Question 2
The considerations from question 1 make it plausible that there will be some value \(\lambda^{*}\) for \(\lambda\) such that \(\left(p_{1}x_{1}\left(\lambda^{*}\right)+p_{2}x_{2}\left(\lambda^{*}\right)\right)=y\). Intuitively, this value for \(\lambda^{*}\) is just the right level of penalty that the consumer should impose himself for spending money.
Consider any other pair \((x_{1},x_{2})\). What do we know about \((x_{1},x_{2})\)?
True: We must have \(u\left(x_{1},x_{2}\right)-\lambda^{*}\left(x_{1}p_{1}+x_{2}p_{2}\right)\leq u(x_{1}\left(\lambda^{*}\right),x_{2}(\lambda^{*}))-\lambda^{*}\left(p_{1}x_{1}\left(\lambda^{*}\right)+p_{2}x_{2}\left(\lambda^{*}\right)\right)\)
False: We must have \(u\left(x_{1},x_{2}\right)-\lambda^{*}\left(x_{1}p_{1}+x_{2}p_{2}\right)\geq u(x_{1}\left(\lambda^{*}\right),x_{2}(\lambda^{*}))-\lambda^{*}\left(p_{1}x_{1}\left(\lambda^{*}\right)+p_{2}x_{2}\left(\lambda^{*}\right)\right)\)