=(\frac{y}{2\ p_{1}},\frac{y}{2\ p_{2}})\nonumber \\ \end{equation}\)
From our reasoning in the previous exercises, we know that \(\left(x_{1}(\lambda^{*}),x_{2}(\lambda^{*})\right)\) is a solution to the original problem of maximizing \(log(x_{1})+log(x_{2})\) subject to the budget constraint \(p_{1}x_{1}+p_{2}x_{2}\). One can also show that there is only one solution to this problem, so \(\left(x_{1}(\lambda^{*}),x_{2}(\lambda^{*})\right)\) is the unique solution.