Question 1
Consider a consumer who can choose the quantities of two goods that he can consume. He wants to find the which consumption bundle \(\begin{equation}(x_{1},x_{2})\end{equation}\)maximizes \(u(x_{1},x_{2})\) among all those that satisfy the budget constraint \(p_{1}x_{1}+p_{2}x_{2}\leq y\). If he did not take into account the constraint that \(p_{1}x_{1}+p_{2}x_{2}\leq y\), the consumer would want to choose
both \(x_{1}\) and \(x_{2}\) as high as possible. How could the consumer
mathematically try to take into account the consideration that his
expenditure \(p_{1}x_{1}+p_{2}x_{2}\) should not be too high? A
natural thing to try is to add a penalty term to the consumer’s
objective. Thus instead of maximizing \(u(x_{1},x_{2})\) without any concern
for his expenditure the consumer could consider maximizing
\(\begin{equation}u(x_{1},x_{2})-\lambda\left(p_{1}x_{1}+p_{2}x_{2}\right)\nonumber \\ \end{equation}\)
Here \(\lambda\) could be called ‘the penalty weight’. Let us denote by \(\left(x_{1}\left(\lambda\right),x_{2}\left(\lambda\right)\right)\) the solution to the problem of maximizing \(u(x_{1},x_{2})-\lambda\left(p_{1}x_{1}+p_{2}x_{2}\right)\).
Intuitively (before doing any computations), which of the following
statements would you expect to be true?
True: The higher the \(\lambda\), the lower the \(p_1x_1\left(\lambda\right)+p_2x_2\left(\lambda\right)\)
False: The higher the \(\lambda\), the higher the
\(p_{1}x_{1}(\lambda)+p_{2}x_{2}(\lambda)\)
True: The higher the \(\lambda\), the lower the \(u(x_{1}\left(\lambda\right),x_{1}\left(\lambda\right))\)
False: The higher the \(\lambda\), the higher the \(u(x_{1}\left(\lambda\right),x_{1}\left(\lambda\right))\)
Explanation:
Choosing a high \(\lambda\) means penalizing oneself a lot for spending
money. Intuitively, if one increases the penalty for the expenditures \(p_{1}x_{1}+p_{2}x_{2}\), one will end up making a choice with lower
expenditures.
Intuitively, by choosing a higher penalty rate \(\lambda\), one
prioritizes more the consideration of reducing expenditure. This must
necessarily come at the expense of the utility.
Both of these intuitively plausible statements can be proved formally.