Question 5
Let us ponder what we have found: For any pair \((x_{1},x_{2})\) we know that if it gives strictly higher utility than the pair \(\left(x_{1}\left(\lambda^{*}\right),x_{2}\left(\lambda^{*}\right)\right)\), then \((x_{1},x_{2})\) is not affordable.
What can we conclude from this?
True \(\left(x_{1}\left(\lambda^{*}\right),x_{2}\left(\lambda^{*}\right)\right)\) is a solution to our original problem of maximizing \(u\left(x_{1},x_{2}\right)=x_{1}x_{2}\) under the budget constraint \(p_{1}x_{1}+p_{2}x_{2}\leq y\).
True No pair \((x_{1},x_{2})\) that is affordable can give a strictly higher utility than \(\left(x_{1}\left(\lambda^{*}\right),x_{2}\left(\lambda^{*}\right)\right)\).